Dedication For Philip, Richard and Ruth Misce stultitiam consiliis brevem: Dulce est desipere in loco Horace 65–8 BC ODES (book 4, poem 12 (a poem on the pleasures of Spring), lines 27-8) Mix a little foolishness with your serious moments Silliness in its place is charming (i.e don’t be po-faced all the time)
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Preface
This book is about the deliberate or accidental cutting of all sorts of materials from ‘soft, compliant and weak’ to ‘hard, stiff and strong’. Examples are drawn from the engineering, physical and biological sciences, history and archaeology, palaeontology, medicine, veterinary medicine and dentistry, and food technology. It is a very broad canvas over which I attempt to demonstrate that separation of one part from another and the factors controlling separation – the mechanics of cutting – are basically the same, whatever the field and whatever the material. There is much to learn by looking at how other people do similar tasks. I am an engineering scientist, interested in why things break, with limited knowledge in biological and other fields, but I have been fortunate for many years to collaborate with members of the Centre for Biomimetics at the University of Reading. Biomechanics is the study of the physical properties of all biological materials (including foodstuffs), and their employment in understanding biological design and function. The subject should not be confused with the complementary field of biomedical engineering whose employment of physical properties of human body parts relates to specific replacements of limbs and organs. Biomimetics (‘mimicking nature’) is inspired by biology and applies biomechanical knowledge to the manufacture of new engineering materials and devices. It may be said, perhaps, that biomimetics was invented at Reading by my predecessor, Jim Gordon, and carried on by George Jeronimidis, Julian Vincent (now at the University of Bath) and Richard Bonser. Professor Gordon also built up contacts with archaeologists, classicists and historians, and that has continued in Engineering at Reading with our group of archaeometallurgists (Henry Blyth, Eddie Cheshire, David Sim and Alan Williams). Having been fortunate to study under Charles Gurney in Cardiff and David Tabor in Philip Bowden’s Laboratory for the Physics & Chemistry of Solids in the Cavendish Laboratory at Cambridge, I am receptive to this sort of wide thinking. The pioneering work on metal cutting at the University of Michigan, Ann Arbor, also rubbed off on me during my tenure in Mechanical Engineering there, as did discussions with Wallace Hirst and Gerry Hamilton at Reading on contact mechanics and wear, for which I am grateful. One difficulty in writing an interdisciplinary text is that what may be elementary and well known to a worker in one field is completely strange to others. I have had to have a dictionary
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next to me when reading non-engineering papers, and have pestered very many colleagues across the world for clarification of lots of topics. I owe a great debt to all those who have advised me on the disparate topics covered in this book and corrected many of my misconceptions. Geoff Rowe once said that I had a habit of ‘setting crossword puzzles for tired minds’. At one stage it seemed that the list of acknowledgements would be longer than the text. Even so, any mistakes in this book are entirely down to me. Even within engineering, some branches have a different language from another: the subject of soil mechanics (geomechanics), for example, developed separately from solid mechanics. Again, I have had to face the problem that different fields use different symbols and different nomenclature for identical things. Some ideas about cutting require mathematics for explanation, and biological scientists and others do not always have the mathematical background of physical scientists. The mathematics in this book is not advanced but, to help those to whom modelling and mathematical derivations may appear mysterious, I have tried to be as basic as possible. Detailed workings may be found in original papers. Equally, specialists in the biological and other fields may no doubt consider that I have over-egged the pudding in many places regarding their subjects. The author’s own inadequacies will be evident to all. Sir Charles Inglis, in the Proceedings of the Institution of Mechanical Engineers in 1947, said that: ‘… Mathematics [required by engineers] though it must be sound and incisive as far as it goes, need not be of that artistic and exalted quality which calls for the mentality of a real mathematician. It can be termed mathematics of the tin-opening variety, and in contrast to real mathematicians, engineers are more interested in the contents of the tin than in the elegance of the tin-opener employed …’. In this book we hope to discuss how the tin-opener itself works with such simple mathematics. The reader will become aware that there is, perhaps, less maths in some of the later sections of the book that deal with biology, palaeontology, medicine, food, etc. With notable exceptions, there are fewer workers performing instrumented experiments to provide data to assess the role of cutting and its interaction with biological microstructure. Sometimes the level of taxonomic sampling is not yet wide enough to determine the role played by cutting in biological design and function. Many exciting experiments are waiting to be performed and analysed. Unfortunately, there is no space for interesting topics from literature, mythology, art and so on that involve cutting of various sorts: driving of stakes through the chests of Dracula’s victims; King Arthur’s magical sword Excalibur; St George slaying the dragon; Beowulf’s cutting off the head of Grendel’s mother; the sword of Damocles; Odysseus killing the sleeping giant Polyphemus by driving a stake through his eye; Hercules’ second labour was to kill the Hydra, a monster with nine heads where, when one was chopped off, two grew back in its place; the oldest of the classical Greek Fates Atropus cut the thread of Lachesis’s life with her shears (Lachesis was the Fate who spun life’s thread and determined its length); the courtly love of Lancelot and Guinevere (King Arthur’s wife) by Chretien de Troyes from the twelfth century involved crawling over a bridge made of swords. A knight with his ‘sword and buckler’ is found in poetry. A buckler is a shield with a boss (from the French bocle, a boss). The word swashbuckler to describe a swaggering bully comes from swash, the noise made by swords clashing or a sword beating on a shield. In Kipling’s The Glory of the Garden, we read that ‘… better men than we go out and start their working lives at grubbing weeds from gravel paths with broken dinner knives …’. Museums around the world have collections of cutting instruments or illustrate tools and cutting in various guises: the knife grinder in the Octagonal Room (Tribuna) at the Uffizi Gallery in Florence; among the lozenge-shaped panels from the old campanile now in the Cathedral museum in Florence, the panel entitled Logic by Gino Micheli da Castello shows
Preface
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shears, and an image of David and Goliath appears in relief in the door of the Duomo. A billhook representing winter cutting of wood may be found in the pediment over Minerva’s temple at the Roman baths in Bath. Among paintings that show cutting is David Teniers the Younger’s Interior, Old Woman Peeling Apples at the Fitzwilliam Museum in Cambridge. A strange pair of scissors that will not function is to be found in Salvador Dali’s painting entitled Guillaume Tell: the blades of the scissors in his left hand cannot close up because the thumb and forefinger are already together when the blades are still open. Caillabotte’s Les Raboteurs de Parquet shows workmen finishing a wooden floor by scraping. A lot of arms, armour and decapitations (Medusa; John the Baptist) have been painted and sculpted over the centuries. In an interactive exhibition of modern art in the 1960s Yoko Ono invited people to take part and cut bits off her dress. Man Ray’s Cadeau at the Pompidou Centre in Paris has fourteen carpet tacks stuck in line along the axis of the lower surface of a ‘Gendarme’ flat iron. It is curious that teeth in a number of sculptures of heads are not always represented accurately. In the Louvre, for example, there is a stone lion dating from 350 BC from a cemetery at Glyphada (near modern Athens): the teeth are ‘human’ dentition, not animal. The same is true of a number of ceremonial lions guarding doors in China. There is also an oil painting entitled Surprised! by Henri Rousseau at the National Gallery in London showing a tiger in the jungle. Are the teeth correct? The cutting of metals is commercially important and there are many admirable books on the subject. There are also books devoted to cutting of wood and plastics (polymers). They contain much practical information and detail of industrial processes that it has not been possible to include here. Apart from numerous empirical formulae for cutting forces, cutting energy and so on, most models of cutting for different materials in such monographs follow what is to be found in the metal-cutting monographs, by and large, namely that the work required for cutting comprises two components: (i) plastic or other irreversible work in forming the offcut or chip; and (ii) work done against friction. Cutting is different from other deformation problems in elasticity and plasticity since after cutting, a single starting body has been separated into a number of entirely separate bodies that are no longer ‘attached’ to the parent body. The work for separation is absent in traditional models of metal cutting because it was believed that it is insignificant. That view is challenged in this book, based on tracing the history of the assumption in original papers (the use of the chemical surface free energy rather than the fracture toughness). Central to the current theme is the idea of separation of parts and that cutting is a branch of elastoplastic fracture mechanics. The cutting of floppy and brittle materials principally concerns fracture: the specific work of separation is not negligible, and calculations for forces and power consumed employ the fracture toughness of the material, not the surface free energy. When significant work of separation is incorporated in analyses for the cutting of ductile materials, as well as the customary plasticity and friction, a number of experimental observations for which the traditional treatment has no explanation, now make sense. Because the work of separation takes place in thin boundary layers contiguous with the cut surfaces, the separation work and remote work are essentially uncoupled, which is why flow fields in the cutting of ductile materials may be estimated without reference to fracture work. However, when cutting forces and power are to be determined, consideration of separation work is necessary. The classification of materials into ductile and brittle is based on the behaviour of laboratorysize testpieces. Yet brittle glass can be machined at micrometre depths of cut and ductile steel behaves in a brittle fashion in large sizes. Such behaviour reinforces the concept that cutting is a branch of elastoplastic fracture mechanics because it is a manifestation of the cube-square scaling inherent in fracture mechanics, where there is competition between energies dependent on volume and dependent on area. In cutting a given material, ductile chips are produced at very small depths, but as the thickness of the slice is increased the behaviour becomes less ductile,
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so that eventually splits form. This is common experience in wood. Ductility (called tenacity in old papers), as well as a material’s strength or hardness, had always been known to influence the ease or difficulty of cutting, but its role was never properly quantified. The use of fracture toughness, as well as yield strength, simply makes that connexion. Both these mechanical properties may be altered independently of each other by different thermomechanical or other treatments so that different samples of the same material may have the same hardness but quite different toughnesses. It is to be expected that they will cut differently. Energy methods are employed throughout this book. They permit solutions of problems that might otherwise be difficult: for example, the explanation why cutting with a knife is always easier when sideways motion is coupled with simple pressing-down, and hence the importance of the ‘slice–push ratio’ in cutting. Materials are considered as continua having reproducible mechanical properties, but microstructure is discussed where appropriate. It comes down to the ‘magnification’ at which a material is being looked. Concrete and salami to the naked eye are coarse heterogeneous microstructures; other solids require inspection under the microscope to see the constituents. All biological materials are hierarchical composites of one sort or another and, in trying to understand behaviour, it is important to know the level at which particular properties are controlled. The subject and sources of information are scattered over many disciplines and published in a bewildering number of journals. Whole books have been written on just parts of the subject. While I hope I have recorded important papers, I expect that I have missed a number, and can only apologize to the authors. There are far too many references to quote even a small selection. My choices of papers from a given author/school of work are, to an extent, arbitrary, but it is possible to trace other publications from the papers that are referred to. In addition to historically interesting references, some of the ‘working’ references are quite old. I make no apology for that. Early researchers thought carefully about what they did and experimental work was carefully and painstakingly done, often under difficult circumstances: insensitive load and displacement measurement devices; no image recognition schemes for flow fields; no data acquisition and manipulation software; algebra rather than finite element methods (FEM), and so on. It seems to be a disappointing trend that some young researchers are not familiar with the old literature, and know only about things that can be downloaded from a search engine. In consequence, they sometimes reinvent the wheel, and often have the view that if they have used FEM or similar techniques, then it must be all the better for it. What is possible with modern elastoplastic computational models is, of course, truly remarkable, but FEM is not a substitute for thinking and experimentation. Furthermore, FEM requires physical property inputs and they come from experiments. Calibration of FEM models has to be done with care: while computational simulations can explain the results of experiments too complicated to be modelled by simple algebra, the real success comes when FEM is able to predict events ‘blind’.
Acknowledgements
Many people have helped and advised me on this book. I must mention David Wyeth, Eddie Cheshire, and all former research and project students; Richard Bonser, Richard Chaplin, George Jeronimidis, Tony Pretlove and other colleagues at Reading; John Frew is especially thanked for experimental assistance over many years. Peter Lillford, Peter Lucas, Gordon Sanson and Julian Vincent have patiently fielded e-mails from me seeking clarification on the biological side. Gordon Sanson has read drafts, made the most helpful comments and prevented my making many biological howlers. What is written down though is entirely my responsibility and I hope that there are few errors. Other people to whom I owe thanks are: Julian Allwood, Hilary Arnold-Baker, Daniel Balint, Dick Bassett, Roger Bentley, Henry Blyth, Malcolm Bolton, Roy Brigden, Brian Briscoe, Andy Brunner, Tim Burns, Byron Byrne, Peter Chamberlain, Maria Charalambides, Chen Zhong, Tom Childs, Brian Cotterell, Matt Davies, John Dempsey, Coen Dijkman, Peter Dunn, Caroline Ellick, Bill Endres, Roland Ennos, David Felbeck, Paul Fenne, Tony Gee, Giacomo Goli, Roger Hamby, Bryan Harris, Linda Holland, Ian Horsfall, Ian Hutchings, Norman Jones, Dirk Keeley, Kevin Kendall, Tony Kinloch, Raja Kountanya, Hans Kruuk, Brian Lawn, Ming Li, Ken Ludema, Yiu-Wing Mai, Remy Marchal, Adrian Marshall, Paulo Martins, Shelagh McKay, Roy Moore, Sue Mott, Barbara Murray, John Nairn, Kazimierz Orlowski, Andrew Palmer, Lucy Peltz, Gill Pittman, Tracey Popowics, Charles Preston, Tony Pretlove, Richard Rahdon, Jenny Read, Steve Reid, Pedro Reis, Peter Roberts, Liz Robertson, Benoit Roman, Pedro Rosa, Marco Rossi, David Sim, Gerhard Sinn, Stefi Stanzl-Tschegg, Roger Stewart, David Stirling, Hew Strachan, Frank Tallett, Bernard Thibaut, Michael Thouless, Chris Tufnel, John Videler, Julian Vincent, Stephen Walley, Celia Watson, Shelley Wiederhorn, Tomasz Wierzbicki, Alan Williams, Gordon Williams, John Williams, Xianzhong Xu and John Yeo.
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Chapter 1
Controlled and Uncontrolled Separation of Parts Cutting, Scraping and Spreading The design of structures and components in nature and in engineering usually aims to avoid fracture – at least during life – but there are circumstances where separation of parts is required. These range all the way from the beast of prey tearing apart its victim with teeth and claws, to the manufacture of a precision surface in metal using special cutting tools. Some processes of separation rely on pulling, bending or twisting an object at regions remote from where the object breaks; others load right at the zone of fracture and this includes cutting. Processes of ‘separation’ that are not cutting include pulling corks out of bottles. Some processes that are thought to be cutting are really not: it is a common misconception that ice breakers cut ice fields by splitting; rather, they ride up on the edge of the ice sheet and break pieces off by bending fracture. Separation of materials is all around us: in the kitchen (e.g. carving meat, coring apples, grating cheese, peeling vegetables), when eating (on the dinner plate, in the mouth), in carpentry and building (e.g. sawing, planing and drilling wood; cutting bricks and paving stones), in the office (e.g. paper guillotining and shredding, pencil sharpening), in manufacturing (e.g. all metal-cutting operations), in agriculture (e.g. ploughing, harvesting of crops, sheep shearing), in medicine and dentistry (surgery; the drilling of teeth), in nature (hunters, raptors, their prey and defences; teeth and chewing), in shaving, in opening packaging and in war (arms and armour: a spear through ancient armour, depleted uranium missiles through modern tank armour). While, usually, the cutting tool remains undeformed, in the latter field both cutter and target are deformed. ‘Cutting’ is interpreted very broadly in this book, but even so we do not consider flame cutting, liquid jets, abrasive water jet cutting, laser cutting, plasma arc cutting, electrodischarge machining and electrochemical machining. Different materials respond differently when cut with a knife, well illustrated by the wide variety of foodstuffs that includes mashed potatoes, boiled potatoes, uncooked potatoes; cooked and uncooked vegetables; stringy vegetables like celery; squidgy food like blancmange or tofu; boiled sweets, fudge and sugar; easy-to-chew high-quality meat, or poor-quality meat with lots of gristle; soft puddings like icecream, or hard puddings like toffee; some are mixtures of hard and soft (crème brûlée); chocolates may have a hard case with a soft inside. Properties may change with time and storage: some fruit has to be stored after picking before it becomes ripe enough to eat. Fresh food and stale food behave differently: when freshly harvested, foods such as carrot or celery are hard and stiff owing to the turgor pressure that pressurizes the composite structure from within (from the Latin for ‘to swell’); turgor pressure is a plant’s internal stressing to keep it erect, among other things. Turgor pressure decreases with time after harvesting, making fruits and vegetables flaccid (from the Latin for flabby) when they become rubbery and bendy. Loss of turgor pressure is why flowers wilt. The condition of food affects how they are dealt with on the plate, their ‘mouth feel’ and how we bite and masticate food. A wide variety of different types of implement is found, ranging from butcher’s knives to cheese graters. Kitchen shops offer strange and ingenious gadgets with pointy bits for doing special cutting jobs in the preparation of food. One of the most exotic, perhaps, is a foie-gras Copyright © 2009 Elsevier Ltd. All rights reserved.
The Science and Engineering of Cutting
cutter. The Swiss have different slicers for potatoes (to prepare roesti) and for apples (for muesli). A mandolin is a device like a wood-plane over the blade of which foodstuffs are sliced, grated or shred depending on the blade. A hachoir is a rocking device for cutting up herbs (from the French hache for axe; hence hatchet). In antique shops may be found old devices such as sugar cleavers (in Victorian times, sugar used to come in big lumps), mechanical apple peelers, nutmeg grinders and so on. Experiments in the kitchen can be very instructive about cutting, and the reader is encouraged to do so and get a feel for stiff/compliant, strong/weak, tough, etc., materials. For example, scrape a carrot with a knife and notice the difference depending on the angle of the blade. What controls the depth of cut in a potato peeler? Why is peeling with a knife more wasteful? Can you skin an orange with a potato peeler? Indeed, can you shave with a potato peeler? Are there differences depending on whether the fruit is hard and stiff, or soft and squidgy? What determines the ease of scraping up a portion of butter on to a knife from a block, or a scoop of icecream? What are those serrations that appear on the back of the butter after scraping? What determines the ‘spreadability’ of butter on toast? What is the best way to take the top off a boiled egg? What are those cracks that appear having scraped the back of a spoon across the surface of a table jelly? These are not flippant questions or suggestions: the answers are central to understanding of the mechanics of cutting. When we eat with the aid of a knife, fork, spoon, chopsticks or fingers, we often separate (fracture) food into smaller pieces to fit the mouth, where further deformation and fracture takes place before swallowing. Why is it easier to cut when we ‘slice’ across the food as well as simply ‘press down’? Food on the plate will have been previously prepared from larger pieces and/or cooked to make eatable and digestible. Cooking alters the properties of food and distinguishes humans from other creatures. To tell whether potatoes or other vegetables have been cooked for the requisite time, we stab the vegetable in the saucepan with a knife and see how easy it is to pierce, or see whether it can be suspended from the knife. The altered properties revealed by the knife must connect with perception in the mouth and what, for example, al dente means. Similarly, to get food from plate into mouth, we often pierce, indent or perforate the food with the prongs of a fork, the mechanics of which are similar to nailing a piece of wood. Cutting may concern big pieces being separated into two or more still-big pieces (sawing logs of timber into planks, slitting metal sheets off rolling mills, cutting wedges from ‘rounds’ of cheese, cutting fruit into segments). In other examples, thin slices or chips are removed from the surface of a larger piece (peeling potatoes, whittling wood, lathe cutting, carving). Sometimes the piece cut off is important (wood veneer, microtomed sections for histological examination); at other times the piece left behind is important (true of most manufacturing processes where the offcut ‘swarf’ is scrap, trimming the edges of bound books); and sometimes both are important (the division of paper sheets into smaller sizes or the slicing and dicing of semiconductor wafers). Sometimes the quality of the resulting surfaces is of particular concern (limits and fits in engineering assembly) but sometimes it does not matter (chopping firewood). Sometimes the same mechanism of cutting may be both undesirable in one situation, yet beneficial in another (erosion versus sandblasting) Different types of cutting include: l l l l l
cutting layers or slices from the surface or edge of a body cutting a groove in the surface of the workpiece dividing a workpiece into sections by cutting through the thickness making profiles (e.g. round shapes on a lathe) making some sort of hole down into, or though, the thickness by penetration and perforation.
Controlled and Uncontrolled Separation
Some separation processes using tools are on the borderline between being under control and not (such as in cutting toenails, where offcuts sometimes fly around the bathroom; and in nut cracking, where the aim is to preserve the kernel uncrushed or unbroken, but where the fate of the shell is unimportant). The flexibility of whatever is holding both the cutter and the workpiece can be important: try spreading butter with a very springy knife; observe the deformation of a loaf of bread on cutting. Sometimes there is little control at all (in crushing/ ball mills or chipping timber to make wood pulp). ‘Loose tools’ are bullets and shrapnel, or the stream of grits in erosion, abrasive cleaning and sandstorms. Cutting during fault or accident conditions is often uncontrolled. Accidental and unintentional cutting, in the form of scratching, piercing, perforating and tearing, occurs when a nail punctures a tyre, and when supermarket plastic bags are torn by sharp objects. A ship getting holed after hitting a rock is a larger scale example of the same thing, as is the defeat of armour by a weapon but where the process is intentional. An understanding of piercing, cutting and perforation enables better armour to be designed that will defeat the weapon, and vice versa. A characteristic feature of some of the controlled cutting considered in this book is that the cutting tool or blade is not deformed when cutting, and remains ready for reuse. The only significant deterioration may be wear and blunting; unless corrected, the quality of the cut deteriorates. Progress in production engineering from the onset of the Industrial Revolution depended on having satisfactory tool and die materials, and heavy rigid machine tools. Wilkinson’s boring machine of 1775 that made possible the manufacture of Watt’s steam engine was a vital development in the history of machine tool manufacture that encompassed lathes and machines to perform drilling, milling, boring, grinding, shaping, planing and so on. Benjamin Thompson (Count Rumford of the Holy Roman Empire) realized the connexion between mechanical work done and heat generated from observing cannon being bored in Bavaria. (Thompson was American but supported the British in the American Revolution and subsequently fled to Europe. He designed the Englische Garten in Munich.) ‘The Multifarious Perforating Machine’ is illustrated in Figure 1-1. An interesting question is how hard a tool should be to avoid itself being deformed in cutting. Mutual cutting is possible where both tool and workpiece deform (bullets and the target). When it is difficult to insert a woodscrew, the high torque will distort the blade of a poor-quality screwdriver and also cut slivers from the side of the slot in the head of the screw. Cutting tools are usually thought of as hard, stiff solid objects. But when one ship collides with another, both hollow structures deform and may fracture, and this is another example of mutual cutting. Many tools can be resharpened and used again. What ‘sharpness’ may, or may not, mean and whether it is an absolute concept is explored later in the book. Sometimes tools are used once only (disposable scalpels) or thrown away when blunt (disposable razors, indexable tool inserts). Hollow needles that pierce the skin and through which liquids may be inserted into the body (hypodermics) or removed (cannulae) may sometimes be reused depending on conditions. Improvements in tool material qualities, and reductions in cost, mean that it is often uneconomic to resharpen tools (few craftsmen these days sharpen and ‘set’ the teeth of woodcutting handsaws, as saws are relatively cheap at DIY shops). Some weapons can be reused (swords, spears, cannon balls, the stones of slingshots); others not (bullets, shells). Sometimes the cutting tool is sacrificial (a bee sting). Sometimes a broken or otherwise defunct tool cannot be repaired or replaced and there must be consequences. What was the effect of their teeth being ground down by sand, picked up with food, on Ancient Egyptians? Animals who lose their teeth in combat or in old age may no longer be able to feed and they die. This raises the whole ‘chicken or egg’ question of the evolution of teeth and animal diet: which led to which?
The Science and Engineering of Cutting
Figure 1-1 ‘The Multifarious Perforating Machine’ that, among other jobs, was used to punch holes in the plates making up the Britannia Tubular railway bridge to Anglesey at Conway, North Wales. Many of the plates in the bridge were 12 feet long, 2 feet 8 inches broad and ¾ inch thick and the rivets were 1 inch in diameter. The machine was highly adaptable as regards rivet pitch and punches were operated by Jacquard-loom type punched cards. According to Fothergill (1848), the operation of changing plates, weighing 6 or 7 cwt each, was performed by half a dozen men in less than a minute. The machine made 11 or 12 strokes per minute, so (with a 4-inch pitch) a 12-foot plate could be punched in less than four minutes, and allowing one minute for changing plates, twelve plates could be perforated per hour. (Courtesy of IMechE)
It is possible to separate a given solid into pieces by methods not involving cutting tools, by pulling, bending, twisting and so on. A sheet of paper may be torn down the middle as well as cut with scissors. A cotton thread can be snapped in tension by jerking, or cut by scissors or even with the teeth. A plank of wood can be snapped in two by bending, or alternatively cut with a saw. Facial hair can be plucked out with tweezers, but is most often shaved off, except for women’s eyebrows. Holes are usually drilled in wood, but if a drill is not available, they can be made by burning through with a red-hot poker from the fire. Sometimes items separated by tearing will not be exactly the same as those cut into nominally the same shape. The edges of a torn sheet of paper will be rough compared with an edge cut by scissors; burrs may form along the edges of a thin sheet of metal torn into two, which will be absent when cut by shears – unless the shears are in bad condition, when the sheet may fold down between the blades even if they have sharp edges. Whether such differences are important depends on what is going to be done with the separated pieces. Many modern tin cans have ring-pull tops
Controlled and Uncontrolled Separation
and may be opened simply by pulling, but food cans still exist that have to be opened with a can opener which indents, pierces and then progressively propagates the initial hole by levering around the rim of the tin, thus opening the metal lid. The quality of the edge in either case probably does not matter; its sharpness does. Sometimes cutting is not the preferred option (the best long bows are made by splitting the yew, even if an irregular cross-section results, in order to ensure continuity of the wood fibres; spoke-shaven bows have ‘exposed’ grain-ends from which cracks may propagate during flexure). Similarly for split-cane fishing rods: when the nodes are machined off bamboo, the fibres become discontinuous. The best rods have the nodes flattened, so fibres lie in the line of the rod (Vincent, 2008). It is possible to cut the feathers from chickens, turkeys and other birds in preparation for cooking, but that would leave parts of the rachises (Ancient Greek for the spines of feathers) in the flesh, so feathers are plucked off instead (Bonser, 2008). In the cutting of solid bodies, the blade creates new surfaces that are exposed from within the bulk of the body. In layered materials, or glued joints or welds, interfaces already exist and, although the mechanics is the same, the specific work of separation along a pre-existing interface will be different from the bulk material. Adhesion between the pages of a book, or between the sides of plastic bags, has to be overcome to permit separation; adhesion is why gauge blocks (Johansson blocks) have to be wrung apart in the workshop. When the adhesion between a deposit and substrate is not too strong, separation is sometimes achieved by scraping (debris adhered on kitchen counter tops, ice on car windscreens, dirt removed from under the fingernails, mud on dock floors, scrapers in sewage plants and in ore dressing buddles). Errors in writing by pen on parchment or paper can, with care, be scratched off and the surface returned to its original state (it is the origin of ‘to gloss over’). Hygienists at the dentist scrape plaque from the surface of teeth. The Roman strigil was an instrument with a curved blade used to scrape sweat and dirt from skin in the bath-house. The scraper used to remove films of paint from window glass is a tool that can be used in both directions, as can a hoe when weeding. Is there a difference in performance depending on direction? When a pile of soil just sits on a concrete base, the effort of a bulldozer or scraper is principally to move soil about with little effort for soil-to-concrete separation; in contrast, more work is required when the machine digs into a pile of soil to cause soil-to-soil separation. Many processes, which may even employ cutting tools, are more processes of ‘dislodgement’ rather than cutting. Is removal of dog hair from sofas by tearing off with sticky tape, or leg hair by waxing, a cutting process? A pin extracting a winkle from its shell is dislodging it rather than cutting, but the pin does have to penetrate the winkle to give a grip. When eating snails, they are held by a plier-like clamp and then are wiggled out with a two-pronged small fork. How the interface is detached (all at once or in stages) is of interest in these situations. Is the device with rollers rotating in different directions, to split open the skins of grapes before pressing, a ‘cutting’ device? Probably not. Is a toothbrush (or sweeping or scrubbing brush) a cutting tool? Is a wire brush a cutting tool when taking rust off a steel component? Strongly adhered debris requires more work to flick away, and this is achieved with stiffer bristles and more pressing down of the brush into the surface: the forces deflecting the bristles increase and the bristles store more elastic strain energy before the instant at which the force for detachment is attained in the deflected bristle. Is the tongue a tool when licking icecream from a cone? Certainly an icecream scoop cuts, as does the device for making whorls of butter: it is controlled chip formation. Instead of cutting or scraping (in which material is removed), a tool may push ahead a standing wave of material, resulting in a more even spreading of material over a substrate (butter on toast, plastering a wall, painting by brush). Some scraping damages the surface.
The Science and Engineering of Cutting
Wax transferred to clothing from blowing out candles (rather than snuffing them) is difficult to remove: a trick is to use the edge of a piece of ice as a stiff tool to scrape wax away without damage. A painter’s palette knife has varying thickness and hence varying flexibility and is used both for mixing colours and for applying paint to canvas: in action it is the same as a butter knife. The way in which words such as cut, scrape and spread are used is not always consistent. The hand scraping of journal bearings is certainly ‘cutting’ into the bearing metal. Engineers’ blue is applied on both halves of a split plain bearing and high spots are taken away by the scraping tool that is shaped like an isosceles-triangle screwdriver. Its three edges are very sharp and make excellent pencil sharpeners (Dunn, 2008). Again, a cabinet maker’s wood scraper actually cuts at the very small depths at which it is used, as does a holystone used for scouring the wooden decks of ships. So why and when is cutting preferred over other methods? Important reasons for using cutting tools is that control can be introduced into the process of separation, and that there is some precision in the resulting dimensions of the cut piece and its surface finish. The end of a piece of wood broken in bending is rough and splintered; the surface obtained by sawing or planing is much better. Cutting is performed at the location required to give a required size, whereas a snapped piece of wood may break uncontrollably and unpredictably at any location. Furthermore, the forces required for pulling, bending or twisting apart may be impracticably large. Trees in a forest could be felled by bending over and snapping, but the forces would be ridiculously large since the whole tree would have to be loaded and there would be no control. Hence the use of an axe or saw, where the tool concentrates the effort into a relatively small region of material, leaving the majority of the tree unstressed. Limiting the zone in which separation occurs is an important feature of cutting. The sharper the blade the smaller the region where cutting is concentrated and the more effective the effort. Hence paper is cut with scissors if a prescribed path has to be followed. Tearing along a prescribed path is only possible if the material has been weakened along that path by perforating with small holes (postage stamps, lavatory paper) or by creasing; but curved creases are difficult to make. Maps that have been continually opened and refolded eventually fail by fatigue. The ‘teeth’ on sticky tape dispensers make perforations across the width of the tape and permit a chosen length to be unpeeled and removed from the roll. Sharpness is important, as seen when opening envelopes with a paper knife or a finger, with different neatnesses of opening. Some solids (called anisotropic materials) have properties that are different in different directions (e.g. paper is not isotropic and attempts to tear across a sheet often result in tears that curve into the machine direction along the manufactured length; try it with newsprint). A well-known example is timber, which is easily split down the grain but virtually impossible to split across the grain. A woven fabric is another example of an anisotropic material: it is easily torn down the warp or weft directions, but impossible to tear at any other angle (if your trousers get caught on a nail, the tear is always L-shaped, and a missed cue gouging the baize in billiards similarly results in an L-shaped tear). So, while the haberdasher may rip fabric to length when selling by the metre off the roll, the dressmaker must use scissors to cut in other directions (known as cutting ‘on the cross’ or ‘on the bias’): tearing, as an alternative to cutting, is a non-starter. The shop assistant may start the rip with a short cut or nick with scissors, which helps to initiate the tear, in the same way that people bite packaging or rolls of sticky tape with their teeth. This suggests once more that the use of sharp tools to concentrate stresses and strains into a local region, rather than diffusely throughout a body, is an important feature in cutting. Other highly extensible materials such as rubber sheet are similarly difficult just to tear, and some thin sheets of plastic used in rubbish bags and packaging
Controlled and Uncontrolled Separation
for magazines sent through the post just stretch and stretch. Once again, however, nicking the edge often, but not always, enables tearing to take place easily. Even when cutting is the preferred method of separating parts, there can be different tools and devices available to perform ostensibly the same task. For example, the Swiss army knife has a variety of blades. Which is the ‘correct’ one for the job? A sheet of metal may be cut by a bench-mounted guillotine, or with hand shears, or by sawing. It may be possible to perform a given operation with different devices; or the same implement can be employed for different cutting tasks. When ‘correct’ cutting techniques are not followed, it may not matter: hedges trimmed with a chainsaw rather than a hedgecutter may still bud out. Choice of tool may relate to the power available. What provides the required effort? Sometimes it will come from the hands and muscles directly: the surgeon in delicate operations; the cabinet-maker with a chisel; the farm hand with a sickle. Feet and muscle power enable digging to be done with a spade in the garden. Sometimes hand forces are increased through the mechanical advantage provided by levers, as in scissors or garden loppers. Various mechanical aids are now available to help aged or infirm people to garden with conventional tools. Aboriginal spear throwers, which project weapons at higher velocity than possible merely by throwing by hand, rely on similar ideas. Work (the same as energy) is done when a force F moves its point of application through some distance , where work F (newton-metres, Nm, called joules with symbol J). The same work is performed when F is high and small, or vice versa. That is why a lightweight person at the extreme end of a seesaw can balance someone heavier sitting much closer to the pivot. Thus, by employing a machine (and the garden shears or any system of levers is a machine), a small force at the handle may be made into a big force at the point of cutting, by making the hand force move through a large distance. This will be reacted by a large force over a small distance where cutting is to occur. It is an example of mechanical advantage or changing gear. Inefficiencies in the machine, principally friction in hinges, pivots and bearings, will reduce the available work, as is well known to gardeners who have left tools out all winter in the rain. In a paper guillotine where the long cutting blade is pivoted at one end, cutting is done progressively along the edge rather than all in one go. To cut through the thickness all at once, a big force moving through a small distance (the thickness of the stack of paper) would be required to perform the necessary work, but the same work is done by a smaller force applied at the handle moving through a much greater distance. The idea of ‘force times distance’ is the reason why tools are sometimes struck by hammers or mallets to get a job done. It could be that the force required to push a chisel, say, by hand into a workpiece was just too large even with sustained great exertion. Perhaps a weight could be lifted into the air and dropped on to the chisel held against the workpiece. The chisel would enter the workpiece a smaller or greater distance depending on the resistance to penetration. If the weight were too large, or lifted too high, or the resistance were too small, the chisel might go right through in an uncontrolled fashion. Nevertheless, the work done by muscles in lifting a weight over some distance and giving it potential energy has been made to do the work of cutting over a much smaller distance. That is not to say that all the energy has been transmitted upon impact (see Section 2.5). When a hammer is used, muscles in the body accelerate the hammer head as it descends, so that more energy is available than if the hammer were simply allowed to fall on the job. How muscles work, and the power that may be expected from them, is discussed in Section 2.4 and in French (1988). When cutting tools are driven other than by muscle power, the drive may just remove the human effort (electric hedge trimmers, powered lawnmowers, powered log splitters, where the jobs could just as well be done by hand tools). Human effort is much reduced by powered devices but the machine has to be purchased and its running costs covered. Animal-hauled
The Science and Engineering of Cutting
ploughs had greater power (i.e. the rate of doing work with units of joules/second, J/s, called watts, W) to cut a deep furrow than did ancient simple scratch ploughs. A driven tool also often speeds things up. Thus a chainsaw is far speedier than a reciprocating saw at felling a tree. The same job done by hand may require a comparable amount of work but takes a far greater time, as human effort is limited by the way muscles work. In addition to eliminating human effort and speeding up cutting, the introduction of power drives may have eased problems of dimensional control of products. Thus steam-driven and later electrically driven circular saws replaced reciprocating hand saws and the saw pit. It is remarkable to recall that wood veneer was once made by sawing by hand, instead of as nowadays peeling from a log in a machine. Machine tools in a sense deskill hand operations but quicken up time. Where once parts were made by cold chiselling and filing, they are now manufactured by milling on computercontrolled machining centres. Nevertheless, we should always remember the tremendous skill required, and the limited tools then available, to produce the first screw threads and first gauge blocks by hand. Opening packaging can be troublesome and sometimes there is a need to stab at it before the contents can be separated, rather like the ‘jab’ of a medical injection. The properties of materials alter when loaded at different speeds (silly putty is an extreme example); they also alter when the temperature changes (the proverbial hot knife through butter) and when the environment changes (a brittle dry biscuit compared with a soggy biscuit after being dipped in a cup of tea). Strain rates and temperatures in some sorts of cutting are high, much greater than in the usual sort of material property testing. They can have subsurface effects. In grinding steel, for example, there are high local temperatures well above 1100°C regardless of cooling. Changes occur to the microstructure within 5–10 m of the surface. In contrast, lapping, polishing and other slow methods of superfinishing affect the subsurface down only to 1 m because the temperatures and forces are low. Other interesting questions arise about the speed at which cutting is performed. Beyond the property changes, something else happens with speed of cutting. Even the sharpest scythe will not cut hay or corn if the implement is merely pressed up against the stalks. There is a minimum speed above which cutting will occur. Why? Again, despite the bluntness of its cutting string, a strimmer impacting blades of grass will cut at a sufficiently high speed: battery-driven strimmers fail to cut at low speeds when the battery runs down. Sometimes energy can be stored in a cutting device before doing a job, such as in a catapult or bow and arrow. Flywheels prevent machines from stalling when energy is taken out of the system to perform some operation. It appears that in the early days of trying to understand the parameters that influenced cutting, it was believed that the forces and power required depended on some sort of ‘cutting stress’. For example, in a series of machining tests performed by the Manchester Association of Engineers in 1903, cutting forces were related to the area of the uncut chip without, it seems, any worry about the combination of width of cut and depth of cut to produce that area. Furthermore, there was an understanding that ‘tenacity’ (what we now call toughness) was as important in cutting as strength or hardness (they are proportional to one another). People did not know how best to quantify the idea or how to incorporate it in models of cutting. Bengough (in the discussion of Turner, 1909) said that ‘… there was the assumption by engineers that the difficulty of cutting or working a metal was a measure of its hardness. Of course, high carbon steel of great penetration hardness was more difficult to machine than a medium carbon steel of smaller hardness;
Controlled and Uncontrolled Separation
but hardness was not the only factor which entered into the question of difficulty of machining. Another was its toughness, that was the amount of deformation a metal would undergo before fracture, and the toughness was of course measured by the elongation in the tensile machine. A tough metal of low penetration hardness, such as manganese steel, might be more difficult to machine than a steel of greater penetration hardness but less toughness. Boynton had put that point well when he showed experimentally that more energy was used in boring (with a rotating diamond) cementite than corundum, since the former had some slight ductility not possessed by the latter. Within certain limits and with certain steels it so happened that ease of machining and penetration hardness varied directly with the tensile strength, and inversely with the elongation. With very soft steels difficulty of machining increased simultaneously with loss of penetration hardness, owing to the increase of toughness. With hard steels difficulty of machining might increase with increase of penetration hardness and loss of toughness …’. Whatever the method of separation into separate parts, the process concerns fracture, and the mechanics of cutting, perforation and so on is a branch of elastoplastic fracture mechanics. Not only yield properties at very high rates of strain (Drucker, 1949), but now also fracture toughness may be determined from cutting experiments, Atkins (2005). The rates and temperatures achievable in cutting can be beyond the range of strain rates achievable by special apparatus such as the split Kolsky–Hopkinson bar. Furthermore, cutting at different rates and temperatures is continuous, in steady state. Therefore, there is time for steady forces to be measured and videos to be taken, at steady rates and steady temperatures. In contrast, most mechanical property measurements are determined in experiments that are transient.
Chapter 2
Fracture Mechanics and Friction Muscles, Impact and New Surfaces Contents 2.1 Introduction 2.2 Fracture Mechanics 2.3 Friction in Cutting 2.4 Muscles 2.5 Impact Mechanics and Hammering 2.6 Work of Formation of New Surfaces
11 11 23 24 28 31
2.1 Introduction This book is aimed at a wide audience. Experience of teaching materials mechanics to students of biology, food science and biomimetics tells me that help is required on mathematical analysis and modelling of problems. The first part of this chapter aims to cover the basics of stress analysis, fracture mechanics and ideas on friction, as these are all central to the ideas in the book. The mathematics should be familiar to engineers, materials and physical scientists, but they, in turn, require help from biologists to understand how biomechanics relates to biological design and function. Later parts of the chapter deal with how muscles function, since the force and work necessary for cutting are often supplied by hand or foot when animals attack prey and afterwards chew food in the mouth. Forces for cutting may be provided as a slow push or a fast blow. In turn, that takes us to hammering and impact. All cutting generates new surfaces and the final part of the chapter discusses whether this process requires significant work or not.
2.2 Fracture Mechanics 2.2.1 Load–deformation curves and stress–strain curves To determine the mechanical (strength) properties of materials, a body (testpiece) may be deformed by applying known loads to it, for example by hanging on weights to the suspended body. Deformation may be pulling, compressing, twisting, bending and so on, including combinations of these different ways of loading. The resulting deformation (extension in tension, reduction in height in compression, angle of twist, rotation in bending) can be measured. Alternatively the same body can be deformed by a known amount, and the loads required to achieve the deformation measured. Figure 2-1(A) shows various types of load–extension diagrams that may result from such experiments under increasing load or deformation. The simplest is a straight line whose stiffness (the slope of the line) is constant. Alternatively, the lines may be curved up or down with uniformly changing slopes and thus increasing or decreasing stiffnesses. The curve Copyright © 2009 Elsevier Ltd. All rights reserved.
11
12
The Science and Engineering of Cutting Stiffening
B
Linear
Rubber
C Load
Load
Softening
Hysteresis D
E 0 A
0
Deformation
Deformation
B
Load
B
Irreversible deformation Yield point
0 C
X Deformation
Y
Figure 2-1 Load–deformation curves: (A) linear, stiffening, softening and the sigmoid-shaped curve for some types of rubber; (B) hysteresis loop formed after a loading–unloading excursion; (C) load–deformation curve for a material where part (XY) of the deformation is recoverable on unloading and part is irreversible (OX).
with increasing stiffness (a so-called J-shaped curve) is characteristic of extensible biological tissues. The sigmoid-shape curve (for some rubbers) is stiff at small extensions, less stiff at intermediate extensions and very stiff before fracture. Non-linear load–deformation curves may be caused by the material itself having non-proportional behaviour, or result from certain geometric configurations in structures made of linear materials (shells and domes at large deflexions, for example). How much deformation is possible before something breaks depends upon the material itself and whether there are defects/flaws in the body, which may range from visible holes in the body to microcracks. The action of deforming a body requires work to be done. Work (also called energy) is given by the product of force displacement, so the energy dissipated by friction, when a brick is slid across a surface by constant force F over distance , is given by F. During the deformations shown in Figure 2-1(A), the force is not constant as bigger loads are required to produce bigger deformations. The work done must be determined incrementally and is given by Fd, i.e. by the area under the load–deformation curve. When an algebraic relation is known between F and , the integration can be performed. For a linear material F (constant), so incremental work done is Fd (constant) d (constant)2/2 0.5F, the area of the
Fracture Mechanics and Friction
13
triangle between the linear stiffness line and the -axis. Some stiffening and softening curves may be represented by F (another constant)n, where n 1 for a stiffening (J-shaped) curve and n 1 for a softening curve. Then Fd (another constant)(n1)/(n 1) F/(n 1). If bodies are unloaded somewhere before fracture, various responses are possible. Whenever load–deformation during unloading retraces exactly the same path as for loading, the behaviour is elastic (reversible). Engineers have been interested mostly in linear behaviour (Hooke’s seventeenth century law for springs is still relevant for returning computer keyboard keys) but reversibility is possible for all the types of non-linear behaviour shown in Figure 2-1(A), as common experience with rubber bands tells us. In these circumstances, all the work put in to deform the body, and stored in the body under load, is recovered. What happens to the energy should fracture occur before unloading is the vital question in fracture mechanics. When fracture occurs within the reversible range of deformation, the material is said to be brittle (so an elastic band is just as brittle as glass or rock in this sense, although brittle is normally associated with ‘hard and stiff’ solids). When different paths are followed during loading and unloading, the classification of behaviour depends on whether load–deformation returns to the origin (no deformation at no load), or whether there is some permanent deformation at zero load. In the first case (Figure 2-1B), the strain energy stored when loaded to point B is given by the area OCBEO. The strain energy recovered is given by area ODBEO, so the work represented by area OCBDO has been lost and means hysteresis, i.e. irreversible behaviour where energy has been dissipated even though the body returns to its original state. Friction between cutting tool and workpiece can generate hysteresis loops (pulling out an axe stuck in a log where friction opposes motion both into and out of the wood), but often hysteresis results from ‘internal’ material behaviour. In the second case, there is permanent deformation (OX in Figure 2-1C), only displacement XY being recovered elastically, and only the work represented by area XBY being recovered from the whole work area OBY just before unloading. Irreversibility is often caused by plasticity, and the yield point where irreversibilities begin is indicated in Figure 2-1(B). The combination of plasticity and some recovery is familiar when opening ring-pull cans of food where the lid is permanently curled (plasticity OX) but is still springy (XY, springback). When OX is extensive before fracture supervenes, the metal is said to be ductile. Irreversibility in other materials may be caused by other mechanisms, e.g. sliding of fibres over one another in paper. When XY is negligibly small, elastic deformations may be neglected, and permanent deformations predominate (rigid-plastic deformation, as in putty, and is an assumption often made in analyses of large-deformation metalworking). Some materials are time dependent in their load–deformation behaviour, meaning that under a fixed load, the deformation does not stop immediately but (in tension, say) stretching continues for some time, perhaps eventually stopping, or perhaps never stopping (viscoelasticity and creep); or that the force to keep a body stretched to a given displacement decreases with time (relaxation). Load–deformation behaviour depends upon the temperature (hard butter from the fridge softens in front of the fire) and the rate at which loading takes place (slow versus impact responses); it also depends upon the environment, water particularly affecting the behaviour of biological materials (it is easier to shave and cut nails after a bath or shower). Polymers respond very differently in the presence of solvents. Deformation behaviour may vary with direction (anisotropy), familiar in the easy splitting of wood along the grain but the virtual impossibility of splitting in other directions. Whatever the load–deformation behaviour, different sizes and shapes of bodies require different loads to produce a given deformation; or the same load gives different magnitude
14
The Science and Engineering of Cutting
deformations in different-size bodies. In consequence, for every body there is a different load– deformation plot. This is not very helpful, but the results may be rationalized by using stress instead of load and strain instead of deformation. Stress is defined as force F per unit area A, so F/A with units N/m2, called the pascal (Pa). If the area Ao over which the force acts changes only imperceptibly in size with increased loading, we divide F by Ao to give engineering or nominal stress (Figure 2-2A). Nominal stress is acceptable throughout the elastic range of engineering metals and some other solids. If deformations are extensive and A changes markedly (as in highly extensible elastomers and particularly in metals undergoing plastic deformation) it is better to divide F by the current A to give the true stress. Strain is geometric and does not depend upon the relationship between stress and strain. There are two major versions of strain: when deformations are small, engineering or nominal strain is given by the change in a dimension divided by the original dimension. For axial extension of a bar of length Lo (Figure 2-2B) to length L, the extension is (L Lo) so the strain is /Lo (L Lo)/Lo (L/Lo) 1. Strain is a ratio and has no dimensions. When deformations are extensive, true or logarithmic strain is used and is given by ln(L/Lo). When the volume of the body V LA remains constant during deformation, true stain may be written in terms of cross-sectional areas A to give ln(Ao/A), that in the case of rods becomes 2ln(do/d), where d is diameter. In tension, A reduces as L increases (and vice versa for compression), so log strains have signs, positive for tension and negative for compression. This is the usual sign convention for stresses as well. (In some fields where it is known that stresses are mostly F Area Ao Lo
L
δ B
A δs F
Area of top Ao
θ L
C
F
Figure 2-2 Stress and strain. (A) A body of cross-sectional area A0 is loaded in tension by a force F to give a tensile stress of F/A0; (B) its original length L0 is extended by to give a current length L. The axial strain is ε /L0; (C) a body of cross-sectional area A0 is loaded in shear by a force F to give a shear stress of F/A0. The top face slides by distance S relative to the bottom face to give a shear strain S/L where L is the height of the body. For small displacements, S/L tan .
Fracture Mechanics and Friction
15
compressive, negative signs are omitted.) The stretch ratio (L/Lo) is often used as a measure of strain in rubber elasticity, rather than engineering or true strains (see Sections 8.2 and 8.3). A 100 per cent tensile engineering strain corresponds to a doubling of length. A 100 per cent compressive engineering strain would require the body to reduce to zero height, which is silly. This makes clear the need for a different definition of strain at large deformations. Intuition suggests that doubling the length should correspond to halving the height and, according to log strains, ln2 ln(1/2) and this gives the sign as well. Log stresses are greater than engineering stresses; log strains are smaller than engineering strains, and there are relations between them (Felbeck & Atkins, 1996). Nominal and true stresses, and nominal and true strains, are often distinguished by the use of different symbols, but in this book we use for stress and for strain for both. It will be clear from applications which definition is meant. In sliding deformations (shear), similar definitions apply for shear stress and shear strain (Figure 2-2C) with FS/Ao and S/Lo tan. [Do not confuse this use of for shear strain with its use for surface free energy; Section 2.6.] In place of a myriad of load–deformation curves for different sizes and shapes of testpiece of a given material, conversion of loads to stresses and deformations to strains gives single stress–strain curves for every different type of loading (tension, compression, torsion and so on). When these stress–strain curves start with linear portions that are reversible, characteristic elastic moduli can be identified for materials such as the Young’s modulus E for tension and compression given by E /, and the shear modulus G given by G /. These single values for moduli replace the multitude of stiffness lines for different-size bodies. The theory of elastic deformation starts with these basic constitutive relations and gives the well-known formulae for simple bending, torsion and so on in linear elasticity. It goes on to deal with complicated problems such as contact stresses (ball bearings). There are theories for reversible non-linear elasticity too. The same sorts of ideas about stress and strain are employed in the theory of irreversible plasticity (Hosford & Caddell, 1983). The area beneath stress strains curves given by d and d may be shown to be the work done per volume of material and will be used in this book to help solve cutting problems. When a balloon is inflated, the stresses in the skin (membrane) which are in equilibrium with the internal pressure p are not simple tension but are biaxial tension (two tension stresses 1 and 2 at right angles). In a spherical balloon, 1 2 pr/2t where r is the radius of the balloon, so that we have equibiaxial tension. In a cylindrical balloon, while axial pr/2t, hoop pr/t (where hoop means circumferential as in a barrel) so we have unequal biaxial tension. In general, bodies under complex loadings may be thought of as being loaded by three mutually orthogonal stresses 1, 2 and 3. How to determine the resulting strains in different directions in the body during elastic, elastoplastic or rigid-plastic deformation is explained in standard texts. Uniaxial stress–strain curves obtained in the way described presume that the sample has no defects such as cracks, that the behaviour of individual microconstituents is smoothed out and that the behaviour is experimentally reproducible. In truth some materials, such as grey cast iron that is full of graphite flakes, are ‘defective’ and full of sharp-ended cracks (the graphite has no strength compared with the surrounding metal matrix). The load–deformation behaviour is, however, experimentally reproducible and in such cases, what is determined as characteristic elastic moduli and quoted in handbooks are really ‘effective’ moduli in the presence of the cracks. Without the cracks the stiffness of the cast iron would be that of the matrix which is higher than that measured in experiment. For this reason, the Young’s modulus of cast iron depends on the flake size and is lower for longer flakes. This leads us to ask what happens when a supposedly flaw-free body, from which we might expect to determine moduli, has a crack in it when tested (which we may not be aware of) and which will affect its load–deformation
16
The Science and Engineering of Cutting
response, so that ‘false readings’ of stress and strain will be given. This, in turn, takes us to the subject of fracture mechanics. Traditional engineering design is based on employing ductile materials, and analysing the stresses throughout a component or structure under normal working loads (and perhaps under accidental ‘overloads’) to make sure that the stresses everywhere remain below some proportion of the yield stress y (y/f, say, where f ( 1) is the factor of safety). In this way, the behaviour of the component or structure remains within the elastic range and is thus reversible; the deformations will be small when materials having large E are used. So when a train crosses a bridge, say, the bridge neither sags excessively nor remains bent after the train has passed over. Such calculations assume both that the materials used in the design have no flaws in themselves and that no flaws will be introduced where components are joined together (welds, glue). In use, should loading for some reason result in stresses that locally exceed the design stress and even attain or surpass the yield strength of the material, the concept is that there will be localized yielding ‘because the material is ductile’, thus locally relieving the overload. Unfortunately this does not always happen: fracture mechanics gives the reasons why. Fracture mechanics also explains some strange ‘scale effects’ whereby large structures may fail in a brittle fashion, even though they have been made of materials that are ductile when tested in the laboratory.
2.2.2 Behaviour in the presence of cracks Traditional design is inadequate when cracks or other defects exist. Traditional analyses are not able to predict, for example, either the safe working stress in the presence of a known flaw or the critical size of flaw just tolerable with a given working stress. Some readers may be puzzled that the topic called Strength of Materials is not about materials but rather seems to be about stress analysis. The title comes from the days of traditional engineering design and so-called theories of strength that specified simple limiting stresses for ‘perfect, flawless materials’ which should not be exceeded. For practical purposes, fracture mechanics is the strength of materials for bodies having cracks. First, we ask what happens when we load a body containing a crack (Figure 2-3A). Cracks reduce the stiffness of a body (or increase its compliance, that is the reciprocal of stiffness). Longer cracks lower the stiffness more than shorter. Thus if OB in Figure 2-3(B) represents the stiffness of a flaw-free body (from which moduli would be obtained), OC, OD, etc., are the stiffness lines for ever-longer cracks a1 a2, etc. Suppose that a body having crack length a1 begins to crack in a controllable fashion at constant load along LM (meaning that we can stop and start it at will) and that we unload the body when the crack has propagated to length a2. Unloading will take place along the stiffness line appropriate for the longer crack length, i.e. down MO. After unloading, the body will be the same as before, except that it has a longer crack. Let us do an energy balance from zero load up to the load at L and back down to zero load again. Just before cracking began, the body had stored elastic energy given by area OLQO. During crack propagation, the constant load moves its point of application over distance LM QN. During this movement, it therefore adds energy to the body (‘external work done’) given by work area QLMN. On unloading, energy given by OMN is recovered. When we balance the energies present just before cracking, put in during cracking and recovered after cracking, we conclude that the energy represented by area OLMO has ‘disappeared’. It has not been lost but has been dissipated as the work required to propagate the crack from length a1 to a2. The mechanical property called the specific work of crack propagation, or fracture toughness, given the symbol R quantifies the work required. R for a material is defined as the energy required to propagate a crack by unit area, and has units of J/m2. Area is employed
17
Fracture Mechanics and Friction F, δ
Crack length B O C a1 S
F
Crack length a
L
M
Q 0 A
D
a2
N δ
B
Ma
L Md Me
F
Mc
S
Mb
L
Crack length a1 a2
F
M
Mf Q
T V C
δ
D
N
δ
Figure 2-3 (A) Body having crack of length a loaded by force F having displacement ; (B) loaddisplacement stiffness lines for increasing crack lengths. Crack of length a1 propagates at constant load along LM to crack length a2 at which point the body is unloaded; (C) the many possible load paths during propagation; (D) the same ideas for a cracked body whose load-displacement is non-linear.
rather than crack length because it is possible to have cracks with triangular or elliptical plan views. Thus in Figure 2-3(B), area OLMO in newton-metres joules, is the work required to generate new crack area of size w(a2 a1), where w is the (uniform) thickness of the body in Figure 2-3(A). It follows that R [OLMO/w(a2 a1)] and loading/cracking/unloading experiments are useful ways experimentally to determine R for materials (Gurney & Hunt, 1967), without recourse to algebraic formulae, finite element methods (FEM) or other techniques. While it is possible for cracking to take place under constant load as assumed above, the locus of (F,) during propagation takes a variety of paths depending upon the geometry of the cracked body and the manner of loading. As shown in Figure 2-3(C), cracking can occur (i) under increasing while F increases or decreases along LMa, LMb, LMc, etc.; and also even (ii) where decreases, along LMf, which may seem odd. In all cases, however, and irrespective of what the load does along the segment LM, the triangular work area OLMO Rw(a2 a1). The path LM depends on whether the energy available at crack initiation is sufficient to feed the toughness work required by propagation (LMe), or whether external work has to be supplied during propagation (LMa to LMd) to keep the crack going, recognizing that when the load changes the elastic strain energy stored in the body is also changing, and everything should balance. The path LMf takes us into questions of crack stability. It was assumed above that cracks propagate in a controllable fashion. If we do an energy balance for path OLMfO, we find that work area LVTMfL is work that can be removed from the system during propagation, yet still
18
The Science and Engineering of Cutting
leaving enough for the work of fracture given by OLMfO. Since most loading systems do not run backwards during cracking, we cannot control propagation. Experiments where weights are hung on bodies are load controlled and cracking is controllable only when an increased load is required during propagation, path OMa and (just) path OMb. Displacement-controlled experiments achieve control of cracking for those paths where increases, but paths where decreases are not controllable. The fascinating topic of crack stability and how to design testpieces to achieve controlled propagation forms the topic of Chapter 8 in Atkins and Mai (1985). We can put the graphical understanding of crack propagation into algebraic form by writing down an incremental energy balance during the period in which the crack area A increases by dA, while the load-point displacement increases by d. We have
(2-1)
Fdδ dΛ RdA
in which Fd is the incremental external work done, is elastic strain energy and RdA is the incremental work of crack propagation. has been left in general form for use with nonlinear cracked bodies, but for the linear bodies of Figure 2-3, are the triangular areas OLQ, OMN, etc., given by F/2 from which d (Fd dF)/2. Substituting in (2-1) gives
Fdδ δdF 2RdA
(2-2)
Dividing throughout by F2 we recognize the left and side as d(/F) so
F2 2R/[d(δ/F)/dA]
(2-3)
Relation (2-3) is called the compliance calibration equation of linear elastic fracture mechanics (LEFM) because (/F) is the compliance – the reciprocal of stiffness – and if it is known how the compliance alters as the crack length alters (either experimentally or algebraically from beam theory and suchlike), the rate of change of compliance with crack area d(/F)/dA will be known, so that (2-3) is ‘calibrated’. The relation may then be used either to predict F at a particular crack length when R is known, or R may be determined from experimental cracking loads F in specimens having known A. Owing to the different ways in which the subject developed, different symbols are used for the same thing in fracture mechanics. In connexion with Eq. (2-3), R is written as GC in older papers. The behaviour of cracked bodies that display non-linear F- behaviour is shown in Figure 2-3(D) for a J-shaped stiffening curve; the same principles apply for cracked bodies that soften. An energy balance for loading/cracking/unloading along path OLMO reveals once again that work areas OLMO Rw(a2 a1), the only difference being that the strain energies OLQ and OLN are bounded by curved, rather than straight, lines. Notice that in the linear case, the area OLQ for the elastic strain energy is equal to the area OLS: in non-linear cases this is no longer true. We therefore distinguish between the elastic strain energy OLQ in Figure 2-3(D) and the so-called complementary strain energy dF [F ] area OLS. Partial differentiation of Eq. (2-1) gives
R ∂Λ/∂A δ ∂Ω/∂A F
(2-4)
Equation (2-4) may be used for non-linear cracked bodies when F is known as a function of and A. Note that when there is no crack propagation, i.e. dA 0 in Eq. (2-1), we obtain
F ∂Λ/∂δ
F
and δ ∂Ω/∂F δ
(2-5)
Fracture Mechanics and Friction
19
which are the well-known Castigliano theorems for loads and displacement in elasticity. In fact, the energy approach in fracture mechanics is really Castigliano’s method extended to include cracks. In place of loads or stresses at cracking from which to obtain R, energies can be employed, using expressions such as
R ηΛ/wb
(2-6)
where is the work area (linear or non-linear) up to the instant of fracture, w is specimen thickness and b (W a) is the length of the remaining ligament. The factor depends on the geometry of the cracked body and the type of loading, and is obtained from the relation between areas such as OLQ and Fcrack. For a single-edge-notch three-point bending testpiece, 2; for other cracked bodies is given in handbooks (see also Atkins & Mai, 1985). A general energy balance equation for fracture is
Fdδ dΛ RdA dΓ dKin dChem
(2-7)
where is plastic work, Kin is kinetic energy within the specimen during cracking (impact fracture) and Chem is chemical energy (corrosion, solvents, etc.). The irreversible work of fracture RA is confined to thin boundary layers contiguous with the crack faces, which is why the unloading lines in Figure 2-3 return to the origin of displacement. The stress state within a body remote from the crack depends on circumstances. It might be elastic, in which case the body behaves in a globally elastic fashion, and the broken bits may be refitted together to regain the original shape and size of the starting component, as assumed in Figure 2-3. In that case, only (d RdA) appears on the right-hand side of Eq. (2-7), resulting in (2-1) and the expressions deriving from it. Griffith, the father of fracture mechanics, was able to evaluate Eq. (2-3/4) in 1921 for a small crack in a large sheet, and arrived at a formula for the stress at fracture which is
σcrack (ER/πa)
(2-8)
where E is Young’s modulus and a the size of the crack that extends through the thickness of the body. We see that crack depends upon both stress and size of crack, not just stress alone. The cracking stress is high when the crack size is small and vice versa. There is a critical crack length below which the crack will not run for a given applied stress, and a critical applied stress below which a crack of given length will not run The lowest possible value for R is the thermodynamic surface free energy that Griffith believed was correct for his experiments on glass, so he wrote in place of R (see Section 2.6). While the energy line of attack is straightforward to understand, mainstream LEFM developed along a different, parallel, line starting with the complicated stress and strain fields around the tips of cracks, rather than global energies. The results for loads or stresses at fracture are the same by both routes, of course, but a different mechanical property parameter for resistance to crack propagation arose, namely the stress intensity factor K due to Irwin. It must not be confused with the much older idea of the stress concentration factor Kt that deals with discontinuities in section where irregularities in uniform stress distribution occur, where local stresses can be far above average values obtaining remote from changes in cross-section.
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The Science and Engineering of Cutting
Thus for a large plate containing an elliptical through-hole of semi-axes a and b, the stress at the ends of the ellipse is
σ σo (l 2a/b)
(2-9a)
where o is the remotely applied uniform stress. The stress concentration factor Kt for the points at the ends of the a-axis is
K t σ/σo (l 2a/b)
(2-9b)
For a circular hole a b, and the local stress is three times the average stress. The stresses drop off rapidly with distance from the discontinuity. Tables and charts for Kt in various geometries are available in standard texts. Geometries include members with grooves, notches, fillets, etc., in axial loading, bending, torsion and combined loading. The magnitude of Kt depends on the ‘abruptness’ or ‘sharpness’ of the discontinuity. Traditional engineering design takes account of stress concentration factors when the geometry of the discontinuity is known, but the danger is that when a b in Eq. (2-9a), as in crack-like defect, Kt can be very high indeed. If the component or structure is proportioned so as to keep the local stresses at severe discontinuities below y, most of the rest of the structure will be operating inefficiently at very low working stresses. Fracture mechanics deals with the whole field of stress and strain around discontinuities, not merely at one point, to determine the critical loads for fracture. What it is doing is working out or and evaluating Eq. (2-4), to arrive at formulae for stress intensity K. They have the general form
K σ (πa).Y(a/W)
(2-10a)
where is applied stress and a is the size of the crack (in general, ‘size’ may be length, halflength, depth of the crack depending on circumstances; for a small crack in a large plate, a is the half-length of the crack that extends through the thickness of the body). W is some characteristic dimension of the cracked body. Y(a/W) is a non-dimensional ‘shape factor’ that depends on the geometry of the body, orientation of crack and the way in which loads are applied. When a cracked body is loaded, a stress intensity appears around the crack tip and if the body is unloaded before cracking occurs, K disappears in the same way that stress (or indeed strain) increases or decreases in a flawless body. However, experiments show that cracking takes place at a critical value of the stress intensity factor, called KC, and this becomes another mechanical property to indicate resistance to cracking. Then, for fracture at stress crack,
KC σcrack (πa).Y(a/W)
(2-10b)
KC has peculiar units, N/m3/2 or Pam, and this makes it difficult for students to get a physical feel for the parameter when fracture mechanics is not taught starting with energy. Comparison of Eq. (2-10b) and the Griffith expression (2-8) shows that
KC2 ER
(2-11)
The reason for the peculiar units is now obvious: KC is a combination of two more easily understood parameters, i.e. Young’s modulus E and the specific work of fracture R. The attainment of a critical stress intensity factor at fracture is the same as satisfying an energy-based
Fracture Mechanics and Friction
21
criterion determined from the integrated stress and strain fields around the crack and in the body generally. Y(a/W) generalizes the Griffith formula to any type of cracked body. Stress intensity factors are often written with Roman number subscripts I, II or III. The subscripts represent the mode of cracking, i.e. the way in which separation occurs at the crack tip. Mode I refers to crack opening by simple tension; mode II to in-plane shear cracking where the crack faces slide along one another; mode III also refers to shear, but to cracking by out-of-plane (twisting) sliding motion across the crack faces. The notation is sometimes used inconsistently, since a subscript I is often employed to indicate tensile cracking in plane strain (thick plates, where the critical stress intensity factor KIC is least owing to high hydrostatic stresses, see below, thus leading to conservative design), whereas it is just as possible to have tensile cracking under plane stress (thin sheets) where toughness is greater owing to less constraint). A further confusion is that critical stress intensity factors are sometimes called the ‘fracture toughness’. In this book we rarely employ K and by fracture toughness we mean the specific work of cracking R. R may also be written with subscripts, but this is rarely done. In any case, much cracking takes place in mixed mode, meaning that the crack opening has both tensile and shear components, as when the path of cracking is curved. There are non-linear versions of Eq. (2-10b) (Atkins & Mai, 1985), but they have rarely been employed for true reversible non-linear fracture. Rather, they have been used for elastoplastic fracture where unloading lines do not return to the origin after some propagation. This procedure is justified by the equivalence between non-linear elasticity and total strain plasticity (Kachanov, 1971). In simple terms, when loading up along a curved path, as in Figure 2-3(D), the body does not know whether on unloading it will be reversible or irreversible. Thus toughness at the initiation of elastoplastic cracking may be calculated using Eq. (2-4), treating all energies beneath the load-displacement curve ( and in Eq. 2-7) as being reversible even if is not. Owing to history, yet another symbol JC is employed in place of R for elastoplastic fracture. However, predictions of loads and displacements during subsequent propagation under decreasing load will be wrong when calculated from non-linear elasticity, because the theory believes that all the strain energy is recovered, whereas only part of it is (cf. Figure 2-1D). Relation (2-6) may also be used for the determination of R in elastoplastic fracture, thinking that and together are reversible, but here again problems arise when accumulated energies during propagation are employed: the crack resistance at initiation is correctly determined but the resistance during propagation (called JR) is overestimated as the energy includes remote irreversibilities that have nothing to do with cracking, and make answers specimen dependent. Should any of these difficulties arise later in the book, they will be explained in the context of the particular cutting problem. In cutting, where the surface has been formed as a crack, the loads are applied by the cutting blade on the faces of the crack. In most fracture problems the loading is remote from the crack surfaces; but a crack on the inside of a pressure vessel, say, is loaded by the pressure and there is a fracture testpiece – the wedge opening loading (WOL) specimen—where a wedge is forced down a pre-existing starter crack to produce cracking, that is really just cutting.
2.2.3 Scaling in fracture mechanics Equation (2-10b) applies to any size of cracked body made of the same material providing that the behaviour is linear elastic. We can write out the fracture stresses for two bodies of different size, one a prototype (p) and a model (m): KC )p (σcrack )p (πap ).Y(a/W)p
KC )m (σcrack )m (πam ).Y(a/W)m
(2-12a)
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The Science and Engineering of Cutting
The ratio of the two fracture stresses is obtained by dividing these two equations. If one body is an all-round magnification of the other, where the ratio of corresponding lengths (magnification, or scaling, factor) is , we obtain
(σcrack )p (σcrack )m / λ
(2-12b)
where ap/am Wp/Wm and assuming that the material property KC does not change with size. Thus larger geometrically similar cracked bodies require smaller stresses for fracture than smaller bodies. This is strange if one is used to events being determined by a critical stress that does not change with size. It comes about owing to so-called cube-square scaling where the energy available for cracking depends on the volume of the cracked body but the energy required depends on the area. The reduction in fracture stress with size may result in a body plastically deforming in small-size pieces but fracturing in the elastic brittle range in large pieces, because crack for the large body has been reduced below the yield strength y, whereas crack for the small body was below the yield strength y. This is dangerous for design and may lead to unexpected failures. Furthermore it means that classification of materials as being ductile or brittle on the basis of laboratory-size testpieces is misleading because, for example, we shall show that glass can be machined at very small depths of cut to give ductile continuous chips, and ductile solids appear to be brittle when in very large pieces. For similar reasons, rocks can be crushed (comminuted) down only to limiting sizes after which they flow rather than fracture. There are scale effects for non-linear elastic and for elastoplastic fracture similar to those given here for linear elastic fracture (see Chapter 9 of Atkins & Mai, 1985). Scale effects occur throughout cutting mechanics where, in the same material, ductile continuous chips are formed at small depth of cuts, discontinuous chips at intermediate depths of cut and brittle chipping at largest depths of cut. Chapter 4 investigates all this sort of thing.
2.2.4 Notch sensitivity Notch sensitivity concerns by how much the load-bearing capacity of a body containing a crack or notch is reduced compared with the uncracked body. There are two definitions of notch sensitivity depending on whether the material is ‘normal’ (like ordinary engineering materials with good shear connexion between elements giving the usual isotropic moduli relations) or whether the material is fibrous and anisotropic where the degree of fibre-matrix coupling may not be perfect. Considering the second case first, a material is notch sensitive if, on reducing the load-bearing area (making a cut in from the side, say) the stress to break reduces by more than the percentage loss of bearing area. Thus a bundle of isolated filaments will be notch insensitive, as will a well-oiled wire rope. But a rusty wire rope is notch sensitive to some degree, depending upon the slippage between elements during loading. For no relative slip between fibre and matrix the notch sensitivity is greatest; that is what fracture mechanics is all about in ‘normal’ materials – a crack reduces the strength disproportionately. That takes us to the second definition of notch sensitivity where three terms are employed for metals: (i) simply brittle; (ii) notch-brittle; (iii) simple ductile, and relates to the behaviour of plain and deeply notched testpieces (usually, but not necessarily, tensile specimens). The idea comes from a time when fracture mechanics was not fully developed, and essentially it all comes down to the ER/k2 ratio, where k is the yield stress in shear ( y/2). A notch-brittle material is one that is ductile in an ordinary laboratory test but brittle in notched tests with
Fracture Mechanics and Friction
23
large constraint factors, i.e. deep cuts. The ER/k2 parameter has dimensions of length, and when different types of chip are produced when cutting the same material, by altering the depth of cut, it is a question of how small or how large the depth of cut is in relation to ER/ k2. The same length scale determines ductile or brittle behaviour in general. Shear connexion may be introduced during use: the fracture of fabric shoelaces may be connected with built-up layers of shoe polish that couple the threads and stiffen up the lace, thus making fracture easier.
2.2.5 Criteria for crack initiation in ductile solids Employment of fracture mechanics requires knowledge of the crack size. When no crack is obvious, in brittle solids the effective starter crack is related to microstructural features. But what of materials that behave in the ductile range and supposedly have no starter flaws? Experiments reveal that crack formation in commercial ductile materials, which have inclusions or hard secondary phases in the microstructure, results from void formation at the particles, growth of these isolated voids under increased load and eventual coalescence of voids to form a crack. Void growth depends upon both strain and hydrostatic stress (mean stress), given by H (1 2 3)/3. Fracture occurs when sufficient microstructural ‘damage’ given by f(H)dε has been accumulated (see McClintock, 1968, for a two-dimensional array of holes; Rice & Tracey, 1969, for the three-dimensional growth of a single hole). Various forms of the function f(H) of hydrostatic stress have been proposed to account for the fact that holes in practical microstructures are not empty and to account for interaction between growing voids in three dimensions (Atkins, 1987). Wierzbicki et al. (2005) have reviewed a number of different criteria and compared the predictions with experimental results over a wide range of H. It is because thick (plane strain) specimens have the greatest H that they result in lowest KIC, and lowest R or JC, and thus give perhaps pessimistic estimations of toughness for use in design formulae. Another parameter for crack resistance is the so-called critical crack opening displacement, C, that is related to R by R myC, where m indicates constraint and is 3 for plane strain and 1 for plane stress. In ductile solids C is also least under greatest H.
2.3 Friction in Cutting When cracking takes place by the crack faces being prised apart by some sort of tool, the general energy balance should include the incremental work done against friction that opposes the movement of the tool over the crack faces, i.e. Eq. (2-7) should have d(friction) on the right hand side
Fdδ dΛ RdA dΓ dKin dChem d(friction)
(2-7a)
Friction between offcut and blade in cutting is modelled, in this book, usually in terms of the well-known Amontons/Coulomb relation ‘friction force along surface (normal force across surface’, where is the coefficient of friction, owing to its simplicity. While this sort of relation may apply in lightly loaded conditions, experiments show that it does not apply under heavy loading as in metal cutting where (or the equivalent angle of friction where tan ) varies systematically with depth of cut and tool rake angle. In ‘ordinary’ friction the real contact area is considerably smaller than the apparent contact area (the area of the bottom of the sliding brick or whatever), because even carefully prepared surfaces are never truly flat
24
The Science and Engineering of Cutting
and at large magnification comprise hills and valleys. It is these surface protrusions that are in intimate contact when surfaces touch – Bowden and Tabor (1954) likened friction to ‘turning Austria upside down on Switzerland’, although the slopes of surface asperities are much gentler than the peaks and crests of mountain ranges, being about 5–10°. The local stresses at the isolated contacts are nevertheless very high and the hills squash plastically and weld together, even if they are dissimilar materials. Even materials that are usually brittle adhere at interfacial junctions owing to the high compressive hydrostatic stresses that suppress local fracture. The friction force F results from shearing all these isolated locally welded asperities, together with a deformation term that occurs when the harder surface of a contacting pair pushes along a prow of the softer component – the so-called ‘ploughing’ of surface asperities (Bowden & Tabor, 1954, 1964), although agricultural and civil engineering ploughing involves cutting as well as flow. A similar story applies for other materials such as rubber and wood where asperities become attached by local interfacial bonding/local molecular adhesion, instead of by pressure welding, and friction results from having to separate these regions in shear. With some materials, subsurface effects are important in sliding friction, and in the case of rolling friction, subsurface effects predominate for all materials. An important realization about ‘ordinary’ friction is that the bulk of the bodies in contact remain elastic: it is only the locally adhered junctions that are irreversibly deformed. Hence when offcuts in cutting are floppy or elastic, they slip over the blade in much the same way as all materials in contact slide. Thus friction between tool and material when cleaving slate or chopping wood can probably be represented by Coulomb friction, and handbook values for may be appropriate for friction against scalpels, corkscrews, razor blades and so on. However, when contact pressures between offcut and blade become large (and particularly in the case of metals cut with ‘chunky’ rather than knife-like tools), the limited numbers of isolated welded junctions between tool and workpiece grow, interpenetrate and coalesce; plasticity, formerly limited to the junctions, now spreads into the bulk of the chip. The real and apparent areas of contact more closely approach one another, and ‘sticking’ friction (where a material finds it just as easy to slip within itself as at the adjacent interface between tool and material) is likely to occur along at least part of the contact surface, with some sort of Amontons/Coulomb sliding elsewhere. Newly cut virgin surfaces, in immediate and intimate contact with the rake face of the tool, are uncontaminated and may give high adhesion and metal transfer near the tip of the tool. It is the changing amounts of sticking and sliding on the rake face with different depth of cut t and tool rake angle that give rise to varying . Note that even when chip material is stuck to the rake face, sliding must still occur somewhere in adjacent regions of the chip, otherwise the chip would not be able to flow onwards. When the sliding zones occur at some distance from the rake face, a ‘built-up edge’ or ‘dead metal zone’ is formed attached to the tool (see Chapter 4). ‘Stickiness’ and ‘high friction’ is a common experience when cutting certain foodstuffs such as cheese. Appendix 2 discusses friction in cutting in detail and explains how Coulomb analyses may be modified to take account of mixed sticking and sliding along the rake face.
2.4 Muscles The forces required for cutting at home, at work and in industry may come from electric or other powered drives, but it may come by hand or foot. Movements of claws, legs, feet, jaws and other limbs of creatures, and the associated forces and work done, are produced by the action of muscles. Muscles are engines that convert energy from biochemical reactions into
Fracture Mechanics and Friction
25
mechanical work. Some knowledge of the work available from muscles and the rate at which it can be supplied (the power of muscles), and the duration for which it can be supplied, is important for cutting interpreted in the widest sense, such as in bite forces and the design of teeth, the forces generated in attacks by knife and sword, in both of which the cutting tool remains attached to the driving hand or limb. Muscle power also determines the velocity at which a free-flying weapon such as a javelin can be directed at a target. Muscles are orientated short fibre composites, the largest volume fractions in which are proteins called myofibrils. Some of these are thick and some are thin, with chemical bonding cross-bridges between filaments. Cross-bridges continually attach and detach as the filaments slide past one another when the muscle is working (cf. Section 12.2). As explained by French (1988), problems arise over peak power requirements in the designs of nature and of humans, where continuous normal behaviour and short bursts of high activity both have to be accommodated. Red muscle (dark meat) owes its colour to the protein myoglobin and supplies normal requirements; white muscle (white meat) contains glycogen and supplies the bursts of energy. Red muscle gets its energy from body fat using oxygen from the bloodstream and the lungs. White muscle can release a limited amount of energy without oxygen (anaerobic) but restoration of that energy, ready for the next burst, takes place by very slow chemical processes, so that bursts are infrequent. The stress–strain behaviour of muscle removed from creatures may be determined in the laboratory using testing machines. It may also be determined indirectly from live animals by measuring bite forces, say, and using simple mechanics knowing the locations where muscles are attached to the jaw and the directions in which they act (Chapter 11). Muscles are anchored at one end and transmit movement to limbs via tendons. Tendons, like muscles, are non-linear in their stress–strain behaviour. Where muscles are anchored and where they are attached to limbs is important because different positions have different lever arms and hence generate different forces. Displacements given to limbs from muscles are often magnified or reduced by appropriate lever movements, and the forces are altered accordingly since the work ( force displacement) will remain constant providing that there are no losses. An obvious example is the jaw, considered in Chapters 11 and 13. Palaeontologists use the space occupied by muscles in the skeletons of extinct creatures to estimate the forces generated by tooth and claw. The loads and deflexions measured in experiments depend very much on whether the muscle is working as a biochemical engine or is just quiescent. The axial stiffness of unstimulated muscle is quite low but it increases considerably when chemical energy transfer within the muscle occurs. Stimulation (twitch, spasm or tetanus) of muscle of anaesthetized animals, and of excised laboratory samples, may be achieved by pulsed electrical discharges from a capacitor connected to the muscle through electrodes (e.g. Thomason et al., 1990). Note that the electric pulsing merely provides stimulation, not the energy derived from the muscle. Physiologists characterize stimulated muscle in terms of applied tensile force F and the resulting contractile velocity v of one end. Such plots of load versus velocity are essentially plots of stress against strain rate. An unrestrained stimulated muscle contracts at the greatest velocity; slower velocities are produced when motion is opposed by forces applied to the end of a muscle; and a big enough applied force will completely prevent contraction of a stimulated muscle; even larger forces produce extension of muscles. Hill, in his pioneering paper of 1938, showed that a good fit to experimental data (except perhaps at greatest forces) is a rectangular hyperbola offset from the axes, given by
(F a)(v b) (Fo a)b
(2-13)
26
The Science and Engineering of Cutting
where a and b are the amounts by which the curve is offset from the abscissa and ordinate, respectively, and Fo is the force that prevents contraction. Physiologists usually plot contractile velocity on the y-axis and force on the x-axis (engineers would probably reverse the axes). Not all muscles behave the same, having different Fo and vmax (where vmax is the maximum velocity at zero load), so it is convenient to non-dimensionalize Eq. (2-13) to give v* (1 F*)/(1 F*G)
(2-14)
where F* F/Fo, v* v/vmax, and G Fo/a vmax/b. Note that there is not a single plot for all muscles even when plotted non-dimensionally since different muscles also display different a and b that together control the curvature of the plot and how close the curve comes to the axes (Figure 2-4A). This factor is important when we consider the power of muscles, i.e. the rate of doing work. Power is (force displacement/ time) or (force velocity), and is therefore given by the rectangular area contained between any (F,v) point on the curve and the axes; similarly for normalized power Fv/Fovmax F*v*. Such areas may be evaluated, giving the sorts of power versus Fo curves shown in Figure 2-4(B) for different values of G. All stimulated muscles pass through a maximum in their power output. It may be shown by differentiation that maximum power is produced when F* v* [(1 G) 1]/G. The value of the maximum power, and the force and velocity at which it is produced, all depend upon the curvatures of the (F,v) plots through the value of G. We may divide both sides of Eq. (2-13) by AL, where A is the cross-sectional area of the muscle and L its length, to obtain Σ(dε/dt b/L) Σo (b/L)
(2-15)
where (F/A) (a/A) (a/A) is a biased stress; o (Fo/A) (a/A); and dε/dt v/L is the conractile strain rate (considered positive by physiologists). Hence dε (b/L)[(Σo /Σ) 1] dt
(2-16)
0.11 1.0 Power: (F/F0) (V/Vmax)
0.10
V/Vmax
G=3 G = 10
0.5
0 A
0.5 F/F0
1.0
G = 10 0.05
0 B
G=3
0.5 F/F0
1.0
Figure 2-4 Behaviour of muscles: (A) normalized velocity–force curves calculated from Hill’s equation; (B) power is force velocity and this figure shows how normalized power depends on normalized force for the two curves in (A): the power maxima, and at what force they occur, depend on the value of G, i.e. on the curvature of the velocity–force relations (after Woledge et al., 1985).
Fracture Mechanics and Friction
27
It is clear from Eq. (2-16) that when o, dε 0; when o, positive (contractile) strains result; when 0, strain increments are supposedly infinite and the velocity is greatest; and when o, d becomes negative, i.e. tensile in the sign convention employed, so that increased loading reverses the contraction back to the original length and eventually produces extension. For known variations of applied stress with time, Eq. (2-16) may be integrated to give the change of strain with time. For constant (a dead weight load), increases linearly with time, the rate being given by (b/L)[(o/) 1]. For F increasing directly with time according to F kt (as in a load-controlled testing machine), (kt/A) (a/A) and the typical J-shaped stress–strain curve of extensible biological tissues is predicted having the following algebraic form
ε (ab/kL) {(G 1) ln[G(F/Fo) 1] G(F/Fo)}
(2-17)
The implication of Eqs (2-16) and (2-17) is that any magnitude of strain may be produced given enough time, and hence that the work output of muscles is unlimited. However, this assumes that the muscle can be continuously stimulated and does not tire out. In practice the strains in muscles working normally are limited to about 25 per cent because lack of oxygen, and the consequent build-up of lactic acid, so that the efficiency of the muscle engine is reduced. Fo for many muscles is some 300 kPa (Woledge et al., 1985) so, with a limiting strain of 0.25 and assuming that Fo is maintained, the work done per volume of muscle in a slow contraction is 300 103 0.25 75 kJ/m3 (Alexander, 1992) or 70 J/kg using 1060 kg/m3 for the density of muscle. The rate of doing this work is very low because the velocity of the muscle is very low, but that is acceptable for activities such as weightlifting. To increase and maximize the power output, F* has to be reduced to 0.3 (G increased to about 3) (Figure 2-4B). Thus the maximum work expected from muscle while functioning at maximum power must be about 20 J/kg. Maximum power is required in activities such as swimming, cycling and so on. An athlete may be able to produce short bursts of energy up to about 1 kW but that fades as soon as the anaerobic fuel supply is exhausted. Within a minute the power drops below 700 W and the burst is virtually over after two minutes, leaving a steady rate of about 400 W; the ‘normal’ rate might be 100 W. The burst cannot be repeated until the system has had time to replenish its anaerobic fuel reserve (oxygen debt) (French, 1988). Having done its work, the muscle has a reduced length. To return the muscle to its initial position, it has to recover (in fact be pulled back since all muscle work is dissipated and there is negligible elastic recovery). Contracted muscles are ‘reset’ by other muscles to which they are attached. During the work stage of the active muscle, the attached retraction muscle is not stimulated, is thus very compliant, and therefore little work is required to move it. Exactly the opposite applies during the recovery stroke when it is the second muscle only that is stimulated. Cramp is when both the original and resetting muscles act simultaneously. A given limb may have a number of separate muscles attached to it. The tendon attachments may not be all in one plane, so that movement is produced by pulls from various directions. Ligaments (sinews) hold bones and organs together: the knee joint is held together by crossed (cruciate) ligaments, so the way the limb itself moves when acted upon depends on the constraints to which it is subjected (e.g. limited rotation of knee and hip joints; the head and neck; and so on). When physiologists perform in situ experiments to see what power a muscle produces by investigating overall movement, it is essential that means be found to isolate the effect of
28
The Science and Engineering of Cutting
the particular muscle of interest. Otherwise the action of all other attached muscles will be included, leading to an overestimate of capacity. The scallop is a bivalve mollusc that swims by clapping the two valves of its shell open and shut, where a single joint is worked by a single muscle so that there is no ambiguity about the muscle’s contribution (Marsh et al., 1992; Alexander, 1992).
2.5 Impact Mechanics and Hammering The force required for cutting may sometimes be more efficiently or more easily delivered by an impact blow rather than by a steady push or pull. It could be that the force generated by impact is not that different from a steady load, but that the material being cut is more easily deformed and separated under a ‘jerk’. Far more often, however, impact is a means of increasing the energy and momentum that can be imparted in the cutting operation. In history, hammers for forging metals were large masses raised into the air by winch, water or steam power, and let fall. The trip hammer was rather like an automated Charpy impact testing machine. A very early paper on lifting and holding the weight (tup) in this sort of hammering was by Kitson (1854), a topic of interest to his grandson, Kitson Clark (1931), from whose paper Figure 2-5 is taken.
2 12
6
14
8
5 11 3 13 7 15
1 10 4
1. Piano hammer 8. Stone-mason’s hammer 2. Boilersmith’s sett 9. Mechanic’s hammer 3. Dentist’s gold stopper 10. File-cutter’s hammer 4. Coppersmith’s hammer 11. Cobbler’s hammer 5. Flint knapper’s hammer 12. Dental plate hammer 6. Stone-breaker’s hammer 13. Dentist’s riveting hammer 7. Indian agate knapper 14. Gold-beater’s hammer 15. Worn disintegrator blade
Figure 2-5 Various types of hammer (after Kitson Clark, 1931). (Courtesy of IMechE)
9
29
Fracture Mechanics and Friction
A mass m raised to a height h has potential energy mgh. When let fall, the potential energy is gradually converted to kinetic energy given by 0.5 mv2, where v is the velocity. The velocity when dropping through the original datum is v (2gh). The same potential energy may be attained by different combinations of m and h, so that the same energy can be delivered at different velocities in an impact by adjusting m and h. However, the momentum at impact, given by mv, will be different even if the energy is the same. Equally, it is possible to produce identical momenta at different energies. In impact problems between materials having different characteristics, it is not always clear whether the energy or the momentum is more important. In hand-held hammers, muscles accelerate the already lifted hammer during the downward stroke, increasing the velocity of the hammer head (and its kinetic energy) above the velocity it would have just from loss of potential energy during free fall. A golf ball is not driven by holding a golf club like a pendulum and allowing it simply to swing. It is moot point whether the implement is almost free-moving by the time of strike: the major acceleration of arm on hammer occurs at beginning of stroke, and later, muscles are being asked to move too quickly to do any work. However, in stabbing attacks (Chapter 8) the ‘run through’ of the knife after first contact depends on the continued pressing home by the hand and arm. The contribution of all joints is important (compare the difference between kicking a ball with the knee locked or not locked). To prevent damage to surfaces that have to be hit, a layer of a softer material is often interposed, or the hammer head itself, or its face, is made of a softer material. ‘Softer’ is relative to what is being struck: lead or leather hammers are used with chromium-plated knock-on wheel nuts that hold in place splines on the wire-wheels of old cars and sports cars. In the piano, the hammers are covered with felt to avoid the shrill harmonics produced when striking the strings with a hard-surfaced hammer. How much energy is transferred into a chisel or other type of tool? Consider the impact between a moving hammer of mass M, moving at velocity VH, with a stationary chisel of mass m. After impact, the hammer has velocity vH and the chisel vC. Conservation of momentum gives
MVH Mv H mvC
(2-18)
The coefficient of restitution e is the hypothesis in collision mechanics which connects the speeds of separation and approach, and is defined in the present problem as
e (vC v H )/VH
(2-19)
When e 1 the bodies are said to be perfectly elastic, and all the initial energy stored in the elastically deformed chisel and hammer is recovered; when e 0, they are perfectly inelastic, all the energy is dissipated and there is no recovery. From Eq. (2-19)
v H vC eVH
(2-20)
Substituting in Eq. (2-18) and calling m/M , we obtain
vC V (1 e)/(1 µ)
(2-21)
The energy of the hammer just at the instant of striking is MVC2/2; the energy of chisel after is mvC2/2. Hence the proportion of energy transferred from hammer to chisel is
α (m/M)(vC /VH )2 µ[(1 e)/(1 µ)]2
(2-22)
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The Science and Engineering of Cutting
Plots of vs for different values of e from Eq. (2-22) always peak at 1 (Figure 2-6). At other (m/M) ratios less energy is transferred. The greatest proportion of a hammer’s energy capable of being transferred to a chisel takes place, therefore, when the mass of the hammer and chisel is equal. Hence the common experience that the mass of a cold chisel is comparable with that of a lump or engineer’s hammer; that wood chisels are best driven by wooden mallets; and that in croquet, the masses of wooden croquet balls and mallets are matched. In all cases, the actual amount of energy transferred depends on e. Note that energy is still transferred when e 0, and is dissipated in the chisel. The mass of a hammer head is not always matched to the nail it strikes, so a smaller proportion of energy is transferred. However, use of an overmatched heavy hammer that will have a larger amount of energy, may transfer more work than a perfectly matched pair. For further details about energy and impulse in this context, see Inglis (1951); for information on ‘mechanical impedance’, which this is all about, see Bishop (1965); and for ‘matching’ generally, see French (1988). A fascinating paper of historical importance on this topic entitled ‘The pressure of a blow’ was given by Hopkinson (of Hopkinson Bar fame) in 1912 (see his collected papers, 1921). Of course this analysis does not address the question of what happens at the cutting edge when a chisel is struck, nor what happens at the interface between the bottom of a pile and soil in civil engineering when the pile is driven down by a blow. That depends upon cutting mechanics, which is the subject of this book. Even so, the analysis tells us that to make most use of the muscle power expended in bringing a hammer into contact with a chisel, the masses of the hammer and chisel should be comparable. However, there is the additional question of control of the cutting process, i.e. is it possible to stop and restart a cut at will? The answer depends upon the constraints under which the load is applied, as is known in the topic of crack stability in fracture mechanics (Atkins & Mai, 1985). Design of nutcrackers well illustrates this sort of thing. Nutcrackers in the form of two levers hinged together at one end, and squeezed until the nut shell fractures,
1.2
Energy transfer proportion
1 0.8
e=1 e = 0.8
0.6
e = 0.6 0.4
e = 0.4 e = 0.2
0.2
e=0
0 0
0.5
1
1.5
2 m Mass ratio µ = M
2.5
3
3.5
Figure 2-6 Proportion of energy in a blow transferred from hammer to struck object plotted against the ratio of the masses of hammer and target (after Pretlove, 2008).
Fracture Mechanics and Friction
31
are usually uncontrollable and the kernel of the nut is often damaged since it is the load, not displacement, that is controlled. Nutcrackers looking like a cup with a bolt threaded through the wall are, in contrast, easy to control since it is the displacement at the point of contact that can be started or stopped at will, not the load. Hitting a chisel with a hammer is a cutting operation under ‘load control’ and, as it is virtually impossible to reduce the load during a strike, operations may be difficult to control. Hence ‘tapping’ a chisel, rather than ‘thumping’, may be preferred in order to make sure that the work is not spoilt. Another aspect of the use of hammers concerns the ‘sting’ that may be experienced through the handle when the chisel is struck. It is the well-known ‘centre of percussion’ problem in mechanics and concerns how to ensure zero impulsive reaction at the point where the handle is held when something is struck at the other end. The impact in a hammer takes place in the limited region of the hammer head, and the lengths and cross-sections of handles can be proportioned so that at the normal place of gripping, no sting is felt. With cricket or baseball bats, the ball may be struck at a variety of locations along their lengths, implying different holding positions along the handle for no sting. Cricket bats and Welsh (but not US) baseball bats have springs in the handles to reduce the effect so that the grip can be in roughly the same place. The design of the so-called ‘bearded’ execution axe, where the cutting edge extends backwards, and thus moves the centre of gravity of the axe up the handle, may be connected with reducing the impulsive reaction at the grip (Pretlove, 2007). In impact problems, questions of stress waves should be considered. That is, when a rod is loaded at one end, it takes a finite time before the other end experiences the load. The velocity of an elastic stress wave is given by (E/), where E is Young’s modulus and is density. For steel, with E 210 GPa and 7850 kg/m3, v 5000 m/s. So if a 100 mm long rod is loaded at its top end, it will take (0.1/5000) 20 s before the bottom end knows what has happened. It is only when the wave front has passed that the material is loaded. The stress wave is reflected back and forth along the rod, and eventually is damped out to leave a uniform state of loading all along the rod. Until then, different parts of the rod experience different loads, and this sets limits on the sizes of testpieces for high strain rate studies of mechanical properties (Atkins, 1970). Most stress analyses in this book presume quasi-static (almost static) behaviour where wave effects are unimportant. Sometimes it is necessary to include wave analyses in cutting problems (see Chapters 10 and 15). At first sight, the impact of a hammer on a nail, tool or pile driver may be expected to be analysed in terms of elastic waves. However, except for long nails and conventional pile drivers, Salem et al. (1975) showed that there would be many wave reflexions during the brief time of a single blow (some 10 ms), so that the problem is practically quasi-static. With a stress wave speed in steel of about 5000 m/s, the distance covered in 10 ms is 5000(0.01) 50 m. In a 150 mm (6 inch) long nail, that means 50/0.15 300 traversals. Salem et al. (1975) analysed the strain pulse in a nail caused by hitting with a hammer and the predictions were favourably compared with the experimental results of striking a strain-gauged nail, which gave the changing strains at that location in the shank for the duration of the blow.
2.6 Work of Formation of New Surfaces Within bodies, chemical bonds are matched and balanced by those of surrounding atoms. On surfaces, the exposed bonds and those of the surrounding vapour do not match. This gives rise to the thermodynamic concept of surface free energy, something possessed by all surfaces including the pages of this book. Surface energy is a short-range parameter (Kendall, 2001). It is given
32
The Science and Engineering of Cutting
the symbol , and has units J/m2. Values of are typically a few J/m2 and are different for different materials, e.g. different for newsprint than glossy art paper. Surface energy may be measured in a variety of ways, the contact angle method employing Young’s equation of equilibrium at the interfaces being well known from observations of water droplets on clean and greasy surfaces (e.g. Sinn et al., 2009). Surface energy is important in adhesive joint design (Kinloch, 1987). Griffith, in his seminal work on the mechanics of fracture (1921, 1924), recognized that when something broke, new surfaces were produced, and that part of the work performed in fracturing the body had to provide the energy of the new surfaces. The rest of the work done deformed the body in bulk and, depending on circumstances, was recovered. Griffith worked on glass that broke within the elastic range (it was brittle) so that the energy required for surfaces was supplied from the elastic strain energy stored in the loaded testpiece just before fracture. The Griffith relation, f (E/a), given in Section 2.2.2 relates the fracture stress f to the Young’s modulus E and surface energy of a sheet of glass containing a crack of length 2a. Later, Obreimoff (1930) cleaved mica in the elastic range and his relation for splitting force also employed . When elastic fracture mechanics was being developed after World War II, and applied to engineering metals such as steel and aluminium alloys, it was found that although crack 1/a, the Griffith relation did not predict experimental fracture stresses when was employed. Rather, something like 1000 had to be employed to make sense of the results. The reason for the orders of magnitude greater value was down to the fact that the concept of relates to unmatched bonds across a single flat surface and, in practice, new surfaces do not form that way. Even brittle cleavage in practice usually occurs on a number of planes to give ‘river marks’ and steps between different cleaved regions, and requires more work/ area to produce than the simple surface free energy. Orowan in 1949 examined ships’ plate that had fractured in cleavage and found a layer about 0.5 mm thick of disturbed material beneath the surfaces. In other cases, fracture takes place not by cleavage, but by entirely different separation processes. All practical fracture surfaces have thin, but severely deformed, subsurface boundary layers. It is in these finite-thickness, highly strained, layers that mechanisms of crack formation take place which permit cracks to initiate, propagate and form new surfaces in solids. Although the volume of the boundary layers may be small, the work/volume is relatively large for the different micromechanisms of separation in different materials. Such microprocesses often require specific surface energies of kJ/m2, and that is why simple cannot be employed in fracture mechanics formulae. Values of kJ/m2 may be independently predicted using models of the microstructural processes by which separation is achieved, such as void growth and coalescence. The thickness of the boundary layers in which separation occurs, and the surface roughnesses produced by practical separation, are both at least an order of magnitude greater than the range over which the forces associated with the chemical surface energy parameter act. Indeed, Bikerman (1965) was critical of the concept of solid (as opposed to liquid) surface energy because the surfaces of solids are elastic or plastic, and the associated forces are so large that they dominate the surface tension, thus making solid surface energy extremely difficult to measure. Consequently, hundreds or thousands of J/m2 are typical of the values of the fracture toughnesses of materials that are used in elastic, elastoplastic and plastic fracture mechanics (R, GC, JC values and so on), and which are employed in fracture-mechanics-based design and in structural integrity assessments under fault or accident conditions (Section 2.5). Even for brittle materials, magnitudes of R rather more than just are employed in LEFM formulae. (That is defined with respect to the areas of both sides of a crack, but R is defined with respect to just one surface, so that R 2, is neither here nor there.)
Fracture Mechanics and Friction
33
Thin films need to remain adhered to surfaces to be of use. Here again, the practical work of adhesion is found to be orders of magnitude greater than thermodynamic work of adhesion (e.g. Krieser et al., 1998). These authors also point out that is very sensitive to the chemical purity of a material. Cutting mechanics in the 1940s concerned metal cutting in particular owing to its commercial importance. As will be described in Chapter 3, models of cutting recognized that new surfaces were produced but it was believed that the work of surface formation was insignificant compared with the plastic work of chip formation and work of friction. The conclusion was reached by estimating the work of separation and of formation of new surfaces using the surface free energy of a few J/m2 and, indeed, if is used in calculations, the work of surface formation is swamped by the works of plasticity of chip formation and friction. This viewpoint, that work of separation is negligible, has been the received wisdom in metal-cutting mechanics ever since. But Atkins (2003) showed that cutting is a branch of fracture mechanics, and that R should always be employed in place of , even when cutting the most brittle materials. Incorporation of R values of kJ/m2 for ductile materials makes sense physically in that yield strength k and toughness R are mechanical properties that can be altered independently of each other, so that the hardest materials do not always require the greatest cutting forces. Thomsen et al. (1953) had noted that there was usually subsurface deformation on cut metals and that the associated work required was not considered in traditional analyses that concentrated on chip formation and friction. Thomsen et al. drew the analogy with the thin shear zones produced in punching and blanking. Insofar as subsurface plasticity, at least with a sharp tool, is a result of the boundary layer in which separation work occurs, Thomsen et al. had identified the third component of work that Atkins (2003) argued was important. Even so, the traditional metal-cutting view, that cracking can have nothing to do with chip formation in the cutting of ductile materials, has common currency. It is argued that the mechanism by which new surfaces are formed is by simple ‘plastic flow’ around the tool cutting edge, rather like water passing around the pier of a bridge. Furthermore, since fracture means cracks, and since cracks are not seen at the tips of tools in continuous-chip machining of common metals at ‘normal’ depths of cut, there has been a reluctance to believe that fracture can have anything to do with continuous-chip machining of ductile metals. Chapter 3 will show that the answer to this question has been confused by not distinguishing the difference between crack existence and crack stability. Furthermore, experiments described in Chapter 4 show that it is perfectly possible to have cracks at the cutting edge under certain conditions. It all depends on the toughness/strength, ER/k2, ratio of the material and the depth of cut. There is a whole spectrum of chip types ranging from continuous ribbons, through varieties of discontinuous chip, to brittle chipping. It was Orowan and Irwin who showed that 1000 had to be employed in fracture mechanics rather than alone (e.g. Irwin, 1948; Orowan, 1949). The origins of the idea that separation work in cutting was negligible are down to Ernst and Merchant in Cincinatti in the late 1940s, and Shaw and his team at MIT in the early 1950s. What is interesting to realize is that Orowan and Shaw were both in the Department of Mechanical Engineering at MIT at the same time. Did they ever discuss the question of which parameter to employ, i.e. or 1000, when estimating the work of surface formation in machining? David Felbeck was a research student in Mechanical Engineering at MIT at that time and his view (2008) is as follows: ‘… Orowan first understood the importance of plastic surface energy, well prior to Irwin. I knew both of them well in the 50s and 60s. Irwin did some of the esoteric mathematics by way of explaining what was happening, and developed a big name
34
The Science and Engineering of Cutting because of that, but the concept was Orowan’s … Orowan’s office was in Building 3, and Shaw’s was closer to the Auto Lab, probably 100 yards away …. In those heady years, the ME Department had a very active Friday afternoon (4:00) seminar, where most of the senior faculty attended and participated actively in discussion, even argument, regarding the subject of the day. I am confident that Shaw and Orowan talked many times, but I suspect you are correct in stating that neither realized that they were examining two aspects of the same phenomenon …’.
Felbeck’s doctoral thesis experimentally confirmed Orowan’s theoretical explanation. The magnitude of the near-surface plastic work so overwhelmed the magnitude of the surface free energy that the surface free energy term in the Griffith equation could be replaced with the 1000 boundary layer plastic work term, and this was found to be consistent with the stress required for brittle fracture in steel below the so-called transition temperature, that is the boundary between cracking in steels by cleavage and by void growth and coalescence (see the 1997 reprint of the original Felbeck & Orowan paper). Further confirmation that cutting is a branch of fracture mechanics, and cannot be based just on stresses or strains, is the scaling phenomenon mentioned in Section 2.2.3 and analysed in Chapter 4. Cube-square energy scaling gives rise to the multiplicity of chip types, and to the fact that ‘brittle’ materials can be machined with the production of continuous ductile chips under the correct conditions, and that ‘ductile’ materials can shatter like brittle materials under appropriate conditions.
Chapter 3
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials Contents 3.1 Introduction 3.2 Floppy Offcuts 3.3 Different Types of Offcut Formation 3.4 Brittle Offcuts 3.5 Ductile Offcuts 3.6 Offcut Formation by Shear 3.7 Finite Element Simulations 3.8 Cutting Through the Thickness of Ductile Sheets and Plates: Shearing and Cropping
35 37 41 43 49 52 67 70
3.1 Introduction In so-called orthogonal cutting, a straight-edged tool overhangs the workpiece and cutting takes place along the surface. The relative motion of blade and workpiece is perpendicular to the edge of blade: the blade may move over the workpiece, as in cutting slices with a kitchen knife or in wood planing, or the workpiece may move over the tool, as in a food grater and in some types of microtome. A microtome is an instrument used by biologists and histologists for cutting extremely thin sections for microscope work (from the Greek tome , a cutting). We assume that there is no sideways spread of the workpiece or offcut during cutting so that the problem is one of plane strain. This is usually achieved experimentally by ensuring that the width w of the workpiece is at least about ten times the depth of cut t. Should sideways spread be difficult to avoid, cutting may be performed with the side of the workpiece up against a sheet of glass; the extra frictional constraint is often presumed not to influence deformation patterns. In analyses of orthogonal cutting, there is supposedly lots of material ahead of the tool so that the far end of the workpiece where the cut ends has no effect. In contrast, when cutting through the thickness of relatively thin workpieces, the far end (bottom of the sheet) is felt almost from the outset as soon as the knife touches the top surface. The cutting mechanics of orthogonal cropping/guillotining is thus different (Section 3.8). To overcome problems of limited length of workpieces for orthogonal cutting in a straight line, cutting may be performed in a lathe on the end of a tube having a large diameter in comparison to the wall thickness (that becomes the width of cut), where the differences in cutting speed between the inner and outer radii of the tube are not too great. Another trick is to cut deep slots around a solid cylindrical testpiece to give circular bands for cutting. The deformation patterns of offcut formation have been investigated by a variety of techniques, namely: grid or circle patterns scribed or etched either on the outer edge of the workpiece or on the inner surface of a split workpiece; in place of such manually introduced coordinates, the motion of individual grains revealed by chemical etching may be used. When Copyright © 2009 Elsevier Ltd. All rights reserved.
35
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The Science and Engineering of Cutting
polymers that stress-whiten are cut, the extent and severity of the deformation is revealed. Deformation zones have also been delineated by microhardness surveys over the area encompassing the ingoing undeformed workpiece through the deformation chip-forming zone to the completed offcut. Video films (cine films formerly) enable the deformation to be studied slowly; pattern recognition software enables the changing strains through deformation fields to be evaluated. Glass or quartz tools have sometimes been employed, through which films have been taken to find out behaviour at the chip–tool interface. References are given later to these different experimental methods. The manner in which the offcut/chip is removed (by bending or shear), and its state after cutting (a continuous ribbon or broken up), depends on material properties, cutting blade geometry, friction and other factors. In this chapter we mostly consider cutting where the offcut remains continuous but we do also consider some circumstances where the offcut breaks. Further consideration of these latter discontinuous types of offcut/chip is taken up in Chapter 4. The blade is modelled as a sharp wedge of included angle (Figure 3-1) (what sharpness may mean quantitatively is covered in Chapter 9). For cutting it is not the included angle that is important but rather the inclination of the rake face along which the offcut travels. The rake angle is defined in this book with respect to the perpendicular to the cut surface; in agricultural engineering and some other fields is often measured from the cut surface. The clearance (or relief) angle between the lower face of the blade and the cut surface is . Hence 90°. When the clearance angle is small, it is sometimes ignored or assumed to be contained within . We assume that the workpiece is always securely clamped and that there is no negligible deformation except in the region of the cut. These assumptions are not always met, as when carving the Sunday joint and when cutting a loaf of bread, where the ‘workpiece’ moves back and forth with the knife; and when cutting unrestrained materials like stalks of corn and other crops (Chapter 10); and when shaving where not only the hair but the substrate skin is flexible. We also assume that both the depth of cut and the motion of the tool are controlled by the cutting device, so that the cutting forces are those that arise from work requirements at that fixed depth of cut (the set-up is displacement controlled and can be stopped or started at will). Hand-held tools are, in contrast, load controlled where the motion of the blade may be difficult to start and stop (which is how accidents arise), and where it is sometimes difficult to achieve a controlled depth of cut. For example, when peeling potatoes with a knife, there is a tendency
Tool
α θ
ζ
Figure 3-1 A sharp tool represented by a wedge of included angle . The rake angle of the tool is ; the clearance (relief) angle is .
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
37
for the knife to ‘dig in’ and increase the thickness of the peel during cutting when the inclination of the blade gives a large rake angle. Proprietary potato peelers, on the other hand, control the thickness of the peel.
3.2 Floppy Offcuts When the offcut has little capacity to store or dissipate energy, there are only two components of ‘internal’ work: (i) that required to create the newly separated surfaces by division; and (ii) work done against friction as the offcut slides along the cutting face of the wedge tool (Figure 3-2). Calling the specific work of surface separation R, the incremental work required is RdA, where A is the area of the cut surface. By convention, only one side of the separated surface is employed in the definition of R, whence dA wda where w is the width of a rectangular sample and a is the increase in cut length. In steady-state controlled cutting, da d where d is the incremental movement of the tool, so dA wd and the incremental work required is Rwd. When the cutting edge moves through a distance d, the friction force F moves through a distance d/cos along the sloping face of the tool, where is the blade angle that includes the clearance angle, and does incremental friction work F d/cos. Assuming Amontons/Coulomb friction in which F N, where N is the contact force on the rake face of the blade, the incremental friction work becomes N(d/cos). The separation and friction work increments are provided by the external incremental work done by the forces acting on the tool. In general, the resultant force on the tool has components FC parallel to the workpiece surface, and FT perpendicular to the cut surface (Figure 3-2a). FC is called the cutting force and FT the thrust force. Of these, only FC does work: there is no displacement in the direction that FT acts, so it does no work. The incremental work done by the forces on the tool is thus FCd.
a
ζ θ
w
Fc,δ FT
Figure 3-2 Cutting floppy materials on a microtome. Cutting force FC has displacement ; thrust force FT has no displacement in its own direction and so does no work.
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The Science and Engineering of Cutting
Equating internal and external works for a rectangular sample of width w, we obtain
FCdδ Rwdδ µN(dδ/ cos θ) = Rwdδ + [µFC /(µ cos θ sin θ)](dδ/ cos θ)
(3-1)
using Eq. (A1.3c) in Appendix 1 for the NFC connexion. Hence
FC /w RH θ / tan θ
(3-2a)
and
FT /w R/ tan θ
(3-2b)
where
H θ [(µ tan θ)/(1 µ tan θ)]
(3-3)
using the sign convention that the thrust force FT is positive when upwards, away from the cut surface. The wedge equilibrium equations have different forms depending on the sign convention employed; positive FT downwards into the cut surface is often used in the metalcutting literature. In addition, they are often written in terms of in place of (Appendix 1). Relationships (3-2) are often inverted so as to make R the dependent variable and then used to give toughness values from experimental cutting forces obtained on microtomes or other devices (Williams, 1998). When friction is negligible,
R FC /w
(3-4)
and for constant R and w, the force is constant. Neither the wedge angle, nor the inclination of the tool, nor the clearance angle between tool and cut surface, nor the thickness of the slice, enters this simple picture. This relation is similar to that used in the ‘trousers test’ for tearing thin sheets when there is no elasticity or plasticity in the legs (Atkins & Mai, 1985). When friction is finite, (FC/w) is very large when the tool angle is very small; it is also theoretically infinite when tan (1/). In between, there is a minimum in (FC/w) at tan [(1 2) ]. For tan1(1/) the wedge cannot cut since, as explained in books on statics, it will try to pop back out of the cut. By way of example, a butcher cutting a slice from a piece of meat 100 mm wide, having R 200 J/m2, needs to exert a downwards force of FC 200 100 103 20 N when the friction is low. (In practice, the butcher would probably slide the knife as well as just press down, for reasons explained in Chapter 5.) When the cutting forces in microtoming are measured, the device is a means of measuring the fracture toughness (Atkins & Vincent, 1984). Microtome samples are often mounted in wax, and friction over the mixed cut face may not be known in experiments to measure R from FC. But note the remarkable fact that, according to Eq. (3-2b), it is not necessary to know the friction to obtain R when FT is measured. This important fact (first highlighted by Williams, 1998) has not been appreciated until recently and, unfortunately, many instrumented microtomes measure FC only. Many authors now use microtomes to determine toughness (e.g. Ericson & Lindberg, 1996). Forces in rectangular samples are expected to be steady, particularly with soft solids. Forces will vary when samples having varying toughness are microtomed. Thus the outer regions
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
39
(cortex) of fresh carrot are tougher than the interior (medulla), and this is shown in force plots. Figure 3-3 shows the differences in force required between a sample and the wax in which it is mounted. Histological samples sometimes show variations in cutting forces in regions that staining will not distinguish. (Staining is where a sample is treated with a chemical reagent or dye to highlight certain tissues, or parts of tissues.) Indeed, for this reason, Allison and Vincent (1990) described the instrumented microtome as a ‘mechanical microscope’. Non-rectangular samples, even for constant R materials, will give varying forces according to the variation of w with . In an unmounted circular sample, the force will increase from zero at the start of the cut, take the greatest value at mid-section, and fall to zero again at the end of the cut. At some from the start of cutting, the geometry of a circle gives (w2/4) (D ), where D is the diameter of the specimen, so FC 2RH[(D )]/tan from Eq. (3.2a). Even though F varies with when cutting a circular cross-section of a constant R material, the total work done given by the area under the F, curve will, in a frictionless case, simply be the fracture toughness times the cross-sectional area, i.e. FCd (D2/4)R, and from this a mean force equal to (D/4)R may be defined. For geometrically complicated cross-sections, where the algebra for cutting forces is cumbersome, the work areas under FC vs diagrams can be evaluated and divided by the total cut area to give an average R. Compensation for friction may be obtained approximately by running the blade over the cut surface after the initial experiment, and deducting the resulting frictional work area before dividing by the new surface area. Employment of such work areas after friction has been removed and is a powerful experimental method in the determination of R in cutting, and requires neither algebra nor computational finite element methods (FEM). The work area method derives from a simple graphical energy balance for displacement-reversible globally elastic fracture (Cottrell, 1964; Gurney & Hunt, 1967), and is illustrated later in the book when describing guillotining, scissor cutting
Force
Direction of sectioning
Displacement
Figure 3-3 Cutting force vs blade travel diagram for microtoming a specimen (shaded) embedded in wax (unshaded). The cutting force is lower across the whole wax section than when cutting the region including the specimen (after Willis & Vincent, 1995).
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The Science and Engineering of Cutting
and so on. With floppy materials, there is no elasticity, but the system is still reversible since on retraction of the cutting edge having made a ‘horizontal’ cut, the offcut falls back under gravity on to the cut surface to the position before cutting. There are many examples in the kitchen where slices of vegetables and so on can all be reassembled to give the original size and shape of the cut foodstuff.
3.2.1 Cutting pretensioned sheet A floppy sheet that cannot store elastic strain energy in bending may be able to store energy when stretched in tension. When cut, the energy is released and feeds the crack so that less external work has to be done by the blade. Cutting forces are smaller the greater the pretension. A sheet of length L, thickness t and width 2w, already containing a cut of length a, will have triangular floppy regions alongside the cut (Figure 3-4). When the cut is extended by da, the floppy regions grow and release elastic strain energy. Thus the work equation becomes
Fdδ Rtda dΛ
(3-5)
where F is the cutting force, the displacement of the blade and is the elastic strain energy. Consider a sheet whose elastic stress–strain behaviour follows the non-linear relation on; for a linear elastic material o E, the Young modulus, and n 1. The energy stored per unit volume when stretched to a strain is the area under the stress–strain curve and is given by d on1/(n 1). The stretched volume is 2 wt(L 0.5a), so [on1/(n 1)][2 wt(L 0.5a)], and d [on1/(n 1)](wt)da. Hence
Fdδ Rtda [σoε n1/(n 1)](wt)da
(3-6)
In steady cutting, where d da, we have
F [R wσoε n1/(n 1)]t
Crack length a Floppy
Floppy
F, δ
Figure 3-4 Cutting of pretensioned elastic sheet.
(3-7)
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
41
so that the effective fracture toughness is reduced. Spontaneous separation will occur when won1/(n 1) R, when no external work is required. If the stretch is produced by extending w by u, the strain (u/w). Alternatively, in highly extensible materials, the stretch ratio 1 (u/w) is employed. Different forms of – or – relation may be employed as appropriate for different materials (see Sections 8.2 and 8.3). How rubber is cut is important for tyres, conveyor belting and so on, particularly when wet (when friction is low). Stretching testpieces enables cutting to be performed under reduced friction (Lake & Yeoh, 1978). Similar ideas apply when materials having residual stresses are cut. It can be dangerous. A glazier never cuts toughened glass: glass is first cut to size, edges bevelled, etc., then sent away for heat treatment. Toughened glass, if hit or dropped, will shatter into ‘sugar’. Friction is readily included in this sort of analysis by adding d given here to Eq. (3-1). Pretensioning is important in surgery (Chapter 11) and in assessments of sharpness (Chapter 9).
3.3 Different Types of Offcut Formation Slices of floppy materials have the same thickness as the depth of cut since they are produced by continuous bending (even though there is negligible bending stiffness). The thickness of a slice is still unchanged when there is elastic, or plastic, resistance to bending. Common experience of peeling potatoes, splitting wood and so on, might suggest that offcuts of all materials should form in bending, and Reuleaux in 1900 said as much (Figure 3-5). However, there is a problem: instead of a constant thickness offcut, there is a tendency for the path of separation at the end of thin beams to curve towards the free surface and to break off giving a series of spalls, or discontinuous chips, particularly in brittle solids where cracks run ahead of the cutting blade. The resultant cutting force Fres on the ‘beam arm’ has components FC parallel to the cut surface and FT normal to it, where Fres (FC2 FT2) (Figure 3-6A). The direction of Fres is given by tan1(FT/FC) and the ratio (FT/FC) depends on the inclination of the cutting blade and friction (Appendix 1). The component FC at the tool tip is statically equivalent to FC acting along the centre of the offcut/beam together with a clockwise bending moment of magnitude FCt/2 (Figure 3-6B). In contrast, FT has an anticlockwise bending moment about the end of the beam given by FTa, where a is the length of the beam. The net anticlockwise moment opening up the cut, and putting the bottom of the beam into tension, is M FTa FCt/2. The tensile bending stress in material just above the end of the cut is
Sp
Stichel
ah n
Figure 3-5 Reuleaux’s idea of 1900 that all cutting was by formation of offcuts in bending.
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The Science and Engineering of Cutting
t
FT
FCt/2
FC
FTa
FC
a
A
B
Figure 3-6 (A) A beam offcut separated in bending by axial load FC and transverse load FT applied by the rake face of the tool at the tip of the cut. (B) The loading is statically equivalent to the load FC applied at the centre of the beam together with a clockwise moment of magnitude FCt/2; and an anticlockwise moment FTa.
6 M/wt2, where w is the width of cut. (Simple beam theory does not really apply in this sort of geometry but it helps to understand what is happening.) Force FC pushing down the centre of the beam produces an opposing compressive stress of magnitude FC/wt. The net biggest tensile stress across the offcut is thus
σ 6M/wt2 FC /wt (6 /wt2 )(FTa FCt/ 2) FC /wt
(3-8)
When FC 3FTa/2t, is compressive (negative). A simple criterion for crack paths is that they run perpendicularly to the greatest tensile stress (this is not really correct but works quite well). Consequently, when FC 3FTa/2t, there is no incentive for the cut to divert from its original path and begin to turn towards the surface. However, becomes tensile when FC 3FTa/2t, and the crack may turn. Aluko (1988) argued that a criterion for the beginning of crack turning in cutting was when 0, i.e. the crack turns when it has grown in length to
a 2FCt/ 3FT
(3-9)
If the subsequent path to the free surface is approximated by a quarter circle of radius t, the length L of a scallop in the surface is
L a t t (1 2FC / 3FT )
(3-10)
and its aspect ratio (length/thickness) is (a/t) 1 (2FC/3FT 1). Experiments show that this simple approach gives the right sort of trends for when offcut formation by continuous constant-depth bending alters to paths that curve to the free surface. A similar problem arises in the design of fracture mechanics testpieces where it is desired that a crack should propagate down the middle of the testpiece. An approximate analysis for turning based on the signs of the stresses along and across the crack path is given in Atkins and Mai (1985). Proper criteria for crack turning are more involved (Cotterell & Rice, 1980). Straight grooves are often cut in the direction of the starter crack in fracture toughness specimens having shallow arms to prevent crack turning. A different reason for sometimes cutting grooves in specimens is to promote plane strain. [Grooves (score lines) are formed in all ring-pull cans to ensure that the opening follows the desired path.] The only way a constant thickness continuous offcut can be formed in elastic bending is when the fracture toughness parallel to the surface is smaller than in other directions, i.e. in an anisotropic material that has different properties in different directions (Section 3.4.2). Mica, slate and wood are examples.
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
43
The result of all this is to realize that continuous offcuts formed in elastic bending in isotropic materials do not occur that often. Offcuts formed in plastic bending are even rarer since an entirely different mode of chip formation occurs in practice in which the offcut is formed by shear on a well-defined plane extending at an angle from the cutting edge to the free surface (Section 3.5.2).
3.4 Brittle Offcuts In contrast to cutting floppy materials, as soon as the offcut displays some stiffness, it stores elastic strain energy, , in compression and bending. This will enter the energy rate balance during cutting and alter the magnitude of the forces required compared with the values predicted for floppy offcuts. That is, the incremental work equation becomes FCdδ Rwdδ d(friction) dΛ
(3-11)
This equation (without the Rwd term) also relates to situations where the surface of a brittle solid is loaded elastically by a sliding or rolling indenter. In such cases, there is no deliberate indentation or depth of cut, and elastic deformations caused by the contact stresses recover when the contact patch passes by, as in roller bearings. However, there are circumstances where a series of cracks forms in the tensile stress field in the wake of the slider and propagates down into the surface, in which case the equation above applies in full. Sometimes these subsurface cracks intersect so that chips are detached and material removed. This sort of problem is usually discussed under scratching (Chapter 6), but brittle fracture under a sliding line contact (Bower & Fleck, 1994) relates to the sort of two-dimensional plane strain orthogonal cutting discussed here. Their line-load solution suggests that there is not much crack interaction and not much wear debris.
3.4.1 Brittle discrete offcuts In cutting brittle solids we assume either (i) that the ‘starter crack’ for fracture mechanics calculations arises within some distribution of small flaws in the microstructure of the workpiece in the region where the cutting edge is applied; or (ii) that the minute indentation into the material by the tool before cutting commences produces a starter flaw. Freund (1978; cf. Barenblatt & Cherepanov, 1960) gives an linear elastic fracture mechanics (LEFM) solution for a body loaded normal and parallel to the surface by forces FC along the crack of length a, and FT perpendicular to the crack (Figure 3-7), which is
KI [4π/(π2 4)] (1/w a) {FT (2 /π)FC }
(3-12)
where w is the thickness of the body. This expression is applicable to cutting by a symmetrical wedge in the way an axe is used. The ratio of (FC/FT) is determined by the angle of the faces of the wedge and the friction. Appendix 1 shows that
FC /FT (µ tan α 1)/(µ tan α) Hα (A1.3e in Appendix 1)
or
FC /FT (µ tan θ)/(µ tan θ 1) H θ (A1.3f in Appendix 1)
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The Science and Engineering of Cutting
a
FT FC FT
Figure 3-7 Splitting by symmetrical wedge.
The geometry is similar to that of the so-called ‘wedge opening loading’ test in fracture mechanics to determine toughness of materials. Vincent et al. (1991) used a symmetrical sharp wedge test to investigate the fracture toughness of foodstuffs, as described in Chapter 12. Note that for FT 0 (loading only parallel to the putative split), KI is negative. This means that the faces of the split jam together at the tip as explained by Bilby (1980), who goes on to show that sideways forces FT are generated in reaction to the faces compressing against one another (see also Benbow & Roesler, 1957; Kendall, 1978). Whether cutting in the geometry of Figure 3-7 can be stopped and started at will (the energetic stability of cracking) may be assessed with the methods of Chapter 8 in Atkins and Mai (1985). That is, the ‘compliance calibration’ expression in LEFM is FC2 2R ÷ d/dA(/F), into which the expression for FC given by Eq. (3-12) may be inserted for given (FC/FT). This gives an expression for d/dA(/F) – having cancelled R (KC2/E) on both sides – and hence d2/dA2(/F). Using these first and second rates of change of compliance expressions enables stability to be determined. It may be shown that under both displacement control and load control, cutting is stable for constant R in Freund’s geometry. In finite-length workpieces, the relation will not apply when the cut reaches the back face, and things will change. Cutting with a wedge inclined to the free surface, but where both faces of the wedge are still in contact with the workpiece, produces an unbalanced force normal to the cut surface which has to be reacted in some way, possibly by friction but usually very diffusely and remotely by the workpiece being held in a vice. The unbalanced force introduces the complication that the problem becomes asymmetrical, i.e. has both tensile and in-plane shear components at the crack tip, in other words both mode I and mode II components of the stress intensity factor K. Stanzl-Tschegg et al. (2008), in their work on the fracture toughness of timber, rotated the wedge in Freund’s geometry deliberately to induce different combinations of mode I and mode II cracking. When the wedge is inclined so much that only one face is in contact with the workpiece, we arrive at orthogonal cutting with FC and FT applied to only one ‘beam arm’. The existence of asymmetrical loading and mixed-mode stress intensities is the ‘proper’ reason for cracks turning to the free surface and for the formation of discrete spalls, which were explained in Section 3.3 in terms of simple bending stresses. Experiments show that the trajectory of the path of separation is not as simple as that implied by simple bending: the path need not start off in the direction of the cut surface and may first dip below the intended depth of cut, before rising to the free surface; alternatively, its starting direction may already point towards the free surface. The aspect ratio of the chips depends on the nominal depth of cut and on the rake angle of the tool. Similar behaviour is found when cutting grooves and grinding brittle solids (Chapter 6).
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
45
Figure 3-8 Crack trajectories predicted for cutting with a zero rake angle tool. The upper scallop (A) shape is for (KIC/Et) 2 102; the middle (B) for 2 103; and the lowest (C) for 2 104 (after Chiu et al., 1998).
Prediction of the crack trajectories by which scallops are formed is a difficult exercise in elastic fracture mechanics. Figure 3-8 shows paths, non-dimensionalized by the depth of cut t, for cutting with a rake angle 0 (Chiu et al., 1998). The three scallops are for different values of KIC/Et, where KIC is the critical stress intensity factor for mode I (tensile) cracking and E is Young’s modulus (KIC2 ER). Each trajectory therefore represents the effect of different depth of cut in a material with constant E and R. The upper plot is for KIC/ Et 2 102; the middle for 2 103; and the bottom for 2 104, i.e. in increasing order of t. It was found that the spall aspect ratio (L/t) was approximately proportional to (Et/KIC), i.e. more brittle materials have longer scallops; or, alternatively, the aspect ratio is only weakly dependent on t in a given solid. Notice that the modelling reproduces an initial dipping of the spall and subsequent rise to the free surface. The spall shape predictions in Figure 3-8 are based on developments of elastic fracture mechanics models for beam fracture, produced by an axial load applied at some arbitrary point (not necessarily at the mid-point) within the ‘single beam arm’ that, as explained in Section 3.3, is equivalent to an axial load and bending moment (see Cotterell et al., 1985; Thouless et al., 1987; Thouless & Evans, 1990). These solutions have found particular application for problems of delamination, promoted by relaxation of residual stresses, of thin coatings in electronic
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The Science and Engineering of Cutting
components and bimetallic layers (e.g. Li et al., 2004). Chiu et al. (1998) showed that when applied to cutting, non-linear effects have to be included in the algebraic expressions for the mode I and II stress intensity factors. Non-linear effects are more important at longer crack lengths and at large values of the material parameter KIC/Et. The criterion employed for the crack paths in Figure 3-9 is that they are trajectories having KII 0 (Goldstein & Salganik, 1974; see also Cotterell, 1965 and 1966, for crack paths in terms of the so-called T-stress). Should KII 0 at some point along the path, the crack is driven deeper; should KII 0, the crack is driven towards the free surface, Chiu et al. (2001). The formation of spalls often occurs in an uncontrolled, but periodic, fashion as shown by the results in Figure 3-9(A,B) for cutting polymethylmethacrylate (PMMA) (Wyeth, 2008). Cutting forces during spall formation fluctuate violently: the load on the tool builds up until the crack initiates, followed by a precipitous drop to zero, the crack having propagated ahead very quickly and curved to the free surface to detach a chip. The load remains at zero until the tool catches up with the new surface formed by the bottom of the spall and then the process repeats itself. The first spall is formed by the inclined blade cutting at full depth into the vertical 400 350 300 250 Force (N)
200 150 100 50 0 –50 –100 –150
30
35
A
40
50
45 Distance (mm)
10 mm
55
60
Tool 1 mm
Sample
B
Figure 3-9 Orthogonal cutting of polymethylmethacrylate by 30° rake angle tool at depths of cut that produce scalloped chips: (A) force vs blade travel for 1 mm deep cut; (B) paths taken by cracks to form chips at various depths of cut.
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
47
edge of the workpiece. All other spalls are formed by the tool cutting into progressively thicker regions of the curved end of the previous spall. Since (KIC/E) is greater for PMMA than for ‘more brittle’ materials like glass, the scallops are ‘short and fat’ in Figure 3-9(B) rather than ‘long and slim’ as in Figure 3-8. (Even so, the scallops are still formed in globally elastic stress fields for which LEFM applies, and the bits can all be fitted together to regain the original size and shape of the PMMA testpiece.) The length of spall increases with depth of cut, so that the gaps between load peaks are further apart as t increases. To predict cutting forces for spall formation (or, inversely, to predict the toughness from the cutting forces) requires a fracture mechanics solution for a beam of very short length (really for a starter crack length of microstructural size) loaded both axially and by a bending moment. No algebraic solution exists at present (Thouless, 2008). However, we know that the critical value of the stress intensity factor will control cutting, and that this quantity has units of N/m3/2 (Chapter 2). It follows therefore from dimensional analysis that the resultant cutting load per unit thickness of workpiece (FC/w) at spall initiation should vary as t. Data for orthogonally cutting 3 mm thick PMMA with 30° demonstrate that (Fres/width) 826t with the numerical constant having units of N/m3/2. Data for PMMA cut with other rake angles and for spalls formed in other materials show similar trends with different proportionality constants (Wyeth, 2008). The behaviour is less well behaved when the rake angle is small. Results confirm that the aspect ratios of spalls increase only weakly with depth of cut as suggested by the (Et/KIC) parameter of Chiu et al. (1998), and appear to be nearly geometrically similar within experimental error. In a study of chipping produced by indentations near the edge of testpieces, Chai and Lawn (2007) also found that spalls were approximately geometrically similar. An unusual example of brittle chip formation is found when ploughing frozen soil. Aluko (1988) and Aluko and Chandler (2004) have applied the present sort of analysis to this problem, as discussed in Chapter 14. Details of the aspect ratios of spalls are of interest in flint knapping and pressure flaking, the method employed by ancient peoples to produce sharp cutting edges in minerals (Fonseca et al., 1971; Cotterell et al., 1985). In knapping or percussion flaking, the workpiece is struck by a sharp blow; in pressure flaking the piece is removed by a steady load. According to Figure 3-8 the aspect ratio of a spall is bigger the smaller the value of (KIC/Et). Hence, at constant t, longer knapped flints are produced when (KIC/E) is small, i.e. in very low toughness, very stiff, solids. Flaking in flint knapping does not rely on well-defined cleavage planes. A major contribution of Cotterell et al. (1985) was the realization that in flaking it is not the direction of the force that determines the path of fracture, but the displacement of the tip. They showed that the crack path remains parallel to the surface when the direction of the blow or thrust varied over quite a wide angle. If flaking depended on accuracy in placement and direction of the force, flaking would never have been successful. Machin (2007) is investigating the mechanics of the Acheulean hand axe, made from knapped flints, as it applies to butchery of animals. [Acheulean is the name for the handaxe-dominated industries/assemblages of artefacts in the Lower Palaeolithic (Old Stone Age) archaeological record.] Materials differ in their knappability: obsidian is volcanic glass that is easily knapped to fine, thin edges, whereas flint is full of impurities and difficult to knap.
3.4.2 Continuous offcut formation by elastic bending Continuous offcut formation by elastic bending, where crack paths do not divert to the surface, can occur in materials with anisotropic toughness, the obvious example being splitting wood along the grain with an axe. The difficulty of propagating a crack across the grain prevents the
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The Science and Engineering of Cutting
crack path deviating towards the free surface and forming a scallop. Other examples concern cleaving crystals and layered materials like slate, and cracking along interfaces such as paint films (Asbeck, 1970). When opening pistacchio nuts and shucking oyster shells, the inserted knife is twisted to lever the shell apart along weakened interfaces; walnuts need not be opened with nutcrackers as there is a weak spot at one end into which a wedge can be inserted. The cracking of glued joints is also investigated in this way. Obreimoff’s (1931) classical experiments on surface energy concerned the cleavage of Muscovy mica by a wedge (mica from the region around Moscow cleaves into large flat surfaces with few steps and faults, and is excellent for this sort of experiment). In all cases, the pieces separated in bending can be refitted to regain the original workpiece. Had Obreimoff’s experiments been better appreciated when the subject of fracture mechanics took off after World War II, the topic of crack stability might have been better understood: Griffith’s testpiece geometry was, unfortunately, inherently unstable, whereas the split cantilever, under displacement control, is not (see Chapter 8 of Atkins & Mai, 1985). Steady-state cutting forces for elastic beam offcuts that run with constant depth have been determined by Williams (1998) and Atkins (2006) for a tool that cuts on only one side as in Figure 3-10. We have
FC Rw/Qbend
FT Rw/ tan θ
(3-13a) (3-13b)
where Qbend [1 µ/ cos θ (sin θ µ cos θ)] tan θ/H θ
(3-13c)
Note that there are other forms for Qbend written in terms of and/or (Appendix 1). Note also that friction has disappeared in Eq. (3-13b) for FT, exactly as in (3-2b). Putting 0 in Eq. (3-13a) gives FC Rw (an expression not involving ) which is the same as that for the simple cutting of floppy materials which neither store nor dissipate energy. This comes about because bending to a circle has been assumed in the analysis so that the incremental elastic bending energy put in as the crack advances is matched exactly by the increment of bending energy recovered as the beam flattens out when the blade moves forward. (The deflexion of a point-loaded cantilever does not form a circle but the difference will be small.) F
N
α θ
FC ρ
FT
t ao
Figure 3-10 Cutting where the offcut is formed in bending.
x
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
49
The greater the cutting blade angle and relief angle, the greater the curvature of the offcut, and the greater the bending strain energy. On the other hand, the smaller the blade angle, the longer the sliding contact length between offcut and blade, and the greater the friction work done. This means that there is an optimum blade angle for cutting where forces are least (Willis & Vincent, 1994). The explanation for optimum die angles in wire drawing is the same, except that the plastic strains in drawing (analogous to bending strains here) are irreversible (Rowe, 1965). The foregoing explains the skill of the microtomist in setting up the instrument to produce ‘best’ (least damaged) sections. The operator is, in fact, finding the blade setting for minimum force. With hindsight, it is not surprising that smallest microtoming force damages the cut sections least (fewest internal tears between different components of micro/macrostructure etc), and achievement of least separation force is the aim of much food cutting machinery (see Chapter 12). Furthermore, the remaining surface on the bulk piece being cut, which forms the top part of the next microtomed slice, is also least damaged, which is advantageous. It is possible to instrument a microtome for forces and fit a control system which automatically finds the optimum blade orientation (Atkins & Vincent, 1984; Atkins et al., 1983a; Willis, 1988). In steady-state beam cutting there will be a gap between the cutting edge and the point of separation (Figure 3-10). It is related to the steady-state radius of the travelling elastic beam; the gap is never zero for separation by bending (unless root rotation occurs; Williams et al., 2009) but can be very small, implying that, once started, a cut could just as well be progressed by a blunt tool and that sharpness does not matter. However, surface damage at the start of a cut caused by a blunt tool may not be acceptable, so tools are kept sharp. Friction opposes the motion of the offcut up the tool face during cutting. If the tool is withdrawn from a cut by reversing the drive, FC reduces and the beam elastically relaxes, still pressing on the blade. Friction between offcut and blade now reverses its direction to oppose the motion of the offcut down the rake face during withdrawal of the wedge. Consequently, unloading lines will follow different paths from loading lines, and hysteresis loops are created, with only part of the stored elastic bending energy being recovered. The FC unloading line may be predicted by changing the sign of the friction force in the analysis for loading a beam by a wedge (Atkins, 2006). When friction is very high, there is no tendency for the blade to be pushed away by the release of bending energy and work has to be done by the drive mechanism to pull out the tool, meaning that FC changes sign and takes negative values. This is what happens when an axe gets stuck in a log.
3.5 Ductile Offcuts In the quasi-static cutting of ductile solids, the incremental work equation is
FCdδ Rwdδ d(friction) dΛ dΓ
(3-14)
where is the plastic work required to form the chip, and is the elastic strain energy stored in the chip. The plastic strains are much larger than the elastic strains in this type of cutting, and elastic strain energy may often be neglected to give
FCdδ Rwdδ d(friction) dΓ
(3-15)
which is the equation for rigid-plastic fracture mechanics. There will be no recovery or ‘springback’ of the offcut when the tool is withdrawn in this case. Being plastically deformed, the cut-away chips cannot be stuck back together to regain the workpiece.
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The Science and Engineering of Cutting
Figure 3-11 Mallock’s 1881 sketch of chip formation when cutting a hardened steel.
Slender tools having small and inclined at large might be expected to produce chips by plastic bending; a PVC washing-up bowl can be cut by a kitchen knife in this way. When cutting ductile metals, however, the forces are high and the fragility of slender tools prevents their being employed (even if a slender blade could be employed with metals, it would have limited capacity to conduct away the heat generated at factory production rates). In consequence, ‘chunky’ tools with large included wedge angles – much greater than in microtoming or chiselling – are used. The angle for the tool is chosen to withstand, without breaking, the cutting forces the workpiece material requires and to give the tool stiffness and strength. Despite the large , chunky tools are still sharp (Chapter 9). The wedge angle may be so large that becomes negative. Such negative rake tools find application both with less ductile tool materials (e.g. ceramics) and with less ductile workpieces, in both cases because, when cutting, compressive stress fields are set up in tool and workpiece which inhibit cracking. As a result of employing strong chunky tools having large that inevitably must operate at small or negative , offcuts in ductile materials are formed not in plastic bending but rather by an entirely new deformation field, namely concentrated shear along a well-defined shear plane inclined to the cut surface (Figure 3-11). The chip is altered in thickness as it passes through this primary deformation zone, and then experiences secondary shear in a flow region around the rake face of the tool. Flow through the complete field produces a continuous chip, curled to a greater or lesser degree (a single primary shear plane predicts ‘straight flow’ along the rake face). Permanently-deformed curly continuous ribbons may be produced in a variety of materials ranging from butter in the kitchen (scrape a knife along the surface of a block of butter) to high-strength metals. In addition to plastic chip formation by bending or by shear, a third plane strain mode of deformation takes place at very negative rake angles when cutting ductile materials. At negative rake angles of about 60° to 70°, Connolly and Rubenstein (1968) showed that a transition to ‘prow formation’ where a standing wave of metal is pushed ahead of the tool with no separation into a chip (Figure 3-12). Prow formation is important when scratching grooves (Chapter 6). Also at large negative , workpiece material can squidge out sideways so that the deformation ceases to be two dimensional.
51
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
Tool
Figure 3-12 Standing wave (prow) propagation in place of cutting at large negative rake angles.
3.5.1 Dimensional analysis for elastoplastic cutting In a general case, we assume that the cutting force FC depends upon the width w of the cut, the uncut chip thickness t, the shear yield strength k and the fracture toughness given either by the critical stress intensity factor KC or equivalently by E and R (KC2 ER), where E is Young’s modulus and R the specific work of fracture or fracture toughness. We have, following Palmer (2008) for example,
FC ∝ ta w bKCc kd
(3-16)
Inserting dimensions (newtons and metres), and solving for two of the four unknown indices, we find
FC ∝ kt2 (w/t)b (KC /k t)c kt2 (w/t)b (ER/k2 t)e
(3-17)
where e c/2. Thus forces in cutting depend upon kt2 directly, but also on (w/t), and on the toughness/ strength ratio non-dimensionalized by the depth of cut. There was an early belief that machining depended on some critical cutting stress (FC/wt) (Manchester Association of Engineers Report, 1903), but this is seen not to be so. An extensive body of data exists that empirically relates cutting forces and ‘unit horsepower’ (FC/wt) with depths of cut t and widths w (ASME, 1952). For example, FC (constant) txwy where 0.75x1.0 and 0.96y1.0. The results were obtained from log plotting experimental data. According to Eq. (3-17) the index of t is (2 be) and the index of w is b. Taking the latter to be unity, the index of t is (1 e). Experimental data give inverse relations between yield strength (hardness) and toughness (Atkins, 1974), so that (ER/k2) may be written in terms of k only, and the empirical relations predicted. Thus for low carbon steel of strength 30 ksi (210 MPa) FC 120 000t0.86w (lbf) is predicted, and Boston (1951) gives for hot-rolled 1020 steel FC 133 000t0.85w0.98 (lbf).
3.5.2 Continuous offcut formation by elastoplastic and plastic bending Williams et al. (2008) present a solution for chip formation by elastoplastic bending, where the relative elastic and plastic proportions of the beam section during steady cutting depend upon the toughness/strength (ER/k2) ratio of the material as well as the rake angle and friction. When plasticity is extensive the cutting force for formation of a continuous plastic chip is
FC /Rw (1/Qbend ){[ (kt)/(n 1)(n 2) R ] 1}
(3-18)
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The Science and Engineering of Cutting
where n is the workhardening index in the stress–strain relation given by on with o a constant, and where Qbend is given by Eq. (3-13c). As for continuous chip formation by elastic bending, it has been assumed for simplicity that the neutral axis of the arm is bent into a circular arc of radius . Hence FC varies as t with an intercept of Rw/Qbend. Relation (3-18) is a special case of the Williams et al. (2009) elastoplastic solution and may be determined either directly (Williams, 2006) or by transition arguments (Atkins, 2006b). In the latter case, the problem of a block of material already having a starter crack situated at depth t below the surface, into which a wedge is driven so as to bend the beam arm and eventually cause propagation (Figure 3-10) is solved. Atkins (2006b) noted that a false solution is obtained to this problem when plasticity is treated as if it were non-linear elasticity (Kachanov, 1971) and a solution for FC obtained using non-linear elastic fracture mechanics (NLEFM). This comes about because unloading occurs behind the wedge-shaped cutting blade as it progresses down the starter crack, and NLEFM presumes that the associated energy is recovered; it is not, of course, since unloading of a rigid-plastic solid is ‘dead’. The solution is of pedagogical interest as it is relevant to problems of crack propagation in elastoplastic fracture mechanics (so-called JR curves and so on) where overestimates of crack resistance can be made (Atkins et al., 1998, 2003). Note that these solutions do not address the question of whether, in practice, the depth of cut will remain at the starting level of t, or will divert to the free surface giving elastoplastic chips in the form of a series of broken-off portions of beam. Calculations along the lines of those in Section 3.3 may be performed for plastic collapse in the beam. The axial stress consists of compression to stress level FC/wt y, together with a plastic collapse moment of [FTa (FCt/2)] ywt2/4, where y 2 k is the uniaxial yield stress. The net tensile bending stress at the tip of the tool is zero therefore when a (3FCt/4FT). Whether the gap between the tip of the tool and the crack front (atrans in the nomenclature of Atkins, 2006b) in steady cutting can attain the length (3FCt/4FT) 3Ht/4 depends on friction, wedge angle , and the beam radius during cutting. The analysis employs engineers’ beam theory where t. Atkins (2006b) shows that chip formation by plastic bending is possible only at small Z R/kt. At large Z, the radius of the plastic beam during cutting approaches the dimensions of the beam and the solutions are inadmissible. This is when chip formation by shear supervenes. Williams et al. (2009) express this in detail in their solution for chip formation by elastoplastic bending.
3.6 Offcut Formation by Shear Mallock, as long ago as 1881 (and others before him), had observed that cutting in a great many materials often occurred by shear (Figure 3-11). Cold chisels remove material in this way. Piispanen in 1937 pointed out that chip formation by shear along a thin lamina in orthogonal cutting is like a deck of cards sliding over one another (Figure 3-13A), and Ernst and Merchant independently employed the same idea in their pioneering modelling of cutting forces. The zone of concentrated shear is idealized as a thin shear plane inclined at angle f to the cut surface and emanating from the tool edge. The shear plane has zero thickness according to slip line field/upper bound analyses; in practice, workhardening and tool bluntness will widen the zone of deformation into a pie-shaped zone emanating from the cutting edge. Figure 3-13(B) shows the hodograph (velocity vector diagram) in which the workpiece is imagined to be flowing plastically with velocity V towards a stationary tool. The material is rigid-plastic with constant shear yield stress k; there is no elasticity. V* is the so-called velocity
53
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
Chip
α
1 2 3
W
Tool
4
Z
V*
Vchip
5
V
6 t
10
9
8
7
α
φ X
φ Vwork
Y
B
A
Figure 3-13 (A) Piispanen’s deck of cards model for chip formation by a single shear plane inclined at angle φ to the direction of cutting. Plastic slip in plane strain along a primary shear band is impossible under constant plastic volume unless a gap occurs in the region of XY. Otherwise ZWV is an increase in plastic volume (adapted from Cook et al., 1954); (B) the hodograph (velocity vector diagram) for material flowing across the simple single shear plane.
discontinuity or ‘sharp shear’ experienced by elements crossing the shear plane that makes them change direction to flow parallel to the face of the tool with velocity Vchip. Geometry gives
Vchip / sin φ V*/ sin(90 α) V/ sin(90 φ α) V/ cos(φ α)
(3-19)
The change in speed results in a change in chip thickness. Continuity of plastic volume gives
tchip t cos(φ α)/ sin φ
(3-20)
After passage through the shear plane, the magnitude of Vchip will not necessarily be the same as V, nor will tchip be the same as t. Only in the special case of sinφ cos(φ ), i.e. when tanφ cos/(1 sin), will Vchip V and tchip t. Hence when chips are formed in shear, the thickness of the chip is generally different from the depth of cut. Since, in metal cutting, φ is usually smaller than about 45° because is usually not greater than about 40°, chips are thicker than the depth of cut. However, at much greater , steeper φ are possible and then tchip t, as happens when some non-metals form chips in shear. The incremental plastic work d in Eq. (3-15) is performed in the shear plane. Plastic flow occurs at constant volume, and Cook et al. (1954, p. 156) realized that it was impossible for the Piispanen pack of cards model to operate in plane plastic strain, without there being ‘new surface’ (i.e. ‘fresh air’) at tool tip. Unless there is a gap of at least the thickness of the shear band, which releases material to be sheared, plastic volume cannot be conserved, and ZWV in Figure 3-13(A) would represent an inadmissible increase in volume of material. The plastic shear work per volume is given by the area under the stress–strain curve, i.e. by kd, where is the shear strain along the primary shear band (do not confuse this use of with that for surface free energy). V*/Vp, where Vp is the velocity perpendicular to the shear band (Johnson & Mellor, 1973). Vp Vsinφ from the hodograph, whence
γ V*/Vp cos α/ cos(φ α)sin φ cotφ tan(φ α)
(3-21)
Experiments show that 10° φ 30°, say, for many ductile metals cut with tools of small rake angle, so that 6 2 for 5° 30°. Such values for shear strain are very high
54
The Science and Engineering of Cutting
indeed and far greater than normally encountered in typical tension or compression tests, and even in most torsion tests. How shear strains ought to be defined in severe plastic flow is discussed by Stüwe (2003). Large strains set cutting apart from other deformation processes and it is an interesting question as to how materials withstand such large shear strains without fracturing (Chapter 4 discusses broken-up, discontinuous chips). Workhardening stress–strain curves eventually saturate out even at the smaller strains achievable in laboratory testing, so the shear yield stress encountered in the cutting of a soft, annealed material may very well be comparable with that for an initially hard version of the same material, since the strains are so high. Shear strain rates in cutting are also very large (d/dt is theoretically infinite for a shear plane of zero thickness). To evaluate d/dt requires an estimate for the thickness of the shear band in practical cutting in order to estimate the time taken for an element of material to cross the slip band. For a slip band average thickness q, the time is (q/Vsinφ) where Vsinφ is the component of the cutting speed perpendicular to the slip band. Experiments show that q is proportional to the length (t/sinφ) of the slip plane, i.e. q ct/sinφ (Stevenson & Oxley, 1970–71; Childs et al., 2000). Values for c depend upon circumstances, and practical deformation zones are not simple parallel-sided slip bands, but c3 is representative. Whence d/dt [cotφ tan(φ )](Vsin2φ/3t). For, say, 0, φ 30°, t 0.1 mm and V 1 m/s, d/dt2 103 s1. Commercial metal cutting may have V of 100 m/s and more, so it is clear that d/dt is very high. Even in very slowspeed laboratory cutting with V 1 mm/s, d/dt is 2 s1. In contrast, quasi-static testing in the laboratory takes place at about 103 s1. Similar high strain rates are found in many commercial metalworking operations (Atkins, 1969). Mechanical properties of materials can be different at high rates. For example, the shear yield stress k of mild steel, some aluminium alloys (but not precipitation-hardened alloys) and titanium is increased at high rates, whereas magnesium and copper alloys are not. Annealed materials are more sensitive than cold-worked samples. It is clear that these higher values of k should be employed in models of cutting, but there is the competing effect of what temperature does to mechanical properties. Increasing temperature usually results in softening in most materials, in contrast to the hardening resulting from high strain rates (Stevenson & Oxley, 1970–71; Dean & Sturgess, 1973; Hashmi, 1980). The work dissipated in cutting can produce large increases of temperature in the cutting zone. For example, the work done per unit volume in shearing steel having a shear yield stress of 200 MPa to a strain of shear 3 is 200 106 3 600 MJ/m3 (the units of stress and work/volume are the same: Nm/m3 N/m2). For a depth of cut of 0.1 mm and width of cut of 5 mm, the cross sectional area of chip is 0.5 mm2 and the metal removal rate is 0.5 106V m3/s for a cutting speed of V m/s. The rate of doing shear work is thus (600 106) (0.5 106V) 300V J/s. For adiabatic deformation all this work is converted into a temperature rise T in the chip, which requires (0.5 106V) 7850 486T 2 VT J, where 7850 kg/m3 is the density of steel and 486 J/kg k is its specific heat. Hence T150°C. This simple calculation does not include friction work that will make T greater. It is crude – more sophisticated methods incorporating the thermal conductivity of the workpiece and tool will be found in standard metalcutting monographs – but it serves to show that cutting can result in large T. Davies et al. (2003) give a valuable review of analytical and experimental methods for the temperature fields at the chip–tool interface, including infrared microscopic measurements. Herbert, in the early years of the twentieth century, was the first to use a thermocouple with a cutting tool (Herbert, 1939). In the case of some polymers the temperature rise heats up the chip to pass through the glass transition temperature Tg and may even melt the chip (Kobayashi, 1967). In what follows, we assume that k remains constant during steady-state cutting. In estimates for cutting forces, k should take a mean value representative of the actual strain, strain rate and temperature in the cutting zone.
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
55
3.6.1 Cutting forces The plastic work done per volume in shear is given by the area under the stress–strain curve kd. The volume flow rate is Vwt so, for constant k, the shear work rate is (kγ)(Vwt) (k cos α/ cos(φ α)sin φ)Vwt
(3-22)
The sheared chip slides along the rake face of the tool at a speed of Vsinφ/cos( φ) (Figure 3-13B). The friction work rate is thus FVsinφ/cos( φ). F may be expressed in terms of FC (Appendix 1), whence FCsec( )sinVsinφ/cos( φ) is the friction power, assuming Amontons/Coulomb friction. The rate of generation of separated area at the tool tip is Vw, so the separation power is RVw. The external work done/time is FCV. FT does no work but is related to FC by the assumed friction condition at the rake face. For Amontons/Coulomb friction FT FC tan ( ), where tan. Equating the external and the sum of the three internal work rates we obtain FC (wkt) [cos(β α)/ sin φ cos(φ β α)][1 R cos(α )sin φ/kt cos α ] (3-23a)
or
FC (kwγ /Qshear )t Rw/Qshear wtk(γ Z)/Qshear
(3-23b)
where
Qshear [1 {sin β sin φ/ cos(β α)cos(φ α)}]
(3-24)
and where
Z R/kt
(3-25)
is a non-dimensional parameter formed from the toughness/strength (R/k) ratio of the material (that has dimensions of length) and the depth of cut.
3.6.2 Traditional metal-cutting theory Given that toughness is the common thread running through analyses of the cutting of floppy and elastic materials, one would expect toughness to be part and parcel of analyses of metal cutting. Historically, this has not been the case. Traditional analyses say forces and energy for cutting ductile solids depend on plasticity and friction only: separation work (the fracture toughness work Rw required to form the gap XY at the tool tip, Figure 3-13A) is either ignored or considered to be negligible (Section 2.6). As to how material forming the chip separates from that remaining on the cut surface, the opinion in the metal-cutting literature seems to be that material ‘just flows around the tip of the tool’. Many workers in metal cutting also query how fracture can have anything to do with cutting, since ‘cracks are not seen at the tip of the tool’ when cutting ductile metals. The answer to this question will be discussed in detail in Chapter 4, but two processes are being confused: (i) the requirement for separation at the cutting edge before the tool is able to move (otherwise inadmissible new
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The Science and Engineering of Cutting
plastic volume WZV is created (Figure 3-15A); and (ii) crack stability, i.e. whether the separated portion XY just keeps pace with the tool, or whether it runs ahead of the tool as a free-standing crack. When the separated region at the cutting edge has the same velocity as the tool, no tool-tip cracks are seen even though they exist; in less ductile solids, steady freestanding cracks can be seen ahead of the tool tip; in extremis, only limited (or negligible) plasticity occurs and the free-standing crack travels all the way to the free surface, scalloping out chunks of material, as described in Section 3.4.1. In between there is the variety of discontinuous chips described in Chapter 4. The toughness and the yield stress (or hardness) are mechanical properties that can be independently altered by appropriate thermomechanical treatments, so that it is possible to have an alloy with the same strength but quite different toughnesses. Such alloys machine differently: Kopalinski and Oxley (1987) observed that experimental forces when cutting hard cold-worked steels were lower than when cutting the same steels in the soft annealed condition. This is not possible according to plasticity-and-friction analyses. It suggests that there must be some other mechanical property playing a role in addition to k. Furthermore, the effects of strain rate, temperature and environment on k and R may not always be in the same direction, which may give rise to difficult-to-understand experimental machining behaviour when interpreted in terms of k alone. Childs et al. (2000) state that ‘the reason for the easier machining of copper and aluminium alloys compared with the elemental metals is not obvious from their room temperature, and low strain rate mechanical behaviours that are very similar’. But the mechanical properties upon which this statement is based do not include toughness. Traditional metal-cutting theories have RZ0, so Eq. (3-23a) becomes
(FC /wkt) γ/Q shear cos(β α)/ sin φ cos(φ β α)
(3-26)
This is the relation given by Ernst and Merchant (e.g. Merchant, 1944). It suggests that values of FC for different depths of cut should plot as a straight line passing through the origin having slope [wkcos( )/sinφcos(φ )] if φ and are invariant with t. To predict cutting forces ‘blind’ using Eq. (3-26), or have to be known, and also φ. While an intelligent guess may be made for , likely values for φ are unknown. Merchant (e.g. 1944) argued that for given , φ will adjust itself so as to minimize FC (or equivalently minimize the work rate). Differentiation of Eq. (3-26) gives for the optimum f
φ (π/4) (1 / 2)(β α)
(3-27)
This relation is shown in Figure 3-14 where φ is plotted against ( ); it is a single line with slope (1/2), and intercept (/4) radians (i.e. 45°). Experimental results for φ, for a wide range of ductile metals by many workers, are superimposed on Figure 3-14. In these experiments, cutting forces FC and FT were measured and converted into F and N on the rake face (Appendix 1) to give ; φ was measured either directly on the sides of samples or by measuring the thickness of the chip that is related to the depth of cut by Eq. (3-20); alternatively, the lengths of orthogonal chips may be used to obtain φ. According to Eq. (3-27), φ is the same for all materials, irrespective of their different mechanical properties (independent of k). The only way φ can alter in Eq. (3.27) is for to be different for different materials, but any differences in for different materials are already incorporated in the φ vs ( ) plot. Figure 3-14 demonstrates that not only do experimental values for φ lie below the minimum work prediction, but also φ is material dependent from the way φ values
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
57
50 Tin
φ (degrees)
40
Ernst–Merchant theory
30
20
Mild steel SAE 4340
10 Aluminium
Lead
Copper 0 –20
–10
0
10 20 (β – α) (degrees)
30
40
50
Figure 3-14 The Ernst–Merchant theoretical relation between the inclination of the shear plane φ and ( ), where is the friction angle along the rake face and is the rake angle of the tool. Experimental results for a variety of metals are superimposed.
group together for different materials. Several authors have given empirical relations between φ and ( ) for a wide range of different materials. For example, Pugh (1958) gives
φ c1(β α) c 2
(3-28)
where c1 and c2 are material-dependent constants. When Eq. (3-27) is substituted in the force relation (3-26), we obtain
(FC /wk) {2 cos(β α)/[1 sin(α β)]} t
(3-29)
that is predicted to pass through the origin of FC vs t plots. Experimental results do follow quasi-linear relations, but the data do not pass through the origin; furthermore, when is determined from F/N, it changes systematically with depth of cut. According to Eq. (3-29) cutting forces depend only on k, hence the many empirical correlations for forces and cutting power with yield strength, ultimate tensile strength or hardness, e.g. Boston (1951), mentioned in Section 3.5.1. The Ernst–Merchant theory has been applied to other materials, notably polymers (Kobayashi, 1967) and wood (Kivimaa, 1950; Franz, 1958; McKenzie, 1960). In the latter case, anisotropy limits its application to particular grain directions.
3.6.3 Historical changes to traditional theory The failure of the Ernst–Merchant model with its single shear band to predict practical cutting was attributed to non-uniqueness of the plasticity solution by Hill (1954), who pointed out that the shape of the free surface of the chip is unknown and itself must form part of the solution, so that the minimization procedure used by Ernst–Merchant to obtain φ is invalid (but see Stephenson & Agapiou, 1997, p. 471). But even when experimental φ are used in Eq. (3-26) to predict cutting forces, the values of and k (particularly) required to make
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theory and experiment agree, do not usually agree with independently determined values. Such differences are attributed to lack of data at the strains, strain rates and temperatures involved in metal cutting. Even special equipment, like split Hopkinson bars, that might produce such data, do not always attain the strain rates encountered in commercial cutting, so the required data are simply unknown: all one can ask is whether the derived and k are ‘reasonable’. It is the case that in metalforming generally, has often been used as a disposable parameter to make theory and experiment agree. This sort of thing remains a difficulty for modern finite element simulations of cutting where friction and mechanical properties that depend on rate and temperature are not known with confidence. To improve the Merchant theory in the expectation that it should come more into line with experimental data, metal-cutting research has followed three major lines: (i) more accurate modelling of the plastic flow fields by which chips are formed; (ii) how better to represent friction; and (iii) workhardening, rate, temperature and environmental dependence of the shear yield stress as elements of material flow through the deformation zone. Under (i) are all the painstaking visioplasticity studies of chip flow, laboriously done by hand in the days before image-recognition software from photographs of incremental experimental flow fields. From these data, curvilinear slip line fields were constructed with great skill and imagination (see Johnson et al., 1982, for a compendium of slip line fields), and great strides were made developing workhardening slip line fields, for both sharp and blunt tools. Conditions for the degree of chip curl were established. Important discoveries were made about stress distributions in cutting, for example that a force-balance for Oxley’s (1989) parallel-sided shear zone says that the hydrostatic stress H is compressive at the free surface, but reduces down the band towards the tool and can be tensile at the tool tip. The high strains given by the single slip line of the simple Piispanen shear band model are confirmed by the magnitude of the strain accumulated by elements passing through complicated slip line fields; likewise the high strain rates, e.g. Atkins et al. (1983b). It is perhaps pertinent to the argument that toughness should be included in cutting analyses, that McClintock (2002) introduced the new subject of ‘slip line field fracture mechanics’ (SLFFM) for combined flow and fracture problems, of which the cutting of ductile metals is one. Plots of FC/w vs t predicted by conventional slip line fields pass through the origin. A simple application of SLFFM at a given depth of cut would be to add to the predicted FC/w value the R value associated with separation, to give a plot that does not pass through the origin, as found experimentally. Of course, the slip line field is itself a minimum energy solution, so a proper analysis should include separation work in the minimization (Section 3.6.4). Even so, where the separation work, and plastic and friction work, appear to be uncoupled (they will not be in a general SLFFM problem) the simple approach might work well. FEM simulations of orthogonal cutting in Section 3.7 perhaps lend support to this idea. Under (ii) is elegant work using photoelastic tools and split tool dynamometers to determine the distribution of stresses along the rake face of the tool, described in Appendix 1. This confirmed that in most metal cutting there is a zone of high adhesion near the cutting edge which often results in metal transfer to the tool (e.g. Zorev, 1958). In consequence, friction in cutting is usually a mixture of sticking and sliding, which explains why coefficients of friction given by (F/N) vary systematically with depth of cut. Better understanding of so-called secondary flow in the chip in the vicinity of the rake face has resulted from all these endeavours. Experimental studies under (iii) are part of a wide interest by many workers investigating mechanical properties at high strain rates and over different temperature ranges. This
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
59
information is vital for proper understanding of impact problems, ballistics and such like. It is suggested elsewhere in this book that cutting itself may be a way of determining mechanical properties at high rates of strain, and at high or low temperatures. Cutting with continuous chips is steady state; most mechanical property tests are non-steady. Finite element simulations of cutting rely on these sorts of mechanical property inputs. The number of papers published, even under the separate headings given above, is immense. Perusal of standard cutting monographs and plasticity textbooks, listed in the references, will give some idea of the scale of work done, and continuing to be done. Students of metal cutting may be affronted at the limited space devoted here to proper consideration of chip flow fields, and that this book continues to model ductile cutting employing the simple single shear band. No disparagement is intended for the numerous researchers who have been, and continue to be, involved in this sort of vital work. But a contention of this book is that the physics of cutting requires the incorporation of fracture toughness, and even a crude modelling of plasticity and friction when coupled with significant separation work, reveals things that plasticity-and-friction-only analyses fail to do, however complicated and realistic the chip flow field. Since inclusion of R is central to analyses of the cutting of all other materials, what is special about ductile materials to warrant its exclusion?
3.6.4 Ductile cutting including toughness Kaneeda et al. (1983) took transmission electron micrographs of thin films taken from sections (i) within the primary shear plane; (ii) at the separation point at the tool cutting edge; and (iii) of the cut surface, from cutting experiments on pure aluminium. They observed that the subgrain cell structure at the separation point was smaller than that of the shear plane, indicating the much greater deformation required for separation. Furthermore, they showed with scanning electron microscopy that ‘two forms of ductile fracture’ occurred in the separation zone: dimple fracture (tension) and void sheet slipping off (shear). Subbiah and Melkote (2007, 2008), picking up Atkins’ (2003) arguments about the need to include toughness, took pictures of chip formation in oxygen-free high-conductivity (OFHC) copper and 2024-T3 aluminium alloy and found fracture zones at the roots of the chip. These observations lend support to the contention that toughness should be included as an additional mechanical property independent of yield stress in algebraic models of cutting. When R is included, we minimize Eq. (3-23b) instead of Eq. (3-26) to find the optimum φ for least FC. We find that f depends upon Z R/kt as well as . Williams (2009) gives the closed-form solution for φ as follows
cotφ tan(β α) [1 + tan2 (β α) Z{tan(β α) tan α}]
(3-30)
so that (FC /Rw) [2cotφ](1/Z) 1
{2 [1 tan2 (β α) Z{tan(β α) tan α}] 2 tan(β α)}(1/Z) 1
(3-31)
and, from the assumed friction condition, (FT/Rw) (FC/Rw)tan( ) as before. (Note that Williams’s Z parameter has a different meaning from that here.) Figure 3-15(A)shows loci of constant Z given by Eq. (3-30) plotted on axes of φ vs ( ). The Ernst–Merchant line corresponds with Z 0 and it appears that the experimental data in Figure 3-14 falling below this line lie on loci of constant Z. The only materials studied by Ernst
α
β = 29° 60 At the same Z, increasing β lowers φ and vice versa
Z = 0.1 Z=1
40
+23°
40
–11.5°
20
–23°
10 10–5
20
C
–20
0 (β – α)°
20
40
A
60
105
α
40
–29°
30
0
20 +29° 10 R/k = 5 x 10–4m 0
Z = 0.1
60
40 1 20 10 100
0 –60 –40 –20
0
20
40
60
Tool rake angle α (degrees)
t (mm)
D
80
15
R/k = 5 x 10–4m
β = 29° 10
Rake angle α
5
–29° 0 +29°
0 0
E
1
0.5
0
Primary shear strain γ
Shear plane angle φ (degrees)
80
B
103
10
β = 29° Shear plane angle φ (degrees)
–40
10–1 Z
Z = 100
–60
10–3
The Science and Engineering of Cutting
Ernst–Merchant theory
Z = 10
β = 29°
+11.5° 0
30
60
φ°
Z=0
Shear plane angle φ (degrees)
80
0.5
1
1.5
t (mm)
Figure 3-15 Predictions of cutting model that includes work of separation in addition to plasticity and fiction: (A) φ vs ( ) at various Z R/kt; (B) φ vs at various Z for a constant ; (C) φ vs Z at various for a constant ; (D) f vs depth of cut t at various for a constant and a constant Z; (E) shear strain vs t at various for a constant and a constant (R/k).
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
61
and Merchant that gave experimental φ close to their prediction were plastics having limited ductility, i.e. having low R, and hence small Z. Pugh’s empirical relations (3-28) for φ are linear approximations of the constant Z loci in the right hand part of the diagram (Atkins, 2003). Figure 3-15(B)shows predictions of φ vs at constant . The plots should be compared with experimental φ obtained at different in different materials, reported by Okoshi (1929), and shown in Figure 3-16, that include the droop at large . Calculations (Atkins, 2003) show that when Z 0.1, i.e. t 10(R/k), φ is practically constant at given and (Figure 3-18). So, at depths of cut sufficiently deep compared with the length scale given by (R/k), φ is constant and so is (Figure 3-18). Qshear is also constant. It follows from Eq. 3-23b that the cutting force per unit width (FC/w) is predicted to vary linearly with depth of cut t, according to FC /w (1/Qshear ) [(kγ) t R ]
(3-32)
The slope is (k/Qshear) – the same as the Ernst–Merchant theory, except that depends upon φ given by Eq. (3-30) rather than Eq. (3-27) – but now a force intercept given by (R/Qshear) is predicted. Such a linear relation is reminiscent of the Cotterell essential work of fracture method for determining toughness in the presence of extensive plastic flow (for a review of this method, see Atkins & Mai, 1985). At smaller depths of cut, Z 0.1, φ is smaller, bigger and Qshear bigger. This means that the (FC/w) vs t plot droops down at small t, but it still does not pass through the origin and has intercept equal to R because Qshear 1 for t 0 or Z . The full non-linear FC vs t relation, including the droop in FC to the smaller force intercept of R at zero t, is given by Eq. (3-31) for , and (R/k). There are plenty of experimental results that demonstrate linear plots with intercepts for metals, polymers and wood (e.g. Thomsen et al., 1953; Kobayashi, 1967). The droop region at very small t was noted by Finnie (1963) and is shown in Figure 3-17 for nylon cut with 0 (Ireland, 2007). Values for R and k derived from such plots are given in Atkins (2003). Analysis of data in Whitworth Taylor (1946) for NE 9445 steel, for example, gives 23 kJ/m2 for R. Results determined for polypropylene cut at various speeds are shown plotted against speed in Figure 3-18.
50 Cast iron 40
φ (degrees)
Brass 30 Aluminium 20
10
Mild steel
0
10
20 α (degrees)
30
40
Figure 3-16 Okoshi’s experimental results for the shear plane angle φ of various metals.
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While both R and k increase with speed, the toughness/strength ratio (R/k) decreases, meaning that at higher speeds the material is relatively more brittle. Values determined for annealed and cold-worked SAE 1112 steel at 200 mm/s (data from Eggleston et al., 1959) are 50 kJ/m2 and 100 MPa, from which R/k 5 104 (annealed samples), and 40 kJ/m2 and 200 MPa, from which R/k 2 104 (worked samples). As expected, cold working increases the strength and reduces the toughness. When R reduces disproportionately with respect to the increase in k, the reduction in separation work outweighs the increase in plastic work, so that a harder material is easier to cut than a softer one. 180000 160000
(FC/w) = 108t + 13,230 (r2 = 0.9966)
Normalized force (N/m)
140000
FC
120000 100000 80000 60000 40000 FT 20000 0 0.0002
0
0.0004
0.0006 Depth of cut (m)
0.0008
0.001
0.0012
Figure 3-17 Cutting force/width (FC/w) vs depth of cut t for nylon 66 using a 0° rake angle tool. ‘Droop’ at small t is evident (after Ireland, 2007).
50
5
k
0.5
0
R k (MPa)
R (kJ/m2)
1
4
40
1.5
30
3 R/k
20
2
10
1
0
R/k (m X 10–5)
2
0 0
100 200 300 Cutting speed (m/min)
400
Figure 3-18 Toughness R and yield strength k for polypropylene determined from data in Kobayashi (1967). Both R and k increase with speed but the toughness/strength ratio (R/k) decreases showing the material becomes relatively more brittle at higher speeds.
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Other methods of analysis are possible. When the tool face is heavily loaded, Coulomb friction may not be appropriate (Appendix 1) and FT FC tan ( ) may not be the right relation between the measured components of the cutting force. For fixed and , tan( ) is just a multiply factor, positive for and negative for . But many plots show FT rising at first as t increases but then curving over and decreasing sometimes to negative values for FT, even though FC is still rising at increasing t. This reflects a changing at constant . In the cutting of polymers, Kobayashi (1967) makes great play of the rake angle that results in zero FT, i.e. when tan( ) 0 or when . The resultant cutting force then coincides with the direction of tool motion and was believed to produce least deformation in the cut surface. Williams et al. (2008) discovered that many plots of F vs N follow ‘F Fo N’, where Fo was identified as an adhesion term, and used that in analyses for FC and FT. Appendix 2 discusses friction in cutting in more detail. A check on the sense of the cutting model that includes toughness, without regard to the detail of friction at all, is as follows. It is due to Williams. The friction work rate of Section 3.6.1 given by FVchip is FVsin/cos( φ). F may be expressed in terms of FC and FT, i.e. F FCsin FT cos (Appendix 1), whence the friction work rate is (FCsin FT cos)Vsinφ/ cos( φ). The overall work equation (3-15) then gives for a rigid-plastic material [(FC /w) (FT /w) tan φ] k[tan φ (1/ tan φ)] t R
(3-33)
Figure 3-19 shows results for high-impact polystyrene (HIPS) and the linear relation of Eq. (3-33) with intercept is well followed. In these data, φ is determined experimentally from chip thickness ratios. A linear fit gives R 1.7 kJ/m2 and k 35 kPa. The toughness is comparable with LEFM results but k is about 2.5 times the quasi-static shear yield stress. Williams et al. (2008) quantitatively attributed the difference to workhardening at the large strains encountered in cutting, and similar considerations will apply to k for other materials determined from cutting data.
3.6.5 Mixity in cutting toughness Wyeth (2008) shows that the values of toughness R, determined from cutting tests on polymers, can vary systematically with tool rake angle , being large at negative and small positive , α (°) 25
–20 0 10 15 20 30
FC FT – w w tanφ
20 15 10 5 0 0
0.1
0.2
0.3 0.4 (tanφ + 1/tanφ)/t
0.5
0.6
Figure 3-19 Plot suggested by Williams et al. (2009) to check the sense of the cutting model that includes work of separation. Data from cutting high impact polystyrene (HIPS) with various tool rake angles .
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and decreasing as increases. There is some evidence that the same effect is seen for other materials, to a greater or lesser degree. The variation seems to reflect the different tension/shear loading conditions produced at the cutting edge by tools of different rake angle, and seems to be a logical extension of the asymmetrical mixture of KI and KII components found in the cutting of elastic brittle materials (Section 3.4.1). Thin knife-like tools have large and would be expected to produce separation at the tool tip predominantly by tension, with mode I toughness RI. At small and negative , separation will have increasing contributions from shear, requiring mode II toughness RII. How to proportion the mixity of modes having different toughnesses is an interesting question (Hutchinson & Suo, 1991). When toughnesses are different, but the modes are the same (cracking of a brittle sheet/ductile sheet sandwich so well glued together that they have a common crack front during propagation), a simple rule of mixtures, based on layer thickness, describes experimental results for the composite cracking mechanics (Guild et al., 1978). But in mixed-mode separation, the modes are different and the crack opening displacements are perpendicular to one another. The displacement (velocity) discontinuity V* along the shear plane may be resolved into components V*cosφ along the cutting direction to give a mode II displacement, and V*sinφ perpendicular to it for the mode I displacement. These displacements are really vectors and a rule of mixtures for the effective toughness measured in cutting is
R sin2 φR I cos2 φR II
(3-34)
that becomes either
R R I cos2 φ(R II R I )
R (R I R II )sin2 φ R II
(3-35a)
or
(3-35b)
These relations give mixity in terms of f instead of but, other things being equal, and φ vary in the same sense. Figure 3-20 shows results for nylon 66 which are fitted to Eq. (3-35) using RI 1 kJ/m2 and RII 10 kJ/m2. Further work is required in this area.
3.6.6 Intercepts and slopes of FC vs t plots The predicted intercept on the cutting force axis in plots of FC vs t has often been noted in experiments, but plasticity-and-friction-only models of cutting all say that FC vs t should pass through the origin. As mentioned in Section 2.6, Thomsen et al. (1953) associated the intercept with work required for subsurface plasticity, a component that traditional analyses do not consider. Thomsen et al. drew the analogy with the thin shear zones produced in punching and blanking. Insofar as subsurface plasticity, at least with a sharp tool, is a result of the boundary layer in which separation work occurs, Thomsen et al. had identified the third component of work that Atkins (2003) argued was important. Nevertheless, the concept that relations between FC and t should be linear through the origin is found in the literature of cutting for all sorts of materials. In the case of wood, for example, Kozhevnikov (1960) argued that any offset in FC vs t plots is due to bluntness in the tool.
Fracture toughness R from cutting (kJ/m2)
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
65
12 10 8 6 4 2 0 0
20
40 60 80 Shear plane angle φ (degrees)
Figure 3-20 Mode mixity of toughness determined from cutting tests. Material is nylon 66. Curve is prediction of Eq. (3-35) employing RI 1 kJ/m2 and RII 10 kJ/m2.
Various reasons, in addition to bluntness, have been put forward to explain away the intercept. These include wear on the flank (clearance) face of the tool and rubbing on the clearance face; workpiece bulging ahead of the tool; and the intercept has even been called ‘the force unavailable for cutting’ (Thomsen et al., 1965). Brown and Amarego (1964) calculate the uncertainties caused by the effect of including, or subtracting, the intercepts in traditional cutting calculations. There is no doubt that all these effects will influence the results, but as discussed in Chapter 9, when all precautions have been taken to eliminate these ‘parasitic’ forces, an intercept remains when cutting is performed with a sharp tool and, according to the new theory, it is a measure of material separation work. When a tool with a fixed included angle is employed, where different rake angles are obtained by rotating the tool in its holder, smaller rakes result in greater clearance angles. If intercepts are caused by clearance face rubbing, they would be expected to reduce at smaller . Yet Wyeth (2008) found that they increased owing to the greater influence of separation by shear over separation by tension fracture at small or negative rake angles (Section 3.6.5). Cutting may be used to find the unknown toughness-to-strength (R/k) ratio of a material, and the individual values of R and k and, indeed, RI and RII if the mixity discussion in Section 3.6.5 is correct. Most experiments seem to produce few data in the droop region of (FC/w) vs t plots where t 10(R/k), so Atkins (2003) concentrated on the linear plot given by Eq. (3-32). The slope is (k/Qshear) so, given enough data where the slope is constant, k is determined with knowledge of , and φ. This is a well-known procedure in traditional analyses but sometimes, instead of measuring the actual slope, people have assumed that the data pass through the origin and have taken pointwise values of (FC/w) from which to get k. Far away from the origin it may make little difference, but at small t the differences can be marked and lead to erroneous increased k. The intercept back-extrapolated from the constant slope is (R/Qshear), with Qshear based on the constant φ value where the plot is linear. Alternatively, the droop in the plot at small t may be faired through data to estimate the lower R intercept, given by Rw (Qshear 1 for t 0). The full equation including the droop is given by Eq. (3-31). The above procedure presumes that φ has been determined experimentally. In the absence of such information, φ may be predicted from Eq. (3-30), except that Z will not be known
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ab initio. A spreadsheet can be set up to calculate various φ for candidate values of (R/k), and Z. The best estimate for (R/k) is determined from the ratio of the intercept of the (FC/w) vs t plot to the constant slope of the plot. The theoretical value is (R/k), which depends on and hence on φ. A series of intercept/slope values will be given by the spreadsheet for different candidate R and k and (R/k). Those that give the same value as the experimental slope/intercept are selected. Instead of considering only the data that follow the linear part of the (FC/w) vs t plot, all data including those on the droop of the curve near the origin may be included and a program written to find the best R, k and (R/k), as done by Williams et al (2009); note that Williams models friction by ‘F Fo N’ (Appendix 2). Another popular way to find k in the metal-cutting literature is to find the force FS acting along the shear plane from resolution of the resultant experimental cutting force, and to plot it against the area of the shear plane (AS tw/sinφ) at different depths of cut. From the well-known forcecircle construction (Appendix 1), a linear plot passing through the origin is expected from plasticity-and-friction-only analyses. However, as with (FC/w) vs t plots, experimental results give an intercept on the resolved force ordinate. Either the slope is measured and a correct k is given or, pointwise values of (FS,AS) are connected to the origin and will give increased k at small t. Atkins (2003) showed that the relation between FS and AS was
FS kAS Rw cos φ
for the special case of 0, which also represented data for positive and negative rake angles in the practical range (FS vs AS plots are relatively insensitive to changes in ); φ is the constant value from the linear part of an (FC/w) vs t plot. The intercept in (FC/w) vs t, and FS and AS, plots flags up the important point that the Merchant ‘force-circle’ construction results from a plasticity-and-friction-only approach. Toughness cannot be represented by a force in an equilibrium equation, so that when it is included in a cutting model (Section 3.6.1) it is not clear how much of the work done by FC relates to plasticity and how much to separation. The force circle construction can still be used for the ‘internal’ stress FS/AS along the shear plane, however, by writing a reduced force [(FC/w) R] in place of (FC/w) alone, when calculating the internal plastic work done. The force circle remains correct for ‘external’ relations between F and N on the rake face (Appendix 1). The ideas in this section are taken forward in Section 9.7.
3.6.7 Apparent machining ‘scale effect’ In the metal-cutting literature there is something that has been called the ‘scale’ or ‘size’ effect whereby experiments show that the so-called specific cutting pressure or ‘unit power’ given by (FC/wt) rises, for no apparent reason according to traditional theories, at small depth of cut (Backer et al., 1952). However, let us recast Eq. (3-23b) into (FC/wt) form, namely
(FC /wt) (1/Q shear ) [(kγ) (R/t)]
(3-36)
It is clear that with the (R/t) term on the right hand side the specific cutting pressure must increase at small t. Also at small t, φ decreases and increases (Figure 3-15d,e). What is happening is that at small t, the toughness work becomes a greater proportion of the total work than at deeper depths of cut. It is the same sort of thing as taking the shear yield stress k from pointwise (FS,AS) rather than from the slope of an FS vs AS diagram, as described above.
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67
Atkins (2003) showed that Eq. (3-36) for unit power was able to represent the experimental data of Kopalinsky and Oxley using constant R and k, and how the experimental data for reduction in f at small t may also be explained by the cutting model that includes toughness. This sort of thing reappears in analyses of oblique machining (Chapter 5) and groove formation (Chapter 6).
3.7 Finite Element Simulations From the 1980s onwards, metal cutting has been simulated using FEM. The programs employed have been the same rigid-plastic or elastoplastic codes that are successful in modelling processes such as forging, extrusion, wire or deep drawing and so on. Cutting is modelled with an inclined tool moving into the corner of a block at a chosen depth of cut. Mackerle (1999, 2003) provides bibliographies of publications on FEM simulations of machining, see also Atkins (2003). A difficulty encountered in all FEM simulations of cutting is that the tool will not move appreciably to form a chip, unless a separation criterion is employed at the tool edge that permits nodes to detach, and the chip to separate from the cut surface and flow along the rake face of the tool. A separation criterion is an essential part of FEM simulations, however all-singing, all-dancing the computer code. Without the separation criterion, FEM simulates indentation by an oblique wedge (the tool) at the corner of a block (a mathematical ‘quarter-space’). Separation does not feature in all the usual types of steady-state metalforming processes that the codes are able to simulate very well, since in those processes only plastic flow and friction requires to be modelled (the only occasion separation may arise in ‘normal’ metalforming is when we enquire what really takes place at the sticking interface between dead metal zones and flowing material, or in the operation of bridge/porthole dies for extrusions of hollow sections where the workpiece separates and recombines). It should be noted that there are FEM simulations in which chip flow around a tooltip with finite radius is modelled without a separation criterion, separation being handled by adaptive remeshing (e.g. Özel & Zeren, 2007). While this approach enables solutions to be obtained, it is unlikely to represent what is actually happening physically at the cutting edge since experiments show how important the inclusion of toughness work is in practical cutting, where behaviour does not depend on just plasticity and friction. Again, Limido et al. (2007) review meshless methods of computation. Cai et al. (2007) argue that at nanometre levels of cutting, where continuum representations of the problem become questionable, molecular dynamics simulation methods may be more appropriate than FEM. In having to employ a separation criterion, FEM workers have rediscovered what Cook et al. had said in 1954 about cutting, and shown by the length XY in Figure 3-13(A), namely that (in FEM parlance) nodes at the tool tip had to be released to permit tool travel. Astakhov (1999) points out that the major difference between machining and other metalforming processes is that there must be physical separation of the layer to be removed from the workpiece. For those who favour separation ‘just by plastic flow around the tip of the tool’, it is worth remembering that in plasticity, elements of material that are neighbours before permanent deformation are the same neighbours after flow. In machining, this means that elements just above, and just below, the putative parting line at the tool tip are supposed still to be neighbours afterwards. So elements on the underside of the chip are still ‘joined’ to elements on the machined surface, however far away from one another they may have travelled. This is implausible and suggests that simple plastic flow cannot be the manner in which chip and machined surface separate (Atkins, 1980).
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The separation criterion employed in FEM simulations of cutting ought to reflect the physics of separation in the microstructure of the workpiece, e.g. in commercial ductile metals, by void growth and coalescence in the boundary layers forming the bottom of the chip and the top of the cut surface, modelled by McClintock/Rice–Tracey/Gurson–Le Rousselier porous plasticity, cohesive zones, etc. (see, for example, Atkins, 1997). There will be a plastic work/volume in the boundary layer and a specific work of separation associated with such models which must relate to the fracture toughness of the workpiece. Such works are not usually given in papers, but calculations by Atkins (2003) showed that works/volume of MJ/m3, and works/area of kJ/m2 were implied by separation criteria that had some physical meaning. In other FEM simulations of cutting, however, separation criteria are arbitrary and seem to be computational fixes to overcome the singularity at the sharp tool tip; they consume negligible work. Nevertheless, whatever the physical sense or otherwise of the separation criterion, good representations of chip flow fields are often predicted by FEM that agree with experiment; Huang and Black (1996) review predictions given by different separation criteria. However, comparisons between predicted and experimental forces are not always good. Two crucial questions are therefore: (i) why are FEM simulations often good at predicting chip flow yet not forces, irrespective of the separation criterion; and (ii) why, if separation is crucial in FEM analyses of cutting, is it absent in traditional algebraic (force balance, slip line field, etc.) analyses? The answer to both questions relates to the comparative thinness of the boundary layers in which separation occurs (Rosa et al., 2007a, b). Rosa et al. (2007c) performed a series of combined numerical and experimental investigation of the transient beginning to machining and the transition to steady-state orthogonal metal cutting. The separation criterion employed was the attainment of a critical plastic shear work per volume dependent on hydrostatic stress, which was calibrated by independent measurements. Such a criterion coincides with void growth and coalescence models of separation. FEM simulations show that extremely fine meshes have to be employed to reveal the boundary layer in which separation takes place. The mesh sizes (about 50 m or greater) usually employed to obtain steady-state solutions in reasonable computer time are too large to isolate the microstructural effects taking place in the boundary layer for which a mesh size of about 5 m is necessary. Figure 3-21 shows how the experimental values of the cutting FC and thrust FT forces evolve over the early (transient) stages of metal cutting for lubricated conditions, with 10° and t 0.5 mm. Four different regions can be distinguished. Region 1 is indentation, the process resembling a wedge hardness test at an oblique angle. The cutting force FC increases steeply as the cutting tool starts to advance into the workpiece. The resultant force Fres is directed into the uncut chip thickness owing to the absence of material separation and the predominance of a compressive force resulting from material piling-up along the rake face of the cutting tool. The end of region 1 is the instant when separation begins at the tip of the tool. In region 2, the ‘bulge’ of material displaced by indentation begins to take up the form of a chip. Even so, a steady state has not yet been reached and the contact length, between displaced material and the rake face of the tool, continues to increase. The resultant force also continues to increase but its nature changes from the indentation of region 1, to a classical machining system based on cutting and thrust force components. Because friction opposing material flow along the tool–workpiece contact interface starts gaining importance, the resultant force Fres starts progressively shifting from rightward to leftward, towards the undeformed region of the workpiece. The end of region 2 corresponds to the instant when the displaced material forming the chip first detaches from the rake surface of the cutting tool owing to chip curl. It might be thought that this point should herald the onset of steady-state cutting, but this is not so. What happens is that a third region of deformation occurs in which chip curl increases a little
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials 1800
40 3
4
30
Cutting force
1200 Force (N)
35
1
1400
Thrust force
25
Resultant force angle
1000
20
800
(β – α)
2
1600
15
600 400
10
200
5
0
69
0
1
2
3 4 Displacement (mm)
5
6
0
Figure 3-21 FEM modelling of the transient start of orthogonal metal cutting showing the boundary shape taken by the displaced material from the instant of tool contact until attaining steady-state cutting conditions. The enclosed figures correspond to displacements of the cutting tool equal to (a) 0.06 mm, (b) 1.54 mm and (c) 4.37 mm (after Rosa et al., 2007c).
and the chip/rake face contact length decreases a little from its value at the end of region 2. At the point of maximum contact length, the thrust force FT reaches a maximum value. After the peak value, FT decreases as the contact length decreases, until both force components level out at their steady-state values indicating the commencement of region 4. During region 3, the cutting force starts growing non-linearly at a considerably lower rate owing to changes in the contact length and normal stresses along the rake surface of the cutting tool. The resultant force continues to reorientate towards its final direction corresponding to steady-state material flow. The existence of region 3 is very interesting. Were the displaced material forming the chip to remain straight and in contact with the tool, the friction work would rise inexorably. There is thus an energetic incentive for the chip to curl away from the tool, even though that curling will consume more plastic work than a straight chip. However, the present FEM calculations predict that the contact length at which detachment first occurs is not the final steady-state contact length. Rather, it is longer, meaning that the incremental friction work over that longer contact length, coupled with the incremental curling work at the curvature with which the chip is initially detached from the rake face, must be greater than the incremental works that result from a shorter contact length but greater curvature of the chip. Reduction of the chip radius to even smaller values does not occur because the extra plastic work is not compensated by a reduction in friction work. This process determines the steady-state radius of chip curl. Carefully performed very slow orthogonal cutting experiments on lead by Rosa et al. (2007a), with independently calibrated mechanical properties, were compared with the predictions of FEM simulations of steady cutting. When a separation criterion was employed that consumed zero work at the tool tip, it was found that the chip flow fields were well predicted but the experimental forces were always greater than those predicted. Furthermore, the force vs depth of cut plots passed through the origin. In contrast, when separation work at the MJ/ m3 (toughnesses of kJ/m2) were incorporated, (i) good agreement was obtained for forces; (ii) the predicted flow fields for the chip were the practically the same as before, the thin boundary layers of separation hardly disturbing the flow patterns; and (iii) there was an intercept at zero uncut chip thickness in the FC vs t plots. In simple terms, the first prediction modelled only the
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chip flow and friction, and thus underestimated the forces. When, in the literature, agreement has been obtained between experiments and FEM simulations not incorporating separation criteria that represent the physics of events at the tool edge, it could be that uncertainty about plastic flow properties at the appropriate rates and temperatures of the experiments permits apparent agreement to be obtained. Also in such studies, a result for only one depth of cut is often calculated. Since separation is confined to thin boundary layers shared between the bottom of the chip and the cut surface, the plastic flow fields of chip formation are virtually uncoupled from the work of separation. This explains why curvilinear slip line fields and the like do accurately depict chip flow fields even though the existence of separation is not acknowledged. FEM modelling also demonstrates that essentially the same chip flow fields are predicted for cutting irrespective of whether the separation criterion absorbs a large work/volume or zero work/volume. What is different are the forces, which are lower when the separation criterion absorbs negligible work than when it is significant. Static experimentation where the mechanical properties are carefully independently determined highlights this difference. So when asking whether plasticity and friction analyses of cutting are adequate, and why bother with separation and fracture, the yardstick against which the assessment is made should be stated: is it flow fields or forces? Davies et al. (2003) comment that results from FEM (irrespective of whether the separation criterion is physically meaningful or not) are highly dependent on the constitutive model employed for the workpiece, and on what is assumed for friction. They urge caution when FEM simulations are used to predict rather than merely interpret machining operations.
3.8 Cutting Through the Thickness of Ductile Sheets and Plates: Shearing and Cropping Cutting through the thickness of a ductile sheet or plate is variously called shearing, cropping, guillotining or blanking. When the cutting blade contacts the whole top surface of the plate, it is orthogonal cropping. When the blade is angled, and shears the cut face progressively, it is guillotining (Chapter 5). If the blade is imagined to wrap itself into a circle to form a punch, it is the related process of blanking or hole forming in plates and sheets, discussed in Chapter 8. In all cases the deformation is confined approximately to narrow regions of intense shear (Figure 3-22). In double bar cropping there is some bending action across the shear zones from the centrally applied force (it depends on the clearance between blade and anvil/baseplate, or punch and die), but there is also constraint across the slug. In the shearing of offcuts, where there is a free edge, constraint is lost and bending caused by the offset loading becomes apparent. This results in a burnished land on the top face of the offcut associated with its progressive rotation as the blade indents the plate. Cropping is a non-steady process, unlike the types of cutting considered earlier in this chapter where a steady state is achieved after a transient start. The transient start in cropping is immediately followed by a transient end (Figure 3-23). This comes about because the zones of concentrated shear are set up from the top surface to the bottom surface right through the thickness almost immediately the sharp blade, the sharp supporting baseplate, and the workpiece all come under load. Such a shear plane is similar to what happens at the end of orthogonal cutting using a tool with rake angle of about 0°: when the tool approaches the end of the workpiece, it becomes easier to form a shear plane along the direction of cutting to the far face rather than along the shear plane inclined at the angle φ. The transition from steady cutting to shearing through to the back face occurs at a distance of about twice the depth of cut.
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
71
Punch Punch
Punch Burnished land
A
C
B
Figure 3-22 (A) Hole punching; (B) double bar cropping; (C) orthogonal cropping of offcuts. There is little bending in (A) and (B) owing to constraint. In (C) the offcut bends and rotates, and a burnished land appears on the offcut. Copper 6 mm thick VPN 83 25
Load (kN)
20
15
10
5
0
A
B
C 0
D
E
1 2 3 4 Blade vertical travel (mm)
Figure 3-23 Load–blade travel diagrams at successively greater blade travel for progressive orthogonal cropping of 50 mm wide 6 mm thick copper plates of hardness 83 VPN following 4400.27 MPa. Displacements uncorrected for rig stiffness.
As with other types of cutting, the transient start of cropping consists of an indentation phase followed by the onset of separation (cracking) in shear. The reason why separation does not start immediately upon loading is that a criterion for ductile fracture has to be satisfied in the sheared band. A criterion for void growth and coalescence will involve both hydrostatic stress H and shear strain around the cutting edge of the blade. The sharpness (edge radius r) of the blade is the major factor in determining H, but depends upon the blade travel since (/h), where h is the effective width of the shear band. This is, of course, an average for the whole shear band: near the tools is likely to be somewhat larger. At some critical cr, attains the critical value necessary for cracking with that blade sharpness. Elements on the surface of the plate become separated from elements in the shear band under
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the bottom corner of the blade. If the edge sharpness of the anvil matches that of the cutting blade, the indentation from below is likely to be the same and cracking will then start from the bottom as well, giving reversed images of smooth and rough regions on the cut edge of the offcut and on the cut edge of the workpiece (Figure 3-24A). The region within cr (sometimes called the burnished area) is smooth and tool marks are replicated on it; the separated region within cr is rough. At different clearances between blade and bottom anvil or baseplate, shear in the band is accompanied by transverse tension, and separation takes place under a mixture of mode II and mode I fracture, the effect being more pronounced the greater the clearance. The critical cr depends therefore not only on the tool sharpness but also on clearance. Experiments show that cr increases when r increases (Figure 3-24B), and becomes variable along the cut edge. As the clearance c increases, cr decreases somewhat (Li, 2000a). The magnitudes of both effects depend on microstructure.
Smooth
Rough
δcr t
δcr
δ
L
a
t
Possible secondary burnish area Burr
A Clearance 0.60 Cutting blade
Burnish depth (mm) δcr
0.50 Pad
0.40
r 0.30 0.20
Cutting angle
Cutting angle
Die
0°
0.10
20° 0.0 B
0
0.005 0.01 0.015 0.02 Blade edge radius (in.)
Tool sharpness
Sheet metal 0.025 C
Figure 3-24 (A) Edge of guillotined surface. Chamfered region at top is followed by indentation region on which tool marks are seen to depth cr, followed below by ductile crack propagation region where tool marks are absent. Crack of length a shown running ahead of the tool during propagation. Total length of crack is L ( cr) a. Clearance results in inclined shear plane and burr on lower edge. A second smooth burnished region may occur should the tool contact already cut bulbous edge. (B) Variation of cr with blade edge radius. (C) Inclining a sheet during cropping/trimming to obtain cleaner edges with no slivers (after Li, 2003).
Simple Orthogonal Cutting of Floppy, Brittle and Ductile Materials
73
Owing to the clearance necessary for practical cropping, the cracks emanating from the top and bottom blades are not in line and this would be expected to produce an inclined shear zone extending from the edge of the top blade to the bottom blade tip. Experiments show that the zone actually extends from the top blade tip to the point of initial indentation by the bottom blade, resulting in the formation of a burr on the lower edge and the production of detached slivers (Li, 2000b, 2003) (Figure 3-24A). The burr height is greater at greater clearances and for duller blades. The shorn edges are, in consequence, rarely flat and have an ogive profile, in the middle of which may be a torn section if the paths of the two approaching cracks were not going to cross. The existence of optimum clearances for best practical cropping, as reported by Chang and Swift (1950) and given in engineers’ handbooks, is caused by a balance between (i) reduced mixed-mode toughness as tensile loading increases; and (ii) severely burred and distorted cropped edges at large clearances. Noble and Oxley (1963) showed how, by judicious choice of tool and die radii for punches, crack paths can be made to go into regions that will become scrap (see Chapter 8). For cr, the force Fshear required for plastic shear in a shear band is given by the product of the sheared area multiplied by the shear stress. Accounting for workhardening according to k ko()n ko(/h)n with ko and n constants, we have
Fshear w(t δ)ko (δ/h)n
(3-37)
where w is the contact length between straight blade and workpiece. Friction must affect Fshear but experiments show that it is not significant (Atkins, 1980). Owing to the increase in strength of the sheared band caused by workhardening, but also the simultaneous reduction in sheared area at increased blade travel, Fshear passes through a maximum. Differentiation of (3-37) shows that it occurs when peak nt/(1 n). This has nothing to do with the start of separation and, depending on the magnitudes of cr and peak, cracking may start before or after the peak in load caused by plastic instability. The tougher the material, the more likely separation will begin after the plastic instability peak. The prediction of cr, and whether the crack runs ahead of the punch or simply keeps pace with the punch, is a problem of ductile fracture mechanics (Atkins, 1988b, 2000; Atkins & Mai, 1985). A transition to cracking takes place even though there are three energy sinks subsequently (plasticity, friction and fracture), as opposed to only two (plasticity and friction) for the previous indentation phase: although the incremental (plastic work/volume) increases because the shear strain increases when the punch travel increases, the volume being sheared is less than it otherwise would be because of the crack; however, crack formation and propagation requires work, and a bifurcation can occur when the possible reduction in incremental plastic work is balanced by the necessary increase in fracture work (Atkins, 1981). It was shown (Atkins, 1988) that
δcr R(n 1)/ko (δcr /h)n R/k mean
(3-38)
where R is the ductile fracture toughness in shear as affected by tension at increasing clearances. kmean ko(cr/h)n/(n 1) is the mean shear stress for the indentation phase between first loading at 0 to cr. The calculation for cr in Eq. (3-38) implicitly assumes an equivalence between reversible NLEFM and Hencky total strain plasticity (Kachanov, 1971) and it may be shown that the same answer for cr is given when ‘R |∂/∂A|’ of non-linear fracture mechanics is used in which Fd where F [(wh(t a)ko(/h)n) and crack area A wa (Section 2.5). Since the Kachanov equivalence is lost as soon as unloading occurs, use of relation (3-38) for cr is suspect should cr peak.
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At some cr, the length of the crack alongside the punch is ( cr). Such a crack, not seen because it just keeps pace with the tool, is the same as the gap XY in Figure 3-13(A). However, a notable feature of cropping and punching is that the offcut or plug can be separated from the parent plate at punch strokes f that are smaller than the thickness t of the plate. Thus in Figure 3-23, the cropping load has fallen to zero at f 4 mm in the 6 mm plate. This is taken advantage of in ‘flying shears’ on plate rolling mills where the stroke of the blade is set to less than the thickness of the plate. Complete severance at displacements smaller than the sheet thickness means that the crack not only lies alongside the tool but must also run ahead of it. In these circumstances, if the length of the crack ahead of the tool is a (measured relative to the cutting edge of the tool), the total length of the crack is L (a cr) and the length of the remaining ligament is (t a) (Figure 3-24A). Final parting takes place when the ligament reduces to zero, i.e. when (t f a) 0, or f (t a). The plastic work done up to cr is Ubefore
∫ Fsheardδ Ushear πDk [ tδcr (δcr
2
/ 2)]
(3-39)
After cracking, the total plastic and fracture work done between cr and f, i.e. between a 0 and a (t f), is
Uafter
∫ Fsepdδ Usep πDk ∫ [{t δ a} (R/k)(1 da/dδ)] dδ
(3-40)
To evaluate Utotal Ushear Usep requires knowledge of a. In Atkins (1980) it was arbitrarily assumed that the length of crack running ahead of the tool was proportional to ( cr). If a is very small or zero, the normalized Utotal becomes
[U total /πDkt2 ] (kπDt2 )[0.5 ∆cr ∆cr 2 ]
(3-41)
where /t. An alternative derivation for a, based on minimum energy arguments, is given in Section 8.5.6 for punching. Transformer steel sheets are highly textured, and experiments show that the shearing load and total energy required are dependent on the orientation of the workpiece (Bhattacharyya et al., 1983). They linked their results for peak cropping load to Hill’s orthotropic theory of anisotropic yielding (R. Hill, 1950), with the novel feature that they were able directly to employ the throughthickness parameters L and M of the theory (most experiments involve in-plane deformation). They also compared their results with the predictions of an alternative approach based on minimization of the work required to activate internal crystallographic slip systems. Both lines of attack gave good agreement. Unfortunately, there is no information in the paper on cr and f. Note that Eq. (3-38) suggests a way of experimentally determining the mixed-mode R just from measuring cr and knowing the hardness. Reasonable estimates for R are obtainable with sharp tools when a material with known hardness H is cropped; kH/(56) in consistent units (Tabor, 1951). Systematic alteration of edge roundness as in Figure 3-24(C) would enable the influence of sharpness on R to be established, and the influence of clearance on mode mixity could also be explored. There are practical problems to be overcome in trimming autobody sheets, with slivers, burrs and rough edges. Li (2000a, b) demonstrated remarkable improvement in all three factors when the sheet was angled before cropping (Figure 3-24C). Whereas zero inclination is normal for cropping steel sheets, finite inclination is necessary for aluminium alloy body panels.
Chapter 4
Types of Chip Load Fluctuations, Scaling and Deformation Transitions Contents 4.1 Introduction 4.2 Energy Scaling 4.3 Variations in Depth of Cut and Rake Angle 4.4 Cutting with a built-up edge 4.5 Sawtooth Profile chips 4.6 Classification of Chips 4.7 Wood
75 79 85 92 94 96 99
4.1 Introduction Chip formation in cutting under globally elastic, and globally plastic, conditions has been described in Chapter 3. Globally elastic corresponds to what was called brittle chip formation where chips break away in a series of spalls, with violent load drops. The broken-off chips can be collected and fitted together to regain the original article. Globally plastic corresponds to continuous chip formation under ductile conditions, with cutting under steady load. The chips are highly distorted and it is impossible to refit them. We shall discover that these two different types of chip are at the ends of a spectrum. What sort of chips are produced, what the cutting forces do, whether the chip is apparently continuous but perhaps partially cracked into loosely attached segments, or whether portions of the chip are completely detached from one another, depends on the degree of elastoplasticity during chip formation, and the stiffness of the tool holder. When bodies are loaded with a cutting tool, they initially experience globally elastic deformation. They may cut within the elastic range; if they do not, the deformation becomes a mixture of elasticity and plasticity. They may begin to cut shortly after entering the elastoplastic range, when the elastic and plastic strains (and elastic and plastic energies) are comparable; or the onset of cutting may not occur until after extensive plastic flow, where the elastic strains are much smaller than the plastic strains. It is possible that cutting may be difficult to achieve even after substantial plasticity. Figure 4-1 illustrates these events for time-independent behaviour. Whether cutting occurs in the elastic, elastoplastic or plastic ranges depends upon the relative magnitudes of the loads (work) required for fracture and flow, and the role played by friction. As will become evident later, great care is required in what we mean by brittle and ductile because the same material can display quite different behaviour depending on the uncut chip thickness: (i) at very small depths of cut brittle materials like glass machine with continuous ribbon-like chips and quasi-steady forces exactly as in workshop cutting of ductile metals; and (ii) at very large depths of cut, ductile materials machine in a brittle fashion with cracks running ahead of the tool and undeformed ‘lumps being knocked out’ with violent fluctuations in load. This sort of behaviour is a manifestation of the scale effect inherent in all fracture mechanics, detailed in Section 4.2. The scale effect explains why it is impossible Copyright © 2009 Elsevier Ltd. All rights reserved.
75
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The Science and Engineering of Cutting
Elastic/floppy deformation
Elastic cutting Elastoplastic cutting Elastic deformation
Plastic cutting Plastic flow Unlimited plastic flow
Figure 4-1 Possible progressive transitions in cutting from floppy behaviour of offcut, through elastic and elastoplastic chip formation, to rigid-plastic cutting and even the possibility of no cutting at all.
to comminute powders below certain sizes and why grinding is employed to cut very hard metals that have low toughness. Wood is a familiar material that exhibits changes in mode of chip formation with depth of cut: the small, thin curly shavings produced along the grain by a cabinet maker’s scraper transform through the coarser shavings of a wood plane, to the subsurface splits produced by an incorrectly set plane, to the gross splits produced by wood chisel and axe at the deepest depths of cut. Transitions between different types of offcut are found with all materials. For some materials, only one type of chip formation will be realized unless special rigs are employed (very stiff machines for micrometre depths of cut in brittle materials; or very powerful machines for deep uncut chip thicknesses in ductile solids). On the other hand, there are materials in which transitions between types can be observed over normal depths of cut and within the stiffness and capacity of common machines. Polymethylmethacrylate (PMMA) is one such material. It was used as an example of a brittle material in Section 3.4.1, but at much smaller depths of cut than those given there it behaves as a ductile material forming continuous chips. Change of depth of cut alters the relative amounts of elasticity and plasticity in chip formation, but it is not only alteration in depth of cut that produces transitions to different types of chip. The conditions under which different types of chip form depend on many factors, including (i) how the mechanical and thermal properties of the workpiece change with rate (cutting speed), temperature build-up in the cutting zone, environment (e.g. moisture content of natural materials) and, in anisotropic materials, orientation; (ii) the geometry of the cutting tool, its sharpness and its inclination to the direction of cut (obliquity; see Chapter 5); and (iii) lubrication and friction between the cutting tool and workpiece. The effect of temperature is well known from the problem of drilling holes in rubber bungs at room temperature, where large reversible deformations inhibit the start of cutting. But when the stopper is first frozen to stiffen up the material (and also to alter the toughness and yield strength of the rubber), drilling becomes easy. Civil engineers sometimes tunnel in frozen ground. Brehm et al. (1979) described single-point turning of glass at temperatures around the ‘American softening point’ (500–800°C, depending on the glass). Continuous chips were formed; at room temperature the glass would have shattered. Dempster (1942) showed that temperature was important when microtoming paraffin wax sections. He discovered that there was a critical temperature above which ‘warm sectioning’ produced continuous translucent wax offcuts. Below the critical temperature, ‘cold sectioning’ produced discontinuous offcuts having transverse prisms of clear paraffin separated by thin, opaque regions where the opacity was caused by minute cleavages and air spaces in highly stressed regions, rather like the stress-whitening phenomenon in some polymers (Figure 4-2).
77
Types of Chip
45° Oblique knife H
81° Rake A
– 45° Rake G
J I 75° Rake
0° Rake–warm
B
60° Rake C
0° Rake F
30° Rake
45° Rake D
E
Figure 4-2 Structure of thick paraffin wax sections for various microtome knife rakes. Stipple and fine lines indicate regions of wax opacity. Arrows show direction of knife or chisel movement. A section cut with a 45° rake angle is shown uncurled at J (after Dempster, 1942).
Every prism slightly overrides the next in sequence, giving perceptible ‘imbrication’ (i.e. serrations like overlapping roof tiles, from imbrex, a type of Roman tile) on the free surface in compensation for the compressive stress on the inner face of the curled offcut. Similar discontinuous serrations can be produced very easily in the kitchen by scraping a knife over butter. Dempster observed that owing to the bad conductivity of paraffin wax, even if the bulk temperature of the wax block were low, the local temperature rise in the cutting region caused by the dissipated cutting work could be enough to alter the cut to a ‘warm cutting’ continuous type. Furthermore, in other materials having bad thermal conductivity (or at very high speeds in some metals), the deformation can be limited to discrete shear bands of softening materials with regions in between that are almost undeformed (quasi-adiabatic discontinuous chip formation). The effect of cutting speed is illustrated when drilling holes in PMMA. Holes are successfully drilled at slow rotational speeds and slow feed rates; but at fast speeds or feeds, cracks radiate from the partially formed hole and severe spalling may occur. In contrast, reduction of speed when cutting most metals results in a transition from continuous to discontinuous chip formation.
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The Science and Engineering of Cutting
Yet Bottone (1873) showed that while a wrought iron disc having a peripheral speed of 10 m/s may readily be cut by a steel cutting tool, the disc itself will cut the steel tool when the disc speed is 30 m/s; at 100 m/s the wrought iron disc will cut quartz (see Chapters 9 and 15). The role of friction in cutting is well known from cutting cheese and cold butter: ‘stickiness’ and high friction against knife steel result in high cutting forces and transfer of material to the knife blade. One way of reducing friction when cutting such materials and altering the type of slice is to make holes in the blade to reduce the area of side contact. Alternatively, cutting may be done with a wire (Chapter 12). The initiation of cracking and hence the production of discontinuous chips can be delayed by improved lubrication (Palmer & Riad, 1968). Contact along the rake face of the tool in many cutting processes is a mixture of a sticking region near the cutting edge where the contact stresses are high, and sliding regions further away where the contact stresses are lower. In the sticking area, friction is limited by the shear yield stress of that part of the chip adhered to the tip of the tool, sliding occurring within a thin boundary layer near the chip/tool interface. Sometimes the boundary layer grows appreciably to form a ‘nose’ (built-up edge; Section 4.4) of workpiece material stuck to the tool and substantial enough to act as its own cutting edge with a different effective rake angle and hence give a different type of chip. Dempster’s 1942 microtome studies showed how microstructure, and the applicability of continuum mechanics, has to be considered in chip formation. His wax crystals were about 4 mm in size and larger than the thickness of cut. Observation of the movement of the grain boundaries enabled flow patterns to be identified when the crystals were divided into a number of slices during sectioning. The elastic effect of compressive forces ahead of the cutting edge, and tensile stresses in the cut surface behind the tool, were also demonstrated in wax by the formation of vertical tensile cracks in the cut surface. Similar cracking patterns are well known in the cutting of brittle materials and wood (Section 4.7). Such effects are reduced at large rake angles and with thin sections. Tensile stresses in the wake of the tool were predicted by the early photoelastic studies of Coker and Filon (1931), and are well known in contact stress problems of both elastomers and stiff elastic solids (e.g. Johnson, 1985). The action of some tools generates discrete chips of varying thickness, for example circular saws and cylindrical milling cutters (Chapter 7). Clearly, a critical depth of cut at which a transition occurs might fall within a chip of variable thickness. Ball end-mills also produce chips of varying thickness but, in addition, have cutting speeds that vary with distance from the axis of rotation. Transitions may be speed dependent and this may result in surface finish problems, as when ball end-mills are used to cut grooves in glass. Matsumura and Ono (2008) obtained much better finish by inclining the ball end-mill to become a negative-rake rotating tool. The action of a ball end-mill is similar to that of Napier’s rotary tool (Section 5.3.1). Whether an offcut comes away damaged is often important commercially. In the food industry, rashers of bacon are required whole with no separation of meat and fat in the slice: as discussed in Chapter 5, this has led to the realization that least damage is caused by least cutting forces and hence the aim in commercial cutting machinery is to design blade profiles that require least force (Chapter 12). While least forces are also desirable in metal cutting, the handling and disposal of long continuous ribbons of swarf is troublesome, so broken-up chips are preferable and, if they do not form naturally, chip-breaking devices may be incorporated in the tool holder. Although it has long been known that transitions between different types of chip depend on the ‘ductility’ of the workpiece (Brooks, 1905; Cook et al., 1954; Palmer & Riad, 1968; Childs & Rowe, 1973; Komanduri, 1993), there has been no quantitative measure of ductility in explanations of transitions. The specific work of separation or fracture toughness
Types of Chip
79
R provides that parameter and transitions depend on the characteristic length scales given by (ER/k2) for elastoplastic deformation, or (R/k) for rigid-plastic deformation. Sometimes continuous chips have non-uniform thickness with wavy top surfaces (variously called serrated or sawtooth chips), as illustrated in Vyas and Shaw (1999). These arise from oscillations in the depth of cut caused by the tool fixture not being sufficiently stiff and deflecting under the cutting forces. The oscillations are called chatter (from the accompanying noise) and can leave marks on the machined surface. The Victorians were aware of the need for stiff tool holders (e.g. Smith, 1866). Because some types of discontinuous chip also have wavy top surfaces, there is confusion between the types. There is a vast literature on discontinuous chip formation and, owing to the range of different rate- and temperature-dependent parameters that may affect chip formation, attempts at rationalization are fraught with difficulties. As remarked by Childs (2006), it is not always clear what is being looked at when studying chips of metals produced at high speeds and high temperatures. The same is true for other materials as well. Figure 4-3 reproduces two of the remarkable series of photographic plates, of different types of chip produced in different metals under different conditions, presented by Brooks (1905). His ‘cutting angle’ is (90 ), where is the rake angle. The photographs illustrate many of the different types of chip found when cutting metals under various conditions.
4.2 Energy Scaling The general idea behind scaling is whether it is possible to predict the unknown behaviour of one size body (the prototype) from the known behaviour (in terms of forces, energy, etc.) of another body (the model). Scaling is familiar in fluid mechanics, aerodynamics and heat transfer, with well-known non-dimensional groups or ‘numbers’: Reynolds, Mach, Nusselt and so on (Palmer, 2008). Scaling is less familiar in solid mechanics but it applies there too. An early discussion of scaling is by Thomson (1875). Scaling is particularly important whenever fracture occurs, as the scaling relationships are, perhaps, peculiar since the energy stored and available (or dissipated and unavailable) scales with the volume of the cracked body, but the energy required for fracture scales as the crack area (for a review, see Chapters 8 and 9 of Atkins & Mai, 1985; Atkins, 1988a, 2009a). Cube-square scaling is well known in other branches of physics: in condensation theory the size of droplets depends upon their mass (volume) and the surface tension (area) holding them together; and in fluid mechanics, the resistance of ships through the water depends on wetted area but the amount of fuel carried depends on volume of hull. Brunel confounded his critics in 1838 with his first steamship, the PS Great Western, which crossed the Atlantic with coal to spare because he made the ship big enough. An example connected with the drag on bullets or arrows and how the range is affected is given in Section 8.5.1. The scaling factor is the ratio of characteristic lengths in the prototype and the model, rather like the magnification of a picture. The lower bound for energy scaling during fracture is 2, when there is fracture but no plasticity (brittle behaviour); the upper bound is 3, when there is no fracture but only plasticity (ductile behaviour). Traditional scaling presumes that the behaviour of the model and prototype is identical, for example that the mechanisms of wave formation are the same for both a model ship in a towing tank and the prototype ship on the ocean. In fracture and cutting, that is not always the case: it is possible that the dimensions of a testpiece can be altered and the mode of deformation remain the same (always brittle or always ductile, say); but it is also possible that altering the dimension of a testpiece moves the deformation into a different regime, owing to cube-square scaling. Thus the cracking stress in a small body may be higher than the yield stress, so that crack propagation takes place with
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The Science and Engineering of Cutting
A
C. A. 55°
D. of C. 0.0125"
C. A. 55°
D. of C. 0.025"
C. A. 60°
D. of C. 0.0125"
C. A. 55°
D. of C. 0.05"
C. A. 60°
D. of C. 0.025"
C. A. 60°
D. of C. 0.05"
C. A. 65°
D. of C. 0.0125"
C. A. 65°
D. of C. 0.05"
81
Types of Chip C. A. 70°
D. of C. 0.0125"
C. A. 70°
D. of C. 0.05"
C. A. 75°
D. of C. 0.0125"
C. A. 75°
D. of C. 0.05"
C. A. 80°
D. of C. 0.025"
C. A. 80°
D. of C. 0.05"
C. A. 90°
D. of C. 0.0125"
C. A. 90°
D. of C. 0.05"
B
Figure 4-3 Pictures, published in 1905 by Brooks, of chip formation in mild steel cut at various depths of cut with tools of various cutting angles (CA) that are measured from the horizontal, so rake angle (90 CA)°. Brooks also studied cast iron, wrought iron, cast steel and copper. (Courtesy of IMechE)
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The Science and Engineering of Cutting
lots of surrounding plasticity, but a proportionately much larger body will crack with no plasticity (brittly) because the fracture stress is now smaller than the yield stress. This is dangerous for design based on ideas of keeping working stresses below some safety-factored yield stress. In cutting, it means that simply altering the depth of cut can cause transitions between ductile and brittle chip formation. Scaling in cutting relates to the formation of different types of chip under varying degrees of combined elasticity, plasticity and fracture. In what follows we shall investigate the scaling laws for cutting (i) when the behaviour of model and prototype is identical (chip formation always by plastic flow, or always by brittle cutting, and so on) as in traditional scaling; but also (ii) when the mechanism of chip formation is different in the small and in the large. In the former cases there are no transitions; in the latter there are.
4.2.1 Floppy materials The cutting of floppy materials is governed by Eqs (3-2a,b), namely FC/w RH/tan and FT/w R/tan. Because the only work done is separation work, FC and FT are independent of slice thickness. For the same R, and in different size workpieces, FC must scale with the width w of the sample, i.e. with the scaling factor . Energies, given by force displacement, must therefore scale with area (2) when cutting floppy materials.
4.2.2 Elastic materials Within the elastic range of cutting, energies also scale as 2 since the only dissipated work is work of separation in the boundary layers between the cut surface and the underside of the offcut. However, forces no longer scale just as because the stored strain energy (in the elastically deformed chip and surrounding material as cutting proceeds) can ‘feed the crack’ as it is released, and its availability depends on size. Consider first the splitting of a brittle solid having a short crack length a by a symmetrical wedge (Figure 3-7). Freund’s 1978 LEFM expression for cracking is given by Eq. (3-12). We write this out twice, once for the large brittle body and once for the small brittle body, and obtain
(KI )large [4π/(π2 4)] (1/w large alarge ) FTlarge {1 (2 /π)(FC /FT )large }
(KI )small [4π/(π2 4)] (1/w small asmall ) FTsmall {1 (2 /π)(FC /FT )small }
(4-1a)
(4-1b)
where w is the thickness of the testpiece. (FC/FT) depends upon the inclination of the cutting blade (rake angle of the wedge) and the friction, and for the same wedge angle and friction in both large and small, the ratio (FC/FT) is constant. In geometrically similar workpieces alarge/ asmall wlarge/wsmall . For the same KC in large and small bodies, division of (4-1a) and (4-1b) gives
(FT )large λ 3/2 (FT )small
(4-2)
A geometrically similar brittle workpiece nine times the size of a smaller workpiece thus requires 93/2 27 times the load to split the smaller one, rather than 92 81 times given by fracture at the same stress. This comes about because in the larger body relatively more stored elastic energy is available to contribute to the work of separation.
83
Types of Chip
Most cutting of surfaces takes place with an inclined wedge that results in asymmetrical loading having both mode I (tensile) and mode II (in-plane shear) components of the stress intensity factor K at the tip of the tool (Chapter 3). Despite the mode mixity, the general form of the linear elastic fracture mechanics relation will still be of the form KC (a) Y(a/W). Thus, from Atkins and Mai (1985), the forces, stresses and tool displacements, for the same type of chip formation, in large and small geometrically similar workpieces will be related by
Flarge λ 3/2Fsmall
(4-3a)
σlarge σsmall / λ
(4-3b)
δlarge λδsmall
(4-3c)
These expressions are modified when scaling is non-proportional, e.g. when the thickness is the same in small and large workpieces (see Section 4.2.4).
4.2.3 Elastoplastic and plastic cutting Scaling relations for elastoplastic cutting may be worked out using expressions for forces given by Williams et al. (2008). Scaling within the rigid-plastic range is obtained by writing out Williams’s (2008) closedform solution (Eq. 3-31), once for a large ductile body and a second time for a small ductile body and dividing, as above. For the same material in different sizes having invariant R and k, Zlarge Zsmall/, and since the primary shear strain depends on φ, that in turn depends on Z, the result is algebraically cumbersome (Atkins, 2008). For illustrative purposes, however, let us assume that . Then Eq. (3-31b) simplifies to
(FC /Rw) {2 [1 Z tanα ] (1/Z)} 1
[(FC /Rw) 1] 2 [1 Z tanα ] (1/Z)
(4-4a)
from which
(4-4b)
That is, (FC /Rw)2 2(FC /Rw) [ 1 (4 /Z){(1/Z) tanα}] 0
(4-4c)
or (FC /Rw) 1 [(4 /Z)( tanα (1/Z))]
(4-4d)
Hence FClarge /FCsmall λ{1 [(4λ/Zsmall )( tan α (λ/Zsmall ))]}
{1 [(4 /Zsmall )( tanα (1/Zsmall ))
(4-5)
FClarge is greater than FCsmall since the numerator of (4-5) is greater than the denominator for 1. We see that the relationship in terms of is complicated, even when we assume .
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The Science and Engineering of Cutting
This analysis of scaling in the cutting of rigid-plastic ductile materials relates to the laws for scaling in plastic fracture given by Atkins (1988). However, that analysis was more concerned with energy absorption in large and small ductile structures under impact, and friction is absent; moreover, the plastic work per volume given by kd was taken as invariant with size. In cutting the primary shear strain cotφ tan(φ ) and, as explained above, will therefore depend upon the parameter Z R/kt that has a length in its definition. In other words, in cutting will vary with size and scaling factor, so that a general comparison between the ratio of (FC/Rw) values for cutting large and small bodies is not simple. In the special case of frictionless cutting with a zero rake angle tool, 0, and is then independent of Z. A comparison can now be made with the predictions of the 1988 paper. For 0, φ 45° and 2, so from (4-5) FClarge /FCsmall λ[(2λ/Zsmall ) 1]/[(2 /Zsmall ) 1]
(4-6)
The ratio of fracture loads in plastic fracture was given by Atkins (1988) in the form Flarge /Fsmall λ(λξ 1)/(ξ 1)
(4-7)
where
ξ (kγ/R)(V/A)
(4-8a)
in which V is the volume of material undergoing plastic flow and A is crack area: is thus the ratio of plastic work done to cracking work done. (Do not confuse this with used for slice–push ratio in Chapter 5.) Now V/A t in cutting, so
ξ (γ/Z)
(4-8b)
Whence, using Eq. (4-8b) in Eq. (4-7) gives
Flarge /Fsmall λ[λ(γ/Z) 1]/[(γ/Z) 1]
(4-9)
This agrees exactly with (4-6) since, as we have shown above, 2 for the special case of 0. Relation 3-23b can be rewritten in the form
FC /w (kγt/Qshear )[1 (R/ktγ)] (kγt/Qshear )[1 (1/ξ)]
(4-10)
The (R/kt) parameter links not only the material (R/k) ratio and the depth of cut, but also the tool rake angle and shear plane angle, via cotφ tan(φ ). Relation (4-10) shows that the numerator is the sum of plastic work plus fracture work, and the denominator is the friction correction factor Qshear that affects both components equally (see Section 4.3.2). This was assumed by Atkins (1974) when analysing cutting force waveforms for different chip formations. In that paper, it was furthermore suggested that the parameter (KCsinφ/kt) determined whether brittle or ductile chips were formed on the basis of a ratio of cracking load to shearing load. It could be, in the rigid-plastic case, that is the appropriate parameter. Despite the analysis being perhaps opaque, the trend is that cutting forces increase at a smaller rate than increases in depth of cut, and that cutting stresses are smaller in larger workpieces. Also, the trends for energy consumed in cutting ductile solids follow known
Types of Chip
85
s caling laws for combined flow and fracture, i.e. follow x where 2 x 3, where x depends on (R/k) (Atkins, 1988).
4.2.4 Non-proportional scaling The above predictions assume geometric scaling, i.e. the bodies being cut are simply different magnifications of the same shape, where both the width and depth of cut change in proportion to . Of more practical interest for cutting is where the width of a workpiece remains fixed and where the depth of cut is altered. This is ‘non-proportional scaling’ (with different for depth of cut, width of workpiece, etc.) (Atkins & Mai, 1985). An example of nonproportional scaling has already been used in Section 3.4.1, where it was shown that Fres/w for brittle scallop formation followed 1/2. The experiments concerned variable depths of cut at constant workpiece width. Had the width increased in proportion to depth of cut t, F 3/2, instead of (F/w) 1/2. The relationships derived above for scaling have also assumed that the material properties KC, E, R, k (and perhaps workhardening capacity), or friction, are invariant. Calculations for cutting forces may be performed when these alter with size (or, indeed, with rate, temperature and environment should those be different in small and large), simply by not cancelling KC, R or k when dividing relations like 4-1. In this way, it is possible to have transitions at a fixed depth of cut. It is thus clear that a variety of different types of chip can be produced, depending not only on the depth of cut, but also on how friction, elastic moduli, yield stresses and toughness may change with rate, temperature and environment and between model and prototype. Friction and tool rake angle influence the deformation of the offcut since, from a wedge analysis, they determine the ratio of effective forces applied across, and along, the putative cut surface and hence the amount of tensile and shear loading near the cutting edge (in other words, the mode I/mode II mixity of separation and the tendency for the chip to curve upwards to the free surface and detach itself). It is no wonder that generalizations about discontinuous chips are difficult to make.
4.3 Variations in Depth of Cut and Rake Angle Experiments show that the same material can exhibit very different behaviour during cutting depending on the tool rake angle , the depth of cut t, cutting speed, temperature and environment. Ceramics, glass and other solids normally thought of as ‘brittle’ and producing scalloped chips can be cut in a ductile fashion when machined at extremely small depths of cut (e.g. Puttick et al., 1989), with removal of ductile ribbons of material, yet glass shatters if put in a lathe and machined in the normal way employed for ductile metals. Similarly, even very ductile materials will machine as if they were ‘brittle’ at sufficiently large depths of cut. This points up that it would be dangerous to predict the cutting behaviour of one from the other, employing the scaling laws of Section 4.2 that apply within a given regime of chip formation. Yet it is these same cube-square scaling laws that are the cause of the transitions in chip formation. The type of behaviour and type of chip depend on which stage of deformation has been reached in Figure 4-1 before cutting reaches steady state (where steady state may mean varying, but periodically repeating, cutting forces). As explained in Section 4.2, the length scale that determines these transitions is ER/k2 (KC/ 2 k) , or just (R/k) for rigid-plastic solids. In the case of glass and similar solids ER/k250 m. Below this depth of cut, ductile behaviour occurs and glass may be micromachined with
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c ontinuous shavings at these depths of cut exactly like low carbon steel in the workshop (see Chapter 6). Again, for low carbon steel ER/k2 0.25 m, say. Above this depth of cut, brittle behaviour occurs. This emphasizes that classification of materials as being ‘brittle’ or ‘ductile’ based on the behaviour of laboratory-size testpieces can be misleading, and lead to unexpected behaviour at sizes much greater and much smaller than some known reference size.
4.3.1 Load variations The load–displacement behaviour during cutting tells us a lot about what is happening from a fracture mechanics point of view. First, it is important to appreciate that variations in cutting load may be real and reflect what is happening during cutting, or may be spurious caused by the characteristics of the force measuring system and method of display (e.g. compliance of load cell – especially a nuisance at high rates, causing ringing in the load signal), inertia of pen recorders, data sampling rates, and so on. The compliance of the tool holder itself is important, even when not recording forces, and can cause troublesome chatter. In what follows we assume that the force signals are all to do with cutting. That is, both the tooling and load cell systems are very stiff. When analysing data from experiments, however, it must be acknowledged that despite the best attempts to make a stiff cutting system, there will always be some compliance and this may influence the results. When the load drops precipitously to zero on first cutting, a crack has run quickly ahead of the tool and will have detached itself as a spall as described in Section 3.4.1. A vertical load drop implies unstable crack propagation (see Atkins & Mai, Chapter 8, 1985) and the triangular work area under the FC vs plot is an upper bound on the work of elastic fracture given by Rx (curved length of spall) (width of workpiece). It is an upper bound not only because the system has an unfavourable geometric stability factor (Gurney & Mai, 1972), but also because additional work has been done against friction during loading. There is more than enough energy for cracking and the excess may send the chip flying across the room. The load remains at zero until the cutting edge has caught up with the surface geometry left behind by the spall, when the process repeats itself. The closer the gap between force spikes in this sort of cutting, the smaller the aspect ratio of the detached chip. As shown in Figure 3-8, the aspect ratio depends on (Et/KIC), and so depends both on the material properties and (weakly) on the depth of cut. While Figure 3-8 relates to 0, the same ideas will apply to other rake angles. In other types of cutting, FC does not drop to zero and oscillates between approximately fixed limits in a quasi-steady state. Here, a crack runs ahead of the tool but does not reach the free surface, and so a chip is not detached. The initially fast-running crack stops because the system ‘runs out of steam’, i.e. the rate of release of energy from the loaded chip just before it begins to run is insufficient to take the crack to the free surface. Depending on tool geometry, depth of cut and material characteristics, the crack may arrest on the curved part of what would have been a scallop, or earlier. In this way chips, while remaining apparently continuous, are internally cracked. Whether such cracks are apparent or not depends on the conditions. An indication of internal damage may be gained by bending back flat a curled chip and seeing whether it breaks apart in transverse fractures; this is readily seen when uncurling shavings from the planing of hardwoods. When such damage becomes extensive, the offcut may become segmented, i.e. still in one piece, but consisting of sections held together along the top surface with cracks in between. The Franz type I chip for classification of chips in the cutting of timber along the grain is of this sort (see Section 4.8). Fleck et al. (1996) machined sintered bronze, a porous material. A tool having 45° produced continuous chips in high-density material at small depths of cut; the same tool
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Types of Chip
produced discontinuous chips in low-density material and at greater depths of cut. In both cases compaction of the workpiece occurs so that in the continuous chip case, the porosity within the shear band is half that of the undeformed material. Figure 4-4(a,b) shows the fluctuating waveform for t 0.4 mm where incomplete scallops are formed. Tests were interrupted at the four stages of cutting labelled A–D. Round about the peak load a crack develops from the tip of the tool and grows in a stable manner round to the free surface as the load drops from A to B. Continued tool travel compacts material, the load increases through point C up to D when the process starts all over again. Other examples of this sort found in the cutting of wood are described in Section 4.7, and Malak and Anderson (2005) produced a variety of different types of chip when cutting polyurethane foams of different densities. Workpieces ranged over open-celled foams to ‘solid’ foams with isolated pores. At very small depths of cut compared with the cell size, the tool scraped along the surface accumulating broken cell walls; this was designated ‘fragmentation’. (This behaviour seems to be a feature
(a)
(b) 150
FC/wt, FT/wt (MPa)
D
A
120
FC/wt
C
90
(c) 60 30 B 0 FT/wt –30
A
0
1
2
3
Displacement v (mm)
(d)
4 B
Figure 4-4 Machining of sintered bronze. (A) The fluctuating load waveform for t 0.4 mm where incomplete scallops are formed. (B) Round about the peak load a crack develops from the tip of the tool and grows in a stable manner round towards the free surface as the load drops from A to B. Continued tool travel compacts material, the load increases through point C up to D when the process starts all over again (after Fleck et al., 1996).
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of compressible materials and is seen in the cutting of wood veneer at very small peel depths; Section 4.7.1.) At somewhat greater thicknesses, continuous chips were formed, and at yet deeper depths of cut a variety of discontinuous chips occurred, including (i) completely scalloped chips (called type 3b); (ii) scalloped chips where the crack does not reach the free surface, as in Figure 4-4 (called type 3a); and (iii) and sheared chips (called type 3c). The force waveforms that accompany the different cuts range from smooth to violently fluctuating. Fragmentation and continuous chip formation have similar waveforms for FC, but differ in the FT behaviour: in fragmentation the tool pushes down on the specimen but in continuous chip formation the tool pushes the chip up and away from the cut surface (at least over the depths of cut investigated). Malak and Anderson (2005) found that the periodicity between force peaks depended on the rake angle . For 60°, the wavelength was roughly the same as the depth of cut for all foams, but at smaller it depended on the density of the foam. At 23°, the wavelength was 7t for 160 kg/m3 material, 3t for 480 kg/m3 and just t for 640 kg/m3 foam. Plastically deformed scallops can, in theory, be produced in plastic bending (Chapter 3), but this type of chip formation is rare in isotropic materials. Cutting force (FC and FT) tool travel diagrams for discontinuous chips from the ductile end of the elastoplastic cutting spectrum are shown in Figure 4-5(A) (EN3B steel) and 4-5(B) (-brass). FC and FT have been calculated from rake face forces in Palmer and Riad (1968). These chips occur at low speeds in the unlubricated machining of ductile materials (data for EN8 steel, gun metal, NE7 Al/Mg alloy and titanium are also given by Palmer & Riad). The sketches above the force plots show how the chips are formed at different stages of loading. As with the porous materials described above, the start of every chip of this type has the tool indenting into the sloping surface left by detachment of the previous chip. A crack, roughly parallel to the motion of the tool, eventually initiates at the cutting edge and there is simultaneous plastic distortion of the thickening chip in front of the tool. With this type of discontinuous chip, there is hardly any additional cracking at first and, instead, under increasing load there is extensive plastic compression of the material forming the chip. Thus, in contrast to elastic scallops, chips are permanently deformed compared with their uncut shape. The reason that cutting forces produce plastic flow is that materials in which this sort of chip is formed are much tougher and have relatively lower yield strengths than the brittle materials that give elastic scallops. In the toughest materials with the lowest yield strengths, i.e. those having high R/k ratios, separation by cracking to the free surface would not occur, the crack direction would remain approximately parallel to the free surface and we should have steadystate continuous chip formation. In Palmer and Riad’s experiments they found not only discontinuous and continuous chips, but also ‘partially discontinuous’ chips in which the crack propagated only so far before the next chip began to be cut, thus giving an apparently continuous, but really periodically cracked, chip. Sharma et al. (1971) concluded that the transition from ribbon-like chips to this sort of discontinuous chip occurred when the shear strain along the putative fracture surface (not the same as along the primary shear plane) exceeded a critical value that depended on the normal stress across it. This accords with ideas of void growth and coalescence (see Section 2.5.4). While alters considerably with cutting conditions, H varies far less and, over a wide range of conditions, Hk in magnitude, becoming more tensile nearer the tool edge (Oxley, 1989). Brooks’s 1905 pictures in Figure 4-3 should be contrasted with Palmer and Riad’s results. Sharma et al. (1971) employed Klopstock (1925, 1926) limited-contact tools and showed that as the rake face length reduced, the frictional restraint reduced, and so did the shear strain. A transition from discontinuous chip formation to continuous thus took place just
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Types of Chip
Work travel 0.005 in
0.050 in
0.020 in
t = 0.2
de
0.080 in
0.100 in
1800 1600
600
Chip separated
800
New chip started
1000
Sliding started
1200
Crack observed
Previous chip separated
Forces N Fc bigger
1400
400 200 0
0
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A
1.5 Tool travel (mm)
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0.045 in
0.030 in
Work travel 0.008 in
3
Beta brass
0.015 in
1800 1600
Chip separates
1200 1000 Crack observed
Fc and FT FC bigger
1400
800 600 400 200 0
B
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Tool travel (mm)
Figure 4-5 Cutting forces FC and thrust forces FT vs tool travel during stages of discontinuous chip formation in metals studied by Palmer and Riad (1968). (A) EN3B steel; (B) -brass. FC and FT calculated from forces F and N on the rake face given in the original paper.
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by reducing the contact length between chip and tool. The transition occurred when the face length was about the same as the thickness of the discontinuous segment produced by a full contact length tool. A characteristic of the type of discontinuous chips studied by Palmer and Riad is that the plastically deformed material that comes into contact with the rake face seems not to slide up the rake face, as might be expected, but instead sticks to the rake face; the shaded regions in the diagrams show the extent of these stagnant regions observed from movies. (Other workers, e.g. Cook et al., 1954, have described this feature as the chip ‘rolling’ on to the rake face or ‘being extruded’.) These events occur as the load increases up to a maximum, with hardly any additional crack motion ahead of the tool. Mallock (1881) noted that ‘… some metals, copper for instance, when unlubricated, actually refuse to slide over the face of the tool, and the metal is then driven before the tool in a growing lump, as stiff mud would be before a board pushed through it. The separation in these cases does not take place at the edge of the tool, but some distance beneath it …’. Eventually the distorted chip begins to slide along the rake face round about the maximum in load or just after, and the crack then moves more quickly to separate the chip. The final shape of the chip seems to be controlled by the workhardening index of the metal (i.e. by n in on). Of the various materials studied by Palmer and Riad (1968), gun metal (a bronze 10 per cent tin–copper alloy) and -brass display precipitous load drops when the chip separates. The arrested load remains finite, however. The other materials, EN3B (0.2 per cent plain carbon steel), EN8 (0.4 per cent plain carbon steel), NF7 aluminium/magnesium alloy and commercially pure titanium, while all showing load drops during chip separation, display continuous load–tool travel records as the load drops and passes through minima to begin to rise again as the next chip forms. In these cases, the tool begins to cut a new chip just before the previous chip is finally detached. Note that the commercially pure titanium gave completely separated chips; many titanium alloys produce chips having a sawtooth free surface (the same as Dempster’s ‘imbricated’ chips in microtoming) that are discussed below. Gun metal and -brass are less tough than the other ductile metals studied, and for this reason are called ‘semi-brittle’ or ‘short’ materials (but their R/k ratio is still considerably greater than glass, PMMA, etc.). It is perhaps not surprising that discontinuous chips in these materials and others such as cast iron ‘ping off’ with a sharp load drop. A difference between them and the other more ductile metals studied by Palmer and Riad is the steeper angle (about 45°) at which the discontinuous chip meets the free surface; for the ductile solids, like copper and iron, cracking planes are almost tangential to the work surface. Steep planes of separation are characteristic of discontinuous chips in other ‘semi-brittle’ metals such as magnesium. The aluminium/magnesium alloy studied by Palmer and Riad shows planes of separation in between the other groups. The ‘tear chip’ of Rosenhain and Sturney (1925) appears to be a chip where sticking is not eventually relieved by slipping along the rake face, where the initial crack parallel to the surface continues to propagate in that same direction as the chip is loaded in compression until the chip buckles and breaks off (Atkins, 2004). Palmer and Riad (1968) plotted tool travel per chip against the depth of cut and obtained a series of straight lines passing, more or less, through the origin each line corresponding to a different rake angle. The slope is an indication of the aspect ratio of their discontinuous chips (recall that the final chip shape is shorter than the tool travel owing to the plastic squashing of the chip before separation). The straight line relationships demonstrate that, for each rake angle, this type of discontinuous chip is formed in a geometrically similar fashion whatever the depth of cut (see also Sharma et al., 1971). The aspect ratio changed not only with but
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Types of Chip
also with material. The aspect ratios for the more brittle metals were only weakly dependent on (4 at 10° to 2 at 20° for gun metal; 5 dropping to 3 for -brass over the same range of ). This reflects the steeper crack separation angles. With the more ductile materials, the chip aspect ratios at large were only a little greater than for the semi-brittle metals, but they were considerably greater at small . EN8 had a ratio of (tool travel per chip/depth of cut) 20 even at 10°. The order of decreasing aspect ratio at any was: EN8, EN3B, titanium, aluminium/magnesium alloy, -brass, gun metal, and it is likely that this corresponds with their ranking in terms of R/k toughness/strength ratios. Ho and Brewer (1965–66) gave a slip line field for a tool incipiently indenting a sloping surface, employing workhardening slip line field calculations (see Oxley, 1989). They showed how the stress state within the slip line field became conducive for crack formation. Maekawa et al. (1996) have simulated this type of discontinuous chip formation with finite element methods (FEM), and predicted the sort of cutting force traces given above. It would be interesting to see whether, when a tool attacks a ramp, there is an indentation phase before separation. If so, alternate plasticity/separation features should be observable on the cut surface.
4.3.2 General waveform The general type of waveform that fracture mechanics would predict for different degrees of elastoplastic cutting is shown in Figure 4-6(A) (Atkins, 1974). The diagram is based on an early attempt to depict continuous chip formation with simultaneous tool tip cracking that
FC
A
E
FT B,F,
A
M L
C D
O A
J
G
K
O
FC
H
O B
δ
Figure 4-6 (A) Schematic variation of cutting force FC and thrust force FT vs tool travel proposed by Atkins (1974) for chip formation in shear. OA is the FC vs tool travel stiffness of chip without tool edge separation; OCE is the FC vs tool travel stiffness with tool tip crack. Area OAC is energy that becomes separation and friction work; area OCEJ is work of plasticity and friction as the tool moves forward to catch up with tool tip crack, after which the process repeats itself. OB is the FT vs tool travel stiffness of chip without tool edge separation; OD is FT vs tool travel stiffness with tool tip separation. Once cutting starts, FT simply oscillates between B and D, doing no work. (B) Essentially the same as (A) but made non-linear. The separation crack does not run uncontrollably ahead as during the precipitous load drop AC of (A), but rather is accompanied by tool travel as the load falls in a more controlled fashion from A to L, after which load rises from L to M when the tool catches up with the separation for the process to start all over again. L would coincide with M during continuous cutting. The same modifications as in (B) can be made to the linear diagram shown in (A).
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kept pace with the movement of the tool (the model of Chapter 3). Elastic loading up OA stored strain energy in the chip, some of which was released during cracking along AC to some arrested length ahead of the tool; in brittle materials, the crack does not arrest and the load drops to zero. When the system remains globally elastic, OCE represents the stiffness of the chip containing the arrested crack, and further loading up to E causes the process to start all over again. The treatment is similar to how ‘crack arrest’ is analysed (see Chapter 8, Atkins & Mai, 1985), except that in cutting the point of application of the load moves with the moving tool and hence with the crack, unlike the usual type of fracture mechanics testpiece where the point of application of the load is fixed. In Atkins (1974) it was argued that loading up CE did the plastic work of chip formation required to restore the system to its state at point A. The partitioning of total work area OACEJ into components of fracture and flow (and in this case friction as well) is a problem of ductile crack propagation mechanics (Section 4.2.3). Taking due account of friction, values for R and k were derived from experimental cutting force wave forms. It may be possible to generalize the diagram as shown in Figure 4-6(b) where, additionally, we have shown load drops during cracking that are not precipitous and gently slope forwards. The areas OAC and OEJ (both corrected for friction) represent the work of fracture, and of plastic flow of the discontinuous chip, respectively. If we knew how the non-linear stiffness of the partially cracked discontinuous chip altered with increase of crack length, we could predict the load at A from R ∂/∂A ∂/∂AF as described in Chapter 3. Figure 4-7 shows schematically how elastic, elastoplastic and rigid-plastic predictions for (FC/w) in the same materials at different depths of cut t, intersect to give different zones where one mechanism requires lower loads than any other mechanism, and hence delineates the different types of chip expected. As all forces act over the same tool displacement, lowest force is equivalent to lowest work.
4.4 Cutting with a Built-up Edge Except in extremely slow cutting, there is little time for newly formed metal chips to be oxidized or otherwise contaminated, so that friction between offcut and tool can approach that of clean material-to-material contact, which is very high. It is not surprising therefore that the undersurface of the chip sometimes adheres to the tool giving a transferred film of workpiece material over a region near the cutting edge (Zorev, 1958, 1963). The transferred film is a thin layer and has been severely sheared. The hydrostatic stress in the layer determines the shear strain at the interface between flowing and adhered material since together they must satisfy the criterion for shear fracture. In this way newly cut material in the chip can continue to slide over the thin layer stuck to the tool (Appendix 2). Within this region, the friction stress is limited by the (workhardened) shear yield stress k of the chip. While called sticking friction, there must still be sliding between the severely workhardened thin film and the bottom of the chip, otherwise the chip could not flow along the tool. The sticking region may occupy a small, or large, part of the whole contact region between the underside of the chip and the rake face of the tool. In the non-sticking regions, where the chip slides directly against the rake face of the blade, the local friction stress is assumed to be given by p, where is a coefficient of sliding friction and p the local contact pressure between chip and rake face. Most cutting takes place with mixed sticking and sliding friction at the rake face. The proportions of sticking and sliding regions change with depth of cut and tool angle. The force F opposing motion of the chip along the rake face and normal force N on the rake face
93
Types of Chip Force
Tool travel
Cutting force/width
Rigid PIFM
LEFM
Elastic brittle chipping
Elastoplastic discontinuous chips Rigid-plastic continuous chips Depth of cut
Figure 4-7 Schematic scaling diagram showing how cutting forces are expected to alter with depth of cut, when governed by different cutting mechanisms, i.e. rigid-plastic chip formation, elastoplastic chip formation and elastic chip (spall) formation. The regime that occurs is that requiring the lowest cutting force (lowest cutting work over the same tool travel).
also change with the mixity. This causes the apparent coefficient of friction given by F/N to vary in cutting, as discussed in Appendix 2. Even with sticking, it is possible to obtain a good surface finish providing that the attached layer remains a ‘film’ so that sliding occurs close to the rake face. Under appropriate conditions, however, the adhered region can grow in thickness, with the sliding interface moving further and further away from the tool rake face, and forming a built-up edge (BUE) at the tool tip. The BUE forms a new tool profile having greater and results in different φ from before. Were BUEs to remain in place attached to the tool, things might not be so bad. But BUEs tend to be unstable, with metal first welding to the tip of the tool and the stuck material building in size; then some or all of the BUE becomes detached, and the cycle repeats. The changing tool angle and shear plane angle results in fluctuating forces. The debris left behind produces bad surface finishes. Furthermore, broken-off fragments from BUEs often find their way into the tool–workpiece interface and the resulting contact pressure forms a wear land on the clearance face. The implication of instability is that the system does not know what to do, i.e. the reduction in cutting force caused by the increase in effective rake angle is eventually balanced by increase in force caused by the greater length of sticking contact on the ‘false tool’.
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Built-up edges may be removed during cutting by adding a few drops of carbon tetrachloride (CCl4) to the deformation zone (Usui et al., 1961; Childs & Rowe, 1973). This means that the adhesion between tool and chip has been destroyed. Carbon tetrachloride was once used as a lubricant in laboratory machining experiments, but is now banned owing to its toxicity. Fang and Dewhurst (2005) present a slip line field and its associated hodograph for cutting with a BUE. Their paper provides a valuable summary of the extensive experimental and analytical investigations that have been carried out over many years. They pay particular attention to the size of the BUE and its effect on chip flow and cutting forces. The new model simultaneously predicts the length and height of the BUE, the forces FC and FT, chip thickness and chip curl, and the tool–chip contact length. The major causes of BUE formation are the combination of small rake angle and high rake face friction. Note that the slip line field is a plasticity-and-friction-only solution. If it is assumed that the necessary separation work takes place in a thin boundary layer along the putative line of cutting, an approximate correction to the cutting forces per unit width will be to add R to the slip line field prediction. In the case of steels it is found that a BUE forms at low speeds in dry cutting, with small rake angles, and at small uncut chip thickness. Built-up edges tend to disappear (i) when increases; (ii) when t increases; (iii) when lubrication improves; and (iv) when cutting speed increases (Pugh, 1958). Built-up edges have been found even at the small depths of cut involved in grinding (Chapter 6). The formation of a BUE may be changed considerably in interrupted cutting if the length of cut between interruptions is insufficient to allow the build-up to develop (Williams et al., 1970). Four different types of BUE have been described by Childs and Rowe (1973), the growth and shape of which depend upon temperature build-up during cutting. The distribution of temperature within the cutting tool is modified when cutting with a BUE. The tool is separated from the frictional heat sources by the BUE, resulting in lower tool temperatures (Trent, 1952). This is important when cutting with carbon steel tools which would otherwise soften in the temperature range in which the BUE forms. Perhaps this is why the Institution of Mechanical Engineers (IMechE) Tools Research Committee (Herbert, 1928) thought that cutting with a BUE was a good thing.
4.5 Sawtooth Profile Chips In the early days of metal cutting, any chip that was not a continuous ribbon was ‘discontinuous’. Included in that classification were not only the sort of broken-up, separated and discrete chips discussed above, but also any other chip that did not appear to be ‘normal’, whether cracked or not. The bottom (underside) of the chip will be smooth, having passed along the rake face, but the top (back) surfaces of chips display features that have been used for chip classification. For example, the free surfaces of some continuous chips display periodic variations in thickness. In some materials under certain conditions the variations are continuous and appear as smooth undulations on the backface of the chip, but in others there are sharp changes in thickness resulting in sawtooth-like profiles. In metalworking generally it is expected that the free surface of any workpiece will display some roughness after working owing to polycrystallinity, unless it has been burnished by passing along a dieface. A wellknown example is the orange-peel effect in deep drawing (Hosford & Caddell, 1983). But the variations in thickness observed on the top surfaces of chips are often far larger than would result from anisotropic deformations between crystal grains.
Types of Chip
95
Periodic variations in thickness mean non-steady cutting. Cutting with a BUE causes v ariations in effective rake angle. In turn, this will cause variations in φ and hence variations in chip thickness. Cyclic movements of the tool tip during chatter will cause changes in depth of cut and hence wavy chips. As we shall see, there are other causes of variation in chip thickness. Unfortunately, both broken-up chips and uncracked chips having sawtooth backface profiles have been lumped together in the discontinuous classification, in the latter case perhaps because changes in thickness of the continuous chip are ‘discontinuous’ and not smooth. This has led to much confusion, not helped when words like ‘segmental’ and ‘serrated’ have been employed by different authors to mean different things (for a discussion, see Vyas & Shaw, 1999). Figure 4-3 shows a variety of these chip forms, and photographs of different types of discontinuous chip in metals may be found in Childs and Rowe (1973) and Komanduri (1993). Cracks in all sorts of chips are shown in Komanduri and Brown (1972). Rice (1961) discussed what he called segmented chips, which were like the discontinuous chips studied by Palmer and Riad (1968) but where the chips had not separated but had slid over one another to leave a continuous chip consisting of ‘definite segments firmly attached together’. They are therefore different even from Palmer and Riad’s ‘partially discontinuous’ chips. An explanation for this type of chip formation in highly worked -brass was given by Nakayama (1974) in which shear cracks begin not at the tool tip, but on the free surface so that compression of the chip as the tool advances slides it along the shear crack to form an overhanging ledge (a Dempster [1942] ‘imbrication’), the formation of which stops when the next crack begins and the next chip begins to form. As discussed in Section 4.6, cracks can start from the free surface when tools are blunt. Komanduri and Brown (1972) assert that segmental chips result from a stick–slip oscillation and fracture (i.e. an interaction between the primary and secondary shear zones), and that this occurs only in certain speed ranges. Zener (1948) showed that during rapid plastic deformation (as in ballistic impact), the usual sort of strain hardening was counteracted by strain softening caused by local increases in temperature. When the rate of increase in strength due to workhardening is exceeded by the rate of decrease in strength due to increase in temperature, the material deforms locally in quasi-adiabatic shear bands, with hardly any deformation in between. In cutting steels at high rates, the workpiece may be locally heated into the austenite region of the phase diagram and then, as it quickly passes out of the cutting zone, be quenched by the rest of the relatively cold workpiece into martensite. This results in shear bands showing up as narrow regions a few micrometres across. The material properties conducive to the formation of such highly localized shear bands in both void-containing, and void-free, solids are discussed in Dodd and Atkins (1983). Williams (1977) used three-dimensional maps of depth of cut, speed and friction to show when adiabatic and other types of chip might be formed. At the rapid deformation rates in commercial metal cutting, the process is effectively adiabatic on the time-scales of interest. Then the deformation can become unstable locally, with the result that ‘catastrophic thermoplastic shear bands’ occur as the chip is formed (Shaw et al., 1954; Siekmann, 1958). Recht (1964) modelled catastrophic thermoplastic shear and related his analysis to cutting experiments on steel and titanium. The difference between steel and titanium was that shear-localized (serrated) chips were produced in titanium at quite low cutting speeds (say 20 mm/s), but to get the same result in steel, it required speeds 1000 times higher owing to the different thermal conductivities, densities and specific heat capacities, and different rates of change of yield stress and toughness with cutting speed and temperature. Subsequent investigations of this type of chip have been performed by many workers (e.g. Komanduri & Brown, 1981; Manyindo & Oxley, 1986; Molinari & Dudzinski, 1992). Vyas and Shaw (1999) elaborated the Nakayama model and applied it to shear-localized
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The Science and Engineering of Cutting
serrated chips, arguing that it was a preferable alternative explanation to adiabatic shear, but Nakayama’s model, incorporating cyclic cracking, is more likely to apply to Rice’s segmented chips, according to Davies and Burns (2001). These authors modelled chip formation in terms of thermomechanical oscillations in material flow, and applied their results to cutting at very high speeds where strain rates may approach 106 s1, and where temperatures may increase from ambient to a large fraction of a material’s melting temperature in milliseconds or microseconds, thus giving large property gradients in the deformation zones. Calculations showed that cutting may be either steady state or oscillatory in nature, depending on the conditions of cutting. They showed that the material flow patterns associated with different chip types correspond to dynamic equilibrium states of the non-linear physical system. Thus, continuous chip formation is a non-oscillatory flow pattern in which the stress, strain, strain rate and temperature profiles remain constant in time, in a frame fixed with respect to the tool (i.e. it is steady state). A continuous chip with a BUE is the result of a dynamic equilibrium state of the system that contains a stagnation zone. Separated discontinuous chips result from a periodic rupture. Segmental chips are characterized by chips that are continuously sheared, but with periodic variations in thickness. According to Davies and Burns (2001), this type of chip exhibits the characteristics of a subcritical bifurcation in the context of dynamical systems. They modelled shear-localized serrated chips and showed that the process at low speed appears to be somewhat disordered. However, the average spacing between shear bands increases monotonically with cutting speed (known from experiment), and the spacing becomes more regular and asymptotically approaches a limiting value that is determined by the cutting conditions and the properties of the workpiece. From experiments and modelling, they concluded that it is shear localization, not periodic fracture, that is the cause of serrated chip formation.
4.6 Classification of Chips Owing to the importance for manufacturing, it was the metal cutters who first classified the different types of offcut obtained when cutting common ductile engineering materials under different conditions. The results, when plotted out on axes of depth of cut and rake angle, would nowadays be called a machining map. The first of these was published by Rosenhain and Sturney in 1925, for cutting normalized mild steel and 61-39 brass, and is shown in Figure 4-8(a,b). They identified three types of chip: (i) the flow type; (ii) the shear type; and (iii) the tear type. Later, Ernst (1938) classified metal-cutting chips into (iv) discontinuous; (v) continuous; and (vi) continuous with a built-up edge. Flow and continuous are identical; shear and discontinuous are identical; but the other two are completely different. Rosenhain and Sturney knew about the BUE but considered it a subset of their continuous flow type of chip, even though the cutting forces fluctuated irregularly as a cap of dead metal attached and detached itself from the cutting edge. The tear chip is a type of scallop that starts off as a crack approximately parallel with the cut surface; it was not mentioned by Ernst. In the Rosenhain and Sturney/Ernst classifications, ‘discontinuous’ meant elastoplastic chips that broke completely apart (therefore producing swarf that is easier to handle and dispose of). Later, in addition to this type of chip fracture, discontinuous metal chips came to include all the various sorts of chip discussed in Sections 4.3, 4.4 and 4.5. Palmer and Riad (1968) plotted the combinations of depth of cut and rake angle that produced different types of chip for all the metals they studied. The boundaries between different types of chip follow the same trends as Rosenhain and Sturney, except that some of Palmer and Riad’s ‘discontinuous’ chips are in Rosenhain and Sturney’s ‘tear’ chip region
Titanium 12
0.04"
10
xA
0.03" Depth of cut
xA
xA xAA
0.02" xA 0.01" xA
xA AxxB xB xB Shear type xB xB xB xB
xC
xC
xC
xC
xC
xC
10°
A
Flow type
–10 15° Top-rake angle
20°
C
25°
xB
xB
xC
xB 0.01" xC
xC
xC
xC
xC
10
20
30
Aluminium alloy
xB xB
0
Rake angle (degrees)
12
xC
x
Flow type
Cx
Depth of cut (in. x 10–3)
Depth of cut
0.02" xB
4
0
Brass
Shear type
6
2
xC 5°
8
Types of Chip
xB
Cx
Depth of cut (in. x 10–3)
Tear type
10 8 6 4 2 0
5° B
10°
15° Top-rake angle
20°
25°
–10 D
0
10
20
30
Rake angle (degrees)
97
Figure 4-8 Combinations of depth of cut and tool rake angle at which different types of chip are produced in (A) normalized mild steel and (B) 6139 brass (after Rosenhain & Sturney, 1925); boundaries between different chip types identified by Palmer and Riad (1968); (C) titanium; (D) NF7 aluminium/magnesium alloy.
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The Science and Engineering of Cutting
of the diagram. Figure 4-8(c,d) shows their results for NE7 Al/Mg alloy and titanium. If the factor that determines which type of chip will form is the material characteristic length scale given by ER/k2 (or better ER/k2, see Section 4.2.3) then data from different materials should condense into one diagram with universal boundaries. That has yet to be explored. Discontinuous chips (also called type 1 by Ernst) are produced at low speeds in ‘more brittle’ metals using tools of small rake angle with large uncut chip thicknesses: these conditions accord with the Rosenhain and Sturney scaling diagram. This type of chip is most easily disposed of, and when the feed is small for tools with more than one cutting edge (Chapter 6), a good surface finish results. Tool wear when discontinuous chips are produced is by blunting and wearing away of the cutting edge itself. Continuous chips without a BUE (called type 2 by Ernst) are produced at high speeds in ductile metals using sharp tools of large rake angle and with small uncut chip thicknesses; low friction helps continuous chip formation (either with a polished tool or using a tool material having inherently low friction such as cemented carbides). Again, these conditions tie in with the ideas of Rosenhain and Sturney. These chips give the best surface finish with ductile solids. Tool wear during production of type 2 chips is partly by abrasion of rake face and partly by blunting. Continuous chips with a BUE (called type 3 by Ernst) are produced at low speeds under conditions of bad lubrication with small rake angle tools. Type 3 chips result in a rough surface finish because of fragments of BUE escaping with the workpiece, as explained earlier. Tool wear is by cratering of the rake face a short distance back from the cutting edge and abrasion on the clearance face owing to fragments passing under (see Chapter 9). Similar combinations of depth of cut and rake angle under which different chips are formed were reported for wood along and across the grain (Franz, 1958; Koch, 1964; McKenzie, 1966) and polymers (Kobayashi, 1967). Wood machining maps are given in Section 4.7. In the case of polymers, chip formation is affected by elasticity and viscoplasticity as well as other material properties. Furthermore, as polymers are bad thermal conductors, shearlocalized serrated chips must also occur under the right conditions. These factors all affect what types of chip result and Kobayashi in his remarkably comprehensive book had to classify additional types of chip, including discontinuous-complex where the chip stuck to the tool owing to rise in temperature (see also Aluddin et al., 1995). Essentially, though, the chip types are those already described earlier in this chapter. Figure 4-9 shows a series of machining maps for cast polyester resins at different cutting speeds (Kobayashi, 1967). Also with polymers, there is elastic recovery of chips (Patel, 2008). Cutting conditions that produce good surface finish for one material may not do so for another even though they may have comparable hardnesses, giving contradictory indications. This is because R is different. The above classifications imply the use of sharp tools. Blunt tools produce another level of complication. The behaviour is then similar to orthogonal cutting with tools so blunt that the tool tip radius is greater than the depth of cut. Two-dimensional experiments with modelling clay by Bates and Ludema (1974), using a cylindrical tool, show that a prow is built up in front of the indenter until at some critical height a shear crack starts to form at the front of the bulge at its junction with the original surface. The shear crack propagates downwards (probably along a slip line) towards the bottom of the slider, during the forward motion of which the chip is simultaneously forced up along the face of the tool. Either the crack propagates through the whole chip thickness to produce separated chips, or the crack arrests within the thickness to produce a continuous, but broken-up, ribbon. An important feature is that the crack propagates from the surface downwards, not from the tool tip upwards. Presumably there must be a tool sharpness at which cracks may start from both tool tip and surface.
99
Types of Chip Cutting speed V (m/min) V = 23.8
V = 1430 0.08
0.08
0.05
0.05
0.023 0.015
0.023
V = 0.8
Depth of cut d (mm)
V = 715 0.08
0.08
0.05
0.05
0.023 0.015
0.02 V = 0.5
V = 362 0.08
0.08
0.05
0.05
0.023 0.015
0.02 V = 136.5
V = 0.2
0.08
0.08
0.05
0.05
0.023
0.02 −20°−10° 0° 10° 20° 30° 40°
−20°−10° 0° 10° 20° 30° 40°
Rake angle α (degrees) Type of chip Continuous shear Discontinuous simple shear
Discontinuous complex Discontinuous crack
Figure 4-9 Combinations of depth of cut and tool rake angle at which different types of chip are produced in cast polyester resins, including the effect of cutting speed (after Kobayashi, 1967).
4.7 Wood In addition to showing all the transitions between different types of chip mentioned earlier, wood displays additional features when cut owing to its anisotropy in physical properties, i.e. wood is easily split along the grain but impossible to split across. Anisotropy can feature in the behaviour of other materials such as at interfaces between thin films and substrates, and it can be important for things like scratch resistance (Section 6.6). Wood is a natural material, a tree’s growth being shown by annual growth rings. The rate of growth is different at different times of the season, a single year’s growth ring being divided into early (spring) wood or sapwood, and late (summer) wood. The cell walls of the tree within the early part of a ring are thinner, so the timber is less dense and ‘weaker’; in late wood the walls are thicker, and that part of the ring is denser and ‘stronger’ (Barnett & Jeronimidis, 2003). Trees that grow
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The Science and Engineering of Cutting
slowly (narrow, closely packed, growth rings) produce timber with the best structural properties. Box wood is chosen for carving as it has very tight rings, and Baltic oak has been found to keep its shape better over centuries than other woods when employed for oil paintings on timber panels (Young, 2009). It is the growth rings that form the grain. Broadly speaking (there are exceptions), so-called hardwoods such as oak are dense and come from deciduous trees; softwoods such as pine come from evergreen coniferous trees. Hard and soft usually mean strength and there are overlaps between different species. Yew, for example, has a low stiffness relative to its high density. Woods of low density are quite compressible and one characteristic type of wood chip in cutting (the Franz type III, discussed later) is a consequence of porosity. Wood is a cellular material formed, as it were, from bundles of helically wound tubes like drinking straws. It has a hierarchy of microstructure, different properties being controlled at different levels (Jeronimidis & Gordon, 1980). The fracture toughness of most softwoods and hardwoods (about 10 kJ/m2) is exceptionally high in relation to other mechanical properties. It cannot be explained on the basis of the ‘pull-out’ contribution to toughness, as this is predicted to be 1 kJ/m2 at most. Indeed, wood fracture surfaces do not have the stubble-like appearance characteristic of pull-out. The helical substructure of the cellulose fibres in the load-bearing walls provides the clue to why this pattern exists. When they are put under tension, the fibres become unstuck and the cell walls buckle into themselves in such a way that longitudinal extensions of the cell wall of up to 20 per cent become possible. So, although the overall elastic fracture strain in the body of the wood is usually quite small, cells close to the fracture surface are able to extend and absorb a great deal of energy as they do so. In experiments on arrays of model cellulose fibres made by winding glass and carbon filaments into hollow helices with resin, Gordon and Jeronimidis were able to achieve toughnesses up to 400 kJ/m2 that are far tougher than most artificial composites. As with other natural materials, physical properties are very sensitive to moisture content, and there are differences between green (wet) wood (as-felled lumber) and dried or seasoned wood. Although wood can reabsorb water, it never will return to its saturated state once dried. In addition, mechanical properties depend in the usual way on rate, temperature and environments other than water. The grain of wood is raised when sandpapered in water, but not with turpentine. Green timber is heavy owing to retained sap and water and is difficult to saw as the temperature rise makes the wood pinch the blade. At the low temperatures of a Siberian winter, it is possible to hammer dry wood without it shattering, as its toughness does not change much even at liquid nitrogen temperature. Wet wood, on the other hand, shatters at those temperatures. There is a commercial problem for logging during cold weather: some timber can only be removed from the forest when the ground is hard, yet frozen wood is difficult to cut. Reviews of the modern understanding of wood science and engineering relating to cutting as a result of the Cost E-35 Action Program may be found in Holzforschung,volume 63(2) published in 2009.
4.7.1 Veneer peeling A practical example of orthogonal cutting is illustrated by the manufacture of wood veneer by peeling from a rotating log by a wide wedge tool (Figure 4-10). Veneer cut from hardwoods is employed to provide a fine surface on a backing of lower quality wood in furniture manufacture; veneer cut from softwood is used to make plywood. The inventor of veneer peeling was the nineteenth century Swede Immanuel Nobel, father of Alfred (with whom he experimented on nitroglycerine), who founded the Nobel Prizes. It is interesting to contemplate the skill of
101
Types of Chip
Pressure bar
Veneer
1 3
1
2
Tool
1 Bolt rotation
4
1: Crushing and rubbing near tool and pressure bar 2: Main shearing plane 3: Tensile rupture in front of the tool tip 4: Elastic displacement far from tool and pressure bar
Figure 4-10 Basic processes in veneer cutting (after Beauchêne, 1996).
those who cut veneer with saws before log peeling was invented; the grain orientation of sawn veneer on old furniture is different from present-day patterns. Rowing ‘shells’ (i.e. not clinkerbuilt boats) relied on veneer for their manufacture. Thibaut (1988, 1995) and Marchal et al. (2009; and see Mothe et al., 1997) have performed pioneering work in this field. Reasonably steady cutting is achieved in veneer peeling since the path of the cutting edge is in a spiral through the growth rings: in contrast, cutting across the end face of a log encounters lots of anisotropy, and alternating rough and smooth surface finish is produced. At first sight, one might expect veneers to be produced by offcut bending since, apart from compressibility effects and the crushing of cells in softwoods, the final thickness of the veneer is essentially the same as that set by the depth of cut: there is only minor shortening in the length of material peeled from a log. Evidence seems to be, however, that a shear plane is involved, the orientation φ of which is nearly along a radial direction. Values of φ that result in tchip t are given by Eq. (3-20), i.e.: sinφ cos(φ ). Thus for a near-radial shear plane, φ → 90° and → 90°. In practice is about 70° in veneer peeling, for which φ is 80° when the thickness is the same as the depth of cut. In metal cutting it is often said that φ is rarely greater than about 45°, but that is because rake angles in machining are much smaller than those in veneer peeling. A problem in peeling is the formation of checks or partially propagated cracks that turn upwards across the grain from the tool edge. These seem to be the counterparts of cracks turning towards the free surface in brittle cutting by bending (Section 3.3). Depending on the species of wood (density, anatomy, presence or not of wood rays, diameter of the cells, etc.) and cutting speed (commercially about 1 m/s) and temperature (logs are steamed before peeling to temperatures in the range 20–90°C), there is a limiting depth of cut above which checks start to appear (e.g. Thibaut & Beauchêne, 2004). This behaviour is in line with the scaling and transitions in chip formation discussed earlier, so practical veneer peeling seems to be on the borderline between offcut formation by a shear plane and by bending. At the free surface of logs during peeling, large transient compressive bending strains (more than 30 per cent) are observed from which water is expelled out of the cellular structure of wood; these are soon recovered by viscoelastic processes. Fully developed checks would break the veneer into fragments, so a nose or pressure bar is used in the manufacture of thicker veneers to inhibit their formation and propagation (McMillin, 1958); Mckenzie and Karpovich (1975b). In industry, production of veneer thicknesses greater
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than about 0.5 mm requires a nose bar. The nose bar compresses the wood just ahead and behind the cutting edge, inhibits checking and encourages shear plane formation. When the pressure bar is acting, the compression rate (defined as the radial penetration of the bar into the wood divided by the nominal thickness of the veneer) is in the range 10–20 per cent. The resulting compression forces can be very high, depending on the wood species and wood temperature, but can sometimes reach values not far removed from the actual peeling forces. The pressure used can be so great in practice that wood cells are crushed near the contact zone, leading to variations in the final thickness of the veneer. It is virtually impossible to produce check-free veneer sheets in thicknesses greater than 3 or 4 mm (Marchal, 2008). Another feature of veneer peeling is that at very small depths of cut, it is impossible to obtain continuous chips, even with the sharpest knives. High-speed films show that what happens is alternate wood compression and sudden relaxation in front of the tool tip with check formation both above and below the cutting edge (the Horner effect). Clearly, the behaviour of the cellular microstructure at that level no longer approximates to a continuum.
4.7.2 Effects of grain orientation on cutting The first systematic studies of how cutting forces varied with grain orientation, and the different sorts of chip formed, were performed by Kivimaa (1952). Franz (1958) cut wood orthogonally along the grain; McKenzie (1961) studied chips formed when cutting end grain. A great deal of experimental work has been done at these orientations. Fewer studies have looked at cutting with intermediate grain orientations (e.g. Stewart, 1971, 1983; Cyra & Tanaka, 2000; Goli et al., 2002). Franz (1958) identified three types of chip when cutting at zero grain orientation. Type I is a chip formed by a split ahead of the tool and snapping off in bending, like Figure 3-10 with a crack across the depth of the beam; type II is a chip formed by shear (as in metal cutting), like Figure 3-13; and in type III, chips are formed by compression ahead of the tool, and look like discontinuous chips. The type of chip produced in cutting depends on timber species, rake angle, depth of cut and moisture content. When continuous shavings are formed in shear, steady cutting forces are found experimentally, and plots of cutting forces vs depth of cut are linear with an intercept (Figure 4-11). Analysis of the linear data using the methods of Chapter 3 gives R 2.1 kJ/m2 and k 1.7 MPa. Similar analysis of Franz’s own data for cutting fully saturated white ash give (i) with chip formation in shear, Rshear 3 kJ/m2 and k 7 MPa from data at small depths of cut; and (ii) at larger depths of cut where chips form in bending Rbend 1.5 kJ/m2 and k7 MPa. Formation of type I chips results in fluctuating cutting forces, the periodicity of which links to the length of the arrested split as the load falls after crack initiation, to be followed by increasing load back up to bending fracture (not necessarily completely through the chip) and chip splitting once more. It is similar to the mechanics of scallop chip formation described in Chapter 3, except that anisotropy inhibits the crack path from curving to the free surface of the workpiece. With very large rake angles, the curvature of the ‘offcut beam’ may be insufficient to cause the fracture in bending that is characteristic of Franz type I chip formation, and then continuous shavings are formed, not in shear, but in bending. Such chip formation is really an additional classification to Franz’s three for cutting along the grain. The cellular structure of softwoods results in recoverable compressibility so, in cutting, the force system ahead of the tool tip, which is not too significant for metals, can affect chip formation and leads to the Franz type III chip, with force oscillations as the material is alternately compressed and released at the cutting edge. Ryvkin and Nuller (1994) have
103
Types of Chip 8000.0 7000.0 FC/w
(FC/w) and (FT/w) (N/m)
6000.0 5000.0 4000.0 3000.0 2000.0 1000.0 0.0
FT/w
–1000.0 –2000.0
0
0.0001
0.0002
0.0003 Depth of cut (m)
0.0004
0.0005
0.0006
Figure 4-11 Cutting of Douglas fir (10 per cent moisture content) on tangential planes along the grain; early and late wood are cut simultaneously. Plots of FC/w and FT/w vs depth of cut when cutting with a 60° rake angle tool and chips are formed by shear. Data are fitted by FC/w 5 106t 3313 (N/m).
presented a model for cutting cellular elastic materials, based on a criterion of fracture under a maximum tensile stress. An example of a machining map for timber is given in Table 4-1. There are many more given in Koch (1964). McKenzie (1960) identified two types of chip when cutting end grain (90° material orientation). Confusingly, he used Roman numerals as Franz (1958) had done in his chip classification scheme, but McKenzie’s type I and II are not the same as the Franz types I and II. McKenzie’s classification is really more related to subsurface damage on the cut face than the chip type: his type I chips form with splits into the grain of the sample (Figure 4-12); in type II the material fails not only in planes parallel to the grain, but also parallel to and below the tool path. Chip formation when cutting at an arbitrary angle to the grain is more complex. Even so, none of the ‘standard’ orientations includes cutting on skew planes, i.e. cutting on diagonal planes. Obtaining good surface finish is a problem when cutting against the grain (where the cutting edge faces grain emerging from the timber) or cutting with the grain (where the emergent grain trails the cutting blade). Surface finish not only concerns appearance in furniture, for example; it also relates to whether glued joints adhere properly between timber components. Forces when cutting the face of a testpiece in a turning operation occur over all grain orientations. In the general case of cutting at an angle to the grain, transverse compression of the putative chip, fibre bending and shear along the grain have a complicating influence on chip formation that is difficult to predict. In some circumstances, the inclination of the shear plane required for continuous chip formation may roughly coincide with the angle of the grain itself. In such a case, anisotropy may cause shear to occur preferentially along the grain direction rather than at the primary shear plane angle given by isotropic continuum mechanics. Should the strains in the grain direction exceed the shear fracture strains of the wood, discontinuous chips are formed (Goli et al., 2009).
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The Science and Engineering of Cutting
Table 4-1 Relation of type of chip formation to cutting angle and chip thickness for sugar pine at various moisture contents. Chip thickness (in.)
Type of chip formationa for the following cutting angles 5°
10°
15°
20°
25°
30°
1.5% moisture content 0.002
B
B
C
C
C
C
0.005
B
B
C
C
C
C
0.010
B
B
B
A
A
A
0.015
B
B
B
A
A
A
0.020
B
B
B
A
A
A
0.025
B
B
B
A
A
A
0.030
B
B
B
A
A
–
8% moisture content 0.002
B
C
C
C
C
C
0.005
B
C
C
C
A
A
0.010
B
C
C
C
A
A
0.015
B
C
C
A
A
A
0.020
B
B
C
A
A
A
0.025
B
B
B
A
A
A
0.030
B
B
B
A
A
–
Saturated with moisture 0.002
–
–
–
–
–
–
0.005
B
B
B
B
C
C
0.010
B
B
B
B
B
C
0.015
B
B
B
B
B
C
0.020
–
B
B
B
B
C
0.025
–
B
B
B
B
C
0.030
–
B
B
B
B
B
Source: From Ref. 14. a A: tear chip; B: discontinuous chip; C: continuous chip.
Figure 4-13 shows the cutting forces for a 0.6 mm depth of cut superimposed on the appearance of the surface obtained after processing Douglas fir at 70° against the grain. The link between force fluctuations during the cut and features on the cut surface is clearly evident. The spindle speeds of commercial woodworking machinery are very high and can give linear speeds of even 500 m/s (Kottenstette & Recht, 1981) (see Chapter 7). Speeds and strain rates in commercial cutting are certainly greater than the speeds and strain rates achieved in conventional testing machines, and may exceed the strain rates achievable using special
Types of Chip
105
Force (N)
Figure 4-12 McKenzie type Ia chip formation. Douglas fir: 0.4 mm depth of cut; 5 mm/s cutting velocity; 20° tool rake angle; 90° grain orientation.
2000 1800 1600 1400 1200 1000 800 600 400 200
0 200
Figure 4-13 Surface and cutting forces when processing a tangential specimen of Douglas fir at 70° against the grain with a depth of cut of 0.6 mm superimposed on the resulting surface finish (after Goli et al., 2005).
devices such as the split Hopkinson bar. It may not be possible to obtain independent mechanical properties at these rates to compare with experimentally derived values, or properties to employ in FEM modelling. While forces can be measured during commercial cutting with high-speed routers and saws, it is difficult to film chip formation. In consequence, controlled experiments in which chip formation is studied with high-speed cameras often use the driver of a Hopkinson bar to shoot a tool against a workpiece, or vice versa. At high speeds, if the cutting is ‘interrupted’ (i.e. done by a series of cutting teeth or cutting edges that are not cutting all the time, as on saws, milling cutters and so on), each cut involves an impact.
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The Science and Engineering of Cutting
8 m/s chip formation
90.00
0,005 m/s 8 m/s
75.00
Tool forces (N)
60.00 45.00
Fc
30.00 Fc 15.00 Fn 0.00 –15.00
Fn
–30.00 –100
–80
–60
–40
–20
0
20
40
60
80
100
Grain orientation (°)
0.005 m/s chip formation
Figure 4-14 Plots of maximum tool forces at different grain orientations, and the different types of chip produced, when cutting at two very different cutting speeds.
Figure 4-14 shows the effect of cutting speed on chip types formed at different grain orientations, for identical tool geometries (rake angle 20°; depth of cut 0.2 mm). The high-speed measurements were taken on a rig adapted to a Hopkinson bar (Goli et al., 2009). A Hopkinson bar/gas gun has been used to cut steel by Sutter et al. (1998) with only one depth of cut (0.4 mm). They found that the cutting force for 10 mm wide workpieces decreased
Types of Chip
107
quite rapidly from an initial value of nearly 9 kN at 0.1 m/s to about 4.5 kN at 10 m/s, after which it remained almost constant up to some 70 m/s, after which it rose a little. The advantage of correctly proportioned Hopkinson bar devices is that the cutting velocity does not drop appreciably by the end of the cut, in contrast to some other types of impact cutting device (e.g. Kottenstette & Recht, 1981). In Figure 4-14 in a 0° direction of cutting, the Franz chip is continuous at low speed (i.e. type II) but becomes the split type at high speed (i.e. type I). At 90° grain orientation, the high-speed chip disintegrates after formation. At other orientations, differences with speed may not be so marked. The variation in maximum cutting forces with grain orientation is also shown. The cutting force FC is a little smaller at 8 m/s than at 5 mm/s and the variations with grain orientation are similar. However, the thrust force FT changes sign.
4.7.3 Plywood and particleboard Plywood is made up of thin layers of softwood, produced by veneer peeling, glued together to give large sheets. Alternate sheets are layered with respect to each other, usually by 90°, to give a more balanced behaviour; and plywood is difficult to chop. Chipboard and particleboard contain recycled wood and hardwood in various proportions, together with an adhesive such as urea–formaldehyde glue. Boards are made on continuous roller presses or single-action presses, with contact stresses of some 2–4 MPa and temperatures of 150–230°C. Pressing time, temperature, friction between particles and moisture content all affect the final properties. A feature of particleboard and medium-density fibreboard (MDF) is differences in density between outside and inside caused by friction between particles (rather like in sintering or hot pressing of powders). The less dense centre material is weaker and comes apart more easily, and this affects the energy consumption in cutting (Wong & Schajer, 2003). Orthogonal cutting of particleboard on a microtome under a wide variety of conditions has been performed by Kowaluk et al. (2004) and Beer et al. (2005). Figure 4-15 shows plots of (FC/w) vs t that were analysed in terms of the shear plane model of cutting of Chapter 3 to give toughnesses R from the intercepts of 23 kJ/m2 and works/volume from the slopes of some 30 MJ/m3. Experiments with blunter tools where 24 m gave values for R of 33.5 kJ/m2 but works/volume remaining roughly the same as for sharp tools. Beer et al. (2005) investigated the properties at different locations within particleboard by cutting and found that the toughness of the outside regions was twice as big as the value from cutting the whole thickness. Sinn et al. (2005) investigated the effect of speed on upmilling (chips starting thin) the edges of 6 mm thick MDF and particleboards. The cutter was 125 mm in diameter and circumferential speeds of 9 to 31 m/s were achieved, all at a constant feed/tooth of 2.1 mm. The maximum feeding force (Ff/w) decreases slightly with speed, but the normal force (FT/w) increases slightly. These results agree with similar trends for solid wood reported by Kivimaa (1950) and McKenzie (1960). All particleboards behaved pretty much the same, but the MDF required double the forces at all speeds. Owing to the shape of the milled chip, there is an almost linear increase in chip thickness t during the time of a cut (see Figure 7-4), over which the force increases again almost linearly with time. The cross-plot of (Ff/w) vs t is the same as that for a tapered cut (Section 5.4.3) and contains, in a single cut, information that would otherwise require many separate cuts at constant thicknesses to achieve. Sinn et al. (2005) found that the intercept value of toughness of particleboards and MDF varied with cutting speed, following a relation of the form (Ff/w)t0 a(velocity) b where a and b are constants. The
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The Science and Engineering of Cutting 9000 8000 7000
(Fc/w) (N/m)
6000 5000 Manufacturing Temp. Pressure 180°C 2.5 MPa
4000 3000
180°C 4.0 MPa 220°C 2.5 MPa
2000
220°C 4.0 MPa
1000 0
0
20
40
60
80
100
120
Cutting depth (μm)
140
160
180
200
Figure 4-15 (FC/w) vs t for particleboards manufactured under different combinations of temperature and pressure (after Kowaluk et al., 2004; Beer et al., 2005).
magnitude of b at zero velocity should correspond with quasi-static toughness values for the materials. R ranged over roughly 1–5 kJ/m2 for different particleboards, and 5 kJ/m2 for MDF, with MDF having the greatest b. The slopes given by a ranged between 0.13 and 0.31 kJ.s/m3. Boards with hard surface coatings (such as melamine) are difficult to router with other than very sharp blades. Blunt blades will damage brittle surface layers by chipping. Out-ofplane deformation when cutting melamine-coated particleboard can be detected by electronic speckle interferometry: the blunter the tool the more the out-of-plane deformation and the worse the surface finish of the product (Stanzl-Tschegg, 2008).
4.7.4 Wood pulp Thin sheets, made from fine strips of the stem of the papyrus reed plant, by soaking in water, pressing and drying, were used as writing material by the Ancient Egyptians, Greeks and Romans. Parchment was a similar material made from animal hides, especially from sheep and goats. Neither of these materials worked well with early printing presses for which paper was found to be better, Singer (1959). Paper is a felt of vegetable fibres. The best paper is made from linen rags that were originally hand beaten out by inmates of prisons. Most paper is nowadays made from wood pulp, with special machines (refiners) to disintegrate wood chips into fibres. In the refining process wood chips are broken down between two discs where one or both rotate (it is a high-speed grinding process). The chips are normally fed into the centre and are transported by centrifugal action to the periphery where they are discharged (Berg, 2001). The first stage of breakdown is into matchstick-like material, then as they move to the outside of the machine, into slender ‘pin’ chips, then into fibre bundles and finally
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Types of Chip
into separate fibres (May, 1973). Longitudinal grinders are also employed where the action in breaking down is similar. Various high-rate devices including pendulums have been employed to study the break-up of wood to form pulp, to determine relevant mechanical properties and energy consumed (e.g. Atack et al., 1961; Eskelinen et al., 1982; Marton & Eskelinen, 1982). Theories for pulping have been given by Atack (1980) and Strand and Mokvist (1989). The specific energy of pulping is of concern to paper makers. It might be expected that work for dividing up wood into very small slivers will be large owing to the inverse dependence of cutting force FC on chip thickness (Section 3.6.2). The specific energy is given by (FC/wt), where w is width of cut and is density. For softwoods employed in pulping, with toughness R 60 J/m2 and shear yield stress k 30 MPa, (R/k) 2 106. A sliver of 0.1 mm thickness is greater than 10(R/k), so the linear relation ‘FC/w (1/Qshear) [(k) t R]’ will be followed, although it is a process of mutual abrasion between wood particles as well as contact with the blades of the refiner rather than cutting with a rigid tool. Even so, the specific energy including density is
FC /wtρ (1/ρQshear ) [(kγ) R/t]
(4-11)
For 3, and r300 kg/m3, this gives about 500 kJ/m3, which has the right order of magnitude. While there is an inverse dependence with t, comparable values of (R/t) to match (k) arise only when t is at the micrometre level. It happens that the final size of wood fibres in pulp is some 30 m, so disproportionate power consumption should be expected in the final stages of pulping.
Chapter 5
Slice–Push Ratio Oblique Cutting and Curved Blades, Scissors, Guillotining and Drilling Contents 5.1 Introduction 5.2 Floppy Materials 5.3 Offcut Formed in Shear by Oblique Tool 5.4 Guillotining Edges 5.5 Drills, Augers and Pencil Sharpeners
111 113 119 123 134
5.1 Introduction In the kitchen or at the dinner table, cutting may be performed simply by ‘pressing down’ with a knife. It is common experience, though, that even with the sharpest knives, cutting seems to be easier when sideways motion as well as vertical motion is incorporated in the cutting action. By easier, we mean that the vertical force is reduced. Even when just ‘pushing down’, without sideways action, angling the cutting blade to the direction of cut is beneficial in some way, for example by giving a better surface finish. Why is there this difference in cutting forces for angled blades, and for blades having sideways slicing motion as well as the normal pushing motion? In the case of a loaf of bread, it might be thought that this is to do with the serrated teeth found on many breadknives, but the phenomenon is just as evident with plain smooth edges on blades. The effect seems disproportionate in that the pressing force is reduced quite markedly by even the smallest sideways motion of the cutting blade. The greater the sliding velocity relative to the pressing velocity the greater the reduction in the pressing force. Captives whose wrists are tied by rope find it necessary to rub their bindings back and forth as well as press hard against the best available edge to cut through their bindings. In orthogonal cutting (Chapter 3) the cutting edge is always at right angles across the workpiece. When a straight blade is angled to the direction of motion of the workpiece, it is called oblique cutting. All the different types of chip described in Chapter 4 are found in oblique cutting. The inclination of the cutting edge need not be constant: it changes as the straight blades of scissors are closed, and in devices with curved blades such as the scythe the inclination continuously changes. Metal-cutting tools often have two cutting edges, both of which are angled to the direction of cutting, and in round-nosed tools the inclination continuously varies (Chapter 6). We shall discover that the slice–push ratio given by (blade displacement or velocity parallel to the cutting edge/blade displacement or velocity perpendicular to the cutting edge) is important in making cutting seem easier, and that greater gives easier cutting. As shown in Figure 5-1, a slice–push ratio is obtained when (a) an orthogonal blade is driven sideways as well as down; (b) driven straight down but at an angle, since the cutting feed velocity has components along and across the inclined blade; and (c) when an angled tool fed into the workpiece with feed f has its own independent motion parallel to the Copyright © 2009 Elsevier Ltd. All rights reserved.
111
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The Science and Engineering of Cutting
ν h f
i
A
B
f h i
C
Figure 5-1 ‘Slice–push’ produced by various blade motions and orientations: (A) an orthogonal blade with displacements (velocities) both into the workpiece (v) and across the workpiece (h); (B) an oblique blade, inclined at angle i to the crossways direction of the workpiece, moving into the workpiece with feed displacement (velocity) f; the blade itself has no motion along its edge; (C) as in (B) but now where the blade has velocity h along its edge as well.
cutting edge. Thin samples fed into a rotating disc cutter along the centreline are an example of case (a), where is given by (wheel peripheral speed/workpiece feed speed). Feeding a sample either above or below the centreline into a stationary wheel is case (b); when the cutting disc rotates as well, it is case (c) where different are given since the cutting edge of the disc is both inclined to the feed direction and has its own velocity. The behaviour in case (c) depends on whether the cut is taken above or below the centreline as, in the one case, the edge speed augments the geometrical effect of the inclined blade, and in the other it subtracts from the geometrically induced . Why the cutting force is reduced when there is slice–push can be explained using energy arguments. It is not surprising that if energy is put ‘sideways’ into the system, less energy and hence a smaller force will be required in the vertical direction. But it is not as simple as that because, as we shall show, a non-linear coupling occurs between the two forces which causes the vertical force to drop markedly as soon as the slightest horizontal motion is introduced. Reduction of forces by slice–push produces better surfaces whatever the material. In the laboratory, the best sorts of junctions with fibre-optic cables and scintillators are obtained when a slicing cut is made with a warm razor blade; there are similar proprietary devices. Lower tractions across a cut surface reduce the tendency for components in the microstructure to separate (fat from meat in bacon slicing). Cutting with slice–push is the only way that some materials
113
Slice–Push Ratio
can be cut easily. For example, attempts to cut highly deformable soft foams by pressing down alone are rarely successful, but slicing with an inclined blade readily cuts such a material. Thus it is difficult to cut foam with nail clippers, but cutting is achieved with scissors (Bonser, 2005). If, in addition, the foam can be prestressed in bending across where a cut is to be made, cutting is even easier. Slice–push is the reason why one’s tongue is sometimes cut when licking envelopes. The reduction in forces when cutting thin sheets with slice–push stops prows and buckles forming ahead of the blade which stop or interfere with the process. Cutting with a steeply inclined blunt blade may be possible when, at smaller angles, cutting fails. Spades and shovels on the continent of Europe often have pointed blades, along which there will be slice–push, in contrast to the square-ended tools found in the UK. They also have long handles and are operated differently.
5.2 Floppy Materials 5.2.1 Frictionless thin blade In Figure 5-1(A), a thin knife (negligible wedge angle) cuts a block of material of width w. The knife blade is long enough always to overhang the workpiece (or it is a ‘band blade’, like a band saw but having no teeth). The blade is orthogonal to the workpiece and, additionally, it moves across as well as down; it is thus case (a) of the Introduction. Forces V (normal to the cutting edge) and H (parallel) have associated displacements v and h, respectively. The incremental work done is [Vdv Hdh]. This provides the fracture work required for the increment of new cut area, which is given by Rwdv, assuming frictionless conditions and that the growth of cut keeps steady with the movement of the blade. Thus
Vdv Hdh Rwdv
(5-1)
The resultant force is given by [V2 H2]1/2 and the resultant displacement is [(dv)2 (dh)2]1/2. When there is no permanent distortion of the offcut, and when the wedge angle of the blade is small, these increments are coincident, so that we may also write
[V 2 H 2 ]1/ 2 [(dv)2 (dh)2 ]1/ 2 Rwdv
(5-2)
The slice–push ratio is given by (dh/dv), whence solution of these two simultaneous equations gives
[H/Rw ] ξ/[1 ξ2 ]
(5-3a)
[V/Rw ] 1/[1 ξ2 ]
(5-3b)
H ξV
(5-3c)
and i.e.
The resultant force is given by (V2 H2), so the non-dimensional resultant force (FRes/Rw) is
(FRes /Rw) (1/[1 ξ2 ])1/ 2
(5-4)
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The variation of normalized H/Rw and V/rw with is shown in Figure 5-2. For 0, H/Rw 0 and V/Rw 1. For → 1, H increases to a peak at 1 (when H/Rw V/ Rw 0.5) and then diminishes as increases. V diminishes for all . The smallest normalized forces occur for largest , i.e. the sideways speed has to be as great as possible to reduce cutting forces so long as R is constant (strain rate effects may very well affect R). The effect of friction, curves for which are also shown in Figure 5-2, suggests that there is no point in increasing indefinitely. The common experience of V diminishing quickly as soon as some sideways motion is introduced is immediately apparent from Figure 5-2. The effect is noticeable because it is disproportionate: the non-linear coupling between V and H is because the vertical blade displacement and the area of new cut both depend upon V. Since a knife failing to penetrate with only a vertical force will be almost at rest, the slightest horizontal motion will cause 0 and hence much reduce V, as found practically. When a workpiece approaches a stationary blade whose normal is inclined at an angle i to the direction its motion, a slice–push effect exists because the approach feed velocity f has components fsini parallel to the edge of the blade and fcosi perpendicular to the edge (Figure 5-1B); in orthogonal cutting f v. A familiar example is planing wood with the plane angled to the length of the workpiece (although wood is not really floppy). It follows that sini/cosi tani so that greatest is obtained with the steepest inclination. Note that the effective width weff of the sample becomes (w/cosi) for use in Eq. (5-3/4) to give H and V whose directions are along and across the inclined blade (not along and across the direction of f). The sign of the inclination angle i is immaterial for magnitudes of forces, the only difference being the direction of H. The forces in the direction of f and across are given by resolution, i.e.:
Falong f Vcosi H sini V(cosi ξsini)
Facross f Hcosi V sini V(ξcosi sini)
4.5
(5-5b)
θ = 6°, μ = 0.3
4 V/Rw and H/Rw
(5-5a)
θ = 12°, μ = 0.3
3.5
Frictionless
3 2.5 2 1.5 1 0.5 0 0
1
2
3
4
5
6
ξ
Figure 5-2 Reduction in normalized force V/Rw at increased slice–push ratio and initial increase in H/Rw up to 1 followed by decrease, when cutting floppy materials. Frictionless case is the same for all included angles of blade but, with friction, predictions depend on both and . Examples shown for 6° and 12°, both with 0.3.
115
Slice–Push Ratio using Eq. (5-3). When tani,
Falong f V/cosi Rw eff /[1 ξ2 ]cosi Rw
Facross f 0
(5-5c)
(5-5d)
Thus, in frictionless cutting with an inclined blade, the force required to cut in the direction of tool or workpiece motion is simply Falong f Rw, with zero sideways force, as expected since the only work is separation work.
5.2.2 Cutting with friction Still with case (a) of the Introduction, the orthogonal cutting edge has displacements dv and dh as above, but the blade now has an angle (including clearance) of . The resultant displacement of the offcut over a flank of the blade has two components: dh parallel to the cutting edge and (dv/cos) along the line of greatest slope of the rake face of the blade (Figure 5-3). This gives a resultant displacement dr on the rake face of magnitude
dr [(dh)2 (dv/cosθ)2 ] (dv/cosθ) [(ξcosθ)2 1]
(5-6)
since dh/dv; dr acts at an angle q tan1[dh/(dv/cos)] tan1[cos] with respect to the line of greatest slope. The resultant friction force between offcut and rake face of the tool is assumed to act in the same direction as the resultant displacement. Hence the incremental friction work for Coulomb friction on one flank of the blade is Ndr and is given by
µNdr µ[V/(sinθ µcosθ)cosθ] [(ξcosθ)2 1]dv
(5-7)
Blade dν Cos θ dh
dh dr
θ
dν
Figure 5-3 Motion of slice over blade has two components: (i) dv/cos along the rake face of the tool having included angle ; and (ii) dh along the cutting edge.
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substituting for N in terms of V from Appendix 1. The expression in Eq. (5-7) may equivalently be obtained by summing the work done by the component Ncosq of N along the line of greatest slope of the wedge times (dv/cos), plus the component Nsinq of N parallel to the cutting edge times dh. Equating external and internal work increments for an orthogonal cut with a sidewaysmoving blade gives
V/Rw 1/{1 ξ2 [(2)µ ((ξcosθ)2 1)/cosθ(sinθ µcosθ)]}
(5-8a)
and
H/Rw ξ(V/Rw)
(5-8b)
The bracketed ‘2’ with is to be used when both sides of a blade are in contact with the workpiece. Figure 5-2 includes curves for V/Rw and H/Rw that include friction according to Eq. (5-8). When a stationary blade is inclined at angle i to the crossways-dimension of the workpiece, tani, and the forces V across and H along the edge are given by Eqs (5-3a,b) noting that w is replaced by the inclined length of contact weff w/cosi. The feeding force Falong f and the crossways force Facross f are obtained using Eqs (5-5a,b) to give
Falong f /Rw eff 1/cosi {1 ξ2 [(2)µ ((ξcosθ)2 1)/cosθ(sinθ cosθ)]}
(5-9a)
and
Facross f 0
(5-9b)
since tani and, in this case, the blade is stationary. There may be optimum inclination angles i for least cutting force owing to the competition at large between smaller forces on the one hand, but larger frictional contact length on the other. It will depend on and . A similar effect is found when cutting materials with an initially slack wire (Chapter 12).
5.2.3 Inclined separately propelled blade: the disc slicer Cutting on a delicatessen slicer involves workpieces of bacon, salami and so on which are relatively thick compared with the diameter of the cutting disc. Here we consider laminae fed into a rotating disc cutter, where is approximately constant across the thickness. Cutting of thick workpieces that cover considerable parts of the blade is considered in Chapter 12. Consider a sheet fed, below the centreline, into a cutting disc of radius (Figure 5-4A). Point P is located at angle i measured from the centreline of the wheel where positive i is anticlockwise. The disc rotates with angular velocity , where positive is anticlockwise. The feed rate of material into the wheel is f from left to right. We bring the workpiece to rest by adding a velocity (f) which means that in addition to rotating in a clockwise sense, the disc now has a forward speed f from right to left, which has a tangential component fsini in an anticlockwise direction, and a radial component given by fcosi. The local velocities (displacements) at P normal to the cutting disc dvdisc, and parallel to the edge of the disc dhdisc, are thus
dv disc fcosi
(5-10a)
117
Slice–Push Ratio ω
ρ
i f
f
P
i fsini
fcosi
ρω
A 0.3 0.25
Facross f
0.2 0.15 0.1 0.05 –180°
–90°
i
Falong f
0 0
i
+90°
+180°
–0.05 –0.1 –0.15 B
Figure 5-4 Thin sheet fed into a disc cutter below the centreline with speed f. (A) Geometry of device where zero for i is along the centreline and positive i is anticlockwise; disc has radius and rotates anticlockwise with angular velocity ; (B) variation of feeding force and vertical force with position i of cutting for /f 5, 0.1 and 6°. Negative values for feeding force mean that the workpiece has been ‘grabbed’ by the cutting disc.
and
dhdisc (ρω fsini)
(5-10b)
where positive dhdisc has the sense of . The slice–push ratio at P in an anticlockwise sense is
ξdisc dhdisc /dv disc (ρω fsini)/fcosi (ρω/fcosi) tani
(5-11)
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For cutting above the centreline, i is negative and tani changes sign. If (/f) sini, the contribution of tool obliquity to the push/slice effect will not be noticeable except at large i (cutting at the top and bottom of the disc). The forces Falongf in the feed direction and Facrossf perpendicular to the feed table are given by Eqs. (5-5a,b) using V and H from Eqs (5-3a,b) that includes friction, i.e.
Falongf V[cosi ξdiscsini ]
(5-12a)
and
Facross f V[ξdisc cosi sini ]
(5-12a)
The variation of Falongf and Facrossf with position above (i negative) and below (i positive) the centreline is shown in Figure 5-4(B) for /f 5, 0.1 and 6°. The negative values of Falongf indicate that the workpiece has been ‘grabbed’ and requires no positive force to push it through the disc cutter. This is familiar to anyone who has used a hand grinding wheel or circular saw. Calculations show that, unsurprisingly, overall cutting forces increase with greater friction and with smaller , but the pattern of disproportionate decrease in V as increases is retained. Atkins et al. (2004) performed experiments with a disc cutting cheese and salami and demonstrated the effect of slice–push in reducing cutting forces as the speed of the disc was altered at constant feed. In that paper, the friction force was modelled not by the Coulomb relation, but rather as a frictional stress that was some fraction m ( 1) of the workpiece shear yield stress, i.e. mk, acting over some finite contact area between offcut and blade (this approach is often employed in metal cutting; Appendix 1). It may be shown that for an orthogonally orientated blade
V/Rw [1 S (1 ξ2 )]/(1 ξ2 )
(5-13)
and H V, in which S (2)mLk/R with L the contact length along the rake face. The bracketed (2) relates to whether one or two faces of a blade contact the workpiece. There are similar expressions employing S for Falong f and Facross f when the blade is both inclined and independently moving. Whichever way friction is modelled, calculations show that there is probably no benefit in increasing disc indefinitely, by increasing the speed, owing to increased work against friction, and experiments confirm this. Instead of determining Falongf and Facrossf using H and V as intermediate values, there are other ways of obtaining the feed and across-feed forces directly, employing the effective wedge angle eff of the disc (not the line of greatest slope in the cutting bevel, rather the slope along which the offcut passes for which taneff cosi tan) and ieff (where tan ieff disc), but these alternative lines of attack are, perhaps, confusing. Similar alternatives occur in modelling the formation of ductile chips during oblique cutting (Section 5.3).
5.2.4 Pizza cutter: disc harrows A similar analysis may be performed for the pizza cutter disc that rolls along the base of the pizza. It may be shown that at the base pizza is infinite as the motion is instantaneously all slice and no push so, in theory, requires no force (rather like an extremely thin sheet cut at the top or bottom of a delicatessen slicer). While it is possible to define a mean slice–push
119
Slice–Push Ratio
ratio pizza*, it is unbounded. The force Fpizza in the direction of cutting with a frictionless rolling disc is simply Rh where h is the thickness of the pizza. With friction, the procedures in Section 5.2.2 may be employed. In practice, there will be additional friction as the bottom of the cutter rolls up and emerges behind the wheel. Pizza wheels are used to cut cork in Sardinia (Negri, 2008). Discs are used in some designs of harrow for improving the tilth of seed beds (Chapter 14). They must act rather like pizza cutters, but in a complicated way, as the plane of the disc is often angled to direction of tractor motion, and the disc itself may be dished, in order to improve disturbance of the soil. Godwin et al. (1987) show that haulage forces arise from two components, namely a passive reaction on concave faces and scrubbing action on convex faces. The associated forces were estimated using pressure-dependent soil yielding mechanics. As explained in Chapter 14, this is equivalent to plasticity-and-friction-only analyses of cutting but, as also explained in Chapter 14, toughness work in soils may be swamped by other components of work done such as lifting the soil.
5.2.5 Reciprocating blades Reciprocating blades have slice–push but, unlike blades moving continuously in the same direction, varies at different positions in the stroke. There is zero slice–push at the ends of the stroke where the blade is instantaneously at rest. The maximum will be at mid-stroke. Owing to the continuing changes in , force plots from a high-speed reciprocating blade are very spiky. If the reciprocating motion is approximated by h hosint, (ho/f)sint, where f is the feed displacement in orthogonal cutting, Eqs (5-3a,b) and (5-4) give for frictionless cutting
[H recip /Rw]=(ho /f)sinωt/[1+(ho /f)2sin2ω t]
(5-14a)
and
[Vrecip /Rw] 1/[1 (ho /f)2sin2ωt]
(5-14b)
and
(FRes /Rw) (1/[1 (ho /f)2sin2ωt])1/ 2
(5-14c)
The variation of the forces in one cycle is shown in Figure 5-5, for (ho/f) 10. Forces are always low in the middle of the stroke, but high at the ends, the more so when h v. Benefits of large are evident only in the centre of the stroke. The analysis is easily modified to demonstrate the effects of friction. Similar considerations apply to hedge cutters, hair trimmers, sheep-shearing comb cutters and electric carving knives (see Chapter 10).
5.3 Offcut Formed in Shear by Oblique Tool When a chip is formed in shear in orthogonal cutting of ductile materials, it has the same width as the uncut chip thickness but a different thickness. It also has curvature caused by the non-uniform width of practical primary shear zones, and also by secondary shear, which together with bending forms the chip into a spiral. When a straight-edged tool is angled to the direction of feed and used to cut a ductile material, there are two main differences from orthogonal cutting: (i) the offcut has not only a different thickness, but also a different width
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The Science and Engineering of Cutting 1.2 1 0.8
Fres/Rw
0.6
V/Rw
0.4 H/Rw 0.2 0
0
0.5
1
1.5
2
2.5
3
3.5
– 0.2 Reciprocating displacement of knife (arbitrary units)
Figure 5-5 Variation of V/Rw, H/Rw and Fres/Rw for frictionless reciprocating cutting. Slice–push varies during the stroke of the blade. It is a maximum in the centre but zero at the ends of the stroke. Benefit of lost except at central portion of stroke.
(caused by primary shear over a longer angled contact length between tool and workpiece); and (ii) more complicated curvature that bends the chip into a permanent helix with the axis of rotation approximately parallel to the cutting edge. The greater the inclination angle i to the direction of feed, the wider the chip and the tighter the curl. In a simple single shear plane model of oblique cutting, the shear plane connecting the cutting edge to the free surface is skewed at the obliquity angle i to the feed direction (Figure 5-6). There are three velocity components: the workpiece approach velocity VW, the shear velocity VS in the shear plane, and the chip velocity VC in the plane of the tool rake face. In orthogonal machining, all three velocities and the hodograph lie in the plane of cutting, the direction of shear is along the line of steepest slope in the primary shear plane, and the direction of chip flow is along the line of steepest slope of the rake face of the tool. In oblique cutting, both the primary shear direction in the shear plane and the chip flow direction across the tool are no longer in the directions of steepest slope. VS is now at an angle S (the shear flow angle) to the normal to the cutting edge in the shear plane; the shearing action at angle S results in the final cocked direction of the chip over the rake face of the tool, which is defined by the angle C (the chip flow angle) to the normal to the cutting edge in the rake face. Since all three velocities VW, VS and VC form a hodograph in one plane they are related by geometry (e.g. Amarego & Brown, 1969, p. 80). It was pointed out earlier that when cutting thin sheets not along the centreline of a disc cutter, it was possible to do calculations in terms of the effective blade included angle eff rather than the usual included angle given by the line of greatest slope. When shear planes are formed in oblique machining, there is again a number of alternative definitions of tool rake angle and of shear plane angle (for a discussion see Shaw, 1984; Amarego & Brown, 1969). That usually employed in analyses of ductile materials is the rake angle n, given by the rake
121
Slice–Push Ratio
Shear plane
Tool rake face Vchip
Vshear Zc
α
Zs i Vwork
Figure 5-6 In orthogonal cutting, the shear velocity VS is along the line of steepest slope in the primary shear plane and the chip velocity VC is along the line of steepest slope of the rake face of the tool. In oblique cutting, VS is at an angle S (the shear flow angle) to the normal to the cutting edge in the shear plane, and the chip flows over the rake face of the tool at angle C (the chip flow angle) to the normal to the cutting edge.
angle measured in the plane normal to the cutting edge. It is the angle of greatest slope and is the same as the tool rake angle used in orthogonal cutting; n is variously called the ‘normal’, ‘oblique’ or ‘primary’ rake angle. Similarly, the angle of greatest slope of the shear plane or ‘normal shear plane angle’ fn is usually employed to define the inclination of the primary shear plane in oblique cutting. The resultant force Fres has components FC parallel with the velocity approach vector VW, FT perpendicular to the finished work surface, and FR perpendicular to the other two. FC is the ‘power’ force, FT is the ‘thrust’ force and FR is the ‘radial’ (sideways) force. These are the forces usually measured by a dynamometer. They are related by force resolution (Amarego & Brown, 1969; Shaw, 1984). It is not clear that the force vectors are co-linear with their respective displacement vectors (Shaw et al., 1952). In the plane of the finished surface, the components of the resultant force and resultant displacement are coincident; it is what happens out of that plane that is uncertain. In what follows we shall assume for simplicity that the force and velocity components are co-linear; it turns out to be an acceptable approximation when comparing theory and experiment. Another approximation often employed is ‘Stabler’s rule’, which says C i. The reader interested in the detail of oblique cutting of ductile materials should consult original papers and standard texts, in particular those by Amarego and Brown (1969) and Oxley (1989). By equating the external and internal work rates that include toughness as well as plasticity and friction, an expression is obtained for the power force FC (Atkins, 2006):
FC (1/Qshear oblique ) [(kwγ oblique )t Rw]
(5-15a)
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The Science and Engineering of Cutting
where
Qshear oblique 1 (F/FC )[sinφn /cos(φn α n )]
(5-15b)
with F the friction force along the rake face, and
γ oblique [cotφn tan(φn α n )]/cosηS
(5-15c)
(Note: there is a misprint on p. 87 of Amarego and Brown (1969) in their Eq. (4-37) for oblique, where ‘cos’ is written in the numerator instead of ‘cot’.) Alternatively,
FC /Rw=(1/Qshear oblique ) [(γ oblique /Z)+1]
(5-15d)
in which Z (R/kt) is the non-dimensional term incorporating the toughness/strength ratio and uncut chip thickness; (1/Z) represents the non-dimensionalized uncut chip thickness. Relation (5-15) has the same form as Eq. (3-32), but with different expressions for the shear strain and the friction factor Q. Without the fracture term, i.e. when R 0, FC (kwoblique)t/Qshear oblique is the plasticity-and-friction-only solution for oblique cutting first given by Amarego (1967). Experiments show that fn is essentially independent of the angle of obliquity, other things being equal, and is almost constant at sufficiently large t (sufficiently small Z) as in orthogonal cutting (Atkins, 2006). It follows that for a given workpiece material, f in orthogonal cutting and fn oblique cutting are the same. It is also found that S does not alter too much with obliquity, whence from Eq. (5-15c), oblique is approximately constant. For a given tool rake angle and friction, quasi-linear plots of FC and FT vs uncut chip thickness should therefore be approximately independent of i. Equation (5-15a) says that there will be a positive force intercept in plots of cutting force vs uncut chip thickness, and that it is a measure of the material toughness R. As in traditional analyses, the shear yield strength k is obtained from the slope of the plots. The power required for cutting over this same range of obliquity is also approximately constant. The corresponding quasi-linear plots of the sideways force FR vs uncut chip thickness do depend on i and, for given tool rake angle, increase as i increases mainly because Qoblique decreases. Experimental data from Brown and Amarego (1964) confirm these dependencies. If, instead of varying i at constant n, n is varied at constant i, plots of FP and FQ vs uncut chip thickness now depend on n, but this time experimental results in Brown and Amarego (1964) show that FR is apparently independent of n. At small t (large Z), fn is predicted to become smaller, so oblique becomes greater; but Qshear oblique increases at smaller t, and the net result is that FC vs t plots droop downwards near the origin and have an intercept of Rw since Qoblique 1 at zero t when f 0, exactly as for orthogonal cutting. For given material (R/k) ratio, tool rake angle, friction and uncut chip thickness, the primary shear plane angle fn may be predicted by minimizing the total work done or, equivalently, by minimizing Eq. (5-15a). Experimental data for oblique cutting give reasonable agreement with predictions (Atkins, 2006). The specific cutting pressure (‘unit power’) given by FC/wt becomes
FC /wt (1/Qoblique )[γ oblique k R/t] (k/Qoblique )[γ oblique Z]
(5-16)
As with orthogonal cutting, (FC/wt) in oblique cutting is expected to rise disproportionately at small t owing to the inverse-dependent final term on the right hand side of the relation (Section 3.6.7). Experiments confirm that the specific cutting power does indeed rise to large values
Slice–Push Ratio
123
at small t. But when data are analysed in terms of the plasticity-and-friction-only theory, the shear yield stress k (given by FCQoblique/wtoblique) must also rise to extremely large values.
5.3.1 Napier’s rotary cutting tool The analysis given above concerns a straight-edged tool that is angled to the direction of motion of the workpiece, in which slice–push results simply as a consequence of i (Figure 5-1B). There is no reason why a tool for cutting ductile materials should not have independent motion parallel to its cutting edge and produce enhanced . Napier invented such a tool in Victorian times. It consisted of a small chunky disc that was driven and could be rotated independently of the motions of the rest of the cutting machinery. Thus, on a lathe, it acted like a round-nosed tool that cut in the usual way with set depth of cut, feed and speed, but additionally revolved. Shaw et al. (1954) analysed the behaviour of rotary cutting tools using an equilibrium approach that, as explained in Chapter 3 and Appendix 1, is acceptable when toughness is omitted. To employ equilibrium for the internal work components requires that (FC/w R) be employed in place of (FC/w). The reduction in forces was not explained in terms of , rather in terms of i, but they are related. In Shaw and colleagues’ work with a driven rotary tool, the greatest was 2.5; given the much greater possible with disc cutters, it would be interesting to know the performance of rotary tools at greater . The ability of the rotary tool to become self-propelled is related to what happens with the disc cutter when the feed force goes negative (Figure 5-4B), the workpiece being grabbed by the wheel. In conventional cutting, the tool tip is in continuous contact with the workpiece, and is perpetually hot at commercial cutting speeds even with coolants and this can lead to tool failure. Some respite occurs with tools taking interrupted cuts: for example, it is possible to use high-speed steel (HSS) tools in interrupted milling if the tool is adequately cooled in the idle phase. In the Napier rotary tool, parts of the rotating cutting edge continuously move out of the hot cutting zone to cool before recutting and should therefore experience reduced wear and longer tool life. However, the device has the extra complication over ordinary tooling of requiring a very stiff holder that allows the tool to rotate and/or to be driven. Since modern tool materials have long lives and can withstand heavy cutting, the benefit of in-built cooling of the tool may no longer be important. Even so, given that slice–push reduces cutting forces, and that surface damage is less with high , there should be applications of the device for difficult-to-cut materials. Chang et al. (1995) remark that because the nose radius of rotary tools is much larger than conventional tools, feed rate has much less influence on the machined surface roughness. There is similarity in action between ball end-mills and the rotary tool (Section 4.1).
5.4 Guillotining Edges Instead of cutting the whole length at one fell swoop, it makes sense to incline the blade in cropping and perform the cut progressively. Although the total work required to cut an edge may be comparable in the two cases, forces in guillotining are lower since a longer stroke is involved to cut the same edge area. Equipment can be lighter and potential damage to the cut edge reduced. In contrast to unsteady blanking and orthogonal cropping, there is a steadystate region for much of the stroke in guillotining. Some guillotines have a cutter in the form of an undriven cutting disc; in others the blade may be straight and move through the workpiece always having the same orientation, as when
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cutting paper with an inclined razor blade and, of course, in guillotines used for execution (Chapter 11). In the case of a disc cutter, the forces may be derived using the analysis given in Section 5.2.3; in the second case, from the expressions in Section 5.2.2. Alternatively, the guillotine may have a long straight cutting edge pivoted at one end which is levered down through the workpiece. In this case the angle changes continuously, although some pivoted blades are curved with the intention, it seems, of maintaining the same angle at the point of cutting. Curved blades are also employed in hand tools like secateurs and scythes. Whether there are optimum shapes of curved cutting edge for least cutting forces is explored in Chapter 10. Whether the offcut is permanently deformed or not depends on the R/k ratio for the material. The gearing equivalence (that the work given by the force on the handle of the blade times its stroke remains constant) is lost when cuts are formed by ductile shear owing to the different planes in which offcuts curl. It is also lost when friction is significant. In orthogonal cropping the offcut is comparatively undeformed, except at the sheared edges, but guillotined offcuts of ductile materials are permanently bent owing to the inclined tool.
5.4.1 Floppy materials An office paper guillotine mounted in the frame of a testing machine was used by Atkins and Mai (1979) to determine the fracture toughness of thin sheets of materials. A graphical approach was employed, as displayed in Figure 5-7, where the forces for cutting are shown together with forces on second cuts, after the material has been parted, in order to establish friction. A third cut would establish the forces required to scrape the ‘set’ blade over the baseplate of the device; a set blade is curved and crosses the baseplate to give clean cuts. See Table 5.1. It is common experience that narrow offcuts of paper form into permanently deformed open helices, but that wider cuts remain flat for all practical purposes. Irreversibilities in paper come about from fibres that permanently slide over one another, as happens in simple tearing of paper where permanent curling may be produced. Analysis of the formation of helices is given in Section 5.4.3. In high-quality bookbinding, the edges are not guillotined but rather ploughed. A plough press in bookbinding is a vice in which the book is held while edges are orthogonally planed. 250
Load F (N)
200 B
A
150
100
C D
50
0
2
4
6
8 10 12 14 Deflection δ (mm)
16
18
20
Figure 5-7 Load–deflexion traces for guillotining a single layer of manila folder, 0.26 mm thick.
125
Slice–Push Ratio
The blade of a book plough has a convex half-round shape and a large rake angle (it is nearly parallel to the cut surface).
5.4.2 Scissors Figure 5-8(A)shows the geometry of a typical pair of scissors. The half-thickness t of the material being cut, the angular opening of the scissors during cutting, and the half-separation of the handles, are related (Atkins & Xu, 2005). Figure 5-8(B) shows the experimental results of Perieira et al. (1997) in which samples of palmar skin were cut by scissors. The specimens were some 1.4 mm thick. The upper of the two experimental force–displacement plots gives the cutting load, and the lower the forces to close the scissors after the cut has been made. The frictional contribution to the total force is about 25 per cent at all displacements. Table 5-1 Fracture toughness values determined from guillotine experiments. Material
t (mm)
R (kJ m--2)
Remarks
Interleaving paper
0.06
16.93
1 layer
0.24
10.32
4 layers
Copier paper
0.36
12.70
4 layers
Cardboard paper
0.48
13.60
Drawing paper
0.27
18.50
0° angle of cut with fibre direction
26.00
45°
29.00
90°
0.26
15.16
1 layer
0.52
13.63
2 layers
Waxed paper
0.18
15.10
2 layers
Rubber reinforced with cotton fabric
1.05
4.35
Aluminium foil
0.105
10.68
8 layers
Shim brass
0.05
53.18
1 layer, rolled direction
0.10
33.38
2 layers, rolled direction
0.05
63.93
1 layer, rolled direction
0.10
42.86
2 layers, rolled direction
0.12
41.45
rolled direction
0.11
41.39
rolled direction
0.11
18.47
rolled direction
0.11
25.95
rolled direction
0.22
21.41
2 layers, rolled direction
0.22
22.59
2 layers, rolled direction
0.16
42.06
rolled direction
0.16
47.15
rolled direction
Manilla folder
Copper foil
Shim steel
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The Science and Engineering of Cutting 4
Force F Displacement δ
3
θ 2t h
j
Scissors load (N)
y
2
1
0 0 A
B
10 20 Scissors displacement (mm)
30
Figure 5-8 (A) Geometry of scissors and cut material; (B) comparison between experimental results of Pereira et al. (1997) for cutting palmar skin with scissors and predictions of theory. The upper experimental curve is the total cutting scissors force; the lower is the force to close the scissors from the same 0 and indicates friction. The thick line is the prediction of the theory using R 2.4 kJ/m2; the experimental value estimated from the work area between the two force plots is some 2.4 ( 0.2) kJ/m2.
The thick line in Figure 5-8 is the prediction of theory for R 2.4 kJ/m2. Pereira et al. (1997), from the work area bounded by the two experimental force plots in Figure 5-8(B), gave R 2.4 ( 0.2) kJ/m2 along the skin creases and 2.6 ( 0.4) kJ/m2 across. The motion of some element in contact with the material along the blade is perpendicular to a line joining the element and the pivot. The slice–push ratio is given by the ratio of velocities along and across the edge of the blade and may be shown to vary along the blade. The biggest variation in will be at the beginning of a cut when is smallest, but even so the range is not marked. The variation is quite regular and it may be shown that use of the value at the mid-point of the blade is adequate in calculations. Large promotes low cutting forces at the beginning of a cut, when the mean is greatest, and the forces at the handles of scissors are low, particularly for thin slices. Later in the stroke, however, decreases, which increases the cutting forces. In addition, the effective lever arm of the cutting force decreases, so that the handle forces increase even more, as shown in Figure 5-8(B). Lucas and Pereira (1991) used both scissors and the guillotine to cut newsprint (sheet thickness 70 m) tested singly and in layers, applying the graphical method to the force– displacement results (compensating for friction and the set of the blade) to obtain toughness. Toughnesses by both methods are comparable but they show that values depend on the number of layers. Lucas and Pereira attribute the different R to fracture mode mixity; there is
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127
probably also an effect from the (blade clearance/thickness) ratio changing as the number of layers change. The effect of bluntness on scissor toughness, discussed in Chapter 9, has been studied by Arcona and Dow (1996) and Meehan (1999). Curved-bladed scissors, secateurs, pruning shears and specialist scissors such as pinking shears (with zig-zag notched edges to prevent fraying of cloth), hairdresser thinning scissors, nail scissors and so on, can all, in principle, be analysed in the way given here. Some scissors have spring-loaded blades whose natural position is open. That may be produced by a separate spring between the handles, but in old-fashioned sheep shears and cloth sampling shears, clever design enabled the two blades to be manufactured from one piece that crossed over in a spring loop behind the gripping positions. When permanent deformation of the offcut occurs, as with tinsnips or devices for clipping coins (in the USA, two bits one quarter), the analysis of the next section is required.
5.4.3 Ductile materials The cut edges of guillotined plates of ductile material are similar whether cropped with an orthogonal blade or with an inclined blade (Figure 3-24A), where a smooth tool indentation region lies above a rough separated region. The critical depths cr at the transition in cropping, guillotining (and in punching, Chapter 8) are very similar, despite questions about shear in different modes, and mode mixity. The critical depth depends on blade sharpness and clearance; similarly, an estimate of mixed-mode toughness is given by R kcr (Section 3.8) when the tool is sharp. In guillotining, deformation occurs only over the small region around that part of the guillotine blade currently cutting. The action in guillotining is a steady-state composite of the sequential actions which occur progressively as increasing travel of the blade in orthogonal shearing, but with additional features. On the one hand, (i) the offcut must bend to conform to the inclination of the blade (bending about EB in Figure 5-9A), which is not found in orthogonal cropping. On the other hand, (ii) the offcut in orthogonal cropping bends about the cut face (about an axis perpendicular to that in (i) above) and the rotation increases with blade travel (Figure 3-22C). Because blade contact in guillotining occurs over a range of indentation depths from first contact to down to f (the blade travel at which separation is complete), this rotation gets progressively greater through the contact zone. The shape BFB’ of the (hidden) crack profile alongside the blade in the deformation zone is very important in determining both sorts of offcut rotation. Since the type (ii) rotations start at B and get progressively greater until F is reached, the offcut comes away as if it were twisted (Figure 5-9B), although the deformation is really the result of plastic bending of cantilever elements of diminishing built-in thickness, not torsion. Twisting in ductile plates is most marked at offcut overhang widths less than the plate thickness in size. Narrow offcuts of paper deform into open helices in a similar manner, where the permanent deformation is a result of irreversible stretching of fibres that permanently slide over one another (paper bent or torn to a tight radius gives permanent curling). Helix formation occurs in most materials that can experience permanent deformation. In guillotines whose cutting angle does not change during the cut (disc cutters), the pitch of the helix depends on the width of offcut and thickness of material; the thinner the offcut the tighter the helix.
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The Science and Engineering of Cutting
α
A F Fʹ
Aʹ E
Bʹ
Tapered burnished land
Uncracked
B F
C Cʹ
Eʹ D Dʹ
A
Cracked B
Figure 5-9 (A) Sketch of plate deformation mode during guillotining. Note triangular zone BFB of uncut material, and interfacial crack profile, beneath inclined blade. Sideways bending rotations (corresponding with those in Figure 3-22C) vary along varying uncut cross-section BFB, and the offcut comes away twisted. (B) Progressive fan-like rotation of elements experiencing sideways bending in the guillotining zone. Burnished land not uniform as in Figure 3-22(C) but now tapers out to its full width.
When the cutting angle changes during the stroke (lever guillotines) the pitch will change along the cut and, with scissors, the range of pitch depends on what ‘gape’ (angle of opening) the scissors had at the beginning of the cut. Whatever the sheet material, the helical deformation is less marked as the offcut becomes wider. A consequence of the progressive twisting with permanently deformed offcuts is that the burnished land seen on ductile metals, which has a uniform width in orthogonal bar cropping, now tapers out to the full developed width at which the offcut parts company with the blade (Figure 5-9B). A work rate analysis for guillotining includes components relating to: (a) shear and fracture in the plane of the cut face (b) bending to the inclination of the blade (c) differential sideways bending and shear (offcut ‘twist’) (d) friction. Experiments on ductile metal plates suggest that these components are uncoupled. An approx imate analysis incorporating all work components was derived (Atkins, 1987a) but, except at small offcut width w, the forces and work associated with component (c) are comparatively small compared with the total forces measured experimentally. Also, the friction component (d) is small compared with other components (dry and lubricated cuts requiring virtually the same load). Consequently, a simplified algebraic expression is adequate to interpret the experimental results. Figure 5-10 shows how the steady-state guillotining force over the whole contact length is predicted from the non-steady orthogonal cropping force element by element. The guillotining force Fguillotine at w t is given by Atkins (1990)
Fguillotine /t (kψ )w (R*/tanα)
(5-17)
129
Orthogonal force/Unit width
Slice–Push Ratio
δcr
δf
δ Tool travel
Blade α
δ δcr x
dx
δf
Figure 5-10 Schematic of how the non-steady force vs blade stroke for orthogonal cropping is used to predict the steady-state force component of guillotining in the cut plane of intense shear. Blade movement to the left. Top surface of sheet is at top of diagram.
where t is plate thickness, R* is the effective fracture toughness in the plane of intense shear (see below), is blade inclination, k is the shear yield stress resisting bending under the blade, and w is width of offcut; ( [(l n)/8n]sin 2, with n the workhardening index in 0n) is a function connected with where the bent offcut becomes tangential to the guillotine blade. We see that a linear relation between Ftotal/t and w is predicted, the slope of which is k and the ordinate intercept is (R*/tan). R* is defined as the mean total work per area performed in the cut face from 0 to f. It thus sums the indentation plastic work up to cr and the subsequent plasticity and fracture between cr and f. Insofar as the cut face cannot be produced with given tooling without some combination of flow and fracture in the cut plane, R* is the effective fracture toughness that may be analysed separately from the accompanying plastic rotations and friction in guillotining. R* is not the same as the ‘true’ specific essential work of fracture (the fracture toughness) R, which is the work of fracture alone devoid of the remote flow component. The use of R* is similar to the effective toughness that applies when tearing ductile sheets (Mai & Cotterell, 1984). A burr is left on the torn edge and separation by tearing is impossible without. The specific work of burr formation is added to the fracture toughness to give the effective toughness for tearing. Figure 5-11(a,b) shows representative plots of guillotining force against offcut overhang width w for 6 mm plates of copper cut with blades of angles 10° and 25°. At w t, Fguillotine varies linearly with w according to Eq. (5-17), the different lines (all with the same slope) corresponding
130
25
Clearance 0.3 mm 20
–3 m
/k y
r y R*
15
20 25° blade 6 mm thick copper 83 VPN 15
–3 m
T
Clearance 0.9 mm
10 Intercepts 12–15-4 kN ∴R* = 352-453 kJ/m2
Slope = 160 kN/m ∴k = 133 MPa
∴(R*/ky) = (2.6–3.4)10–3 m
5
ry
o he
Guillotine force (kN)
/k y =
r y R*
Theo
10 2.6 x
–3 m
0
ry eo
/k y
R*
1 5x
The Science and Engineering of Cutting
Guillotine force (kN)
Theo
x 10 = 3.4
Increasing clearance
10° blade 6 mm thick copper 83 VPN
–3 m
0
/k y
R*
1 4x
Th
10
Slope = 375 kN/m ∴k = 125 MPa Intercept = 6.5 kN ∴R* = 505 kJ/m2
5
(R*/ky) = 4 x 10–3 m 0 0 A
6
12
18
Offcut overhang width w (mm)
0
24 B
0
6
12
18
24
Offcut overhang width w (mm)
Figure 5-11 Experimental results for guillotine force vs offcut overhang width at various clearances for 6 mm thick copper plate of 83 VPN following 440 0.27 MPa. Dashed lines give theoretical predictions. Linear approximation to theory at w t shows how R* and k may be determined from Eq. (5-17). (A) 10° sharp guillotine blade; (B) 25° sharp blade.
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Slice–Push Ratio
to different clearances between blade and baseplate. Increasing clearance at constant offcut width required less force, giving smaller back-extrapolated intercepts. Quoted on the graphs are the derived k from the slopes and R* from the ordinate intercepts. At small w, Fguillotine increases rapidly owing to the increasing importance of work of twisting of the offcut. Also shown in the figures as dashed lines are the theoretical plots using the full relationship for Fguillotine (Atkins, 1987) for the appropriate ranges of R*/k. The overall agreement between theory and experiment is satisfactory: that was the case also for other materials (mild and stainless steel, and brass) not shown here. The value of k for copper determined from Figure 5-11 (about 130 MPa) is that to be expected, at the level of bending strain under the blade, both from the 0n relation for copper and, simply, from the hardnesses H using the approximate relation H (56)k in consistent units. That is, for copper’s 83 VPN we expect k 830/6 140 MPa. In other experiments on 6 mm brass plate guillotined with both 10° and 25° blades, k 220 MPa; its hardness was 110 VPN, and we should expect k 1100/6 200 MPa. The clearance-dependent R* values of 350–500 kJ/m2 for copper (and 500–650 kJ/m2 for brass) are what would be expected on the basis of independent orthogonal cropping of the same material, as discussed in Chapter 3. R* decreases at increasing clearance owing to a changing mode of fracture from nearly all shear to a combination of shear and tension, with increasing bending across the clearance gap. The true mixed-mode specific essential work of fracture (the fracture toughness) R is given approximately by R kcr (VPN/6) cr in consistent units. For 83 VPN with cr 2 mm in 6 mm thick plate we expect R (830/ 6)2 277 kJ/m2. The larger values for R* reflect the inclusion of remote plastic work, which has nothing to do with the process of fracture. Experimental results for cr from guillotining thin sheets with sharp workshop shears with negligible clearance between blade and anvil are given in Table 5-2. Guillotining of thin sheets is related to the slitting of sheets, and its finite element method (FEM) modelling, discussed in Section 5.4.4. Guillotined cuts taken on plates tapering in thickness give linear traces of Fguillotine vs blade travel , increasing uphill and decreasing downhill (Figure 5-12). In both cases, (Fguillotine/t) is (Fguillotine/) are constant since t is proportional to the horizontal length of cut or equivalently to the vertical blade travel . This suggests a way of obtaining lots of results from one experiment and is a useful experimental trick for orthogonal cutting too. The idea has also been employed in milling by Sinn et al. (2005) (see Section 4.7.3). Inspection of the edges of cut tapered plates shows that cr increases with increasing plate thickness (and vice versa)
Table 5-2 Experimental results from guillotining thin sheets. Material Low-carbon steel
Thickness (mm)
Hardness (kg/mm2)
cr (mm)
R (H/6)cr (kJ/m2)
2
200
0.17
57
5
200
0.40
133
1
158
0.30
79
3
158
1.00
263
Brass
6
110
0.95
171
Stainless steel
2.4
175
0.40
114
Soft low-carbon steel
Source: Atkins (1988b).
132
The Science and Engineering of Cutting Tapered copper 10° blade w = 18 mm
Guillotine force (kN)
15
25° blade w = 18 mm
10
0 5
10
20 30 40 50 Blade travel (mm)
60
B
0 0 A
10
20
30
Blade travel (mm)
Figure 5-12 Guillotine force vs blade travel for cutting tapered plates with increasing thickness: (A) 10° blade; (B) 25° blade. Theory predicts a linear variation. Discontinuity in slope for the 10° blade concerns a change in the relative sharpness between blade and current plate thickness; early in the stroke there is much shear before separation.
but always remains the same proportion of the thickness, i.e. cr t. Were cr a fixed size, it would suggest that plates whose thickness was smaller than cr could not be cut. But changing cr has strange implications for the toughness in shear: if hardness is fixed, and cr alters, then R ( kcr) must alter with thickness. It might be expected that R ought to have one value characteristic of the material and its thermomechanical state, as in ‘normal’ fracture mechanics testing. (The well-known variation of RI with plate thickness caused by different plane stress/plane strain constraint is something different; see Atkins & Mai, 1985.) It is believed that changing cr and hence changing R comes about because the plastic deformation zone (shear band) is set up through the whole thickness from the outset of cutting. In the usual type of fracture mechanics testpiece, the crack tip zone is limited in its forward extent and does not reach the back face of a specimen until late in the test. Metallographic examination of guillotined tapered sections reveals that h, the width of the shear band, changes with thicknesses in the same way as cr, i.e. h t (Atkins, 1988). The thickness of primary shear bands in orthogonal cutting is also found to be proportional to the length of the slip band (Stevenson & Oxley, 1970–71; Childs et al., 2000). Thus when both cr and h vary directly in proportion to t, cr (cr/h) remains constant for all t. An alternative, equivalent, interpretation of R varying directly with plate thickness is that R/h is constant since h varies in proportion with t. (R/h) represents a critical plastic work/volume for cracking for particular blade sharpness. (There is an assumption of uniform deformation in the shear band which will not be quite correct as the local strains near the cutting edge where the fracture process zone forms will be greater than average.)
Slice–Push Ratio
133
The variation of R with t has implications for punching holes and for the scaling of energies in plug formation in armour penetration, and perhaps for scaling teeth, and is discussed further in Chapters 8 and 13.
5.4.4 Slitting and shredding sheets A type of office paper guillotine consists of an undriven sharp cutting wheel fitted to a block that slides on a long bar parallel with the direction of cut. Sheets are cut by pushing the block holding the wheel along the bar to cut off paper hanging over the edge of the baseplate. On production lines, thin materials of all sorts (floppy to ductile) are slit into different widths by pulling the sheet through a similar sort of wheeled device. There can be multiple cutting wheels to produce many strips from a wide roll. Paper shredders operate in a similar fashion. The problems of slivers and burr on the edges of trimmed sheets of metal, which were discussed in Chapter 3, can occur in slitting. Ma et al. (2006) and Lu et al. (2006a) apply Li’s ideas of slitting at an angle to eliminate this difficulty, and it is found that the arrangement is relatively insensitive to clearance and gap between the rotary knives. The process has been simulated with FEM (Ghosh et al., 2005) in which a variety of separation criteria were tried. Gurson–le Rousselier porous plasticity models were not successful at replicating the experimental results for the orientation of the shear plane, height of burr and so on. What worked best was a criterion of critical equivalent plastic strain (affected by hydrostatic stress in the separation zone). The work ought ultimately to be able to predict the critical depth cr , at which separation begins in a ductile sheet, for different combinations of clearance, tool sharpness and material properties.
5.4.5 The can opener A canister is any sort of small container with a removable lid for storing things such as tea or coffee (from the Latin canistrum for wicker basket). The familiar ‘can’ or ‘tin’ in which food or drink remains fresh for a long time is completely sealed and has to be opened in order to consume the contents. Sealing was originally done by soldering, later by wrapped-around joints. Steel sheet for canning was plated with tin against corrosion, and helped soldering. Advice on the opening of Victorian hand-soldered big cans (that could weigh up to 7 lbs empty) was to use a hammer and chisel. Many modern cans have ring-pull devices by which the lid, or a portion of the lid in the case of beverage containers, is removed. Steel corned-beef cans once had a flap on which a key could be wound to open them. The torque required for this elastoplastic fracture mechanics operation is given in Atkins and Mai (1985); the solution for ring-pull cans following prescribed crack paths is similar. The reason why the parallel tear sometimes runs to a point before completion around the circumference of the can is discussed in Chapter 15; other types of sealed tin do not have in-built release devices and a can opener has to be used to reach the contents. There are various designs but all involve first making a hole followed by propagation of the slit around the rim of the can. A basic type of can opener has a thin sharp point and cutting edge that is used to stab a hole in the lid by hitting with the palm of the hand, after which a series of discrete leverings cuts around the lid to give a wavy edge to the removed lid. In another design, the knife-indenter is attached to one of a pair of hinged handles and the initial piercing is made by squeezing the handles. At the same time the device latches on to the underside of the
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rim by means of a toothed wheel. Winding a handle attached to the wheel drives the cutting edge around the rim progressively detaches the lid. A similar device works in a horizontal plane to remove the lid by cutting around the top of the cylindrical wall of the can rather than cutting the lid just inside the rim. The mechanics of cutting around the rim of a can made of ductile sheet is similar to that of guillotining where the length of cut is the circumference of the lid. However, there is more constraint across the lid than for the overhanging free edge of simple guillotining. A simplified analysis might go as follows. The deformation of the lid around the initial pierce consists of plastic bending under the inclined knife (i) around the circumference (as in guillotining) and (ii) in the radial direction of the lid. Both plastic bend radii are determined by the sloping geometry of the knife, and may be represented by a single effective radius of curvature . Observation of can opening suggests that the radial distance over which plastic bending takes place is approximately equal to . A rotation d of the toothed wheel gives a circumferential movement (D/2)d, where D is the diameter of the lid, and hence an incremental fracture work requirement of Rt(D/2)d, where t is the thickness of the lid. The incremental volume of material plastically bent is [(D )t/2]d. The mean bending strain is (t/4), so for a yield strength y, the incremental bending work is y(t/4)[(D )t/2]d. These two components of internal work are provided by rotation of the toothed wheel. Hence for torque T
Tdθ Rt(D/ 2)dθ σy (t/8)[(D ρ)t]dθ
i.e.
T Rt(D/ 2) σy (t/8)[(D ρ)t] or (T/RtD) 0.5 (σy t/8R) [(1 (ρ/D)]
(5-18)
The torque is a steady value. This treatment ignores friction and indentation of the toothed wheel into the underside of the rim, and is unable to predict the value of . However, it may be possible to couple the fracture work and the plastic bending work via the critical crack opening displacement as described in Section 8.6.1 and thus obtain a more comprehensive answer.
5.5 Drills, Augers and Pencil Sharpeners Shards from flint knapping in the form of parallelepipeds were once used for drills. They were inserted into a cleft in stick, and glued in place with tree resin heated up with charcoal, and then bound tightly with thread made from nettle stems (Sim, 2008). The string of a bow was wrapped a few times round the stick and, by reciprocating motion, rotated the drill clockwise and anticlockwise in succession. Glass may be drilled using a copper rod with emery powder, and when holes are required through gemstones (e.g. beads), a small rotating rod or tube with a diamond tip is used to drill through the stone, sometimes aided by a slurry of silicon carbide and coolant. The process is ‘rotating scratching’ and depends on the factors described in Chapter 6, including point geometry, and the toughness and hardness of the material. A wimble is a marbleworker’s brace for drilling; the word is also used for a device in mining for extracting spoil from the drilled hole. A proposal by Jagger (1897) to determine the hardness of minerals used drilling. A diamond ‘point’ (of undefined geometry) rotated on an orientated mineral section under uniform rate and uniform weight. The number of rotations to penetrate to a given depth was
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135
found to vary with the resistance of the mineral to abrasion by diamond. What was being measured appears to have been some composite behaviour of the toughness and strength. The simplest type of drill having a controlled geometry is the spade drill, as used in a braceand-bit woodworking. The drill, of full radius rdrill, consists of a spike or screw thread at the tip which indents or screws into the workpiece and keeps the hole central, with flat pieces extending sideways, the bottom edges of which cut the base of the hole and the corners of which cut the sides. Every element of the bit cuts as if it were a zero rake angle tool (the edges of the spade are angled underneath to provide clearance for the normal direction of drilling, i.e. clockwise from above). The length of an elemental chip produced by one cutting edge of the spade depends on how far it is from the centre, being given by r, so is zero at the centre and rdrill at the outside. The tangential speed of cutting, and the steepness of the helical path followed by an element of the cutting edge, also vary with radius. Since the elemental chips are all joined up, the short portions of the chip from near the centre of the drill are stretched (and may fracture), while those from the outside are compressed and may buckle. The whole chip therefore must curl in space. Experiments where a spade drill is used on a block of butter demonstrate that the offcut rises in a circular fan shape to cover the face of the drill. When drilling wood with a spade bit, the torque will fluctuate owing to grain orientation and anisotropic mechanical properties. In a single rotation, a drill will encounter a variety of orientations and the effect of this is revealed in the surface quality of a drilled hole. The growth direction of the tree from which the workpiece has been cut may be identified from the appearance of the surface of a drilled hole (particularly large-diameter holes): a smooth surface is produced where the cutting tool has turned in the same direction as that in which the tree grew, and a rough torn-out surface results on the opposite side of the same hole. Drilling energy alters as the drill passes through the early and late wood in the growth rings. Such effects are experienced in other woodworking operations such as turning on the lathe. As pointed out by Effner (1992), however, the effect is noticeable only for a hand-held tool such as a brace and bit. An auger is an ancient drill made by twisting a narrow strip of steel into a helix. One end is sharpened like a chisel: there is a point in the middle to mark the centre of the hole and spur knives to cut cleanly round the outside of the hole in advance of the main blade. The spur cutters act like the coulter on a plough (Chapter 14). Material removed from a hole made by an auger is lifted out along the helix to the surface. Unlike augers, spade bits do not have spur cutters on the outsides, resulting in rough holes, and removal of debris from a deep hole made with a spade drill can be difficult. The holes for wooden pegs or trenails, which held together wooden-framed buildings and ships, would have been made by augers. Unlike a spiral (properly called helical) staircase, there is no central core to stiffen the tool in the simplest sorts of auger. Also, simple augers with a single helix cut on only one side of the axis so that the action is unbalanced. With two or more helices balance is restored and the auger runs centrally. The present-day parallel-sided twist drill was a development of multiplestart augers: debris is discharged along channels (flutes) formed out of the solid body of the twist drill. The point of a simple twist drill has two main cutting edges, each of which has a rake angle and relief angle (with indexable drills the cutting edges are formed of separate carbide inserts). Were these edges orientated radially, they would meet at a point, but such a point would be likely to break off in use. In practical drills the cutting edges are arranged as shown in Figure 5-13, and have a secondary cutting edge (the straight chisel edge web) at the centre. The secondary edge, while solving the problem of tip strength, creates other problems. The rake angle of the secondary edge varies from perhaps 60° at the centre to say 10° at its ends (Black, 1961), and the cutting speeds similarly vary from very low near the centre to faster at the outsides. The centre of a drill appears to pierce the workpiece by a different
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Web thickness
Figure 5-13 Tip geometry of flute drill.
action from the actual cutting edges (Shaw, 1984). Built-up edges are often observed in drilling operations with steels and other metals (e.g. Ventakash & Xue, 1996). All these observations indicate that drilling is overall inefficient. Again, because the ends of the secondary edge of twist drills are never perfectly parallel to surface of the workpiece, the drill tends to wander and so produce non-round holes that are also not straight. This problem is absent in a trepanning saw, but not in a hole saw with its own centre drill, nor in flycutters. A pencil sharpener is an inside-out drill that cuts a taper on the end of a pencil to reveal the pencil ‘lead’. In the days of 78 rpm records, gramophone needles were made from steel or from bamboo (fibre needles) and were sharpened with a pencil-sharpener-like device. (All needles tended to wear the grooves, as discussed in Section 6.4.) The angle of the taper of a pencil sharpener corresponds to the point angle on a drill. The nearest device to the hollow drill equivalent of the pencil sharpener is perhaps a type of gimlet. As with drills and augers, the action of a pencil sharpener relies on both axial and rotary motion with associated thrust/ feed force and torque. The combined work done in one revolution by the axial force Faxial and the torque T performs (i) toughness work in separating the shaving; (ii) work against friction; and (iii) deformation in the shaving. Experiments show that the wood shaving is thin, floppy and broken up, so for simplicity we neglect work component (iii). Figure 5-14 shows a pencil sharpener with a taper angle p. The radius of the pencil is r and the feed per revolution of the pencil into the device is f. The thickness t of the shaving is t fsinp (from equating the volume of pencil r2f consumed in one revolution to the conical surface area r2/sinp times the thickness t). In a pencil sharpener, the same depth of cut is a bigger percentage of the circumference at the conical tip than further back. The feed f has components fsinp perpendicular to the cutting edge and fcosp parallel to it. There is therefore slice–push of magnitude cotp. Insofar as the motion can be considered similar to the oblique cutting in Section 5.2, an element of cutting edge of length ds dr/sinp experiences a force dV perpendicular to the blade (circumferentially to the device) and dH along the blade. The incremental component dH may be resolved into an axial component dHcosp and a radial component dHsinp. For simplicity, consider frictionless cutting where the deformation in the shavings is negligible. We have
dV Rds/(1 ξ2 ) Rds/sinp Rdr/sin2 p
(5-19a)
and
dH Rξds/(1 ξ2 ) Rcospdr/sin3p
(5-19b)
137
Slice–Push Ratio Faxial, f
Cutting
edge
T, ω
V H
p
Figure 5-14 Geometry of a simple pencil sharpener having a single blade aligned along the surface of a cone of slope angle p. The pencil is fed into the device with force Faxial and feed f, and rotated with angular velocity by torque T.
where R is the toughness of the wood of the pencil. The axial component dHcosp is dFaxial, where Faxial is the thrust force feeding the pencil into the device. The radial component Hsinp is reacted by the walls of the conical cavity. It follows that Faxial Rrcos2 p/sin3p
(5-20a)
ignoring the difference between wood and graphite at the tip. The element of torque dT rdV, whence
T
2
∫ Rrdr/sin p Rr
2
/ 2sin2 p
(5-20b)
These results have been obtained without writing out the work equation because the use of gives a connexion between Faxial and T. The expression for Faxial suggests a minimum value at p 51° (very high) in this special case. Friction may be included by employing the equations of Section 5.2.2 that will modify an optimum blade angle. When drills cut ductile materials, the drillings are skewed across the cutting edges of the drill and permanently curled into helices, because of obliquity. The slice–push interpretation of pencil sharpening may be applied to the drilling of ductile solids in terms of the obliquity angle i using tani and cotp, using the relations for cutting forces at the oblique edge given in Section 5.3. This would be expected to predict the sorts of Faxial and torque relations depicted in Figure 5-15 where thrust and torque vary almost linearly with feed for drilling polyethylene and acrylic resin with twist drills (Kobayashi, 1967); similar results are found when drilling cancellous bone (Shuaib & Hillery, 1995). Flutes of drills may vary in number from one to three and must be free-flowing so as not to become clogged. Drills intended for cutting metals have ‘slower’ flute helix angles, i.e. are less steep than the flutes of woodworking drills since, other things being equal, wood is easier to remove from the cutting face. Typical flute angles are about 36° (quick helix), 27° (normal)
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The Science and Engineering of Cutting 0.8
Torque (Nm)
0.6
Rotational drill speed N (rpm) 2000
0.4
1000
Thrust
4000
2000
0.2
1000
Torque
4000 0
0.05
0.1
0.2
Feeding distance per revolution f (mm/rev)
Figure 5-15 Change in torque and thrust with feeding distance per drill revolution when polyethylene having a density of 0.95 g/cm3 is drilled. Drill diameter was 8.1 mm, helix angle was 27°, and point angle was 120° (after Kobayashi, 1967).
and 14° (slow). Since the cutting edges of drills remain in contact with the workpiece until the job is completed and are down at the bottom of the partially drilled hole, cooling may be difficult and flutes will clog. As with the clearing of saw gullets (Chapter 7), periodical removal of a drill out the hole helps to unclog flutes. The temperature rise may be significant if the feed rate is too high. The need to have adequate channels along which swarf can escape and not jam, yet to retain sufficient torsional strength in a tool, causes problems in operations such as reaming (enlarging previously bored holes), tapping (cutting threads) and broaching (cutting slots), where cutting forces may be high. This means that some of these operations have to be performed relatively slowly. To ensure a ‘clean’ hole when making a hole right through a workpiece, the design of the point angle of drills is important. With wood, in particular, there is the danger of leaving a torn edge on the undersurface of the workpiece when the drilling thrust force is high: a total included tip angle of not more than 60° minimizes this effect. With hand-driven devices, when withdrawing the drill, it is customary to keep the rotation the same as when making the hole. The type of woodworking drill having a flat end, a centre point and spur cutters on the periphery was originally intended to cut dowel holes in the furniture industry but is now widely used for all sorts of jobs. To force blunt drills to cut requires a high thrust force with its associated problems of uncontrolled breakout when drilling a through hole: instead of all the displaced material being removed as chips, some may appear in a lip around top edge of the hole as a result of having to press so much – it is like a hardness impression with a rotating indenter. Miller et al. (2006) discuss so-called friction drilling in which a rotating conical disc simultaneously pierces and flanges sheet material. Owing to the action of drills by which the whole cross-section of the hole is deformed, the material drilled out of a hole is waste, unlike punching holes in materials where either the hole or the disc or both may be used later. There is some waste when using hole saws, but
Slice–Push Ratio
139
the cut-out discs can be used for other purposes. A form of hollow drill is the cork borer that removes a plug whole (Chapter 8), like an apple corer. The sharp bottom edge of these devices separates material by splitting/cleavage, not by shear plane formation through the chip. Cork borers have slice–push given by /f, where is the tube radius and the angular velocity. It is possible to drill apples, but rather silly as they are easily punched through by hand. Geological core samples are made with hollow drills. Hollow (cannulated) drills are used in surgery (Chapter 11) but for a slightly different reason, where they slide on prefixed stainless steel guide rods that direct the cutting edges to the right place. Some materials, such as cellular foams, present special problems for drilling. With metal foams, to make blind holes into which studs or screws can be inserted, flow drilling is employed where a polygonal ‘drill’, rotating at over 200 rpm, is pressed into the face of a sheet or plate. The cell material ‘plastifies’ and becomes easily formable, so much so that material from the face flows into the hole. Depths of holes are typically about three to four times the thickness of the face material (Seeliger, 2001).
5.5.1 Civil engineering drills Fence posts can simply be banged into the ground or a hole dug with a post-hole digger. This device is a steel rod, turned by hand, at the bottom of which is fixed a flat disc having two cutting edges and flaps that permit spoil to pass up through the disc; when the device is lifted up out of the hole, the flaps close and the debris comes away. It is not always possible to lift out spoil in this way and special narrow spades to reach down into deep holes form part of the kit. Augers are used to make holes for telephone or electricity poles. They do not need the circumferential cutting spurs found in wood augers because the helix just scoops out loose earth like an Archimedean screw. Civil engineers use large augers for taking foundation samples, preparation of some types of piling and in tunnelling. In large-scale drill-like machines, a cutting head rotates about a central axis while penetrating parallel to that axis. Cutting edges can variously be blades, discs, abrasive grains, studded or toothed rollers. In order to cut a hole, material has to be removed across the whole radius. Instead of a continuous cutting edge as in a drill, large machines employ a series of smaller tools arranged in a staggered pattern. Overlapping knives give smooth and unbroken cutting like the wide cuts of a cylinder lawnmower. As the cutting head rotates and penetrates at constant speed, the trajectory of every tool point is a helix, the angle of which becomes steeper at locations closer to the axis of rotation; along the axis, it is theoretically infinite (Mellor, 1975–81). Since tool linear speed varies with radius, a finite-width tool located close to the axis of rotation has different speeds at its two ends that travel along helical paths of quite different inclination. Furthermore, since the depth of cut of every individual tool should strictly be taken with respect to its helical trajectory, it follows that the depth of cut varies with radius, and that tool penetration normal to the work face also varies with radius. The behaviour of a cutter near the centre of the hole bears little relation to orthogonal cutting on which modelling is usually based. In practical equipment, a central core of uncut material is sometimes left and allowed to break off periodically or some sort of ‘pilot bit’ like the point of a spear is employed. As modelled earlier in this chapter, and with cutting devices for earth that are on chains (Chapter 14), the resultant force in civil engineering drills is usually resolved into components that are normal and parallel to the work surface, even though the head thrust down the axis of drilling FT is not strictly parallel to the axis of rotation since the tool cuts a helical path. FT is provided either by deadweight and friction, or by reacting the machine against the surfaces
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of the hole. When deep drilling in rock, the weight of the drill string can exceed the desired head thrust, so the drill string has to be held in tension at the upper end. For devices such as rock augers (where tools follow shallow helical paths), the ratio of axial velocity to the cutting velocity is about 1:100 (rather like a pencil sharpener) so that the head thrust is given by the sum of the axial components of force on every cutter. The broken-up debris/spoil of some materials occupies a greater volume than the solid from which it was cut; for example, loose crushed stone has about 45 per cent voids. Thus in deep drilling with augers, the helical flight channels may have to accommodate a greater volume than that removed by the cutter. In reports published by the US Army Cold Regions Research and Engineering Laboratory (CRELL), Mellor (1975–81) reviewed the relationship between head thrust and cutting head diameter in actual large machines, to see whether there was consistency in design. Since installed power often drives the systems for clearing and hoisting spoil, however, it was difficult to get a clear picture between, for example, the relative cutting power and hole size. In oil-well drilling, the spoil is usually removed by circulation of special fluids (muds), with up-hole velocities between 25 and 60 m/min. In such conditions rotary power is consumed largely in shearing the drilling fluid and by hole-wall friction, rather than for actual cutting.
Chapter 6
Cutting with More Than One Edge Scratching, Grinding, Abrasive Wear, Engraving and Sculpting Contents 6.1 Introduction 6.2 Scratching of Low ER/k2 Solids 6.3 Scratching of High ER/k2 Solids 6.4 Grinding and Abrasive Papers 6.5 Scratch Hardness/Scratch Resistance 6.6 Scratching of Thin Films and Coatings: Pencil Hardness 6.7 Erosion 6.8 Definitions of the Coefficient of Friction 6.9 Engraving, Writing Tablets and Polishing
141 143 144 150 157 160 163 165 166
6.1 Introduction Cutting in previous chapters has concerned blades that overlap the cut surface. Blades narrower than the surface will form a slot or groove. A ball bearing rolling over the hard surface of its races forms a minute elastic groove that recovers after the ball passes by. The much larger deformations when one walks or rolls around on a mattress, water bed or air-filled ‘bouncy castle’ also recover. Above some limiting contact stress, but still within the elastic range, various types of crack appear to the side, and below, the track on the surface of brittle bodies. The behaviour depends on the geometry of the contact zone (‘round’ or ‘pointed’). Unless the cracks intersect there is no material removal but, even so, the surface may be deemed to have been ‘damaged’ by formation of regular patterns of cracks in the wake of the slider. When cracks do intersect, chips are detached by spalling and so result in wear, either on first passage of the contacting bodies or after many traversals, aided by additional cracks forming by release of residual stresses on unloading as the slider/roller moves on. Instead of multiple cracks, a permanent groove may be produced when scratching with a pointed tool, as relied on by the glazier when marking the line along which glass is snapped to size by bending; similarly in the scoring of silicon discs preparatory to snapping (dicing) into chips. A diamond ring can be used to produce graffiti on glass by scratching. Rough coatings of plaster are scratched before being quite dry to aid mechanical adherence of the next coat. Permanent grooves involve elastoplastic deformation, to which the scaling laws of Chapter 4 apply, so that grooves are formed by ductile mechanisms at depths of cut small compared with the length parameter (ER/k2); at deeper depths they are formed by elastoplastic mechanics giving discontinuous chips; at even deeper depths elastic chipping will occur (Taylor, 1949). Production of a V-shaped groove involves cutting with two straight edges; a round-shaped grooved is formed by a tool having a continuously curving edge. Tools with more than one edge have application beyond cutting grooves. Consider reducing the thickness of a large ductile plate. Cutting the waste material off in a single orthogonal (or oblique) cut, or even in a series of thinner cuts, would require a special wide tool and could involve big forces. Instead, it is found Copyright © 2009 Elsevier Ltd. All rights reserved.
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The Science and Engineering of Cutting
better to use a tool having two cutting edges and to cut off a series of ‘ledges’ progressively from one edge of the workpiece to the other (Figure 6-1), the tool being fed across the width between each stroke (shaping). Such tools do not overlap the workpiece or pass completely through thickness. A spade used to dig the garden does not overhang the plot and cuts on its sides as well as bottom; similarly for a trowel, and for a knife used for surgery or stabbing. Tools with more than one cutting edge also often solve problems in ductile cutting where offcuts foul the tool, and where the handling and removal of swarf become inconvenient. Other familiar devices having more than one cutting edge are saws, the ribbed heads of orange juicers, dentists’ drills, milling cutters and so on. In Figure 6-1 the sloping (transient) surface is cut by the side-cutting edge of the tool, and the finished surface by the end cutting edge. Should the cutting forces be high, chipping of the tool may occur at the junction between the two cutting edges of the tool, so a nose radius is usually employed between the two. The finished surface comprises a series of scallops having the radius of the nose of the tool. The apparent flatness of the surface is improved the smaller the feed compared with the depth of cut. The shapes of tools having a number of cutting edges are defined by the tool signature, which is the combination of angles to which the tool is ground. Galloway (1953) discusses the minefield that is the nomenclature of cutting tool geometry. The geometry of the chip when cutting with more than one edge is complicated because of the connected three-dimensional flow arising from two edges and a nose (Colwell, 1954; Young et al., 1986; Nakayama & Arai, 1992; Arsecularatne et al., 1995). This is the case even when the feed of a round-nosed tool is linear (as on a lathe); it is even more complicated when the tool itself follows a curved, or three-dimensional, path, and/or is rotating as well. This sort of thing has received wide discussion in the metalforming literature (see standard monographs). A scratch is the same as a groove, but ‘scratch’ often has the implication of ‘unwanted’ and ‘damaged’. Scratches are not wanted when the appearance of the surface is marred (furniture; car paintwork; grooves left across wooden floors when heavy furniture on castors is moved) or when damage results (non-stick coatings on frying pans gouged by too-hard kitchen utensils). Yet tampering with cricket balls relies on scratching with the fingernails or scratching with dirt in the pocket. By what mechanism do grooves form over time in the runners of drawers in wooden furniture; are they cut or rubbed? In contrast, making permanent grooves and other shapes is the aim when machining threads, carving wood, sculpting stone, decorating pottery, engraving, pargetting (exterior
Transient surface Finished surface
Depth of cut
Fe e
d
Figure 6-1 Cutting with two cutting edges (shaping in this case).
Cutting With More Than One Edge
143
plasterwork on houses bearing designs in low relief) and so on. Highly polished surfaces have closely spaced shallow grooves differing only in degree from the grooves produced by sanding, grinding and abrasion. Samples for metallographic investigation are prepared with abrasive papers and slurries of progressively diminishing coarseness. Ancient writing on clay tablets was formed in this way: tabula rasa (a word now used for an empty mind or place with no history) is the Latin for a writing tablet that has been scraped clean, and a palimpsest (from the ancient Greek ‘to rub again’) is a tomb brass from which the original writing or design has been erased to make room for something new. Again, a codex (from caudex, a tree trunk) is a book of wooden tablets with words inscribed into a block of inlaid wax – not carved out of stone as averred by Asterix, Mount (2007). Interestingly, the use of ‘chasing’ to describe various forms of engraving comes from ‘enchase’, meaning to place gemstones in an ornamental setting.
6.2 Scratching of Low ER/k2 Solids Under static axisymmetric Hertzian contact (ball indentation), a small zone of tension develops in the form of an annulus on the surface and, at sufficiently large loads, induces a characteristic ring crack that may subsequently develop into the frustum of a cone (Roesler, 1956). The magnitude of the tensile stresses around a ball indentation is increased when the contact is slid along the surface (Hamilton & Goodman, 1966). Preston (1922) demonstrated that the Hertz surface ring crack becomes a series of regularly spaced partial ring cracks (horseshoe-shaped) in the wake of the slider that propagate down into the body. This is easily demonstrated by drawing the back of a spoon over the surface of table jelly. Hills et al. (1994) showed that the spacing of such cracks is not dynamically excited (by loading systems that are soft in the direction of sliding) and hence are independent of sliding speed. Indentations and grooves may be made by pointed indenters in brittle solids that would shatter under tension, owing to the significant compressive hydrostatic stresses set up beneath the indenter, which suppress crack formation (Marsh, 1964). The transitions of Chapter 4 between elastic, elastoplastic and plastic chip formation are altered since whereas yield criteria for plastic flow are unaffected by hydrostatic stress state, cracking is affected. The material ER/k2 is still a controlling parameter, as shown by the difference in behaviour between glass and rock salt (Tabor, 1956). Eventually with pointed indenters localized crushing occurs, and lateral cracks form with plastic flow beneath the indenter. This induces residual stresses and residual strain energy on unloading, resulting in the formation of additional median cracks. A relation for abrasive wear by brittle lateral cracking is given by Hutchings (1992). Crush behaviour is very sensitive to local conditions near the point of application of the load. A crater of loose fragments is formed, and the normal and lateral forces developed as loading is increased further may be sensitive to local asperities and depressions of the specimen and also of the tool. (When comminution down to very fine sizes is achieved by crushing, the competing processes of agglomeration may occur: limiting sizes below which materials cannot be reduced further in crushing are a consequence of energy scaling; Chapter 4.) Whatever the shape of a tool, if indentations and scratches are made too close to the edge of a brittle workpiece, chips of material are spalled away, since the induced crack system propagates to the edge when it is sufficiently close. Chai and Lawn (2007) investigate edge chipping from sharp contacts by indenting with a Vickers pyramid at varying distances from a free edge of a brittle plate. The process of nibbling with a plier-like tool when cutting awkward shapes in ceramic tiles is similar. Scieszka (2005) describes a rig for assessing wear by
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edge chipping as it may relate to the performance of rock drilling bits and other types of cutting tools. The chiselling of brittle materials at various angles to the surface is discussed in Section 8.2.1, along with the doubled-edged ‘hack axe’ used by masons in mediaeval times. Scratching experiments, using pointed tools at various inclinations on polymethylmethacrylate (PMMA) at different controlled depths of cut, replicate the different types of elastoplastic chip, and transitions between them, that were seen for orthogonal cutting in Chapter 4 (Frew, 2008). Brittle spalling occurs at deep depths of cut relative to the length scale given by (ER/k2); discontinuous chips at smaller depths of cut; and continuous ribbon-like chips at very small uncut chip thicknesses. At large negative rake angles and small depths of cut there is a tendency towards prow (standing wave) formation, as discussed in the next section. Vanishingly small prows may be connected with the experimental observation by Briscoe et al. (1996) that sliding tracks on surfaces produced by spherical contactors seem to be merely smoothed or burnished at the smallest depths (or rather at the smallest loads in their loadcontrolled experiments). A similar observation was made by Malak and Anderson (2005) in orthogonal cutting of foams (Chapter 4). In complete contrast, a large-scale example of scratching is ‘ice gouging’ on Arctic beaches, where onshore winds drive large ice masses over the surface to produce continuous grooves and gouged-out scars on the seabed and shoreline. Such grooves can be 5 m across by 100 m long, Palmer (2009). There is some similarity with how glaciers gouge out tracks (Croll, 2008). To perform controlled cutting of low (ER/k2) (‘brittle’) materials, very small depths of cut have to be employed so that material is removed by the ductile deformation processes that occur at much larger depths of cut in materials having larger (ER/k2), i.e. as explained in Chapter 4, the same deformation mode occurs when the (ER/k2t) values are comparable in the two cases. This is the basis of micromachining glass, ceramics, and hard crystals like silicon and gallium arsenide (e.g. Puttick et al., 1989). Instead of controlling depth of cut, it is often more convenient to alter the load. There has long been interest in the turning of optical glass (e.g. Shinker & Döll, 1987). High-resolution aspherical lenses, which are used for placement of components by photoresist techniques on silicon microchips, are manufactured in this way. The alternative of multielement spherical lenses is costly, and their manufacture by free-abrasive grinding and polishing operations is time consuming (Gee, 2009). Furthermore, only flat, cylindrical and spherical surfaces can be produced by lapping in the traditional manner. Needless to say the cutting devices have to be extremely stiff to obtain satisfactory results when taking micrometre and nanometre size depths of cut (Whitehouse, 1993; Hancock, 1996).
6.3 Scratching of High ER/k2 Solids The formation of permanent grooves by pointed tools takes place by two distinct modes of deformation shown in Figure 6-2, depending on the geometry and attack angle of the tool. In the upper illustration (that is not cutting), material displaced by the tool forms a prow above the original surface, through which material flows up from the groove to form ridges alongside the groove: a groove formed by such ‘rubbing’ or ‘ploughing’ is consequently wider than the intersection of the tool and the original surface. In the other, the tool cuts away the material in the form of a chip that may be continuous or discontinuous, as in Chapter 4. The width of such a cut groove is essentially that of the intersection of the tool and the original surface. (Note that ‘ploughing’ in the agricultural sense, Section 14.4, is completely different and is cutting.) The two modes of deformation in scratching match those in orthogonal cutting (Chapters 3 and 4), where at very large negative rake angles material is not cut; rather, a
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Cutting With More Than One Edge
Cutting
Ploughing
Figure 6-2 The two extreme modes of deformation that produce grooves in sliding experiments in ductile solids. Material is removed from the groove in the upper drawing by a process of cutting; in the lower drawing it is displaced through a prow of material in front of the tool into ridges alongside the groove. When a groove is formed by cutting, not only continuous shavings may be formed as illustrated, but also a variety of types of discontinuous chips depending on the attack angle of the leading face and the depth of scratch relative to the characteristic length scale for the material given by ER/k2 (after Samuels, 1978).
standing wave of material is pushed ahead of the tool. Note that the transition here is between alternative modes of deformation at constant depth of cut with constant material properties, and the same friction. It arises from different easier modes of deformation with angle of attack. It is not an energy scaling transition in the sense of Chapter 4. But if the depth of cut is increased sufficiently at fixed attack angle, energy scaling transitions with chip formation by elastoplastic and eventually fully elastic deformation take place (Frew, 2008). Other things being equal, the transition is controlled by a critical attack angle crit of the tool, defined with respect to the approaching surface; (90 )° where is the tool rake angle as conventionally defined from the vertical in metal-cutting theory (Sedriks & Mulhearn, 1963, 1964). For many metals (copper, -brass, lead, nickel and aluminium)
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experiments showed that 45° crit 85°, i.e. 45° 5°. Data on polymers suggest similar transition angles. Figure 6-2 shows a facet-first pyramid scratching a surface. When a pyramid is slid edge first, the transition between prow formation and cutting still exists, but cutting prevails at lower attack angles: material is divided on either side of the leading edge and Torrance (1987) termed the removal of material by chip formation in these circumstances ‘side wall stripping’ (see also Challen & Oxley, 1979; Gilormini & Felder, 1983, and Abebe & Appl, 1988). Similar transitions in sliding are found with other tool geometries: experiments have been performed with standard hardness indenters, such as the Vickers pyramid and the round-nosed Rockwell hardness test cone, and with pin-on-drum instruments (e.g. Buttery & Archard, 1970–71; Sakamoto & Tsukizoe, 1977; Kita & Ido, 1978; Murray et al., 1979). Maan and van Groenou (1977) used squared-based pyramids of various angles. Transitions occur not only with metals, but also with other monolithic materials such as polymers (Briscoe et al., 1996). Tabor, in his 1954 paper that put a physical interpretation to the Mohs minerals hardness scale (Section 6.5), also performed experiments on metals in which grooves were formed both by rubbing and by chip formation. Transitions occur with spherical-ended tools, but depend on the depth of cut as it determines the effective angle of attack (Bates & Ludema, 1974; Kayaba et al., 1986). At very light loads, the surface is burnished (Briscoe et al., 1996). At deeper depths of cut, a standing wave is formed, and ridges appear alongside the groove by sideways flow. Penetrating to even greater depths increases the attack angle and promotes cutting. Ribbon- or platelet-like chips or wear debris are formed only when a sufficient depth of indentation relative to the tip radius is exceeded (see also Hokkirigawa & Kato, 1988). The behaviour is then similar to orthogonal cutting with tools so blunt that the tool tip radius is greater than the depth of cut (Chapter 9). Sometimes when the tip of a tool does not have cutting edges, a dead metal zone (DMZ) or built-up edge may be formed in front of the tool and chips may be produced by the DMZ acting as a ‘sharp’ tool (Lortz, 1979). Since a point should always be formed by the intersection of three faces, the Berkovich (1951) indenter is often used in scratching investigations to avoid problems from blunt indenters. The Berkovich indenter has a 65.3° nominal angle between faces and axis, giving an angle of 139° between an edge and the opposite face. In practice there will be a tip radius, of perhaps 50 nm from scanning electron micrograph (SEM) pictures. Imperfections in manufacture resulting in chisel edges or bluntness can become significant in nanoindentation and nanoscratching (Jadret et al., 1998). Castors, even under very heavy pianos, do not indent floors too far, and hence have small attack angles and do not cut. The radius of cylindrical or ball-ended castors should be designed to suit the ‘terrain’ – similarly for the size of all wheeled vehicles where the ground is deformable. Big wheels enable soft ground to be traversed; a child’s buggy with small wheels is difficult to push over gravel, but an old-fashioned pram is easy. Ruts along farmtracks are formed by tractor wheels partially cutting and partially displacing material out of the groove (noting that the rolling wheels act downwards into the ruts, not upwards). A body moving along a sinusoidal surface experiences vertical accelerations proportional to the height of the wave above its mean (Inglis, 1951). The maximum vertical reaction force is thus at the bottom of a hollow and the least on a crest. Insofar as wear depends on contact force, maximum wear occurs on upwards slopes and minimum on downwards. Hence the wave surface will gradually move forwards in the direction of motion. Inglis goes on to explain that this pattern of wear is very noticeable on an ice toboggan run. Humps do not get worn away, but become emphasized by a hollowing away of the surface before the hump and it is the way in which snowboarders make life difficult for skiers on the same slopes.
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The same sort of thing can happen on ski-jumps where grooves are deliberately cut in the ice by machine so that skiers keep straight down slope. Similar hump effects are seen on road surfaces, particularly in places where brakes are applied such as bus stops and corners. The phenomenon also gives insight regarding the way a wind, blowing along a horizontal sheet of water, creates waves that have a velocity of progression. Explanations for the existence of a transition between rubbing and cutting have been given by a number of workers, using the idea that the mode of deformation that requires the smaller force will prevail. Atkins and Liu’s (2007) explanation for the transition, while superficially similar, is different as their model for the cutting regime includes work of separation (ductile fracture toughness). Previous authors modelled the cutting regime using only plasticity and friction and sought different flow fields that gave lower forces than for prow formation. For continuous chip formation, Atkins and Liu (2007) modified the Amarego (1967) plasticity-and-friction-only solution for cutting a V-groove to include work of surface separation. The force to cut a groove of depth t is Fcut (1/ Qshear )[kt2 {tanδ/cos(α φ) sinφ} 2R t/cosδ]
(6-1a)
i.e. Fcut /kt2 (1/Qshear )[{tanδ/cos(α φ) sinφ} 2Z/cosδ ]
(6-1b)
and
FT Fcut tan(β α)
(6-1c)
where Z R/kt as in orthogonal cutting, and where the Coulomb friction correction factor Qshear is the same as for orthogonal cutting in shear since plane strain deformation is assumed in the Amarego model. As for orthogonal cutting, the relation for FC is minimized to find the optimum value for φ for given Z, tool point semi-angle and or . Figure 6-3 shows how (Fprow/kt2) and (Fcut/kt2) vary with , from which the transition at some crit between the two modes is clear. Rubbing requires greater sliding forces as the attack angle increases for the same k and t, the effect being more marked for broader tool point angles. The increases in sliding force are not appreciable until 45°. The cutting forces are greater for greater friction, as expected, but the major influence on force is Z, rather than : materials with greater toughness-to-strength ratios require greater cutting forces. Normalized scratch cutting forces are fairly steady for positive rake angles but they increase rapidly for negative rakes. For the same friction, and at the same attack angle , the theoretical (Fcut/kt2) values are lower the smaller the Z. This suggests that, other things being equal, the critical attack angle will be smaller at deeper groove depths; alternatively, at the same groove depth, crit is smaller the less ductile the solid. The (Fprow /kt2) curve for frictionless prow/ridge formation enables estimates to be made of the accumulated shear strain involved in ridge formation since, by work, (Fprow/kt2) tan. For a triangular tool having a total point angle of 90°, 2 for attack angle 10°; 5 for 20°; 18 for 40°; 25 for 60°; and 62 for 80°. These are very large at greater , and in some circumstances might lead to fracture unless an alternative mode of deformation supervenes. Predictions for the ridge heights h normalized by the depth of the groove below the original surface t are higher the broader the tool point (Atkins & Liu, 2007). The ridge height approaches the groove depth when the attack angle is about 50° with a tool having a total
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Normalised Fcut and Fprow
80
μ
Z = R/kt
0.8 60
0.3 0.1
15
40 High
20
Friction Low
0 0
20
40
60
80
100
120
140
0.8 0.3 0.1
5
all μ
0.5
160
180
Attack angle λ (degrees)
Figure 6-3 Normalized Fcut and Fprow vs attack angle for various frictional conditions and values of Z (R/kt). Transition between modes where curves cross.
point angle of about 130° and operating under high friction. The ridge heights produced by sharply pointed triangular tools hardly change for all attack angles. For the 90° tool for which the accumulated shear strains were estimated above, (h/t) increases from about 0.2 to 0.4, over the same range of . Amarego (1967) cut 65ST6 aluminium alloy (i) orthogonally with a tool rake angle of 20° at various depths of cut and different widths of workpiece; and (ii) triangular grooves of various depths with a facet-first triangular tools having various face angles and at different rake angles. Amarego used a variety of workpiece widths and all force data normalized by width fall on a single line, as predicted by Eq. (6-1a). In neither case does the normalized cutting force vs t plot pass through the origin, and there are positive intercepts at t 0. The orthogonal data may be analysed using Eq. (3-23b) with 20°, φ 30° and 30° (given in the paper), from which 1.9 and Qshear 0.74. The slope is 111 000 psi, from which k 43 000 ksi or 300 MPa; the intercept is 69 lb/in., from which R 51 lb/in. or 9 kJ/m2. The groove data may be analysed with Eq. (6-1), using 0, φ 21° and 25°, giving Qshear 0.82 from which k 40 000 lb/in. or 283 MPa; the intercept is 400 lb/in., from which R 100 lb/in. or 18 kJ/m2. While k agrees well from the two methods of cutting (and agrees with Amarego’s value), R does not agree too well, owing to the errors of back-extrapolation in both plots. In the same paper, Amarego gives the solution for deepening, with the same tool, an already cut V-groove. Hankins (1923) attempted to relate scratching force data to the square of the width of the groove by an equation of the form (P p) K(w2 q), retaining Hankins’s symbols. P is the load and his w t tan in our nomenclature; p and q are empirical offsets. According to Eq. (6-1),
[Fcut (2Rt/Qshearcosδ)] (1/Q shear )[kt2 tanδ/cos(α φ) sinφ]
(6-2)
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Therefore the offset p is given by (2Rt/Qshearcos) and cannot be constant as it varies with t. Direct analysis of data in Hankins (1923; his Table 2) in terms of Eq. (6-1) may be performed, determining k from the hardnesses given in the paper. Good linear plots are obtained. The following values for toughness are given: 34 kJ/m2 for mild steel and 11.2 kJ/m2 for various alloy and hardened steels. Mechanical properties may be estimated from scratch tests using Figure 6-3. In the Sedriks and Mulhearn (1963, 1964) experiments, crit, and k were known (k from hardness tests). Figure 6-3 gives the different Z at the different crit for given , so given the depth of groove at the transition, we may calculate R/k. Then, for known k, R is determined as given in Table 6-1. In other experiments on annealed copper, it was found that 55° crit 60° and F 160 N at t 0.3 mm. Figure 6-3 gives (F/kt2) 20 for a transition at that depth, so k 160/20 9 108 89 MPa (which agreed with independent tensile and hardness values). Figure 6-3 also gives Z 2 for a transition at 0.3 mm. Thus R is estimated to be 2(89 106)3 104 53 kJ/m2. Independent experiments on compression testpieces, cylindrically notched along the compression axis, gave R about 70 kJ/m2. The problem of determining ‘valid’ fracture toughness, from specimens too small to satisfy the minimum size requirements of national and international standards, is well known and is part of the wider problem of scaling in fracture mechanics. It is a problem not only in structural integrity but in other fields also. One such is archaeometallurgy, where only very small samples are usually available. Such samples are usually mounted and metallurgical micrographs taken to identify the microstructure. Although small, the samples are often big enough to enable microhardness measurements to be taken from which the strength may be estimated. In investigations relating to the performance of ancient weapons, armour and tools, microstructure and hardness alone are incapable of telling the whole picture, since it is really desirable to have knowledge of the fracture toughness as well (Chapter 8). While it is arguable that modern copper alloys can replicate the behaviour of ancient bronze, the same is not true for Roman and mediaeval steels and this lacuna in knowledge proves to be a drawback for proper interpretation. Measurement of the critical attack angle in scratching might be used to estimate the fracture toughness of small mounted metallurgical samples when other, more conventional, methods are impossible to perform. The sensitivity of such a procedure may not be very high, but in the absence of anything else, it will be better than nothing. Again, in assessments to extend the life of existing old structures/components coming to the end of their design life, where there is uncertainty about mechanical properties, scratching
Table 6-1 Toughness determined from scratching experiments. Material
critical
k
(all 5°)
Z
t
R/k
R
(MPa)
(mm)
(mm)
(kJ/m2)
Lead
0.6 (5)
55
7.5
4
0.17
0.6 (8)
51
Aluminium
1.2
85
90
25
0.05
1.2
100
Copper (w/h)
0.4
45
200
1
0.04 (7)
0.05
9
-Brass
0.5 (5)
55
340
4
0.03 (2)
0.12
43
Nickel
0.7
65
530
5
0.02 (6)
0.13
69
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could be employed as a non-destructive test for yield strength and toughness. It also suggests that insofar as continuum mechanics applies at very small scales, scratching could be employed to determine toughness and yield strength at the nano level. Similarly, the analysis is applicable to those scratch resistance tests of monolithic and coated materials in which the load is increased with slide length. Sedriks and Mulhearn’s 1960s’ experiments were performed under constant deadweight load. At the largest angles of attack, the pyramidal tool ‘digs in’ and eventually seizure occurs. Seizure is peculiar to deadweight loading: under constant groove depth, the rubbing/ cutting transition still takes place, but there is no seizure. The equilibrium vertical load on the indenter during cutting is given by Eq. (6-1). During sliding at the same depth as produced by the initial indentation, the equilibrium vertical loads given by Eq. (6-1) are not the same as the deadweight load W; but in a sliding experiment where the depth is maintained constant, the vertical force adjusts itself to the changed requirement. It follows that the equilibrium depth of groove during sliding under deadweight loading is generally different from the initial indentation depth determined by hardness. At very small attack angles, it can be smaller than the static indentation depth below the surface, but as the attack angle increases, so does the depth of groove formed by cutting. Illustrations in Sedriks and Mulhearn (1963, 1964) and in Buttery and Archard (1970–71) give broad support to this sort of behaviour between the size of the initial indentation, and the subsequent width and depth, when sliding in the chipforming mode. Such differences are central to differences observed between ‘scratch hardness’ and indentation hardness (Section 6.6). According to Eq. (6-1) the downwards vertical force reduces and can even change sign and become an upthrust. It follows that under deadweight loading the slider must dig in under such conditions. During sliding under a constant indentation load W, the depth can be worked out by putting FT W in Eq. (6-1).
6.4 Grinding and Abrasive Papers In the grinding of corn, grains of cereals are crushed to powder to make flour. Grinding in this sense is comminution and means reducing everything to a fine size and is a sort of automated pestle and mortar (comminute from the Latin to lessen, as in the word ‘minor’; pestel from the Latin pinsere to pound; and mortarium the vessel for pounding or grinding; pestel and mortar together are the ancient symbol of the apothecary). Flour mills used rollers rather than stone discs after the introduction of steam power. The different sense in which grinding is employed in manufacturing is removal of material in very small amounts from the surface of a larger piece by simultaneous multiple scratching. Grinding is employed in a number of different ways in the workshop, such as a finishing operation when parts require particular dimensional accuracy. It is used for sharpening tools and for shaping very hard materials that would otherwise fracture were attempts made to remove material at normal depths of cut (owing to the fracture mechanics scale effect). In contrast to corn grinding, rotational speeds are far higher. For a depth of scratch of 5 m and a grinding wheel operating at 4000 rpm, the time of contact between an individual grit and the surface being ground may be 1 s in every revolution. Grinding wheels are described in Chapter 9 in connexion with sharpening. Small grits of artificial abrasives are bonded within the material of the wheel, those exposed on the surface starting the job of scratching the surface, and as grits break off or wear down, new edges or particles are revealed. Eventually grinding wheels have to be ‘dressed’, which is sort of reprofiling and resharpening after clogging up. Tools employed in cutting usually have defined cutting edge geometries, but the grits in grinding are geometrically undefined. They may be
Cutting With More Than One Edge
151
thought of as a mixture of pointed, ridged and truncated pyramids with all orientations possible from edge-first to face-first. Jones (1886) discovered that attempts to grind annealed steel or iron or other soft metals were not successful. ‘… Emery wheels will grind hardened steel freely, but will scarcely act on soft steel or iron, and much less on copper. Hardened steel will cut with a crisp feel under a diamond tool or emery wheel, whereas the softer metals will cling tenaciously and drag under the treatment, the particles yielding slowly, being torn off with great force rather than cut …’. This is a very early reference to the formation of a built-up edge in grinding, and to prow/ ridge formation rather than cutting (see also Connolly & Rubenstein, 1968). Abrasive papers and belts have similar grits bonded to their surface and, like grinding wheels, come in different grades reflecting the size range of the abrasive particles. Emery boards for finger nails and corns are similar. Finer grits produce shallower and narrower scratches than coarser. Finer yet are polished surfaces over which there are closely spaced shallow grooves differing only in degree from the grooves produced by sanding, grinding and abrasion. The fuzzy ends of fibres on cloth, leather or other materials, drawn up by ‘napping’, are employed on polishing pads and dusters that are used with polishing compounds of various sorts. Buffalo hide was used not only as body armour (Chapter 8) but also for polishing, hence ‘to buff’. Stropping with a leather belt in sharpening is similar. Shark skin makes an excellent abrasive cloth as its surface comprises series of tooth-like dermal tentacles Southall & Sims, 2003). Paradoxically, it is this roughness that lowers resistance through the water by forming turbulent boundary layers, rather like dimples in a golf ball in air (e.g. Ball, 1999). There is great interest in biomimetic non-smooth, yet low friction, surfaces inspired by the knobs on the surface of Chinese dung beetles (Ren et al., 2003; Tong et al., 2009). Abrasive papers, like grinding wheels and indeed files, clog up and lose their ability to remove material. Mercer and Hutchings (1989) looked at the deterioration of papers during the wear of various metal alloys, including iron and Ti-6V-4V alloy. Finer SiC papers (below a mean particle diameter of some 15 m) deteriorate principally by adhesion of metallic debris to abrasive particles. There is an overall reduction in penetration into the surface and individual grits have reduced cutting ability. Coarser SiC papers with grits of mean particle diameter up to 50 m have a greater effective load (fewer grits per area for same load) and fragmentation and attrition contribute more to deterioration. These same wear mechanisms apply to all grades of Al2O3 because iron does not adhere as strongly to Al2O3 as to SiC. More material is removed by larger grit sizes since they do not deteriorate as rapidly as finer grits. Hand-held abrasive papers and belts operate under load control; when a cutting tool is shaped and sharpened by hand on a grinding wheel, there is no proper control over depth, unless special fixtures are employed. On grinding machines where the workpiece is clamped to a feed table, however, the depth of cut is closely controlled. The two different modes of deformation, in the scratching of solids where ER/k2t results in ductile deformation, are crucial for both wear and polishing of materials. In the one case, prow/ridges are formed where, in theory, no material is removed unless ridges get knocked off on repeated passes (Adams et al., 2001). In the other, cutting occurs where material is removed. The efficiency of grinding and abrasion produced by wheels, papers and slurries is not particularly high, often lower than 20 per cent. This is because the abrasive points in contact with the workpiece have a variety of orientations from facet-first to edge-first and only a proportion have attack angles suitable for cutting a chip (Mulhearn & Samuels, 1962). Johnson (1970–71) remarked that the low efficiency of material removal in abrasion and
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The Science and Engineering of Cutting
grinding is also connected with the fact that blunt grits produce small imposed strains that can be accommodated by elasticity within the hinterland of the ground body. Surface finishing of ceramic components is typically achieved by grinding that is a process of multipoint scratching at varying depths of cut, produced by hard particles (usually diamond) embedded in the surface of a wheel. The protrusion of some hard particles will be small enough to produce ductile chips; others will result in a variety of elastoplastic chips; and those protruding most will give brittle chipping. Material removal is thus by a complicated mixture of flow and fracture (Marshall et al., 1983). The complicated nature of material removal was revealed by Puttick et al. (1989), who showed that in the case of glassy fused quartz (spectrosil) material removal takes place partly by a component of residual stress that is compressive and directed along the machined groove. This peels off ribbons of glass behind the tool. Workers interested in grinding often perform experiments where a single grit is attached to a rotating wheel (fly cutter) or at the end of the arm of a pendulum (e.g. Kenneford, 1946; Grisbrook et al., 1965; Brecker & Shaw, 1974; Vingsbo & Hogmark, 1984; Briscoe et al., 1999; Wang & Subhash, 2002). Should the workpiece be stationary, both methods result in a lens-shaped groove. However, should the workpiece be fed towards a continuously rotating flycutter, material is removed from along the arc formed by the previous pass; in this way, the device functions very much like a milling cutter. The geometry of an arc is part of a trochoid (from the Greek for ‘round like a wheel’), but when (i) the feed velocity is much less than the tangential speed of the grinding wheel, and also (ii) the depth of cut t is much smaller than the diameter D of the grinding wheel, the boundaries of the chip are nearly circular arcs and the length L of the long slender triangular-shape chip is given by L (Dt), and the average thickness of the undeformed chip may be used in calculations (Backer et al., 1952). Marzymski (1961) and Staniszewski (1969) both used a pendulum to study wood cutting (see Sinn et al., 2006). The relative direction of rotation of a cutter and that of the feed determine whether the milling action is up or down. In up-milling chip thickness starts from zero and increases; in down-milling, the other way. In these sorts of experiments a single tool may displace material by prows/ridges or cuts, or a sequence of both, since not only does the depth of cut alter through a groove or along an arc but also the attack angle changes continuously to more positive rakes. The behaviour is then a composite of what is observed from experiments where the grit is slid parallel to the surface. (Swinging a tipped pendulum into an already cut arc results in almost constant depths of cut.) Vingsbo and Hogmark (1984) used a standard Charpy machine having a tungsten carbide tip in the form of a truncated square-base pyramid with 90° face angles (one of Hankins’s diamonds of 1923 was of similar shape). The tool was mounted radially on the arm and chips were cut in a variety of materials in order to study abrasion resistance. Depending on conditions, they found that all the same sorts of chip were produced (continuous, discontinuous, quasi-adiabatic, etc.) as described in Chapter 4. The machine was instrumented for forces as well as obtaining energy in the usual way. A quick-stop device enabled metallographic sections to be made which demonstrated that thin layers of hardened material were produced alongside and beneath the groove in the specimens. The specific energy e for grooving (pendulum energy loss E divided by mass loss M from the specimen) was related to the mass loss as illustrated in Figure 6-4(A). Similar plots are given by other authors such as Kenneford (1946). Their data can be interpreted in terms of the scratching analysis given above, which includes toughness. The force to cut an arbitrary-shaped groove of depth t will have the same form as Eq. (6-1), and may be written as Fgroove (1/Q)[At2 Bt], where the friction factor Q, and A and B all depend upon the geometry of the tool tip, and A relates to the plastic work/volume and B to the fracture toughness. The energy consumed in cutting a groove is E FgrooveL,
1.2 10 1
Structural steel HSLA steel
5
e (MJ/kg)
Hadfield steel
3
57/39 brass 66/24 brass
0.6 0.4
1
0.2
0.5
0
0
50
100
150
250
300
350
400
1/ M 1/ kg
B
0.3
200
7
A
100
6
200
M (mg)
Figure 6-4 (A) Specific energy e for grooving (pendulum energy loss E divided by mass loss M from the specimen) vs M, given by Vingsbo and Hogmark (1984). Pendulum grooving data according to the cutting analysis that incorporates work of separation should plot as e vs (1/M). Results are shown for (B) 66/24 brass; and (C) Hadfield’s manganese steel.
5 e (MJ/kg)
0
4
Cutting With More Than One Edge
e (MJ/kg)
0.8
Tool steel
3 2 1 0
200
400
600 1/ M 1/ kg
800
1000
1200
153
C
0
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The Science and Engineering of Cutting
where L, the length of the groove, is given by L 2(2pt), where p is the length of the pendulum (t p); p 830 mm in Vingsbo’s machine. Whence E (1/Q)[At2 Bt]L. The prism employed by Vingsbo and Hogmark, having a 90° triangular face and truncated with a 1 mm flat, produces a groove of width w [0.001 2t] m. The mass M of the chip is M wtL, where is the density of the workpiece. Owing to the (w,t) relation, a cumbersome expression results for the specific energy e given by E/M. However, it may be shown that
e E/M ≈ (1/Q)[(kγ/ρ) (0.4R/ρ1 / 2 )(1/ M)]
(6-3)
i.e. e should plot approximately linearly against (1/M), with the slope relating to the toughness R and an intercept relating to the plastic work/volume k. Figure 6-4(b,C) shows data for brass and Hadfield’s manganese steel from Figure 6-4(A) plotted in this way and the equations of the regressed lines. The agreement is good and there is agreement between the intercepts in Figure 6-4(B,C) and the asymptotes in Figure 6-4(A). We do not have a flow field for cutting with a truncated pyramid, but for Q 0.8, the intercepts give k 1 GJ/m3 (57/39 brass), 1.5 GJ/m3 (66/24 brass), 2.5 GJ/m3 (structural steel) and 430 MJ/m3 (Hadfield’s manganese steel). Dynamic shear stresses at the strain rates involved are not known, but estimates seem to show that the truncated tip results in 5 6. Likewise, from the slopes, R 190 kJ/m2 (57/39 brass), 500 kJ/m2 (66/24 brass and structural steel), but 1.3 MJ/m2 for the Hadfield’s steel. These values cannot be exact, but the remarkable toughness suggested for Hadfield’s wear-resistant steel in comparison with the other metals shows that toughness plays a crucial part in its wear and scratch resistance. Hadfield’s steel has a comparatively low hardness (about 380 kg/mm2) in the as-received condition but its wear-resistant performance results from the production of stress-induced martensite on loading. Reports of experiments with flycutters often calculate the ‘specific cutting pressure’ (‘specific energy of cutting’ or ‘unit power’) for grinding, given by (Fgroove/wt) and plot it against depth of cut. The shapes of such graphs are the same as the specific energy plots in Kenneford (1946) and Vingsbo and Hogmark (1984), the ordinates being scaled by density. The significant rise in experimental specific energy at smaller and smaller grit depths of grinding led Backer et al. (1952) to talk of a ‘size effect in metal cutting’ wherein the calculated shear stresses in grinding 1112 steel at small depths of cut (50 m) approached the theoretical shear stress (about G/10 or 10 GPa). But as explained for orthogonal cutting in Section 3.6.7, and elsewhere, the reason for a steep rise in specific energy at small t is because the contribution of the toughness term is inversely dependent on t: when k is calculated from the plastic work term only, anomalous k must result. The mechanics of scratching are important in the production of diffraction gratings. While small gratings can nowadays be made by photo-resist methods, large gratings rely on scratching. A diffraction grating consists of a band of equidistant parallel lines about 1 m apart and 500 nm deep ‘ruled’ (scribed) on glass or polished metal to produce diffraction (i.e. the bending of light waves around obstacles in their path to produce optical spectra, as exemplified by the Young’s slits experiment). Since perhaps 95 per cent of the light is scattered by transmission gratings, reflexion gratings are preferred. Although grooves have a 90° angle in cross section, the grooves are asymmetric with respect to the surface of the polished metal, i.e. the tool axis is angled to the surface. Diffraction gratings are produced with diamond tools having very negative rake angles (small angles of attack) (Chao & Gee, 1991). Instead of pyramidal tools, so-called ‘hatchet’ or ‘boat-shaped’ tools may be employed having a curved edge like an axe. Since only a small part of the edge is used during scratching, when a fresh edge is required the tool is simply incremented round the curved edge. Because diffraction gratings were made with
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tools of small attack angle and on the surface of polished metals, the fact that brittle materials could be scratched with tools having a positive rake angle to produce continuous ribbons was not widely known or investigated until the 1980s. Gee (2009) recalls the party trick of one researcher in earlier years who would balance a glass cutter on his hand and lightly sweep out an arc on glass. Under the microscope shavings could be seen, like fluff on a gramophone needle. It is an interesting question why the stylus of a gramophone does not cut away the grooves. Certainly if there is fluff on the needle and one foolishly tries to blow it away while a record is playing, the arm will move radially inwards across the grooves and form a scratch even under the low contact load of a few grams. Of course, when a scratch is made across the grooves, the stylus is going over hills and valleys, and it is easy to see how parts of crests could be removed. Along a groove there are sideways undulations and the stylus oscillates according to these modulations and transmits the mechanical motions to the transducers in the pickup arm; in a stereo record the groove has a 90° V-shape with the left track undulating on the left flank, the right track on the right. The needle then transmits its two separate oscillations to two separate piezo crystals. The wiggles are thus both side-to-side and up-anddown and there are two of them. The tip of a diamond stylus must be rounded otherwise the softer vinyl would be cut away. In the days of 12-inch shellac records, it was known that steel needles did wear out the record grooves. Fibre (bamboo) needles did not, but performed better if sharpened after every playing. Polished steel needles have a higher coefficient of friction than a polished diamond, and this must play a part. Steel needles wore and left steel particles in the record grooves. The next playing then allowed friction between steel and steel particles embedded in the shellac, which would have an even higher friction coefficient, and lead to more wear (Harris, 2008; Pretlove, 2008). Studman and Field (1984) analysed the general problem of two rigid surfaces approaching each other with small brittle particles in between. Applied to the problem of a dust particle in the groove of a record, they showed that the stylus can only push the particle aside if the diameter of the dust particle is more than about 0.25 times the diameter of the stylus. Otherwise the stylus must ride over the particle or fracture it. In either case, a relatively high load will be applied through the dust particle to the record’s surface, since the stylus in effect impacts against the particle. Krushschov and Babichev (1954) performed landmark experiments in which materials were worn against abrasive papers. They showed that the wear resistance (inverse of wear rate) of many ‘pure metals’ increased in direct proportion to hardness H. As they employed abrasive papers, the results are some composite of the behaviour of a range of grits of different geometries, different attack angles and different orientations to the direction of sliding. Were all material from the grooves and scratches removed as debris, and were the depth of the grooves the same as for simple indentation hardness, the wear rate would depend directly on the load W and inversely on the hardness H, as suggested by Eq. (6-1), and embodied in the Archard (1957) equation: wear rate KW/H, where K is a constant for a particular combination of materials. Krushschov and Babichev found, however, that in hard metals the rate of increase in wear resistance fell off so that there was a departure from the linear relation for much harder materials (Figure 6-5). The idea of scratch hardness (discussed in Section 6.6 later) is based on the width of grooves and the loads required to produce them, and must therefore relate also to the performance of papers. It is pertinent that Turner (1909) seems to anticipate Krushschov and Babichev when he came to the conclusion that ‘while there was some relation between indentation hardness and scratch hardness in the case of relatively pure metals in their cast or normal state, yet when the resistance to deformation is due to tempering or mechanical treatment, no comparison is possible’.
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60 W 1.2% C Steel 50 0.8% C Steel
Be
Relative wear resistance
40
Mo
0.4% C Steel
Ti St St
30
1.0% C. 11.7% Mn Austenitic manganese steel
St St Co Fe Ni
20
Ni
Anneoied Hardened by quenching and tempering Workhardened
Zr Cu
10
Cu Zn
0
Al Sn Pb 0
250
500
750
1000
Hardness (kg/mm2)
Figure 6-5 Relative wear resistance as a function of indentation hardness for metals rubbed against abrasive papers (after Khrushchov & Babichev, 1954).
The departure of harder materials from the Krushschov–Babichev relation was qualitatively explained in terms of ‘material brittleness’ by Murray et al. (1979) and by Atkins (1980a), who noted that the harder metals are relatively less tough or more brittle and have a lower (R/k) ratio than the pure metals, all of which are soft and ductile with similar large (R/k) ratios. In principle, the depth of a cut groove and hence the volume of the track can be established from Eq. (6-1) by putting FT W, the dead weight. In this way, the influence of (R/k) on a Krushschov–Babichev plot may be determined and the departure from the initial linear relation explained quantitatively. As remarked in Atkins and Liu (2007), however, the optimum φ that minimizes Fcut depends on the non-dimensional parameter Z (R/kt) that in turn depends on the unknown t, so some iteration is necessary to obtain converged solutions. The (R/k) ratio will be altered by different thermomechanical treatments subjected to a given alloy. In the report of the Hardness Tests Research Committee (IMechE, 1916), it
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Fully elastic deformation
Ironing
Cone angle (degrees)
150
120
90
Ductile ploughing + Elastoplastic deformation
Ductile ploughing
60
Ductile ploughing + Edge cracks formation
45
n
atio pag
Pro 35
le ritt
ion itiat
0.5
In 0.8
m
B 1.2
1.5
ng
ini
h ac
1.8
hip + C
a for m
tion
Wedge formation ♦ Brittle machining ♦ Deep grooving
2
3.5
Normal load (N)
Figure 6-6 Deformation map for polycarbonate. The picture shows results from scratch tests performed for a range of cone angles and normal loads at room temperature and at a scratching velocity of 2.6 mm/s (after Briscoe et al., 1996).
was noted that a steel specimen tempered at 500°C gave greater wear than that tempered at 600°C, even though it was harder (453 vs 120 BHN). Childs (1988) discusses the mapping of metallic sliding wear in terms of the different chip types (including standing waves) on to axes of surface shear strength and the slope of surface roughness. In the case of wear of surface films on metals (oxides usually) mapping of wear was found best presented on axes of nominal contact pressure and sliding speed. Figure 6-6 shows a deformation map for polycarbonate scratched by cones of various angles under various loads. As with the machining maps of Chapter 4, all sorts of deformation mechanisms are identified.
6.5 Scratch Hardness/Scratch Resistance Scratch hardness is thought of in two different ways: (i) Standardize a series of materials of graded relative hardness and use them to try and scratch an unknown material. The hardness of the unknown solid will be located in the interval between the two standard materials that just do, and just do not, result in a scratch being formed. The Mohs (1822) hardness test is the most well-known of these; there were earlier similar scales but with fewer intervals (O’Neill, 1967). The softest material on Mohs’ scale is talc at number 1; the hardest is diamond at number 10. An idea of what the different hardness numbers mean was given by Lange (1939, quoted by O’Neill, 1967): ‘Talc very easily scratched by finger nail; calcite scratched by brass pin or copper coin; a knife easily scratches fluorspar but scratches apatite only with difficulty; a
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file easily scratches felspar but quartz hardly at all, but quartz will scratch window glass’. A copper coin is about 3 on the scale; ductile iron 4.5; and a file 6.5. The modern pencil hardness scale (Section 6.6) is, at first sight, of this sort but matters are not quite so straightforward, as explained below. (ii) Use a very hard scratching tool, indent the unknown material with a fixed load W, and then produce a scratch. Seebeck (1833) and Franz (1850) employed this method, using a sharp steel point on soft minerals and a diamond on hard minerals, and Turner (1886) was the first to apply the idea to metals and alloys, using a diamond point. (Turner’s paper is interesting in that it debates differences between ‘hardness’ and ‘tenacity’ – what we would now call toughness; a paper by Ludwik, 1922, is entitled ‘Cohesion, hardness and toughness’ and debates similar issues.) Turner thought of scratch hardness in terms of the load that would cause ‘a just-perceptible groove’. Martens (1890) – after whom the very hard microconstituent martensite in steels is named – made the test somewhat quantitative by saying that scratch hardness was the load in grams required to produce a scratch of average width 0.01 mm. The variation of scratch width was first used to study anisotropy in single crystals by Exner (1873). O’Neill (1928) concluded that scratches should be separated by about five times their width to avoid interference. He found that lubrication made no measurable difference to scratch width. O’Neill (1967) continued to use scratch tests to investigate the anisotropy of single crystals, polycrystals (scratching over grain boundaries) and electrodeposited films. In the first quarter of the twentieth century, it was appreciated that Brinell hardnesses above 600 were suspect because of elastic flattening of the indenter. It was thought that scratch hardness of some sort might be a substitute and there are many correlations between scratch hardnesses and Brinell hardnesses in the literature of the time. However, as explained earlier, since the depth of a groove during sliding is not the same as the depth of the initial indentation, scratch hardness is not just ‘travelling indentation hardness’ having the same value. The colour of material transferred by scratching to a touchstone (a black siliceous stone) was used in ancient times for identifying and assaying metals, especially precious metals. Touch needles were gold or silver needles of various known degrees of purity that when rubbed on a touchstone would leave different-coloured marks. The colour of the streak left by an unknown specimen was compared to those left by the standards. Many people know about Mohs hardness, but how many have actually performed the test? There are problems to do with the geometry of the scratching corner; the inclination of the corner when scratching by hand; the direction of scratching; the fact that some minerals on the scale are difficult to find with corners; and so on. Williams (1942) quotes extensively from Mohs’ 1822 book, where it is found ‘… it may be difficult to find specimens having the requisite features such as smooth faces and solid angles or corners of the same form and be durable … different geometry corners give different and uncertain results, but if we take several specimens of one and the same mineral and pass them under a fine file, we shall find that an equal force will everywhere produce an equal effect, provided that the parts of the mineral in contact with the file be of similar size, so that one does not present to the file a very sharp corner, while the other is applied to it by a broad face. It is necessary also that the force applied in this experiment be always the least possible. The degree of hardness of an unknown mineral is compared with the degrees of hardness of the members of the scale not immediately by their mutual scratching but mediately through the file …’.
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(Files are discussed in Chapter 7.) All these problems were highlighted by various authors who were trying find correlations between hardness scales. Bottone (1873), in trying to establish a hardness scale for all materials, found that the Mohs scale was ‘utterly unreliable, as a softer body was found to be able to scratch a harder one, provided a certain angle of the scratching surface were presented to the surface to be scratched’. Tabor (1954), by scratching a metal bar having a gradient of hardness with a pointed tool of intermediate hardness Hp, found that the transition between scratching and not scratching took place when Hp 1.2 Hs, where Hs is the local hardness of the surface at the transition. This suggests that a scratch hardness scale, where the intervals just do not overlap, is related to indentation hardness by Hindent (const)(1.2)Hp. Semi-log plots of logHindent vs HMohs showed that the factor was 1.6 rather than 1.2 for Mohs’ chosen minerals. There does not appear to have been any systematic study of mutual scratching of metals of comparable hardness having different geometries of contact. It would be interesting to see the modes of deformation induced; the work would be relevant to how hard and tough a tool has to be to cut something else without itself being deformed, as discussed in Chapter 9. It is likely that cutting is possible with a deformable tool but, for example, is a steady state attained and can a constant depth of cut be maintained? Scratching with a hard indenter is like Mohs but there is no deformation of the indenter. Unlike Mohs, the geometry and inclination of the leading face of the scratching tool is well defined and, unlike Mohs, the force Fscratch parallel to the surface is sometimes measured as well as knowing the load W vertical to the surface. It has been supposed that these two loads are borne by those parts of the indentation area projected parallel, and vertical, to the surface. If, in consequence, only the front half of the indentation bears W, it might be supposed that the scratch hardness Hscratch 2Hindent, as pointed out by Meyer (1908a). However, as already shown, the width w (and depth t) of the scratch is usually different from the width across the initial indentation and its initial depth, so that Hscratch 2Hindent. Scratch hardness for rigid-plastic materials is usually defined in terms of the vertical load W as Hscratch 2W/ (w2/4) 8W/w2 for a ball or cone tool in ductile metals; mutatis mutandis for pyramidal indenters. In viscoelastic and viscoplastic materials, the factor (8/) is smaller (Johnson, 1985; Briscoe et al., 1996). Scratch hardness is often determined on deadweight machines where the force parallel to the surface is not measured. However, the ‘mean flow pressure resisting sliding’ pm given by 4Fscratch/w2cot, where is the semi-angle of a conical or pyramidal indenter, was used by Brookes et al. (1972), who found that H Hscratch pm. Furthermore, their experiments revealed that Hscratch H for workhardened copper and steel, but Hscratch H for annealed copper and steel. This demonstrates not only differences in width of track from the size of the initial indentation but also that the width is greater for workhardened samples and smaller for annealed samples. Differences between Hscratch and H were noted by Shires (1925). A definition that depends only on load W or Fscratch is easy to employ, but it makes no distinction between the manner in which the groove is formed (prow/ridging or cutting). In the early days of scratch hardness, the standardization of indenter geometry did not seem to cause concern, nor did the way in which material was removed from the visible groove. Turner (1886) definitely talks about ridges alongside the groove, but Hankins (1929) was of the opinion that older tests produced scratches by ‘abrasion’ (i.e. cutting). The manner of groove formation when scratching with spheres depends on the depth of groove relative to the diameter of the ball (Chapter 4), so that care must be taken when determining scratch resistance from ball sliders (e.g. Xiang, 2000).
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In the same way that H cY (Tabor, 1951), in which c represents the constraint of the surrounding material to plastic indentation by different-shaped indenters, we perhaps ought to write Hscratch Fscratch/w2 c1Y (prow/ridge) and Hscratch Fscratch/w2 c2Y c3R (cutting), where c1, c2, and c3 are appropriate constraint factors for particular-shaped tools inclined at given attack angles. The differences between these two definitions are relevant to the assumptions made by Sedriks and Mulhearn (1963, 1964) when employing the Bowden and Tabor ‘ploughing’ analysis for both the rubbing and the cutting regimes, and their assumption that the ‘cutting pressure’ was a simple multiple of H (Atkins & Liu, 2007). Hadfield and Main (1919) state that scratch hardness was used to test British steel helmets during World War I at Hadfield’s factory in Sheffield. In that paper an inverse relation was found between BHN and (width of scratch)3; that is surprising as it is dimensionally incorrect. In 1923 Bierbaum employed his Microcharacter scratching instrument to discover the differences in hardness of microconstituents in alloys. His instrument had a carefully controlled indenter in the form of an edge-first cube with a 35° attack angle that produced ridges along grooves in ductile solids. The microhardness was defined as 104/w2, where the width w of the scratch is in m. Merigoux and Minari (1964) showed that the shape of ridges formed alongside scratches varied systematically with crystal orientation, and Brookes et al. (1972) showed that their pm could distinguish anisotropy on different crystal faces scratched in different directions that simple indentation could not (see also O’Neill, 1928, 1967; Puttick & Hosseini, 1980). Whether the indenting is done first followed by sliding, or whether the depth of penetration is kept constant, or whether sliding takes place under increasing load, makes a difference in scratching. It is a question of whether sliding takes place under load control (as in all deadweight loaded and hand-held) devices, or whether the depth of cut is maintained constant. The latter is only possible when using a stiff machine. Under deadweight loading, the depth of scratch is usually different from the depth of the initial indentation.
6.6 Scratching of Thin Films and Coatings: Pencil Hardness Surface properties of monolithic bodies may be modified by thermomechanical treatments such as shot peening or case hardening that improve wear and corrosion resistance. The subsurface microstructure is affected and residual stresses set up. Alternatively, separate films are deposited whose microstructures gradually meld into that of the surface. In both cases there will be a gradation of properties from the outside to the inside. In yet other cases the coating remains a separate entity adhered to the surface with an obvious interface. Various combinations of properties of coating and substrate – stiff/compliant, weak/strong, tough/brittle – are found, with one or more layers of different thicknesses ranging from millimetres to micrometres. The general patterns of surface and subsurface cracking, discontinuous and continuous chip formation, prow/ridging, etc., found when scratching monolithic bodies, are also found when scratching materials with modified surfaces and coatings, except that the presence of close-by layers of different properties modifies the patterns. Thus a brittle coating on a ductile substrate will show partial ring cracks on the surface under sliding spherical contact, but the partial cone cracks may stop at the interface if the substrate is much tougher (Bull, 1991, 1997). Clearly, depth of scratch is important: very small depths may show just the behaviour of the surface layer. Deep grooves, going through one layer to another, result in a composite behaviour between the two, perhaps like the simultaneous cracking of sandwiches made up of brittle and ductile materials (Atkins & Mai, 1985). The mechanical properties of thin films (and indeed of near-surface layers of monolithic solids) may be determined by indentation hardness and scratching, using instrumented
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microhardness and nanohardness machines, the latter capable of continually measuring loads and displacements to N and nm resolution. Elastoplastic models for spherical and pointed indenters relate the applied load W to the depth of indentation. Stillwell and Tabor (1961) showed that elastic recovery after removal of the load takes place principally in the depth of indentation, and elastoplastic models permit not only yield (hardness) properties to be determined but also the Young’s modulus from unloading (Oliver & Pharr, 1992). Scratch testing is often employed to determine the adhesive strength of coatings, using fracture mechanics models having cracks at the interfaces (Tellier & Benmessaouda, 1994; Bower & Fleck 1994; Xu & Jahanmir, 1995; Krieser et al., 1998; van Vliet & Goldstone, 2001; Holmberg et al., 2003). It was Strong (1935) who first proposed the use of sticky tape to assess adhesion of films. In scratch tests on thin films, the normal load is either kept constant or arranged to increase progressively with length of scratch. The aim of an increasing-load test is to induce, during a single experiment, a critical point of damage somewhere along the track, such as coating delamination, coating cracking or spallation (brittle materials) or coating whitening (polymer coatings). The critical load, or scratch hardness at the critical load, is used to compare the performance of different coatings (Benjamin & Weaver, 1960; Malzbender & de With, 2000; Vercamnen et al., 2000; Dasari et al., 2007). Chen and Wu (2008) provide a valuable review of the contact mechanics in these sorts of experiments with layered bodies and contrast the predictions with experimental measurements. Figure 6-7, taken from their paper, shows the various patterns of damage that occur around tracks. Chen and co-workers have investigated the scratch resistance of hard, brittle thin films on compliant substrates with applications to sol-gel protective coatings on plastic optical lenses, safety windows, and flexible electronic devices and displays. The quality control test used by industry is the ‘pencil hardness test’ (ISO 15184, 1998), originally introduced for the assessment of paint and varnish coatings. Pencils leads of different hardness grades (9B to 9H)
1. Coating – cracking related Hertz tensile cracks
Tensile trailing cracks
Forward chevron tensile cracks
Scratch direction
Scratch direction
Scratch direction
Conformal cracks
Scratch direction
2. Coating – delamination related Buckling spallation
Wedge spallation
Recovery spallation
Scratch direction
Scratch direction
Scratch direction
Gross spallation Scratch direction
Figure 6-7 Scratch failure modes for a hard coating on a compliant substrate (after Chen & Wu, 2008).
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are used as scratching styluses under the same constant load, rather than increasing the load on the same hard stylus as the scratch develops to precipitate failure of some sort. Rather like the Mohs scale, there will be a pencil grade that just does not scratch the surface and one that just does. Different pencil grades result from different clay–graphite mixtures before extrusion into leads, before encapsulating in wood. The connexion between pencil ‘hardness’ and conventional ideas of indentation hardness is shown in Figure 6-8. Following Tabor’s comparison between the Mohs hardness numbers and indentation hardness (Section 6.5), semi-log plotting of the results in Figure 6-8 gives Hindent (const)(1.8)Hpencil. The ISO standard applies a vertical load of 7.5 0.1 N at the tip of the pencil that is inclined at 45° in a holder that is pushed forward over the surface to be tested. The end of every pencil lead is flattened to a circular cross-section before commencing a run, so that locally the surface is subjected, as it were, to a forward-moving disc having an attack angle of 45° (45° rake angle). Table 6-2 shows results for sol-gel coatings, of 5 m constant thickness, having different silica contents deposited on polycarbonate substrates. The table contains not only the pencil hardness data, but also independent mechanical properties of the films. Young’s moduli and hardness were obtained from nanoindentation tests, and the interfacial fracture toughness from controlled buckling tests. The pencil hardness of the PC substrate without any coating was 6B and its indentation hardness 0.19 GPa, so even with no added silica, the sol-gel coating having a pencil hardness of B provides substantial protection. At fixed silica content of 30.5 per cent, increasing film thickness gave increasing pencil hardness (Table 6-3) and increased scratch resistance. The failure mode in this type of coating was brittle cracking at the trailing end. Pencil grades harder than the pencil hardness number caused substrate gouging. What is curious about the results is that scratch damage occurs when the pencil lead is generally much softer than the coating. As some sort of mutual scratching test, it might be expected that at pencil hardness grades softer than critical, the surface would deform all pencil points; and at pencil hardness grades harder than critical, the pencil points would always deform the surface. Under constant load, the tips of softer pencil grades deform so that the contact stress is reduced (and graphite crumb is release in the track of the scratch); in contrast,
0.70
Hardness (GPa)
0.60 0.50 0.40 0.30 0.20 0.10 0.00 6B 5B 4B 3B 2B B HB F H 2H 3H 4H 5H 6H 7H 8H 9H Pencil grade
Figure 6-8 Nanoindentation hardness of pencil leads from grade 6B to grade 9H (after Chen & Wu, 2008).
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Table 6-2 Pencil hardness, indentation hardness, Young’s modulus and fracture toughness of coatings with different silica content. Colloidal silica content (vol. %)
0.0
6.7
17.6
30.5
36.0
Pencil hardness grade
B
HB
F
H
H
Indentation hardness (GPa)
0.59
0.62
0.66
0.71
0.88
Young’s modulus (GPa)
2.65
5.26
5.43
8.30
9.99
4.9
5.2
4.6
5.5
5.2
8.5
8.5
8.7
9.4
10.2
Coating thickness (m) 2
Fracture toughness c (J/m ) coating fracture toughness Coating thickness is within the range of 5 0.5 m.
Table 6-3 Increase in pencil hardness with coating thickness for 30.5% colloidal silica content films. Coating thickness (m)
5.5
7.9
10.5
13.9
20.5
25.1
Pencil grade
H
2H
2H
3H
3H
5H
harder grades retain the flat-ended, sharp, geometry and hence maintain the contact stress. A number of unresolved questions remains about this test: it appears to be some sort of Mohs measurement, yet clearly there are differences. Related questions concern whether all pencil hardness grades greater than critical produce the same damage.
6.7 Erosion Erosion occurs when material is worn away by impact of abrasive particles in a fluid stream. Sometimes erosion is a source of trouble (e.g. ingestion of dust into engines), but at other times it is employed intentionally to remove material (grit blasting) (Tilly, 1969). The dulling of car paints in dusty atmospheres is a form of erosion; the way different materials react to sandstorms is seen by the high erosion of the sphinx in Egypt but the comparatively low erosion of granite obelisks. Erosion can take place under wet or dry conditions. In grinding, grits follow a prescribed path but in erosion they do not. Their translational and rotational motion depends on the resistance presented by the target to grooving and material removal. The path taken, and whether a particle lodges in the target or passes through, is determined by the equations of motion, rather like bullets passing through, ricocheting off or embedding in a target. Finnie (1958) investigated erosion of ductile (high R/k) materials by analysing the motion of a single rigid abrasive particle interacting with the surface and estimating the volume removed from the trajectory of the particle. Finnie’s analysis for the resistance presented to a grit impacting the surface at a glancing angle was a two-dimensional Ernst–Merchant ductile cutting treatment in which certain assumptions were made to simplify the complicated cutting action of a tumbling particle. Chamberlain (2008) extended Finnie’s analysis by not only making the analysis three-dimensional but also incorporating work of separation via Eq. (6-1) for the resistance presented to a grit. Figure 6-9(a,b) shows that the R/k ratio has a significant effect on erosion at different incident angles, and the skewing of the curves to the left for more ductile solids accords with experiment (cf. Figure 6.33
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The Science and Engineering of Cutting κ=0 κ=0
κ = 0.1
L = 0.125
Erosion
Erosion
κ = 0.1
L = 10
κ = 0.5
κ=1 κ = 0.5
0 A
45 Impact angle (degrees)
90
0 B
45
90
Impact angle (degrees)
Figure 6-9 Erosion depends on a material’s toughness as well as its strength (hardness), and the volume of material removed by erosion depends on the impact angle. These results are from Chamberlain (2008), in which the parameter L is a measure of plastic work (small L meaning extensive flow) and the parameter is a measure of toughness work (increasing meaning increasing material fracture toughness). (A) Skewing of graphs when large plastic flow takes place so that maximum erosion rates occur at smallest impact angles for least tough materials; and (B) more symmetrical curves when only limited plastic flow occurs at different toughnesses. In both cases there is less erosion the greater the fracture toughness. The scales on the ordinates are arbitrary and different. For L 5 0.125, full scale is about 0.1; for L 5 10, it is about 1.
in Hutchings, 1992). Earlier references to be found in Papini and Dhar (2006) demonstrate that erosion depends not just on hardness but on toughness as well. Erosive particles alter their orientation along the trajectory and may even roll over in some circumstances (Hutchings, 1977). Even so, most analyses of scratching are concerned with the sliding of a hard indenter, rather than the sort of ‘three-body wear’ in which debris becomes trapped between the principal contacting surfaces. Note that Chamberlain’s analysis assumes for simplicity some average rake angle for the particle throughout its trajectory. The actual situation concerns a variable rake angle as the grit rotates and thus variable shear plane angle at different locations along the path. Scale effects of the sort discussed in Chapter 4 are shown in erosion. Sheldon (1966) showed that erosion of nominally brittle solids by very small particles is by ductile mechanisms whereas debris formation is by ‘knocking lumps out’ when the same material is eroded by larger particles. Hutchings (1992) remarks that when the parameter KC2/rH2 is small, material response is dominated by fracture and when it is large separation will be by ductile modes (r is the dimension of the grit). Finnie et al. (1967) found that the volume removed depended on (velocity)n, where 2 n 3. Finnie’s 1958 model predicted that n 2. To explain the discrepancy, the so-called ‘scale effect’ described in Section 3.6.7 was invoked, but the 1958 paper does not include toughness. The erosion and deformation of polyethylene by solid-particle impact is described and analysed by Walley and Field (1967). All the different chip types described in Chapter 4 were found in their comprehensive experiments from which deformation maps were constructed. A candidate model for wear by thermal mechanisms was shown to require much greater flux rates than used in their experiments. In another paper, Walley and Field (2005) review the long-standing work on erosion of all sorts of materials at the PCS Cavendish Laboratory in Cambridge.
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It is easy to visualize a hard sphere that forms its own ductile track by rolling. As with sliding spherical indenters, under light loads the attack angle is low and prow/ridging should occur; under heavy loads the attack angle is increased and material may be removed by cutting. Herbert invented a machine where a sphere was indented and then oscillated about the original contact point, elongating the indentation. The device was an inverted pendulum and has no connexion with the sort of pendulum used in Charpy testing and described above in connexion with grinding. With his pendulum, Herbert found that already hard metals could be workhardened further, and this led him to the idea of superficially workhardening material surfaces by putting them into ‘an atmosphere of hard steel balls flying about at high velocity bombarding the workpiece’. His first experiments had steels balls running down an old speaking tube from his office into a biscuit tin in the workshop below (Herbert, 1939). This was Herbert’s ‘Cloudburst’ process, which must be a very early example of shot peening; sandblasting to clean surfaces was already known before World War I. In erosion it is often assumed that the impacting particles remain undamaged and that only the target surface is affected. But mutual damage can occur in, for example, ice (hail) impact. Laboratory experiments with ice are inconvenient and Wang et al. (1995) have shown how nylon spheres may be substituted, with the proviso that the impact velocity is adjusted so that the nylon spheres fail in the same way as the ice particles. The vast majority of pebbles found in nature are shaped by the mutual abrasion of other like pebbles rolled with them in water courses and on sea beaches. How the shapes we find come about is discussed by Rayleigh (1944).
6.8 Definitions of the Coefficient of Friction Friction between unlubricated surfaces arises from two main factors: (i) junction shearing (adhesion) and (ii) (when one surface is harder than the other) so-called ploughing (Bowden & Tabor, 1950). The attack angles of surface asperities in practice are quite small, so that ploughing is most likely to be prow formation rather than cutting. The manner in which toughness contributes to the separation of locally welded surface asperities to give friction has been discussed by Atkins (2003). Many papers have modelled contact between surface asperities of different geometries in order to predict the coefficient of friction between surfaces (e.g. Goddard & Wilman, 1962; Childs, 1970). The coefficient of friction at these asperity contacts is defined as the sliding force F parallel to the surface divided by the force W normal to the surface. In papers on contact mechanics the relationship between forces N normal and F parallel to surfaces is occasionally written in terms of some F/N that is an input to the analysis. In models of scratching in this chapter (and indeed in other types of cutting), friction at the tool/workpiece interface is modelled as in metalforming. Where material flows over the surfaces of tooling, resistance to interfacial sliding is conventionally represented by Coulomb friction, or a frictional stress mk that is some fraction m of the shear yield stress k (sticking friction when m 1), or some mix of the two as discussed in Appendix 2. Such ‘bulk’ values of friction are the summation of all the microevents between chip and tool along the contact length. In turn, every microevent is the result of a local summation of smaller contacts and so on, rather like Jonathan Swift’s ditty in A Rhapsody: ‘So, naturalists observe, a flea Hath smaller fleas on him that prey And these hath smaller fleas to bite ‘em And so on ad finitum’.
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In scratching, it is clear that a coefficient of friction defined as Fscratch/W is not the same as the bulk friction between material and the face of the tool. Instead, is a measure of Fprow or Fcut under a given normal load. Under deadweight loading where the thrust force FT W, the ‘coefficient of friction’ given by Fcut/W is equal to tan( ), from Eq. (6-1), where is the rake angle, and where the presumed interface coefficient of friction tan. Rather like the excruciating screeches sometimes produced by chalk on a blackboard, noise can be produced when scratching the surfaces of superhard nanocomposite materials. Lu et al. (2005) analysed the phenomenon using a stochastic stick–slip friction model.
6.9 Engraving, Writing Tablets and Polishing The difference between groove formation by prow/ridging and by cutting is central to the various processes of engraving in which designs are incised on metal surfaces. The work of decorating silverware, ‘chasing’ weapons, putting names of winners on sports cups, is often done by hand. Engraving is also done with computer numerically controlled (CNC) milling machines and is automated in the production of rotogravure cylinders in the printing industry. However, an engraving usually means a picture (including bank notes, share certificates and formerly postage stamps) that has been printed, using ink, from a zinc or copper plate, on which the reverse image in a sunken design has been incised. Before photography, engraving was used to reproduce paintings: engravings in books and newspapers lasted into the early years of the twentieth century; photogravure was a means whereby negatives from early photography were transferred to metal plate. The skill of the engraver is seen in some of the very old illustrations scattered through this book. There are different techniques for forming engraved lines which give different results (Brown, 1996). The line may be formed by hand directly in the plate using a sharp, faceted tool (a graver), angled almost along the surface. As may be expected from the idea of attack angle in Section 6.3, different results are obtained depending on whether the motion of the tool is away from, or towards, the artist. The traditional manner of engraving is to hold the tool (a burin) in the heel of the hand and push it away, removing material out of the groove in the form of chips, to form a reverse image on the printing plate. Lines engraved in this manner have variable depths and tapered ends, made as the burin settles down into the metal or as it comes up out of it. Some skill is required by the engraver since his hand ‘load’ controls the position and depth of the burin that, with a large attack angle, will want to dig in. Roundended gravers are employed with metals such as steel and nickel that are difficult to cut, to avoid tool point breakage. Another name for an engraving tool is scorper (related to scalpel). When the tool is pulled towards the engraver, the technique is called drypoint, and material is displaced up into ridges (burr) alongside the line since its attack angle is now below Sedrik’s and Mulhearn’s crit. A groove formed in this way will be shallower than a groove of the same width cut by a burin moving away from the engraver and cutting. The heights h of ridges produced by facet-first triangular tools producing grooves of depth t given in Atkins and Liu (2007) are important for successful dry-point engraving. Sometimes a small metal wheel with protruding knobs (a roulette) is rolled across engraving plates to form interrupted lines that have a patterned burr resulting from the material displaced up around every small hardness indentation – so workhardened metal plates that will produce localized ridges are better than annealed (Tabor, 1951). This type of roulette is not to be confused with the one that perforates sheets of paper, etc., with patterns of cuts rather than holes (Chapter 8).
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Instead of forming lines directly, the plate may first be covered in wax or resin, and the required image cut into the wax deeply enough to expose the metal below. The exposed metal is then etched away by acid, thus forming lines. The depth of lines in an etching is determined by how long it sits in the acid, not by how the grooves are formed; the tips of etched lines are generally blunt rather than tapered like engraved lines. There are so-called hard-ground or softground waxes. Drawing through a hard ground with a pointed tool creates thin, even, wiry lines. Any tool sharp enough to penetrate the wax may be employed but usually a so-called metal needle is used, which looks like a pencil with a long, fine point. The tool may be moved both away from and towards the engraver (indeed it is possible to draw with the tool on the wax as one would sketch with pencil and paper). Varnish is sometimes used as a hard ground, but it can be too brittle and susceptible to chipping as the tool is taken across the surface, exactly as described earlier for scratching on brittle coatings (Section 6.3). Rembrandt combined etching techniques with drypoint and was the first artist to make drypoint engravings working directly into copper plate without an etching ground. Instead of metal plates, lithography employs stone plates coated with wax on which lettering or designs are incised for printing. To print both engraved and etched plates, the plate is inked with a roller, the excess wiped off and the surface wiped clean, thus leaving ink only in the grooves. The plate is then run through a press with a sheet of paper in contact. What results is a series of inked lines on a ground colour of the paper (so-called intaglio printing, from the Italian for ‘cutting in’ or engraving). The character of the lines, particularly the density of ink and the clarity of the edges, depends on how the reverse image was made on the plate (by ridge formation or by cutting). Cut lines (tool moving away) produce sharp images. Printed drypoint lines (tool moving towards engraver) look irregular and fuzzy because the ink is held by the burr which, sitting above the surface of the plate at the sides of the lines, plays a bigger role. Instead of the colour of the paper, the background of an intaglio print can be the colour of the ink, with the picture appearing as lines the colour of the paper. The result is called a mezzotint. The effect is achieved by first roughening or texturing the plate surface with a hand-held rocker having a curved bottom serrated with miniature spikes. As the tool is rocked all over the plate, the spikes dig into the metal like the roulette mentioned above, and displace material up into burrs on the surface as conical indenters do (so, again, workhardened metal plates are best for mezzotints too). The lines of the image are created by smoothing out the texture by a burnishing tool. Complete removal of the texture, by burnishing or scraping away down to virgin metal, gives a line that will not retain ink. However, different ink-retaining abilities, and hence graded colours and the ‘velvety tones’ characteristic of mezzotints, are achieved by leaving different depths of burr and pit after burnishing. The main use, though, of burnishing and scraping in intaglio printmaking is to correct mistakes. The existence of the resulting hollow regions in the plate is accommodated by placing a soft, compressible, layer (blanket or swan skin) under the paper in the printing press. Instead of metal, blocks of wood or pieces of linoleum backed by wood may be used for the printing ‘plate’. Although the technique and tools are different, woodcuts and linocuts are like mezzotints, i.e. they print in ‘relief’ with ink retained on the surface and the incised lines appearing as the colour of the paper. There is no ‘overflow’ (meaning the fuzzy edge associated with burr) since all lines are cut. Tools employed are essentially wood chisels and gouges but, like wood carving tools, some have curved shanks to reach into awkward recesses. An important characteristic of wood for engraving, carving and sculpting is closeness of grain; box and lime are often used. Figure 6-10 shows a Chinese craftsman cutting a ‘chop’ or name stamp. Wood is said to be engraved in the end grain and cut in other directions.
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Figure 6-10 Name being cut in relief by Chen Linlong, a Chinese craftsman.
When an engraved plate, woodcut or linocut can no longer print a satisfactory image, the plate is discarded. What constitutes ‘satisfactory’ is subjective. Drypoint plates wear out quickly because the raised burr breaks down under the pressure of printing. However, wornout plates can be reworked or recut. This gives rise to different editions of a print, and these may have different market values. The plates of engraved postage stamps also wore and philatelists make great play of different stamps printed from original and reworked plates, as inspection of stamp dealers’ catalogues will show. There also you will find detailed differences between different plates made by different engravers of nominally the same stamp. Engraved banknotes are often the work of more than one engraver in an attempt to thwart forgery. Scratched decoration in wet pottery, wet plaster and so on comes under the general heading of scraffiti, a term also used where gold is revealed by scratching through paint (fifteenth century Siennese painting). Fresco is painting into wet plaster. (J.M.W. Turner and his contemporaries scraped and scratched the surface of their wet paintings done with water colour and gouache.) Scraffito is known from ancient times. Artists contrast the ‘static’ decoration achieved by tools simply impressed into a pot, and the freedom of movement achieved by scratching. The width and depth of lines depends upon the size and shape of a tool, whether it is pointed or round-ended, and the angle at which the tool is moved across the surface to cut or ridge-up material. Tools may have a single point or multiple points (artists scratch with hacksaw blades or finger-comb with the hands). The result depends on the (R/k) ratio of the clay which changes with drying-out: different conditions of clay when scratched are known to potters as ‘wet’, ‘cheese hard’ or ‘leather hard’, but what those mean in terms of (R/k) is not known. Pargetting is the incision of designs in the cement renderings of the outside walls of houses; it is a feature of many houses in East Anglia in England. The technique relies on
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a correct cement mix that will have an optimum (R/k) so that (as in ordinary plastering of walls) the design does not end up around the amateur plasterer’s feet. Tools similar to those employed by engravers are used in finishing work by sculptors, stonemasons and wood carvers (Brett, 2000). With epitaphs cut into gravestones by light taps on a chisel, the displaced material comes away as small chips or powder owing to the ‘more brittle’ character of minerals than engraving plates at comparable depths of cut: the cutting of very hard stone involves more pulverizing than cutting. The cleanness of the cut edge depends on the material and the sharpness of the chisel. Limestone, for example, is full of what once were shells: a blunt tool will make a feather/fuzzy edge by knocking out whole shells rather than cut through them. Lettering may be done freehand (there is great beauty in eighteenth century hand lettering on tombstones) or may be done by tracing with a pantograph from an alphabet of larger prototype letters, using a rotating cutting tip. The outline of the letter is made first at progressively deeper cuts, then the middle is taken out. Instead of being cut, lettering could be scratched on clay tablets before being baked for preservation. Roman legionnaires received metal tablets on retiring from the army that gave the record of their service. Here the message was punched (indented) into the sheet from above to give recessed letters. Statuary has all aspects of cutting and polishing found in general manufacturing: flat regions, ledges, curves in two dimensions and in three, re-entrant regions, protruding regions and so on. Michelangelo is said to have broken off Christ’s left leg, left arm and the Madonna’s elbow in his Pieta (Museo dell’Opera del Duomo, Florence) owing either to dissatisfaction with his work or during sculpting ‘owing to an imperfection in the grain of marble’. It may be possible to resolve this question using interactive light techniques (Wassermann, 2000) in which original tool marks on polished statuary may be discovered using interactive light techniques. Between 1811 and 1818, Sir Francis Chantrey (the leading portrait sculptor of his age) corresponded with James Watt on developing sculpture machinery (Peltz, 2006). Chantrey was concerned with a ‘pointing machine’ and Watt with a ‘carving machine’ for the production of reduced-size replicas of statuary in durable materials. (Watt’s pantograph is preserved in the Science Museum, London.) Designs in gems and jewellery are made in a similar fashion to carving and sculpting. Designs incised below the starting surface are called intaglio, as used in engraving; when material is cut away around the design so as to leave it proud of the background surface, it is called relief (French relever, to raise), rather like a rubber stamp. The depth removed below the original surface may be small (to give a bas-relief sculpture, as in Ancient Greek pediments, from ‘bass’ meaning low) or great (to give high-relief). In jewellery, low-relief designs are called cameos, where often a second stone of a different colour lies below the upper stone in which the design is cut, to form the background as in onyx, agate and shells. Polished surfaces differ only in degree from scratched surfaces. Something to be polished, whether a metallurgical sample or gemstone, is first sawn and ground to the desired shape before being subjected to ever-finer abrasive papers that remove rough marks left by coarser grits. Metallurgical samples need to be ground to one flat surface and are usually held within a mount; gemstones are attached with adhesive wax or glue on the end of a metal dopstick, which is then inserted in a fixture that allows precise control of positioning. Depending on the finish required, lubricated/cooled lapping follows which is performed on a rotating flat disc made of cast iron, steel or a copper–bronze alloy. Further processing on similar wheels with slurries of different polishing compounds – very fine powders of diamond, metal oxides or other materials – results in a polished surface. Metal polishes for brass, copper and silver used with polishing cloths are similar. Jeweller’s rouge is ferric oxide. Materials like chalk make an excellent polishing medium for hard materials like quartz (Hollnagel, 1923). Owing
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to the slow rates employed in polishing, the degree of subsurface deformation is quite limited; that is not the case for cutting at high speed. When polishing to optical requirements at even finer scales, the trick for finishing is to use fine grit embedded in pitch, the role of which is to even out the overall load among active abrasive particles. Highly polished flat, cylindrical and spherical surfaces of glass are made in this way. The grits rotate and ‘find their own level’ so as not to gouge the surface. In electronics, silicon wafers are polished by grits held in an elastomer backing for the same reason (Gee, 2009). In addition to reflexion of light from a polished face, transparent gems refract light through the stone. So-called cabochon jewels have rounded surfaces rather than flat (from the French for rounded head or cabbage); they are often polished on cloth, cork, felt, leather or wood. Huygens (1728) seems to have been the first to report on crystal anisotropy and Babbage (1832) points out that some faces of diamond are easier to work than others on a cast iron lap, owing to anisotropy (see also Wollaston, 1816). This is important in the cutting of diamonds and it is also relevant to why diamond chips will cut other diamonds. A paper of historical interest on cutting, grinding and polishing of crystals is that by Tutton (1895).
Chapter 7
Sawing, Chisels and Files Contents 7.1 Introduction 7.2 Knives and Chisels 7.3 Saw Teeth 7.4 Files
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7.1 Introduction The principal difference between separating a solid by sawing and by cutting with a knife is that there is waste (sawdust) after sawing. In a plain knife the cutting edge is continuous and in the same plane as the motion of the knife. Separation takes place by a wedging action of the blade that moves into (and may be also along) the material to be cut. Pieces severed by knife (a sheet of paper cut through the middle with scissors or guillotine, for example) may be reassembled to regain the original piece; the permanently deformed helical shape taken up when a narrow strip is cut from the edge of a sheet of paper cannot. In both cases, however, no material is lost. A saw consists of a series of narrow single cutting edges (teeth), arranged either along a straight edge (reciprocating hand saw, continuous band saw) or around the circumference of a disc (circular saw). Specialist saws include hole (trepanning) saws where the teeth are arranged around the end of a tube; the corresponding ‘hole knife’ is the cork borer, apple corer or hollow cheese sampler. The cutting edges of saw teeth are perpendicular or oblique to the direction of motion of the saw and this distinguishes saws from knives. In sawing, every saw tooth acts like a small cutting edge that moves perpendicular to the direction of cutting; every tooth removes a small amount of material, thereby cutting a slot, and material is lost as sawdust. Usually parts of the body on either side of a saw cut are not permanently deformed and when sawn pieces are refitted together, they are shorter or narrower than the original. In plan view, saw teeth are bent (set) sideways alternately to right and left, and therefore cut a path slightly wider than the thickness of the blade. This reduces contact between blade and workpiece, hence reducing friction, heat generation, possible overheating and damage of the blade. It prevents the saw being pinched and possibly jamming in the slot; it also lets the sawdust escape. The sideways splay of the teeth is called the kerf and it determines the width of the slot and volume of material lost. It is desirable to have the smallest kerf possible in order to increase the volume recovery from the starting workpiece. In the eighteenth century, felling of timber for mining and for the thousands of small water-driven sawmills in Norway was rather wasteful. In 1739 a forest administration (the Generalforstamt) was established to regulate timber harvesting. To save timber, only blades of the so-called Dutch type and less than 1/6th of an inch thick were allowed (Gjerdrum, 2006). Some modern saw blades are made with the back of the blade thinner than the working edge so that no kerf is necessary; these blades are designed to operate on the pull stroke. Copyright © 2009 Elsevier Ltd. All rights reserved.
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Since saws go deep into workpieces, there are problems about disposal and clearance of sawdust and problems about friction and heating up between the sides of a blade and the sides of a slot. Gullets are an important feature of saw design for this reason. Gullets are the spaces into which the chips travel from the cutting edge. Gullets fill up and sawdust can only empty when the adjacent part of the saw blade emerges from the workpiece into fresh air. The function of gullets is similar to that of the flutes on twist drills. As in drilling deep holes, it is sometimes advantageous to remove a saw from a cut to clear out the spaces between teeth, particularly when cutting soft/sticky materials that transfer to the surfaces of the cutting blade. Gullets, as such, are not found on carpenters’ saws, but are present on the blades of bow saws and logging saws, where weird and wonderful tooth patterns may be found. There are different tooth profiles to suit different materials; and different profiles for different directions of the same material when anisotropy is pronounced. A well-known example is wood, for which rip saws with coarse teeth are employed for cutting along the grain and cross-cut saws with smaller teeth and pitch for cutting perpendicular to that. Chain saws perform badly when cutting along the grain because the teeth are cutting off the tops of the ‘drinking straw’ wood cells (Chapter 4) rather than separating between them. Knives may be used in carpentry on thin sheets. Allowance must be made for the width of the saw blade when cutting to a predetermined size, but that is hardly necessary with a knife. Different tooth profiles are used in combination along blades especially for ‘coarse’ cutting. In so-called ‘skip tooth’ band saw blades, alternate teeth are deliberately omitted to provide more room in which to remove chips. There are saws whose ‘teeth’ are abrasives, such as abrasive cut-off circular saws and angle grinders, which are thin grinding wheels with grits on the sides. They operate at high speed and are able to cut hardened materials that are difficult to cut by conventional methods. Power-driven saws have a long history. There is a picture of a sixteenth century water-driven flour mill in Lester (1975) that has a masonry cutting saw driven by the outflow from the overshot water wheel. Stone is first cut in quarries then split by wedges. Molari (2007) refers to an ancient water-driven saw illustrated in the Basilica at Urbino that has a mechanism by which logs are indexed forward at each stroke. A gang saw has several parallel blades for making simultaneous cuts. There is, perhaps, confusion between the action of saws and scalloped or serrated knives, such as some bread knives, steak knives and so on (Figure 7-1). Many such blades have a pitch (distance between points) of about 5 mm: when the depth of cut-away is smaller than
Scalloped
Serrated
Figure 7-1 Scalloped and serrated knives.
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the pitch, it is a scalloped edge; when the depth of the recesses is greater than the pitch, it is a serrated edge. Inspection in shops shows that there are numerous varieties. What is intended by these forms? Is the effect on cutting real or illusory? When manufacture of such knives produces a continuous sharpened zig-zag edge along the whole blade, it is still a knife and should cut without waste. If the zig-zag teeth of a knife are sharp at the tip but retain the thickness of the blade, the action is part knife and part saw, and waste will be generated. If the zig-zag teeth retain the blade thickness at the tip as well, and sharpening is at right angles to the direction of motion, it is a saw. Even so, such knives rarely have any kerf, and are not really proper saws. Any sort of cutting action that involves sideways reciprocating motion, as well as motion into the material to be cut, is commonly called ‘sawing’ by the person in the street (e.g. string players ‘sawing’ in tremolo passages) but that is not really correct if by sawing one means cutting that produces waste. It is a moot point as to the purpose of zig-zag knife edges in terms of cutting mechanics. A plain knife that is sharp will cut bread just as well as a serrated bread knife. Soft foods will compress into the sides and bottoms of serrations and give a longer cutting edge which may be thought of as desirable. But, in practice, there is snagging and tearing of the workpiece material sitting in the hollows, which not only results in a poorer quality cut but also absorbs more energy in doing so. However, it is difficult, for example, to cut a banana skin with a blunt plain knife that just slips over the surface, and scallops and serrations indent and give grip. A blunt point will perform better than a blunt edge. Camping shops sell saws in the form of coils of flexible wires notched to create ‘teeth’. They look like abrasive wires, but are not. To make a saw for firewood, etc., a stick is used as a bow. It might be thought that they would be useful for cutting high branches in a tree. A fishing line attached to the wire would be fired over the branch with a catapult. With a length of rope on each end of the wire saw the idea is to cut the branch from ground level. In an experiment, however, friction increased the load in the wire so much that it broke in tension (Chaplin, 2008). As with ‘machinability’ (Chapter 9), there are different meanings of ‘sawability’, i.e. best combinations of feeds, speeds and depths of cut. But ‘best’ for what? – production rate or tool life, or something else? What are the fundamental controlling factors for sawability given by the theory of cutting? What properties of tool/workpiece make a material difficult to saw?
7.2 Knives and Chisels Why are saws employed rather than knives or chisels? While it is possible to cut soft materials with knives, and even cut slivers from harder solids (e.g. the whittling of wood), the forces required to cut big sections of hard materials with knives become very large. This comes about not only because of the hardness and toughness of the workpiece, but also because of friction. Thin knives have flat sides that give a large area of sliding between tool and offcut, and hence high frictional forces (a wire is preferred over a knife to cut cheese owing to high friction; Chapter 12). Furthermore, to maintain size control of the offcut when cutting under high forces with a knife, the blade must have high stiffness, and that means that the blade has to be thick and deep. In consequence, there are high side forces on the blade as it is wedged down into a cut, thus further increasing friction. Even if all these problems are overcome, wear of the thin cutting edge of a knife is more rapid under high cutting forces. A chisel is a type of knife but unlike a hand-held knife where ‘sawing’ often occurs, the principal motion is perpendicular to the cutting edge. Bolsters are chisels splayed out at the edge for cutting bricks. Chisels are tools widely used in woodworking and, years ago, the cold
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chisel was a basic tool of the fitter and turner to cut metal. Things such as slot links in railway valve gear were once prepared with hammer and chisel, later by shaping or milling machine. Apprentices were taught how to shape pieces of steel by hammering with a cold chisel (in the author’s case, an exercise of chiselling a round bar into a square bar). The variety of chisels in use at the Midland Railway workshops at Derby just before World War I is shown in Figure 7-2 (Fowler, 1916). The length of a cold chisel is typically 250 mm; in former times two-handed chisels, perhaps 5 feet long, were employed striking with a 14 lb sledge hammer when no machine tools were available. Chisels are held against the material to be cut and, in the case of wood chisels, either pushed by hand or tapped with a mallet; cold chisels are struck with a hammer to achieve the required force (Chapter 2). A woodworking plane is a chisel installed in a frame with a flat bottom to control the depth of material removed. Chisels can have cylindrical blades with the sharpened bevel on either the convex or concave side (gouges). Different materials respond differently to chisels: a large piece of timber can be split down the grain by a blow from a chisel (or by the wedge of a log-splitter) but it is impossible to split a block of metal by hand with a cold chisel owing to the much higher toughness of metals and the fact that the deformation is in the ductile range (see scaling in Chapter 4). All that can be achieved is removal of small amounts from surfaces, which is the basis of all metal-cutting processes. An axe is a chisel on the end of a long shaft that directs the axehead to where it is aimed. Throwing an axehead at the target would not be as effective, even assuming that the correct location could be hit every time. Even so, guillotines used for execution are essentially axeheads let fall under gravity through guides (Chapter 11). Muscles raise the axe on high and accelerate the head on to the workpiece. To withstand the impact, axeheads are chunkier than chisels and have considerable mass which increases the force produced (see Chapter 2): many blows are required to fell a tree with an axe since cutting is across the grain where toughness is greatest. The design of the tool is important for the application: not many trees will be felled by a razor blade attached to the end of a haft, but such a device can form a nasty slashing weapon. Axes are used these days principally with wood, but in mediaeval times the battleaxe and poleaxe were weapons used for fighting on foot (Chapter 8) and masons trimmed stone with axes. The poleaxe, once used for stunning animals prior to slaughter, had a hammer opposite the cutting edge. Accepting that not all cutting can be done by knives, separation of an object into smaller parts could be achieved by a wood plane or a shaping machine in which the blade was made to go deeper and deeper in a series of single passes to cut a slot. Friction and heating up of the system could be reduced by relieving the sides of the tool. But the blade/tool depth would have to be reset each pass. (It would be like the process of parting-off on a lathe, except that continuous rotation of the workpiece and the ability to feed the tool mean that it is not necessary to keep going back to the beginning of the stroke on a lathe.) To reduce the width of slot, and hence material lost as scrap, it would be desirable to have a ‘thin’ tool, but then problems would arise of tool stiffness, etc. In place of the above ideas, it is found more expeditious to have a ‘saw’ that cuts deeply into a workpiece, and usually completely through the thickness.
7.3 Saw Teeth The title of a book on woodworking by Effner (1992), Chisels on a Wheel, aptly brings out the connexion between saws and chisels. Every tooth of a saw, whether a straight-edged, circular or a hole saw, is essentially a little chisel that removes material in the way described for orthogonal or oblique cutting in Chapters 3 and 5 with tools having negative or positive rake
175
Sawing, Chisels and Files Diamond point for jagging, etc.
Heavy brass work.
3" 1" 2
1" 116
4'
9"
9" 7" 16
11°
Long cross cut.
Heavy iron and steel castings.
3 3" 4 1" 116
3"
25°
9"
1'.0" 1" 2
17°
Round nose. 4'
Cylinder repairs. (Right hand.) 2"
Flats 9" 16
1" 4
9"
3" 8
7" 8
6"
Gouge tool. 5" 8
3" 7" 16
1"
Side tool. (Right hand.) 2"
1.0"
9"
25°
White metal.
Square nose.
1 1' 2
1'.0"
2"
3" 8
3"
3" 4
9" 10 1° 2
Figure 7-2 Various types of chisel used at the Midland Railway works at Derby before World War I (after Fowler, 1916).
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The Science and Engineering of Cutting
angles. There are many different varieties of saw tooth profile for cutting different materials but all aim to give a blade along which, in normal use, the teeth do not break while cutting, that does not wear too quickly and remains sharp, that permits the sawdust to escape, which does not jam by friction during cutting the slot, and where cutting can be performed with least effort. Clearance or relief angles on saw teeth are different for different materials: very dense or wet woods, for example, require bigger reliefs than dry softwood. Rate and temperature effects on mechanical properties will affect cuttability. In certain forests, logs can only be extracted when the ground is hard during winter. Cutting of frozen logs presents several problems for sawmills: there are difficulties of debarking, an increase in unit power is required, and the resulting high cutting forces may damage saw blades (e.g. cracks in saw blade gullets) (Orlowski & Sandak, 2005). For reciprocating saws, the geometry of tooth depends on the direction of cut. Hand saws in the West usually cut on the forward stroke, but in earlier times in some Asian countries cutting was designed to take place on the pull stroke (teeth pointing backwards towards the operator), e.g. Japanese azebicki saws (Lee, 1995). As mentioned in Chapter 9, the names of some special Japanese knives incorporate ‘pull’. As buckling is not as important as in a push saw the blade can be thinner, so that there is less sawdust. Its behaviour is related to wasp ovipositor ‘drilling’ (Chapter 11). Some designs of reciprocating saw blade are intended to cut on both strokes: bow saws and pruning saws used in the garden fit this category, as did large two-man saws used in sawpits to cut planks from logs and to fell trees. (The expression ‘top dog’ may have been associated with the sawyer at ground level at the top of the pit.) Other saws operate in one direction only, such as band saws, circular saws and chain saws. The latter are portable band saws in the form of ‘cutting caterpillar chains’; chain saw tooth profiles are different again from most other types. There is a specialist machine employed in the furniture industry for cutting mortises (female slots to receive male tenons in joints) that is like a miniature chain saw. Power hack saws are among the slowest cutting devices: they cut only on the forward stroke and time is wasted on the return (as also in a shaping machine and in filing). While it is possible to produce in sawing all the different chip types described in Chapter 4, saw blade design aims to avoid continuous ribbons owing to disposal. The shape of the gullet, and its relation to the cutting edge, determines how chips are broken into segments, compressed (in the case of wood) and possibly repeatedly recut to form the sawdust. Blades are manufactured with different intertooth spacing (or ‘pitch’), which is characterized by its reciprocal, i.e. the number of teeth per inch (tpi) in the imperial system. It is not clear whether different-size teeth are intended by manufacturers to be geometrically similar (i.e. blades with different tpi are all different magnifications of the same shape), but it seems to be so in many cases. Choice of tooth pitch depends upon the thickness being sawn. It is desirable to have at least three teeth cutting at one time to produce a successful cut: a pruning saw will not successfully cut thin sheet metal. The number of teeth in engagement can be altered by angling a straight saw, by cutting at the top or bottom of a circular saw rather than at its mid-position, or by altering the orientation of the workpiece such as cutting plates on edge rather than on the flat (Herbert, 1939). It is possible to cut thick pieces with blades having closely spaced teeth, but the rate of material removal is reduced. The quality of the sawn surface (roughness, etc.) is, however, improved with finer teeth (e.g. wood cut with a hand hacksaw blade). When it is intended to saw a flat surface, the depth of a blade is much greater than the material being cut (for accurate surfaces the hand-held tenon saw in woodworking is stiffened on its back by a steel or brass band). When curved cuts are required, the depth of the blade has to be small enough to permit cutting round bends (as in a fret saw).
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Sawing, Chisels and Files 7.3.1 Action of a row of teeth in a straight saw
The cutting action of every individual saw tooth is essentially the same as described in Chapter 3 for orthogonal cutting and in Chapter 5 for oblique cutting (and in Chapter 6 if the teeth are ‘bevelled pointing’, i.e. V-shaped in plan view). In the past, the mechanics of the cutting action of a single saw tooth has been modelled in terms of the Ernst–Merchant analysis for continuous orthogonal cutting originally proposed for metals. It is sounder to include the fracture toughness as well as yield properties and friction. A saw will not cut unless there is motion into the workpiece as well as motion parallel to the surface. The same is true for files and food graters. Consider Figure 7-3. Let the velocity of the blade along the cutting direction be h and that down into the material be v. Usually h v, so the ratio h/v is large. For teeth of pitch p, the time taken for the tooth at B to travel to the original position of the next tooth ahead at A, is (p/h). In the same time interval, the tip of the tooth has moved into the material a distance (p/h)v to reach position A that is therefore distance (p/s) below the position of A. The ratio s h/v is superficially similar to the slice–push ratio of Chapter 5, but here it is not slice–push since the teeth are perpendicular to the plane of the diagram. Only when the teeth are angled to the direction of h (i.e. oblique teeth in-and-out of the paper) is there any with saw teeth, and then it is geometrically produced , not one resulting from independent motions along and across the blade). The resultant displacement along AA is (h2 v2) v(s2 1). The cutting action of the moving saw is as if it were travelling along BA with an ‘effective rake angle’ * given by * ( ), where is the rake angle of the tool with respect to the perpendicular direction across the blade, and tan (1/s). Since s is usually large, the difference between * and may be neglected; similarly for the difference between the effective depth of cut t and the value of (p/s). The material cut by the single tooth over a stroke p is parallelpiped with volume (p2/s)w, where w is the width of the slot cut by the saw. (Practical teeth, set to each side of the centreplane of the blade, cut the total slot width in two parts, one to the left and one to the right, where the spacing between teeth on each side is 2p and the width of each cut is (w/2), but this also results in an intertooth volume (p2/s)w being removed.) The rate of material removal per tooth is (p2/s)w ÷ (p/h) (pwh/s) pwv. For teeth of pitch p engaged over a cutting length L, the number of teeth in contact N is given by (L/p), whence the rate of material removal is wvL, as expected. h w
v
A
B
p
Aʹ
Figure 7-3 Motion of the teeth of a saw blade during a cut. Inclination of the saw blade to the horizontal is .
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The Science and Engineering of Cutting
The difference between simple orthogonal cutting and the production of a slot by a saw tooth is that surface work is required to separate the sides of the slot as well as its bottom. The height of the sides of the slot per cut is the same as the effective depth of cut, so the new surface area is given by (2t w)p. (In set blades, owing to the staggered formation of the slot, this expression must be modified to (3t w)p.) From Eq. (3-23b), the force along AA on a single tooth is given by
FAA (1/Q){[k w γ ] t R(2t w)}
(7-1)
in which φ (in and Q) is the optimized value based on , t and . Hence
FAA (1/Q){[k w γ 2R] [p/s] Rw}
(7-2)
and
FperptoAA FAA tan (β α)
(7-3)
These are the forces per tooth along and perpendicular to AA during the cutting portion of the stroke. The total force for N teeth actively cutting is simply N times the force per tooth, and would be the forces measured by a dynamometer. Equation (7-2) suggests that FC per tooth should plot linearly against (1/s). The slope of such a plot is (p/Q){[k w 2R] and the intercept is (Rw)/Q. Orlowski (2007) applied these ideas to cutting pine with a single saw blade and predicted the behaviour of multiple blades in a sash gang saw.
7.3.2 Action of a tooth in a circular saw Crescent-shaped chips are formed by the teeth of circular saws owing to the trochoidal path swept out by the tip of the tool resulting from the combined rotary motion of the saw and linear movement (feed) of the workpiece; or, in hand-held rotary saws, by the rotary and linear motion of the saw blade when the workpiece is stationary. The trochoidal motion leaves shallow scallops in the cut surface (Figure 7-4A). When the direction of the circumferential velocity of the saw is opposite to the feed of the workpiece, chips ‘start thin’ and become thick; when the saw has a peripheral speed in the same direction as the feed, the full depth of cut is taken on first entering the workpiece. Circular sawing is similar to routering and cylindrical milling of metals, with similar ‘upmill’ and ‘downmill’ connotations. Having cut, a tooth passes a second time through the slot just cut in the workpiece. The two motions produce two sets of cycloidal marks on the cut edge (Figure 7-4B), the spacing of which depends on the number of teeth on the saw. Residual marks on surfaces cut by a band saw or gang saw are different. The angles of the teeth on a circular saw are defined with respect to the disc of the saw. Because of the combined rotary and linear motions, the rake angle of a tooth with respect to the workpiece changes progressively during a cut (Figure 7-4C). When cutting wood, problems of splitting and tearing of the lower surface of a plank can be caused by unfavourable (large positive) rake angles as the blade emerges from the stock. Typical rake angles for wood-cutting circular saws are 10° for chip boards having brittle coatings, through 0° for hardwoods to
179
Sawing, Chisels and Files
A
B
C
Figure 7-4 (A) ‘Upcutting’ circular saw showing thickening chip and scallops left on the cut surface by combined motion of feed and rotation of blade having spacing between teeth. (B) Cycloidal marks left on a cut edge owing to the teeth of a circular saw passing twice through the workpiece, once when cutting and a second time through the trailing cut slot. (C) The rake angle of the teeth of a circular saw is fixed with respect to the radial direction of the blade. The rake angle with respect to the workpiece alters as the tooth passes through the thickness of the work, (after Effner, 1992).
20° for ripping softwoods. One solution for bad undersurfaces is to alter the height of the saw appearing above the worktable. As far as wood is concerned, the advantages and disadvantages of the two ways in which circular saws may be used are listed by Effner (1992). Climb cutting (i) gives a smooth finish when cutting along the grain because the leading edge rubs against the material and burnishes the surface as the saw tooth completes its arc (ii) minimizes the ‘grain tearing’ effect when cutting end grain or at an angle to straight grain (iii) permits greater feed speeds to be employed while maintaining surface quality. The disadvantages of climb cutting are that: (iv) the material must be very securely held while being fed to the cutting tools to avoid grabbing of the workpiece by the saw
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The Science and Engineering of Cutting
(v) more power is required to cut in comparison to downhill sawing since short, blunt chips take more energy to form than longer, slimmer chips (vi) the greater power requirement loads the machine more severely and causes impact vibration problems (vii) the saw teeth require more frequent sharpenings owing to the rubbing/burnishing at the end of the cutting arc. A feature of power-driven woodworking saws (and indeed woodworking tools generally) is that they travel at very high speeds, far higher than corresponding metal cutting saws and tools. For example, a 300 mm diameter circular wood saw might have 70 teeth and run at 4000 rpm. The rim speed is about 65 m/s (240 km/h or 150 mph) and 70 4000/60 nearly 5000 saw teeth enter the workpiece every second, sending chips into the gullet which are thrown out again as the saw leaves the workpiece. Band saws run at 10–60 m/s. At these high speeds it is arguable that the work to accelerate the chip, given by the kinetic energy imparted, ought to be included in analyses of cutting. In metal cutting, the chip momentum change is customarily ignored. At these very high speeds, thin blades often result in ‘wandering cuts’ that give thickness variations in the product, owing to loss of saw blade dynamic stability (Orlowski, 2003). Dynamic balance of large circular saws is vital and circumferential tension rings are regions of the saw disc that are mechanically hardened by pinch rolling to promote wobble-free running. A steam-driven circular saw for cutting hot steel blooms at the LNWR locomotive factory at Crewe is described by Webb (1875). The saw was 7 feet in diameter and ran at 100 rpm, giving a circumferential speed of 25 mph. The teeth were 5/16ths of an inch thick and had a 1 1/2 inch pitch.
7.3.3 Routers A router in woodworking originally meant a moulding plane for producing profiled surfaces. (The word router comes from the same source as ‘root’ meaning to scoop up; pigs root up soil, as discussed in Chapter 14.) Nowadays routers are the woodworking equivalent of milling cutters employed in metal cutting. One or more cutting edges (straight or helical) are mounted on the cylindrical body of the bit with gullets or flutes to permit removal of chips. When cutting edges are mounted on the ends, routers plunge-cut like end-mills in metal cutting. However, routers are intended only for shallow holes: there are no flutes on routers and debris removal in plunging relies on air movement. Routers having profiles on their ends usually operate on edges of workpieces so that debris removal and overheating are not a problem. In routers having long working faces, a single cutting edge may be divided into two, one angled one way to the generator of the cylindrical body, one to the other, rather like herringbone gearing. This balances the side forces. Helical cutting edges result in slice–push action, and cutting forces are reduced over straight-edged routers. Lower forces may be employed to produce better surface finish; they are also an advantage in cutting particleboard laminated with hard and often brittle polymer surfaces. The cylindrical bodies of routers are relatively small. If fast linear feed rates are to be achieved in manufacturing without inferior surface finish (scalloped surfaces, for example), routers have to rotate at very high speeds. Routers up to 20 mm diameter rotate at anything between 16 000 and 40 000 rpm, resulting in peripheral linear speeds of some 20 m/s.
Sawing, Chisels and Files
181
7.3.4 Empirical relations As in metal cutting, many empirical relations for cutting forces and power in sawing have been published. Empirical relations for forces typically take the forms
FC K1t
FC K3 K2 t
FC K 4tn
FC K6 K5tn
Values are tabulated for the constants K for different materials in different states (metals having different thermomechanical treatments; wood having different moisture contents; plastics having different molecular weights), all cut at different speeds, etc. The range of sawing parameters over which different formulae have been obtained may very well have been limited, and it is far from clear that the formulae can be used predictively. Also it is not clear that such formulae recognize the different chip types that occur in practice, and lump all data together. In the case of timber, the McMillin–Lubkin (1959) equation has wide currency for predicting the power P involved in the circular sawing of wood. It predicts P t(C Df)
where t is depth of cut and f is feed rate, and C and D are constants. Other empirical formulae for wood can be found in Ettelt (1987) and Maier (2000). Empirical relations for cutting soils are given in Chapter 14.
7.4 Files The action of a file is similar to that of a saw (the surface has to be indented and there is motion into the workpiece as well as along for the device to function), but a file is intended to cut surfaces (round as well as flat) rather than slots. The action of files is somewhat similar to that of food graters, but most graters permit the debris to pass through the body of the device to escape on the free surface, which cannot happen with files. There are grater-type tools intended for wood and soft metals which have replaced files in some applications. Like many saws, files do not cut on the backstroke. Files were essential hand tools for the fitter in the days before mass production. The earliest screw threads would have been cut by filing. A right-angled triangular template was cut out in which the shortest side is the pitch, and the intermediate side has the circumference of the bolt; the template is wrapped around the rod and the thread marked out by the template being shifted along the rod; and filing is started. Leonardo designed a screw cutting machine; as with similar sketches it is not certain whether it represents reality or imagination. The Penny Magazine Supplement (1844) reported that ‘… [files], simple and unimportant as they may seem, are the working-tools by which every other kind of working-tool is in some degree fashioned. Whether a man is making a watch or a steam-engine, a knife or a plough, a pin or a coach, he would be brought to a stand if he had not files at his command …’.
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Files still have a place in the workshop today and are also used for manicure, for removing hardened skin from the feet, and for removing burrs on the edges of skis that have passed over rocks. The working surfaces of a typical file are covered in ‘teeth’ that may be either a series of ridges or a criss-cross pattern of pyramidal points, the sizes, spacing and depth of which reflect the intended coarseness or fineness of cut. The action of a file having ridges is very much like that of negative rake cutting tools; the action of files with points is scratching. A file with points is like an abrasive paper with a very stiff backing and cuts at a larger scale, and some files have teeth that are abrasive coatings. Filing is slow, but Victorians developed filing machines, and Napier invented a mechanism for filing the teeth of hand saws which raised each file during the return stroke (Jones, 1930). Most files are hand tools and the depth removed depends upon the downward load imparted by the hands, one at the handle and one at the far end of the file. Filing of flat surfaces is a very skilled operation, since it is easy to rock the file. Depths of cut of 0.1 mm are perhaps typical but it depends on the hardness of the workpiece. The material removed by filing has to sit in the regions between the teeth until motion of the file brings the debris to an escape point (the end of the workpiece perhaps) when it can fall away. There is thus always the danger of a file becoming clogged up – like the gullets of a saw – after which its performance is diminished. ‘File cards’ or wire brushes then have to be employed to restore the cutting surface of the file. To avoid clogging up, particularly when filing aluminium and its alloys, chalk or washing-up liquid may be used as a lubricant. Files with parallel rows of single ridges were usually manufactured by hand using a cold chisel of the required profile successively indenting the working surface of the file before hardening. In Victorian times, a number of machines for chiselling files was described (e.g. Greenwood, 1859) (Figure 7-5). According to Ross (1856), the main difficulty in designing a machine was how to alter the blow of the automated hammer for various depths of tooth and width of file. Leonardo sketched a file cutting (teeth raising) machine. Some modern files are made by milling or knurling. The criss-cross pattern on files results from the intersection of two sets of ridges, again originally made by hand. The geometry and inclination across the face of the file of the two sets of ridges need not be identical: the series of pyramids left from the intersection of the two lots of grooves might be flat-topped, or even rounded, for example. The terms ‘first cut’ (chief, or up, cut) and ‘second cut’ (over cut) employed with files arose from the way of manufacturing; in more recent times, a second cut file came to mean a finer file to be employed after the first, coarser, cut has been taken. A file differs from a rasp in having the furrows made by straight cuts of a chisel, either single or crossed, while the rasp has coarse, single teeth, raised by the pyramidal end of a triangular punch. Curved surfaces of files were still cut with straight chisels but as only short indentations were possible, the filemaker went round the file by degrees, making several rows or ranges of cuts. There may have been over 20 000 cuts, each made with a separate blow of the hammer. A file-maker’s hammer is shown as number 10 in Figure 2-5. Kitson Clark (1931) quotes results from Frémont who measured the work done by a workman using a file-cutting hammer (Table 7-1). An excellent paper on the action of files was presented to the Manchester Association of Engineers by Herbert (1909). In addition to describing systematic experiments on the testing of files and tool steels, it is one of the first papers to discuss machining economics: ‘it is far cheaper to buy and use well-cut files made of inferior steel and throw them away, if it is the case that the expensive fitter needs to file off the maximum amount in the minimum of time’.
183
Sawing, Chisels and Files File-cutting machine.
O E
I U
F
G E
K
C
B
A
A
R
H U I T
O
Q
A
Scale 1/10 th. 0
5
10
20
30 Inches.
Diagrams of hand and machine file cutting. (magnified.)
1st cut.
2nd cut.
B
Figure 7-5 A file cutting machine of 1859 (reproduced by permission of IMechE).
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The Science and Engineering of Cutting Table 7-1 Work done by a workman using a file-cutting hammer.
Poids du marteau utilisé (kg)
5.0
4.5
3.9
3.6
3.0
2.6
Hauteur moyenne de chute (m)
0.225
0.225
0.255
0.260
0.285
0.300
Nombre de coups à la minute (travail continu)
88
90
92
96
105
114
Nombre moyen de coups à la minute
50
53
60
62
67
75
Travail pour lever le marteau (kg/m)
1.125
1.012
0.995
0.936
0.855
0.780
Travail par coup de marteau (kg)
4.40
4.00
3.80
3.75
3.60
3.45
Effort moyen d’appui de la main sur le marteau (kg)
14.50
13.30
11.00
10.80
9.65
8.90
Vitesse d’impact du marteau (m)
4.15
4.15
4.36
4.52
4.85
5.09
Travail produit par seconde (travail continu) (kg/m)
6.45
6.00
5.83
6.00
6.30
6.55
Travail moyen produit par seconde (kg/m)
3.66
3.53
3.80
3.88
4.02
4.31
Source: after Frémont, given in Kitson Clark (1931).
Owing to the lack of understanding of how files worked and wore, there were files being sold at that time that were made of the best steel that did not wear, but whose teeth had a bad geometry and did less work than a file made of inferior steel but whose teeth were cut well. There were files sold especially for working on brass (the first cut perpendicular to axis of file; others for wrought iron had the first cut sloping at a very acute angle to the axis). It was all a bit ‘smoke and mirrors’. Herbert talked to Sheffield file makers and was surprised that they believed that the front face of file teeth had to have ‘lean’ or ‘lead’ (i.e. positive rake). His examination of files showed that in practice teeth leant backwards by about 15°. It was thought that positive rakes would be satisfactory for soft metals, but for harder metal such teeth would be mechanically weak and break. However, in soft metals, the teeth sink in and result in great resistance to filing, but in hard steels there is hardly any penetration at all and the file slips over the surface with little danger of breaking a tooth. Taylor (1906), in his tool life experiments (see Chapter 9), did not measure cutting forces, and looked at cutting speed and tool life at given depths of cut on the lathe. This was before the days of appropriate hydraulic gauges for forces (Stanton & Hyde, 1925) and long before the introduction of strain-gauged dynamometers. In Herbert’s file and cutting tool testing machine (1909), the workpiece was in the form of a vertical rotating tube, against the lower end of which the cutting tool or file tooth was loaded by a dead weight on a lever. That is, the thrust force FT was set and the depth of cut adjusted itself to suit. A later addition was to install a device that measured the cutting force FC. Thus, Herbert’s pioneering machine gave not only wear and life, but also the associated forces. Variations in tool rake angle and clearance, tool obliquity, cutting speed and so on were studied. At the time, the cross-sectional area of the cut (w times t) was employed by many as a basis for comparison of tool performance, with a belief that there was some ‘critical stress’ by which cutting took place (in some of the series of cutting tests performed by the Manchester Association of Engineers reported in 1903, w and t were often altered simultaneously). Herbert’s results demonstrated that this was not correct. Herbert performed some of the earliest
Sawing, Chisels and Files
185
force measurements of historical significance and interest, and it was he who initiated the practice of scientifically testing files. Herbert’s file testing machine recorded the number of strokes made, and the amount of metal reduced to filings, before it became too blunt to cut. He discovered remarkable differences in the quality of files, but it was not always possible to decide whether the superiority or inferiority was due to shape and sharpness of the file teeth, or to the quality of the file steel. It was noted that tools taking fine cuts at very slow speeds did not last long. Herbert proved that this apparent anomaly was intimately connected with edge temperature and that only the hardness at the hot operating temperature mattered. Herbert’s achievements are, perhaps, not as well known as they should be: among other things he had invented what later became the Rockwell method of measuring hardness with major and minor loads, but because he was aiming for a portable instrument where forces were applied by hand, the indentations were very shallow and demanded too great a refinement in the measurement of depth, so that the project was not fulfilled (Herbert, 1939). According to Herbert (1909), the factors affecting the cutting efficiency (material removal per stroke) of files are: (i) (ii) (iii) (iv) (v) (vi)
sharpness of teeth slope of the front face of the teeth (the rake) slope of the back face of the teeth (the clearance) angles at which the two cuts lie relative to the axis of the file the different pitch of the two sets of teeth the ratio between the two pitches.
The front face of file teeth is sometimes vertical (i.e. 0) but usually has a backward slope that may be anything from 3° to 25°, i.e. a 65° attack angle. Sedriks and Mulhearn (1963, 1964) (Chapter 6) showed that there was a critical attack angle, different for different materials in different thermomechanical states, at which a transition occurred between cutting chips and forming a standing wave where no material is removed. It seems clear that if file manufacturers got their rake angles the wrong side of critical, their files would perform poorly. They seem also to have discovered empirically that different materials have different critical attack angles. Herbert said that common experience with lathe tools would suggest that a file with a considerable negative rake would be inefficient on most materials, and that it was surprising that such a tool would cut at all under the light pressure that can be exerted by hand. Continuous chips are nevertheless cut by sharp files with negative rakes, and he noted that most files are presented to the work at an angle, giving a slicing cut (slice–push ratio ; Chapter 5). Even so, that is impossible when filing against a shoulder, or in a slot. ‘Draw filing’ is used to produce very smooth surfaces. The file is placed at right angles across the workpiece. The success of this process is down to altering the inclination angle of the file teeth so that is increased and the filing forces are reduced. Herbert’s experiments showed that the chief factor influencing durability and output of work was the clearance angle or slope of the back of the file tooth. This controlled the gullet space for debris, but too great a clearance resulted in the tip of the cutting edge breaking off. A clearance of about 25° on a file with zero rake (a vertical front face) gave best results. Figure 7-6 shows the material removed by filing plotted against the number of revolutions of a hard steel tube workpiece (1/16th in. wall thickness) cut in Herbert’s machine. Beside every curve is the number of inches of tube turned away before the tool was too blunt to cut at that load, and the number of revolutions made by the tube. The efficiency or average rate of cutting is the quotient of those values, given as (in./1000 revs). The duration of the test, on the abscissa, is what Herbert calls the ‘output’ of the file. Beneath each curve is the profile of the imaginary file
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D Inch 2.35 2.5 1 Revs. 3,550 .66 3,550 .702 930 1.08 B
A
20° R-15
C
C20° 90°S
R-15°
8.6 1.3 6,600
C20° 70°S
C20°
R0°
90°S
70°S
R0°
Inches 9.2 1.61 Revs. 5,700 G E C20°
R+15 S90°
H
10.4 1.70 5,800
F C20°
R+15 S70°
C5°
R+15 S90°
C5°
R+15 S70°
Figure 7-6 Effect of different profiles of file teeth on cutting performance. C is (negative) rake angle of front face of tooth; R is angle of rear face of tooth; and S is the obliquity of the file teeth with respect to the crossways direction of the file (after Herbert, 1909).
tooth and a plan view of the inclination of the teeth to the axis of the file. Three different front face rake angles were studied: 15° (commonly found in file teeth), 0° and 15° (never found in file teeth). The teeth were aligned either straight across the axis of the file (90°) or at 70° (typical of files for general work). All tests were carried out with a 25 lb load, at 30 fpm with a 20° clearance, using a single tool to simulate the action of a file tooth. The effect of angling the teeth of the file is shown in D. Tests E and F are for the positive-rake tool; their small duration was because the edges of the tools broke off. When the clearance angle was reduced from 20° to 5°, however, curves G and H were produced. The ratio between the pitches of the two cuts is very important for file performance as it determines the relative positions of the teeth and how much material is presented to a following tooth to cut. Equal pitch results in every tooth standing immediately behind a tooth in the row in front. Such a file would produce a series of furrows with uncut material in between. (Grater-type planing tools can result in grooves rather than smooth surfaces for the same reason, depending on the angle across the workpiece at which they are used.) Changing the pitch ratio places teeth where uncut material was in the equal-pitch arrangement. In this way, ridges left by teeth in the first row are attacked by teeth in the second, third and subsequent rows, resulting in a better surface finish. The relation in which teeth in different rows follow
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one another is altered when the file is pushed at an angle to the workpiece and craftspeople adjust their action to suit different materials. Loss of sharpness took place continuously from the beginning to the end of Herbert’s test ‘until pressure of the tool against the tubular workpiece is no longer sufficient to cause it to penetrate the metal’. (Files were traditionally resharpened by immersion in sulphuric acid.) Files having blunt teeth required increased downwards pressure to indent the workpiece before pushing the tool forward, after which they would resume cutting. Herbert found that there were minimum loads below which sharp tools would not cut. Such loads would seem to correspond with the intercepts of FT displayed in diagrams of cutting force vs depth of cut, which are a measure of the fracture toughness of the workpiece.
7.4.1 File hardness There is a limit to the downwards impressed load that can be exerted on a file by hand, and if a workpiece cannot be indented during attempted cutting, the file simply skids over the surface, however sharp the teeth may be. Being impossible to ‘catch hold of’ very hard workpieces by file led historically to the idea of file hardness. The results of tempering steels were tested with a file in factory smithies well into the twentieth century. A gemstone was once defined as something that could not be scratched (not ‘attacked’) by a file (Barba, 1637). Rather like scratch hardness tests, file hardness is based on the idea that a material that cannot be cut with the file is as hard as, or harder than, the file. The hardness of a file itself is typically about 40 RC (Rockwell C scale). There is a presumption that the teeth are sufficiently hard so that they remain undeformed whatever the workpiece properties, but there may very well be chipping of the hard file teeth that have relatively low toughness. Hence recommendations for storing files so that they do not touch or cross one another. The toughness of the workpiece also determines the depth removed in filing for the reasons given in Chapter 6.
7.4.2 Similarities between processes Rowe and Wetton (1965) pointed out that deformation patterns in several processes (grinding, broaching, strip drawing, machining, prow formation, built-up edges, sliding wear) are very similar whether or not the aim is material removal. Sections through partially completed cuts showed the similarity between broaching and sawing. In strip drawing, material is reduced in thickness by pulling a sheet through tapered dies. But under certain conditions chips can be detached. When the die mountings are insufficiently stiff, drawing proceeds intermittently rather like chatter in cutting. Rowe and Wetton show that the metal has been heavily sheared in the immediate vicinity of the tool, and that deformation and strain hardening extend well into the metal ahead of the tool face. The resilient die mounting permitted the dies to spring over the peaks of the hardened zones. If a less tough material is used, the extensive strain hardening does not occur. The drawing still proceeds intermittently but complete fragments (the scallops of Chapters 3 and 4) are detached and are left as acicular debris or chips on the surfaces of the drawn material. In different circumstances, depending on materials and rake angles, a standing wave is formed ahead of the die that acts like a negative rake cutting tool (Chapter 3). Sometimes the wave breaks off in some sort of discontinuous chip. Shaw (1965) talks of a scallop-type
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Figure 7-7 Accidental cutting rather than plastic flow during single point incremental forming (SPIFFING) of aluminium alloys (reproduced by permission of Bay and Martins).
chip in metals cut at low speeds. Similar behaviour has been observed in spinning, spin forging, wire drawing with detached rings of material, and in the ‘scalping’ of rod that aims to improve surface finish. An example from single point incremental forming (SPIF) is shown in Figure 7-7. SPIF is a comparatively new dieless method of sheet forming in which a computer numerically controlled centre is used to drive a tool in prescribed paths to make complicated shapes. The figure shows how the forming tool becomes a cutting tool when the material and conditions of operation are wrong.
Chapter 8
Punching Holes Piercing and Perforating, Arms and Armour Contents 8.1 Introduction 8.2 Quasi-static Piercing with a Pointed Tool 8.3 Quasi-static Circular Punching 8.4 Hollow Punches 8.5 Arms and Armour 8.6 Penetration and Perforation of Armour
189 192 199 201 202 209
8.1 Introduction In the drilling of holes (Chapter 5), material is cut away and removed as highly distorted swarf, and only the object containing the hole is usable. The hole may go right through the workpiece, or may be ‘blind’ and go only partially through. In this chapter we consider a variety of processes where holes are made by different processes. For example, making holes for plants in the garden with a pointed dibber, making a starting hole with an awl in wood for nails or screws, the hole made by a bullet, and so on. Wider holes are made with hollow or solid punches. A pastry cutter for gingerbread men is a simple example of a hollow punch, where dough on the kitchen counter is separated by a thin, knife-like, tool that divides material with (theoretically) no waste. A similar tool is a bulb planter in the garden, but here the tool penetrates only to a limited depth and, to detach the plug of soil from the earth below, the device is twisted, separating the plug by shear after which it is lifted out with the tool. Like the dough left over from cutting gingerbread men that is balled up in the hand and rerolled, the plug of soil is placed back in the hole over the bulb. A cork borer is a rotating hollow punch, where the material displaced from the through-hole comes away as a solid plug, not as swarf as it would if the hole were drilled. To preserve the sharpness of hollow punches employed in precision work, a block of wood or soft pad is placed beneath the workpiece so that when the punch emerges it is not damaged (Chapter 9). Solid punches driven into the surface of a brittle solid produce craters with the displaced material removed as radially broken-up lumps like slices of a cake. The action of flat-ended solid punches when making holes in floppy or ductile materials is by pushing a plug of material from the near (proximal) surface through to the far (distal) side of the workpiece. Separation of the centre portion from the rest of the sheet can either be by scissor-like cutting (punching holes in paper for ring-binders where the punches have angled bottoms and guillotine the paper) or by ductile shear in a narrow band around the separated edges (holes for rivets in ships’ plates used to be punched in this way). In the canning industry, immense numbers of discs are stamped out of sheet metal to provide the ends. In these processes, sometimes both the separated part and the remaining stock remain usable even though, in practice, one or the other may be discarded. It depends on whether the hole is wanted or the disc. Noble and Oxley (1963) showed that by having cutting edges of different sharpnesses on the punch Copyright © 2009 Elsevier Ltd. All rights reserved.
189
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and die (an anvil with a hole in it for the disc to pass into), cracks could be directed into the waste material thus producing edges (of discs or holes as appropriate) that would be less susceptible to failure from fatigue in applications. As with most fracture problems, sometimes you want perforation to happen (making holes with a leather punch); at other times you do not (a puncture in a tyre); with weapons and armour, whether you do or do not depends upon your point of view. The introduction of Bessemer steel in 1856 revolutionized the whole of industry as, before, wrought iron (and, to an extent, malleable cast iron) was the only ductile constructional material available in tonnage quantities. Bessemer steel was cheaper and, when consistently made with proper metallurgical control, had better properties. Smith (1875), in the quest for quality control tests on Bessemer steel rails, showed that the loads required to punch holes for fishplates at the ends of the rails were a better indicator than bend tests, and would indicate rogue properties that drilling the holes would not reveal. He tested worn, as well as new rails, some of which had been manufactured in the early days of Bessemer steel and were harder than intended; other old rails he tested did have the design hardness. He showed (contrary to expectation) that softer rails wore much less than the harder, hinting that ‘ductility’ (toughness) was as important as ‘strength’. There is, perhaps, confusion in the way different words are used in this field. The word perforate is commonly used to mean ‘make holes through’ and comes from the Latin perforare, which actually means to bore a hole and thus implies some form of rotary cutting tool, even though most perforations are not made that way, being pressed out as in the manufacture of lavatory paper and postage stamps. A dictionary definition of pierce and penetrate says that both mean an object passing through another, but that pierce is a quick action and penetrate slow. Puncture is another word with similar meaning. To pink means to pierce, perforate or notch (Brewer, 1981) and in the seventeenth century commonly meant ‘to stab’ (Brewer refers to Scott’s Peveril of the Peak, Chapter 23). Pinking shears, used by dressmakers, are scissors that cut zig-zag (notched) edges to minimize fraying. In metalforming blanking and piercing mean the punching of holes in sheet or plate: in blanking it is the part punched out that is required while in piercing it is the remaining stock which is the finished component. But piercing in metalforming is also employed to describe indenting a surface with a punch to produce a cavity which may be the final operation (as in bolt heading) or as a prelude for subsequent forming operations (this is also called hubbing); such formed cavities do not involve separation/fracture at all and are just plastic flow. A process which does involve fracture yet uses the word piercing is the manufacture of seamless tubes. When a solid round bar is compressed and rotated under load, a crack appears in the centre. (To convince yourself, squeeze and roll between the fingers the little cylindrical eraser from the end of a pencil, in the way cigarettes can be rolled.) In the Mannesmann process of rotary tube piercing a pointed mandrel is inserted into the opening crack to expand the hole to its required internal diameter. There is no wastage of metal. Where the displaced material goes as a tool penetrates is of importance for the mechanics: thin tools have to have some wall thickness. In woven or knitted fabrics, small holes can be made with even blunt pins where fibres are displaced out of the way. A fid is a heavy-duty needle used in, for example, sailmaking with canvas, or with leather, and adapted to industrial sewing machines. A pin used to pop a balloon or blister probably causes fracture not by cutting but by facilitating, through its sharpness, crack initiation in the stretched membrane at the strain induced in the balloon by the particular inflation pressure. The path of the crack is often wiggly (see Section 9.10). It is possible to sit on an inflated balloon without bursting
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191
it. In contrast, it is perhaps puzzling that if you prick a hole in a balloon before blowing it up, it is possible to inflate it: it will stay inflated for some time even though there is a small hole through which air escapes (Chaplin, 2008). When solid ductile or compressible materials are penetrated, by a nail for example, the process begins as if it were a hardness test but, at deeper penetrations, separation must begin and continue at the sharp tip. Separation permits the material to move out of the way of the oncoming tool. Sometimes the volume of the penetrator can be accommodated by elastic compressibility of the material (nail through a rubber tyre); in more ductile materials, by radial elastoplastic flow (driving in piles in civil engineering or fence posts; and in seamless tube-making). When a punch is passed through an already existing hole in a ductile sheet, the displaced material forms a ‘circular wall’ around the hole on the distal face; this is the process of hole-flanging in manufacturing which is just plastic flow. When solid punches having hemispherical or other profiled noses pass through a workpiece, material can be displaced out of the way by a variety of means. In addition to the plug pushed out by flat-ended punches, sometimes a number of radial cracks is produced and the adjacent material is bent out of the way in the form of petals. This sort of thing is of particular concern in ballistics – whether spears, arrows, bullets and shells can ‘defeat’ plate armour. In ballistics, also, targets are not always as restrained as in the workshop. Unsupported targets deform with considerable ‘dishing’ in the direction of projectile motion with bending and in-plane stretching in addition to the expected deformation. In the study of perforation into thin sheets in the laboratory, different behaviour will result depending on whether the workpiece is just placed on top of a piece of pipe or clamped all around the circumference. The existence of dishing is relevant to what we mean by ‘sheet’ or ‘plate’. Normally, material thinner than about 2 mm is considered a sheet or strip, and thicker is a plate. When plate widths are also narrow, it is barstock. However, in ballistics, when the extent of dishing is much greater than the target thickness (so that there is considerable stretching beneath the penetrator), the target is considered as a sheet whatever the thickness. The behaviour during penetration and perforation depends on the relative magnitudes of tip radius and sheet or plate thickness, and the mechanical properties of the sheet: (i) With a tool where the tip radius thickness t of the sheet, the punch behaves as if flatended when indenting a ductile material and makes a hole by through-thickness shear (‘plugging’). (ii) With a bluff tool ( t) and a ductile sheet there is in-plane stretching under the nose of the tool followed by local necking around a ring, the location of which depends on contact friction. Fracture in the circular neck results in ‘discing’ where a cap of material is detached. Further penetration produces petalling from the resulting hole. This sort of behaviour is employed in tests for formability in sheet metalworking (e.g. Marciniak et al., 2002). (iii) With a bluff tool ( t) and a brittle sheet, an unstable circumferential crack rapidly forms (push the blunt end of a pencil through a sheet of paper). (iv) With a pointed tool ( thickness of sheet) and a ‘ductile’ material, permanently deformed petals alongside the radial splits are pushed away to permit the tool to pass through (poke the sharp end of a pencil through a sheet of paper). (v) With a pointed tool ( thickness of sheet) and a ‘brittle’ material, the sheet is pushed away elastically, radial fractures are produced as the tool passes through and triangular segments are broken off.
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8.2 Quasi-static Piercing with a Pointed Tool 8.2.1 Piercing stiff brittle solids Experiments with sharp wedges vertically penetrating large pieces of rock show that as the load is increased the wedge fragments, crushes or pulverizes material in the vicinity of the wedge tip where the hydrostatic stress is highly compressive. Simultaneously, lateral cracks begin to extend stably under increasing load from the tip of the tool into undamaged material; their paths slowly curve upwards. At some critical load, the cracks accelerate and run unstably to the free surface, dislodging material as chips to either side. The load falls (but not to zero) as the cracks propagate to the surface (Figure 8-1a,b). The cycle then repeats itself under everincreasing oscillating loads since the surface area of successive dislodged material increases with penetration, requiring more work, i.e. a bigger force over the same displacement. Sikarskie and Cheatham (1973) modelled the behaviour using the Coulomb–Mohr failure envelope (see Chapter 14) to represent pressure-dependent fracture along the inclined cracks between the tip of a wedge and the surface. The analysis took into account friction on the wedge faces and used an empirical expression for the crushing load given by Pcrush (const) ( o), where is penetration (o is the penetration when the first chips form). The vertical wedge-tip separation that releases chips to slide up to the surface, and the work associated with it, were not considered. A hemispherical-ended round chisel driven vertically into a block of a brittle solid produces the well-known Hertz–Roesler cone crack. No chip is produced and there is no material removal but inclination of such a tool results in inclination of the cone crack, and after a critical angle parts of the cone divert towards the free surface, thus removing material (Lawn & Wilshaw, 1975). An axisymmetric pointed tool behaves in a similar way to the wedge (e.g. Lundberg, 1974). In practical cases of stone breaking, the tool displacements far exceed the very small penetrations at which indentation plasticity is possible and the behaviour is globally elastic. Material from the saucer-shaped craters that result is broken by bending into ‘slices of cake’ as the axisymmetric chip is detached. Rhyming et al. (1980) and Cooper et al. (1980) discuss problems in excavating rock when attacking a working face in quarrying, tunnelling and ‘stoping’ (the excavation of a thin tabular body). A common solution is to bore a hole and to apply pressure to the walls by explosive or mechanical means. In a pressurized borehole, cracks tend to propagate in planes perpendicular to the axis of the hole, so that if the hole is drilled normal to the face (as is common practice), not much material is released as chips. Johansson and Persson (1970) showed that, to remove the same quantity of rock, only one-tenth of the amount of explosive is required when the hole is parallel to the free surface rather than normal to it; see also Tilert et al. (2007). What is required for efficient removal of material from rock faces are outwardly directed tensile forces, and these may be achieved by applying shear loading to the walls of the hole via a device like a modified anchor bolt. In this way, craters of material are removed from the surface. Rhyming et al. (1980) describe experiments on polymethylmethacrylate (PMMA) models and Cooper et al. (1980) on full-size machines, giving a full fracture mechanics analysis of the process. They point out that the breaking force depends on the fracture toughness of the workpiece, not on other parameters such as tensile or compressive strength. They observed a scale effect, as would be expected of a brittle fracture problem, whereby the breaking force was proportional to h3/2, h being the hole depth (see Chapter 4). Percussive jack hammers are employed in road mending, and are used extensively for fragmentation of rock and similar materials. Karlsson et al. (1989) explain the working
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Punching Holes Force
Wedge θ Chip
Chip Fracture surface Crushed zone
Extension fracture
A
Applied bit force, lbs (� 10�3)
14 12 10 105° 8 90° 6 75° 4 60° 2 0 B
2θ � 120°
0.01
0.02
0.03
0.04
Bit penetration, inches
Figure 8-1 Cutting of a brittle material by a wedge: (A) model of penetration; (B) characteristic force–penetration curves for a charcoal grey granite cut with 25 mm wide wedges having the angles indicated (after Sikarskie & Cheatham, 1973).
of these devices: a low-level force accelerates a hammer during a time that is much longer than the transit time for elastic waves through the hammer. The work performed is almost completely converted to kinetic energy. The hammer impacts on the material either directly or via intermediate members (when down deep holes, for example). The impact generates a high-magnitude force which has a duration of the order of the wave transit time. It is this short-duration force that does the fragmenting. Not all the kinetic energy is converted and after the blow, energy in various forms remains in the percussive system, so that there is an efficiency given by W/WK where W is the actual work done and WK is the initial
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kinetic energy. W is the area under the load–displacement diagram similar to Figure 8-1(b). WK (1/2)mv2, where m is the mass of the hammer and v the impact velocity. Lundberg and co-workers have published many papers on the dynamics of percussive systems (e.g. Lundberg & Okrouhlik, 2006).
8.2.2 Piercing of soft compliant solids Compliant soft solids such as some rubbers, gels and layers of skin/fat, are brittle in the sense that their behaviour after piercing is globally elastic despite the extensive deformation during penetration. A thin flat tapered knife (Figure 8-2) has four contact areas that continuously increase with penetration . The increments of crack area are (i) d(2tan1) 2tan1d; and (ii) d(22) 22d. Hence from the work equation, and the force equilibrium relations in Chapter 3, Finsert 2R(tanα1 α 2 )δ 2µp(tanα1 α 2 )δ2
≈ 2Rtanα1δ 2µptanα1δ2
(8-1)
since 2 1, where p is the contact pressure between the material of the ‘target’ (clothing, flesh) and the knife blade. This analysis is relevant to investigations of the resistance of skin and flesh to attack (Chapter 11). In practice, the target is deformable so that there may be some sinking in before the knife penetrates; it depends upon knife sharpness. Covering of the target by armour aims to defeat penetration all together, or at least limit it. Deep penetration of highly deformable soft solids was studied by Shergold and Fleck (2005) in connexion with medical injections into the human body. Experiments show that a sharply pointed punch results in a planar crack formed in mode I. Indenting was modelled
F, δ
2α1
Figure 8-2 Triangular knife blade and penetration .
2α2
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Punching Holes
as shown in Figure 8-3. For pointed indenters, the incremental work of penetration performs work of incremental plane crack advance plus the sideways incremental elastic compression of the material which occurs as the shank of the tool displaces material from the core of the hole. The penetration pressure is greater for greater material toughness, greater elastic shear modulus and greater non-linearity; it increases as the punch radius decreases. These sorts of analyses are relevant to acupuncture, in which the needle is also twisted during insertion. This may be to do with frictional effects. An extensive knowledge of anatomy is required by the practitioner in order to know where to put needles to prevent bleeding. The analyses are also relevant to unpleasant happenings such as stabbing attacks with pointed weapons (by the stiff knitting needle called an ‘ice pick’ in US police and prison parlance). Dracula and Vlad the Impaler are known for staking victims, and there are illustrations in the Duomo museum in Florence of various imaginative and ghastly tortures involving spikes and orifices. Shergold and Fleck’s modelling of indentation for a blunt flat-ended penetrator into soft solids is given in Section 8.3.2.
8.2.3 Piercing ductile solids When an awl or similar pointed tool is pushed into a thick block of material to make a hole, the start of the process is similar to cone indentation where the pressure p is given by p p′(1 µcotα) 2.8Y(1 µcotα)
(8.2)
where p is the pressure in the absence of friction and is the semi-angle of the cone (Hankins, 1925); p 2.8Y where Y is the uniaxial compressive yield stress (Atkins & Tabor, 1965). The material displaced by the point is accommodated by local compressibility and/or plasticity and may rise up on the surface exactly as in indention hardness (Tabor, 1951). At deep indentations of a tool where the point continues to push apart material at its tip, the mode of deformation ceases to be simple indentation, because the displaced material has to go somewhere. It might be accommodated by compressibility in materials like softwoods, PP Punch, diameter D z
D
Opened crack
y
h
B
x
Closed crack 2a A
C
Figure 8-3 (A) Penetration of a soft solid by a sharp-tipped punch; (B) crack opened to allow punch advance; (C) crack closed after punch removal (after Shergold & Fleck, 2005).
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but is more likely to be accounted for by local radial plastic flow surrounded by an outer region of cylindrical elasticity (Johnson & Kudo, 1964). This mode of deformation for deep penetration is employed in geotechnics to analyse piling, and in agricultural engineering to analyse the motion of tines moving through soil (Chapter 14). The expansion of both cylindrical or spherical cavities requires p (4 or 5)Y (Bishop et al., 1945). If the hole does not close up completely when the awl is removed, some permanent radial deformation must have taken place. French artillery experiments at the turn of the eighteenth/ nineteenth century showed that wood recovers much of its original volume after being penetrated and thus cavities close up after passage of projectiles (Johnson, 1986a,b). Both compressibility and plasticity/elasticity produce radial compressive stresses on the sides of the penetrator, and hence frictional forces, both when pushing in and during extraction of the penetrator (such friction holds nailed joints together). Consider a pointed cylindrical tool at depth below the surface. Let the radial pressure on the parallel shank be p. For Coulomb friction the frictional stress is thus p, and the frictional force pD (4 or 5)YD. The crack area created by penetration is given by D. The incremental external work done by force F in a further incremental displacement d is Fd. The incremental separation work is RDd and the force for separation is DR. Using Eq. (8-2), the total force Fpierce required for piercing into a yielding material is thus
Fpierce 2.8Y(1 µcotα)(πD2 / 4) πDR µ(4 or 5)YπDδ
(8-3)
An instantaneous piercing force given by {2.8Y(1 cot)(D2/4) DR} is expected, followed by a linear increase with penetration at the rate (4 or 5)YD. The work Upierce required to insert a straight conically pointed penetrator of diameter D to depth is obtained by integrating Eq. (8-3):
U pierce {2.8Y(1 µcotα)(πD2 / 4) πDR}δ µ(4 or 5)YπD(δ2 / 2)
(8-4)
This is typical of nailing without splitting; nailing with splitting is given in Section 8.2.4. These relationships apply to forks of various sorts having different numbers of tines, forklike weapons such as tridents, harpoons, barbs and fish hooks, eel spears, graveyard sounding rods, piles in construction, and spiked hammers used to make holes in roofing slates. And to nasty things like a caltrop, which is a ball with four spikes at the points of a pyramid, so that when on the ground one spike always points upwards and was used to obstruct the passage of cavalry; caltrop comes from the Latin for ‘tread trap’. The effort required to insert a corkscrew into a cork in order to open a bottle may be determined along these lines. The coordinates of a point on a right cylindrical helix in r,,z coordinates is {r,,(p/2)}, where p is the pitch of the screw. The length of the helix over one turn of the screw is 2[r2 (p/2)2]. After N revolutions, the torque T has done work given by 2NT. The work done Upierce by the helical corkscrew in N revolutions is given by Eq. (8-4), in which is replaced by 2N[r2 (p/2)2]. Equating the two expressions for work gives the torque T after any N. For a cork of length h (corks are about 40 mm long), the initial frictional force F opposing withdrawal of the cork from a bottle is F Dhp, where p is the sideways contact pressure between cork and neck of bottle, and the coefficient of friction. The contact pressure is determined by the interference fit between cork and bottle. It may be estimated from the expansion of a cork on removal from the bottle and the modulus of cork. During withdrawal,
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the force decreases as the length remaining in the neck of bottle decreases. The work done in extraction over length h is (Dhp)dh (Dh2p/2). The pitch of a corkscrew is important for its performance. Too slow a pitch (a flat helix) may not leave enough material between the coils to resist force to pull out the cork against friction between the cork and the neck of bottle. Too fast a pitch might permit unscrewing, although in practice it is possible to suspend a bottle from a corkscrew inserted into the cork; the bottle does not spin and undo, so friction is high. It is a moot point whether cork is really a ductile material. Dartboards were originally made from rounds of tree trunks but now are made from bristle or sisal, and recover after penetration.
8.2.4 Splitting of wood during nailing Owing to the anisotropy of wood, where toughness varies considerably with orientation to the grain, driving in a nail often causes splitting alongside the nail. A single planar crack is formed after the initial indentation. Splitting relieves the sideways pressure on the shank of the nail and thus reduces the friction force opposing the nail coming out and hence its holding ability. Figure 8-4 shows a nail of diameter D penetrated to a depth with a surface split of length c to each side. The split takes the form of a deep semi-elliptical cavity as shown. Owing to the splits, the nail is in circumferential contact with the timber only over angle 4o rather than all the way round. It may be shown that the fracture work required to form the semi-elliptical cavity is negligible compared with the total work of penetration. However, fracture is important in that it controls the length of split, and hence the area of contact with the nail and the friction force. The linear elastic fracture mechanics expression for a deep thumbnail crack loaded on part of the crack faces by pressure p is
KI p(2 / π)sin1(b/c) (πδ)Y(c)
where o (2b/D) and where Y(c) is the factor accounting for the finite width of the splits. This relationship is independent of depth for deep indentations (c ). The friction force on the sides of the nail is p(4o/2)D. The nip pressure will be given approximately by (8-1), so the normalized total pressure is
Fnail /(πD2 / 4) 2.8Y(1 µcotα) 4µp(4θo / 2π)(δ/D)
(8-5)
The appropriate value for p depends on o and c. There are limits on the magnitude of p as explained in Atkins and Mai (1975) and, with knowledge of KIC for wood in the relevant grain direction, it permits 2c/D to be determined. Relation (8-5) predicts a linear increase in force with penetration, with an intercept representing the nip force. Salem et al. (1975) performed experiments that displayed this behaviour. Taking due account of the anisotropy in the wood, quite good agreement was obtained between the above theory and experiment. Salem et al. also showed that the process of hammering a nail in is essentially one of rigid-body dynamics rather than elastic wave propagation (see Chapter 2). Nails are proportioned to try and avoid buckling, in the same way that the wire diameter of staples is proportioned. A village fête event in Britain involves seeing how far a participant is able to drive a nail into a baulk of old oak without the nail buckling. Nailing forces may be calculated as above (Eq. 8-5), noting that oak is hard and tough, and does not split. Nailing
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2c
x
2b
2θ
° p
Figure 8-4 Frictionless nailing modelled as propagation of a ‘part-through’ semi-elliptical crack of semi-major axis x (the nail penetration) and semi-minor axis c (the length of split accompanying nailing) (after Atkins & Mai, 1975).
forces may be compared with the Euler load for a strut encastré at one end (2EI/4L2, where E is Young’s modulus, I is the second moment of area [ D4/64 for a round nail of diameter D] and L is the exposed length of nail). The nailing force goes up as the nail goes in, but so does the buckling load at a different rate. What actually happens is affected by the difficulty of aiming straight, which results in bending/buckling. Salem et al. (1975) also considered buckling of a nail, since the dimensions of long penetrators are chosen to avoid column buckling either initially or as a result of the increased load after some penetration; dynamic buckling is suppressed by a high-rate increase of E, so that buckling occurs at 1.5 times the quasi-static Euler load (Johnson, 1972). Stapling is piercing where the punch is wire that, after penetration, is bent under the stack of sheets so as to bind the layers together. When attempts are made to staple too thick a pile of papers, the initially vertical sides of a staple buckle and collapse since the piercing load after some penetration exceeds the Euler buckling load. The aspect ratio (exposed height/wire diameter) of the staple when it collapses gives an indication of the loads involved. For thicker layers, thicker wire is used.
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8.3 Quasi-static Circular Punching 8.3.1 Punching floppy materials A hole punch for paper illustrates cutting holes in a floppy material. Consider a sharp punch, of diameter D, to be at some penetration into the sheet of thickness t. The separated surface so far formed has area D, assuming that the cylindrical crack does not run ahead of the punch. For a further movement d, the incremental external work done is Fd, and the increment of work required for separation is RDd. Hence
F RπD
(8-6)
and the total work done is RDt. Relation (8-6) is the equivalent of Eq. (3-4). In practice, the punch is probably not flat-ended and has a vee that permits more gradual penetration of the paper at lower forces since not all the periphery of the punch is in contact with the paper at once. While the force is lower, the work required is the same of course as the punch stroke is longer, exactly like the action of a guillotine (Chapter 5). Rows of holes are made by reciprocating punches operating over sheets periodically indexed forward, or with miniature punches at the appropriate pitch around rollers. Sometimes such rows of holes are there to do a job (punched cards or tape for computers, pianola rolls, Jacquard loom cards); sometimes they are there to permit subsequent separation of sheets into parts (postage stamps, labels, lavatory paper). The distance between stamp perforations is usually the diameter of the holes. Perforations are important to philatelists: Henry Adams first perforated British penny reds in 1850 (he had rouletted some sheets in 1848, i.e. made short discrete cuts in patterns between rows and columns of stamps to facilitate separation). It is a diversion on trips to foreign countries to see which nationalities roulette their lavatory paper and which separate by rows of holes. Why are there perforations on British self-adhesive stamps that are already separated? The forces required to separate parts of perforated sheets depends on whether they are pulled apart by stretching in the plane of the paper, or torn by lifting one side up out of the plane of the sheet. In either case, the external work done by one’s hands has to provide the appropriate fracture toughness work. In the early days when computers were few and not always on the same site as the people using them, stacks of punched cards had to be sent through the post to the machine and the results returned by post perhaps a week later. People involved became accurate typists because mistakes in punching caused inordinate delays in obtaining answers. The above analysis presumes that the punch is sharp enough to cut into the paper. When the edge is not uniformly sharp, parts of the perimeter may not be cut and a flap is left still attached to the partly formed hole; it is the same as discing in ballistics. The notorious ‘hanging chads’, in the 2000 Florida US Presidential election, were holes not completely punched through. A hemispherical-ended punch is likely to bend and stretch the paper over the matching hole in the baseplate and produce a hole by radial tearing across the circumference; with appreciable clearance between punch and hole, perforation may not occur. A punch ending in a point would not be much good, even with the sharp edge of the matching hole, as it would result in radial cracks exactly as happens when a hole is made in sheets by poking through with a pencil; remanent radial tears tend to propagate when sheets are put into ring binders and the sheets fall out. Even neatly punched holes are reinforced sometimes to avoid the ring-binder itself acting as a blunt cutter to shear out a slot to the edge of the sheet. When holes are made in pressure vessels for nozzles, the hole is reinforced so that the resistance to widespread yielding is made approximately equal to that of the plain, unperforated, shell; see example 10.8 (due to Ruiz) quoted in Atkins and Mai (1985).
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Gaskets for old vehicles can be made by placing plain gasket material over the engine block, or other component, and lightly tapping with the pein of a hammer around the edges of the holes to cut out discs or other shapes, through which bolts will tighten up and seal the joint between components.
8.3.2 Punching of soft compliant solids Indentation by blunt punches, as well as pointed probes (Section 8.2.2) was studied by Shergold and Fleck (2005). Experiments showed that a cylindrical crack forms in mode II of fracture mechanics beneath flat-bottomed punches (Stevenson & Abmalek, 1994) (Figure 8-5). Formation of a cylindrical crack here is a compressive version of the tensile ‘pull-out’ problem in fibre-reinforced composites (e.g. Kelly, 1964), but unlike the stiff composite materials to which pull-out analyses have been applied, soft solids are non-linear and deform at constant volume. Consequently, it is not only the energy stored in the squashed column of material beneath the punch, but also the energy stored by the radial displacement of material around the punch, that makes up the elastic strain energy . The non-linear behaviour was modelled by Shergold and Fleck (2005) using an Ogden strain energy density function (Ogden, 1972). The work is relevant to injections of all sorts (Chapter 11). The line of attack is to write down the energy of the material disturbed by the punch as a function of cylindrical crack length, and employ standard elastic fracture mechanics to find the load for further penetration, i.e. R ∂/∂A, where R is the mode II fracture toughness, A is the cylindrical crack area and is the punch displacement (Gurney & Ngan, 1971). Calculations show that the penetration pressure for a flat-bottomed punch is two to three times that for a sharp-tipped punch (Figure 8-5). For both geometries of punch, the penetration pressure on the shank is greater for greater material toughness, greater elastic shear modulus and greater non-linearity; it increases as the punch radius decreases.
8.3.3 Punching ductile materials The workshop leather punch that looks like a rowel spur can cut different size holes by punching out solid ‘plugs’. When punching ductile solids, cutting and separation do not start straight away, unlike simple punching of floppy materials. As explained in Section 3.8 for cropping, and Section 5.4.3 for guillotining, a shear band (width w) is set up from top to bottom of the sheet between the edge of the punch and the rim of the supporting baseplate. There is simple plastic indentation from both edges until cr at which a transition to separation takes place. The transition leaves a well-marked border between ‘smooth’ plastic indentation (on which will be replicated tool marks) and ‘rough’ surfaces separated by shear cracking (see Figure 3-29a). The normalized forces and energy in plugging a rigid-plastic material are given approximately by the relationships in Section 3.8 but with the length of contact w between a straight blade and workpiece replaced by D. The punch for making holes in horseshoes is called a pritchel, the blacksmith punching redhot steel with a chisel over the round pritchel hole in the anvil (from the old English ‘to prick’); the square hole in the anvil is the hardy hole. The shoe is relatively thin and the hot metal relatively weak and easily deformed: holes cannot be made that way when the steel is cold. (In passing, a ‘hot chisel’ was used by the blacksmith to cut through heated thick plate that could not be cut cold.) Other features of a shoe such as calkins (blunt, downturned ends to the heel of each shoe cut off square to give grip on ice or slippery roads) are fabricated at the same
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Force (N)
60
D = 2.0 40
D = 1.0
20
D = 1.0 D = 0.5 0
0
5
10
15
δ (mm)
Figure 8-5 Punch load–punch displacement diagram for the penetration of 12.00 mm thick Sil8800 rubber blocks by sharp-tipped and flat-bottomed punches (after Shergold & Fleck, 2005).
time as the holes. The farrier’s nails are hammered through hoof, bent over and twisted off. The earliest known horseshoes, instead of having the familiar smooth outer and inner curves, presented an outline of a series of six or more wavy bulges resulting from punching the nail holes (it is the business of where the displaced material goes). Similar bulges around punched holes are seen in old jewellery (e.g. the copper headband on display in the Rembrant Museum in Amsterdam, which is seen in the painting Woman with Cap). Early smiths did not have the knowledge to eliminate the bulges without compressing up the holes. The first types of horseshoe nail were domed and protruded below the shoe. Then smiths began to fuller the underside of the shoe. The wedge-shaped fullering tool indents and forms a groove right around the underside of the shoe, in which the nail holes are punched. (The fuller groove does not involve cutting, the displaced metal being merely moved around on the shoe: punching the holes does.) Nails are narrow-headed and fit within the fullered groove, thus preserving the life of the head of the nail and shoe. (Shoemakers replicate these ideas by cutting a diagonal slit all round a leather sole, just inside its periphery, to take the row of stitches. The leather then closes down on the stitching and the waxed thread is preserved against abrasion until the leather itself is worn down to the base of the slit.)
8.4 Hollow Punches A very thin-walled hollow punch acts like a thin knife blade and can cut, in principle, by being pressed against the workpiece: like a knife, the punch separates material by the wedging action of its cutting edge (not by shear). This is how apple corers work. While there are constraints against motion inwards towards the centre of the disc/plug, and outwards towards the boundary of the workpiece, reasonable estimates of forces are given by Eq. (8-6) when using hollow punches with floppy materials, with modifications for friction as necessary. In stiffer materials, the contact pressure p will be greater and thus increase the frictional resistance. Stiffer workpiece
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materials probably require a stronger punch with a thicker wall to withstand the perforation forces, and hence a steeper wedge angle, and this exacerbates the friction problem. Even so, variations of hollow punches are used for cheese sampling; extraction of rubber, maple syrup and other liquids from trees; and extraction of core samples from trees and bore samples from soil. Hollow punches, like knives, benefit from ‘slice–push’, so rotation is helpful as in a cork borer. Slice–push can also be obtained without rotation by angling the ends of tubular penetrators such as cannulae (Chapter 11). Hollow punches like large cookie-cutters are used to stamp out cardboard jigsaws. The first jigsaws were made of plywood and cut by hand with a ‘jig saw’ (like a fret saw, not the modern hand-held electric jig saw; ‘jig’ here means up-and-down as in the dance of the same name). Hollow punches, as such, are not used with ductile materials owing to the forces required, but scalping dies that clean up the outsides of drawn rod act in essentially the same way. In can making an ironing ring is employed to thin down the wall below the base thickness by a process of pushing a standing wave of metal up the side of the can (Hosford & Duncan, 1994). The angles and reductions of area in an ironing ring are chosen so that the mode of deformation remains simply plastic flow and friction. However, with different angles and reductions of area, the standing wave transforms into a chip and we have axisymmetric negative-rake machining. The wall of the can would be thinned were this to occur, but the final height would be the same as the drawn cup, and the excess material be removed as a chip in the form of a ring, as sometimes happens in the process of metal spinning (Section 7.4.2). A scalping die is designed deliberately so that cutting occurs.
8.5 Arms and Armour The history of arms and armour is a fascinating story. The leapfrogging between weapon design and armour design continues to this day (Doig, 1998), and archaeometallurgy is a thriving research field (e.g. Blyth, 1977, 1993; Blyth & Atkins, 2002; Cheshire, 2005, 2007, 2009; Sim, 1995, 2002; Sim & Ridge, 2002; Fulford et al., 2005; Williams, 2002, 2003, 2007; Williams et al., 2006). The game of conkers seems to be something where arms and armour are equally matched. Armour in the widest sense includes reinforcement around bird box holes to prevent rodents gnawing their way in, and thick coats slung over the top of a spiked fence, or brokenglass-topped wall, or over barbed wire in war. In history, protection has been provided not only for the bodies of people, but also for buildings (castles), towns (fortified cantonments), for land, sea and air vehicles, etc. Weapons start with fists and feet. (The strange backwards-leaning poses of fighters seen in Victorian illustrations are because they usually aimed for the body only: they did not strike the opponent’s head in bare-knuckle fighting since it was as hard as their fists and it hurt their fists.) Hand-held weapons might remain in the hand during combat – knives, swords, axes, tomahawks (a Virginian Algonquin word) – or be thrown (spears, javelins). Assailants might be on foot, on horse or in a vehicle, and the latter may have had additional weapons (the Persian king Cyrus in the sixth century BC had metre-long scythes fixed to the ends of the axles of his chariot, as did Queen Bodicea). Means of giving extra energy to thrown weapons are seen in slingshots, catapults, aboriginal spear throwers (Cotterell & Kamminga, 1992) and bows and arrows – the long bow relying on the strength of the bowman, the crossbow having some mechanical advantage in its wind-up mechanism. Later, firearms converted chemical energy in the powder to kinetic energy of the bullet that was originally a lead ball in a smooth bore; afterwards, rifled barrels appeared. Later again, missiles exploded on impact. Even so,
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some modern weapons rely on ballistics (i.e. just kinetic energy) to penetrate targets: the highvelocity fin-stabilized armour-piercing discarding-sabot shot is the arrow reborn. Improvised hand weapons have always been around (pitchforks, billhooks, and in World War I soldiers found that a pointed spade, sharpened all around and swung, was more effective than a bayonet; it is reminiscent of a battle axe). Beyond hand-held weapons, there were large trebuchets and ballistas which were large catapults flinging rocks at the enemy’s fortifications in ancient times. Mobile artillery field guns, or similar guns in emplacements, originally fired cannon balls loaded at the muzzle and later shells loaded at the breach. Weapons for ships and aircraft developed similarly. In the era before guns, battering rams (so called because the impacting end was formed in imitation of a ram’s head) were employed on land; at sea the whole ship became a ram. World War I destroyers were designed to ram submarines on the surface. Early body armour in Mesopotamia in 2500 BC consisted of a flexible backing of layers of woven fabric or leather to which metal platelets were attached (a design which is perpetuated today) along with bronze helmets. Chain mail was worn by the Celts in 400 BC and in the twelfth to thirteenth century appeared hardened rawhide (cuire bouilli, literally boiled leather, but actually not leather in the sense of being tanned, but rather boiled rawhide) (Cheshire 2007). Iron and steel replaced bronze and, in mediaeval Europe, armour employed a few large plates to make a stiff, articulated, ‘exoskeleton’ – the familiar ‘suit’ of armour, the proper word for which is ‘a harness’ owing to all the leather straps for attaching, and to die in harness meant to die in battle. Typical thicknesses of armour were 1 mm (legs and arms), 2 mm breast plate and 3 mm for the front of the helmet. The hardness of mediaeval plate armour was 100–200 VPN. In comparison, surviving arrows are in the range 120–400 VPN. Shields went out of use when full suits of armour came in. In earlier times the Dacian battle scythe or falx (Latin for sickle) proved to be a weapon that could, literally, get over the top and behind Roman shields. The lettering on the Trajan column in Rome is beautiful and is the basis of many modern typefaces, but it tells the story of how the Romans returned to Dacia (in present-day Rumania) and got their own back. The horses ridden by mediaeval knights also had some protection, in the same way that in bullfighting the horses ridden by the picadors have padding to minimize damage from goring by the bull; the bull has no protection. It was beneficial if armour could not only prevent injury to the wearer but also blunt the attacking weapon; armour should ‘quench and dissipate the force of any stroke that shall be dealt it, and retund the edge of any weapon’ (retund from the Latin to strike or beat again). After about 1500, the weight of armour required to protect against firearms became so great that it limited a soldier’s ability to function. There is some evidence of chicanery by the makers of plate armour to show how good it was against lead bullets, since the power of the gunpowder used to fire the proof shot appears to be less than would be seen on the battlefield (Williams et al., 2006). The conflict between wearability and protection, comfort and mobility of the wearer, how much of the body to protect, remains today (Cork & Foster, 2007). Lightening the weight of the Corinthian helmet was central to the development of mobile infantry tactics used by the Greeks to defeat the Persians at the battles of Marathon and Thermopylae, whereby the Greek soldiers were able to run over long distances into battle. The beaten-out new helmet was about half as thick as before and nearly twice as hard but, being only half as tough, was more vulnerable (Blyth & Atkins, 2002). However, the altered shape and cant of the helmet, which gave increased distance between the skull and the helmet, seems to attempt and compensate for this. Humans are also warm-blooded, so that body armour must also be well ventilated (Fenne, 2005). Between the introduction of firearms and World War I (when ‘tin hats’ were issued to protect against snipers’ bullets when the head appeared above the trench, and against shrapnel from overhead bombardment),
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footsoldiers of most armies usually wore no helmet or body armour. Even so, helmets and cuirasses (armour for the torso consisting of a breastplate and backplate, and originally made of leather) were still being worn by cavalry at Waterloo (1815). Throughout history there has been debate over whether a cavalryman should cut or thrust with his sword. The 1908 pattern British cavalry sword was introduced as a thrusting weapon, whose reach was as far as that of a lance: cavalry charges took place with the sword pointed forwards, with a locked elbow joint (Strachan, 2008). It would seem, however, that for a given effort, more damage should be done by a combined ‘thrust and slash’ than by thrust alone. There was always the problem of removing the weapon from the victim as the rider galloped by. ‘Tent-pegging’ and other such exercises aimed to improve such skills. The Industrial Revolution in Great Britain enabled large iron plates to be manufactured and this led to the iron-cladding of ships, originally with plates bolted on to a teak hull, later to all-steel warships. Trains armoured with steel plate were employed in the Boer War and the tank is an armoured mobile gun platform. Some modern armours are explosively reactive: the armour is layered, the outer surface of which blows out in reaction to an incoming charge. A generic modern armour on a tank, say, has a series of layers, often spaced away from one another, the outer of which disturbs the incoming missile, trying to break it up, behind which is the actual energy-absorbing material. On the inside of the vehicle there will be a spall shield to minimize damage from fragments released from the tank hull itself. Developments were different in different countries: there was Japanese armour that stood off the body so that there was considerable ‘give’ before a weapon reached the body; of course, too great a distance would result in the warrior looking like Mr Bibendum®, the Michelin tyre man, and having limited mobility. The Japanese also had papier-mâché armour; feather armour was used by the Japanese and the Aztecs (Vincent, 2008), in the latter case against slashing attack with weapons made from obsidian, where the toughness of the feather spines (rachises) made cutting difficult (Chapter 11). The effectiveness of feathers against shot guns and air rifles is of concern to game birds (Bonser, 2008). Crusaders had straight swords that relied on brute force, but the Saracens had swords with curved blades that performed more efficiently owing to push–slice along the blade (Chapter 5). When metallurgy was an art, not a science, much depended on the skill of the smith, the source of the iron ore and the inclusion population (particularly slag) in the microstructure. The key to the establishment of armour industries in the Tyrol and southern Germany at the turn of the fifteenth/sixteenth century was down to an understanding of hardening by slack quenching (that is a delayed or interrupted quench where the steel is not cooled quickly enough to form an all-martensitic microstructure, but instead forms a mixture of martensite, pearlite and perhaps bainite; although a lower hardness may result, there are fewer residual stresses and less chance of brittleness). The human body is not very resistant to attack (a 50 mm deep wound in the thorax is likely to cause death in fifteen minutes) and many vulnerable areas exist. Chapter 11 discusses what damage is likely to be trivial or very serious, and whether wounding is short term or long term. The penetration that might be allowable by a knife, for example, is determined by the thickness of skin and fat, and the closeness of vital organs to the surface of the human body. Over the chest, elbows and shins, permissible penetration is very little; over the gut it is greater. In consequence of this vulnerability, some form of body armour has been worn since ancient times, at least by those who could afford it. Even if armour is not ‘defeated’ (not pierced through by the weapon), the impact and blow are still felt. It could not have been pleasant for the wearer of even a well-padded suit of mediaeval armour to have been on the receiving end of a severe blow by a mace or other heavy weapon. So, while the primary aim
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of armour is to prevent perforation, an important secondary consideration is to minimize ‘behind-armour’ damage (blunt trauma), by absorbing as much energy and momentum as possible even if perforation does occur. It is not good when the deformed fabric of a body armour itself causes injury through blunt trauma (Liden et al., 1988), nor is it acceptable that an armour, in defeating a bullet, deforms so much into the wearer that it proves fatal. This is particularly relevant where only partial protection is in place, as in personnel body armour (flak jackets), protection of ammunition and equipment inside armoured fighting vehicles, the storage of munitions in static emplacements, the protection of communications equipment, and the infantryman in a trench (Hetherington, 1996). It is not always necessary to penetrate armour to inflict damage: it is well-known that in hard, not-so-tough materials, a cone crack develops beneath an indenter from the initial ring crack – what happens when a stone hits a glass window (Chapter 6). A conical scab can similarly be detached from the inside of tank armour and cause great damage. (Conical holes can be of great significance in forensic investigations as they indicate the direction from which the bullet came.) Johnson (1986a, b) remarks on the dangerous fast-moving giant splinters that were released on impact of wooden ships by cannon balls, which caused terrible wounds to sailors. There are different types of armour to resist different modes of attack from different types of weapon. It is usually impossible to design one armour that satisfies all demands, as illustrated by the different requirements between dagger attack and bullet attack in body armour. Perhaps surprisingly, flying debris and shrapnel are more likely to be a danger to the military than bullets, whereas police body armour has to protect against stabbing and shooting. The three main elements in modern body protection are (i) textiles (woven materials, including leather); (ii) plates (generally metallic) and (iii) chain mail. Flexible ballisticresistant panels are made up of a series of layers of a woven fabric of a very high strength fibre (aromatic polyamides – aramids – or long-molecule polyethylene). The bullet deforms the pack and stretches the individual fibres, which absorb energy in tension. A 9 mm pistol bullet will require about 20–25 layers of woven aramid to absorb the energy without penetration. However, this type of aramid pack will not necessarily provide protection against a stab with a sharp knife, which requires hard-surfaced plates to cause the knife tip to fail by chipping or blunting, or chain mail to deform and absorb energy. Note that armours are bullet or dagger ‘resistant’, not ‘proof’. Similar considerations about what happens when the human body is bumped or hit forcibly, what is felt and what injury may result, apply to sports. Protective clothing for motorcyclists against abrasive injury with the ground fits this same general category. Cushioned shoes give protection from impact as the feet hit the ground during running or gymnastics; it is a sort of blunt trauma isolation from repetitive strain injury. Assessment of the main risks in different sports, and the possible injuries, informs decisions on what type of protection should be provided and at what level. There are Olympic, international and national standards and regulations. Thus there is head protection in cycling, riding, amateur boxing, skateboarding and so on; torso protection is provided in American football; shin protection in soccer and hockey; foot/leg/joint protection when batting in cricket, and boxes; masks in fencing. Fencing is an example of where the weapon is reduced in potency by having a protective knob on the end of the sword. Knobs are also found on the prows of boats at Oxford and Cambridge (see Chapter 15). Boxing gloves diffuse blows as compared with bare-knuckle fighting. In mediaeval jousting, lances were flat-ended, the aim being to unseat the opposing rider rather than to pierce. In jousting practice with a quatrain (a turnstile-like target used for training) one idea was to break the tip off the lance. In battle, of course, lances and other weapons would be deliberately sharp.
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8.5.1 Energy imparted by weapons The force and energy imparted to a weapon by hand depend on muscles and fitness. It is not always the circus strong man who is pulling the bow. Jones (1984) points out that in studies of the effectiveness of arms and armour, it is important to set a historical context (materials for weapons and for armour, tactics in battle, health and strength of those involved, training, morale, geography of the battlefield), all of which can contribute to the eventual outcome. A stone in a sling being swung in a circle of radius 0.5 m at a rate of 1 revolution per second has a peripheral tangential speed of (0.5)1(2) 3 m/s. For a stone of mass 200 g, its kinetic energy on release is 0.5(0.2)32 1 J. A typical longbow arrow has a mass of 70 g and leaves the bow at some 45 m/s with kinetic energy of 63 J. Handgun bullets leave the barrel of a pistol at, say, 350 m/s. A 9 mm calibre (diameter) bullet has mass 8 g, and so has an initial kinetic energy of (1/2)0.008 3602 518 J. Some weapons (rubber bullets and the like) are designed to incapacitate, through blunt trauma, rather than kill. They are fired from a gun with low muzzle velocities (100 m/s) compared with lethal weapons. Tranquillizer darts used by animal conservationists are discharged at even lower velocities of some 50 m/s (Jones, 1983). The darts are in the form of a syringe having an impact detonator on the piston to inject the drug into the animal through a needle in the nose of the dart. Too high an impact velocity leads to a so-called apple-coring wound (Tawell, 2008). All free-flying missiles will be slowed by air resistance so that when the target is reached their effectiveness has been reduced. It is not a case of shooting ‘point blank’ (meaning that the gun is not elevated). Robins (1742) noted that a 3⁄4 inch (19 mm) ball fired from a musket at 1700 feet/s (540 m/s) at an elevation of 45° (to give the well-known parabolic trajectory its maximum range) would reach 17 miles (25 km) in a vacuum; in practice, the range was less than half a mile (800 m), owing to the loss of energy due to air resistance. At velocities below about 0.6 of the speed of sound (below Mach 0.6 or about 225 m/s at sea level), the air drag is approximately proportional to the frontal area of the missile, i.e. to the square of the shot diameter, d2. At higher speeds, however, the air resistance increases at a far higher rate, the result of which is that within tens of metres of leaving the barrel, the muzzle velocity of highspeed bullets is reduced to one-half (and the initial kinetic energy to one-quarter). Cheshire (2005) discusses the trajectory of a weapon taking account of the energy lost to air drag. In broad terms, the range is shortened and the arrow, bullet or shell descends more steeply after passing the peak height. Since the mass of shot is proportional to its volume (d3), its kinetic energy must also be proportional to d3. Thus, at the lower velocities where drag is proportional to d2, we have the same sort of cube-square scaling discussed in Chapter 4. It explains why a shot gun pellet has far less range than, say, a cannon ball fired at the same velocity, and why bigger guns can engage the enemy at longer ranges. However, for firing against personnel, a large number of missiles increases the likelihood of hits. Grape shot was a number of small cannon balls fired simultaneously from a field or naval gun and was effective at intermediate ranges. The Englishman Henry Shrapnel (1761–1842) invented a hollow cannon ball containing bullets and the like that could be detonated above the enemy, setting free a shower of missiles and shell fragments at high velocity. The word has come to include any type of flying fragments from whatever source. Cheshire (2005) also discusses the Magnus effect whereby round shot, fired down a smooth barrel, bounces from side to side and emerges spinning from the gun to cause deviation in the trajectory. The effect of aerodynamic lift (gliding) that might extend the range of arrows, may also be incorporated in analyses but that seems to be relevant only for competition arrows.
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The angle at which a missile strikes the target (normal or oblique incidence), and whether the motion of its centre of mass is in the same direction as its axis (yaw, etc.), affects its potency. Missiles may be deflected or skid, or dig in and tumble (some Israeli armour deliberately aims to tumble incoming shells). Armour on the Russian T34 tank of World War II was cleverly inclined to the likely direction of incoming shells: it was not better metallurgically, but was more effective. Figure 8-6 shows damage on German gun emplacements at St Malo resulting from deflexion of shells. The velocity of a weapon still held in the hand at impact is affected by the dynamics of the combined motions of arm and weapon and whether the attack is overarm or underarm (Jones et al., 1994; Chadwick et al., 1999; Horsfall et al., 2005), and the grip (Connolly et al., 2001). Furthermore, as distinct from a free-flying weapon, muscles can drive a hand-held weapon onwards into the target with additional energy after the first contact (the so-called run through). Here the presence or absence of a guard on the handle of the weapon is important: the hand may slip down a handle when resistance to penetration is felt. Several studies have been undertaken to measure the physical characteristics of force and velocity during stab motion. Instrumented knives have been used to record movements and loads on the blade during a series of simulated stabs by a number of different volunteers on to a prepared target surface. With the hand up against the guard, there is an initial group of force peaks between 12 and 16 ms associated with momentum transfer from the knife itself to the target. A broader peak follows at 25 ms caused by deceleration of the hand and arm. The initial force peaks are the same when the hand is not initially up against the hilt (Figure 8-7), but the later broad peak is now divided into two distinct peaks at 25 and 40 ms. The first is from forces on the handle as the grip slides down the handle, the second from the forces on the hilt. Stab impact velocity measured in the laboratory ranges from 5–10 m/s and the maximum energy applied by any particular individual may range from 10 J to over 100 J (Horsfall et al., 1999; Chadwick et al., 1999). French (1988) discusses the design of the human arm. Connolly et al. (2001) investigated the differences when gripping spears in overarm, underarm and shoulder-level positions, all of which can be found on Ancient Greek vases and
Figure 8-6 German cast steel gun emplacement at St Malo showing damage by attacking Allied forces. (Courtesy of Eddie Cheshire)
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The Science and Engineering of Cutting 2000 Total force Guard force Handle force
Force (N)
1500
1000
500
0 0.01
0.02
0.03
0.04
0.05
0.06
Time (s)
Figure 8-7 Stabbing with knives: force data from a test with the hand held away from the finger guard. The impact velocity was 5.2 m/s with a total energy of 38 J of which 30 J was delivered through the handle (after Horsfall et al., 2005).
Roman sculpture. Instrumented laboratory tests showed that the overarm grip delivered the greatest impact velocity, peak force and energy to a target (some 6 m/s, 1 kN and 40 J); underarm attacks were about 40 per cent less and shoulder-level grips about 60 per cent less. The results are consistent with the type of action involved, overarm motion allowing more time to accelerate the weapon and maintain the force. Pictorial evidence from vases always shows the overarm grip when the Greek phalanx is shown, and hand-held spears were depicted predominantly in the overarm position in Roman times. The underarm was illustrated being used mainly in loose formation for individual combat or against cavalry. Although the shoulderlevel grip can improve the reach of a spear, there was no pictorial evidence of this grip being used in combat. All this sort of information informs designs of body armour and there too knowledge of what organs will be encountered at what depth into the body, and what damage will be fatal or may be acceptable, determines permissible depths of penetration of a weapon through body armour. Such depths are then incorporated in design standards for vests and their quality control. Many tests rigs to assess body armour for stab resistance are in the form of dropweight devices in which a 2 kg mass, to which a standard blade is attached, falls from different heights (0.5 to 3 m) to give a range of impact energies on to candidate armour. Below the armour are layers of material to simulate the manner in which a body gives way when stabbed. These rigs were designed on the basis of results from instrumented knife tests. In practice, when a knife comes into contact with body armour, the tip of the blade may get caught and the blade then bend sideways (a complicated problem linked to the compliances of the attacker’s hand at the knife handle, the wrist, elbow and shoulder). Conventional proof testing arrangements for candidate body armour constrain blade motion to the original
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direction of impact (Fenne, 2005). Furthermore, the time of contact from the deceleration of a single rigid mass in a drop tower is likely to be less than the relatively extended deceleration period during which energy is transferred from knife/hand/arm to the target (Horsfall et al., 2005). In the Japanese art of Iaido (using Samurai swords), the motion of the arm is almost horizontal and circular. The sword is sliced forward and brought back. It is difficult to do because in the horizontal plane there is no arm acceleration downwards. Execution by sword is not always perfect. According to McAll and McAll (1987), a group of Eighth Route Chinese soldiers was ambushed by the Japanese during World War II near Siaochang and executed. One, who was very tall and at the end of the line, refused to kneel before the much shorter Japanese soldier. The latter, whose arm must have been tired by that time, failed to do a clean job. Nevertheless, the man, with his head partially severed from his body, fell into the mass grave and was left for dead. Eric Liddell (Olympic runner – Chariots of Fire – and at that time internee) was able to smuggle the victim into hospital from the temple where villagers had hidden him. Kenneth McAll repaired the wound and the man survived. Finite element methods (FEM) simulation of stabbing has formidable problems (Ankerson et al., 1997): (i) non-linear behaviour, and rate- and environment-dependent anisotropic properties and so on; (ii) the physical properties of skin, flesh and bone-layered bodies are not yet properly known; and (iii) the physics of the penetration are not fully defined (e.g. criteria for penetration should be established from fracture mechanics, not the use of mere critical stresses or strains).
8.6 Penetration and Perforation of Armour There are many different ways in which a plate target may be perforated by bullets and shells. The type of failure depends mainly on (i) the projectile nose shape; (ii) the ratio of the projectile diameter to target thickness; (iii) the material properties of the target; and (iv) the impact velocity. Energy and forces for different modes of failure scale differently, so that extrapolation of formulae can be misleading. Behaviour of armour over all speed ranges, including non-normal incidence, moving targets, etc., is covered by the excellent reviews on projectile impacts by Backman and Goldsmith (1978), Corbett et al. (1996) and Goldsmith (1999). There is a host of empirical correlations for energy absorbed in impact penetrations that stretch back into the eighteenth century, but many take no account of material properties or variations in failure mode which must have occurred. Many expressions use the idea of ‘sectional pressure’ (weight of a projectile divided by its principal cross-sectional area): the greater this value the more effective the missile (Poncelet, 1829). The way in which a target is held and restrained (simply supported, encastré, stiffened, etc.) affects response and may result in additional deformation (e.g. ‘dishing’ of targets). Of various types of damage, we consider only petalling and plugging as they are relevant to spears, arrows, bullets and shells. They are all separation processes in which cracks are initiated and propagated within extensive ‘remote’ plastic flow fields either to displace parts of the target out of the way so as to permit the projectile to pass through, or to shear out a plug of material from the target ahead of the projectile as it passes through, in both cases to defeat the armour. Necking and radial cracking around perforations in thin sheets produced by spherical and conical indenters are considered in Atkins et al. (1998). With ball indenters, discing precedes necking and cracking (Palomby & Stronge, 1988). We limit ourselves in this chapter to impacts at those ballistic velocities where quasi-static analyses with rate-dependent properties may possibly adequately predict behaviour. The
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behaviour of penetrator, target and armour, the damage produced and the manner of perforating the target, may all change significantly at impact velocities from that under quasi-static conditions (because of, for example, strain rate and temperature effects on mechanical properties). Also, at high speeds inertia effects come into play, i.e. forces and energy are required to accelerate a stationary target to the impact velocity of the projectile. Yet again, stress waves may become important (see Chapters 10 and 15). Whether the effect of weapons is primarily to do with energy or momentum is discussed by Hetherington (1996). The debate harks back to the Newton/Leibnitz controversy in the late 1600s about the measurement of force and who invented the calculus. It was not until d’Alembert in 1743 clarified the difference between Fdt (impulse) and Fds (work) that the problem was resolved. A given force produces different accelerations in bodies depending on their mass. The forward momentum given to a bullet has the same magnitude as the backward momentum given to a gun, since the forces are equal and opposite. They act for the same time, but the two forces act over very different distances because the bullet is lighter than the gun. Hence the energy given to the bullet is much greater than that given to the gun (Inglis, 1951). A parameter often used in armour is the ballistic limit, which is the impact velocity required just to perforate a target with a given projectile [there is, in fact, a number of different definitions (Goldsmith, 1999) but this will do for present purposes]. It is determined experimentally from multiple testing: the velocity that perforates a target on 50 per cent of firings is designated V50. Most empirical correlations of the ballistic properties of a target are with hardness, yield strength or ultimate tensile strength alone with few, if any, quantitative measures of a target’s crack (separation) resistance, i.e. fracture toughness. Usually only qualitative descriptions of the target material are employed, using ill-defined terms such as ‘ductile’ or ‘brittle’, or perhaps that a target material has a certain ‘tensile strain capacity’. The inadequacy of empirical correlations and models that employ plasticity only is reflected in the comments of Sangoy et al. (1988), who point out that, while a basic requirement of an armour is high hardness, there is no simple correlation between hardness and resistance to perforation, as measured by a structure’s ballistic limit. It is clear that resistance to separation should play a part in how well an armour performs, yet fracture toughness is not found in specifications for armour. The following analyses show how toughness influences the performance of armour.
8.6.1 Spears and arrows Is there a best shape for the head of a free-flying spear or arrow? Should shapes be different for direct thrusts and slashes in hand-to-hand combat? The Latin for spear point is cuspis (from which the mathematical term cusp comes), but is that shape best for the head of a spear? The Latin for spear is hasta, hence ‘hastate’ triangular leaves in botany that have two spreading lobes at the base, looking like the halberd, a mediaeval shafted weapon incorporating axe and spikes. Of the many surviving mediaeval arrowheads, two types predominate: a small broad head and a thin dagger type (the so-called long bodkin). The former produced great wounding in humans and animals; the latter was designed for attacking armour. When the fur of an animal was of great value, hunting arrows with wide heads were used, the idea being to knock down rather than penetrate the tissues of the victim in an indeterminate region. Study of the parameters controlling the resistance of armour to attack by knife, spear and arrow may be investigated in the laboratory using tools with pyramidal points. The
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force–penetration behaviour, and energy required, to pierce metal sheets at normal incidence were studied by Rossi et al. (2008), under both static and dynamic loading (in the latter case, adapting a split Hopkinson bar to fire arrows); see also Blyth and Atkins (2002) for penetration by a single-edged knife. Pyramids and cones, with a range of ‘point angles’, were employed including diamond-based (lozenge) profiles typical of spade-like spears and arrows. Defeat of metallic armour occurs by petalling, each edge of a pyramid cutting into the sheet at an angle, thus releasing metal to be bent into spirals. For a given number of sides, the most pointed pyramids required lowest loads for perforation. Petal radii are smallest at the commencement of perforation in all cases and gradually get larger as perforation proceeds. Pyramids with a large number of edges did not crack at every corner and required large penetration loads. Experiments with conical indenters confirmed that the number of radial cracks seems to reach a limiting number (5 for a cone having 15° semi-angle; 4 for both 30° and 45°). Similar behaviour has been found when conical penetrators have been used to model teeth (Chapter 13). Cones and many-sided pyramidal arrowheads are not efficient puncturing weapons. The ‘field’ arrowhead of conical form is good aerodynamically but has poor penetration (Cheshire, 2008). In the case of lozenge-based tools, fracture of the sheet occurred only at the sharper corners, with just stretched (but uncracked) zones at the blunter corners, with a single petal encompassing two sides of the tool. (Sometimes limited cracking occurs at the blunt corners owing to in-plane anisotropy in mechanical properties.) The forces required for perforation with lozenge tools were the lowest of all the tools investigated. Penetration forces were predicted using a modification of the model by Wierzbicki and Thomas (1993) for the loading of a thin plate edge-on by a wedge, in which the wedge indents and cuts the plate, displacing the material occupied by the wedge in a pair of outof-plane cylindrical flaps. The general idea for analysis of pyramidal perforations is that enlargement of the hole at each corner in the plane of the sheet, by movement of the pyramid in a direction perpendicular to the sheet, may be likened to wedge cutting at each corner into an inclined plate. Each out-of-plane cylindrical flap in wedge cutting may be viewed as forming one-half of a curled petal in pyramidal perforation; there is no geometrical mismatch along the centre line of the complete triangular petal. The total force and energy required for an n-sided regular pyramid with n corners may then be obtained from the force and energy required for a single corner. At the point of separation along every edge of the pyramid, there is (i) membrane work in each stretched zone at the tip of a crack and (ii) bending work for petal curling. The membrane work rate increases as 2/3, but the bending work rate varies as (1/), where is the current radius of the emergent petal. There is therefore an optimum for minimum work done. The in-plane wedge force is obtained by equating the external to the internal work rates, from which the piercing force perpendicular to the plane of the sheet may be obtained, and from that the energy absorbed by integration of the force with respect to displacement. For a given pyramid and target material,
Fpierce / Yt2 (constant)(u/t)0.4 (δc / t)0.2
(8-7)
where Y is the uniaxial yield strength, u is the weapon displacement in the direction of Fpierce, and t the thickness of the target. The parameter C is the critical crack opening displacement (CTOD), which is an alternative measure of the fracture toughness of the target material (R YC for plane stress where the fracture process zone is long and ‘flame-like’;
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or R 3YC for plane strain where the plastic zone is almost circular, like the constrained flow field beneath an indentation). Using C rather than R enabled Wierzbicki and Thomas to couple the far-field petal work to the near-field separation work and thus minimize the total work done. The energy absorbed U is
U/Yt3 (constant)(u/t)1.4 (δc / t)0.2
(8-8)
The functional dependence of U on (u/t) and (C/t) given by Eq. (8-8) is followed by experimental results. The analysis helps to answer the practical question of whether there is a ‘best’ number of faces on a pyramidal weapon in order to make penetration easier. For symmetrical-based tools, the three-sided pyramid, for which 90[1 (2/n)]° 30°, where n is the number of edges, requires least work. The forces required for lozenge-shaped implements depend upon the asymmetry. Obviously, when is 45°, the performance is the same as a square-based tool and it requires more effort to perforate with a four-sided than a three-sided penetrator. As angle reduces, the effort required to perforate with a lozenge-based pyramid decreases and when 35°, spade-like tools require the lowest forces of all. Figure 8-8 shows that the Tudor arrowhead has 35–36°. Note that the shape of the Tudor crossbow bolt is not a symmetrical lozenge, however, since such a shape is difficult to make by hammering on to a flat anvil; likewise, triangular-section arrowheads are difficult to produce. While lowest requires lowest forces, razor-thin lozenge weapons are not practicable, as they will buckle and bend upon impact rather than penetrate. The forces required to bend plastically the tip of a lozenge-shaped penetrator having semi-angle may be estimated (Rossi et al., 2008), and Figure 8-9 shows this buckling/bending relation superimposed on the graph of the force for
Figure 8-8 Types of arrowhead showing changes in shape in response to improvements in armour. The three on the right are modern reproductions used in shooting trials: the leaf type for hunting (far right) had little armour-piercing ability; the long bodkin (second right) was used against chain mail; the third replica is a more robust bodkin with improved resistance to buckling. Both bodkins have been damaged in laboratory trials. The arrow on the left is a Tudor crossbow bolt intended to pierce ferrous armour; its semi-angle of 35° is remarkably close to the minimum angle below which, in theory, lozenge-headed arrows require least force for perforation for all polygonal-based arrowheads (courtesy of Eddie Cheshire).
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perforation for two geometries of arrowhead: (i) a thin lozenge-type tool with 4° and average tip thickness of 2 mm; and (ii) the Tudor crossbow bolt having 35° and average tip thickness of 4 mm. In both cases 0.15. All forces in Figure 8-9 are normalized by Yt2 of the sheet material, using Yarrow 200 MPa, typical of a mild steel arrowhead. The perforation forces are calculated for (u/t) 0.1, which was chosen to represent the start of perforation, when the competition between buckling/bending and perforation determines which deformation occurs. We see that (for the particular values chosen) the thin leaf-type arrowhead buckles unless 28°, but the much stockier Tudor arrowhead should never buckle or plastically bend. Thin blades have low bending stiffness in the transverse direction, and the buckling resistance is poor. A deformed weapon point has an impaired ability to penetrate armour because the damaged tip reduces the energy concentration per unit area. Augmenting the second moment of area by fattening up the lozenge to improve buckling resistance paradoxically necessitates more energy to puncture a wider hole in the armour. A more violent deceleration causes higher stresses in the weapon, which must be resisted by a further increase in bending stiffness. Historically, a vicious circle of cause and effect led from thin hunting arrowheads to the lozenge section of crossbow bolt heads of Tudor times. The fact that oblique impacts are more common than normal incidence also influenced these developments. When an arrowhead strikes a target obliquely, the tip will either indent the surface (without necessarily perforating the armour) or be deflected violently sideways. The latter condition imposes a large bending moment on the weapon, which may smash the wooden shaft of an arrow or bend the head. A large second moment of area is necessary to resist deformation. Lozenge with θ � 4°
Lozenge with θ � 35°
2.5
4.5 4 Non-dimensional force
Non-dimensional force
2
1.5
1
0.5
3.5 3 2.5 2 1.5 1 0.5
0 A
0
10
20 30 40 Point semi-angle
50
0
60
0
B
10
20 30 40 Point semi-angle
50
60
Pentration force Buckling force
Figure 8-9 Forces (all normalized by Yt2 of the target material) for perforation of the target, or buckling of the arrowhead, as they both depend on the point semi-angle of the arrow. The graph on the left is for a leaf-type arrow with a lozenge corner angle 4° and average tip thickness of 2 mm; that on the right is for a Tudor bolt having 35° and average tip thickness of 4 mm. The curves for the leaf arrowhead on the left intersect at a point semi-angle of about 28°, meaning that buckling will occur at smaller point angles, but the curve for buckling for the Tudor bolt always lies above the curve for perforation.
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Historical source materials and modern injury experiments (Karger et al., 1998) confirm the results that thin lozenge weapons are the most effective against unarmoured targets such as animals and humans. The classical leaf-shaped arrowhead is of this form and it was able to penetrate deeply into flesh (US Surgeon General’s Office, 1871): ‘To-Kah K-ten, or “he that kills his enemy”, an Indian scout, in a quarrel with a fellow scout at Fort Buford, Dakota Territory, January 3, 1870, received a penetrating arrowwound of the pelvis and abdomen. … The shaft of the arrow having been withdrawn before he came under surgical observation, the exact direction of the arrow could not be determined, but, as the blood marks on the shaft showed that it had penetrated about twelve inches, and the arrow-head would make at least three inches more, it is supposed that the arrow had passed up through the pelvis into the abdomen … peritonitis supervened, and death ensued, January 18, 1870.’ The Mongols employed wide, thin arrowheads with sharp edges, but again only at close range against the unprotected parts of their enemies (Ricci, 1931): ‘… we will tell you how the Tartars go to battle. You must know that it is each man’s duty to take sixty arrows to battle; of these thirty are small, and used for transfixing the foe, and thirty are bigger, with large points; these are shot at close quarters, and strike the enemy in the face or in the arms, cut bow-strings, and do other such damage. Once they have shot all their arrows, they take their swords and maces, and exchange terrible blows with them.’ Wearing armour reduces the penetration of wide, thin, blades because of the energy required to make the long slit as the blade perforates the armour. While, in general terms, the mechanical properties required of armour are both a high yield stress and a high toughness, resistance to perforation varies only with (toughness)0.2 when petalling is the mode of deformation. This means that the bending of the petals is the principal energy sink, and that there would be some resistance even if preformed slits existed in the armour (i.e. were C 0). Thus, other things being equal, greater benefit is obtained from higher Y. Nevertheless, the resistance is still improved when C is as great as possible.
8.6.2 Chain mail as armour Present-day chain mail (more correctly just ‘mail’) is made from stainless steel or titanium with wire diameter of some 0.5 mm and ring diameter of some 5 mm. Modern uses of welded chain mail are for gloves and aprons in abattoirs, and as protection to diving suits for those lowered in cages among sharks. Chain mail also forms part of ‘body armour’ intended to resist both bullets and stabbing. Bullets are resisted by aramid fibres (Kevlar®) that stretch around the nose of a bullet, and efficiently resist penetration, but a knife will find its way through woven fabric relatively easily. Aramid cloth placed behind chain mail helps to take up the inevitable first perforation of the point of a knife through a ring. Mail is lighter than plate armour. The areal density is about 2 kg/m2 for 7 mm diameter rings made from 0.7 mm diameter steel wire; a 2 mm thick steel breastplate has about 16 kg/m2. Furthermore, mail is very flexible and drapes well over the body. It is particularly good at resisting slashing by weapons. It is not quite so good at resisting penetration, in that the point of a knife will easily pass through a ring before its edges contact the circular ring, deforming it into an oval. Clearly, slender weapons go through further than broader ones: the stiletto is an Italian assassin’s knife that will penetrate deeply. To protect against penetration, mediaeval warriors
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wore a quilted cotton jerkin, called an aketon, beneath chain mail (aketon from the Persian for cotton). The performance of mail depends on whether the joints in the rings are plain butted, riveted or welded, or whether some of the rings have been punched from the solid. The force required to deform a continuous circular ring into an oval may be estimated as follows. Figure 8-10(a) shows a ring of mean radius r loaded across its diameter by opposed forces F generated by a knife moving perpendicular to the plane of the ring. Ghosh et al. (1981) shows how a circular ring loaded in compression begins to collapse by the formation of four plastic hinges. The tension version is shown in Figure 8-10(b) but when loaded in tension, experimental observations show that the sides of an extensively deformed ring are straight being formed by travelling plastic hinges moving away from the horizontal centreline along every quadrant towards the loading points; the change in radius of curvature along the sides of the ring is from r originally to . At the loading points, there are stationary hinges that monotonically rotate with loading so as to keep the top and bottom of the ring perpendicular to the applied loads. (A stationary plastic hinge produces increasing rotation at the same location in a structure; a travelling hinge produces limited rotations over progressively extending regions of a structure.) The in-plane forces F are given by F 4M p / r(1 sinβ)
(8-9)
where MP is the bending moment to produce a plastic hinge and is the rotation of a quadrant. For a wire having a rectangular cross-section (chosen for simplicity), with radial thickness w and depth t, Eq. (8-9) becomes F/Ywt (t/r)/(1 sinβ)
(8-10)
Hinges
F,y Mean diameter D Mean radius r
y r(1-cosβ)
F
β
β Δr
F
Figure 8-10 Deformation of chain mail ring by forces F applied outwards across a diameter: (A) undeformed; (B) deformation mechanism of four plastic hinges (when forces F are applied inwards across a diameter) is modified when F act outwards, into two plastic hinges at the points of application of F and two travelling hinges that keep the sides straight.
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where Y is the rigid perfectly plastic yield strength. The total movement y of the point of application of the force F is y r(sinβ cosβ 1) r(β sinβ) r(β cosβ 1)
(8-11)
the second term in which recognizes that the straightened arc r on the sides is longer than the space available given by rsin, so that the top and bottom of the ring move bodily outwards. All plots of (F/Ywt) vs (y/r), irrespective of the cross-section of the wire from which the ring is made, will take an identical shape, the only difference being the scaling of the force axis through (t/r) in Eq. (8-10). Initial plastic deformation of a circular ring into an oval is relatively easy as considerable normalized radial deformation (y/r) is obtained for only small increase in (F/Ywt). Only when (y/r) approaches about 0.55 does (F/Ywt) increase, but then it does so almost vertically, the system stiffening up very rapidly. Given that the perimeter of a quadrant is r(/2), the maximum displacement of one of the loading points is r[(/2) 1] 0.57r, so stiffening up occurs very late in the deformation. The forces F push both edges of the knife against the inside of the ring at the opposite ends of a diameter and, depending on the geometry and sharpness of the edges, an indentation of some sort into the ring will occur. The wedge hardness H (defined as load over the projected area of contact) is related to the yield strength Y of the material by H cY
(8-12)
where c is a constant (1 c 2, say) depending on the semi-angle of the wedge and friction (Grunzweig et al., 1953). The same force F, for bending of a ring into an oval, produces indentation into the wall of the ring. The depth of indentation is determined by Eq. (8-12) using F and the geometry of the cutting edge of the blade. The force F moving its point of application by distance y does the ring bending work; the same force moving its point of application the further distance performs the work of indentation. The shapes of F vs (y ) curves are similar to those of (F/Ywt) vs (y/r). The presence of the indentation of local depth reduces the thickness of the ring from t to (t ), implying a reduction in plastic collapse moment at that section (Figure 8-11a). However, the presence of the knife-edge completely filling the V-notch prevents the easier rotation implied by the reduced MP. Indeed, it prevents rotation altogether within the beam to depth . The blocking of rotation means that the reduced ligament ahead of the tip of the edge of the knife is stretched in bending around the point of the wedge with greater tensile stresses than expected. This aspect of the problem turns out to be important for fracture of a ring. The flanks of the wedge are subjected to a sideways compressive force. For a knife edge having 30° total included angle, the sideways compressive load is 5.6Yw (Atkins, 2008b) . The overall bending stress distribution across the ligament of depth (w ), together with the compressive stress on the flanks of the knife edge, must result in zero force across the section. Hence Ywh Yw(t h δ) 5.6Ywδ
Or
2(h/t) 4.6(δ/t) 1
(8-13)
where h is the distance of the neutral axis from the top of the cross-section. The compressive region resulting from bending disappears when (t h ) 0, i.e. (h/5.6) 0.18h.
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Tension
t
h
Compression
5.8 Y
5.8 Y
δ
15° 15° Knife
A
��
δ ρ
Tensile strain
h Neutral axis
S ζ
B
Compressive strain
δ
radius ρ
Figure 8-11 (A) Stress distribution for plastic bending over the ligament that remains beneath the indented blade; (B) the strain distribution over the remaining ligament is altered by the indented knife jamming the rotation and thus increasing the tensile stresses on the outer part of the ring.
When just reaches that depth, the induced tensile loading caused by bending, and given by Ywh, exactly matches the sideways loading of 5.6Yw caused by indentation. At greater , it is impossible to maintain equilibrium with a rigid-perfectly plastic material since (i) the tensile stress across the ligament cannot rise greater than Y in value; and (ii) the area over which it acts, given by (t ), is continuously reducing. Unless workhardening is permitted, fracture is implied as soon as 0.18h. Calculations (Atkins, 2008) show that when the ring material can workharden appreciably, ring fracture is delayed, but not prevented. It so happens that stainless steel, from which chain mail is manufactured for hygienic reasons, has a high workhardening index of about n 0.5 in on. The in-plane pushing apart of a ring along a diameter is produced, in practice, by a tapered knife moving normally to the plane of the chain mail down through the ring. The associated forces (including friction) may be established by resolution. The energy absorbed at some deformation (where is related to y by Eq. 8-11) is given approximately by
U tot [1 (µ/tan2α)]wt2Yβ
(8-14)
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Of course, fracture of a single ring does not produce failure of the chain mail as a whole. Furthermore, note that mail does not begin to act until it is in a state of in-plane tension.
8.6.3 Ballistic plugging In simple punching (Section 8.3.3), the workpiece is supported from below and cutting in shear from the lower anvil usually occurs simultaneously with cutting by the upper punch along the shear band extending from top to bottom surfaces. Static and dynamic blanking and punching of metals is investigated by Balendra and Travis (1970), and by Dowling et al. (1970). In ballistic perforation, the target is not supported from below, Langseth & Larsen (1990). Although inertia of the target helps to keep it in place relative to a fast-moving projectile, the deformation zone is not a simple shear band but is spread out towards the distal surface of the target by a complex combination of shear, bending and tensile loading (Teng & Wierzbicki, 2005). Should extensive dishing occur, tensile cracking may initiate on the distal surface and produce different failure patterns. Nevertheless, the same principles of indentation followed by cracking plus flow will apply and quantitative measures of a target’s resistance to defeat, by whatever weapon and by whatever means, should include toughness. Separation can be complete before the bullet passes through the distal surface of the target. A cross-section of a plug produced ballistically in a titanium alloy (Figure 6a in Woodward, 1984) shows cracks running ahead of the projectile. Since energy absorbed is particularly important for armour, knowledge of the length of cracks a running ahead of the punch is important. Atkins (1980b) assumed that a ( cr) (Section 3.8). Experiments on the different, but related, problem of shearing prenotched ductile steel bars (Atkins, 2000) demonstrated that the length of a crack running ahead of the punch increased at a continuously increasing rate rather than the constant rate implied when a ( cr). This result inspired a different approach to determine the [a, ( cr)] relationship in punching based on minimum energy (Atkins, 2009d). At some punch displacement cr, the volume V of material being sheared by a circular punch is,
V [t δ a]πDh
(8-15)
where h is the width of the shear band and a is measured relative to the end of the punch; the current total crack length L is given by [a ( cr)]. In the next increment d of punch displacement, the increment of plastic shear strain d (d/h) and the increment of plastic work done per volume dW ko(/h)n(d/h). The increment of plastic work at cr is thus
VdW [t δ a]πDh ko (δ / h)n (dδ/h)
(8-16)
The increment of crack area is DdL D(da d), so the increment of fracture work is
πDR II (da dδ)
(8-17)
where RII is the fracture toughness in shear mode II (in practice separation is probably in a mixture of modes I and II). The increment of total work done dU at cr is therefore dU [t δ a]πDh ko (δ / h)n (dδ/h) πDR II (da dδ)
πDko [{t δ a](δ/h)n (R II / ko )(1 da/dδ)] dδ
(8-18)
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We argue now that the evolving a, relationship during perforation is such as to minimize all increments of total work done. That is d[{t δ a}(δ/h)n (R II / ko )(1 da/dδ)] 0
(8-19)
Atkins (2009d) shows that minimization for a rigid-plastic solid where n 0, where k is written for ko, gives a (R II / k)exp[(k/R II )(δ δcr )] δ δcr exp[(δ/δcr ) 1] δ
(8-20)
Complete separation of a plug from the plate occurs when the crack running ahead of the tool reaches the distal surface, i.e. when af f t, where f is the punch displacement at complete separation. How f depends upon cr at complete separation can be determined from Eq. (8-20) (Atkins, 2009d); f is smaller in materials displaying smaller cr (i.e. smaller RII/k). It may be shown that the total work of perforation, in normalized form, is [U total / πDkt2 ] [∆f (∆cr 2 / 2)]
(8-21)
Figure 8-12 shows how [Utotal/Dkt2] varies with cr (RII/kt). The upper curve is for the assumption that a 0. The lower curve is a plot of Eq. (8-21). Of great interest in both cases is the prediction that there is an optimum in normalized energy absorbed in plugging at a particular cr. [Utotal/Dkt2] 0.73 when cr (RII/kt) 0.6. The cause of the maximum is that should cracking occur early in the process, the crack area encompasses nearly the whole thickness of the plate so, depending on the individual RII and k values
0.8 0.7
Utotal/πDkt2
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6
0.8
1
Δcr
Figure 8-12 Variation of normalized energy absorbed in plugging with normalized cr. [Utotal/Dkt2] varies with cr (RII/kt). The upper curve is for the assumption that a 0. The lower curve is a plot of Eq. (8-21). In both cases there is an optimum in energy absorbed at a particular cr. This suggests that the fracture toughness of armour is as important a contributor to total energy absorbed as is strength or hardness.
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that determine the depth cr at which the transition takes place, toughness work is a bigger proportion of the total work; vice versa when cracking occurs late in punching. The competition between incremental plastic work and fracture work leads to an optimum cr for absorption of greatest energy. Given this, it seems surprising that toughness does not appear in specifications for armour plate material.
8.6.4 Scaling Scaling within the range generally combined plastic flow and fracture (not merely cutting) was discussed by Atkins (1988, 1990). Energy scaling in geometrically similar prototype (p) and model (m) follows
U p / U m λ 2 (λξ 1) / (ξ 1)
(8-22)
where is the ratio of the plastic work done to the fracture work done, and which, for cutting, may be written as (k/R)t Z/, where Z is the non-dimensional parameter that governs scaling within ductile cutting (Chapter 4). Relation (8-22) is the ‘true’ form of x-type empirical relations with 2 x 3, with the scaling factor, when combined plastic flow and cracking occur simultaneously. Scaling governed by Eq. (8-22) assumes that the material displays a given R not, as in plugging, where RII in shear is not constant but varies with thickness (Chapters 3 and 5). For all thicknesses of a given material in a given thermomechanical state, cr (RII/kt) is constant and f is a simple multiple of cr, different for every cr. Thus, the square-bracketed term in Eq. (8-21) is just a number depending on cr and independent of thickness. For example, f 0.33 when cr 0.1, so [f (cr2/2)] 0.33, and so on. Irrespective of these thickness-independent values, inspection of Eq. (8-21) shows that Utotal in geometrically scaled bodies should scale as 3, from the lengths D and t2. This is unusual in a fracture problem, but is down to R being thickness dependent. Nielson’s scaling expression that combines the SRI and BRL formulae (see Corbett et al., 1996) also implies geometrical scaling. The results of Duffey et al. (1984) for low-velocity punch impacts showed that 3 scaling applied within experimental error. In practical ballistic scaling experiments it is not always convenient to scale geometrically both the projectile and target together. For example, in Børvik et al. (2003), the target plate thickness t is varied but not the diameter, length and mass of the projectile. In these circumstances, the energy required for perforation must scale as t2, from Eq. (8-21). The impact energy provided is increased by firing the projectile at increased velocities, so the supplied energy must scale as (velocity)2. The experimental ballistic limit velocity must therefore depend directly on plate thickness, and plotting the ballistic limit against plate thickness demonstrates that this seems to represent the data reasonably well. Be aware, however, that these scaling relations apply only within the regime where plug formation is governed by rigid-plastic fracture mechanics. The simple dimensional analysis for cutting forces in Section 3.5.1 may be adapted to plugging to give for Utotal
U total kt3[D/t]a [R II / kt]b
(8-23)
For constant [RII/kt], Utotal should scale as t3, i.e. as 3 as before, but if (D/t) is not constant (as in Børvik et al., 2003) scaling ceases to be geometrically similar.
Chapter 9
Sharpness and Bluntness: Absolute or Relative? Tool Materials and Tool Wear Contents 9.1 9.2 9.3 9.4 9.5 9.6 9.7
Introduction Tool Materials Manufacture and Sharpening Geometry of the Cutting Edge Measurement of Sharpness Retention of Sharpness: Tool Wear and Machinability/Cuttability Effect of Bluntness and Clearance Face Rubbing on Cutting Forces, FC vs t Intercepts and Subsurface Deformation 9.8 Cutting Edge Sharpness and Workpiece Critical Crack Tip Opening Displacement 9.9 Compensation for Bluntness 9.10 Wiggly Crack Paths Produced by Very Blunt Edges
221 224 227 230 231 233 237 240 242 242
9.1 Introduction George Orwell (1933), in Down and Out in Paris and London, remarked that the only way to run a successful restaurant is to have sharp knives. What is sharpness? Perhaps it is easier to define the bluntness (or dullness) of a cutting edge, or even easier to answer the metaphysical question regarding how many angels dance on the head of a pin. How important is sharpness to a cutting operation? Is ‘sharpness’ an absolute quantity or is it relative to the dimensions of the object being cut, or to the sizes of microstructural features of the material being cut? The conventional definition of sharpness is geometrical and refers to a cross-section through the cutting edge. Even after the finest sharpening treatments of grinding, honing and polishing, the cutting edge cannot be a mathematical ‘line’ and must have some finite thickness, owing to the finite site of atoms (rather like the end of a crack always having some crack tip radius). The tip of a cutting edge, like the end of a crack, is not necessarily circular, but the concept of an effective radius of the nose contained between the sloping flanks of the tool that forms the ‘wedge angle’ is a useful parameter in considerations of sharpness (Figure 9-1). References to the ‘roundness’ of a cutting edge, meaning the effective radius at the tip, as an indictor of sharpness should not be confused with a ‘round’ tool whose cutting edge is profiled into a curve; curved tools can, of course, be very sharp. Basing sharpness on edge radius alone may lead to difficulties. A garden spade does not have to be razor sharp to permit easy digging, but the same edge radius would be useless on a chisel intended to cut metals, on a delicatessen slicer trying to produce thin slices, and in dissection scalpels. Is the concept of sharpness related even to the method of cutting? In Chapter 5 it was explained how one’s tongue can be cut by paper when licking envelopes because sliding along the edge of the ‘paper blade’ reduces the forces to cut. So the edge of the paper flap of the envelope does not appear to be sharp when it is pressed against the tongue, but the same edge does seem to be sharp when the paper is slid along the tongue as well as pressed Copyright © 2009 Elsevier Ltd. All rights reserved.
221
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Width
Angle
Figure 9-1 Conventional depiction of a blunt tool as having a tip radius .
into it. When a boiled new potato is cut in half, the restrained skin stays in place. If the two halves are then cut in two by the same knife, the skin may debond from the surface (there is less restraint) and be taken down into cut. Was the knife sharp the first time, but not the second? Fingernails tend to be of the same thickness whatever their length, so ‘sharpness’ in scratching seems more to do with length and overhang. If very short or chewed nails are used to scratch, is it said that they are not sharp? Of course broken nails, or freshly pared fingernails made in a series of straight cuts like a polygon, may have pointed corners that give the impression of sharpness. Sharpness is not necessarily synonymous with a small included wedge angle: the cutting edge of a thin knife could be truncated or rounded. Even so, tools properly ground to a small wedge angle are likely to be sharp. To withstand high cutting forces without breaking and/or wearing away, many metal-cutting tools have large included angles in order to provide sufficient stiffness and strength, particularly during interrupted cuts with impact loading on the edge. They are quite ‘chunky’ compared with knives, but their edge radii can in principle have the same cutting edge radius as a knife and, in that sense, they are equally sharp. Blades that are sharp may appear to be blunt when a cutting device is wrongly adjusted, as in scissors or guillotines that have excess clearance between the blades. This is why the blades of most scissors are bent (or set) so that they scrape across one another even though the pivot may not be tight; similarly for some lever-type guillotines. Hairdressers’ scissors are usually set only very lightly and are probably the best ordinary scissors available for experimental work. Sharpness is particularly important when the cutting edge is always in contact with the newly separated material ahead of the tool. When cutting ductile metals with offcut formation by shear (Chapter 3), the tool remains in intimate contact with the workpiece at the point of separation between cut surface and chip. During the transient indentation start to a cut, tool sharpness will concentrate the stresses and strains in the tip region as the load is increased, leading to the build-up of localized material damage. Eventually, at the appropriate load, the hydrostatic stress/effective strain criterion for fracture will be satisfied and a bifurcation in deformation mode occurs, from the plasticity and friction of inclined indentation to the plasticity, friction and separation of cutting (Section 3.6). Thereafter, at the cutting edge, material has to be continually separated by reinitiation of the ‘separating crack’ during steady cutting. Should the cutting edge of the tool be blunt, the highly deformed region around the tool nose is less localized and more diffuse, so that greater loads are required to effect separation. In addition, a dull tool gives a tool of variable rake angle, with local negative rakes near the cut surface that locally require greater loads (Bitans & Brown, 1965). Such large cutting forces ‘push back’ upon the rake face of the tool, have deleterious effects on friction forces,
Sharpness and Bluntness: Absolute or Relative?
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possibly encourage built-up edge formation, tool wear and so on, and result in bad surface finish. Blunt tools also produce more deformation beneath the cut surface than sharp tools and hence greater residual stresses, which can be a problem in precision engineering. In wood cutting, blunt edges are troublesome because wood is compressible and there can be appreciable recovery under the tool over long lengths of cut, making dimensional tolerances difficult to maintain. In extreme cases, where the tip roundness is comparable in size with the uncut chip thickness, the situation becomes quite different from ordinary cutting. In this sense a cutting edge which is ‘sharp’ for large depths of cut may be ‘blunt’ for far smaller depths of cut. Sharpness is very important in micromachining (sometimes called hard machining) where the depths of cut may be comparable with the tool tip roundness of even the sharpest tools. Sharpness of tools is important in guillotining and punching of ductile metal plates, not only in determining load levels but also in controlling the direction of the paths along which separation occurs and hence the quality of the cut edge (Noble & Oxley, 1963). An example where sharpness is, in a sense, important only at the commencement of a cut, concerns the splitting of wood by an axe along the grain, where the offcut is formed in elastic bending. The crack once initiated jumps ahead and arrests some distance in front of the axe blade. The axe’s sharpness will determine the load to start a split but since the arrested split will have the wood’s ‘natural sharpness’, the sharpness of the axe then no longer matters and further splitting takes place as the flanks of the axehead prise apart the timber. But of course for repeated jobs with an axe, it has to be sharp. For precision manufacture of components in low R/k brittle materials, uncontrolled cutting must be avoided. This is achieved by keeping the uncut chip thicknesses small enough so that offcut formation is by ductile shear (as in grinding, and in the way glass may be machined at sufficiently small depths of cut). The smaller the depth of cut, however, the more important tool sharpness becomes. If the extent of the unwanted cracking can be controlled, deeper cuts may be taken initially when machining brittle solids, and the damaged areas removed in subsequent cutting at progressively smaller depths of cut. This is the strategy traditionally employed in rough, semi-finished and finished machining of metals (Chiu et al., 2000). The need for sharper tools to cut hardwoods than softwoods is well known and relates to the R/k ratio rather than k alone. Green (wet) woods have yet different ratios and require different cutting conditions. Effects of edge bluntness in cutting wood include fuzzy and raised grain when planing, and surface roughness and thickness variation in veneer peeling. Sharpness of tools is particularly important for cutting highly extensible and flexible solids that are very difficult to separate by simple tearing. Whereas a stiff sheet of glass will break at a surface scratch, a sheet of some highly extensible material will not, as its deformation characteristics limit the degree of stress and strain concentration that is achievable. Highly extensible solids have J-shaped non-linear stress–strain curves in which there is initially very little shear connexion between elements, thus giving large extensions for little increase in load (Chapters 2 and 11). Only later do they stiffen up. Tearing of both rubber sheet (reversibly elastic) and plastic rubbish bags or chewing gum (irreversible) is extremely difficult even in the presence of a sharp crack precut by razor blade. This is not because the materials have large fracture toughnesses but because the concentration of stress and strain at the end of the nick becomes very diffuse and ‘runs away’ as deformation proceeds (Gordon 1978; Mai & Atkins, 1989). Cutting such materials with blunt blades is very difficult. In contrast, some materials tear easily once nicked, while being virtually impossible to tear without. Such materials seem to be a favourite for packaging. Woven fabrics display stiff stress–strain curves in the warp and weft directions, but a J-shaped curve along a diagonal, so that fabrics cannot be torn other than in the orthogonal thread directions and must be cut in any other direction.
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Dressmaking ‘on the bias’ (i.e. cut at an angle to the warp and weft of a woven fabric) makes a frock drape well over the body without the use of darts (internal pleats). How the shape of the stress–strain curve of skin affects recovery from cuts and wounds is discussed in Chapter 11. In medical applications generally, sharpness is important for surgery and things like hypodermic needles, cannulae, drips and so on. Similar considerations apply for tranquillizer darts fired into wild animals and to the use of pins to pop blisters. Sharpness of edges is important in skiing and snowboarding. A skier makes a ‘V’ with the skis and angles them to the ground so that they act as negative rake tools when ‘snowploughing’ to slow down. The ‘depth of cut’ (and hence resistance to motion) depends on the skier’s weight and on the push exerted by the legs; deeper penetration into the snow is made easier with sharp edges. Pushing more with one leg than the other produces a turn. Edge sharpness is just as important in more advanced parallel turns, where both skis are inclined in the same direction. Modern ‘parabolic’ skis have curved sides in plan view. It is an interesting question what the best shape is to get the best performance out of such ‘carvers’. The magnitude of the sideways resistance generated by the displacement (cutting) of the snow depends upon the mechanical properties of the snow: extremes are (i) skiing in icy conditions where it is difficult to indent and cut the ice; and (ii) skiing in deep powder snow where there can be too much resistance. In ice skating, the sharpness of the edges of the narrow blades is also vital for success. When a skier or snowboarder catches an outside edge (the one that is downhill), disaster looms. Similar questions about penetration coupled with sideways traction in snow and ice occur with crampons, climbing picks and so on (Chapter 14). An interesting question is whether there are ways whereby existing sharpness can be enhanced, or ways in which bluntness can be compensated for in some way in order to permit cutting with a dull tool. ‘Slice–push’ (Chapter 5) is one way. In wood cutting obliqueness of the tool compensates for bluntness (McKenzie, 1966): nevertheless, cutting of wood across the grain, even with sharp tools, nearly always results in failure down into the cut surface below the cutting edge (Section 4.7). Prestraining elastically extensible materials such as rubber sheet is another way of reducing cutting loads with a given tool (Section 3.2.1). It is a fact that strimmer cords are blunt and round to start with, yet successfully cut grass. Rotary lawnmower blades become blunt soon after first use, but continue to cut. The strimmer wire and blunt mower blade, if just pressed against a blade of grass, would achieve very little in the way of separation. Clearly the high operational speed must be important and experiments show that there is a critical speed below which strimmers fail to cut and that, in variable-speed strimmers, the speed has to be increased when cutting thick brambles and similar undergrowth. So can inertia of a workpiece compensate for bluntness in dynamic cutting? This sort of thing is discussed in Chapter 10.
9.2 Tool Materials There are many different types of knives and blades, including knives for professional chefs, butchers and so on; hunting, combat and sporting knives (including folding versions); razors; scalpels; trimming and craft knives; needles, surgical, dental and other sharp-pointed instruments; chisels, planes and other hand tools; scissors, shears, secateurs; metal-cutting tools of innumerable varieties; grinding wheels of various sorts including angle grinders used for cutting, and so on. Most have metal blades, including stainless blades for food and other hygienic applications, but disposable cutlery includes plastic knives, forks and spoons, employed not only for cheapness but also on aeroplanes and in airport cafes for security reasons. The design of both the handle and the blade is important to the overall performance of all implements.
Sharpness and Bluntness: Absolute or Relative?
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Blades come in a variety of thicknesses: the disposable blade of a craft knife is about 0.5 mm thick whereas a survival knife could be 4 mm thick. Some soldiers in the Terracotta Army at Xian are shown with knives on their belts with which to correct writing on bamboo strips (before the invention of paper); whetstones for sharpening also hang from the belts. The first tools employed by prehistoric man for cutting were flints, knapped from larger pieces to give sharp edges (Section 3.4.1). Arrow tips were made this way and axes were made by mounting larger pieces on a handle. Reproduction obsidian knives may be purchased at tourist shops in Mexico; they were used by the Aztecs in sacrificing enemies and other Aztecs. (Obsidian, named according to Pliny after Obsius, its discoverer in Ethiopia, is a volcanic glass rather like granite but transparent in thin sections and readily knapped.) Nephrite is the only jade mineral found in New Zealand. It can retain a very hard, sharp cutting edge, so was used by Maoris for wood-cutting tools fitted into carved wooden handles lashed together with plaited flax fibre cords. Improvements came once tools could be manufactured from metals, the softer alloys of the Bronze Age being overtaken by harder metals of the Iron Age for cutting tools, arms and armour. Rates of agricultural production improved tremendously when Ransome in Ipswich, England, discovered chilled cast iron in 1804 and used it for plough blades. Previously ploughs wore out rapidly when they cut furrows as they were abraded by stones and hard particles in the earth. Although what might now be called ‘steels’ were known in mediaeval times, their manufacture was rather hit and miss owing to differences in the composition of ores from different places and differences in skill of smiths (cf. Chapter 8 on the centres of excellence for armour manufacture). Development of steels especially for tools took place in the nineteenth century, particularly as the science of metallurgy began to be understood. What was required was high hardness so that softer materials could be cut, and high hardness also usually meant lower tool wear. As explained in materials science texts, the hardness achievable in plain carbon steels depends on the carbon content, and the higher the better. So 0.8–1.3 per cent carbon steels were favoured. However, these have poor hardenability (i.e. a high rate of cooling is required to produce the microconstituent called martensite that is very hard) so they had to be quenched in water before tempering. Tempering is reheating (i) to relieve residual stresses in the as-quenched material and (ii) to put back some toughness since the as-quenched martensitic material is brittle. Quenched-and-tempered plain carbon steels were the workhorse tool materials for much of the nineteenth century. They were sharpened on grindstones (Chapter 6 and below). However, improvement in ductility by tempering comes at the expense of strength and hardness, and the higher the tempering temperature, the lower the strength and hardness. Problems arose in Victorian factories when attempts were made to speed up production by increasing cutting speeds because the tools became hot, and even hotter than the temperature at which they had been tempered, so they became soft again (running the temper). For slow operations with hand tools and weapons, overheating is not a problem. A paper in the Philosophical Transactions of the Royal Society (Anon., 1673) on stone materials for building complained about the lost art of hardening and tempering steel, in that the ancients had managed to chisel and decorate hard stones that contemporary tools found too hard. As explained by Webb (2004), Mushet, in the Forest of Dean in Gloucestershire, discovered in 1860 a steel composition that would harden without quenching (air hardening). He was experimenting with additions of manganese to steel, and also found that there was tungsten in the local iron ore that clearly had something to do with it. (The symbol W for tungsten comes from the German Wolfram, meaning wolf’s cream or wolf’s soot, the name of an ore – a pejorative term referring to the ore’s inferiority compared with the tin with which it occurred;
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blacksmiths in Cornish mines used to harden picks by throwing wolfram into the fire; Fowler, 1916.) The alloy additions enabled Mushet steel to withstand higher cutting temperatures than tools made from plain carbon steel. According to Rollason (1956), the possibilities of Mushet steel for increasing production were not fully appreciated until the 1890s when Taylor and White developed the forerunner of present-day high-speed steels (HSS) showing that, in addition to tungsten, chromium was an essential ingredient. Such a tool steel was demonstrated at the 1900 Paris Exposition, where it machined mild steel at unprecedented speeds, feeds and depths of cut. The chips were blue, and the tool was red hot but remained sharp, resisting tempering up to 600°C. The practical result of Taylor and White’s work was that a turning operation that took 100 minutes with a quenched-and-tempered high plain carbon steel in the mid-nineteenth century took only twenty-six minutes at the beginning of the twentieth. Modern HSS contains other alloying editions to improve performance; lots of tools, saws and so on are currently made from HSS. Non-ferrous cast alloys called stellites that contained cobalt (Co), chromium (Cr), tungsten (W) and carbon (C) became available in the 1920s. They resist tempering up to 800°C but are brittle and have had only limited use as tools. In a way, they were the forerunners of modern sintered/cemented carbide tools consisting of, say, 90 per cent WC, TiC and other carbide powders held in a binding metal (often Co); sintering is necessary as such carbides decompose if attempts are made to melt them. At first, carbide tool tips were brazed on to ledges on normal tool shanks and the tips could be reground. Saw blades are still made like this, but in the late 1950s separate clamped inserts of carbides were introduced in the form of flat squares or triangles: eight or six cutting edges are available and the inserts may be moved around (indexable). A given insert may have different rake angles on different edges. They are used until worn and then thrown away. The economics of using disposable tools and not having to equip and run a tool room for resharpening makes sense with modern machine tools and very high metal removal rates. The teeth of some saws are ‘tipped’ (the cutting edges are similarly separate attachments to the body of the blade). In addition to carbide tools, there are also ceramic tools (alumina, Al2O3; silicon nitride, Si3N4; cubic boron nitride, CBN) and polycrystalline diamond tools, often bonded to cemented carbide inserts, for particular applications. Many tools have micrometre-thin coatings, intended to reduce friction and to improve wear, applied by physical and chemical vapour deposition methods. Lu et al. (2006b) discuss superhard nanocomposite coatings. Wollaston (1816) was one of the first to investigate the cutting ability of a diamond edge, distinguishing between the cutting edges of natural diamonds and those formed ‘by the art of the lapidary’. Data from photoelastic tools used to cut lead suggest that the greatest shear stress within a tool when cutting is near the tip, but the greatest tensile stress is found further along the rake face outside the contact area between chip and tool. Estimations of the maximum shear stress max in a tool from the normal and shear stresses on the rake face suggest that k max 2 k in the tool tip region (Childs & Rowe, 1973). Tool strength was considered by Prandtl (1921). While tensile stresses in tools are usually much smaller than the shear stresses (and are smaller for large included angles of the tool), they may still be large enough to promote edge chipping in more brittle (ceramic and diamond) tools. For this reason, such tools are employed with negative rakes and rounded noses. How the fracture toughness of tool materials responds to shock loading during interrupted cuts (thermal shock and fatigue, etc.) was considered by Atkins and Mai (1973); see also Tönshoff and Kaestner (1991). It is perhaps obvious that a tool has to be harder than the material being cut, but how much harder? And how sharp ought a tool to be? Development of ever-harder cutting edges
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has, perhaps, been aimed more at reducing tool wear on long production runs, but what is the least hardness ratio between cutting edge and workpiece that will preserve the rigidity of the tool, and hence the dimensions of the cut and the character of the cut surface? Cutting is indentation with sideways movement and a similar question arises in indentation hardness: Tabor (1951) showed that a ball indenter has to be about 2.5 times harder than the softer material to remain ‘rigid’. When an indenter is softer than this, mutual indentation occurs where deformation takes place in both bodies, the relative amounts depending on the hardness ratio and the geometries of the contacting bodies. The use of mutual indentation to measure hardness was first proposed by de Reaumur in 1722 in his book (see Plate 10, where a crossed wedge test is shown in which a diamond-shaped saddle indentation is formed; crossed cylinders give circular impressions). Hardness is given by the load divided by the projected area of the impression and may be related to the yield stress by H cY in the usual way (Atkins & Tabor, 1966). Mutual indentation in which testpieces having different geometries are made of materials with different ratios of Yhard/Ysoft, has been studied and found useful in forensic investigations (Atkins & Felbeck, 1974). It was only with the advent of accurately shaped diamond tools in the 1920s that mutual indentation fell out of favour to measure the properties of really hard solids (Smith & Sandland, 1925). Even so, for measurement of hardness at those very high temperatures at which no materials remain rigid enough to be indenters, mutual indentation has to be used (Atkins & Tabor, 1966). What about mutual indentation coupled with sliding? Tabor (1954) made the connexion between the Mohs scratch hardness test and indentation hardness tests. As explained in Chapter 6, the Mohs scale is a sort of mutual scratching test used by geologists and crystallographers. Tabor slid a pointed indenter over a plate having a hardness gradient. The hardness of the indenter Hp had a value in between that of the soft and hard ends of the plate. He demonstrated that the transition between scratching and not scratching took place when Hp 1.2 Hs, where Hs is the local hardness of the surface at the transition. There seems to be no work on scratching when both the indenter and the workpiece deform. Plate 9 in de Reaumur (1722) shows a bar, having a gradient of hardness obtained by heating one end and quenching, that was used to assess the hardness of chisels made of different steels. The best chisels were those that cut closest to the quenched end without chipping or spalling and without having their edges fold back. The saying ‘True as Ripon Steel’ referred to steel spurs made at Ripon in Yorkshire, England, which were the best in the world. The spikes of a Ripon spur would strike through a shilling coin without turning on the point (Brewer, 1981). How the ‘soft’ aluminium alloy wings of an aeroplane can indent and cut through the ‘hard’ steel exterior of a skyscraper is discussed in Chapter 15.
9.3 Manufacture and Sharpening The cutting edges of thin blades are manufactured by cutting tapers on the edge of metal sheets. The tapers in the hard sheet are formed by grinding (Chapter 6). Grinding by hand is also the traditional way of generating the tool signature on lathe tools in the workshop. Proper execution of grinding requires skill and training even with appropriate jigs and fixtures; the sharpening of gouges and of moulding plane blades for curved surfaces in woodworking well illustrates this. Early grinding wheels for sharpening were made of sandstone (the material of millstones) and used on carbon-steel tools. Sharpening at home could be done on stone window cills. Figure 9-2 shows how razors were ground in Victorian Sheffield. In the grinding of scythes, the stone revolves towards the grinder, which is different from most other grinding operations. The phrase ‘show us your mettle’ for a strong-armed person is said to come from grinders having metal particles lodged in their arms.
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Figure 9-2 Grinding razors at Sheffield in Victorian times (after Lloyd, 1913).
Later, wheels containing emery and corundum (natural Al2O3) and quartz (natural SiC) bonded together were used. Emery/corundum has a Vickers hardness of some 2100 HV0.05; quartz about 2700. Modern wheels employ artificial versions of these ceramics and also polycrystalline diamond held together in vitrified or silicate bonds. In continued use, the hard grains exposed on the surface simply wear down to the level of the bond, or debris fills the valleys between exposed grits to bring the surface of the wheel up to their tips, in both cases to prevent a wheel or oil stone functioning properly. While sometimes the grains fracture to produce new sharp grains, in most cases grinding wheels have to be ‘trued’ and ‘dressed’ by a diamond wheel or by other procedures to expose new tips from below. Meyer and Koch (1993) review dressing methods for alumina, CBN and diamond wheels. Care is required in grinding even of initially hard starting material, since the tool will heat up and the metallurgical structure may be changed. This is especially a problem at the thin cutting edge of knives where a region within 3 or 4 m of the edge may be softened. This is why knife blades are manufactured in the as-hardened state, rather than hardened afterwards (as files are). While it is possible to measure the nanohardness close to tip, it is not possible to get to the region right at the tip, so consistent manufacture must rely on proven procedures. Cutting tests according to International Standard ISO 8442.5 (2005) for knives identify edge softening. Edges overheated, decarburized and softened in grinding will wear, become dull and require further grinding. The very thin regions near the apex of the wedge forming the cutting edge are easily bent into burrs during grinding; burrs either remain attached to the ‘feather edge’ or will have been broken off in grinding to leave an edge that is irregular and rough on a fine scale. Harder blades have brittler burrs and are therefore more easily broken off in sharpening. The burr is always on the opposite side from the last grinding operation. In the case of knives, the direction of grinding is important. Grinding along the edge is not good, since the formed edge (the wedge flank) is easily rolled over, giving a blunt edge: the best grind is perpendicular to an edge or perhaps at a slight angle to an edge. In so-called hollow-ground knives, the back of the blade is thicker near the handle than at the tip when looking down on the knife and is thus more flexible at the tip. To achieve the shape requires twist or taper grinding and is
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more expensive to produce than just grinding an edge all the way along a blade of uniform thickness. Grinding produces minute hills and valleys with micrometre spacings (a certain raggedness) at the cutting edge, and many people believe that it plays a part in cutting where slice–push occurs (Chapter 5). The edge produced by grinding may provide a sufficiently sharp edge for some operations and tools are sold in that condition, but all types of tool can be further sharpened (whetted) before sale with an oil stone or diamond stone, or leather strop, and perhaps by lapping and honing where the surface is additionally polished. (Strop comes from strap but may also be connected with the Ancient Greek strophe and antistrophe, that is a back-and-forth movement in chorus and dances; lap is old English to fold or wrap; and hone means stone). Both lapping and honing employ fine abrasive powders such as rouge; honing is usually applied to cylindrical surfaces and lapping to flat surfaces, but the usage is not consistent since the fixture to sharpen blades of wood planes is called a honing jig. The purpose of extra sharpening is to ‘chase the burr down to as small a size as possible’ (Hamby, 2007). The surface finish of a knife blade (whether left in rough-ground condition or polished) makes a difference to its performance and classification. Note that the use of an oil stone, butcher’s ‘steel’ and so on during the life of a blade is for maintenance of an edge, to break off, abrade off or rub off burrs produced during cutting or through contact with hard surfaces. It is a misconception that regular steeling or stropping can resharpen an edge. Eventually, even with correct use, the cutting edge will become rounded and it is not possible to re-create the edge of the as-purchased blade without further grinding or by hand sharpening on a stone for an extremely long time. The shape of tapered burr may be represented by
(t/to ) 1 cx n
(9-1)
where t is the half-thickness of the burr, to is the half-thickness at its tip and x is measured into the edge from the tip. When n 1, the shape is cusp-like; when n 1, the change of thickness has the opposite sense. A force F applied across the cutting edge produces, at a section distant x from the tip, a bending moment Fx and an elastic bending stress on the outsides of the beam given by
σ (3/ 2)(F/w)(x/t2 ) (3/ 2)(F/w)[x/to2 (1 cx n )2 ]
(9-2)
where w is the length of the burr. Differentiation with respect to x demonstrates that the maximum bending stress occurs where xn [1/c(2n 1)], i.e. where (t/to) 2n/(2n 1). For n 0.6, (t/to) → 6; for n 1, (t/to) → 2; for n 2, (t/to) → 1.3. Thus, using the idea that the burr breaks off under tension, cusp-shaped burrs break off closer to the tip than those having a convex shape, and leave a broken-off edge whose thickness is not too dissimilar from the starting tip thickness. In an attempt to short-circuit the time and trouble involved in resharpening, there are so-called self-sharpening blades. One variety has a box or sheath in which the knife is kept, and every time the knife is inserted and removed, some sort of abrasive device ‘sharpens’ the edge. In the days before stainless steel was used for cutlery, it would rust and had to be cleaned. An admirable abrasive for this purpose (‘bathstone’, named after Mr Bath, the inventor) used to be made from mud in the River Parrett in Somerset. The best Sheffield cutlery in those days was made from ‘double-shear’ steel welded from plates of blister steel by hammering (Kitson Clark, 1931). Examples of Victorian proprietary knife-cleaning machines using emery, which may have achieved some sharpening, may be found in antique shops. Other present-day types
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supposedly sharpen the blade, during use, according to the rabbit’s tooth principle, in which ‘designed controlled wear’ continuously exposes a fresh, sharp edge (Chapter 13). Blades having a layered structure work on this idea. Layering may be either through the thickness of blade (e.g. Damascus steel) or from one side of blade to the other (bimetallic hacksaw blades). Damascus steel was famous in antiquity as a tool material. It was a layered microstructure made like flaky pastry by repeated rolling and turning over. When ground, different layers were exposed, giving grey–silver coloured patterns that etched differently in acid. The reputation of Damascus blades was based on the fact that they were said to retain their sharpness. In practice, behaviour was inconsistent (Hamby, 2007): one blade might but another did not, which suggests that it must depend on whether the edge came from a high or low carbon region. Bimetallic blades for hacksaws and band saws may consist of hardened HSS electronbeam welded to 0.5 per cent carbon steel. (A very good knife can be made by grinding an edge on an old hacksaw blade.) Other layered blades may have, for example, a thin (15 m) coating of WC on one flank. Some knife blades have repeating patterns of scallops (shallow cutaways) or serrations (deep cutaways) along their length (see Figure 7-1). As explained in Chapter 7, providing that they have a continuous sharp edge from one end of the blade to the other, they are not ‘saws’ as they cut by cleaving with no waste (and they have no kerf). What the zig-zags actually do may be a moot point, but what is true is that the apparent life of such a blade is increased: the points may be only 5 per cent of the edge and it is only these parts of the whole edge that are likely to be damaged by contact with hard cutting boards, hard storage media or other knives. While some serrated knives can be sharpened by passing through a ceramic-vee, most cannot be resharpened.
9.4 Geometry of the Cutting Edge As illustrated in Figure 9-1, there are three characteristics of a knife cutting edge: (i) its included angle (ii) the radius at its tip (iii) the thickness of the blade 1 mm back from the cutting edge within the bevelled tapered region (the bevel is called a ‘cannel’ in the cutlery industry). Typical values of the included angle are about 15° for razor blades and veneer cutting knives, 20–30° for microtome knives and 30–40° for kitchen knives. Typical effective radii of cutting edges are 5 m for a scalpel and 17 m for a new safety razor blade (34 m when worn). Lucas et al. (1991) had tailoring scissors with edge radii of 2–3 m. Surface profilometers (Talysurf) tips are some 2 m. Atomic force microscopy (AFM) employs micromachined cantilevers of single crystal silicon with tip radii of less than 10 nm to study surfaces and surface layers. The fragility of these probes is examined by KopycinskaMüller et al. (2006). Japanese knives are often ground with a bevel/chamfer on one side only, usually the right side for right-handed people. Translations of the Japanese names for some special knives are, for example, ‘octopus-pull’ and ‘pufferfish-puller’, implying that the cut is taken towards the body of the chef; it was noted in Chapter 7 that some Asian saws cut on the pull stroke. Maximum sharpness of knives is obtained from all the following three factors: (i) a small included angle for the edge (included angle of wedge); (ii) a small tip radius; and (iii) a thin blade. While these criteria may be appropriate for one-off disposable blades, other considerations come into play when it is desired to retain that sharpness in continued use, or for some
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specified time. In addition, the blade has to withstand the cutting forces imposed upon it, so that the thinnest blade may be inappropriate for certain applications (a razor blade profile is no good for an axe). Retention of sharpness depends not only on what is being cut, but also on the surface on which it is being cut. Contact with bone during the cutting of meat, game and poultry will damage the edge of a knife, and contact with a hard cutting block after the cut is finished can also cause damage (glass cutting surfaces for food, while hygienic, are particularly bad in this respect). Even how knives are handled and stored will affect sharpness, rather like the craftsman’s rule that files should never be allowed to rest on one another. Withdrawal of a sword that rubs the inside of a scabbard probably causes damage to the edges. Damage to cutting edges is usually in the form of bending over the very thin tapered regions along the cutting edge, after which sharpness and cutting efficiency are lost. The effective edge radius of a damaged kitchen knife may become up to 500 m. Nevertheless, even a damaged edge may be capable of adequately performing some jobs, and this raises the question of how sharp an edge need be. Metal-cutting tools are not slim like knives, but the same ideas of sharpness apply. Typical edge radii are about 5 m. However, in experiments to study the influence of tool sharpness on cutting forces, Albrecht (1960) showed that it was difficult to attain the same sharpness with negative-rake tools as with positive-rake tools: while an edge radius of 2 m could be produced on a 30° rake tool, only 20 m was achievable with a 20° tool using the same method of honing.
9.5 Measurement of Sharpness One way of looking at sharpness is to ignore the geometrical considerations of Section 9.4 and simply say that, for a given set-up, sharpness is related to the effort (force) to perform the cut: the greater the effort the duller the blade, the smaller the effort the sharper. Section 9.7 shows that the concepts of machinability, sawability and cuttability are also connected with the ease of cutting, but from the point of view of the workpiece rather than the cutting blade. For quality control purposes, traditional methods for assessing sharpness include cutting materials such as corncobs, combing the hair at the nape of the head, rubbing the thumb or finger over the edge of the blade; and in surgical instrument factories they try to slice a layer of epidermis from the thumb with a scalpel (Hamby, 2007). This technique is employed by shohets (persons certified to slaughter animals for food in the manner prescribed by Jewish law) to test for knife sharpness; another technique is to grow the nail of the middle finger somewhat longer than normal and to run that along an edge to feel for nicks after whetting (LeJeune, 2008). All such methods for sharpness are subjective. The ability to cut different thicknesses of gauze is the basis of BS 5194 (1985) for the sharpness of surgical instruments. Controlled cutting of particular materials that display repeatable behaviour is the basis of ISO 8442-5 (2005) for sharpness. Materials employed for cutting include specially made silica-containing paper and silicone rubber (the latter often pretensioned, which opens out the cut so reducing friction between blade and material (Section 3.2.1). The amount of material cut under a fixed load is taken to indicate sharpness. Sometimes actual cutting forces and energy consumed are determined. Silica in paper is abrasive and can even produce scratch marks on blades (which is why dressmakers do not like their scissors to be used to cut paper as they may become blunt), so that repeated test cuts with the same blade give some indication of blade wear and durability. The condition of the edges of the test paper after cutting will also shed light on the state of the cutting edge; for example, a dull edge produces a rough surface with torn-out regions. While objective values
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for relative sharpness are given by cutting tests, the results strictly relate only to that particular test. Thus makers’ claims about knife sharpness, based on data from cutting paper, will not necessarily be repeated when cutting carrots on a hard surface. Also it should be recalled that the cutting force depends on circumstances such as slice–push ratio and so on (Chapter 5). With this in mind, it should not be surprising that one test for sharpness involves slicing through a sheet of paper, held vertically in the hand, in a downwards direction. The significance of the downwards direction is that it increases , thus reducing the force and thereby implying that the blade is ‘sharper’. Actual determinations of edge angles, edge condition (tip radius) and blade thickness are based on: (i) indenting the edge of the blade into lead or other soft material such as dental wax, or into replica material such as vinyl polysiloxane, and viewing from the side at large magnification in an optical microscope having calibrated circles on the eyepiece graticule (e.g. McKenzie, 1966; Arcona & Dow, 1996) (ii) making replicas and viewing in a optical microscope (e.g. Arcona & Dow, 1996; Meehan et al., 1999) (iii) viewing cross-sections of a blade by scanning electron microscopy (SEM) [same references as (ii)] (iv) laser beam reflexions from the facets of the cutting edge (the CATRA goniometer). In the CATRA goniometer, (iv) above, a low-powered laser beam is aimed at the vertically mounted cutting edge. Reflections from the tip and edges shine on to a circular graduated scale. Whether the reflected image is in focus or not gives information about the edge. Flat regions give a focused (‘sharp’!) image but rounded regions give diffuse reflexion. Information given by the goniometer includes the tip angles resulting from grinding and stropping or honing (and whether the edge is symmetrical), surface condition, likely performance, and even method of manufacture. A pioneering experiment to determine sharpness (of a cut-throat razor) was performed by Mallock (1896) – the same Mallock who discovered the shear plane mode of cutting (Chapter 3) – using interference fringes to determine the radius of the cutting edge that he concluded could not be greater than 5 millionths of an inch (13 m). He noted that a well-sharpened razor would cut hair, when merely pressed against it, at a distance of about an eighth of an inch (3 mm) from the place where the hair is held. He cut a human hair of 64 m diameter (hair varies in different individuals from about 50 to 100 m) in this way and, to estimate the cutting force, he loaded hairs at the same location to the same bending angle (about 30°) at which cutting occurred. He found that the load was half a grain (Troy or apothecaries’ weight), i.e. 65 mg. From that he worked out a line contact stress of 3 tons/square inch (46 MPa). If the force for cutting (65 103 g 65 106 kg 650 106 N) is maintained over a distance equal to the diameter (64 m) of the hair, the work done is 650 64 1012 42 nJ. The cross-section of the hair is (64 106)2/4 3.1 109 m2. The toughness is thus 42 109/3.1 10914 J/m2. (In terms of a fracture toughness we have R 2 65 mg/(32 m)40 J/m2) A measurement of blade geometry at one location may, or may not, be representative of a whole cutting edge and it is sensible to make a series of measurements along the blade. Estimation of relative sharpness from cutting tests, in which an appreciable length of blade performs the cut, perhaps smooths out sharpness variations from point to point. When cutting non-homogeneous materials, tools may wear more at some positions along an edge than at others, as in the peripheral milling of multiple-layered wood particleboard where the density varies greatly through the thickness (Chapter 4).
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Another way of showing which edges are sharp and which are dull, is by angling a Biro and resting the blade against the plastic body of the pen: a sharp edge will grip under the weight of the blade and ‘hang’; a dull edge will slide down the body of the pen.
9.6 Retention of Sharpness: Tool Wear and Machinability/Cuttability It is all very well having a sharp tool with which to start a cut, but not much use if the sharp edge breaks off soon into the cut, or if the tool rapidly loses its cutting edge by wear. Some operations require more sharpening than others: wood cut across fibres rapidly wears out edges; in scything, farmhands would carry abrasive stones with them to regularly whet the blade; and long-handled flensing knives could cut no more than 3 feet of skin owing to the toughness of the dermal tentacles of whales (Bertram, 1992). A kitchen knife may last for three to twelve months before the need to resharpen, but the same knife in an abattoir may have to be resharpened every three hours. How long a tool will ‘last’ (whatever that means) is important for the economics of machining all types of material. In factories and sawmills, components have to be made at predictable rates: stopping a line to change tools or saw blades not only increases production costs but also may stop production elsewhere for lack of parts in the supply chain, which additionally affects manufacturing economics. Wear of the tool (loss of material from near the cutting edge) leads to bluntness or dulling. A variety of criteria for the end of tool life is used, depending on the application: these include deterioration of surface finish of the product; when the work becomes oversize because the point of the worn tool does not cut deeply enough; and when the power consumption rises sharply. Blunt tools may be resharpened, or simply thrown away and replaced by new. The rate of tool wear depends on the cutting conditions, the material being cut and the material of the tool. Tools wear for a variety of reasons, such as abrasive particles in the microstructure of the workpiece, frictional conditions on the rake face, attrition, diffusion and even chemical reactions on the rake face at the high temperatures that may be generated in cutting (Trent, 1959). For example, wear of carbide tools in high-speed machining of Inconel 718 superalloy is caused by diffusion of nickel and iron from the workpiece into the cobalt binder of the tool (Liao & Shiue, 1996). Titanium-based alloys present similar problems (Jahanmir, 1998). Tool wear is a serious problem when continuously machining polymers reinforced with hard and abrasive fibres and particles. The wear rates of CBN are different between continuous and interrupted cutting (Diniz et al., 2004; Halpin et al., 2008). Different types of wear seem to occur on different parts of the tool surface. For example, in metal cutting, wear on the clearance face appears to be caused by abrasion; but so-called crater wear on the rake face appears to be controlled by diffusion, and is displaced up the rake face away from the tip often by the formation of a built-up edge. Such differences imply that contact between chip and rake face is more intimate, and this is reflected in Zorev’s zone of high adhesion near the tool edge (Appendix 1). Sometimes the geometry of the tool itself has a greater effect on its wear than the material from which it is made (e.g. the detail of the point geometry can be the determining factor in the wear of drills). Whatever the cause, a workpiece material that causes relatively rapid tool wear is said to have bad machinability. The term ‘machinability’ seems to have been introduced in the 1920s by Herbert, Rosenhain and Sturney, all members of the Institution of Mechanical Engineers (IMechE) Machine Tools Research Committee (see Herbert, 1928). It means different things to different people. Improvements in machinability of components may come about from (i) using better tools to cut the original material; (ii) using the same tools but substituting a different easily machined
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material that has comparable mechanical properties to satisfy design specifications; or (iii) again using the original tools but substituting a free machining version of the specified metal in the case of metals, for example, such as leaded brasses, steels, - nickel silvers containing micrometre-size globules of lead, or resulphurized steels. The importance of machinability in manufacturing is highlighted by Taylor’s pioneering work in the USA at the end of the nineteenth century (Taylor, 1906). The aim of this factorybased research was to recommend cutting speeds, feeds and tool point geometries that would give the ‘best’ cutting conditions for a given material. How long a tool would last (time, t) when used to cut continuously at velocity V, is given by Taylor’s empirical relation
Vtn constant
(9-3)
Experiments show that 0 n 0.2 for common engineering metals. Consequently, relatively small changes in V have a big effect on tool life t. For cutting green (i.e. wet) red cedar wood, 0.85 n 1.18 depending on whether the cutting teeth were thermally insulated from the saw wheel (Inoue & Mori, 1983). Versions of Eq. (9-3) to include feed rates, depths of cut, and so on may be found in Ludema et al. (1987), for example, where it is shown how the equation may be used to determine best machine settings for high rates of metal removal for a desired tool life (and explains why roughing cuts are traditionally performed at slow speeds and finishing cuts at high). Dimensional analyses of machinability incorporate parameters such as thermal conductivity (Bisacre & Bisacre, 1947; Datsko, 1966). Herbert’s pioneering tool and file testing machine of 1909 is described in Chapter 7. In that device cutting forces were measured as well as lives. Other ways of expressing the tool life interpretation of machinability include the volume of material that can be removed before a tool has to be replaced; in the wood-cutting industry particularly, this becomes the ‘cutting distance’ before replacement. Neither Taylor’s line of attack nor these other ways investigates what has happened to the cutting edge: they simply quantify how long the tool has lasted. Details of various parameters used to characterize worn tools in wood cutting are given by Sheik-Ahmed and McKenzie (1997). The tip radius is not employed in wood machining as an indictor of sharpness as experiments show that it tends to stabilize to a constant value even though the rest of the tool is wearing (Pahlizsch & Sandvoss, 1970; McKenzie & Karpovich, 1975a). There are many other interpretations of machinability. The simplest concerns the ease with which different solids may be cut or sawn. Section 9.5 explained how edge sharpness of tools may be assessed by the qualitative ease of cutting, and this idea can be turned on its head to assess machinability where, for fixed sharpness (and adequate machine capacity), materials are ranked for ease of cutting based on cutting forces. Another way of classifying a material’s machinability concerns whether it is possible to produce a good surface finish and/or minimize the level of residual stresses in components. That machinability is not a material property as such is shown by the fact that least cutting forces produce least surface damage, and least residual stresses, for a given tool, but the cutting geometry of even a sharp tool can sometimes be beneficially altered to produce an even better surface finish. Another criterion sometimes used to define machinability is the ease of disposal of offcut (swarf, sawdust): broken-up chips are preferable, not long ribbons. So, production of an undesirable type of swarf may result in a low ranking even though the cutting power required is low. The problem can often be overcome in practice by including chip breakers in the cutter design. Even so, the removal of swarf in the drilling or reaming of deep holes, or in operations such as tapping threads and broaching, can pose problems owing to the need to have adequate channels (flutes) along which swarf
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can escape and not jam, yet to retain sufficient torsional strength in the tool. Such considerations mean that some operations have to be performed relatively slowly. Design of circular saw blades is a compromise between too large a gullet giving too weak a support for the adjoining teeth, and too small a gullet becoming rapidly clogged by swarf. The heat generated in cutting can affect machinability. Temperatures of a few hundred degrees Celsius can be generated in metals at modest cutting speeds (Chapter 3). The even higher temperatures generated in high-speed cutting may cross phase boundaries in metal alloys, so that workpiece mechanical properties will alter, possibly making cutting more difficult. Cooling of the cutting zone is therefore often employed, but in materials having a low thermal conductivity such as titanium and its alloys that is not easy. Conventional cooling cools in bulk, rather than at tool–chip interface which is only imperfectly lubricated by limited capillary action (Williams, 1976). Many traditional lubricants are ecologically unfriendly, and are costly to dispose of. An old favourite in experiments, CCl4, is now banned. Dry machining, at the high production rates required commercially, causes overheating of tools and wear, and does not necessarily produce the required surface finish. In consequence, ‘minimum quantity lubrication’, in which compressed air is mixed with a very small amount of liquid lubricant in a venturi, is advocated as an environmentally friendly method of cooling and lubrication. When cutting polymers at high rates, it is easy to melt the chips. In consequence, to achieve acceptable machining with such materials, cutting speeds may have to be limited. Then, because they cannot be cut at the higher speeds that other materials can be cut, and so have lower production rates, such materials are said to have bad machinability. Problems in the other direction of temperature occur in winter in sawmills: difficulties of debarking, increase in unit power required, and high cutting forces that may damage saw blades (e.g. cracks in saw blade gullets) (Orlowski & Sandak, 2005). Machinability (particularly when defined in terms of cutting forces) has often been correlated with workpiece indentation hardness [that is (2.5–3)Y or (5–6)k in consistent units, with Y the uniaxial yield stress and k the shear yield stress]. Were toughness not important in machining, the softest materials would always be the easiest to cut and the hardest materials the most difficult. But Figure 9-3 shows that the softest plain carbon steels do not have the highest machinability (defined in terms of lowest cutting forces): that occurs at a hardness of about 200 Brinell. Iron has a hardness of 80 Brinell, at the left of the diagram. Steels have increasing amounts of carbon to form the microconstituent pearlite in addition to the soft and ductile iron (ferrite). Pearlite increases the hardness but reduces ductility, and this initial loss of ductility in steels up to a carbon content of about 0.4 per cent improves machinability despite the increasing hardness. Beyond 0.4 per cent, the increased hardness wins out and machinability deteriorates (Lane et al., 1965). The shape of the machinability plot in Figure 9-3 may be explained as follows. It is usually the case that ductility or toughness R is inversely proportional to yield strength or hardness in some way, so that Eq. (3-32) might be written
F const1 k (const2 /k)
(9-4)
Increasing strength k increases F by the first term, but reduces F by the second. The resultant of these two opposing trends will pass through a minimum in F for some hardness. Lowest F means best machinability in this context so a plot of (the reciprocal of F) vs k or H corresponds with Figure 9-3. The precise form of F vs H plots depends on the detailed variation of R and k.
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Machinability index
100
75
50
25 Approx. carbon content 0.1 0
100
0.4
1.0
200
300
400
500
Hardness, HB
Figure 9-3 Relation between ‘machinability index’ and carbon content/Brinell hardness of plain carbon steels (after Lane et al., 1965).
The cutting of soft metals can be facilitated by ‘reducing the impact toughness’ (Rollason, 1956). This is achieved by (i) the presence of brittle or weak constituents, for example, graphite in cast iron, slag in wrought iron, MnS or lead in free-cutting steels, selenium in stainless steels and lead in brass (ii) cold working (iii) the presence of hardening elements. Kenneford (1946) showed that additions of a given amount of sulphur improved machinability somewhat more than the same amount of lead, other things being equal. Kenneford’s criterion for machinability was the energy absorbed per unit volume determined on the Woolwich Arsenal pendulum machine (Section 6.4). He showed that the addition of 0.5 per cent lead reduced the energy consumption of plain 60/40 brass by one-third, and 3 per cent reduced it by half; with 70/30 brass the effect was even greater (0.5 per cent Pb reduced the energy to one-half and 3 per cent to one-third). As regards (ii), Kenneford (1946) showed that cold drawing and an increase in grain size of 60/40 brass improved machinability. It follows that something in addition to hardness must be controlling cutting, because increased wrought strength should require increased energy/volume; it is the fracture toughness that decreases with cold working. As far as (iii) goes, pure copper and its dilute alloys exhibit poor machinability because they possess low strength but high ductility (high R/k), as do those aluminium alloys in which alloying elements form solid solutions; in contrast, elements that alloy with aluminium to form intermetallic compounds or separate phases machine well. When aluminium oxide (which is abrasive) is present, however, problems arise with tool wear. Improvements in machinability of medium and high-carbon steel components are obtained when the pearlite in the microstructure is spherodized; of all the types of pearlite, spherodized has the lowest hardness (ca. 150 BHN). Such treatment is an additional manufacturing cost, and components would have low strength, but should heat treatment be required after machining anyway, it might be acceptable.
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All these methods reduce (R/k) in some way but there are few independent measurements of R from which to check. In the historic literature, fracture toughness had never been heard of. However, in Herbert (1928) Izod values are given for a good free machining steel (about 17 ft-lbs) but 53 ft-lbs for an otherwise identical ‘bad’ free-machining steel. Rosenhain (1928) noted that experience in World War I suggested that hardening by cold work improved machinability whereas hardening metallurgically did not. Different thermomechanical treatments of the same alloy may reduce (R/k) at different rates, and if R is reduced disproportionately compared with increase of k, it will be easier to cut a harder version of a solid than a softer, Kopalinsky & Oxley (1987). Research into machinability continues not only on existing difficult-to-machine alloys such as Inconel (e.g. Ezugwu et al., 2003; Krain et al., 2007) but also on new solids such as hydroxyapatite-based bone replacement and implant materials (e.g. Chelule et al., 2003) and metal-reinforced composites (e.g. El-Gallab & Sklad, 1998; Übeyli et al., 2008).
9.7 Effect of Bluntness and Clearance Face Rubbing on Cutting Forces, FC vs t Intercepts and Subsurface Deformation If a wedge has a blunted tip so that the local included angle is much greater than the nominal wedge angle, not only are larger forces required to start a cut (since the applied stress at the tip is lower for the same force), but the effect of friction is also increased. When a sharp blade with nominal included angle is inclined to the direction of cutting by an angle greater than (/2), the offcut passes along only one side of the blade, there is a clearance angle between the underside of the blade and the cut surface, and there is extremely little contact between the clearance face and the cut surface. However, when a wedge with a blunt edge is tipped up, the contact area is increased. Cutting forces are thereby increased because the local blunted angle has a smaller effective rake angle than the sharp wedge, and because there is frictional rubbing along the contact at the clearance face. So, it must be the case that bluntness and clearance face rubbing increases cutting forces overall, irrespective of the mode of chip formation. How significant are these ‘parasitic’ forces and, when best efforts have been made to use the sharpest tools with least rubbing contact on the clearance face, what forces remain to be measured? And, what remains to be measured when, additionally, best efforts have been made to reduce friction on the rake face? Are the forces expected to be zero? If so, that would imply zero resistance to separation. In fracture mechanics theory, a body having a crack with zero radius of curvature would similarly have ‘no strength’, but blades and cracks must have finite radii of curvature at the separation edge owing to the finite size of atoms, so that a finite force is required to produce separation by cutting or cracking. The work done by the force provides the energy required to separate the surfaces. When chips are formed by shear, a positive intercept is found on the force axis in plots of experimental FC vs t. Historically, the intercept has been explained in terms of bluntness and rubbing (Section 3.6.5), and this point of view is used to refute any connexion between cutting mechanics and material toughness or work of separation. The picture is fogged by the fact that intercepts in FC vs t plots must inevitably be greater the blunter the tool. The important questions to answer are (i) whether, when the effects of bluntness and clearance face rubbing have been reduced as far as possible, an intercept still exists; and then (ii) whether the magnitude of the intercept means more than just the remanent effects of bluntness and rubbing. Some idea of the forces associated with bluntness and rubbing may be gauged as follows. When a tool is blunt and has a round nose, the local rake angle presented to the workpiece
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Chip
Tool α Uncut surface of workpiece
� Z
α
D E αc
F
Cut surface of workpiece
Figure 9-4 Cutting with a tool of finite cutting edge radius. Direction There is of a stagnation point in the flow in front of the tool with some material flowing below the material tool as flow well as most flowing into the chip. Separation takes place at the stagnation point since this is where material divides.
varies with position around the nose. When bluntness is drawn schematically as a circle joining the straight rake and clearance faces (Figure 9-4), it is clear that the most negative local angles are found at the cut surface. Rubenstein and co-workers (Connolly & Rubenstein, 1968; Haslam & Rubenstein, 1970) showed that in two-dimensional cutting with sharp tools, a transition to ‘pushing a standing wave of metal ahead of the tool’ rather than chip formation occurred at negative critical rake angles c of about 60° to 70°. (Note that the shear plane angle φ is almost zero at large negative rake angles in Figure 3-15a,b, particularly for tough solids.) The same type of transition to prow formation occurs in scratching (Chapter 6). If a tangent to the circle is drawn at the transition angle between the formation of a standing wave and a chip, it was argued that material below the tangent line passes under the tool nose to become the top of the cut face, and that above goes into the chip (Figure 9-4). For a rake angle of 65°, say, at the transition, the thickness h of material passing below a tool having nose radius is (1 sin65°) or about 0.1. A sharp metal-cutting tool will have 7 m, say, so the thickness becomes about 0.7 m, which is small compared with typical cuts of tens or hundreds of micrometres for blunt tools. Zorev (1963) and Greenhow and Rubenstein (1969) showed that there was still a force intercept at zero uncut chip thickness when the wear land on the tool was zero, and Childs and Rowe (1973) state that when cutting metal with sharp tools, the forces on the clearance face can be made negligibly small when the clearance angle exceeds about 5°. While the above arguments gloss over what the actual flow field is around the nose of a blunt tool, many authors have shown that there is a stagnation point in the flow field at which the flow separates and divides. Sometimes that division occurs right on the tool nose but sometimes separation occurs at the tip of a wedge of dead metal adhered to the nose (such dead caps are intimately linked to the built-up edge (Chapter 4). Slip line fields having subsurface deformation for round-nosed tools have been presented (e.g. Lortz, 1979; Karpat & Özel, 2008). In his cutting experiments, Mallock (1881) noted that in one or two cases separation took place not at the edge of the tool, but some distance beneath it.
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Material passing beneath the tool rubs on the lower part of the nose, and on part of the clearance face by elastic recovery, and may be significant with highly deformable soft solids and compressible materials such as wood. In veneer peeling the nose bar compresses the workpiece around the cutting edge (see Figure 4-10). Relaxation of the veneer on to the bottom of the tool on exit from the cutting zone must increase clearance face rubbing. The associated friction results in increased cutting forces and will contribute to the positive intercept in plots of cutting force vs depth of cut. Orthogonal cutting experiments have been performed with tools having artificial wear lands of different lengths on the clearance face (e.g. Kobayashi & Thomsen, 1959). At constant uncut chip thickness, cutting forces increase with land length as expected but, significantly, the forces are always finite at zero land length. This implies that although rubbing on the underside of the tool does contribute to increased cutting forces, rubbing forces are negligible for a sharp tool. This evidence suggests that for sufficiently sharp tools, the positive intercept found in FC vs t plots when chips are formed in shear is not down to bluntness and clearance face rubbing, and that the intercept should be identified with toughness/ separation work. Cutting experiments have been performed with tools having different edge radii on a variety of materials, e.g. Dempster (1942) on paraffin wax; Nosovskii (1963), McKenzie (1966) and Kobayashi and Hayashi (1989) on different timbers; Dalziel and Davies (1964) on coal; Harrison (1982) on soils; Arcona and Dow (1996) and Doran et al. (2004) on thin polymer films; Careless and Acland (1982) on skin; Albrecht (1960), Finnie (1963), Hsu (1966), Adbelmoneim and Scrutton (1974), Williams and Gane (1977), Kountanya and Endres (2004), Karpat and Özel (2008), Woon et al. (2008), Karpat (2009) on metals, many of the more recent references being on micromachining where the depth of cut and tool tip are comparable. Lucca and co-workers (e.g. 1998) have investigated the effects of tool tip radii down to nanometre levels of cutting. It is possible for depth of cut to be comparable with crystallographic grain size even with relatively sharp tools (Section 4.1), and experiments can be done on single crystals (e.g. Ramalingan & Hazra, 1973; Williams & Gane, 1977). Williams (1993) shows how local crystal orientation relative to the cutting direction affects the inclination of the primary shear plane. When grain size, depth of cut and tool sharpness all interact, the problem is complicated. In a number of cases, forces at fixed depths of cut and fixed rake angle were measured for different tool tip . In broad terms, blunter tools require greater forces for the same depths of cut, other things being equal. In guillotining, the critical depth cr at the transition between the indentation phase and the cutting phase was approximately 10–15 per cent greater for a blade deliberately blunted to an edge radius of 1 mm compared with the originally sharp blade (Atkins, 1981). Allison and Vincent (1990) showed a 50 per cent increase in microtoming force between sharp and dull blades when slicing 5 m thick sections of kidney. Figure 9-5 shows results for 1040 plain carbon steel having a Rockwell A hardness of about 60, cut at 3 m/s (Kountanya & Endres, 2004). The cutting forces/width for a fixed uncut chip thickness of 37 m level out below a critical tool tip radius of some 40 m. Results from Finnie (1963) for cutting electrolytic tough pitch copper with a 45° rake angle at two different depths of cut and various tip radii give a similar plot. Here the critical radius is some 7 or 8 m, above which the rate of increase of cutting force is the same for both depths of cut. (Finnie remarks that chips cut with dull tools at small depths of cut curl in the opposite direction from normal.) A possible meaning for the tool edge radius at which the cutting forces level out is given in Section 9.8. Tool bluntness, and the existence of stagnation points in the flow field whereby some of the separated material flows below the tool into the cut surface, result in residual stresses in the cut workpiece. None of the classical deformation fields includes deformation below the
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FC/ w
120 100 80 60 40 20 0 0
20
40
60
80
100
120
140
Cutting edge radius �
Figure 9-5 Cutting with tools of different edge radius. Plot of (FC/w) vs for 1040 plain carbon steel having a Rockwell A hardness of about 60, cut at 3 m/s. Below about 40 m, the cutting force levels out (after Kountanya & Endres, 2004).
cut surface. Residual stresses come about when non-uniform plastic flow fields are unloaded. Residual stresses exist in machined surfaces of ‘brittle’ materials as the local deformations are plastic during grinding [small depths of cut with respect to (ER/k2), e.g. Dalladay, 1922; Frank et al., 1967; Johnson-Walls et al., 1984]. Recent measurements of residual stresses below machined surfaces have used synchrotron X-ray diffraction measurement coupled with threedimensional eigenstrain analysis (Yang et al., 2008). A correlation between the maximum residual in-plane shear stress and plasticity-induced diffraction-peak-broadening suggests that the very near-surface residual stress state is induced by shear deformation during cutting. Blunt tools (caused either by round cutting edges or by built-up edges on sharper tools) may also alter the shear/tensile loading at the tool edge and hence the mixity of RI and RII that makes up the measured intercept toughness. Connolly and Rubenstein (1968) and Subbiah and Melkote (2008) show that bluntness promotes tensile stresses in front of the tool and therefore a greater proportion of separation by mode I.
9.8 Cutting Edge Sharpness and Workpiece Critical Crack Tip Opening Displacement As explained in Chapter 2, a crack propagates at its own ‘natural sharpness’, given by the critical crack tip opening displacement (CTOD, C) which is twice the radius 2C of the end of the running crack. When fracture mechanics testpieces have specially prepared crack tips with smaller than the critical, all crack at about the same load, other things being equal, and hence give the same values for toughness. The reason why sharper cracks do not give earlier cracking is that while they produce more intense stresses and strains, they do so over a volume too small to encompass the microstructural features that control fracture. Specimens with greater require bigger loads to initiate fracture because, while the zone of concentrated stress and strain at the end of the crack is larger and will encompass the appropriate microstructural features, the stresses and strains do not reach the necessary levels (Knott, 1973).
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Such higher fracture loads, converted into fracture toughness values, imply erroneously larger resistance to initiate cracking and this is not conservative for design purposes. Critical stress intensity KIC values for H11 steel increase as increases, but below some minimum , KC is approximately constant (Irwin, 1964). Irwin plotted KC against as suggested by the strength of the singularity according to elastic continuum mechanics. In ductile materials the strength is likely to be less; the Hutchinson–Rice–Rosengren (HRR) field equations of 1968 give 1/(N1) for a material following (/y) (/y)N where N 2 (i.e. the power n in the Ludwik expression on is smaller than 0.5). Protocols for monotonic toughness testing often specify that crack tips should be sharpened before testing (by razor blade or by fatigue) to diminish the possibility of calculating erroneous toughness values. With tough polymers, sharpening the crack tips of toughness specimens by tapping a razor blade perpendicularly into an already prepared crack is not as effective as sliding the blade across the notch owing, perhaps, to the slice–push effect described in Chapter 5 (Blackman, 2009). Crack propagation in fracture mechanics testpieces having blunt starter crack tips is often initially unstable, with the loading dropping dramatically, because of the greater strain energy stored at the greater loads just before fracture is released, and may even be enough to break the testpiece in two during the precipitous load-drop. Alternatively, the excess energy may be enough only to propagate the crack a limited distance, after which it arrests because it has ‘run out of steam’ (pop-in during toughness testing is a well-known example of crack arrest). Further propagation requires an increased load but, because the arrested crack has its own natural sharpness, that load is the ‘correct’ load for further cracking from the arrested crack length and would give a valid estimate for the toughness. Readers interested in energetic stability of cracking are referred to Chapter 8 in Atkins and Mai (1985). The value of CTOD is related to the microstructure of a material, and to the toughness and yield strength by R myC, where m 1 for plane stress and m 3 for plane strain is a constraint factor. The CTOD will vary with mode of cracking, i.e. tensile opening mode I, in-plane shear mode II and out-of-plane shear (twisting) mode III. Most of the available data from fracture toughness testpieces are for mode I. For stiff brittle solids, R may be some 50 J/m2 and, with y in MPa, C is in tens of micrometres; for ductile metals, C is about a mm or even greater (for soft steel with, say, R 200 kJ/m2 and y 200 MPa). The radius of a tool edge in cutting plays a similar role to the starter crack tip radius in toughness specimens. In fact, the role is more important in the sense that in cutting there is no starter crack to help separation of surfaces to occur. Separation takes place as one or more cracks form in the initial indentation deformation field produced by the edge of the tool, where the distribution of hydrostatic stress and effective plastic strain ahead of a round cylinder or sharp wedge (in two-dimensional cases), and ahead of a ball, pyramid or cone point (in three-dimensional cases) will be important, not only during the transient start to cutting but also during steady-state cutting. As in toughness testing, unstable separation can happen in cutting with blunt tools. In Figure 9-5 the cutting forces/width for a fixed uncut chip thickness of 37 m level out below a critical tool tip radius of some 40 m. Using R yC, with k350 MPa, gives R(2 350 106)(40 106) 28 kJ/m2 at least. For Finnie’s results the critical radius is some 7 or 8 m, above which the rate of increase of cutting force is the same for both depths of cut. We are not told the mechanical properties of his copper, but y140 (annealed) or 240 MPa (wrought) (Rollason, 1956). Then R becomes either 1 or 2 kJ/m2. Malak and Anderson (2005) found that an edge radius smaller than 35 m was required to cut polyurethane foam successfully. These data suggest a C connexion, but results for other solids are less clear-cut, not always levelling out below the smallest edge radius employed in the experiments. Crossplotting Albrecht’s 1960 results (his Figures 10 and 11) in the form of FC vs at different t gives
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a series of parallel lines, the highest of which corresponds with the greatest t, but they do not level out at smallest . Any link between sharpness and CTOD must therefore remain speculative. Even so, the idea that the rate of increase of (FC/w), for a given rake angle, is the same for all depths of cut accords with ideas in McKenzie (1966), where the effect of different bluntness is to increase the cutting forces by constant amounts at all depths of cut [except that McKenzie says, following Kozhevnikov’s (1960) experiments on wood, that force is proportional to depth of cut, whereas we would say that there are positive intercepts in plots of cutting force vs uncut chip thickness for all materials.] It was noted in Section 3.6.5 that toughnesses R derived from cutting forces can vary with tool rake angle (really shear plane angle φ), and that this may reflect mixed-mode separation caused by the asymmetry of loading. Since varying R means varying C, it follows that critical sharpnesses should vary with and φ.
9.9 Compensation for Bluntness It was remarked earlier that dull blades can sometimes cut effectively despite their bluntness. The case of cutting by a rotating cord (strimming) is discussed in Chapter 10. Using a blunt tool obliquely can sometimes produce acceptable finishes owing to the reduction in forces caused by ‘slice–push’. That is, an oblique blunt tool may require the same cutting force as a sharp orthogonal blade; similarly for blunt disc cutters. This sort of thing is noticeable when cutting timber, where compressibility and distortion of the workpiece by the cutting edge of the tool is a problem, resulting in local splits, bad surface finish and lack of dimensional consistency. Indeed, splitting can even be avoided in very low-density woods by using low tool rake angles with quite blunt edges. Thus, balsa having an air-dry density of 150 kg/m3 may be successfully cut at about 1800 with the comparatively blunt edge of radius 14 m. Sharper tools do not require such a large obliquity to avoid subsurface fractures: white birch with a density of 640 kg/m3 is cut successfully when is about 1 when the edge has a radius of about 1 m. The importance of edge bluntness relative to fibre wall thickness is shown by the interactive effects of wood density and edge radius on the blade obliquity at which splitting below the cut surface is incipient (see Chapter 4). Orthogonal cutting tends to produce damaged sections even with the sharpest tools. Out-of-plane deformation when cutting melamine-coated particle board can be detected by electronic speckle interferometry: the blunter the tool the more the out-of-plane deformation and the worse the surface finish of the product (Stanzl-Tschegg, 2008). Many foodstuffs have coarse microstructures (e.g. salami and black pudding) or open microstructures (bread) and it is difficult to obtain undamaged slices unless a knife is sharp. Even with sharp blades it is advantageous to introduce some slice–push by angling the knife or by driving the blade (delicatessen slicer) independently of the feed in order to reduce forces. Once again, oblique orientation of a dull blade that gives ample permits successful cutting, but only so far.
9.10 Wiggly Crack Paths Produced by Very Blunt Edges Thin polymer films, typically 30 or 40 m thick, are used in packaging for foodstuffs, CDs and so on. When such films are cut with an extremely sharp blade (edge radius less than the thickness of the sheet), the crack path follows the direction of the cutting edge. When cut with a blunter edge, however, a remarkable and most beautiful effect is produced: oscillating paths develop in the direction of cut, with experimentally reproducible regular waveforms, the details of which depend on the material, and anisotropy in its stiffness, strength
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and toughness. The effect depends on the film being locally unsupported where the tear will run, i.e. the film is able to deflect locally out of its plane, the path progressing by a series of flaps (half wave-forms) coming above, and descending below, the plane of the sheet. Wiggly paths are easily demonstrated in commercial packaging (of multiple bars of chocolate, for example) by first making a small hole in the film with the tip of a Biro, and then pulling the pen tip along the sheet. Wiggly paths are produced at both slow and fast speeds of pulling. There is, perhaps, similarity between the different results of opening an envelope with a paper knife and one’s thumb. Roman et al. (2003), Reis et al. (2006) and Atkins (2007) have reported this phenomenon, and Deegan et al. (2002) investigated fast-running oscillating fracture paths in rubber, resulting from pricking a balloon. The effect is quite different from the oscillating crack paths sometimes found in gas pipeline fractures (e.g. Freund & Parks, 1980), nor is it connected to the wavy ‘telephone cord’ fractures found at interfaces in electronics components (e.g. Moon et al., 2004). Furthermore, the effect is not caused, as might be thought, by sideways instabilities of the moving tool. Why paths should oscillate in some materials, yet converge or diverge in others (Chapter 15) is an interesting question. It depends on how the starting edge of the sheet responds to increased loading from the blunt edge. Rather like a ship passing through the water, an outof-plane ‘prow’ may be built up and whether such deformation is reversible or irreversible will be important. Whether a split initiates ahead of the edge or whether splitting is delayed until after the prow has collapsed against the edge, and what causes the split to deflect sideways, etc., will all be crucial events. Nehammer et al. (2000) showed that oscillations in forces, when cutting thin acetate films with blades of different sharpness, were caused by prows building up which then relaxed after sufficient strain energy had been stored locally to send out a crack running ahead of the blade, the process repeating itself in a regular pattern.
Chapter 10
Unrestrained and Restrained Workpieces Dynamic Cutting Contents 10.1 Introduction 10.2 Minimum Speed for Cutting Unrestrained Workpieces 10.3 Comb Cutters 10.4 Optimum Shape for a Curved Blade 10.5 Cylinder Lawnmowers 10.6 Rotary Mowers and Strimmers
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10.1 Introduction During all cutting, the cutting forces have to be reacted in some way. When pushing down with a knife in the kitchen, the vertical force is reacted by the kitchen table top. Very often reaction is achieved by gripping the workpiece in some sort of vice on the bench (chuck on a lathe). Friction may be able to provide sufficient restraint to enable baulks of timber to be adzed by a carpenter simply standing on the log lying on the floor of the workshop. Wood to be planed is often pushed up against a spring-loaded stop on a carpenter’s bench. When a farmer ploughs a furrow, the whole field ahead of the blade reacts the cutting force. In the kitchen, foodstuffs being cut are usually held in place by the non-cutting hand. Full constraint is not always achieved, particularly when workpieces are highly deformable. A loaf of bread will wobble when cut, however well part of it may be gripped. Items too hot to handle, like the Sunday joint, can be held by a fork while being carved. The meat may still wobble, and further workpiece stability and control of slicing are achieved when the joint is placed on a plate having spikes. Food can slip around on the dinner plate so we use a fork to steady things. The fork did not enter into regular use until the beginning of the eighteenth century. Until then diners ate food pierced on the end of a knife or used their fingers, the latter a practice common in some cultures still. Alternatively, entirely different implements such as chopsticks are employed. It is well known that chopstick means ‘quick stick’ in Chinese, but why ‘quick’? Why not ‘eating stick’ or somesuch? According to legend, a seer could not wait for hot food to cool enough to be picked up by hand and took two sticks from a nearby tree with which to pick up the food; this was quick eating. Hedgers and ditchers when using ‘knives’ (sickles) will restrain the target by using the other hand to grasp a bunch of the material to be cut, or otherwise constrain what is to be cut with a stick held in the other hand. When laying hedges, notches (pleaches) are cut in branches that are to be bent (the bent bits must go upwards, otherwise the sap will not run). It is also possible for an initially restrained workpiece to become less restrained as cutting proceeds. For example, the stiff skin of an orange can be scraped or cut to get the zest, and eventually the pith at the interface between skin and fruit is reached, after which the surface of the fruit deflects and gives a bow wave that is difficult to cut with a controlled-depth device. A lot depends on the flexibility of the workpiece. Some materials have large Young’s moduli, others low. But the stiffness depends upon geometry as well as material elastic moduli: a 12 inch Copyright © 2009 Elsevier Ltd. All rights reserved.
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wooden rule is bendy one way but not the other. If possible, workpieces to be cut are clamped to present the stiffest bulk resistance. Sometimes it is very difficult or impossible to achieve this. For example, thin branches of trees are whippy and move with the saw, particularly if the spacing of the saw teeth is not much smaller than the diameter of the branch. Change of tool, to secateurs or loppers, which are devices having curved cutting blades (parrot-beak blades) that wrap around much of the circumference of the branch, makes cutting more controlled. The delightfully named arunculator has those sorts of blades at the end of a long pole for difficultto-reach boughs. Loppers are much easier to use than saws and appear to require less work to finish the job. Extremely bendy workpieces are cereal crops that literally ‘blow in the wind’. Cereals have a peculiar ‘habit’ of upright stems with uniform height with seed in a compact head at the tip of a stalk. They are unrestrained except for their built-in fixity where they emerge out of the earth (and even that depends on the root system). At least the earth from which the stalks emerge is stiff: facial and other hair emerges from a surface (the skin) that is itself flexible. Shavers contort their faces to tense the skin or otherwise stretch it; sheep shearers similarly prevent skin from wrinkling during removal of wool. One method employed by machines for cutting flexible crops is to ‘gather and restrain’ the stalks before the cutting blade takes over. Thus combine harvesters have zig-zag comb cutters (like barbers’ hair clippers, sheep shears and hedge trimmers) where stalks are gathered into the V-shaped notches between the reciprocating blades. Cutting is made difficult when cereal crops become flattened (lodged) by some complicated aerodynamic action between the wind and the stalks acting together as a fluid. Linseed can only be combined with very sharp cutters (angle grinders are the favourite tool of the farmer for sharpening blades rather than oilstones). To illustrate the difficulties of cutting unrestrained materials, try cutting the bristles off brushes with a knife. Cutting with a scythe was the ancient method of harvesting cereal and grass crops (aftermath is new growth after one or two mowings – math from mow). Ancient stone sickles in the Henan Museum in Zhangzhou, China, have serrations laboriously cut into the working edge. Yet an unrestrained blade of grass will simply bend out of the way when a sharp blade is pushed up against it. This is so even under the optimal conditions stated by Gordon (1968): cutting early in the morning when there is dew on the grass. (The supposition is that the grass is maximally turgid. Therefore the grass is at its stiffest, with cells extended and passing strain energy to each other more readily, and will resist bending as the scythe contacts it; also, the cell walls will have more stored strain energy and so will require less energy to take the cellulose fibres to fracture.) A cylindrical lawnmower has a fixed cutting blade at the base of the machine up against which the helical blades gather and press grass for cutting. The wetness of grass has a large effect on how it cuts. Grass sticks to blades when wet, but not when dry, and clippings stick together to become a mush, making cutting very difficult. When ordinary wheat straw gets damp of an evening it becomes difficult to cut and the farmer has to stop combining (Willcocks, 2008). Dampness is crucial in hay making, and leads to the problem of ‘hay shatter’ where the grass comes apart and is difficult to bale. Figure 10-1 shows a unique steam-driven lawnmower; could the clippings be enough to power it? It is shown below that severing of weakly restrained workpieces is only possible through ‘inertia cutting’ (Persson, 1987). That is, if the cutting blade can pass through the thickness of the unrestrained stalk before the part above has been accelerated to the speed of the blade, cutting will be successful. The speed of the blade, and the inertia of the stalk above the cut, are therefore important. There is a minimum ‘sweep speed’ to achieve cutting with a scythe. Out in the field it is important not to let cutting blades scrape the ground because they will rapidly become blunt, owing to contact with abrasive earth and stones, and resharpening of comb cutters is time-consuming. Even experienced scythe hands who rarely let the blade touch
Unrestrained and Restrained Workpieces
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Figure 10-1 The rare Victorian steam-driven lawnmower at the Museum of English Rural Life (reproduced with the permission of the University of Reading).
the ground carried a whet stone for regular sharpening. In contrast, shaving aims to produce a ‘clean shave’ as close as possible to the skin, unless ‘designer stubble’ is in vogue. Obtaining the closest of cuts is not always easy since the face is contoured but blades are flat, even those in electric razors: flexible shaving heads, multiple and vibrating blades are attempts to improve the situation, helped by tensioning the skin. Pressing on to the skin will increase the frictional drag on the razor. Hair on creatures usually lies in preferential directions, like grain in wood. This is clear when stroking a cat or dog (an exception is the Rhodesian ridgeback dog, which has a strip of hair along the top of its back which unusually faces the head). The same is true for facial hair and it is an interesting question as to what direction of shaving produces best results, i.e. down or up the face. One argument for cutting down is that a sharp edge will dig in and put hair into tension. But is tension the way hair is cut? In scything when trying to rescue a lodged crop, cutting is against the grain and shaving upwards certainly seems to give closer cuts in wet shaving. Electric shaving heads rotate and so have up-and-down motions. The thickness of the guard between rotating cutters and the skin would be expected to limit the length of the shortest hair attainable, but some electric shavers pull up facial hair with one blade so as to be cut by the next. A difference between wet and dry shaving is the effect of wetness on the properties of keratin: it is easier to shave, and cut nails, having stepped from a hot bath (Section 11.2.4). A safety razor is essentially a potato peeler: it is possible to shave (with some difficulty) with a potato peeler and you can peel a potato with a safety razor. As with a potato peeler, there is a gap of limited size through which cut-off material passes, so using a safety razor on a head of hair only flattens it. The gap is too small to permit hair removal from a beard. Other devices used for unrestrained plants include the strimmer in which a rapidly rotating cord of polymer or metal wire does the cutting. The need for speed is revealed by battery-driven strimmers that fail to cut when the speed slows below a certain level after the battery begins to
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run down. Electric shavers also fail to cut satisfactorily below some limiting speed and most have a cut-out that prevents operation if the speed is too low. Of some curiosity, however, is that while electric razors are sharp, the cords in strimmers are quite blunt in cutting-edge terms, so why do they cut? Indeed, the blades of most rotary grass cutters become blunt soon after use, and they too cut adequately. Can speed compensate for bluntness? In food mixers blades are not really sharp, yet they function. Moreover, the bits to be cut up are completely loose within the container in which the blades spin. What happens in such devices at slow speeds? Do the contents just move out of the way? Certainly in some devices having revolving cutting tools, such as turnip and chaff cutters, the feed material had sometimes to be forced into the machine, rather like meat bobbling about in a mincer. The forces for cutting by hand are sometimes generated by impact with a hammer on the tool (e.g. wood chisels, cold chisels for metals, driving in of fence posts or piles in foundations). Wave effects may become important, as in the way a bricklayer can divide off part of a brick from another (to give a brickbat) by striking the brick with the edge of a trowel. Hands are used as cutting tools by those martial arts experts who can fracture planks of wood. Some aspects of wave effects in cutting and fracture are discussed in Chapter 15. The conventional design of a bread slicer has rows of reciprocating saw blades, one for each slice. Not only is there waste (breadcrumbs) from the slots cut in a loaf, but loaves tend to be unrestrained and they move up and down during slicing, a problem compounded by the compressibility of many types of bread. Traditional bread slicing machines are not at all efficient (Marshall, 2006). There is an assumption in all the above that the cutting device is under control even if the workpiece is not restrained. This may not always be the case. For example, a remotely operated submersible vehicle (RoV) is equipped with tools according to the work being performed, which can include cutting. While high levels of power may be available there is very little available in the way of reactive force beneath the waves (only what thrusters can generate), and the challenges are usually in providing the reaction to any forces developed in the cutting process.
10.2 Minimum Speed for Cutting Unrestrained Workpieces When slowly pressed against by a cutting blade, a stalk is deflected in the direction of push. The force required increases with deflexion, and is governed by the elastic bending stiffness of the stalk (Figure 10-2). The stiffness depends on the Young’s modulus and second moment of area of the stalk. There will be a force that should produce cutting which is determined by the sharpness of the blade, the local ‘slice–push’ , friction and the toughness of the material. If that force is reached after some stalk deflexion, cutting will commence (recall Mallock’s experiments on cutting human hair with a cut-throat razor; Section 9.5). Another possibility is that before reaching the conditions for cutting, the stalk will fracture near its base by the bending moments induced by the blade forces. But if neither event occurs, the blade simply pushes over the stalk, which ultimately passes under the blade and no cutting occurs. If the blade speed is increased, however, it is found that cutting does commence. An approximate estimate for the minimum velocity for a blade to cut may be established as follows. First, forget any ‘cantilever stiffness’ for the crop and merely think of a vertical free-standing stalk of mass m. Let the approach velocity of the scythe be V0 and let the (changing) velocity of the stalk be Vg after being struck by the blade. The force Fg experienced by the crop is md2s/dtime2 m(Vg)d(Vg)/ds, where s is distance in the direction of motion of the normal to the edge of the blade. The force required to cut is Rw, where R is the
Unrestrained and Restrained Workpieces
249
Stalk Vo
Cutting blade
Figure 10-2 Cutting of unrestrained stalks where inertia of the target becomes important.
fracture toughness (J/m2) and w is the dimension of the stalk in contact with the blade. Any slice–push will lower this value, but that can be ignored for the moment, as can friction. If, during the impact, Vg attains the value of V0 within a distance equal to the thickness t of the blade of grass (the dimension of the stalk perpendicular to the cutting edge), or in a smaller distance, then cutting will occur. Hence, for cutting,
Rw mVg dVg /ds
(10-1)
or
∫ Rwds ∫ mVg dVg
(10-2)
with limits of 0 and t on the left hand side, and 0 and V0 on the right hand side. Hence
Rtw mVo 2 / 2
Vo2 2Rtw/m
or
(10-3)
i.e. when the kinetic energy transferred to the stalk is just equal to the fracture toughness work required to sever the stalk. A blade of grass is about 1 mm wide (w), 0.1 mm thick (t) and has mass about 0.1 g (m). Thus for R < 200 J/m2, V0 → 0.6–0.7 m/s (which seems reasonable for a sharp scythe, or similar instrument like a slashing tool used to clear ferns). If there is bending stiffness from the blade of grass, the force on the left hand side of Eq. (10-1) becomes (Rw Ks)ds, where K is the (linear) elastic stiffness (it probably is not linear). Whence
[Rtw Kt2 / 2] mVo 2 / 2
and
Vo2 [2Rtw Kt2 ]/m
(10.4)
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The limiting velocity is smaller when the stalk has stiffness owing to the release of strain energy as the bent stalk springs back during and after the cut. Friction will increase Vo, (see Section 10.6). Further discussion on minimum cutting speeds for weakly restrained workpieces will be found in Persson (1987).
10.3 Comb Cutters Comb cutters are devices such as hedge cutters, hair trimmers, electric sheep shears and combine harvesters that have reciprocating blades in the shape of overlapping zig-zags, the triangular gaps between every pair of inclined blades opening and closing every stroke. Nowadays these devices are powered, but multiple garden shears having five or six crossing Vs were once worked by hand with just two handles like normal shears. Each pair of inclined blades acts like a guillotine having a blade of fixed inclination. As the device is fed into the material to be cut, material is gathered into the open V-shaped recesses and cut by the sideways movement. The angle of the V-shaped recess is chosen so that when the gaps close, friction will hold the workpiece in place and not simply push it forward. In practice, the ‘workpiece’ may be a series of discrete stalks, twigs, etc., but here we think of it simply as a uniform body. Electric carving knives, which are perhaps similar in operation, do cut into solid material. In Figure 10-3 the gap is shown fully open with length wo. At time t later the length has reduced to (wo w) [wo (ht/tani)] since wtani ht, where h is sideways speed, assumed constant for the moment. The effective width of the workpiece perpendicular to the blade is [wo (ht/tani)]/cosi. The gap completely closes at t wotani/h. Resolution of the feed velocity f and the blade velocity h gives for the slice–push ratio
ξ (fcosi hsini)/(fsini hcosi)
(10-5)
The forces V perpendicular to the edge of the blade and H along the blade in these circumstances are given in Chapter 5, from which the forces X in the feed direction and Y perpendicular to it may be determined. We assumed that h was constant, but for a reciprocating blade we should write h hosint with the length of gap after time t being [wo (hocost/tani)]. The phase of the sideways velocity and the degree of opening of the V-shaped gap will give different results.
10.4 Optimum Shape for a Curved Blade The sickle for cutting reeds for thatching has more of a 90° bend but most sickles and scythes have continuously curving blades. When cutting grass or other crops, a single swing of the tool is made and a swath cut before the person steps forward to make another cut. Cutting could be performed by arcing round a radially straight blade with a circular motion, but the slice push ratio would be virtually zero and the forces high. With a curved blade (either concave or convex), however, beneficial values of can be produced by simple swinging of the blade. Figure 10-4 shows a blade with variable curvature rotating with angular velocity d/dt about a fixed axis through point O. At some point P along the edge of the blade where the polar coordinates are (r,) (Atkins & Xu, 2005):
ξ ({1/2}) [r (d θ/dr) (1/r)(dr/dθ)]
(10-6)
251
Unrestrained and Restrained Workpieces Material remaining to be cut
i wo
h h sin i
h cos i w
i Y f sin i f
f cos i
X
Figure 10-3 Geometry of a comb cutter. Pivot
O
Y θ ω
X
r N Blade ϕ Ψ
P Vh Vc
ϕ Vv
Vr
Figure 10-4 Geometry of a rotating flat blade with variable curvature.
A workpiece of given dimensions will be cut within a single swing of the curved blade, providing all the material is within the sweep of the blade. During a cut, different parts of the blade will progressively be in contact with different parts of the workpiece. By considering the local along the length of the cutting edge momentarily in contact with the workpiece, the local forces and torque on an element of blade of length ds can be determined using the relations for
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-dependent forces in Chapter 5. The distribution of forces and torque along the instantaneous arc of contact are thereby given, from which the instantaneous loads X and Y and torque T on the system are obtained by integration. Repetition of this exercise over all arcs of contact between start and finish of cutting will give the pattern of changes in force and torque in one complete revolution of the blade. Such calculations require the geometry of the blade to be known beforehand in order to give (dr/d), from which is obtained, using Eq. (10-6). Insofar as least cutting forces are associated with greatest , the question arises as to what curved blade profile will give greatest as this will produce a given cut with the least forces; or, vice versa, what curved profile will permit the biggest cut for given forces. Is the curvature in a scythe or a sabre based on any design principle (apart, in the case of the weapon, from the limited length for a straight sword that can be quickly drawn by hand from a scabbard)? In the general case, the motion of a blade comprises various linear and rotary motions. The action of a scythe consists of an almost circular sweep of the blade in an anticlockwise direction from behind a right-handed operator all the way round to the left, before the person steps forward to take the next swath. In the case of a cavalryman, the horse has a forward velocity and the sabre is rotated in a complicated way by the shoulder joint, elbow and wrist. Whether cut or thrust was the better sword action is discussed in Chapter 8. It could be too that hand and wrist motions adapt to given blades. In the simple case of blades rotating about a fixed axis, we differentiate Eq. (10-6) and set the result to zero in order to seek optimum . We obtain
r (d 2θ/d r2 ) (dθ/dr)[1 (1/r)(d 2 r/dθ 2 )] (1/r2 ) (dr/dθ) 0
(10-7)
which may be solved for (dr/d) to give
dr/dθ C1r
(10-8)
whence
ln (r/r0 ) C1θ or r r0 exp (C1θ)
(10-9)
where r r0 at 0, and C1 (1/ro)(dr/d)rro. The value of is obtained by substituting (dr/d) Cr in Eq. (10-6), to give
ξ (1/C1) C1
(10-10)
which is constant all along the blade. In fact the algebraic expression for (Eq. 10-6) turns out to have no maximum or minimum and the only way to get a stationary value is for r(dr/d) or its reciprocal to be stationary, which occurs with a logarithmic spiral. (The industrial bacon cutter analysed in Chapter 12 has a linear spiral the choice of which, as far as is known, was arrived at pragmatically.) From Eq. (10-10), C1 should be small to obtain large , i.e. ro should be large and (dr/d)rro should be small. In the limit when (dr/d) → 0, a circle is predicted but then there is all slice and no push – we have assumed that the blade spins about a fixed axis – and the device will not function. When the axis of rotation itself moves there are, in theory, optimized shapes. Imagine the centre of rotation of the blade in Figure 10-4 moving with linear velocity V in the direction of the y-axis. We bring the axial motion to rest by adding a velocity V in the direction of
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Unrestrained and Restrained Workpieces
the y-axis. The circumferential velocity about the centre of rotation is vc r(d/dt) r and the radial velocity vr dr/dt (dr/d). The net velocity vh along the tangent to the rotating blade at P is therefore v h v c sinϕ v r cosϕ V cos (θ ϕ)
(10-11a)
Similarly, the net velocity vv perpendicular to the edge is v v v c cosϕ v r sinϕ V sin (θ ϕ)
(10-11b)
Coordinate geometry gives sin r (d/ds) r/[r2 (dr/d)2]1/2 and cos dr/ds (dr/ d)/[r2 (dr/d)2]1/2 with s the distance along the curve (Blakey, 1958, Chapter VII). The slice–push ratio is given by the quotient of relations (10-11a) and (10-11b) and, when V 0, gives Eq. (10-6). But when V is included the resulting expression is extremely difficult to optimize. Instead let us investigate the effect of imposing a linear velocity V on the instantaneous centre of rotation of a blade having the log spiral form given by Eq. (10-9), which is optimum for rotation only. This gives
ξ [rω (1 C12 ) V(C1 cosθ sinθ)]/[2C1rω V(C1 sinθ cosθ)]
(10-12)
Large values of the parameter (ro/V) mean slow linear velocities and is essentially the rotation-only value constant along the blade, e.g. 5 for C1 0.1. Lower values of (ro/V) produce different values of at different positions long the edge of the blade. The variations in encompass smaller and greater values than the rotation-only value, the variations getting greater as (ro/V) becomes smaller. While increased C1 smooths out these variations, does not regain its rotation-only value.
10.5 Cylinder Lawnmowers The origins of cylinder lawnmowers (‘reel’ mowers in the USA) reside in a machine invented in the early nineteenth century by George Butting of Stroud in Gloucestershire, England, to cut off the nap (the short fuzzy ends of fibres raised by teasles) on the surface of cloth. Figure 10-5 shows the general geometry of a cutting edge which is arranged along a helix around the cylinder. Gearing from the wheels of the machine causes the cylinder to rotate at a faster speed than the machine as it is rolled forward, and the cutter gathers the blades of grass in a downwards direction, trapping them against a straight cutting edge (cutting bar) fixed along the bottom of the mower, where they are cut by a guillotining action. The motion is similar to ‘down-milling’ but the very flexible grass workpiece is not cut until it encounters the cutting bar. The equation of the helix is
z rθ coti
(10-13a)
dz/dt rω coti
(10-13b)
and its axial speed is
where r is the radius of the helical cutting edge, is the angular speed of the cylinder, and i is the inclination of the helix to the axis of the cylinder. The machine moves forward with velocity f perpendicular to the axis of the cylinder. Resolution of velocities, after bringing the
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ω
i w
z θ r f
Figure 10-5 Helical geometry of cutting blades in a cylinder lawnmower.
machine to rest by imposing a negative velocity vector f, and division of the velocity along the cutting edge by the velocity perpendicular to the edge, gives
ξ (rω coti cosi f sini)/(rω coti sini f cosi) (N cos2i sin2i)/(N 1)sinicosi
(10-14)
where N (r/f) is the ‘gearing’ of the cylinder with respect to the forward speed of the machine. When N is very large, → coti; when N 1, → cot2i. The greatest occur at smallest i and, by implication, the least resultant cutting force is given. However, greatest ‘slicing ’ at i just below 0° means an essentially straight cutting edge aligned along the generator of the cylinder. Such an arrangement will produce a scalloped grass surface, the wavelength of which may be reduced, but not eliminated, by having more than one cutting blade spaced around the cylinder (a ‘multiple start’ helix). Flatter surfaces are produced by blades with lower helix angles, but will require higher forces as will not be so advantageous. The compromise by which the actual inclination, and number, of helical cutting blades is determined, may be investigated as follows. Consider a helical cutter that is just crossing the fixed cutting bar on the mower at the start of cylinder rotation. The time taken for it to traverse from one end of the cylinder to the other is [W/(dz/dt)] (W/rcoti), where W is the length of the cylinder (width of the sward cut by the machine). The rotation of the helix in that time is
θ ωt W/rcoti
(10-15)
Unrestrained and Restrained Workpieces
255
and the distance moved forward by the mower is
ft fW/rωcoti W/Ncoti
(10-16)
A single helix which just completes a full rotation during passage from one end of the cylinder to the other has 2, so that the helix angle to achieve this one rotation is given by coti W/2r, using Eq. (10-15). It may be shown that the helix inclination angle i for a cylinder having n partial helices is given by coti nW/2πr
(10-17)
Equation (10-17) also applies for helices which complete more than one revolution from one end of the cylinder to the other, e.g. a helix which rotates twice has n ½. Similar calculations may be made for multiple-start helices. For given W/r, and n, the helix angles for multiple blades may be evaluated, along with the for every blade configuration (Eq. 10-14), from which the total cutting force per unit length of cylinder may be determined using Eq. (5-13), i.e.
Fres /Rw (1/ √ [1 ξ2 ]) S
(10-18)
where S (2)mLk/R (Section 5.2.3). The minimum number of helical blades on cylinder lawnmowers is about five and the maximum is ten. Machines with the larger number of helices reputedly give finer cuts, as required for the greens on golf links. For small-width mowers, W/r 6, and with n 5, coti 15/, i 12° and 3.9 from Eq. (10-14), assuming N 5. (Fres/Rw) (0.25 S), and with n 5, about two edges are cutting at any one time, so the total normalized force is 2(0.25 S). For more blades in a small mower (n 10, say), i 6°, 7.9 (when N 5) and (Fres/Rw) (0.13 S). For n 10, about four edges cut at any one time, so the total normalized force is 4(0.13 S). For wider mowers, W/r 10 and with n 10, coti 50/, i 4° and 4 (assuming N 5). Then (Fres/Rw) (0.24 S). With about four edges cutting at any one time, the total normalized cutting force is 4(0.24 S). There is not much difference in cutting force per blade, but the greater total force required in wider mowers is caused by more blades cutting at once. Increasing the gearing ratio N between cylinder speed and the forward speed of the mower does not improve matters much owing to N appearing in both the numerator and denominator of Eq. (10-14). Calculations show that increase of the gearing N serves to flatten curves of total cutting force vs number of blades so that at large N there is less difference between the total forces for different numbers of blades. Inspection of practical machines shows that the angle i 20° whatever the W/r ratio. With a greater i, is reduced and the normalized cutting forces per blade are increased to some [0.4 S]. Of course, S refers only to the friction between cutting blade and grass; the friction within the drive system of the lawnmower is on top of this and is likely to predominate. It is salutary to note that the efficiency of a typical petrol-driven garden lawnmower moving at walking pace is about 1–2 per cent (Gordon, 1968): transverse fracture toughness of grass blades is a few hundred J/m2 and the motor size is about 5 HP, i.e. 4 kW.
10.6 Rotary Mowers and Strimmers It is possible to cut with blunt tools when the cutting speed is large, but impossible to perform the same operation slowly (like the candle that may be shot through a plank of wood, but not slowly pushed through). This is the case not only when tools that ought to be sharp have become blunt (rotary lawnmower), but when tools are blunt to start with (a strimmer
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cord). Yet both cut grass and, with wire cord, a strimmer cuts undergrowth and small twigs adequately. How long do the blades of a rotary mower remain sharp? Why bother to sharpen the sides of rotary blades since it is only the tips that cut in normal mowing? (Only when attacking long grass with the base of a rotary mower inclined upwards would it seem that the sides cut.) Sanson (2008) performed experiments on cutting grass with a strimmer whose 2.7 mm diameter nylon cord cut between radii of 46 and 82 mm. The grass selected was a broadleaved species, Penisetum clandestinum, kikuyu. At the lowest speed that was controllable (800 rpm) the grass blades were merely tickled. At about 2500 rpm the grass blades were tangled and torn and twisted into groups but not cut until about 3200 rpm. At 3400 rpm the grass was cut, when the inner circumferential speed was about 18 m/s and the speed at the tip of the cord was about 36 m/s. The machine was run at 3400, 4200, 5200, 6000 and 7200 rpm. Five separate patches within one area were cut at the different speeds, with no possibility of contamination from other cuttings. At least 20 grass blades were collected from each patch cut and immersed in Evans Blue stain to show up the damaged areas. Leaves were then removed after sixty minutes, rinsed in water, blotted dry and scanned. The area of blue was measured by image analysis. In addition, the perimeter of the damaged end, and the width and thickness of the blade were measured at the same scale. While there is a slight trend, there is no significant difference in damage whatever the speed, and the variance is so high that there are no statistically significant results. This is surprising, but perhaps not if the increasing energy of the strike as the speed increases is taken into account. In the presence of both a blunt blade and friction, Relation (10.3) for the minimum velocity to cut becomes
Vo2 (2Rtw/m)(1 M)
(10.19)
Figure 10-6 Photograph still from a high-speed film of a strimmer wire just having struck a target (Courtesy of Chris Ellick & Tony Pretlove, 2007).
Unrestrained and Restrained Workpieces
257
on the supposition (probably dubious) that Kamyab and co-workers’ 1998 correction factor can be applied to grass (see Section 12.3.3). M 2ry[1]/R, in which r is the cord diameter, y is the yield strength of the grass and m is the coefficient of friction. What we mean by yield strength in this application is unclear, but across the blade of grass, 3–4 MPa is not unreasonable (Jeronimidis, 2008). For a 3 mm diameter cord, and 0.5, M → 85, and Vo 6 m/s. This is rather smaller than the linear speed of 20–30 m/s corresponding to Sanson’s 3400 rpm minimum rotation speed. Of course, grass is probably rate sensitive, with both R and y being affected. Figure 10-6 shows a high-speed picture of a deflected strimmer cord just having stuck a target. Experiments in which a strimmer was used to cut strips of balsa wood demonstrate that fracture at the base of the stalk is possible in stiff stems and that there is a particular rotational speed at the transition, dependent on the circumstances.
Chapter 11
Cutting in Biology, Palaeontology and Medicine Contents 11.1 Introduction 11.2 Biology 11.3 Palaeontology 11.4 Medicine
259 260 269 272
11.1 Introduction Beyond the employment of cutting in simple dissection, cutting has been used as a means to an end in various areas of biological and medical research. For example, the effects of nerves being cut or crushed by knives and whether they regenerate, were studied by Sanders and Young (1974) on octopus, Johnston et al. (1975) in axolotl (salamander) limbs, Raisman (1977) on adult rats, Lehouelleur and Schmidt (1980) on frogs, Waite and Cragg (1982) on innervating rat whiskers, Matthews and Cuello (1984) on guinea pigs, and Maxwell et al. (1990) on the cerebrum (anterior of the brain) of neonatal rats. A different example is where systematic cuts in different parts of the hypothalamus (the floor of the median ventricle of the brain) in Japanese quail enabled Davies and Follett (1975) to determine which part controlled photoperiodically induced testicular growth (the photoperiod is the interval in a twenty-four hour period during which a plant or animal is exposed to light). Taxonomy is a hierarchy of classification that describes nature and the reasons why certain creatures have certain features at present. How they evolved from prehistoric times is often unclear and theories are sometimes speculative. There are many questions in evolution, botany and zoology on which the mechanics of cutting, piercing and scratching may be able to shed light. The fruit, flowers and foliage of plants are eaten as food by animals. How is separation from the plant achieved, by claw or by mouth or other means? What role is played by the physical properties of the plant, particularly its fracture toughness? What of the role played by spines, thorns and other features as defences against predation? If the food has to be chewed before swallowing, what is the best form of teeth for a given food and how do teeth masticate the food? How important is tooth sharpness? Carnivores attack prey with tooth and claw. What is the best design for least effort? How do potential prey defend against attack? What of creatures that have no teeth? What of the different sorts of beak and bill found in birds? What about the different types of claw or talon used by raptors to capture prey? What of fangs in poisonous snakes and the ability of some snakes to swallow small animals whole, all digestion being by chemical action in the gut? Which came first, the diet or the teeth? Properties of food are discussed in Chapter 12, and teeth as cutting tools forms the subject of Chapter 13. Similar questions may be posed about the mechanical properties of soils and how plant roots push past particles of soil to lengthen and expand as the plant grows, and how worms travel through. Digging creatures that make burrows have, in addition, to remove soil from the site by various means. How are such animals equipped to do the job? The feet of creatures such as Copyright © 2009 Elsevier Ltd. All rights reserved.
259
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poultry have evolved to scratch the surface of the ground efficiently. Chapter 14 discusses this sort of thing along with different types of ploughing and other earth-moving activities. The reader will become aware that experimentation in biomechanics and cutting of biological materials, and interpretation of data in engineering science terms, is not as widespread as for traditional engineering materials. How to interpret data is reasonably well understood. There is much exciting new work to be performed.
11.2 Biology The study of the physical properties of biological materials, and particularly how they affect attempts to cut them, is a young science but as we shall see it is already clear that connecting biological design and performance to mechanical properties can help to explain many things. Textbooks in the field include Vogel (2003) on biological structure, Vincent (1990) on biological materials and Nelson (2004) on biological physics. Cotterell (2010) has written a splendid text entitled ‘Fracture and Life’.
11.2.1 Biomechanics: physical properties Biomechanics concerns the physical properties of all sorts of biological materials, and their employment in the interpretation of biological design and function. Biomimetics is the inspiration given by nature towards new engineering materials and devices with improved combinations of properties (Jeronimidis & Atkins, 1995). The subject should not be confused with the complementary field of biomedical engineering whose application of physical properties of human body parts relates to specific replacements of limbs and organs, e.g. artificial bone, or to an understanding of why Bunsen burner pipe cannot be used as replacements for arteries when aneurysms are by-passed, and so on. Clearly there is overlap between the subjects. Knowledge of the mechanical properties of biological materials and how they relate to performance is central to biomimetical investigations. This applies to all aspects of biology, not only to cutting-related operations. Study of the mechanical behaviour and structure of biological materials is very revealing. From a structural point of view materials in nature are safe and they have evolved to give the best performance possible wherever stiffness, strength and toughness are metabolically expensive. Many have properties that engineers seek to mimic, and that point the way for biologists to breed improved varieties of plants, are based on desirable mechanical properties to make harvesting easier and so on. By understanding such structures food technologists, too, can process their materials in better ways. Two facts are immediately obvious. First, the mechanical properties of natural materials have evolved to give the best compromise that the organism can afford to make for particular tasks. Second, natural materials are structures with a hierarchy of detail ranging from the basic macromolecules, through microfibrils, fibrils and structured fibres to the final material. Most animal and plant materials are cellular composite structures formed of fibrous or plate-like components, based on very few chemicals. The principal structural polymers employed are proteins (polymers of amino acids) and polysaccharides (polymers of sugars) that make up materials as diverse as infinitely soluble lubricants, such as mucus, to extremely stable rubbers, such as elastin, that can withstand temperatures of 120°C. There is often a mineral phase present, especially when compressive strength is needed, such as the calcium salts in skeletons of animals that live in the sea. Most important of all is that many biological materials contain water; it has a profound effect on mechanical behaviour. The importance of turgor pressure has been mentioned in Chapter 1.
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In the past, measurements of ‘properties’ have not always been performed and interpreted in the ways required by engineering stress analysis and solid mechanics. For example, reporting of fracture loads without regard to the size and shape of bodies, and the method of loading, does not provide a fundamental property that can be applied to other problems where the same material is involved. Similarly, there have been many ‘penetrometer’ tests, particularly in the subdiscipline of properties of foodstuffs, where what is measured is a complicated mix of basic mechanical properties. Even when it is appreciated that things such as elastic moduli, yield stress and fracture toughness should be measured in the ways of materials science – at different rates, temperatures, environments, etc. – it is soon realized that many biological materials do not conform to traditional engineering solids for which testing methods and protocols have been developed. While bone, shells, teeth and other ceramic-based materials in biology, and timber, may be stiff and have linear stress–strain curves like materials employed in engineering, many other biological materials are soft and/or highly extensible and may be very hysteretic (displaying time-dependent recovery on unloading). Gordon (1968, 1978) noted the J-shape of the stress–strain curve of extensible tissues like skin, blood vessels, tendons and worm cuticles (see Figure 2-1), where there is considerable extension with hardly any increase in load until, at some large strain, microstructural orientation results in the material stiffening up (e.g. Rowe, 1983; Purslow et al., 1984). In all biological composite materials (irrespective of the shape of their stress–strain curves) there are far more hierarchical levels than in artificial composites: indeed, in many instances, it is difficult to identify ‘a fibre’ or ‘a matrix’ or even ‘a material’ as opposed to ‘a microstructure’. Very often in attempting to understand behaviour in such hierarchical systems, the crucial thing is to try and discover at what level to look at the structure, i.e. to identify the level that controls the material property of interest (Jeronimidis & Atkins, 1995). Soft tissues are complicated structures containing the cells, blood vessels, lymphatics, nerves, etc., typical of living tissues but, in addition, there are large quantities of extracellular material such as collagen, elastin fibres and protoglycans (Barbenel et al., 1978). Mechanical properties depend principally on these three constituents, with their relative proportions and structural arrangements in different tissues producing detailed differences. A dense network of coiled fibres of collagen forms about 60 per cent of the dry weight of human skin and the straightening out, and subsequent stiffening up of the fibres, produces the characteristic J-shaped stress–strain curve of an extensible tissue (Doran et al., 2004). Tendon has a more ordered structure than skin, in which collagen appears as a coherent bundle of fibres (to form 80 per cent dry weight) interspersed with ground material, some elastin and flattened tenocytes. This structure also displays a J-curve but after about only 3 per cent strain, it rapidly stiffens up and thereafter behaves in an almost linear fashion. The J-shape is the cause of people getting ‘caught short’ and having to run to the lavatory, since the bladder can extend while filling without much stress being generated until almost full. Gordon reported that although such materials are difficult to tear, it is not because they have high fracture toughness but rather because of the J-shape of the stress–strain curve with its absence of shear connexion between elements of the material. For example, the toughness of aluminium foil is some 200 kJ/m2 and that of rat skin some 10 kJ/m2, yet baking foil is readily torn but skin not (Mai & Atkins, 1989). Only where fracture is wanted in nature are tissues stiff with strong shear connexion that will give fracture after only small strains (e.g. amniotic membranes and eggshells). The mechanical properties of the hymen do not seem to have been reported. Biological materials present many challenges to experimentalists: many are squidgy and difficult to grip and, in bending, some may deform under their own weight. Nevertheless,
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using ideas of linear and non-linear elasticity and plasticity, and linear and non-linear fracture mechanics, there is a growing body of sound data on mechanical properties, collections of which may be found in Vincent (1991), Lucas (2004) and Sanson (2006), for example. In some cases, cutting proves to be a very useful way of determining properties that may be difficult to measure in traditional ways (e.g. Pereira et al., 1997; Bonser et al., 2004). Ethical issues may prevent human skin and flesh from being tested, for example, in investigations on stabbing and the design of armour, so that attempts have to be made to find other materials to substitute as simulants (Section 11.4.3).
11.2.2 Drilling and rapping Vincent and King (1995) studied the mechanisms by which female wasps make holes in pine trees to deposit their eggs and to impregnate the wood with a fungus. Although commonly called ‘drilling’, the action is really one of sawing. The ovipositors have diameters of about 0.2 mm and, in one case, a length of 10 mm (Sirex noctilio) and, in another, 50 mm (Megarhyssa nortoni nortoni, a parasitoid of the larva of the other). There are teeth/barbs at the tip of both ovipositors, standing about 7 m proud which, having penetrated, hook over a section of the wood cell wall and break it by pulling, thus producing sawdust. In addition to these ‘pull’ teeth, S. noctilio has ‘push’ teeth further up the ovipositor (Figure 11-1). The work rate (power) required to cut the wood cells by pulling was estimated by Vincent and King (1995) to be about 2 mW/g of muscle; for comparison, 18 mW/g is required of the flight muscles in bumblebees (Heinrich, 1975). Biological control of bracken has been studied using moths that bore into the fronds of the plant (Lawton et al., 1988). There are many insects, loosely termed sap suckers, that have mouth parts, which form a syringe-like structure that they use to penetrate into xylem tissue, phloem tissue or into the cell contents of individual cells. Some penetrate by secreting enzymes that chemically provide access. Others force their stylets into the tissue. Peeters et al. (2007) showed there was a relationship between the density of sap suckers that force their stylets in and the toughness of the plant leaves, but this was different for those sap suckers that chemically penetrate leaves.
T
Tension cutting valve
C = T + Pcrit
Compression cutting valve Sliding support Wood cell wall
Figure 11-1 Three-part ovipositor cutting into a wood cell. The left valve is pulling on a piece of wood cell with force T. The right valve can add this to the compressive force, limited by the critical Euler buckling load, giving a total force for the penetration of (T Pcrit) (after Vincent & King, 1995).
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The teredo or ship worm eats into timbers of wooden ships in tropical waters. Indian shipbuilders used to clad vessels with leather as a means of protection. The hulls of European ships up to the end of the eighteenth century were painted with poisonous stuff then clad in soft wood sheathing that was expected to be eaten by the teredine, which would be killed in trying to penetrate the oak. Once copper nails were affordable, copper-clad ships began to appear which were both teredo-proof and smooth-surfaced, that did not foul quickly and enhanced a ship’s performance (Johnson, 1986a,b). The woodlouse spider has specially adapted fangs for piercing woodlice shells. Woodpeckers use their rapping on trees (and on metal drainpipes, wooden eaves of houses, etc.) as a means of communication and to declare their territory, as well as searching for food. In 1995 green woodpeckers (Picus viridis) drilled 200 holes in the foam insulation of the external tank of the shuttle Discovery, delaying the launch (Mitchinson & Lloyd, 2007). The way holes are made by woodpeckers has been somewhat of a mystery. How does the bird avoid brain damage from the sudden deceleration (some 1200 g on impact)? High-speed films show that its beak strikes the tree at some 3.6 m/s: how can the bird produce such a high velocity over such short distances (about 70 mm )? Why does the woodpecker not fall off the tree with such high decelerations? As explained by Vincent et al. (2007), there are various sophisticated cushioning devices in the beak, neck, skull and behind the eyes that permit the bird to function. For example, immediately preceding and during impact, the woodpecker closes both eyes so that the lids restrain them and to stop them from popping out. Grip on the bark of the tree is improved by the bird having two toes pointing forward and two backward (most birds have three forward and one back). A model of the action of a woodpecker as a low-inertia hammer shows that it operates resonantly and that, since the power required to accelerate the head (about 9 W) is much greater than can be provided by the neck muscles alone, the body muscles must also be involved. Angular momentum of the body, generated by contraction of its leg muscles, accelerates its head towards the tree. These findings have led to the design of a novel lightweight hammer that has a number of advantages over conventional design. For example in space exploration, it has no net inertia until it comes into contact with an object (Gao et al., 2005). Early work by, for example, Tyler (1969) on the strength of eggshells has now been interpreted in terms of both the Weibull statistical theory for the strength of brittle solids and modern fracture mechanics by Entwistle and co-workers (e.g. Entwistle & Reddy, 1996). Such knowledge helps us to understand the mechanics of ‘pipping’ by which a chick breaks the egg with its beak in order to leave the egg. Pearl oysters are predatory gastropods that ‘drill’ holes in other molluscs with the minute chitinous horny teeth of their radulae (see Chapter 13).
11.2.3 Bites and stings A bite may be the closing of a jaw with teeth to grip something (possibly with the intention of inflicting pain or to inject venom). Some venomous snakes have fangs that are long, sharp and hollow, by which poison is injected. Other snakes have grooved teeth and if a person is bitten through thick socks or trousers much of the venom is scraped off and does not penetrate into the wound. Other snakes have their poisonous fangs at the back of the mouth and their mode of hunting is different. Snake venom partially predigests a victim’s flesh and rots it, as well as kills the victim. The etymology of fang comes from the old English for ‘something caught’. The word bite is also used to mean sting, where a plant or creature pricks or wounds with sharp-pointed, often venom-bearing, hairs or organs. Stinging nettles (Urtica dioica) are a
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well-known example. Birchall (1989) showed that the sting has a silica cap that breaks off leaving an edge like the end of a syringe needle. Squid are able to shoot out lots of stings when threatened, by increasing sac pressure. If they miss and the stings fall on the seabed, the stings are harmless as they have to pierce flesh to be effective. Sometimes the biting creature, such as a tick, ‘latches on’. Vampire bats have peculiar teeth so sharp that they can make an incision without waking the sleeping prey. They need to do this because having found a suitable victim they need to feed for long enough to fill their stomachs, then they must crawl away without disturbing the animal and risking injury. Diseases (malaria, dengue fever, yellow fever and so on) are transmitted through bites from disease vectors such as mosquitoes. The word mosquito is Portuguese/Spanish for little fly. As discussed by Budiansky (2008), at the end of the proboscis are two pairs of stylets that slide against one another (some parts sliding inside a sheath made of other mouthparts that provide support). These cut through the skin of the victim, looking for a blood vessel. Inside the proboscis are two hollow tubes, one that injects saliva into the microscopic wound and one that withdraws blood. The saliva contains agents that disrupt clotting of the victim’s blood and it also inhibits pain to the victim during the minute-and-a-half or so while the insect is feeding. Only later does the left-behind saliva produce the allergic reaction that results in the bite mark. Budiansky goes on to explain that, in the wild, only those species of mosquito that attack humans carry the Plasmodium parasite, which causes malaria, although in the laboratory all species can be infected with the disease. In order to transmit the disease from one infected human to another, a mosquito has to have a high probability of biting an infected human, and also live long enough to bite an uninfected human after the parasite has developed in the mosquito’s gut. A mosquito that ‘wastes’ bites on non-humans is a much less efficient vector of a human pathogen. The pepperpot fungus (Myriostoma coliforme) has natural holes (ostioles) through which its spores are released. The holes are typically regular, more-or-less circular, with fringed (fimbriate) raised margins, and are evenly dispersed. The earthstar fungus (Geastrum sp.) is very similar (though unrelated), but has a single centrally placed ostiole, the edges of which may be fairly simple or more complicated. However, earthstars may end up having multiple holes through weathering or insect bites, and it is sometimes difficult to distinguish between a real pepperpot and the more common earthstar. The superfluous holes caused by insects or other invertebrates are irregular, unevenly distributed, and with margins that display, on careful inspection, their formation by cutting (Roberts, 2008).
11.2.4 Claws, nails and grip Birds of prey attack with feet and talons, tendons snapping them shut to break the neck of prey. The claw power comes from a special tibial muscle, and the tendons also have a ratchetlike mechanism to hold the contraction. [The same sort of idea applies to perching (passerine) birds so that they can lock on to a branch while sleeping.] The pangolin (scaly anteater) has claws to break open nests of termites and ants (little is known about the deformation mechanics of termite nests) but no teeth, and crushes ants against the roof of its mouth with its tongue. Food is ground up in the anteater’s stomach from swallowing small stones and sand, exactly as birds fill their gizzards: birds, of course, have no teeth. Moore (related by Sanson, 2009) examined the effect of gizzard stone size on plant cell content release in geese. The idea was that smaller stones were ‘sharper’ than larger stones but had a lower probability of trapping material between two stones and causing damage. Larger stones have a higher probability of breaking material, but may require more
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gizzard muscular force than gizzards with small stones. In addition, there is more dead weight in larger stones, which is problem for birds that fly. It is not a surprise that most birds that are herbivorous and rely on stones tend not to fly much, if at all. Moore found that compression forces by the gizzard muscles had little effect on damaging plant cells and that herbivorous gizzards tended to have asymmetrical muscle bands that generate shear between the gizzard plates. Insectivorous birds have symmetrical gizzards. Some claws are double and pincer-like in action, such as the chela (from the Greek for claw) of crustaceans such as lobsters. Similarly, the chelicera is the first pair of pincer-like appendages of spiders and other arachnids (spiders, scorpions, mites, ticks), deriving from the Greek for spider. On the front legs of lice are powerful pincers used to anchor themselves close to a host’s skin by grabbing the hair or feather barbs. Bite force has been measured for teeth (Chapter 13), and it would be interesting to study the forces that can be generated in pincer claws in animals. An interesting example of whether claws of extinct animals were ‘slashing cutlasses’ or were used to establish a foothold on the flanks of prey is given in Section 11.3. Fingernails and toenails are characteristic features of primates. Unlike the claws of other mammals, nails are flattened, almost straight in their longitudinal axis and cambered transversely. They evolved with changes in the locomotor habits of primates away from movement along large-diameter tree trunks and branches to much smaller branches and twigs. Grip on such supports was improved by the development of broad, soft, apical pads having epidermal ridging (fingerprints), backed by the stiff nail. In addition, the nails can be used to prise open cracks, lever up objects, scratch and fight. In most of these actions, nails are loaded from below and have to resist upward bending forces. As explained by Farran et al. (2004), the structure of human nails has been well studied: the nail is a composite consisting of three layers of keratinous tissue in which long, slender -keratin protein fibres are embedded in an amorphous protein matrix (Fraser & MacRae, 1980) (Figure 11-2). Farran et al. (2004) showed that while the outer layers have tile-like cells and are reasonably isotropic, the thick intermediate Dorsal (upper) Intermediate (middle) Ventral (underside)
A
P
L
B
Figure 11-2 (A) Schematic diagram of a human fingernail showing the dorsal (upper), intermediate (middle) and ventral (lower) layers. (B) Schematic diagram of a nail clipping where the dotted lines indicate an example of a sample taken for clipper testing and L and P are the lateral and proximal directions, respectively (after Farran et al., 2008).
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layer has long narrow cells and is very anisotropic; the energy required to cut this layer transversely with scissors was only a quarter of that to cut it longitudinally. For the composite as a whole, the fracture toughness was about 3 kJ/m2 across, and 6 kJ/m2 along, the nail. In consequence, when nails are torn, cracks always divert to the transverse direction parallel to the free edge of the nail (crack paths in anisotropic solids are discussed in Chapter 3 of Atkins & Mai, 1985). These cutting tests were carried out when the nails were saturated with water. More recent work by Farran et al. (2008), using nail clippers rather than scissors (Bonser et al., 2004), shows that the toughness varies with humidity and water content, with peak values of some 11 kJ/m2 (across the nail) and 20 kJ/m2 (along) at about 55 per cent relative humidity. It was noted that the degree of anisotropy in toughness also varied with relative humidity. At extremes of humidity, nails are brittle when fully dry; they split when wet. There is competition between the requirements of high bending stiffness and not wanting cracks to propagate along the nail towards the cuticle and so damage the delicate nail bed. A nail designed to have highest bending stiffness would have cells and fibres orientated longitudinally, but it would readily split longitudinally, and vice versa for transversely orientated cells and fibres (Baden, 1970). The three-layered structure would appear to give the best compromise for nails, with the isotropic material furthest from the neutral axis of bending. Furthermore, the top and bottom layers wrap around the lateral edges of the nail, producing a smooth covering that prevents potentially dangerous cracks from forming there. Curvature across the nail increases stiffness against upwards bending, but also acts against forces from scissors when cutting that try to flatten the curvature. Surplus elastic energy is stored prior to completing the cut, with attendant clippings flying across the room. Alternatively, the contact length between the arc of a nail and the scissors is adjusted to enable a cut to be made with the available force from the hand. In contrast to nails, horses’ hooves are loaded mainly in compression, not bending, so that most keratin fibres can be orientated parallel to the edge to prevent cracks running up the hoof (Bertram & Gosline, 1986). The toughness of hoof is 5–8 kJ/m2 (Kasapi & Gosline, 1997). Farran et al. (2004) go on to discuss those drawbacks of nail design, with which humans are all painfully familiar. When one tears, bites or chew nails, the outer tissue will often delaminate from the intermediate layer and cracks in it will run towards the base of the nail. This problem becomes particularly acute towards the lateral edges of the nail where the outer layers are thicker so that cracks can start running into the quick, causing damage to the soft skin along the lateral nail fold, bleeding and pain. The delamination of the tile-like top surface also allows nails to chip relatively easily, spoiling their appearance and reducing their strength. As with shaving, it is easier to cut nails after a bath, the soaking in hot water reducing the stiffness of nails and the toughness; saliva wets nails when chewed by people. Tiny babies have relatively long fingernails with which they scratch their faces. The traditional practice was for mothers to chew or nibble them off, but it is easier to cut the nails using baby scissors. Chiropodists (from the Greek for hands and feet) use special scissors to deal with in-growing nails. It seems that Chinese chiropodists traditionally do not use scissors, rather scalpels. In the West, it seems that scalpels are used only to cut back corns. The principles of the mechanical design of fingernails also have important implications for nail care in humans. In many manicures, nails are filed into a point rather than being cut parallel to the lunula (cuticle). From a mechanical viewpoint, this should not be a good idea, because towards the point, the edge would not be protected by the wrapping round of the upper and lower layers; cracks would therefore more readily form, and the nail would be more likely to break. Fashions change, though, and some manicures have become ‘straight across’.
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Right-handed people have difficulty in cutting the nails of their right hand, not only because of the unfamiliarity of holding scissors in the left hand, but also because (when looking from above) the blade on the left of most scissors is the one that rises when the scissors are closed. When such scissors are used to cut the nails of the left hand, this blade becomes an anvil that supports the part of the nail that remains after cutting. But when used on the right hand, it is impossible for the corresponding other blade to serve as an anvil as it is above the part of the nail that remains. It is possible to turn scissors right round when held in the left hand so that the tips are facing the person and make a more successful cut of the nails on the right hand, but it is easier to use nail scissors that meet edge-on-edge.
11.2.5 Barbs, spines and thorns Sharp, pointed needle-like objects appear on the exterior of plants. They also appear on the bodies, inside the mouth and oesophagus (gullet), and inside the body of many animals. Some arthropods have spines or teeth in their crop. There are three principal reasons for these features: (i) in leaves or fruit as a defence against being ingested by animals or, in small creatures, against the animal itself being eaten; (ii) to stop food escaping; and (iii) to attack prey. In addition there are other, perhaps unusual, uses. For example, many nectar-feeding animals (mammals and birds) have spines or barbs that act as a sponge to increase the surface area for absorbing nectar onto the tongue. The long-beaked echnida (spiny anteater) eats earthworms speared with special spines on its tongue. Many types of moth feed on the tears of other animals which they lick up to gain salt. The moth Hemiceratoides hieroglyphica from Madagascar has harpoon-shaped proboscises covered with hooks and barbs which they insert under a bird’s eyelids as it sleeps. As birds have two eyelids, unlike mammals, the Madagascan tear feeder needs the extra weaponry to get through to the surface of the eye. The shrike is called the butcher bird because it impales its prey (small animals) on thorns. Thorns are defences on roses, brambles, gorse and so on, and are a threat to hurt. There is some evidence that some thorny plants produce fewer thorns when grown in the absence of herbivores. Thorns are known to increase handling time by the predator and so act as a deterrent. Animals sometimes or even often find ways around defences. New Forest and Exmoor ponies eat gorse; other ponies and horses will not. What this means in terms of having a ‘hard mouth’ – not responding to the bit when being ridden – is not clear. Goats prefer to eat thistles, brambles and twigs rather than plain grasses, so their mouths must be insensitive to spike damage. The acacia tree is very thorny, yet it is a favourite food of the giraffe, whose head can be raised to a vertical position and thus browse on young thornless leaves right at the top of the tree. Giraffe will use the tongue to wrap around a small branch and strip the leaves and thorns. Other animals leave acacias alone. Catfish have toxic spines, as do many other fish, especially bottom-dwelling fish. The oesophagus of the leather-backed turtle has backward-pointing spines to stop food escaping. Backward-pointing teeth are also found in reptiles to prevent prey escaping and as a rachet mechanism for swallowing large prey (Section 13.2). Blackthorn is the ‘viper of thorns’ as it will go through a leather glove and has poison that will cause bruising and throbbing for days, as gardeners know all too well. Hedgehogs have some 5000 spines reinforced with keratin, and porcupines some 30 000 quills covered in backward-pointing scales, as defences. Of animals living with hedgehogs, only badgers have claws powerful enough to prise open the rolled-up hedgehog ‘spiny-ball’ defence. When in danger, tiny erector muscles in the skin of porcupines make the quills stand up. They attack by lunging backwards, swiping their tails violently (Pliny the Elder, it
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seems, thought that the quills could be fired like arrows). According to Mitchinson and Lloyd (2007), if you have a quill sticking in you, first cut off the end sticking out before removing, so as to equalize the air pressure in the wound. While the existence of needle-like objects has been recorded in the biological literature, there does not seem to be much information on the actual geometry and sharpness of the objects, or whether their action is passive or activated by muscles. Clearly, the deterrence and damage inflicted will depend on the forces required for indentation and piercing that, in turn, will depend upon the geometry and sharpness of the barb, spine or thorn. The mechanics given in Chapter 8 will be relevant, and apply just as much to hooks used in fishing. Chaplin (2008) remarks that punctures in tyres on dirt roads in the African bush are caused mainly by thorns, which can be extremely large and stiff. In New Zealand there are plants, of which the lancewood tree is one, which are said to owe their unusual form to the extinct moa, a flightless bird. When young, the tree has long narrow stiff leaves up to 1 m long, pointing downwards like a partially opened umbrella. In adulthood the top of the tree branches out with upright leaves, but retains the downward leaves below to a height of about 3 m. Moas, emus and ostriches are ratites that browse by a clamping and tugging action, and it is said that the lower part of the lancewood tree makes this difficult (Bond et al., 2004). Such defences are, however, not so good against mammalian browsers that bite off shoots by cutting. Sanson (2008) remarks that it would be interesting to measure the toughness of the leaves of the lancewood tree; its appearance suggests that the lower leaves are much tougher than the upper. Barbed wire and razor wire are defences in war. In World War I ‘barbs as thick as a man’s thumb’ were to be found in wire belts often 40 yards deep between opposing trenches. Gun barrages preparatory to going ‘over the top’ were supposed to clear the ground of barbed wire, but rarely did so adequately, managing to produce shell craters and chewed-up ground that was difficult for the soldiers to traverse.
11.2.6 Beaks: birds and cephalopods Birds have no teeth, and swallow whole insects, seeds, berries and so on. In the case of birds of prey, which eat flesh, their beaks grip and pull off bits of suitable size for swallowing or for feeding their young from a creature too large to swallow whole. Vultures devour carrion (i.e. dead and putrefying flesh) but only the lappet-faced vulture can cut skin. Other vultures cannot, and have to wait, unless a body has been opened up, e.g. by lions. The Egyptian vulture eats only soft carrion and has a beak of a different pattern. Most vultures have no means of cracking bone but the lammergeyer (the bearded vulture and the only vulture with feathers on the head) takes bones aloft and drops them on rocks to break them, so as to release the marrow (even so, the name comes from the German for ‘lambs’ vulture’, from its preying on lambs). Kruuk (1967) describes the different bill systems of the six vultures found in the Serengeti in Kenya: some separate prey into pieces by pulling, some by twisting, some by tearing with appropriately shaped beaks. While beak morphology is usually interpreted in relation to the critical role of feeding, the beak also plays an important role in preening, which is the first line of defence against harmful parasites such as lice, fleas, ticks, mites and so on. Rock pigeons have a 1–2 mm overhang at the end of their upper beak. Clayton et al. (2005) showed that trimming off this overhang had no effect on feeding but triggered a dramatic increase in feather lice. During preening, the beak nips into feathers some thirty times per second and the action generated by the overhang against the tip of the lower beak considerably damages parasite exoskeletons.
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Only a few birds (e.g. geese) are plant eaters. Vegetation is low in energy and slow to digest, so there is a lot of it in the gut, which makes a bird heavy (French, 1988). Geese spend most of their time on the ground. Nevertheless they do migrate over long distances and it seems that they evacuate their guts and their gizzards of stones before they set off. The largest birds (e.g. ostriches) that have given up flying are all herbivorous. Cephalopods are molluscs having tentacles attached to the head, and include the cuttlefish, squid and octopus. The only hard part of their structure is the beak, which is used to bite prey into small pieces before swallowing. Research of stomach contents to establish diet is hampered, and results are possibly biased, since cephalopods often reject hard parts of prey (Rodhouse & Nigmatullin, 1996). Similarly, analysis of the stomach of penguins often reveals only the undigested beaks of celaphods (Sanson, 2008).
11.3 Palaeontology Palaeontologists assess the mode of life of extinct species from their fossil remains and the sediments that contain them. Traditionally, they employed the sciences of botany, zoology and geology to interpret their findings, but rarely appealed to engineering science (Alexander, 2006). This has changed with the growth of the subject of biomechanics, and is a logical development since whether plants and animals can withstand the stresses and strains required of them, when stationary or in motion, determines whether they can survive. Furthermore, deoxyribonucleic acid (DNA) analysis permits entirely new connexions to be made (e.g. Kearney & Stuart, 2004); and it can refute long-held beliefs. How extinct creatures fed and how they defended themselves against predators are of interest in the context of cutting. If a fossil gives a full description of structure, an animal’s strength and locomotion can be calculated and the ecological conditions in which it could survive can be estimated. Conversely, when a fossil gives an uncertain description of structure, engineering guides the choice between a number of possibilities; and if it is not mechanically feasible, it is unlikely to be correct. In a comprehensive study of Pteranodon (the largest known aerial animal, an extinct flying reptile), Bramwell and Whitfield (1974) applied engineering principles to its structural strength and aerodynamic gliding ability, and consequently to its life. Pteranodon was a fish eater and fished ‘on the wing’, probably plucking prey from just beneath the surface. (It could not possibly have plunged into the water to feed as it did not fly quickly enough and could not fold its wings completely; nor did it probably float on the surface and dip its beak into the water as fast-swimming fish would escape; and also it was vulnerable to predators from below and would have had to take off and land on the water.) Pterosaurs (of which Pteranodon is one sort) are often postulated in the biological literature to have been ‘skim-feeders’, where the bird flies low over the water with the tip of the lower mandible continuously immersed, seizing prey on contact. This idea is based largely on supposed convergences of their jaw anatomy with that of the modern skimming bird Rynchops. Humphries et al. (2007) investigated the drag of a pterosaur’s immersed lower jaw, modelling it as a surface-piercing strut, and showed that skimming is considerably more energetically costly than previously thought, even for Rynchops. Pterosaurs weighing more than 1 kg could not have skimmed at all. Bramwell and Whitfield (1974) were right all along, as plucking is very distinct from ‘skimming’. This sort of thing demonstrates the difficulty of using limited morphological convergence to interpret the ecology of extinct forms without proper biomechanical input. It has long been recognized that there is a general correlation between tooth form and type of diet, but it is only recently that the mechanical properties of foods and the pattern of their comminution have been considered in the evaluation of tooth form (e.g. Lucas, 2004).
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Following Rensburger’s (1973) study of occlusion and wear patterns in fossil and contemporary rodents, and comparable work on hyraxes, more attention has focused on the mechanics of food breakdown and the relationship between wear and enamel structure (e.g. von Koenigswald, 1982). Microwear on the tooth surface is a valuable indicator of both direction of tooth movement and of diet. The precision and regularity of movement during the power stroke of chewing produce a distinctive pattern of wear facets on the surfaces of the teeth, which can be used to trace relative tooth movement in a given animal. This pattern of wear has also proved valuable in tracing the phylogeny (i.e. evolutionary development) of tooth form, since the fundamental occlusal (contact) relations of the primary feature of the crown in primitive mammals appear in many cases to be maintained throughout evolution (Crompton, 1971; Crompton & Jenkins, 1979). The action of the incisor teeth is of great interest to palaeontologists regarding the evolution of tooth form in response to variation in food properties. Among living mammals, spatulate incisor teeth in both upper and lower jaws are unique to the Anthropoidea, a taxonomic group containing the apes, Old World and New World monkeys. These teeth are relatively large in fruit-eating species (Hylander, 1975), and cursory consideration may suggest that they evolved in association with this type of diet. However, most fleshy fruit in tropical forests are covered only by a thin skin and can be easily opened without such teeth. It is more likely that spatulate incisors evolved, at least 40 million years ago, in tandem with the emergence of a class of fruits with thicker mechanical protection (Lucas, 1989; Yeakel et al., 2007). When the fruit is ripe, a protective covering, usually in the form of a peel, can be separated cleanly from the underlying flesh. Competition from birds and most other mammals for the same food is limited because they lack the means to remove it quickly. These fruits are, however, usually the minority of the fruit consumed by anthropoid primates and ‘skinned’ fruits prevail in the diet (Janson, 1983; Leighton, 1993). However, the use of the incisors by anthropoids, both in peeling fruit and in other tasks, has been poorly studied. Osborn et al. (1987) and Ungar (1994) recently introduced a descriptive classification of incisal activities that has spurred comparative studies (Yamashita, 2003). In a study of the teeth of the mesosaur (small marine reptiles from some 300 million years ago), severely broken and healed mandibles were found suggesting a violent lifestyle in either predation or fighting, or both (Lingham-Soliar, 1995). Fossils of thylacine – an extinct marsupial (animal with a pouch) – often have lots of broken teeth, but why? It is true that ancient teeth can become brittle post-mortem so that tooth fracture is then more likely. Many carnivorous dinosaurs had curved claws that are usually suggested to have functioned as devices for disembowelling herbivorous dinosaurs during predation. Manning et al. (2006) made a hydraulically driven robot model of a dinosaur (dromaeosaurid) hindlimb and employed it to attack pig carcass at low and high speeds. Puncture wounds 30–40 mm deep resulted but claw motion parallel to the surface of the target produced pile-up of flesh in waves, with no cutting owing to the lack of edge sharpness coupled with the extensibility (Gordon’s J-shaped stress–strain curve) of the flesh. The behaviour is reminiscent of scratching a highly deformable material such as rubber. Noting the similarity of these claws of extinct animals with extant birds, it was concluded that instead of such claws being used as ‘slashing cutlasses’, they were used to establish a foothold on the flanks of prey by piercing and gripping the flesh, the victim then being attacked by the teeth, rather like the way big cats hunt. Biomechanical studies of this sort, combining engineering science and biology, seem to be the way forward for improved understanding of so many aspects of creature behaviour.
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The extinct fossil rodent Ceratogaulus had nasal horns, the function of which has been the subject of wide speculation among palaeontologists. Since the creature descended from ancestors who dug by head-lifting (Chapter 14), it would be thought that the horns helped in digging, but Hopkins (2005) argues that the horns were for defence: they evolved to offset increased predation costs associated with spending more time foraging above ground as their body size increased. Patterns of evolution of freshwater and terrestrial pulmonates (gastropod molluscs – snails, slugs, whelks – that breathe through a lung-like sac) from their marine ancestors was provided by a study of Chilina (Brace, 1983). Chilina plough submerged or semi-submerged through surface layers of soft substrata, burrowing into sand with a stereotyped digging cycle, involving extensions and contractions of body sections, rather like worms or caterpillars. Analysis of the way animals can use limbs to strike one another in fights, and the damage that can be caused, is sometimes part of ‘cutting’, but at other times not. For example, Alexander (1999) reported fractures in the carapace (shell) of extinct glyptodonts; glyptodonts were toothless mammals like armadillos, sloths and South American anteaters. But these fractures were probably caused simply by blows from the tail of another. New palaeontological findings receive wide publicity in the popular press. For example, Henderson (2006) reports that ‘the fish with the most powerful jaws in history’ appears to have been the ancient sea monster Dunkleosteus terrelli, estimated to be able to produce a closing force of 50 kN. Of course, the jaws and jaw joints would have to be capable of bearing such loads, and that requires investigation. Cells within the walls of plants extend axially along the stem like sausages and are under turgor pressure that stresses the cell walls axially and circumferentially to keep plants vertical. Jeronimidis and Vincent (1991) observed that in the case of the dandelion, cells on the inside of the stem had large diameter but thin wall thickness, with a progressive change to small-diameter cells with thick walls on the outside. The cell-radius-to-wall thickness (r/t) ratio is about 15:1 on the inside but only 2:1 on the outside. For the same turgor pressure p throughout, this means that the axial stress in the cell wall (given by pr/2t from thin-cylinder theory) is greater on the inside than the outside of the stem. For p10 atmospheres (1 MPa), axial inside is 8 MPa and axial outside is some 1 MPa. Since axial is different, the axial strains must also be different and be greater on the inside of the stem. This stress state, of itself, would cause the stem to bend, so stems constrained to be straight must have a superimposed bending moment in order to end up with a state of uniform stress and strain. Evidence for such locked-in stresses is given when strips are cut from the stems: the strips curve, the amount of curvature depending on the turgor pressure just before cutting (which can be altered from the as-picked value by immersion in distilled water or sugar solutions of different concentration). From the viewpoint of palaeontology, fossils show that plants whose stems had the (r/t) ratio going the other way have become extinct (Jeronimidis & Vincent, 1991), probably to do with plants evolving from living under water where gravity is of less importance than living in fresh air. Aquatic feeding in lower vertebrates is discussed by Lauder (1985), where it is pointed out that terrestrial methods of prey capture have evolved from aquatic creatures that functioned in a medium 900 times as dense as air and 80 times as viscous. A word of caution is necessary when, in palaeontological studies, estimations of magnitude of forces are made from the size of muscles, or from the space that would have been occupied by muscles, in extinct creatures. The predictions can be misleading, because not all muscle fibres are ‘fired’ at once. For example, the short, thick masticatory muscle (the masseter) that assists in closing the jaws has seven layers with each layer angled slightly differently to maximize leverage as they fire in turn as the jaw closes (Sanson, 2008).
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11.4 Medicine Cutting is performed on live bodies (surgery on humans and animals) and on dead bodies (autopsies, post-mortems, anatomical dissections). It is curious that post mortem is the word usually used in the UK, although it simply is the Latin for ‘after death’, whereas autopsy is from the Greek for ‘seeing with one’s own eyes’. Most surgery cuts are only so far into the body in order to explore what is there, or to remove, repair or replace organs. Dissection of dead bodies opens up the interior for inspection for teaching anatomy, to establish causes of unexpected death of healthy animals, and for forensic purposes. The excision of a piece of tissue from a living body for diagnostic study (biopsy) can be performed in various ways, of which punch biopsy is insertion of a hollow punch to separate a cylindrical column of tissue (Chapter 8) that is then stretched away from the surface and snipped off. The instrumented microtome can be employed as a ‘mechanical microscope’ in pathology (Allison & Vincent, 1990); sturdier sledge microtomes are often used to cut decalcified bone. In surgery, dissection and so on, efficient cutting is required with low forces and ‘clean’ cuts; the victim of a stabbing attack by knife would rather have inefficient cutting, but the mechanics are the same whether the blade is wielded benignly or malignly. The make-up of all mammals is similar so there is a great deal of similarity in surgical procedures, and similarity in the instruments used in surgery on humans, and by vets and dentists. The medical spoon device with sharp edges used to remove tissue is no different, in its way, from the manner in which spoons are used to cut fruit, or cake in a pudding. Amputation cuts limbs off completely, taking place either deliberately by a surgeon, or in accidents or war. The removal of the whole or part of a diseased or malfunctioning limb, or the trimming of the stump of a limb that has been accidentally severed, is one of the oldest surgical operations. There is some archaeological evidence that it was used in Neolithic times: the skeletal remains of a late Neanderthal man, unearthed in a cave in Iraq and dated to 43 000 BC, indicate that the man had lived after having a limb amputated (Orr et al., 1982). The tools used in amputation are very similar to those used by the butcher in cutting up meat – knives through the skin and flesh, followed by sawing through the bone – the difference being that on the butcher’s block, the carcass is dead. ‘Sawbones’ was a nineteenth century name for a surgeon. When not pulling teeth, barber-surgeons performed dissection for the training of doctors under the supervision of a physician who did not himself operate. According to Mason (1961), dissection in mediaeval times was somewhat discouraged by the dictum that ‘the Church abhors the shedding of blood’, which was promulgated by the Council of Tours in 1163 in connexion with the practice of dismembering and boiling down dead crusaders for ease of transportation on the journey home. Throughout surgery there are similarities with other processes more familiar to the person in the street: hip replacement operations, for example, use techniques well known in carpentry. In turn, many of the tools in surgery and dissection have complementary versions in carpentry and butchery, and may be found in different types of workshop. Proper use of all these devices requires great skill. The first part of surgery is to gain access to the organ to be operated on; the second to perform the operation. Once it was said that ‘the longer the cut, boldly not timidly done, the better the surgery’, and that the knife should ‘sweep’ through the cut: learners tend to use only the tip of the scalpel, which eventually blunts. All surgery causes trauma (from the Greek for wound), which is physical damage to flesh and perhaps nerves and organs, and also psychological damage to the patient. In recent times there has been a trend towards minimal access surgery (keyhole surgery) because the large incisions required by traditional surgery in
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accessing the organ often resulted in far greater trauma than the operation itself. For example, the trauma of the few stitches required to repair a hole in the heart is much less than the trauma of getting there in the traditional way and, with minimal access surgery, it is found that patients recover in far shorter times than before. Such innovation required contributions from engineering such as the Hopkins endoscope, video cameras, robots and probes with miniaturized surgical instruments (Darzi, 2008). Knives and scissors attached to the end of endoscopes enable cutting to be performed remotely. The new procedures and techniques have resulted in virtual reality training methods for surgeons, with stereo images, enabling interventions to be planned with great accuracy. A region particularly amenable to keyhole surgery is the abdominal cavity. In former times the use of the trocar risked damage to the aorta, the bowel and other organs. [The trocar is a three-sided cutting point enclosed in a hollow tube (cannula), the name coming from the French trois quatres.] Not all surgery can be of the keyhole variety, though, and traditional methods remain for operations where access over a wide area is required and where extensive manipulations take place, as in orthopaedics; similarly for a Caesarean operation in obstetrics (from the Latin obstetrix for midwife). In plastic surgery cutting is used to excise both flaps of skin (that have blood supply) and grafts (that do not). A difficulty at present with virtual reality training programmes is that the response of tissue depends upon knowing accurate constitutive equations of the soft materials making up the body, and these are not always known. There is also a problem that computation should be in real time but the constitutive models that permit speedy computation are not necessarily realistic or accurate (Charalambides, 2008). Accurate modelling of probe–tissue interaction is necessary for planning the trajectories to be taken by probes in operations on live patients. The path may be through tissues having properties so different that, if not taken into account, accidental damage might result of a vascular structure during surgery, with dire consequences for the patient. Whatever the field, the basic action of a cutting device will be similar, so that redesign or improvement in one area may very well have benefits in another. A ‘do-it-yourself’ amateur who hits his thumb with a hammer may end up with blood trapped below the nail. It may be relieved by drilling down through the nail. The dentist’s ‘drill’ acts more like a milling cutter as it shapes a cavity for a filling. The trephine is essentially a hole saw used to cut circular discs of bone from the skull. The Ancient Egyptians performed this operation to relieve pressure under the skull from wounds inflicted in battle. They employed bronze hole saws of the same sort as they used to drill stone pillars on which to fix pintles to carry the weight of heavy doors at the entrance of tombs (Sim, 2008). Swans on the River Thames in England are owned by the Crown, and by the Dyers’ and Vintners’ Livery Companies. Up until 1998, cutting nicks in the beaks of swans in a ceremony called ‘Swan Upping’ was the way ownership was marked (none for the Queen, one for the Dyers and two for the Vintners). Now they are leg-ringed. Earmarking sheep with nicks is still done in the West of Scotland. Shortening of the tails of horses and dogs (docking) by cutting has gone out of favour and while sensible in the case of horses, working dogs can suffer when caught in brambles and undergrowth. In the British Army, the farrier of the Household Cavalry when on ceremonial parade carries a ceremonial axe. The axe had two functions: the spike on the axe was used humanely to put severely injured horses out of their misery, and the sharp axe blade was used to chop off the deceased horses’ feet, the purpose of the latter being to account in regimental records for animals killed in action (the horse’s number was marked on the hoof). The piercing of ears, bellybuttons and other parts of the anatomy for decorative purposes is no different, in principle, from how a bradawl works, nor from what happens when the skin is pierced and indelible patterns are made by inserting
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pigments (tattooing, from the Tahitian word tatau). Tattooing was originally done with bamboo needles by hand, but now a stainless steel vibrating pricker is used as there is a need to get the ink sufficiently deep below the skin because the epidermis sheds. Tests for allergies wherein the skin is scratched and an allergen is applied to the area (an inflammatory reaction indicating an allergic condition) are no different from all the types of scratching described in Chapter 6, other than that they take place on an extensible and flexible ‘workpiece’. In surgery it is common practice to stretch tissue before cutting as this reduces the forces required to cut or penetrate (Section 3.2.1). Too great a tension across the line of cut may possibly lead to unstable tearing where the cut runs ahead of the knife or scissors. This can be a problem in episiotomy where the vagina is lengthened to permit an easier birth. Dijkman (2008) has designed scissors for this purpose that cut backwards from where the cut is intended to stop, thus starting the incision where the stress is lower. Tearing may also occur in dissection: there are two sorts of dissection, sharp and blunt, which relate to how a cut is lengthened. In sharp dissection, scissors are used directly to enlarge a cut; in blunt dissection the cut is lengthened by closed scissors being put into the cut and then opened so as to prise apart the cut (McIndoe scissors for this purpose are angled and have specially blunt ends). Tissue pulled apart in blunt dissection springs back and blood vessels seal; if cut, they carry on bleeding (Tufnel, 2006). In addition to parallels between cutting operations in engineering and medicine, there is the separate subject of medical engineering that covers a diversity of topics, one of which, keyhole surgery, has been mentioned already. Areas that relate to cutting concern the development of improved instruments and equipment for surgery (e.g. scalpels where the angle of the blade can be rotated so as to cut in regions inaccessible to a straight blade, as in cuts required to free heart valves where disease has thickened and narrowed passages), quantitative determinations of the mechanical properties of anatomical materials and development of new materials (replacement bone as pioneered by Bonfield, e.g. 1988), manufacture of aids for the disabled (shaping of prosthetic implants for reconstructive surgery by computer numerically controlled machining, using computer axial tomography and nuclear magnetic resonance machines to give the geometry of bones and soft organs within the body), heart valves, and many more. Miniature machine tools that are computer numerically controlled on the basis of images taken in the mouth are now used in dental restoration for crowns, etc., in place of the old mould-making techniques. Thring (1972) made the perennially valid point that progress in medical engineering comes about when there is a true partnership between medics and engineers: there must be something in it for both parties. In interdisciplinary fields, collaboration is often synergistic but, without collaboration, data from quite extensive investigations are sometimes inadequate, and opportunities are sometimes lost, because experiments may not be as comprehensive as really required or may not have measured the right things, or data may have been interpreted in an incorrect way. In an attempt to understand head injuries, and for accident investigations, experiments on the fracture of human cadaver heads under static and dynamic loading were performed by Yoganandan et al. (1995). A large radius ball-ended contactor was used and axial computer tomography was employed to show up the locations of cracks. These sorts of test would benefit from stress analysis and links into fracture mechanics. There is no cutting as such in these experiments, but tests in which angled and spiked impactors were used would be relevant to forensics, and to arms and armour (Chapter 8).
11.4.1 Surgery A number of the early letters in the Philosophical Transactions of the Royal Society concerns surgical techniques for removing stones from the bladder (lithotomy) (e.g. Bussiere, 1699, commenting on an operation performed by Brother James, ‘an hermit in France’; Douglas,
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1722; Le Cat, 1744; Warner, 1758). A curious incident is reported in a letter of 1695 communicated by Molyneux relating to the cutting of an ivory bodkin out of the bladder of Dorcas Blake, a young woman in Dublin. The ‘full bodied sanguine maid’, much troubled by hoarseness, having failed in an attempt to make herself sick by thrusting her finger into her throat, drew a four-inch long ivory bodkin from her hair and used that instead. It worked, but in her subsequent gasping for air, she swallowed the bodkin that eventually passed into her bladder and made it very painful for her to pee. The letter explains how Mr Proby (‘Master Chiruegon’) eventually extracted the bodkin ‘in the manner of the higher operation for the stone’ nine weeks after she had swallowed it. The girl fully recovered after a month. Not all patients survived such operations, to which many other communications testified. The need to be able to ‘extirpate excrescences inaccessible to the fingers’, rather than crush them, was pointed out by M. Le Cat, Surgeon to the Hotel Dieu at Rouen and Royal Demonstrator in Anatomy and Surgery to the King of France, in a letter of 1751. To perform the operation he designed a new design of cutting forceps. An early operation on animals to discover the function of certain parts of the body was reported in 1683 ‘by the ingenious William Musgrave, Fellow of New College’, in which the caecum (a cul de sac at the entrance to the large intestine) was cut out from a bitch to see what would ensue. One dog died at the first attempt but the second operation was successful and ‘in a little time (running up and down the college) she grew fat and proud and the last summer brought a litter of whelps’. No side effects were perceived by the loss of the caecum. Brain operations are particularly difficult. The skull and brain contains over 600 km of blood vessels, and well into the twentieth century the fatality rate for operations was about 70 per cent since there was a high risk of the patient simply bleeding to death, as so many vessels could not be clamped off. Things began to change with the pioneering work of Cushing at Yale University in 1902. Even then, having opened the skull to remove tumours, some other vital part of the brain may have been inadvertently removed because there was no ‘map’ to inform the surgeon what was being looking at nor, indeed, proper knowledge of how the brain worked. Again, only four minutes were available to perform the operation, owing to the brain being deprived of oxygen, a problem later overcome by reducing the patient’s temperature to very low levels. Nowadays operations are performed in which the top of the skull is temporarily removed while the patient remains awake, to expose the brain in order to perform neurological work. Furthermore, stereotaxic (three-dimensional) surgery for deep brain simulation can now be performed in which holes are drilled in the skull through which fine probes and implants may be passed, and guided to the region of concern. Such holes are made with slim electric hole saws held in one hand, not the hand-cranked trepanning tools of old. One type of brain operation that is now disreputable was lobotomy. In its original form, holes were made on opposite sides of the skull and a special knife was passed through to cut the connexion between the frontal lobe and the thalamus (the part of the brain relaying sensory information and acting as the centre for pain perception). It was believed that this would moderate patients’ difficult and uncontrolled behaviour. Later, transorbital lobotomy involved insertion of an ice pick (a steel knitting needle device with a handle introduced in the USA between the wars to break up blocks of ice) above the eye socket to enter the brain. Waggling around the ice pick supposedly detached bad bits of the brain from good bits. Drilling of bone requires skill, and how a drill is lined up and guided by the surgeon during the initial drilling of pilot holes for further enlargement, reaming and screw-tapping is very important. The majority of holes drilled preparatory to the insertion of screws are made in the shafts of long bones. Burton (1976) describes experiments in which pig femurs (similar in many respects to human bone) were drilled with 6.35 mm diameter drills having a variety of point angles between 80° and 150° and a variety of flute angles. One set of measurements
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was taken at a constant feed rate of 0.1 mm/rev, and another under a constant axial load of 19.6 N, all at 456 rpm. It was found that the feed force was bigger, and the rate of penetration slower, the greater the drill point angle (see Section 5.5). A (total) point angle of some 85° seemed to be the most efficient (cf. Bechtol et al., 1959) but such drills tend to jam when passing out of the bone because of viscoelastic recovery that reduces the diameter of the hole and grips the drill. Thermal damage (bone death, necrosis) can be caused by drilling at too high a speed (Thompson, 1958). Joint remodelling after operations may be affected by the drilling techniques used. Yamasaki et al. (2008) investigated the effect of multiple drillings on early joint remodelling. They found that there were biomechanical and anatomical changes after drilling into the socket in the hip bone that receives the head of the thigh bone (rotational acetabular osteotomy) which resulted in increasing stress on one side. Guidewires are used for directing drills to the desired locations, particularly in orthopaedic surgery. They are stainless steel rods about 2 mm in diameter and have a pointed end, sometimes with a thread to drill into the bone preparatory to the main drilling. Hollow drills slide on the guidewires. One problem that may arise is that bone debris may get trapped between the inside of the hollow drill and the guidewire. Thus, in hip fracture operations, if trapped debris causes jamming so that drill and guide rotate together, the guide will penetrate into the articular cartilage at the head of the femur and on into the pelvis. Shuahib and Hillery (1995) describe experiments in which the torque and thrust were measured for a variety of drills and types of guidewire (Section 5.5). Sawing through bone is sometimes part of surgery. It also sometimes features after murders when bodies may be macabrely dismembered to prevent identification of the victim, to make transport and disposal easier, or to try to make it difficult to determine cause of death (Hainsworth, 2008). Hainsworth goes on to explain how a particular saw can now be linked to a particular crime from characteristic marks produced on bone by the teeth of the saw, rather like linking the firing of a bullet to a particular gun. Not only surgical instruments, but also fitments left in the body have to be sterile and biocompatible. Hence the use, for example, of stainless steel and titanium alloy surgical bone screws. A special asymmetrical buttress screw threadform for surgical implants was invented by Macanor in France (Hughes & Jordan, 1974). The frictional forces during insertion are smaller than for conventional threads, yet higher for removal, which helps to prevent loosening. Special taps are required to cut the thread.
11.4.2 Sharpness Sharpness is obviously necessary for surgical cutting instruments and it is important too for the various types of tube inserted into flesh and blood vessels to put in or take out liquids and deliver medication (such as drips, self-administered adrenaline dosage needles for those allergic to bee stings, hypodermic needles, and so on). The cannula (also spelt in other ways and meaning ‘reed’) and the catheter (meaning ‘tube’) are widely used. Sharpness at the end of such tubes is produced by tapering the wall thickness to as fine a radial dimension as possible. The ends of cannulae and the like are advantageously cut off at an angle to introduce ‘slice’ as well as ‘push’ (Chapter 5), thus reducing the impressed force and permitting the insertion to occur progressively. Since skin and flesh give way when pressed upon slowly (Chapter 8), and since contact with sharp needles causes pain, the trick is to do the job quickly (hence the colloquial ‘jab’ for an injection). The high rate of application stiffens the viscoelastic skin and flesh. Popping blisters with a pin is also done quickly for the same reasons. During in vivo fertilization (IVF) treatments, thin hollow glass tubes are injected and retracted under the
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microscope to extract foetuses; similar procedures take place in stem cell experiments. The high deformability of cells is shown by the degree to which even sharp hollow glass tubes have to be pressed up against the cell wall. Sometimes it is difficult to find veins prominent enough from which to extract blood, and sharpness helps to prevent bruising in preparatory poking around to find a suitable vein. The person who takes a blood sample is called a phlebotomist (from the Greek phlebos, vein); that same name was given to those who bled patients when such procedures were thought to be good for a cure. The fleam (same root) is an instrument used by vets for bleeding cattle and horses: it is like a feeler gauge with a half-round sharp region sticking out of the side, hit with a mallet. The sharpness of tranquillizer darts fired into wild animals (Chapter 8) is again crucial for the proper working of the device, as it is for the solid needles used in acupuncture. Woolley (2007) notes what improvised tools may achieve, whatever their sharpness. In the very cold winter of 1608/09 the English settlers at Jamestown, Virginia, had problems of survival, not only because of food but also because the local Native Americans were not sure whether to welcome them. Woolley reports that one colonist was taken by the natives and flayed alive with sharpened mussel shells. Surgical instruments become blunt; amputation knives used to be resharpened and hygienists sharpen their scraping instruments. In Dupuytren’s disease of the hands, thick layers of hard and tough skin build up between the fingers and palm. Surgical knives employed in Dupuytren’s disease wear out quite quickly (Rhadon, 2008), rather like the flensing knives once used in whaling (Chapter 9).
11.4.3 Wounds and trauma Pierced eardrums are not wanted; pierced ears sometimes are. The one is an unwanted ailment; the other is wanted and has to be deliberately made with some sort of tool. Knowing a body’s anatomy is crucial to the work of medical, veterinary and dental surgeons when performing planned operations. Precise knowledge of the location of the body’s vital organs is, perhaps, of even more importance when dealing with attacks with knives and other weapons, and planning how to defend against them (Chapter 8). How deep are they beneath the surface of the body? How much penetration by a knife is permissible before the wound becomes lifethreatening? Serious injury is unlikely if an assailant’s knife fails to penetrate the internal organs, even if the blade breaches overlying soft tissue. A letter in the Philosophical Transactions of the Royal Society relates how one of the King’s messengers was attacked and stabbed in Lisbon in 1756, receiving a serious wound in the stomach (Huxham, 1757). A description is given of how the patient eventually recovered after careful attention and the letter is worth reading if only for the descriptions of the medicines used and the twice-daily bleedings of the patient. Bleetman and Dyer (2000) used ultrasonics to identify minimum skin-to-organ distances: the most vulnerable organ was the liver, which was only 9 mm deep to the skin in the thinnest individuals. Critical depths were found to depend on posture and breathing cycle. As explained in Chapter 8, knives having instrumented handles are used in experiments to establish the energy and forces involved when a hand-held knife is pressed against, or plunged into, a target. A slender triangular knife (see Figure 8-2), having point semi-angle , will have created an area of separated flesh equal to A 2tan when it has penetrated to depth . When friction is small, equating the incremental work done by the force Fd to the incremental fracture work RdA shows that F 2Rtan for insertion into skin/flesh having average toughness R. Knight (1975) performed pioneering experiments on cadavers, where he found that the sharpness and geometry of the tip of the knife made the biggest difference to the force.
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Parts of the body displayed different resistance to penetration from other parts and this Knight attributed to variations in underlying tissue rather the skin itself. Regions where the skin could not deflect much (either large muscular areas like the thigh, or where the skin was naturally pretensioned as in the spaces between the ribs) could be penetrated more easily than looser abdominal regions where the skin deflected before being penetrated. However, when the abdominal wall was artificially pretensioned, the penetration forces dropped considerably. All this fits in with general understandings in fracture mechanics. Very blunt knives result in bruising before penetration, and similar spread-out damage is caused by a blow with a ‘blunt object’ (unless it has sharp corners). Once the tip of a knife had penetrated the dermis, Knight found that the rest of the weapon followed easily through the subcutaneous tissues and any underlying organ, unless bone or calcified cartilage intervened. This demonstrates that initiation (what Knight calls the ‘threshold phenomenon’ of skin resistance) is the crucial event rather than propagation of the cut. Even so, Jones et al. (1994) later showed that significant secondary resistance exists in deep muscle layers. Knight’s work also demonstrated that a victim could fall on a knife accidentally and be transfixed, providing that the knife is sharp. Sharpness was presumably very important in ceremonial suicide (hara-kiri, from the Japanese to cut one’s belly). Green (1979) reported stab tests on clothed as well as naked cadavers and found that naked bodies could be penetrated by short rigid sharp knives with a force of about 10 N whether commencing the test up against, or 150 mm away from the cadaver. For a knife semi-angle of 10°, say, penetrating 10 mm , the average toughness would be R F/2tan 10/(2 10 103 tan20) 1.4 kJ/m2, recalling that friction has not been accounted for, which may be small for a knife that is sharp on all edges. Clothed targets required over 150 N in contact tests and 70–100 N when stabbed from a distance of 150 mm. Larger daggers, kitchen knives and the like required larger forces, and some weapons bent and broke off at the hilt before any penetration. In practice, many knives used in attacks of domestic violence are quite blunt (Fenne, 2008). Green (1979) also performed knife withdrawal tests on both clothed and naked cadavers. These demonstrated that greater forces were required to pull out a knife after stabbing and that rocking or twisting was necessary in some cases. This latter point is relevant in forensic investigations as it suggests that an assailant must be ‘aware of what he is doing while he is doing it’ when two or more stab wounds are inflicted. There are obvious ethical difficulties in investigating how human tissue resists penetration by a knife: the early experiments by Knight (1975) were performed on cadavers immediately before routine medicolegal autopsy and later workers have used pig tissue and Plastelina® (a modelling clay for sculptors dating from the nineteenth century), among other substitutes. The problem of finding a satisfactory skin/flesh/bone simulant has still not really been solved: various polymers, foams, gelatines and chamois leather have been tried in addition to pig tissue and clay, and most have been found wanting in some aspects of performance. Again, friction may be different between a given blade and different flesh stimulants (Plastilina clay shows much greater friction than gelatine), and different again when the same knife blade is used to penetrate raw meat (Ankersen et al., 1998). There are also questions about what sort of knife should be used in experiments: what geometry, what sharpness of the point, what sharpness of edge, and so on? Sharpness and protection are also crucial in the handling of hypodermic needles, leave alone being used by an assailant in attack. The behaviour of skin tissue when loaded by a sharp edge is also relevant to (i) road traffic accidents, where a person’s soft tissue is impacted against a windscreen or hit by flying shards of glass; (ii) some sports injuries; and (iii) cases of high-velocity impact, by shrapnel for example.
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Careless and Acland (1982) compared the performance of post-mortem skin (taken from the upper thigh) and chamois leather (as a skin stimulant), subjected to normal impact by falling wedge-like tools of different wedge angle, but all with comparable edge sharpness. Chamois leather is sheepskin tanned with cod oil, and is often used as a skin simulant. In car windscreen safety testing, chamois leather is thoroughly soaked in water and excess surface moisture removed before testing with, presumably, appropriate choice of thickness to replicate the mechanical properties of skin as far as possible. For both the in vitro skin and leather, smaller wedge angles (‘more pointiness’) required less force to penetrate than more obtuse wedges, as might be expected from the relation F 2Rtan given above. (In some papers in the literature it is not always clear whether ‘sharpness’ refers to how pointed a knife is, irrespective of edge sharpness.) Careless and Acland found that the use of chamois leather as an indicator of injuries could overestimate tissue damage, particularly for cutting in a ‘compressive’ contact, e.g. a head-to-windscreen impact where facial tissues are compressed before cutting, compared with impact of a body by flying glass. Although the experiments were not interpreted in terms of fracture theory (critical shear strengths for different wedge angles being employed as the criterion of perforation) the response, like Knight’s experiments above, fits the expectations of fracture mechanics where pretensioning releases energy on cutting and so reduces forces, whereas precompression in the location of cutting increases forces (Atkins & Mai, 1985). They also found that hairy skin from males appeared to be slightly more resistant to impact in their experiments than hairless skin from both males and females. It was uncertain whether this dependence was due to the presence of the hair itself or to the morphology of the skin around the hair follicles. Actual collision damage to humans involves oblique impacts, and here the combination of slice and push in cutting (Chapter 5) becomes crucial, whereby a slash may produce more damage than a push. Instead of being cut, skin is sometimes unpeeled from the underlying flesh in an accident, when some corner of an object indents and scrapes along the body. Characteristically shaped converging or diverging tears are detached from skin. The reason for this shape is given in Chapter 15, where it is shown that similar shapes of damage can be produced in ship grounding and collision accidents. It is, perhaps, of interest to determine the effective toughness of the neck from the energy required for guillotining. Bindman (1989) gives the history of the guillotine and of the various types of ‘maiden’ that preceded the design now associated with M. Guillotine, an anatomist in the University of Paris at the time of the French Revolution. (The last execution in France by guillotine took place in 1977; the last execution by beheading in Great Britain was in 1747.) The Scottish ‘maidens’ often crushed, rather than severed, necks and one notes that mice are not decapitated by mousetraps. From many illustrations, the drop height of the guillotine blade appears to be about 2 m. The blade is a 45° triangle of side about 500 mm and perhaps 15 mm thick. The mass of the blade is thus 0.5(0.5)(0.5)(0.015)7850 15 kg. Its potential energy before being released is 15(10)2 300 J. The diameter of a human neck is, say, 125 mm, so its cross-sectional area is about 0.0125 m2. Whence, an overestimate for the effective toughness is 300/0.0125 2.4 kJ/m2. The calculation is an upper bound because friction has been neglected both during the drop and through the flesh, and we have assumed that all the potential energy has been used up to severe the neck. To prevent the angled blade pushing the head sideways, the head was placed into a half-round depression in the block. How the blade is slowed down and caught after decapitation determines how easy or otherwise it is for the blade to be lifted aloft for the next victim. The efficient employment of some weapons was also affected in a similar way, the lancer removing his weapon as he gallops by on his horse (Chapter 8).
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Wounds can be produced by other means, such as shockwaves, for which there is a critical level of overpressure of about 3 kPa beyond which damage is produced in humans. However, it was not the direct but the indirect effects of blast (blowing over of people, and flying and falling debris) that were responsible for the vast majority of air-raid casualties in World War II (Zuckerman, 1975). Buildings are far more vulnerable to blast than people. Zuckerman tells how a team was set up in the UK to investigate the ways in which shockwaves and highvelocity metal fragments produced their effects on the body: ‘… We were in terra incognita, in areas where the problems and questions had not been formulated before, leave alone answered …’. Are there differences between ballistic blunt trauma (relatively low masses and high velocities over short times – as also happens with the large angular accelerations in sports and car accidents) and blunt trauma resulting from high mass and comparatively low velocities acting over a long time?
11.4.4 Healing After surgery or wounding, the separated skin, flesh, arteries and veins, nerves, etc., are expected to heal. Healing depends on the size and nature of the wound: bullets make holes, knives slits; wounds caused by falling off a motorbike at speed are likely to produce glancing wounds in which flesh may be gouged out. It also presumes that the wound does not become infected. Gas gangrene wounds result from germs in the soil that has been fertilized with dung; gangrene was prevalent on European battlefields (Agincourt, etc., and much later) but not in the nineteenth century ‘Indian’ (Native American) wars in the USA since the ground had never been cultivated and the climate was hot. Gangrene was little or no problem in North Africa for the same reason (Cheshire, 2008). Common experience tells us that comparatively small wounds will heal without needing to line up the edges of the severed parts and fix them together, but larger wounds have to be stitched or stapled together, or stuck with quick-acting adhesive, otherwise relative movement between parts would hinder recovery. Purslow (1991) examined the relationships between fracture stress and crack length, and fracture strain and crack length, as they depend on the shape of the stress–strain curves. In J-shaped curves, as the crack length increases, the fracture stress drops off very rapidly but the decrease in fracture strain is much smaller. This behaviour resists fracture in displacement-controlled conditions. The kneecap, for example, is strained by an amount fixed by flexure of the knee. Experience indicates that a person can walk and run normally without the likelihood of knee flexure resulting in the propagation of a cut unless the cut is extremely long, providing that it is the extension of the skin and not the resulting load generated that is the critical factor. The word suture means a seam, and in medical stitching there are, again, similarities with dressmaking, tailoring and sailmaking except that the needle and thread now go through skin rather than fabric. The mechanics of stitching, the forces needed to pierce skin and flesh, the role of sharpness of the needle, etc., are governed by the relations of Chapter 8, as are the mechanics for the penetration of staples. Many sutures become absorbed given time, so that stitches no longer have to be removed. This is of benefit in those circumstances where significant surgery and its associated trauma is necessary to access the removed or repaired organ. The skin around a healing wound has heterogeneous mechanical properties, something revealed by cutting forces during microtoming (Willis & Vincent, 1995). Figure 11-3 shows various force–displacement traces at various stages of healing of the wound. Initially, the outer
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(a) Scab
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Figure 11-3 Force of sectioning of skin at various stages of wound healing: (a) unwounded skin (heavy line); (b) newly produced wound (shaded area); (c) early scab formation; (d) scar tissue prominent in centre of wound (after Willis & Vincent, 1995).
layer of skin which has disappeared through ulceration is revealed as a reduction in force (b); as the scab is formed the force for cutting increases (c), so that the edge of the scab (not perfectly joined to the intact skin) leaves a dip in the force trace (d). Thus healing can be followed in the force trace as well as in histological sections. Yue et al. (1987) studied the strength of linear wounds (straight cuts) in normal and diabetic rats. Diabetic animals showed reduced strength compared with normal controls but strength could be improved by insulin treatment. Toughness was not measured by cutting but it seems likely that similar patterns apply. Izmailov et al. (1989) reported that wounds produced by sharp blades healed better and more quickly than those produced by blunt weapons. Wound healing, scarring and regeneration mechanisms were addressed by Ferguson and O’Kane (2004), who point out that modern sutured surgical wounds, healing in a clean or sterile environment, were not previously encountered in nature, and that the old evolutionary wound-healing responses are quite different. What happens in practice should be contrasted with how things are sometimes portrayed by Hollywood. According to Bryson (1994), ‘C19th bullets were so slow, relatively speaking, and so soft, that they almost never moved cleanly through the victim’s body. Instead they bounced around like a pin ball and exited through a hole like that created by a fist punched through paper. Even if a bullet miraculously missed the victim’s vital organs, he would almost invariably suffer deep and incapacitating shock and bleed to death within minutes’. Therefore movie gunfights of people taking wounds and carrying on are fanciful. Again, what is best practice with arrows lodged in bodies? Should they be extracted or left in place? Cheshire (2008) points out that Native American stone arrowheads were generally bound to the shaft with rawhide lashings. In the body the moisture would swell the rawhide, thus loosening the binding so the head would stay behind if the shaft was pulled out. The arrowhead
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was also likely to detach from the shaft if it were stuck in bone, as it took greater force to remove. Arrowheads made of hoop iron (from old barrels) were soft and sometimes formed a hook if they hit bone, again making removal difficult. Before the days of X-rays a lost arrowhead took a lot of finding and it was almost certain to be fatal if left in the body. But if the shaft was left in place, it could be followed down into the wound to locate where the head had lodged. The resulting open wound was flushed clean by blood flow. Silk vests were favoured by the Mongols as, it was said, arrows could more easily be withdrawn from the body when the silk was taken through by the weapon.
Chapter 12
Food and Food-Cutting Devices and Wire Cutting Contents 12.1 12.2 12.3 12.4 12.5
Introduction Properties Food Texture The Delicatessen Slicer Wire Cutting
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12.1 Introduction In this chapter we discuss (i) the mechanical properties of various foods eaten by humans and other animals; and (ii) the methods and machinery for cutting foodstuffs consumed by humans and other animals. Many devices for cutting foods have been invented empirically but, in the end, they must depend on physical properties and the basic principles of cutting. Food has metabolic value only when the products of eating enter the bloodstream. ‘Food’ here includes the diet of creatures in general, not only foods for humans. In order to utilize plants as a source of food, herbivores have to select what is to be eaten, gather material into the mouth, and break open the cells to release the nutrition (Wright & Vincent, 1996). Carnivores attack prey and, after the kill, begin to bite off pieces of the carcass before devouring. Chemical actions in the gut complete the process. Sometimes the gut can do it all, with no need to chew: reptiles tend to swallow food whole with little or no processing in the mouth. Not maintaining a high metabolism, a snake can go to sleep while digestive enzymes do their work, whereas the high metabolic rates of mammals and birds mean that they have to eat regularly and often, and chewing before swallowing helps to speed up the throughput. Wild creatures, domestic animals and humans sometimes eat the same food (fruit, for example) but often they have different diets even within their own group, leave alone between different groups. The significant difference is that humans cook food which alters mechanical properties; humans also freeze food for long periods to preserve it. Natural and artificial foods are composite biomaterials materials with a hierarchy of microstructural components at different scales. Whether it is a carrot or a convenience snack, understanding their texture presents formidable challenges. Dunn et al. (2007) remarked that soft solids and foodstuffs can be strangely behaved materials. On the one hand, their flow and fracture behaviour at large deformations can often be well described by the usual solid mechanical properties of elastic modulus, yield stress and fracture toughness, all properties being rate, temperature and environment dependent (e.g. Luyten, 1988; Kamyab et al., 1998; Dobraszczyk et al., 1987; Atkins et al., 2002). On the other hand, some very soft foodstuffs behave, perhaps, more like very viscous materials (e.g. Scott Blair & Coppen, 1939; Scott Blair et al., 1947; Wilkinson, 1960). Cheese, for example, is a sticky gel. Cutting in the harvesting of food crops for humans has been discussed to some extent in Chapters 5 and 10. Breeders can now develop plants that have an altered ‘habit’ and the Copyright © 2009 Elsevier Ltd. All rights reserved.
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right mechanical properties so that the crop is more easily harvested and/or can be harvested by machine. How plant breeding has affected cereal crops is vividly illustrated in Breughel’s painting of harvesting a corn field. The crop is as tall as the farm workers (who are shown using sickles, not scythes). Modern cereal crops come up only to the waist, if that. Some cutting (of vegetables, for example) is done by hand in the fields in preparation for sale. Special tools are used in the field such as the gralloching knife to disembowel deer out on the mountain. Further cutting of food takes place at the retailers and at home, again with special tools such as bread knives, grapefruit knives, and game shears used for cutting up duck and pheasant. Large wholesalers and supermarkets have their own factories where special machines cut foods to make up prepackaged foods. The quality of a cut surface may affect appearance and shelf-life (Clark, 2008). Sawing produces waste so many food items are guillotined by grid (cheese, for example). Food mixers and blenders, mincers and so on, whether large in the factory or small in the kitchen, involve cutting. Food is cut in the kitchen before being served at table, or cut before cooking (peeling potatoes before boiling) for subsequent serving up. To test whether potatoes have been boiled long enough, a knife is used to pierce them in the saucepan. How far in it goes, and how easily, is the subjective test. Some people also lift a potato out of the water to see if it stays on the knife. What is being measured in these empirical tests? At the table lots of cutting happens on the plate before ingestion. How animals harvest their food and the action of teeth is considered in Chapter 13. It is curious how butchery and cuts of meat vary between different countries, and the different uses to which the same foodstuffs may be put in different countries. The case of sugar beet is interesting. Indigenous to Spain and Portugal, it was imported into Belgium in the sixteenth century. It was not extensively cultivated until the continental blockade of the eighteenth century forced Belgium to manufacture its own sugar. Melin (1883) commented that the shape of the knives in beet cutting machines was very varied and even fantastic. Some had wavy edges or with sharp angles, some have a lozenge or roof shape, etc. Root choppers were similar devices to cut up turnips, beets, swedes, neeps, mangel-wurzels and so on for cattle feed. Chaff is chopped straw, fed to cattle as roughage, and chaff cutters had knives to do the job, as do machines for cutting potatoes into real chips (not the sort of fries extruded from potato mash, but even these are chopped to length). Thinking about what happens when one prepares food, and during eating, helps to elucidate what is going on, and the interaction between cutting implement and properties of different foods. There are many subjective terms – juicy, dry, stale, tough, chewy and so on – that ought to be explicable in terms of physical properties. Is one’s thumb a cutting tool when peeling an orange? Why does the skin sometimes come away in large pieces, but at other times only in small bits? Thin peel on other fruits cannot be removed in that way and requires a knife. Why? Could fingernails be employed? Why is the outer covering of an orange called a skin but that of an apple a peel, yet other fruits a rind, and bark for a tree? The equipment used is also worthy of thought: what is the best way to take the top off a boiled egg? How knives become blunt by contact with ‘hygienic’ chopping boards is mentioned in Chapter 9. Beech is the preferred wood for butchers’ blocks and chopping boards, as it does not break up or splinter, and displays hardly any grain in which bacteria may get trapped. The surface of plastic boards is relatively easily cut into and provides a home for bugs. Other problems of the food industry that food biomechanics helps to solve concern things such as the spreadability of butter, margarine, jam and so on, which is similar to the mechanics of negative-rake cutting, and bulldozing soil on a building site (Chapter 14). The way in which icecream is scooped and the formation of butter and so on into balls or whorls, all concern cutting.
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12.2 Properties What, to a person eating, makes cake different from a biscuit or meat? Lillford (2000) describes experiments in which the sequential breakdown of various different foodstuffs was tracked by volunteers who spat out the samples after a set number of chews. The objective of chewing is to reduce the initial material to a swallowable state; its structure must be reduced sufficiently, and appropriate lubrication must be available, to reassemble the particles into a soft, swallowable bolus (from the Greek bolos, a lump or clod). The important fracture particle sizes of foods, and the subsequent reassembly mechanisms, reside at the microscopic level. Cooking may mean heating raw flesh, boiling vegetables or baking. Most baked cereal foods form a common class of composites that can be described as gas-filled cellular foams: pizza crust, bread and cake have very similar microstructures (other gas-filled foams are icecream and confectionery). The cell walls are themselves a composite of protein, which for bread is gluten, whereas for cake, it is commonly egg white and starch. When uncooked or undercooked, these agglomerates appear to behave like notch-insensitive pastes, but when baked correctly behave as connected solids. The Gibson–Ashby (1988) formulae for overall elastic modulus and strength in cellular composites (in terms of the properties of the cell walls) apply to cake and bread. Studies of mixtures of the starch and protein that make up the walls of the cells show that elastic stiffness (modulus) is directly related to the level of the plasticizing water present in the wall material. For baked products, the properties directly relate to the water content, with a change in the fracture mechanisms when sufficient plasticizer is present to change a glassy material to a pliable, rubbery structure. Freshly baked bread is soft; toasted bread is brittle as moisture has been driven out; stale bread is brittle for the same reason. In reverse, the loss of crispness (and the origin of soggy wafers, rubbery pizza crusts and stale potato crisps) is primarily a problem of water mass transfer into the cell walls (Attenburrow et al., 1989), as seen when dunking a biscuit in tea. Whether a gingerbread man is a cake or a biscuit is considered by Fforde (2007), who noted that cakes go hard when stale, but biscuits become soft. Stale gingerbread goes soft owing to absorption of water, so gingerbread is a biscuit. However, Sanson (2008) remarks that ships’ biscuits (the hard tack of Nelson’s navy) became harder the staler they became! While liquid water is always available during chewing, residence times are short (minutes). How quickly properties of food change while sitting in the mouth depends on both capillary flow into the structure and wall rehydration. As explained by Lillford (2000), at one extreme some thin-walled cellular structures ‘melt in the mouth’ owing to rapid mass transfer into the cell walls; at the other extreme, if the capillary walls are fat-coated, then both capillary wetting and hydration of the wall material are slowed, allowing the crisp qualities of fried bread and croutons to be maintained during chewing. Fresh raw fruit and vegetables are cellular structures in which the cell walls are biologically active and in which the filler is not air, as in cakes and bread, but a cytosol containing water and dissolved salts (cytosol means the aqueous part of the cytoplasm that fills cells). Fresh fruit and vegetables contain an intracellular, semi-permeable membrane that maintains a higher than atmospheric internal pressure (turgor), thus prestressing the cell walls (Chapter 1). When it is broken, rapid crack propagation through the cell walls occurs (transgranular fracture) and gives the perception of crispness. With lapse of time after harvesting, water is lost and turgor and cell wall prestressing gradually reduce. The structure becomes limp (its bulk modulus is reduced) and a higher breaking strain is measured in tensile tests. For example, an apple behaves mechanically as a fluid-filled foam and the difference in behaviour between fresh and stale fruit is evident simply upon the depth of bite required to cause fracture.
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Cooking rapidly destroys the semi-permeable membrane of fresh vegetables, and turgor is again lost as in ageing. The mechanical properties are now controlled by quite different processes that are dependent on the architecture of the original specimen, which relates to its biological origin. At least three classes of behaviour can be identified (Lillford, 2000): (i) The cellular architecture is maintained (interconnected) and the cell walls remain stiff. In this case, little changes. On biting, the water cannot flow out of the structure quickly enough to prevent the build-up of wall stresses in the network, and brittle behaviour ensues, as if it were still uncooked. An example is cooked water chestnut. (ii) The cellular architecture is maintained but the cell walls are softened. The behaviour is now critically dependent on the rate at which liquid can escape. That depends on the tortuosity of its flow path from the inside to the outside of the structure. If biting rates are high and/or the cells are small, then brittle fracture can still occur. At lower biting rates and/or with large cells the behaviour is different and becomes less crisp. An example is cooked carrot, where fast biting gives a crisp texture, but slow chewing does not. Try it! (iii) The cells separate from each other. The plant cell wall is itself a composite structure of cellulosic fibres and other more soluble polysaccharides, and each cell is glued to its neighbours by means of a network of pectin. Cooking degrades the pectin network, allowing the cells to separate relatively easily. When loads or displacements are applied, the fracture pathway tracks around the cells through the viscoelastic ‘glue’ (intergranular fracture). This is the usual process in well-cooked vegetables. Occasionally the individual cells are detected in the mouth, giving a ‘mealy’ perception. Meat that is eaten by humans is derived from the contractile, striated muscle tissue of animals, birds, reptiles and fish. This muscle tissue has a hierarchical structure, aligned in the direction in which tension is created by the living animal (Chapter 2). At the cellular level, this tissue is constructed from filaments of myosin and fibrous actin, supported on elastic fibres of titin. Individual cells are then surrounded by a sheath of collagen to form the muscle fibre. Several cells are bound together as a primary muscle bundle, and these bundles are then further organized into secondary and even tertiary bundles, each delineated by collagenous sheaths of slightly different composition (Hutchings & Lillford, 1988). A piece of meat teased apart under high-magnification scanning electron microscopy (SEM) shows these features. Some muscles are constituted of parallel fibres and others are pinnate in that the muscle fibres are at an angle to the resultant vector; pinnate comes from the Latin for ‘feathered’ or ‘winged’. Such muscles ‘waste’ some contractile energy by pulling at an angle to the force vector. However, more muscle fibres can be packed into a given length. It is not known whether such muscles have different textures when chewed. However, it is known that carving meat at different directions to the muscle fibres changes the texture and well-trained carvers can provide different ‘cuts’ to suit different preferences. Some muscles are rich in myoglobin and very low in stored glycogen as a fuel source and rely on a continuous blood supply. These are the red muscles and are generally fatigue resistant and characteristic of endurance performance (Section 2.8). Postural muscles that are continually working, such as the fillet steak, are highly prized. The muscles of a chicken leg are also aerobic and dark, as are the flight muscles of pigeons. Conversely, the white breast muscles of chickens are anaerobic and produce high forces but are rapidly prone to fatigue. Chickens use their flight muscles rarely, only to flap into a tree to avoid a predator, but they walk around all day. It would be interesting to investigate whether these muscles have different resistances to cutting. Texture and mouth-feel of meat are determined primarily by the muscle fibres. After chewing the primary muscle fibres remain largely intact. The composite structure is distorted within
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fibre bundles and fails through the connective tissue linking these bundles, resulting in flakes or fibrous particles. Tough meat has connective tissue (collagen and elastin) that is not easily destroyed by cooking and leaves a highly extensible matrix material with a high fracture toughness. After identical cooking treatments, all varieties of meat fall on a unified trend line when taste panellists’ ratings of flakiness (an early breakdown event) are correlated with the tensile fracture strain of meats pulled at right angles to the fibre orientation (Lillford, 2000); see Section 12.3. No doubt there will be a connexion with fracture toughness too. The aim of tenderizing meat is break up the connective tissue before cooking: this may be achieved to a greater or lesser degree, either mechanically (bashing with a hammer) or chemically (papaya juice has a good effect on pork; Chaplin, 2008).
12.2.1 Measurement of properties In principle, physical properties of foodstuffs may be determined by the same test methods used for engineering solids, i.e. tension, compression, torsion, creep and relaxation experiments, etc., on unprecracked and on cracked samples. As touched on in Chapter 11, there will be differences, however, in that many foods are soft and squidgy: not only is it difficult to grip them for tension tests, but they may fracture under their own weight in beam tests. The behaviour may be time dependent. It will certainly be more sensitive to water content, outside humidity and temperature than typical engineering materials. An important feature also is that the forces and energy involved in deforming food and biological materials are small, so that sensitive load cells are required where, in addition, the signal is as large as possible compared with the noise. Having performed successful tests, the results have to be interpreted and this, again, may be different from procedures in conventional engineering tests owing to large-deformation reversible and irreversible behaviour, and conspicuous non-linearity. For example, if a stress– strain curve displays continuously changing slope, is it correct to talk about a Young’s modulus? What is not done as much as it should be in these sorts of tests, is unloading followed by reloading, followed by further unloading and so on. This would distinguish between reversible (elastic) and irreversible deformation. When simple monotonic loading to fracture of a testpiece gives a non-linear curve, it will not be clear what is permanent deformation and what is recoverable. By way of example, the stress–strain curve of cheddar cheese may be obtained by compressing cylinders using polytetrafluoroethylene (PTFE) films to reduce platen friction (Dunn et al., 2007). In that case, the cylinders were 20 mm diameter by 20 mm high and made with a cork borer. As the load–displacement curves remained quite steep after initial yielding, yield stresses defined as the departure from linearity were noticeably lower than those based on a 0.2 per cent-offset definition, i.e. y 10 kPa (deviation from linearity) but 22 kPa (offset). Notched three-point bend testpieces, nominally 20 mm by 20 mm with a 60 mm span, using 10 mm diameter rollers, were employed to determine the fracture toughness R using standard energy methods (e.g. Atkins & Mai, 1985). R was about 15 J/m2 at 20°C. Typical values for other cheeses range over 2–14 J/m2 for R, 30–80 kPa for y and 100–300 kPa for E (Kamyab, 1998). The toughness : strength ratio, given by R/y, for cheeses is therefore about 0.07–0.14 mm. The effects of ageing on the compressive stress-strain curves of Gouda cheese are shown in Figure 12-1. Valuable reviews of test methods for the mechanical properties of foodstuffs may be found in Alvarez et al. (2000b) and Lucas (2004). Figure 12-2(a,b) shows load–displacement diagrams for compression and for single-edge-notch-bend (SENB) tests for various fruit and vegetables.
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200 Old Gouda with cumin seeds
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Mature Gouda Young Gouda Young Gouda with cumin seeds
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Figure 12-1 Compression test results for cheeses (after Vincent et al., 1991).
Agrawal and Lucas (2003) determined the fracture toughness of various foods in terms of the critical stress intensity factor (KC) of linear elastic fracture mechanics. They give the following values for different cheeses: mozzarella 4 kPam, cheddar 11 kPam and parmesan 23 kPam. Other determinations were 44 kPam for raw carrot, 74 kPam for brazil nut and 83 kPam for macademia nut. Vincent et al. (1991) used a symmetrical sharp wedge test to investigate the fracture toughness of foodstuffs. Figure 12-3 shows schematically the sequence of events as a wedge penetrates samples of apple. After some displacement up to a maximum load, a crack runs ahead of the tip of the wedge to separate the apple into two pieces. Critical displacements are smaller, but loads higher, for wedges having larger included angles, and vice versa. Average fracture toughnesses were determined by measuring the work area under the complete wedge– load vs wedge-displacement curve and dividing by the fracture area. Table 12-1 gives some representative data for different types of apple. Note the anisotropy caused by the radial orientation of air spaces in apple flesh (Figure 12-4). There are no pre-existing starter cracks in this wedge test as usually performed and unloading is not usually done, so strictly speaking there is uncertainty about irreversibilities and friction during the initial indentation up to the critical depth. Even so, providing the experiments pass the ‘can you fit the broken bits back together?’ test (meaning that there must have been negligible remote irreversibilities) the procedure is sound. The wedge test has been employed
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Figure 12-2 (A) Load–displacement curves from compression tests on cylindrical specimens of fruit and vegetables. (B) Load–displacement curves from fracture tests on single-edge-notch-bend (SENB) specimens of fruit and vegetables (after Alvarez et al., 2000b).
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4 3 5 5
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Figure 12-3 Sequence of events as a wedge enters a fracture testpiece (apple is illustrated) (after Vincent et al., 1991). 1: The wedge just about to enter the specimen. 2: The top of the specimen deflects storing elastic energy. 3: The wedge cuts into the specimen; the elastic strain energy is fed to the cut surface and the force drops. 4: As the wedge enters further the two halves of the specimen are forced apart storing strain energy. Small peaks in the curve show the wedge is cutting through cell walls. 5: Sufficient strain energy is available to start a free-running crack propagating ahead of the wedge. The force falls. 6: Propagation of the crack stabilizes and it propagates at the same velocity as the wedge. Table 12-1 Effect of orientation on toughness of apples. Fracture toughness (J/m2) Parallel to air spaces
Across air spaces
Cox
69
103
Gloucester
194
341
Norfolk beefing
668
1044
to determine other aspects of foodstuff behaviour, e.g. anisotropy in the toughness of apple flesh (Khan & Vincent, 1993) and mechanical damage induced by controlled freezing in apple and potato (Khan & Vincent, 1996). For truly brittle behaviour in this test, Freund’s equation in Section 3.4.1 could be employed having determined FC and FT from the wedge geometry,
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Radial
Force
Tangential
Distance moved by wedge
Figure 12-4 Orientation of apple samples for sensory and machine testing. The two directions for crack propagation are (A) parallel and (B) orthogonal to the radial air spaces in the apple. The way in which this orientation affects crack propagation and the force applied to the wedge are also shown (after Vincent et al., 1991).
i.e. FC H FT. In more ductile foods, the beginning of the test will be a wedge indentation hardness test with no cracking. Eventually cracking will ensue but, depending on the size of the sample, there may be considerable distortion of the separated halves. The mechanics of the double wedge test (opposed wedges like pliers) is discussed in Chapter 13. Cutting is often used to determine the fracture toughness of highly extensible foodstuffs, since the large deformations that occur in the rest of a testpiece when pulled, for example, (and their interpretation) are avoided. Lucas et al. (1995) point out that the toughness of plant cell walls (about 3–4 kJ/m2), for example, would be difficult to determine if not by cutting. Scissors (Darvell et al., 1996), inclined razor blades (Ang et al., 2008), nail clippers (Bonser et al., 2004), microtomes (Atkins & Vincent 1984), guillotines (Atkins & Mai, 1979) and punch-and-die (ticket clipper) tests (Choong et al., 1992), all instrumented for cutting forces, may be used. Hand-held devices for fieldwork have been manufactured (Darvell et al., 1996; Aranwela et al., 1999). Analysis of force–displacement diagrams may be done graphically
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Fracture toughness/(J/m2)
800
600
400
200
0
0
30
60 Angle (degrees)
90
Figure 12-5 Values for the fracture toughness of a single lamina (t 0.35–0.4 mm) of Calophyllum inophyllum for a varying angle (after Lucas et al., 1991).
(accounting for friction) to give fracture toughness as explained in Chapters 3 and 5; alternatively, the algebraic formulae of those chapters may be used. Cutting of foodstuffs by wire is discussed in Section 12.3.3. Lucas et al. (1991) was able to show anisotropy in toughness in leaves from cutting tests (Figure 12-5), where orientation of veins is important. They argued from these data that classifications of leaves (sclerophyll, pachyphyll, etc.) should pay attention to the venation of the leaf and the structure of the veins (sclero is Greek for hard and phullon for leaf; pakhus means thick). They went on to propose that the thickening of the walls of the smaller veins to form a network is a defence against invertebrate herbivores. A thorough examination of methods of assessing leaf fracture properties is presented in Aranwela et al. (1999) and further refined by Sanson et al. (2001), who recommended protocols for testing. Ang et al. (2008) recommend cutting leaves and thin-film sheets with an inclined razor blade to overcome difficulties in tests such as scissoring and shearing involving double-bladed cutting. Comparisons are made of fracture toughness R values for the same material determined by different methods. The inclined razor blade test gives considerably lower toughnesses, owing to greater blade sharpness and hence reduced damage to the specimens during cutting. An interesting outcome of the study is that although absolute values of R are lower, the ranking of materials remains the same. This suggests that blunt blades proportionately increase the width of the boundary layers alongside the cut edges in which the microstructural damage required for separation occurs. This is similar, perhaps, to the concept of an effective toughness in guillotining (Chapter 5). This sort of thing must be relevant to sharpness of teeth (Chapter 13) and how teeth perform. Knowing the mechanical properties of food enables lower-bound estimates for bite forces to be made, as explained in Section 13.3.1 (assuming that the forces are not great enough to break the tooth).
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Mechanical properties of foodstuffs are rate, temperature and environment dependent, particularly water content in the case of the last. Dobraszczyk et al. (1987), for example, investigated the toughness and dynamic mechanical properties of frozen beef (m. semitendinosus). How the stress–strain curves, moduli, yield strengths and toughnesses varied with temperature were established, and whether water (ice) was bound with the microstructure, or was unbound to form a separate network was studied. Cryo-SEM confirmed that at temperatures above 15°C, behaviour was mainly ductile, but below 15°C, brittle behaviour prevailed. How leopards in the Russian Arctic cope with frozen carcasses of dead deer that they eat is not clear. Indications of food mechanical properties occur at parties where ‘dips’ are offered, where pieces of carrot or celery, or crackers are to be used to transfer to the mouth. Depending on the properties of the dip and the amount taken, and the properties and dimensions of the other foodstuff, deformation and fracture of one or both pieces can take place. Interesting experiments can be performed by altering the dimensions of the ‘picking-up’ food and where it is held in the hand.
12.2.2 Properties after storage Holt and Schoorl (1983) studied fracture and bruising in potatoes and apples loaded in compression, and the hysteresis in load–displacement diagrams upon unloading. They were concerned with damage in handling, processing and transport, but different modes of deformation (cleavage/splitting and shear) were identified. The associated fracture toughnesses were about 210 J/m2 for cleavage and 770 J/m2 for shear in Sebago potato; and the very low values of about 6 J/m2 for both in Granny Smith. These data help to estimate the behaviour of fruit and vegetables in storage. As already mentioned, size and shape of cells, volume of intercellular spaces and thickness of cells walls all affect the mechanical properties of fruit and vegetable tissue. In addition, the turgor pressure controls, to a large extent, the stiffness of the liquid-filled cells by altering the water potential. Typical turgor pressures for fresh fruit and vegetables are about 100 kPa, which decreases with time, making bodies more compliant. Turgor pressure may be artificially altered for experimental purposes by immersion in mannitol solutions of different concentration, e.g. Alvarez et al. (2000a) studied the effect of turgor pressure on the cutting energy of stored potato tissue. There was a general reduction of toughness from some 100–150 J/m2 to some 50–100 J/m2 at greater molar concentrations of mannitol, patterns being maintained over time of storage. Alvarez et al. used confocal scanning laser microscopy, which has the advantage of tissue being able to be viewed fresh (so not subject to preparative techniques) and being able to be viewed at different levels by focusing (essentially ‘optical sectioning’).
12.2.3 Properties after cooking Lucas (2008) avers that a quantitative measure of the effectiveness of cooking concerns ‘connectivity’ between elements and the solid mechanics concept of ‘notch sensitivity’, discussed in Chapter 2. The fracture load of a material containing a notch must be lower than that of a sample without a cut owing to the smaller cross-section that bears the force, but it can be even lower and that is what fracture mechanics is all about. Disproportionate reduction in cracking stresses happens because of the shear connexion between elements. Raw meat and
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many animal soft tissues are difficult for humans to chew with their blunt cheek teeth because meat lacks this shear connexion between structural elements (and thus has a J-shaped – curve) but roasted meat has a linear – (Purslow, 1985). Lucas (2008) hypothesizes that (i) cooking acts to increase the notch sensitivity of foods so as to promote their comminution in the mouth; and (ii) this has led to an order of magnitude reduction in the relevant mechanical properties of foods now ingested by humans (stress–strain gradients, yield stress and toughness) compared with the food still ingested by apes and primates living in similar habitats to early Homo. Experimental results on the effect of boiling time on the mechanical properties of noodles are shown in Figure 12-6. Experiments on notched tensile testpieces showed that uncooked noodles displayed crack-tip blunting and resistance to fracture whereas, with cooked noodles, crack-tip sharpness was retained, permitting much easier crack propagation. Noodles are soft before cooking; in contrast, pasta is hard and stiff. Experiments (Frew, 2008) show that the fracture toughness of pasta decreases almost linearly with time of cooking. How people perceive that a particular food is undercooked or overcooked is a fundamental problem with which the food industry struggles (what, precisely, does al dente mean scientifically?). When cooking detoughens food, it becomes possible to break food down in the mouth that would otherwise be impossible to chew. Changing connectivity explains why the skins of tomatoes and oranges are easily removed (and the pith of oranges too) after immersion in boiling water for a minute or two. Blanching of vegetables before freezing ensures the destruction of certain enzymes that can continue to operate at low temperature and produce oxidative rancidity and therefore flavour problems during storage. Unfortunately, blanching also damages the vegetable structure, allowing ice to form, and can make things even worse, if adventitious thawing occurs. Modern commercial processing does not always blanch and, instead, rapid freezing is done immediately after harvesting (‘farm to freezer in minutes’). Then, provided storage is at constant low temperature (18°C), rancidity is minimal (Lillford, 2009). The partial boiling (parboiling) of potatoes before roasting relates to obtaining a desirable final texture. It is possible to roast without parboiling, but this must be done slowly and at a lower heat, otherwise the heat transfer characteristics mean that the outside is burnt and dry by the time the temperature in the centre of the potato reaches 70°C (the gelatinization or ‘cooking temperature’ of starch). Very rapid roasting gives burnt and dry on the outside and
Dissipated energy ratio
.8
.7 WSN
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Figure 12-6 Dissipated energy as a multiple of total work done at 2 mm displacement (3 per cent engineering strain) for (left) unnotched white salted noodles and (right) unnotched yellow alkaline noodles (after Lucas, 2004).
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raw and hard in the centre. During controlled roasting Maillard reactions between sugar and protein at the external surfaces produce colour and introduce ‘roast flavour’ to the potato (Lillford, 2009). It is the same for meat, where the roast flavour is only in the external couple of millimetres or so. Cutlery fork tines rely on friction not to come out once pierced into food, and food has to be pulled off with teeth in the mouth. The effort required to pierce food on the plate is related to whether the food is undercooked or overcooked. This, in turn, is determined by the cook by piercing or stabbing potatoes and vegetables with a fork or knife while the food is still in the boiling water in the pot. It would be interesting to perform experiments with tools instrumented for force and displacement to see how the piercing resistance changes with time of cooking and ties in with toughness (Figure 12-7a,b). Knives or forks are also used to prick the skin of potatoes before baking in order that moisture can escape so as not to burst the jacket. Forks and spoons are used on edge to separate soft solids: the runcible spoon with which the Owl and the Pussycat (who went to sea in a beautiful pea-green boat) ate mince and slices of quince, is a fork curved like a spoon and having three broad prongs, one of which has a sharp edge. ‘Runcible’ is a nonsense word used by Edward Lear. Potato raw
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Figure 12-7 (A) Development of a curvilinear J-shaped stress–strain curve in potato flesh with boiling; (B) effects of boiling and roasting potato tubers on their toughness. Toughness reduces with boiling beyond about four minutes to levels lower than with roasting (after Lucas, 2004).
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12.3 Food Texture Most people are able to describe the process of eating, often related to a specific food type, with a degree of common language using words describing mechanics (tough, tender, crisp, soft), flow (thick, thin, juicy, creamy), and size and shape (flaky, fibrous, gritty, smooth). It would appear that all the neural signals from pressure sensors in the tongue and soft palate, together with the forces and displacements of the muscles, are being combined to produce a sensory response (Lillford, 2000). Different constituents in different foods require different mixing agents in different amounts to form a swallowable bolus. Two types of saliva supply these different needs: (i) mucous saliva for emulsification of fats and oils; and (ii) serous saliva for the hydrolysis of carbohydrate polymers (Bender & Bender, 1997). Tough, dry meat requires extensive structure reduction and lubrication by saliva; the process therefore takes longer. Dry sponge cake requires little work mechanically to reduce the structure, but extensive saliva mixing is required before reassembly takes place. An oyster is already close to the swallowing state, provided that the overall size of the initial piece is not too large. Liquids are immediately swallowable. The common method of measuring the mouth feel (texture) of food is to ask someone to eat the food and report their experience. Based on the results of panel tasting, the food industry fabricates new foods that should appeal to consumers in various ways. A lot of time and resources are spent in training taste panels (organoleptic panels, i.e. whose decisions are based on the bodily senses) to assess food as objectively as possible. But objectivity is difficult, and instrumental methods involving fundamental mechanical properties would be far better. As explained by Vincent et al. (1991), there are many empirical tests across the food industry that purport to measure ‘properties’ relevant to food condition and with which food texture has been linked (some mimic the action of teeth). A penetrometer test for ripeness of foodstuffs is not too different from pressing one’s thumb into an avocado in the market. Volodkevich (1938) describes an ‘apparatus for measurements of chewing resistance or tenderness of foodstuffs’: it is a blunt wedge compression test aimed at mimicking the action of molar teeth. The loading of samples in these pragmatic sorts of tests is a complicated mix of compression, tension and shear. Even if the tests are not merely qualitative but measure loads and displacements, it is difficult to extract fundamental mechanical properties: while some idea of yielding in compression may be gained, not much can be established quantitatively about the important property of toughness. Unless basic properties are known, it is impossible to determine what other factors (including size of sample) contribute to the final assessment of texture. Again, perceptions concerning fracture involve the teeth. The first bite with the incisors seems particularly critical in formulating opinions (Bourne, 2002) (see Section 13.3.3). Most uses of the teeth involve the fracture of foods. However, food scientists have, by and large, not quantified fracture in their mechanical characterizations, even though any separation of foodstuffs into parts must involve fracture mechanics. Again, crispness (sudden isolated drops in bite force during chewing) and crunchiness (continual load drops during eating) must relate to the energetic stability of cracking in foodstuffs being bitten in the mouth (Rojo & Vincent, 2008). Vincent et al. (2002) found that the critical stress intensity factor KIC of foods, a parameter related to crack initiation, was linearly related to perceived measures of ‘hardness’ and ‘crunchiness’ from anterior bites as evaluated by trained taste panellists. It must be sounder to demonstrate cause and effect directly, and link basic materials mechanics parameters (determined in the consistent ways of engineering science) to the perceptions of taste panels. Results then should be free from cultural and linguistic limitations: in some languages there appears to be no word for ‘tough’ meat (such meat being said to be
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‘hard’, which is not the same thing). What do the terms al dente for pasta, and well done, medium and rare for meat, mean in terms of basic mechanical properties? Why some fruits can be frozen and thawed successfully without losing ‘mouth feel’, yet others cannot, is explicable in terms of changing mechanical properties with freezing and thawing. Rather similar to taste panel observations, tactile measurements are made for the surface quality of wood (Sinn et al., 2009). The psychological terms are hedonics (pleasurable and unpleasurable states of consciousness) and haptics (cutaneous sense data).
12.4 The Delicatessen Slicer Cutting laminae with a sharp-edged disc was described in Chapter 5 when introducing the idea of the ‘slice–push’ ratio . Delicatessen slicers cut through thick diameters of salami or substantial sides of bacon where considerable parts of the circular blade are in contact with the foodstuff as it is fed by hand through the cutter. Machines are often angled to one side so that the slices fall under gravity to a pile (the way they stack – shingling – is commercially important for ease of handling and packaging). Figure 12-8 shows a partially cut cylinder of salami, say, on a slicer. Both sides of the cutting bevel on the edge of the circular blade are in contact with the workpiece. If the machine is angled, there will be the additional contact area LPQM between the upper side of the circular blade but negligible contact area on the lower side as the slice has fallen away. As with wire cutting (Section 12.5), every length ds of the circumference of the slicing disc that has contact with the foodstuff behaves locally as a miniature inclined blade, having slice–push that depends upon the angle i (Section 5.2.3). Integration of the incremental forces will give the feeding forces and the torque on the disc, as illustrated for a non-circular cutter in Section 12.4.1. Very thin slices cut on a delicatessen slicer are called carpaccio; the word comes from the Italian for the carp fish but the connexion with thin slices of meat is not clear at all. A disc cutter is useful for cutting the end-grain of wood that is difficult to cut without damage. Clearly, cutting with high slice–push ratio helps and a straight cutter at high obliquity should achieve this, since tani (Chapter 5), but there are experimental difficulties of working with steeply inclined blades. According to McKenzie (1966), even when a microtome blade is angled at 60° to the direction of cut difficulties persist in less dense woods such as radiata pine. However, at 60°, is not that great (tan60° 3 1.73). They point out that use of a disc overcomes the problem, which is confirmed by results for cutting end-grain, tangentially to the growth rings, of messmate wood (Eucalyptus obliqua) in the air-dried and saturated conditions. Experimental slice–push ratios were altered in the
ω
P L
ƒ
M
–i +i
Q
Figure 12-8 Delicatessen slicer cutting a round workpiece.
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range 0.25 80 000 by changing the normalized rotational speed ( r/f). While forces reduced as the rotational speed of the disc increased, they eventually levelled out owing to friction; above a peripheral speed of about 3 m/s burning became evident and surfaces became increasingly glazed. The type of chip varied with speed, from slivers at low speed to tightly curled spirals at high speed.
12.4.1 Progressive cutting of flitches of bacon In some types of commercial high-speed food cutting machinery, the workpiece is fixed and, unlike the delicatessen slicer, has no feed in the plane of the blade. Penetration of the workpiece to cut a slice is achieved by the blade being non-circular and having increased radius of curvature with increased rotation (Figure 12-9a). In this way, the blade cuts progressively into the workpiece as the blade revolves. Cutting of a slice is completed in one revolution of the blade and, in a ‘non-cutting gap’ in the blade profile, the workpiece is indexed forward the required thickness of slice. In one design, the radius of the blade increased linearly with angle, i.e. followed a linear spiral, given by r a b, for which (dr/d) b is constant. Equation (10-6) gives at all points along an instantaneous arc of contact during cutting. The forces X and Y in the x and y directions, and torque T, at any position of the blade when cutting can now be obtained by integration, i.e. r2
X 2R ∫
r1
r2
Y 2R ∫
r1
r a r a r r cos b sin b b dr r 2 b2
r a r a r b cos r sin b b dr r 2 b2 r2
T 2R ∫
(12-1)
r1
r3 dr r b2
(12-2) (12-3)
2
For a spiral blade of given dimensions cutting a workpiece of known size, the limits of integration, r1 and r2, may be calculated by geometry for a series of angular positions of the blade (Figure 12-9A). For purposes of illustration the workpiece is a rectangular block of material 380 mm by 61 mm in section cut by a blade having a 203.2 mm and b 36.383 mm/ radian. The distribution of X, Y and T along the arc of contact with the workpiece in particular positions of the blade can then be calculated. Figure 12-9(B) shows the results for the position given in Figure 12-9(A). From a series of such calculations, the variation of forces X, Y and the variation of torque T during one rotation of the spiral cutting blade may be calculated, as shown in Figure 12-9(C) (Atkins & Xu, 2005). In the presence of friction, there will be a multiplying factor involving S [Lf/R] exactly as in Chapter 3, or alternatively Coulomb friction may be incorporated. Unfortunately, as these sorts of commercial device have never been instrumented, it has not been possible to test the analysis against experimental measurements of forces and torques.
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Y
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Variaton of the forces for spiral blade at lower blade position during a cycle
80
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Figure 12-9 (A) Spiral blade and intersection of blade with workpiece at an arbitrary angular position. (B) Variation of forces X and Y, and torque T, along arc of contact with workpiece in position shown in (A). (C) Variation of forces X and Y, and torque T, in one revolution of spiral cutter in which a single slice is cut.
It is clear that, as discussed in Chapter 10, interesting questions are raised regarding optimum blade plan forms that would give least cutting forces. Least cutting forces are important in the food industry since under such operating conditions, least damage occurs to the offcuts.
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It is possible to envisage machines in which the centre of rotation of a cutter moves according to some preferred path (there are, for example, so-called orbital cutters in which a rotating circular blade revolves at a different speed in a circular orbit about a central axis, Atkins & Xu, 2005.
12.5 Wire Cutting Some material/cutting blade combinations display high friction and cutting is often performed with a fine plain wire in order to reduce the contact area between blade and material, and so reduce cutting forces and obtain a better surface finish. The separation of blocks of cheese into smaller pieces is a well-known example; in porcelain factories, extruded clay is cut similarly. The cutting action is that of ‘separation by dividing’ with, in theory, no loss of material. Single crystals and semiconductor wafers are cut by a plain wire in a slurry of abrasive material, or by wire itself coated with abrasive particles. A large-scale example of cutting by a diamond-coated reciprocating cable concerned the removal of a sunken ship in the English Channel which was a danger to navigation (Anon., 2004). The wreck was cut into nine sections with diamond wire to ease removal. It is said that the Ancient Egyptians cut stones for pyramids with papyrus ropes using sand as an abrasive and milk as a lubricant (Yeo, 2008). These latter processes are different from cutting by a plain wire and, like sawing in Chapter 7, produce waste. Teischinger (2006) has shown how timber can be cut using a hot wire. In factories, mass-produced cheese is typically made in 20 kg blocks, which are cut into 5 kg and 2.5 kg blocks using wire, from which smaller portions of 200 g and 600 g are cut. The usual sort of grocer’s cheese cutter has a fixed length of wire, one end of which is fixed to the base of the device, the free end of which has a handle. The thin wire is pulled by hand through a piece of cheese to divide it up. At different times during the cut, the wire takes up different curved shapes within the cheese, the lengths of wire outside being straight (Figure 12-10). What determines the curved shapes will be discussed in a moment, but a far more interesting question is what path does the hand take when pulling on the end of the wire to cut the cheese? It is similar to the question of what the hand does when opening a beer can or any other sort of tin opened by a ‘ring-pull’ device. Playing with a cheese wire cutter will demonstrate that different combinations of pull on the wire and swing of the hand require different effort. Alteration of the pulling force and/or its direction alters the shape taken up by the wire. Common sense says that the path taken by the hand of the grocer is probably the one that minimizes the work done, but it seems that no experiments have been performed to find out. It is illuminating to see the different ways in which the device is operated at delicatessen counters. The simplest type of wire cutter has the wire held taut in a stiff frame like a hack saw. Kamyab et al. (1998) performed experiments of this sort on various types of cheese and measured the cutting forces. The study was later extended to include gruyere and different diameter wires (Goh et al., 2005). While friction from the sides of a knife was avoided, there was
Cheese
Figure 12-10 Grocer’s wire cheese cutter.
Handle
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still friction around the thin wire itself. This was modelled on the assumption that there was a thin flow zone existing around the bottom half of the wire, which presses radially on the wire with the yield stress y, and produces circumferential friction stresses y, where is the coefficient of friction. Integration around the lower half of the wire gives 2ry[1 ] for the frictional force per unit length of wire of radius r. Using work, Kamyab et al. showed that the cutting force is
F/Rw [1 M]
where M 2ry[1 ]/R in which R is the specific work of separation (toughness), and w is the width of the material. (The model gives a value for friction even when 0, the residual friction force/length of 2ry representing the formation of the thin yielded boundary layer around the wire, in which the micromechanisms of material surface separation come into play.) Independent determinations were made of the toughness, yield strength and friction of the various cheeses by various methods, and the cutting data proved to be consistent with the model and the independent data. Typical values for different cheeses ranged over 2–14 J/m2 for R, 30-80 kPa for y, 100–300 kPa for E and 0.5–1.25 for . The surface finish obtained depends on the wire diameter; the appearance is reminiscent of the ‘abortive, evanescent’ cracks found on the surfaces of aluminium cropped, with zero clearance, by Chang and Swift (1950); and when cutting pastes (Benbow & Bridgwater, 1993; Böhm & Blackburn, 1995). Rough fracture surfaces in cheese are caused by its microstructure (Luyten, 1988). There are two length scales of interest in the microstructure of cheddar cheeses: 102–101 mm (unevenness of network) and 1–10 mm (curd grains). Smooth surfaces result when the crack passes through the curd grains; rough when around the curd grains. The crumbly nature of many cheddar cheeses is down to weak binding between curd particles. In the same way, crumbly soils have limited cohesion between particles. Wire cutting experiments on alumina pastes, similar to those of Kamyab et al. (1998), are described in Chapter 14. At fixed speed the local strain rate in the material contacted by the wire diminishes as the wire diameter increases. It is well known that the strain rate in simple tension or compression tests is given by v/L, where v is the crosshead velocity and L is the effective gauge length of the testpiece, so tall specimens have smaller average strain rates. The volume of material deformed by a wire of diameter d, contacting the workpiece over width w, depends on (d2w/4), so the representative dimension of the equivalent testpiece is (d2w/4)1/3. For the same wire speed and specimen width, the strain rate is thus expected to vary as d2/3. Goh and colleagues’ smallest wire was 0.25 mm diameter, and the largest 2 mm. At the same wire speed, the strain rate of the smallest wire is expected to be (2/0.25)2/3 4 times the largest. Finite element methods (FEM) simulations gave a factor of about 6. The study showed that rate effects caused higher forces to be required at higher speeds, and to take this into account, non-linear constitutive relations for stress as a function of strain and time were determined and incorporated in FEM simulations. A critical strain criterion was used for crack initiation and a cohesive zone model simulated propagation. As with knives, there is a lowering of cutting forces for all materials when ‘slice–push’ is introduced with wires, but simple reciprocation of the frame holding the taut wire gives only limited reduction because the frame is at rest at the each end of the stroke and maximum occurs only at the mid-stroke (Section 5.2.5). Unpublished research at Reading shows that when a taut wire is rotated (by an electric drill) and moved directly into the material to be cut, there are reductions of force that are greater the higher the speed of rotation. The action is not really slice–push and the effect may be down to reduction in friction.
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Instead of reciprocating the frame containing a taut wire, a continuous band wire (like a band saw but with no teeth) may be employed, looped around two pulleys, one of which is driven. When the wire is perpendicular to the feed table, the slice–push ratio wire speed/feed speed, and by altering the speed of the driving motor, can be made very high. As explained in Chapter 5, though, friction limits the reduction in forces that are achievable and there is no benefit in 50, say. Nevertheless, experiments (Clark, 2008) show that cutting forces are much reduced in wire cutting and the surface quality of slices of food is improved. For example, cheese is more easily cut, and soft bread rolls are ‘more cleanly’ cut than by serrated knife, the lower wire cutting forces much reducing the compression of the bread which usually occurs beneath a knife (even a reciprocating knife) in normal cutting. Wire in a frame or in a continuous loop deflects only slightly during cutting, and operates under high tension. As mentioned above in connexion with the grocer’s cheese cutter, if the wire were initially slack, and then employed to cut, it would take up a curved shape within the workpiece, and remain straight outside. What determines the curved shape? What are the forces, and are there benefits in cutting with a slack, rather than taut, wire? The simplest to analyse is the symmetrical shape obtained by hanging weights on each end of a wire slung over a block of material. To avoid the complications of wire cutting into the sides of the block, the ends of a length of wire may be attached to the ends of a stiff bar in experiments so as to form a bowstring when cutting (Figure 12-11A). As a wire cuts a material, it is loaded with forces FT along and FR across the wire. These induce a tension T in the wire. The cutting forces will vary along the curved length of the wire and so will T. Assuming that there is negligible bending resistance (as in a catenary), the equations of equilibrium of an element Q1Q2 of wire of length ds shown in Figure 12-11(B) give the following relation for the variation between FT, FR and T (e.g. Dunn et al., 2007):
dT dFT dΨ T dFR
(12-4)
where is the angle of inclination of the element with respect to some chosen direction. Integration gives the variation of T with , for particular dFT and dFR, in terms of the unknown tension in the wire T0 at the apex of the curve. The behaviour of the element of wire in Figure 12-11 is locally the same as that of an element of a stationary knife at inclination to the motion of the workpiece. From resolution of the velocity of the block along and across the wire,
ξ (velocity along wire)/(velocity across wire) tan Ψ
(12-5)
Using the relationships in Chapter 5 we have, for frictionless cutting,
dFR Rds/(1 ξ2 ) Rds cos2 Ψ Rdx cos Ψ
(12-6)
and
dFT Rξds/(1 ξ2 ) Rds sin Ψ cos Ψ Rdx sin Ψ
(12-7)
It follows that
T T0 sec Ψ
(12-8)
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O
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D
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Figure 12-11 Cutting through a block of material with an initially slack wire: (A) wire fixed to a bar wider than the block to prevent cutting into the sides of the workpiece; (B) forces acting on and in wire.
and
T0 tan Ψ Rx
y (R/ 2T0 )x 2
(12-9)
Replacing tan by dy/dx, we obtain
(12-10)
so that the shape of the wire within the workpiece during frictionless cutting is a parabola. As explained by Dunn et al. (2007), the unknown T0 at the apex depends on the length of the wire.
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It may be shown that increments of horizontal force dX along the wire are zero, so that X To; also that Y, the vertical component of force in the wire, is
Y To tan Ψ Rx
(12-11)
using Eq. (12-9). Hence the vertical component of the frictionless cutting force at any point along the wire is given by the fracture toughness times the span of horizontal distance to that point from the apex of the parabola. This simple relationship is reminiscent of the property of the catenary, that the tension at any point in the chain is equal to the weight of a portion of the chain that would extend in a vertical line from that point to the x-axis (Inglis, 1951). These relations may be modified appropriately for friction both for forces and the shape of the wire (Dunn et al., 2007), using the Kamyab et al. (1998) factor M ( 2ry[1 ]/R) for Coulomb friction in wire cutting. For a given exit height (y/w), the parabolic shape for frictionless wire cutting is made flatter as M increases, reducing along the wire, and giving different wire exit angles from the block. Theory predicts that, for all M, the tension Texit on exit from the block is high at small exit because the wire is ‘tight’ (as in a fretsaw frame). At greater exit the forces fall but then increase again, passing through minima that are more marked the greater the M. Minima in wire force arise because of the competition along the wire between (i) the local reduction in force caused by the increased slice–push ratio on elements of wire having high inclinations; and (ii) the increase in local frictional force caused by the longer incremental contact lengths ds ( dx/cos) between wire and material at higher wire inclinations. For frictionless cutting by wire, there is no minimum in wire tension: it decreases continuously as exit increases with longer wires and, as expected, the force component resolved in the direction of cutting is always equal to Rw as in simple microtoming. For finite M, the minimum in force is at 62° for M 0.1, 57° for M 0.6, 53° for M 1, 48° for M 3 and greater. The absolute values of the non-dimensional loads increase with M. Experiments on symmetrical slack-wire cutting of cheddar cheese performed by Dunn et al. (2007) are shown in Figure 12-12, where the experimental variation of the steady-state tension in the wire on exit from the cheese blocks is plotted against the exit angle exit. There is a shallow minimum in force, at an angle between about 50 and 60° using a 0.5 mm diameter wire. Independent determinations of , y and R for the cheeses at various temperatures, and at rates comparable with those in the wire cutting experiments, were made in order to calculate the expected M, which turned out to be 0.7 M 1.4. Dunn’s model for M 1 predicts that for there should be a shallow minimum in Texit at about exit 55°, which agrees well with experiment. The minimum experimental force is about 1.5 N, so with specimens 84 mm in width, and with R 15 J/m2, (Texit/Rw) 2.4; theory predicts that (Texit/Rw) should be about 2. Cutting, with a wire, of foodstuffs that behave like very viscous liquids was also studied by Dunn et al. (2007). Here, the process of cutting might be thought of as viscous flow past the wire, with the material not recombining behind the wire (some rehealing may occur, particularly with ‘sticky’ materials like dough). Dunn et al. were curious as to how the path of a wire passing through a very viscous medium (flow at very low Reynolds number) differed from that for a medium having solid mechanical properties. They also pointed out that the same sort of symmetrical cutting by wire took place during school regelation experiments where a thermally conducting wire passes through a block of ice, leaving no trace of its path. The different catenary-like shapes paths swept out in all three cases were analysed and compared with experiment.
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10
Force in wire on exit from block (N)
8
6
4
2
0 0
20
40 Ψexit (degrees)
60
80
Figure 12-12 Experimental results from cutting cheddar cheese with a 0.5 mm wire. There is a shallow minimum in force between 50° exit 60°.
The optimum entry and exit angles for smallest cutting force given by the above analysis cannot be maintained towards the end of a cut on a grocer’s wire cheese cutter, so forces increase and the cut surface is not so good. It is known that cheese wires tend to break towards the end of a cut on the handle side of the wire where the tension is greatest. An alternative to a band wire is a device in which a long length of wire is spooled from one pulley to another. With appropriate control systems to control either tension or slack in the wire, high- cutting may be achieved with any degree of slack in the wire (Clark, 2008). The shape taken up by a moving wire is different from that in Dunn et al. (2007) since the different tension on each side of the sample, caused by the frictional drag along as well as across the wire, causes the angle to be different on each side. This means that further reductions in cutting force are achievable. For example, experiments on various types of melon showed that, in comparison with cutting with a taut wire, forces are reduced by using an optimum slack between the anchor points of a stationary wire, and then reduced even further by introducing lateral motion. For the particular rig and specimen sizes employed by Clark (2008), a cut with a taut wire required roughly 1.5 N vertical force; with the optimum slack in a stationary wire, 1 N; and 0.5 N when the slack wire was moving with a slice-push ratio of 30. Further experiments cutting under similar conditions, but with a knife of comparable sharpness, revealed that up to 15 N was required when pushing straight down. As expected, the specimens were difficult to cut by knife, and the product was compressed, causing damage to the flesh near the surface. As soon as reciprocating motion was introduced with the knife, the cutting became a lot easier (about 9 N), owing to slice-push, but clearly it was not as easy as when wire cutting with both slack and motion along the wire.
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Many manufacturers have problems with soft fruits and vegetables drying out after having been cut. Packets of diced fruit available from supermarkets often have a large volume of juice at the bottom of the packet and this has commercial implications for the shelf-life of prepacked foods. Experiments demonstrate that cantaloupe melons and courgettes, for example, cut by wire under high retain their moisture in comparison to cutting by knife. This occurs when the diameter of the wire (say 0.1 mm) is smaller than the cell size of the fruit or vegetable.
Chapter 13
Teeth as Cutting Tools Contents 13.1 13.2 13.3 13.4 13.5 13.6
Introduction Jaws and Bite Force Occlusion and Contact Mechanics Sharpness and Wear of Teeth Attack and Defence Scaling
307 310 312 320 322 323
13.1 Introduction There are different-shaped teeth to perform different functions connected with eating and attack and defence. The ‘mix’ of different types of teeth along a jaw, and the size of jaw, vary with different creatures and strongly influence the way they ingest food or attack prey. Teeth are employed to bite off, grip and sometimes chew food before swallowing. The same teeth are used by creatures for other jobs: in attacks on prey; sometimes by males defending territory and harems against their own kind (Drews, 1996); sometimes oestrous females bite over-attentive males (Cresswell et al., 1996). The jaws of creatures vary in size and shape and in different proportions to body size: a dog’s jaw is longer and narrower than a cat’s; a crocodile jaw is very long. The lower jaw, sometimes referred to as the mandible, is composed of a number of bones in all vertebrates except for mammals that only have one bone, the dentary. The upper jaw is composed of a number of bones, with the premaxilla and maxilla the toothbearing bones in mammals. In other vertebrates other bones of the palate may also bear teeth. Mandible is also used for the lower part of a beak. Most people probably think of ‘teeth’ in terms of the familiar rows of separate teeth with separate roots distributed along the jaws of humans and domestic animals, but ‘teeth’ in the sense of pointed bristle-like protrusions or serrations are found on mandibles of arthropods (insects, spiders and crustaceans) and many vertebrates including fish, amphibians and reptiles, but not modern birds. The tuatara is a sphenodon or ‘wedge tooth’, now found only on islands off New Zealand where rodents have not reached. It does not have separate teeth but all-inone serrated jawbones. The feeding behaviour of gastropod molluscs (snails, slugs, whelks) ingesting relatively large pieces of vegetation, is highly dependent upon the efficient manipulation of a broad radular scoop. (The radula is a chitinous band in the mouth of most molluscs, set with numerous minute horny ‘teeth’ that are drawn back and forth to break up food.) While insect mandibles and vertebrate teeth both perform a biting and processing function, they are very different materials with very different properties which operate at different scales (Wainwright et al., 1976). All chewing parts must be stiffer, harder/stronger and tougher than the food, otherwise they break, wear, or are themselves deformed rather than the food; the same criteria are required for a good cutting tool (Chapter 9). Nevertheless, it so happens that the mandibles of tiny insects have properties not too different from those of their cell-wall food; in contrast, the mouthparts of large vertebrates are less prone to wear. Copyright © 2009 Elsevier Ltd. All rights reserved.
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Teeth are used to break off and break up food in order to increase the surface area before swallowing for the gut to finish the job by chemical action. Ingestion (the taking of food into the mouth), mastication (the breaking down of food in the mouth not only into sizes that can be swallowed but also to create surface area to encourage chemical action) and deglutition (food transport and swallowing) are all discussed in detail by, for example, Hiiemae and Crompton (1985), where X-ray movies were used to follow the passage of food into the mouth and beyond. The physical properties of a food should determine the ‘ideal’ tooth shape required to divide it up (Lucas, 2004). The diversity of mammalian diets includes hard, but brittle, bones; tough, viscoelastic meat; high- and low-fibre leaves; and fruits that may be either hard and strong, or weak and juicy. The best method of separating a food will be different for foods having different physical properties. In consequence, as Evans and Sanson (2003) point out, it is expected that a variety of tooth shapes should be found in mouths. However, some animals are omnivorous and include a variety of different food types in their diet. Determining the critical function that controls the optimum tooth shape is complex and little studied. Even though a number of creatures do not chew food, they may still have teeth: anglerfish, for example, along with some animals, and especially snakes, have backward-pointing teeth in their mouths and throats to prevent prey escaping. Sanson (2008) remarks that it is interesting that animals that have lost the use of manipulative limbs have such teeth. The job all teeth do, the forces and energy involved, and how effective they are, will depend upon the mechanics of piercing and shearing given in Chapters 8, 3 and 5. Teeth are arranged symmetrically along a jaw, locations being referenced within upper and lower, left and right, quadrants. The classification systems employed by anatomists and dentists are confusingly different, but do not really concern us (see Lucas, 2004). In mammals, there are incisors (from the Latin for cut) at the front of the jaw, canines (from the Latin for dog) next inline to the rear, and premolars (molar-like teeth) and molars (from the Latin for mill) at the back. The broad-bladed (spatulate) upper and lower incisors of humans are found particularly in the higher primates. Humans use their incisors on a much more diverse range of foods than just fruits, the main food of chimpanzees and similar animals. All mammals that have canines (‘Dracula teeth’) have only one in each quadrant of the mouth, i.e. four canines in all. Elephant and seal tusks are very large canine teeth. The shapes of canines vary across mammalian groups. In some carnivores, particularly canids (dogs, wolves, foxes), the upper canine is strongly oval; in felids (the cat family) it is less so. A fang is a long, sharp hollow or grooved canine tooth by which poison is injected. So-called wisdom teeth in humans are the third molars on each side of the upper and lower jaws. The hyena mouth is characterized by carnassial teeth (molars that are adapted to cut skin and other tissues by a shearing action, although the etymology comes just from ‘flesheating’). Such teeth are towards the rear of the jaw, with blunt bone-crushing teeth in middle; further forward the canines are big and incisors are small. Crushing teeth are also found in the African wild dog. Scavengers like hyenas consume bone, whereas top predators like lions and large cats avoid bones. Carnassial teeth act like the curved blades of secateurs, preventing the escape of the ‘workpiece’ by the cusps at each end of the tooth digging into the food to prevent it escaping. Figure 13-1 shows the jaws of a canid (dog family) in the stages of (A) gripping food; (B) fracturing soft tissues; and (C) breaking a bone. There are many variations on the basic arrangement of teeth, but perhaps not as extreme as that in one of Arthur Koestler’s books, describing how he saw someone in Central Europe in the 1930s with no teeth but instead a continuous strip of stainless steel (Palmer, 2008). Badgers have no carnassials, and their incisors are small, but they have big canines. Some of
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Using the anterior teeth 50 degrees
(A)
Using the carnassials 12 degrees
(B)
Using the molars 25 degrees (C)
Figure 13-1 The jaws of a canid (dog family) in three staged poses: (A) gripping a prey item with its incisors and canines; (B) fracturing soft tissues with its carnassials; and (C) breaking a bone with the molars lying behind the carnassials. Note the different gapes involved in each of these activities (after Lucas, 2004) (Reproduced by permission of Cambridge University Press).
a hippo’s teeth very large, but the rest are relatively small for the size of the animal. Komodo dragons (monitor lizards) are carnivores with flat serrated teeth (more like a shark’s than a reptile’s). Herbivores such as zebras and the horse family (hindgut fermenters) have big incisors in both the upper and lower jaws. In contrast, ruminants are foregut fermenters (that chew the cud, sending food back from a first stomach, or foregut, to chew again) and have incisors only in the lower jaw, with opposing ‘pads’ in the upper. Ruminants include most cattle, buffalo, deer, antelopes, giraffes and camels. Some teeth are peg-like: the aardvark belongs to the Tubulidentata (tube-toothed) order and its teeth consist of twenty flat-topped pegs made up of hexagonal tubes, right at the back of the mouth. Common usage says that grazing is eating herbage (non-woody vegetation) growing on the ground, and browsing is eating any vegetation off the ground. Strictly though, grazing means eating only grass, from where the word comes. Thus the tapirs (ancient forest relatives of the horses now found only in South America and Malaysia) are browsers both on the forest floor for fallen fruit and also in the air for green twigs and ferns. Grazing mammals such as cows, horses and buffalo (but not kangaroos) have a very broad muzzle and the incisors are both splayed out and broader to make an efficient cropping organ; such teeth are also described as spatulate. In sheep the canine teeth (usually vestigial or lost in herbivorous mammals) look and act like an incisor so effectively that they have eight functional lower incisors (four on each side). Grazers can then ‘mow’ a broad swath and increase their intake of relatively
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omogeneously distributed food compared to the discrete leaves of bushes that browsing anih mals feed on. Browsing mammals have narrower muzzles and are more precise in their manipulation of the food (Gordon & Illius, 1988). Grazing kangaroos are unusual in this regard in retaining narrow muzzles. Sanson (2006) is an excellent review of the biomechanics of browsing and grazing. Cows wrap their tongues around blades of longer grass and pull to ingest, leaving an appreciable length of grass; on shorter swards the lower incisors are used. Sheep, rabbits and geese crop very close to the ground. Sheep snatch at grass rather than pull evenly as cows do, and King and Vincent (1996) have postulated that this may be connected with rate effects in the feedstock making fracture easier. The different grass-eating habits between sheep and cows, and the consequent rivalry between ranchers, featured in the development of the ‘wild west’ of the USA.
13.2 Jaws and Bite Force The relative motion of jaws is important in the analysis of how teeth work. In some creatures (reptiles, for example) digestion is done all in the gut with no chewing, prey being swallowed whole. Anacondas and pythons can open their mouths wide enough to swallow deer, goats and baby tapirs, and a python is a match for a leopard or crocodile. Jaws of constrictors do not dislocate as is often averred. The tooth-bearing bones of the upper and lower jaws do not form a simple hinge and are what would be called linkages in engineering. They are highly mobile and are connected to the braincase only by ligaments and muscles, allowing the mouth to open to 150°. The sides of the lower jaws move independently of each other, alternately ratcheting prey into the throat (the pterygoid walk). This intracranial bone mobility (cranial kinesis) exists in lizards and birds as well as snakes (Bramble & Wake, 1985), and permits more jaw motions than in mammals. Mammals have only opening, closing and sideways motion where chewing takes place with rotary motion in the plane perpendicular to the jaw. Mammals, in turn, have more degrees of freedom than arthropods, whose mandible can only move about a fixed hinge, with a circular arc closing motion. There do not appear to be torsion (twisting) motions in mastication. French (1988) discusses the design of the human jaw. There are three degrees of freedom: the jaw can be lowered or raised; it can be thrust forward or back; and the chin can be moved left or right. All this is possible because the hinges between jawbone and skull are capable not only of flexing, but also of sliding. Unlike the bearings of a railway locomotive (with which French compares the jaw) only one side of the slider (axlebox horn) exists and the jaw has to be held in place against the skull by ligaments – what an engineer calls ‘forced closed’. Furthermore, to permit sideways chin motion, the slides are held in cartilage. Jaw design is different in different animals and motion analysis of jaw movement during chewing appears to relate to the types of food ingested (Heath, 1991). Thus the large sideways movements of the jaw in ruminants are associated with the grinding of food between large flat-topped molars; carnivores have a simpler (and stronger) ‘bearing arrangement’ for the jaw. The jaw joint is primarily a compression-resisting structure that can take only modest tensile forces. One way to ensure that the jaw joints are subjected only to compressive forces, so that the joint does not disarticulate, is to position the jaw-closing muscles so that the resultant muscle is always between the joint and the last tooth in the row. A consequence of this is that the output force at the teeth is always smaller than the muscle force. Even so, the forces at the jaw are expected to be as efficient as possible and Greaves (1988) has shown that jaw muscles are located in the position that produces the largest average bite force for any jaw
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length. It is 30 per cent of the way along the jaw from the joint for mammals but only 20 per cent for reptiles owing to a different jaw shape (triangular in mammals and pentagonal in some reptiles). This fundamental arrangement appears to be independent of diet. It seems that evolutionary changes in primitive mammals were to increase the bite force. In turn this affected the design, and number, of molar teeth. Studies have measured actual bite force in creatures by getting the animal to bite on straingauged devices looking like tuning forks (DeChow & Carlson, 1983); Freeman and Lemen (2008) measured bite forces in North American rodents with a piezoelectric sensor. Heath, Wright and their co-workers have used sensors in human teeth (e.g. Xu et al., 2008) to determine occlusal loading. Estimations of bite force based on the associated muscles have been performed. Wroe et al. (2005) predicted the bite force quotient (BFQ bite force/mass of associated muscle) of a wide sample of living and fossil mammalian predators. Bite force may be converted to average stress on teeth: the opossum is said to generate on the order of 170 Pa on its post-canines in a static bite. Most bats feed on insects or fruit, or both; there are in addition those with highly specialized feeding habits (nectar, fish and blood). The evolutionary relationships between bite force, head shape and the diet of bats are reported by Aguirre et al. (2002). The bite force performance was essentially the same for all the bats, except for the specialists where it was lower. Binder and van Valkenburgh (2000) both measured and estimated bite forces for hyenas. Daniel and McHenry (2000) used finite element modelling to estimate bite force in alligators. Crocodiles have a remarkably high speed and power of jaw closure, yet the jaw of a crocodile is apparently easy to keep shut by squeezing from the outside as its opening muscles are weak (Chaplin, 2008). Hoh et al. (2000) discovered that superfast myosin is present in the jaw-closing muscles, but not in the jaw-opening or locomotor muscles. Knowing the mechanical properties of food enables lower bound estimates for bite forces to be made as described in Section 13.3. Chewing forces of 50–150 N are reported for human molars, with the potential to reach more than 1000 N in Inuit males (Jenkins, 1978). Linked to bite forces, there are questions about cranial ‘strength’ required to withstand such forces, for both living and extinct creatures (e.g. Rayfield, 2004), and also interesting questions about the link between bite force and body size (e.g. Thomason et al., 1990). There are far fewer investigations of the connexions between the magnitude of bite forces and the physical properties of the objects or materials being bitten. Wroe et al. (2005) argue that the predicted BFQ is a broad indicator of relative prey size and feeding ecology. Do estimates tie up with the actual experimentally measured bite forces? In reverse, knowing the bite forces (or estimating them from muscle size), are they able to provide the effort that comminution of the food requires? Do both sorts of analysis shed light on the design of jaws and teeth? Dean (2006) argues that the microstructure of teeth ‘tracks the pace of human life evolution’. Jaws become bigger as creatures grow, and the original deciduous (‘milk’) teeth are usually replaced once only by new teeth. The manatee (a sea cow) and the Little Rock wallaby are the only two mammals with continually replacing molar teeth at the same location in the jaw. Elephants, kangaroos and the dugong (another sea cow) have molar progression, where the molars erupt at the back of the jaw and physically drift along the jaw until they are lost at the front in a worn state. Teeth, like cutting tools, are designed not to break. When teeth are not replaced wear has to be avoided, if at all possible, for the teeth to function efficiently. Wearing out or loss of teeth can lead to death by starvation when a creature can neither feed nor defend itself. There is a letter by Slare (1713) in the Philosophical Transactions of the Royal Society explaining that all his grandfather’s teeth fell out at the age of 80, but a complete set of new teeth grew immediately after, which he put down to his grandfather’s great consumption of sugar
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throughout his life. It seems that he put sugar or honey on the butter on his bread, into his ale and beer, and in all the sauces he used with meat!
13.3 Occlusion and Contact Mechanics Mammalian teeth usually close upon one another (occlusion), and different surfaces of a tooth that contact under normal load, or slide past surfaces of adjacent teeth during closing, determine the contact mechanics of teeth. Some contact surfaces are convex-to-convex, some concave-to-concave, and some convex-to-concave. The latter act like a pestle and mortar in breaking down food. The occlusal morphologies of teeth are categorized as primary (the tooth shape is functional upon eruption) and secondary (significant wear is required for the shape to be functional, and the functional form is substantially different from the unworn state) (Fortelius, 1985). Farriers and vets profile the teeth of horses to improve ingestion and mastication, employing the same sorts of file as a carpenter, and the same sorts of high-speed disc grinder used to grind valves in an internal combustion engine. A dentist correcting a human’s teeth for better chewing does the same, although perhaps with more refined tools. What should be the desired shapes? A clinical speciality in dentistry called orthodontics deals with the adjustment of tooth orientation. Although the orthodontic treatment of incisors is indicated usually by aesthetic considerations, functional analysis suggests that a vertical incisal inclination, generally the desired clinical outcome, may be out of alignment with the general bite force direction (Osborn et al., 1987), particularly when large gapes are required (Hylander, 1978) because the direction of muscle force is then directed either vertically or even slightly posteriorly (Paphangokorit & Osborn, 1997). In other creatures, such as fish, the teeth fit into matching V-shaped recesses in the row of opposite teeth. In the case of fish eaters such as seals, their pointed incisors also fit into one another when closed, so that they can grasp slippery fish. Teeth are not always employed for chewing when feeding: the teeth of crabeater seals that devour krill are like interlocking Christmas trees that form a most effective filter. The seal takes a mouthful of sea water, shuts its mouth, squeezes out the water and swallows the krill (Bertram, 1992). It is like the baleen filtration of the great whales. Whalebone (baleen) is an elastic horny substance growing in place of teeth in the upper jaw of certain whales, forming a series of thin parallel plates on each side of the palate. The closest thing to baleen in terms of texture and morphology is rhinoceros horn and possibly hoof. In section it looks like a uniaxial composite of hollow hairs stuck in a matrix. It is an epidermal structure made out of keratin (Vincent, 2008). Thin strips of baleen were once used for stiffening corsets; it was not the skeletal ‘bone’ of whales. When food or prey is wholly or partially within the mouth, what happens when teeth close down on other teeth with food in between depends on (i) the mechanical properties of the food (stiffness, hardness, toughness, strength); (ii) the mechanical properties of the teeth (stiffness, hardness, toughness, strength); (iii) the geometry of the teeth; (iv) the relative approach velocity; and (v) the loads that jaw muscles can generate during closing and contact. When an item of food is partially outside the mouth and partially inside, how is it bitten off? Is tension or shear more important? Within the mouth are pieces of food opened up by cutting in tension or shear, or by crushing? In most mammals mastication is unilateral, i.e. it occurs on one side of the jaw at any one time, but food is also transferred from one side to the other during chewing. The relatively constant rhythm of jaw motion suggests fairly constant occlusal velocities V between teeth. This means that the strain rate (V/some characteristic length) imposed on food is greater for smaller pieces than larger. The actual method of reduction to a swallowable bolus depends upon the
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physical properties of the food, the time-scale of the action of enzymes in saliva and the characteristics of the tooth surfaces acting upon it. Many mammals, such as rodents and primates, hold a food object (a nut, fruit or a leafy twig) in their hands. They then cut or tear portions from it using the incisors as much for grasping as cutting; the portion of the food in the mouth is torn away by head or hand movement. Rodents gnaw the fibrous shell of a nut using the incisors and then chew the kernel. Many fruits have tough, fibrous skins and soft, fleshy pulps. Many primates peel a fruit using their incisors to grip the skin and the action of the arm and hand to pull the fruit away. Whatever method is used to take material into the front of the mouth, it is then transported to the post-canines for processing. The incisors of anthropoid primates vary greatly in their size, shape and occlusal relationships. While incisors are usually used for biting, some methods of ingestion of food depend on using these teeth in other ways. For example, there is ‘stripping’ in which a branch with leaves is taken into the mouth, the head moved back and the leaves scraped off by the incisors. Chimpanzees are reputed to sharpen, with their teeth, sticks to insert into termite mounds; it seems to be the only case of an animal modifying a tool to such an extent. Alaskan sea otters float on their backs and crack, with stones, the big sea urchins they have brought up from the bottom. The bones of sea otters are dyed purple by a pigment originating in the sea urchins (Bertram, 1992).
13.3.1 Mechanical properties of teeth A mammalian tooth is composed mainly of dentine surrounding sensitive pulp, with cementum forming the outer surface of the root below the gum line; the working surface (the crown) is covered with enamel about 1–2 mm thick (in mammals generally the thickness ranges from about 0.05 to 5.0 mm). Contact mechanics demonstrates that soft foods tend to smother the tooth surface and redistribute tensile stresses in teeth to the margins, suppressing radial fractures that otherwise form below the contact region. This led Lucas et al. (2008) to suggest that a function of the cingulum (the ridge surrounding the crown, from the Latin for girdle) is to protect the neck of the tooth from damage sustained in chewing of soft foods. Freeman and Lemen (2006) gave the strength of teeth in terms of an ultimate tensile strength (UTS) without regard to fracture properties. Automated nanoindentation has been employed to map out variations in Young’s modulus E and hardness H within tooth sections (Cuy et al., 2002). Microindentation has been used to measure the resistance to crack propagation (in terms of the critical stress intensity factor KC) from the size of cracks emanating from the impression corners (Xu et al., 1998). KC is a parameter of linear elastic fracture mechanics (LEFM), and is related to the specific work of fracture or fracture toughness R via R KC2/E. The ‘strength’ S of teeth (i.e. the stress to cause fracture according to the relevant fracture mechanics formula) may be estimated using the observation that, in enamel, starter cracks or flaws are most likely associated with lamellae, 100 m in size, in the rodlike microstructure. Typical values of E, H, S and KC are given in Table 13-1. The functional requirements of a well-adapted enamel cap are considered by Lucas et al. (2008). They show that life-limiting damage in enamel initiates and propagates not only from the occlusal contact surface, but also from the lower interface between enamel and dentine. Thick enamel is favoured to extend the life of teeth in mammals that feed on large, hard objects (nuts, seeds) by providing resistance to fracture from radial cracks that form at the enamel–dentine junction. In mammals that feed on small, hard objects, or whose teeth encounter large amounts of grit or phytoliths (plants having grit on them), thick enamel protects the teeth against excessive wear at the cap surface. In the case of those mammalian herbivores (buffalo, etc.) that use transverse jaw movements to break down thin sheets of
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The Science and Engineering of Cutting Table 13-1 Representative properties of materials.
Material
Modulus E
Hardness H
Strength S (ER/a)
Crack resistance KC
(GPa)
(GPa)
(MPa)
(MPam)
Human enamel
90
3.5
30
0.9
Human dentine
20
0.6
–
3.1
Source: Lucas et al. (2008).
vegetation, thick occlusal dentine does not confer direct benefit. However, in these cases, differential wear between the relatively soft dentine and the more resistant enamel exposes a series of folded enamel ridges separated by dentine basins on the occlusal surfaces of molars, that are vital to the animals’ feeding performance (see Figure 13-3, later). The topics of fracture, ageing and disease in both bone and teeth are being increasingly analysed in biomechanical ways, linking the microstructure and fracture mechanics (Ritchie & Nalla, 2006). While ceramics are often used in dental restoration owing to appearance and biocompatibility, they are more brittle than metals and problems arise when they are employed in multiunit bridges (i.e. spanning three teeth, say). Li et al. (2006) model the accumulation of damage and subsequent cracking in tooth bridges. There is good agreement between predictions, experimental and clinical observations. Siegel and von Fraunhofer (1997) examine the effects of load and diamond grit size when machining glass-ceramics employed in dental restoration. The same elastic contact mechanics as employed for teeth can be applied to foods such as fruits or seeds protected by a hard shell designed to resist fracture. Clearly, the forces required to break such shells must be lower than those to fracture enamel. Orangutans have strong jaws, and can break macadamia nuts with their teeth. Macadamias are roughly 25 mm in diameter with 2 mm thick shells. The forces required to break such shells, using flat metal platens and replicas of orangutan teeth, are some 1.7 kN. Lucas et al. (2008) estimated the load for failure by radial cracking from contact mechanics to be 1.9 kN. Such forces are considerably greater than those achievable by modern humans, which are about 700 N. They noted that continual feeding on ultrahard objects, however enticing, could well produce a proliferation of stable radial cracks within the enamel structure of the tooth and irrecoverable damage.
13.3.2 Wedge and double-wedge indentation Indentation of a solid by a single V-wedge or, better, opposed V-wedges mimics, perhaps, the action of some teeth (Figure 13-2). The response of a brittle (low ER/k2) material will be like that described in Section 3.4: there will be minimal penetration before the material cracks into pieces and, owing to the jaw being ‘load controlled’ it will be extremely difficult to stop fracture. This is well known to humans from attempts to crack nuts between the teeth, for example, and what happens is relevant to the concepts of crispness and crunchiness (Section 12.3). The carnassial teeth of hyenas crunch brittle bones into successively smaller shards in this way. In a ductile solid gripped between teeth, initial penetration will not cause fracture and is simply plastic indentation where the contact stress is a multiple of the material yield stress
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Centreline of workpiece (A)
Zone 2
Zone 1
Zone 3
Local indentation material displaced to surface No sideways movement
(B)
Deformation right through workpiece Material displaced sideways (C)
Figure 13-2 Double-wedge indentation. Early during the contact, the upper part of the workpiece will not be strongly aware that there is also contact on the bottom surface. If brittle, the specimen may fracture after only a small penetration, but if plasticity supervenes, again the top and bottom will have separate flow fields with material displaced from the indentations rising to the surface and with their being no sideways movement of the workpiece. At deeper indentations, the two flow fields eventually interact, the displaced material now moves sideways and tensile stresses are set across the ligament remaining between the two wedges. (When cutting with a single wedge, the lower surface directly below the wedge locally rises upwards at this second stage.)
(Grunzweig et al., 1953). Removal of the wedges merely leaves grooves. In these early stages of penetration, the deformation fields of indentation do not overlap (the top surface of the food does not know that the bottom is also being loaded) and there is little, if any, horizontal extension of the foodstuff perpendicular to the line of action of the wedges. At deeper penetrations, the fields interact and horizontal tension is set up, leading eventually to fracture (Kudo & Tamura, 1967). This is how plier-type cutters for wire function (Mahtab & Johnson, 1962) and how engineering nut-splitters are used to break open nuts seized on threads. Applied to teeth and foodstuffs, the depth of indentation at which separation occurs in a foodstuff would be an indicator of toughness/strength (R/k) ratio from which toughness could be determined if the strength or hardness of the food is known. This sort of thing is familiar in biting a chocolate bar, where tooth marks give the critical depth at which separation occurs. Differences in critical depth between turgid and flaccid carrot, for example, demonstrate differences between a hard, brittle response and a soft, pliant response. Penetrations are deeper in tough foods before separation, and in chewy foods no separation occurs as the compressive
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forces required are so high that food is squeezed out of the occlusal contact instead. Strictly, if the teeth actually touch when the mouth closes, separation (at least in the form of a hole), ‘must’ happen – but at what force? What is the limit to muscle power in closing the jaw? Kudo and Tamura also investigated the effects of sideways tension and compression on the depth at which cracking takes place in double-wedge indentation of ductile solids. A whole new variety of crack types was observed (tension originating from the tips of the wedges, or tension originating at the centre of the workpiece; combined shear and tension, again originating either at the top of the tool or in the middle of the workpiece). Such differences should be relevant to pulling and biting at the same time. The two approaching wedges are parallel in Figure 13-2, but they could be oblique and inclined to one another, and this seems to occur in occlusal contacts between ridged teeth, as shown in Figure 13-3 for the hartebeest (Sanson, 2008).
13.3.3 Teeth in action Unlike cutting tools in engineering, teeth do not have one or two simple cutting edges. The posterior teeth, in particular, have various bumps (cusps, crowns) and hollows (basins); various crests and ridges; as well as ridge-like marks at a finer scale on the tooth surface. This leads to different subgroups of teeth, such as selenodont (having crescent-shaped ridges on crowns) and lophodont (having transverse ridges between cusps). Inspection of teeth suggests that mouth action can produce a variety of different types of deformation at different rates; its mechanical action is almost certainly under continuous feedback control. Inspection of human dentition shows that our teeth evolved primarily to break structures under compression, quite different from the dental requirements of a raw flesh eater (Lumsden & Osborn, 1977). As discussed in Chapter 12, cooking muscle tissue not only sterilizes our food, but also renders its material properties more appropriate to comminution in the human mouth. Food is ‘loose’ in the mouth and mastication relies on compressive events. It is interesting to contemplate what would happen if the food were attached to the teeth (as sometimes happens with chewy foodstuffs like toffee) when the jaws were closed and the mouth then opened. Miniature tensile tests would occur. Given enough force, and sufficient attachment to the teeth, brittle foods should break with little displacement. For ductile materials, and especially those with J-shaped - curves, the available displacement in the mouth, for a given gape, may be too small to produce separation. Hartebeest
Figure 13-3 Contact points where ridges cross in hartebeest (after Sanson, 2008).
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Lucas and co-workers (e.g. 2004) discuss whether it is force (stress) or displacement (strain) that is controlling in ingestion and mastication. Stress limits appear to apply to biting by the incisors and displacement limits for mastication by the molars. Lucas et al. (2000) point out that plants and animals can employ these strategies as a defence to avoid consumption. Foods that avoid displacement-limited separation tend to have high toughness R but low modulus E. Those that avoid force-limited separation have low R but high E and high y (that is high hardness). Should fracture occur within the linear elastic range, the controlling parameter is (R/E) for displacement limitation and (ER) for force limitation. If the food properties that govern the fracture response differ between incision and mastication, then this would go a long way towards explaining why the shape of anterior and posterior teeth is different in mammals. Foods with both high R and high E are the most difficult to ingest; those with high R and low E are difficult to chew. Fracture alone (in the sense of separating into smaller pieces as achieved by incisors) may not be an efficient way to make foods in the form of sheets (leaves) digestible. Crushing may be a better deformation process to open up leaves. Crushing, like grinding, can be achieved by simultaneous pressing and shearing. Pandas, being bears, do not have any lateral movement so there is no transverse shear as in most herbivores. Sanson (2008) points out that crushing flat leaves generates very little or no damage, but he observed that pandas roll up bamboo leaves into a cigar-like tightly rolled bundle that they then crush with their teeth, causing more damage. Too few leaves in the roll results in some damage but not enough bulk is processed, but if the number is too large, the leaves in the middle of the bunch are not damaged. Thirty leaves seemed to be optimum. It would appear that crushing without rolling does not do the job and that shear in the mouth or out of the mouth is very important. Hiiemae and Crompton (1985) (see also Lucas, 2004) notes that fractures are generated by teeth either (i) in the form of blades, which are very narrow in one dimension and are usually, but not exclusively, arranged in opposing pairs (Figure 13-4B); or (ii) by pestle and mortar combinations, where the pestle is a blunt convex surface and the mortar a usually larger concave surface (Figure 13-4A). A third tooth pattern consists of low-profile blades that act crossways in shear (Figure 13-4C). Foods that are hard and brittle (nuts) or turgid (fruit pulp) are most effectively crushed by a mortar and pestle system (Figure 13-4A). Turgid food requires that the cell walls fail, but no further comminution is needed; a mortar and pestle can crush many cells with a single stroke, blades only a few. Foods such as muscle and skin that are soft but tough are most efficiently cut by blades (Figure 13-4B). The simplest blades are found in the carnassials of various carnivores. Tough, fibrous foods such as grass are best processed by the combination of compression and sideways motion (Figure 13-4C). According to Lucas, insects as foods for other creatures can be considered as ‘fluid-filled sealed tubes’, the limiting factor in their reduction being the nature of the tube. A soft-skinned larva readily bursts in a mortar and pestle, but an adult with its chitinous exoskeleton is more efficiently opened by a blade. Materials such as these can be most effectively reduced by a dentition with a combination of both elements. To investigate the biting action of human incisors, Agrawal and Lucas (2003) performed experiments in which subjects were asked to bite into food (cheese, raw carrot, and kernels of Brazil and macadamia nut) until the food just gave way. These materials are all quasilinear in their behaviour before fracture and so it is appropriate to employ LEFM for analysis of the results. The different depths at which the food samples began to split were recorded along with the contact stress using pressure-sensitive films. Independent determinations of the Young’s modulus E and the fracture toughness R were made. It was found that a KC (ER), as would be expected from LEFM.
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(A)
(B)
(C)
Figure 13-4 Mechanical principles of tooth design in relation to the nature of the food: (A) hard, brittle or turgid; (B) soft and tough; and (C) tough and fibrous (after Hiiemae and Crompton, 1985).
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(A)
(B)
(C)
(D)
319
Figure 13-5 Schematic sections and three-dimensional reconstructions of opposing crescentic enamel ridges about to shear strands of food lying in basins bounded by ridges. (A) is a section through (B). (C) is a section through (D) when shearing is nearly complete. Straight arrows indicate relative direction of movement of the teeth. Curved arrows in (B) illustrate potential escape paths of food being increasingly compressed in (A) (after Sanson, 2006).
Sanson (2006) points out that the geometry of teeth having folded enamel ridges (Figure 13-3) moving into contact produce cutting edges that have motions along, across and perpendicular to the line of action of the bite. Figure 13-5 shows how the action of such teeth shears strands of food lying in the basins bounded by ridges. ‘Food escape’ between steeply inclined teeth is possible when the resultant force between the tooth surfaces wants to push the food away from the contact region, rather like a wedge popping out from a contact. Potential escape paths of food being increasingly compressed in A are indicated by the curved arrows in B of Figure 13-5. The relative velocities between approaching molars, resulting from their geometry and jaw motion, will produce localized strain rates that, in rate-dependent foods, may affect chewing performance. A system of enamel ridges and dentine troughs is formed and maintained by the different rates of wear of these materials. Such teeth are typical of ungulates (animals with hoofs), subungulates and rodents. Although a newly erupted tooth may have simple blade structure along the slopes of its cusps, in many mammals the enamel cover is rapidly worn away, leaving two enamel edges with a softer dentine surface between. This is the most efficient condition for a parallel array of low-relief blades that are used to reduce grasses by cutting on a transverse rather than vertical stroke. It would be interesting to interpret these observations in terms of slice–push (Chapter 5).
13.3.4 The ideal tooth Although teeth are not as simple as most engineering cutting tools, various workers have drawn parallels between them (points/cusps that act locally like conical tools; straight or curved blades/ crests that form cutting edges) as mentioned in the previous section. Links have been made between tooth geometry and rake angles, clearances, obliquity (approach angle), etc., and the effects of such parameters on the forces required to separate materials. Of the different types of
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Figure 13-6 Comparisons between Evans and Sanson’s ‘model ideal teeth’ and real mammalian tooth forms. Model tools shown left-to-right at top: (A) single-bladed tools, symmetrical and asymmetrical; (B) double-bladed tools. Mammalian tooth forms shown below: (C) lower carnassials of Felis catus (Carnivora: Felidae) and Mustela frenata (Carnivora: Mustelidae); (D) premolar of F. catus; (E) lower molars of Tenrec ecaudatus (Insectivora: Tenrecidae), Didelphis virginiana (Didelphimorphia: Didelphidae) and Chalinolobus gouldii (Chiroptera: Vespertilionidae), and upper molar of Desmana moschata (Insectivora: Talpidae). Scale bars 1 mm (after Evans & Sanson, 2003).
cutting action, the relative motions of teeth produce more of a cropping or guillotining action through the thickness of food, rather than continuous removal of layers from the surface of food. An important function of teeth is to direct food away from the teeth after fracturing, so as not to trap it and thus slow up chewing. Good tooth design has channels and exit structures to facilitate that sort of thing. Evans and Sanson (2003) investigated ideal tooth shapes using virtual reality computer modelling. The shapes of real teeth were disregarded in the modelling, but Figure 13-6 shows the remarkable correspondence between predictions of ideal tooth shapes and real mammalian tooth forms.
13.4 Sharpness and Wear of Teeth What about tooth sharpness, defined in terms of the radius of curvature of a cutting edge of a tooth? Does it scale with creature size? If teeth were geometrically similar (isometric allometry (Section 13.6), sharpness in terms of tooth radii would be greatest in the smallest teeth. Does it vary during life owing either to developmental changes and/or to wear and, if so, what is
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‘representative’? Teeth cannot be really sharp in the razor-blade sense or the mouth would bleed continually, particularly owing to slice-push between teeth. Biting the flesh of one’s cheek, or the lips, can be very painful, but sharpness in teeth may be as much to do with included angles of crests, ridges and other protuberances as with cusp or edge radii. Evans and Sanson (1998) demonstrated the effect of ‘cusp sharpness’ (semi-angle of a conical tooth) on foodstuffs. Lower forces were required with slender teeth, as expected, but does the food come apart having been pierced or stay together with a puncture hole? Whether separation across the direction of piercing takes place probably depends on the size of the piece of food; larger pieces may not be separated. Furthermore, if the object is holed but does not come apart, does the tooth remain stuck in the food by friction when the jaw is opened? Dogs certainly seem to bite on sticks that they have to pull off their teeth by using a paw to hold the stick down. If resistance to separating food is thought of in terms of a critical crack opening displacement cr (where fracture toughness R ycr), there seems little benefit in having teeth sharper than the cr of the food (Section 9.8), and many foods are very tough with very large cr. In fact, pieces of brittle foodstuff in the mouth (having small cr) are broken by secondary tensile stresses arising from compression, rather than from ‘spiking’ (cf. Section 13.3.2). The teeth of some creatures never stop growing, and this can result in a useful self-sharpening action. When rabbits crop grass and beavers gnaw trees, the soft dentine layers are removed more quickly than the hard brittle enamel, thus resulting in self-sharpening; it is, perhaps, similar to the behaviour of Damascus steel and tools made from bilayers (Chapter 9). Sea urchins have five self-sharpening teeth with which they detach the ‘roots’ of under-sea kelp forests attached to rocks off California. Data on edge sharpness compiled in Evans et al. (2005) suggest that at small body sizes (1 kg) sharpness is more closely correlated with body size since geometric properties that alter with developmental processes, such as enamel thickness, may have a greater influence on the final tooth sharpness giving a smaller risk of tooth fracture. For larger animals (1–2500 kg), the degree of attrition appears to be the main factor in determining sharpness: for animals in which high attrition of teeth is characteristic, tooth sharpness is close to constant, and when there is lower attrition, tooth sharpness is lower for the same body size. Wear therefore seems to be the dominant determinant of sharpness within size classes. In the literature, silica in grass and other foods is often stated as the cause of tooth wear. Sanson (2008) points out that it is well known that silica exists in a number of different states of different hardness from water glass, to silica gel, to opal, through to classic quartz, of which only quartz is harder than teeth. It is true that plant silica can be oxidized to ‘quartz’ but it probably takes more than one season, and animals do not eat years-old decayed grass. Nevertheless, such oxidized silica becomes dust and may coat surrounding grass that is eaten. It is worth noting that grass silica is harder than insect mandibles. Wear that blunts cutting edges may result in tooth radii that may be more scale independent. However, the additional constraint of risk of tooth fracture in larger animals will prevent tooth sharpness from being too high. From this, we expect that maintaining high tooth sharpness is more important in small animals than large ones. Tooth sharpness may then be relatively constant in large animals. This gives larger animals relatively sharper teeth than small animals. It is not known which of developmental or wear processes could produce the sharper crest. Evans and Sanson (1998) quantified the relationship between penetration force and tooth sharpness in a series of experiments in which insects were punctured by artificial teeth, from which they concluded that both the radius of the tip of the tooth and tooth size were important. Freeman and Lemen (2007) extended this sort of study employing fresh pig and deer hide. They were interested in the evolutionary trade-off between the shape of canine teeth best able
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to penetrate prey and the shape least likely to break. They employed axisymmetric teeth generated by power-law curves with a conical tooth of the same length as a reference, and found that the maximum force required to push a tooth through the hide to the tooth’s base was greater for sharp cones than for blunter teeth. The fat and muscle beneath the hide is much softer than the skin and offers little resistance to penetration compared with the force for expanding the hole in the hide. Similar findings have been reported by workers investigating the stabbing action of knives (Section 11.4.3). The sharpness of teeth is important for vampire bats, which make a small incision in their prey and then lap up the blood with their tongue (they do not really suck). All leeches are carnivorous but only a few suck blood. The European medicinal leech (Hirudo medicinalis) has three muscular jaws, each of which has a row of tiny teeth. These pierce the skin severing the capillaries beneath and then suck up blood. They leave a Y-shaped wound like a Mercedes car badge. Leech saliva contains the anaesthetic anticoagulant birudin so the flow does not clot during sucking or in the gut. It is advised that if you get a leech attached inside the mouth, you should gargle with vodka (Mitchinson & Lloyd, 2007). Arthropods (insects, spiders and crustaceans) are segmented invertebrates having jointed legs and an exoskeleton (shell or carapace). In order to grow they have to moult. Thus, for example, when a lobster sheds its shell it has to pull out the lining of its throat, stomach and anus to free itself, and its mouthparts are replaced, sharp and unworn, every time. Higher animals typically rely on calcification to harden bones and teeth (scleroty). But in a wide variety of arthropods and members of other groups, elevated concentrations of zinc and manganese in their fangs, teeth, jaws, leg claws and other ‘tools’ achieve the same result (Hillerton, 1980; Hillerton & Vincent, 1982; van der Wal et al., 1989; Schofield & Nesson, 2002). Shrews have iron deposits on or in teeth in high wear locations (Bonser, 2008). Considerable quantities of zinc also occur in the jaws of the marine polychaete worm Nereis (Lichtenegger & Schöberl, 2003). Miserez and Li (2007) find inspiration in the hard tissue comprising the beak of the jumbo squid (Dosidicus gigas). Its main constituents are chitin fibres (15–20 wt per cent) and histidineand glycine-rich proteins (40–45 per cent). Notably absent are mineral phases, metals and halogens. Yet despite being fully organic, beak hardness and stiffness are at least twice those of the most competitive synthetic organic materials (engineering polymers) and comparable to those of Glycera and Nereis jaws. Nevertheless, hardness alone does not guarantee wear resistance: toughness is also very important, as discussed in Chapter 6.
13.5 Attack and Defence When deer rut, they lock antler racks and the loser is driven away from the other’s harem of hinds. It is a trial of strength in which injury or death of an opponent rarely happens. However, a skull has been found that had been pierced by an antler tine (L. Kruuk, 2007). In other animals, the same teeth and claws are used to kill creatures for food, and then to devour them. Are the requirements for attacking prey, and eating prey, different in the two cases? It might be thought that carnivores having pronounced incisors kill their prey by puncture wounds. But this is not so and large carnivores like lions, leopards and cheetahs either weaken the animal until it bleeds to death (or shock kills it), or kill large prey by strangulation, crushing the windpipe and blocking the airflow, having attacked the throat of the victim and gripped on to the body. Even a lion’s canines are only 50 mm or so long and cannot penetrate far enough into a prey to cause vital quick damage. The risk to the predator of slow death of the prey is that the predator might get kicked. The attacker must be able to hold the strangulation bite long enough to kill, which can be many minutes. Small cats, in contrast, kill
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by biting at the base of a prey’s skull. In the attack, when a cat extends its paws, it can move its digits independently and form particular grips before pouncing on the prey; dogs cannot. It so happens that cheetahs cannot form their claws, which therefore have to be permanently extended like running-shoe spikes. David Livingstone (1857) relates an encounter with a lion during his early days of exploring in Africa. The village in which he had set up a missionary station was troubled by a group of lions that leapt into the cattle pens at night and destroyed its cows. It was understood that if one of a troop of such marauding lions was killed, the others would depart the area, so Livingstone and the villagers went after them. Livingstone shot one of the lions, but it did not die immediately, attacked the explorer and bit him in the shoulder. ‘Growling horribly close to my ear, he shook me as a terrier does a rat. The shock produced a stupor … that caused a sort of dreaminess in which there was no sense of pain, nor feeling of terror, though quite conscious of all that was happening. It was like what patients partially under the influence of chloroform describe, who see all the operation, but do not feel the knife. The shake annihilated fear, and allowed no sense of horror in looking round at the beast. This peculiar state is probably produced in all animals killed by the carnivora …’. Livingstone escaped and recovered. The extinct sabre-toothed lions and tigers had long upper canines sometimes extending below the margin of the lower jaw, with reinforcement of the skull to hold them. It is doubtful whether such teeth could accommodate lateral forces, and such extinct creatures are thought to have been solitary predators of young mammoths. To eat prey, many large predators enter the body by the softer belly. Lions often burrow into the prey avoiding as much skin as possible; the fur holds a lot of grit that would wear their teeth, Sanson (2008). Hyenas attack zebras, wildebeest and antelope with their canines, and tear into the belly and legs of prey while it is still alive but exhausted. They devour everything: concentrated hydrochloric acid in the gut enables skin and bones to be digested. Bones after crunching form sharp shards, yet these are swallowed apparently with no ill effect, and no piercing of the wall of the gut. Does this mean that tissue of hyena and similar gut is particularly ‘stab resistant’? How extensible is gut? J-shaped stress-strain curve arguments rely on large displacements during which the material ‘escapes away’. Are such large movements possible within the body of such an animal? There is one reported case of a dead otter found with a gut pierced by fish bones (H. Kruuk, 2007).
13.6 Scaling An interesting question is whether the masticatory system (or parts of it such as teeth or the jaw) scale in proportion to body size (volume or weight are more or less interchangeable parameters to use, as the density of animals is fairly constant). This leads on to considerations of the size of foodstuffs ingested by animals of different size and how the properties of different foods may change with size. The size of creatures varies from insect to elephant. The sizes of plants are relatively fixed. Plants in general are poor nutrient resources for animals, being relatively low in nitrogen and readily accessible energy. Owing to the need to extract from food enough energy consummate with an animal’s size and metabolic rate, large mammals are bulk feeders ingesting whole leaves many at a time, while the smallest herbivorous mammals select plant tissue relatively rich in cell contents. The smallest leaf feeders live within a leaf and eat individual cells or
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parts of cells. Parasol or leaf-cutting ants cut green leaves into pieces and take them to the nest, not to eat as they cannot digest cellulose, but to make up into compost, and then eat the fungus that grows on the compost. It is common practice in engineering to test small samples to establish mechanical properties that are then employed for the design of much larger bodies of the same material. However, owing to the cube-square scaling effect in fracture mechanics (explained in Chapters 2 and 4) fracture stresses in large bodies can be much smaller than fracture stresses in laboratory-size testpieces, i.e. ‘ductile’ materials can behave in a ‘brittle’ fashion in large enough sizes, and vice versa. Hence structural steels will yield before they break in the laboratory, but may fracture in the elastic brittle range in a large structure. In the laboratory testing of engineering materials, it is important in analysis of experiments to remove the work of remote plastic flow from the total work when testing ductile solids, in order to obtain the size-independent fracture toughness, since it is this mechanical property that will determine fracture stresses in large structures. Toughness derived from the total work in a laboratory test overestimates the basic toughness because of the remote flow, and gives a false sense of the material’s resistance to cracking. What does this mean for foods that have a limited size range but may be eaten by differentsize animals? Alternatively, a range of sizes of foodstuffs may be eaten by one-size animal. Behaviour and scaling must depend on the mode of deformation leading to fracture. Scaling in biology is called allometry (from measurement of growth) (Schmidt-Nielsen, 1984). What engineers call geometrically similar scaling (where different-size objects are all different magnifications of one object) is called isometric scaling in biology. Cutting with a sharp blade in the laboratory is controlled by the basic toughness of a material, but cutting with a blunt blade will require more work and will give larger toughnesses owing to the more diffuse deformation pattern. The same is true for simple tearing of ductile materials in tension where a burr is formed along the torn edges. It might reasonably be argued that if it is not possible to fracture an item of food without such remote irreversibilities, one might as well think in terms of an ‘effective toughness’ that includes both the fundamental toughness and the normalized remote work. This concept arose in connexion with tearing and guillotining of ductile material in Chapter 5. There is an argument that the need to know the fundamental toughness occurs only when the mechanical behaviour of a body is going to be predicted from information on an entirely different size body of the same material (the properties of big leaves from little leaves, say). If the same size leaf or whatever is being consumed, why bother? However, this line of argument presumes that the same deformation pattern occurs on the one hand (i) with a fixed size of food whatever the size of the tooth; and on the other hand (ii) with a fixed size of tooth on different sizes of foods. Neither of these assumptions may be true. We need to remember too that the mathematics of scaling presumes that materials behave as uniform continua, and that microstructure plays no role other than setting an overall value for the property of interest. The effort produced by muscles seems to vary as their cross-sectional area, so bite force would be expected to scale as 2, where is the scaling (magnification) factor. If tooth contact area also scales with 2, this suggests that the average contact stress given by load/area is independent of size. But what is required by the food to result in separation, and what must be avoided to prevent tooth fracture? The answer to both of these questions involves non-proportional scaling if the microstructural features that act as starter cracks do not change with size (in enamel, for example, the lamellae, 100 m in size, in the rod-like microstructure; Section 13.3.1). Non-proportional
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scaling with four independent scaling factors (for height h, width W, thickness B and crack length a) is discussed in Atkins (1995b), where it is shown that
FL / FS λ Bλ W / λ a
δ L / δS λ a3 / 2 / λ W
σL / σS 1/ λ a
(13-1)
where L is large, S is small, and where F is load, is displacement, and is stress. When B W h but a 1, we have
FL / FS λ 2
δ L / δS 1/ λ
σL / σS 1
(13-2)
This says that the stress required to fracture a large tooth is the same as that for a small. Is the size of a piece of food taken into the mouth of a larger animal the same as for a small animal or does the amount depend on the gape, meaning in the latter case that the size of the item increases as ? For brittle foodstuffs with size-invariant flaws that follow LEFM, the same conclusions as for the tooth apply, i.e. the stress required to fracture a large piece of brittle food is the same as that for a small. For ductile foodstuffs the question is more difficult to answer, particularly because tooth geometries and relative motions are so complicated. If the cutting action is by vertical ductile cropping or guillotining, the force required to start cracking is given by F w(tk R), where w is the length of blade contact, t is thickness of the food, k is its shear yield strength and R is its toughness; the relation may be deduced from Section 3.8. Because in cropping or guillotining a shear band is set up through the whole thickness, the toughness itself is curiously not constant, rather (R/kt) is a constant C, say, as explained in Section 5.4.3. Whence F wkt(1 C). Insofar as the mechanics of shearing of ductile metals may be applied to the large deformation shearing of foods, FL/FS 2. In the related discussion of punching in Section 8.6.3, the total energy for separation given by Fd scales as (Dt2), where D is the diameter of the punch and t the target thickness. For geometrically similar bodies, it follows that the work required scales as 3. The work done by a tooth is the product of average force during the bite times the tooth displacement. So, if the energy scales as 3, and the forces as 2 (above), it means that the average tooth displacement ratio (L/S) ought to depend on 3/2 , suggesting a proportionately larger vertical shear displacement in larger mouths when larger pieces of the same food are eaten. These arguments are much oversimplified and much remains to be done in this field.
Chapter 14
Burrowing in Soils, Digging and Ploughing Contents 14.1 14.2 14.3 14.4 14.5
Properties of Soils Roots Earths, Sands and Burrows: Hole-making and Scraping by Animals Earthmoving and Ploughing Picks and Crampons
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14.1 Properties of Soils Soils are assemblies of particles that are stuck together to a greater or lesser extent. Soil is a ‘natural’ material. The particles are ground-down rock (stones are also present) and humus (dark organic matter produced by the decomposition of vegetable and animal matter and essential to the fertility of the earth). Soils also contain air and water. The water may be chemically bound to the soil particles or may be free within the network of particles (this is similar to how water is found in the microstructure of biological materials). The amount of water depends on the weather and changes with the climate. In a field, properties can vary with depth: wet on top and dry below, or vice versa. Owing to the way they were formed, soils are not homogeneous and are found in stratified layers as revealed by deep boring; it is rare to find isotropic deposits. Soil scientists talk about different peds, a ped being a structural unit in undisturbed soils (from the Greek pedon, meaning earth). The properties of soils vary markedly from loose particulate behaviour (dry sand) to hard, stiff paste-like behaviour (clays). In the first case the behaviour is that of a free-flowing medium having interparticle cohesion; in the second it is more akin to plasticity. The same soil under different conditions of water content can behave very differently, as illustrated by potter’s clay (a slurry) at one end of the spectrum, and dried mud at the other. The whole range of behaviour from ductile to brittle is found among soils. Cottrell (1964; Sections 7.6 and 10.4) gives an excellent introduction to the behaviour of viscous solutions, suspensions, pastes and soils. Among the materials cut by Mallock (1881) was clay. He remarked that clay ‘was found extremely useful in examining the formation of shavings, for by altering the amount of water it contained, its behaviour under the tool could be made to resemble almost any of the others, and at the same time the forces required to take large cuts were not greater than could be conveniently applied by hand’. ‘The others’ were wrought iron, steel, cast iron, gun metal, brass, copper, lead, zinc, hard paraffin (wax), and soap. The usual ideas of stress and strain, elasticity and plasticity, and so on are employed with soils but since geomechanics developed independently of solid mechanics, there are differences in nomenclature. For example, what is called the coefficient of compression in soils is Copyright © 2009 Elsevier Ltd. All rights reserved.
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what others would call the bulk modulus, a rebound modulus in soils is the modulus obtained on unloading a sample, and so on. One difference between soils and many other materials is that the self-weight of a mass of soil is often comparable to the forces required to deform it. Another major difference between soils and most other materials is that yielding is pressure dependent because of the porosity. (The yielding of ice, snow and some polymers is also pressure dependent and employs similar yield criteria; e.g. Raghava et al., 1973.) The more densely soil particles are packed together, the harder it is for them to slide over one another. Treating the problem as one of increased frictional resistance within a plane of sliding caused by a pressure p acting across the plane, the shear resistance contains a term of the type p or ptan, where is the coefficient, and the angle, of ‘internal friction’ between the particles (not to be confused with the ‘internal friction’ of polymers and creepy metals determined with torsional pendulums, or with friction in cutting; Appendix 2). Thus, the yield criterion becomes
τ c p tan β c µp
(14-1)
where c is the cohesion between the grains, i.e. the shear that the soil can sustain at zero normal stress. In soil mechanics, compressive p is usually taken as positive. Since soils lack tensile strength, the criterion has no physical meaning when p is a tension. Also, we use in (14-1) for the friction angle to conform with the rest of this book; in most soil literature φ is used instead of , but we want to avoid confusion with the use of φ for the inclination of the primary shear plane in cutting; in geomechanics, is often employed for friction. Two extreme cases are (i) soft wet clay, where there is good cohesion but 0, in which case the formula reduces to Tresca’s yield criterion c k used for metals; and (ii) cohesionless powders such as dry sand, for which c 0 so ptan. When c 0, is equal to the angle of repose, the steepest angle a sand-hill or earth-bank can sustain without slipping; for dry sand 30°. It is rare to find cohesionless soils in the natural state with 30°. When c 0, a ‘vertical cliff’ can be sustained, at least temporarily which, as explained by Cottrell (1964), is important for the modelling properties of clay. (Try piling up dry and damp flour, sugar or salt.) When c 0, as in waterlogged soil with grains supported by the buoyancy of the water, the material is then a fluid. [The water table is the depth below which the ground is saturated with water, so Eq. (14-1) applies only to soils above the water table.] When loaded in simple compression, cylinders of soils having cohesion and internal friction yield, and subsequently separate, by sliding on a plane, often called a slip plane or line, at an angle to the horizontal (to the line of action of the smaller compression). Resolution of forces shows that the axial stress ax produces a normal stress on this plane given by axcos2, and a shear stress given by axcossin. The yield criterion (14-1) becomes
σax cos θ sin θ c σax tan β cos2 θ
(14-2)
Yield will occur first on that plane for which (cossin tancos2) is maximized, i.e.
tan β tan 2θ 1; or θ (π/ 4) (β/ 2)
(14-3)
For 0, 45° as for pressure-independent solids. For 45° ( 1), 67.5°, so that the plane on which shear occurs in such a pressure-dependent material is not at 45 ° to the principal normal stresses. To determine the unknown c and of a soil, the testpiece is enclosed in a waterproof membrane that can be externally pressurized by water, and experiments are repeated at various
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hydrostatic pressures (the triaxial compression test). The compressive yield stress ax now increases as the lateral pressure p increases. A Mohr’s circle diagram is constructed for every test, marking off and p along the axis of normal stress, and semicircles are drawn through every pair of points so obtained. The tangential line to these semicircles represents the yield criterion; it has a slope and an intercept c on the axis of shear stress. In terms of the stresses ax and p actually measured, the yield criterion is
σax 2c tan θ p tan2 θ
(14-4)
Because of the friction-like term in the yield criterion, two equilibrium states are possible in a given situation depending upon the direction of loading. In a vertical-sided trench, for example, the sideways pressure to stop soil sliding down a slip plane into the trench is smaller than the self-weight of the soil (the active Rankine state of plastic equilibrium); the sideways pressure required to cause the soil to slip upwards is greater than gravity loading (Rankine’s passive state). The inclination of the slip plane in the former case is with respect to the horizontal but in the latter is with respect to the vertical (Jaeger, 1962). The two states are named after Rankine who, in 1886, first studied the design of retaining walls. The shear box test of soils is rather like cropping (Section 3.8), but with superimposed stress across the shear plane (Atkins, 1980b). For a given normal stress, if yielding occurs at constant shear yield stress k, the shear load drops from the outset; if the soil initially workhardens, the load increases at first and passes through a peak caused by a plastic instability. (This is the case for continual workhardening and is not to do with worksoftening.) There seem to be few, if any, reports of what is seen on the slipped surfaces from shear box tests. Perhaps smearing occurs, but is there a characteristic slip distance with the same meaning as cr? It would be interesting to perform such tests with starter cracks in the plane of shear and ‘clearance’ (packing between the two halves of the box) and use ductile fracture mechanics for analysis. Cropping analysis (Atkins, 1980b) predicts that the displacement peak along the shear plane at peak load is given by nL/(1 n), where L is the length of the box and n is the workhardening index in k kon, where the shear strain (/h) with h the thickness of the shear plane. Thus longer shear boxes are expected to experience increased displacement but, it turns out, the same peak stress. Greater displacements are predicted for stress-softening materials in longer shear boxes (Palmer et al., 2003), but lower peak strengths as observed by Garnier (2002). There are interesting questions too regarding the scaling of such measurements when the shear plane extends from one end of the apparatus to the other right from the outset of testing. The same considerations apply to failure by slip circles, and when complete slip-off and separation occurs, the conditions must involve the fracture toughness of the soil (Palmer & Rice, 1973). The water content of foundations affects the mechanical properties and it is possible for slip planes to be preferentially weakened by water. The book by Jefferies and Been (2006) shows how dangerous liquefaction of soils can be. Relation (14-1) in its various forms has been employed above to describe the effect of hydrostatic pressure on the shear stress for yielding in materials deforming in a ductile manner. Confusingly, a similar relation exists between hydrostatic pressure and the shear stress for fracture in solids behaving in a brittle manner. Many brittle materials fail in tension on planes that are approximately perpendicular to the greatest tensile stress: in bending, a stick of blackboard chalk breaks across its length, but when twisted, in a helix since in torsion the planes of greatest tension are at 45° to the axis of the piece of chalk. Similar behaviour is found with carrots. In compression, brittle materials like rocks fail on shear planes, as exemplified by geological faulting. It is well known that a compressive stress system can inhibit fracture since
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any pre-existing cracks are squeezed up, hence the ability to perform microhardness tests in glass. The compressive strength of rocks increases when subject to hydrostatic pressure; Karman was first to show that at sufficiently great confining pressure marble and sandstone begin to exhibit ductile behaviour (see Nadai, 1950). The way in which the shear stress at failure in brittle materials increases as the hydrostatic pressure increases obeys the same sort of relationship given by Eq. (14-1) (Jaeger, 1962), noting that with rocks, unlike soils, tensile stresses are usually considered positive. Like shear flow planes in soils, the fracture shear planes are not at 45 ° to the principal stresses. When Eq. (14-1) is employed for shear fractures in rock, the failure envelope can extend into the tensile hydrostatic stress region. Application of criteria of the same algebraic form both to flow in shear, and to fracture in shear, has perhaps led to some confusion. It is down to using the word ‘failure’ in a general sense to include both fracture and flow, and comes from so-called ‘theories of failure’ found in Strength of Materials texts that give criteria for fracture or yield under multiaxial stressing that may be used in stress-based design (e.g. Timoshenko, Part II, 1959; for a discussion see Polakoswki & Ripling, 1966). The pressure-dependent criterion given by (14-1) is variously known by the names of Coulomb, Navier and Mohr (the latter considering general curved failure loci tangent to the stress circles, not only straight lines). Not all soils yield according to Eq. (14-1) and even the same soil may follow it under one state of wetness but not another. Because yielding is pressure dependent, the mathematics of soil plasticity is more involved than metalforming plasticity. Particular complications arise in the stress–incremental strain relations, owing to problems with the plastic potential, ‘normality’ and non-associated flow rules (e.g. Johnson et al., 1982). Design of large-scale earthworks, using mechanical properties determined on small-size laboratory specimens, raises the issue of scaling. Centrifuge modelling is a powerful tool in geotechnical engineering to help answer that question, in the same way that towing tanks are employed with model ships and wind tunnels with model aeroplanes. The idea is that the same stresses induced by gravity in the large prototype are replicated at corresponding points in a small model of the same material spun in the centrifuge, by the greater acceleration of centrifugal loading. This is achieved by making the scale factor for acceleration, given by G/g where G is the acceleration in the machine, equal to the inverse of the scaling factor for length as between model structure and prototype. That is, a 1/100-scale model is loaded by G 100 g. The method was originally applied to the solution of classical slope stability theory for materials such as clay and sand that are ductile and whose behaviour is controlled by plastic flow stresses. However, what of brittle soils whose behaviour is controlled by fracture mechanics, where the controlling parameter is a critical stress intensity factor KC whose dimensions are [stress(length)]? Palmer (1991) showed that the scaling is different and that the acceleration scaling factor should vary as (length scale factor)3/2, so that a 1/100-scale model should be loaded by 1000g acceleration. The mechanical properties of soil are complicated even more by the behaviour of the water it contains. Free water may be frozen at low temperatures and tends to make a soil brittle. When the water is liquid, a soil’s response to loading depends on whether it is squeezed out, as if from a sponge, or whether water is retained. Clearly that must depend upon the amount of water initially present, the rate of loading and the time available for passage of water. Water is not compressible but the soil structure is. When an unsaturated soil is put under a steady load, stresses are transmitted instantaneously to the grains and the soil behaves like a pseudoelastic medium. In saturated soils, load is initially transmitted to the water which is then forced out of the soil mass while the load is gradually transferred to the grains. Water drains out at a rate depending on the soil’s permeability. After some time, all the load is carried by
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the soil structure from grain to grain and pore water pressure returns to its original value. The properties of the same soil in these so-called drained and undrained states are different. Shear deformations in sand and silt usually produce a local volume increase (dilation) so that particles move apart and create increased void space in the microstructure. Water flows into these larger voids from the surrounding soil and, when the rate of loading is slow, there is plenty of time for this to occur (familiar around footprints on wet sandy beaches where the area around one’s foot dries out). At faster rates of loading, the water pressure in the dilating pores has to drop more rapidly to create sufficient pressure gradients to permit water ingress. The resulting suction in the fluid thereby presses the soil particles more firmly together, thus increasing the resistance to deformation (this is familiar as the ‘welly boot effect’ when walking in swampy ground, where one’s foot pulls out of the gumboot that is stuck in the boggy ground). In submerged saturated sand, where the voids are already filled with water, the response of the soil to loading is modified, as it is also by increased hydrostatic pressure in deep water (Palmer, 1999). Different answers are obtained in triaxial compression tests depending on whether drainage is permitted: indeed, sometimes the soil sample is pressurized within the membrane. Contraction and expansion happens not at the commencement of test but when the yield surface is reached. The Cambridge critical state model relates pressure-dependent yielding to the consequent pore volume changes (Roscoe et al., 1958). The critical state is when a soil under continuous shear does not change its volume, and the line dividing expansion and contraction in diagrams of deviatoric (shear) stress vs hydrostatic stress is the critical state line. Soils denser than critical expand upon shearing: those looser than critical, contract (Kirby et al., 1998). These differences are qualitatively like those between relaxed and unrelaxed (or dynamic) properties of polymers. Similarly, in wood cutting, free water contained in wood cells that are under high stress is transported to regions of lower stress. In slow cutting, the resistance to flow is not noticeable but at high cutting speeds the so-called Maxwell effect comes into play, whereby there is an apparent increase in rigidity of the material (Costes & Larricq, 2002). The properties of frozen wood are different again. A problem in testing soils is that the act of disturbing the ground to remove a testpiece alters the confining conditions. The in situ stress state at the site of an excavation is always disturbed, and may end up greater than, or smaller than, before. Laboratory testing of undisturbed samples taken from ground is possible but very difficult (Sanglerat, 1972). Customarily, properties are measured by direct-shear triaxial, and unconfined compression, tests. However, it may be difficult to relate laboratory results to in situ properties owing to differences in confining pressure of the overburden and altered water content. In situ testing of soils uses various types of penetrometer that are like large indentation hardness testers (often with conical ends), but are forced far deeper into the ground (to below the water table) by static or drop-weight loading, or by screwing. To reduce frictional effects on the sides of the hole the shank may be of a smaller diameter, or some devices have a central rod working within an outer casing. Soil resistance is measured by load divided by the projected area. The ‘ultimate bearing pressure’ given by the penetrometer at the working depth is used for design of foundations and so on, with some safety factor. There are various formulae for the design of foundations (footings) and retaining walls based on Terzaghi’s (1943) pioneering work with various coefficients relating to (i) the stagnant wedge of soil below the footing; (ii) overburden of soil; and (iii) to cohesion of soil. Settlements may be evaluated employing elastic theory. The behaviour of penetrometers is like the piercing problems discussed in Chapter 8. The same questions asked there at deep penetrations are apposite here: (i) if the behaviour is brittle,
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a crack ahead and around the tip permits penetration with elastic (and perhaps locally plastic) flow fields surrounding the device; but (ii) if the behaviour is supposedly ductile, with similar elastic locally plastic flow fields surrounding the device, there must be separation down the centre of the hole to permit the transition from simple indentation to piercing. So what does a soil penetrometer actually measure? Holes for piles are either drilled out and filled with concrete or the pile itself is driven in to the ground. Having reached the required depth, the loads needed to make the pile penetrate further determine the bearing capacity of the foundation. There are various slip line field solutions for how further incipient penetration may take place, Caquot (1934) being the first to apply Prandtl’s slip line field solution for punch indentation to soils. Similarly, it was Skempton et al. (1953) who first applied Bishop and colleagues’ (1945) model for the expansion of cylindrical and spherical cavities to deep penetration in soils. There is a difference between slip line fields for constant k materials and for pressure-dependent materials: the fans comprising radial lines and circular arcs for constant k materials (Johnson et al., 1982) are replaced by radial lines and logarithmic spirals (e.g. Drucker & Prager, 1952). In addition to constrained and unconstrained compression tests, beams of soil may be loaded to obtain properties, taking care to account for the effect of self-weight on the deformation (e.g. Thusyanthan et al., 2007). When the beams have starter cracks, it is possible to determine fracture toughness using linear or non-linear fracture mechanics, as appropriate, for brittle and ductile behaviour. Thus Chandler (1984) determined values of the critical Jintegral for clays in various states (JC is the same as R). He found that as the shear yield stress k ranged over 23–100 kPa, R ranged over 1–6 J/m2, with R/k ranging over 0.02 to 0.2 mm. Hallett and Newson (2005) employed the crack tip opening angle parameter (CTOA) for fracture in clays: for pure kaolinite it was 0.23 radians, but samples with silica sand mixed-in displayed smaller values at fracture, e.g. 0.12 for 40 per cent sand/60 per cent kaolinite. When soils are brittle it is appropriate to measure fracture toughness in terms of the critical stress intensity factor KC (but only then). Aluko and Chandler (2006) showed that there was a linear relation between KC and soil density for samples of sandy loam, a clay loam and a cemented sand soil. Values obtained ranged over 1.5 kPam when 1.15 Mg/m3, and 5.5 kPam when 1.5 Mg/m3. Harrison et al. (1994) describe how a ring test may be employed to determine toughness of cohesive soils. Their results show the importance of water content on the results. Although their calculation method is described in terms of the Jintegral, their results are given in terms of KC. Load–displacement diagrams (e.g. their Figures 4 and 5) show that their testpieces were not displacement reversible, so that KC is inappropriate and the values quoted (50–100 kPam) reflect the inclusion of remote work. The impact resistance of soils is discussed by Corbett et al. (1996) in connexion with bullets being fired into defence dugouts and trenches. Cutting soils with a wire is another way of determining properties. Experiments in which mixtures of different particle size -alumina pastes were cut by a plain taut wire held in a frame are described by Benbow and Bridgwater (1993), who interpreted their data in terms of a mean cutting pressure given by (F/wd). They found this was linearly dependent (but with an intercept) on a ‘rheological’ yield strength of the different mixtures, obtained from extrusion experiments. Kamyab and co-workers’ (1998) analysis for cutting by a taut wire, given in Chapter 12, predicts that the force F across the wire, normalized by the width of cut w, should plot linearly against the diameter d of the wire; the intercept on the F/w axis is the fracture toughness and the slope is 2(1 )kd, where k is the shear yield stress and the coefficient of friction. Figure 14-1 shows Benbow and Bridgwater’s results for china clay plotted in the way suggested, from which R 126 J/m2 and 2(1 )k 48 kPa. No independent
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Cutting force/width
350 300 250 200 150 100 50 0 0
0.001
0.002
0.003 0.004 Wire diameter (m)
0.005
0.006
Figure 14-1 Cutting china clay with a wire. (FC/w) should plot linearly against wire diameter according to Kamyab and co-workers’ (1998) analysis.
values are available for , but were 1, k → 12 kPa and R/k 10 mm. Most of Benbow and Bridgwater’s other results for -alumina pastes follow the model acceptably well despite there being only three wire diameters. Toughnesses of some 20–40 J/m2 are found. They noted that bigger diameter wires caused greater surface roughnesses of the cut faces, the wavelength of which was about three wire diameters. This is reminiscent of features on cheese cutting and bar cropping. Thakur and Godwin (1990) cut soils with wire with the aim of producing adequate soil disturbance without the wear of traditional tillage implements. The wire was mounted between radial arms that rotated as the device was moved along, and so cut along trochoidal paths exactly like a cylindrical milling cutter or router. Analysis was done in terms Mohr–Coulomb pressure-dependent incipient flow fields. It would be of interest to analyse the data using the Kamyab model. Everhart (1934) reported that wires cut through clay columns more easily and with less deformation of the workpiece when the wires were negatively charged. Böhm and Blackburn (1995) showed that cutting forces could be reduced, and surface finish improved, with the application of 5–30 V at cutting rates of 5–500 mm/min. The optimum speed depended on wire thickness, applied voltage and material composition. When the voltage is too low, tearing of the cut face was observed, again like the cutting of cheese (Chapter 12). When the voltage was too high and the speeds too low, electrolysis of the binder system gave rise to gas bubbling defects.
14.2 Roots Plant roots have to push past particles of soil to lengthen and expand. How roots penetrate the soil is of great interest to soil scientists. Soils need to have sufficient mechanical strength to provide anchorage for the plant throughout its life and to sustain the system of pores that bring water and oxygen, and remove carbon dioxide, which are essential for plant growth. As explained by Gregory (2006), almost all roots growing through soil experience some degree
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of mechanical resistance (impedance in the language of soil science), and if continuous pores of appropriate size do not already exist then the root tip region must exert sufficient force to deform the soil. Few plants have roots smaller than 10 m in diameter and most roots are much larger. This means that roots are often larger than the water-filled pores at ‘field capacity’ (i.e. those pores, typically of diameter 60 m, filled with water when drainage has ceased). Growth in these freely draining pores will be inhibited unless the soil is sufficiently compressible. Regions of dense soil with high ‘strength’ (either occurring naturally, or caused by compaction by heavy machinery, or even during cultivation) limit root growth, as does encounter with stones. The strength of most soils increases as they dry, so that shortage of soil water inhibits growth. Roots form and grow when cells behind the root tip elongate longitudinally and radially, to push the root tip forward. It has long been known that rates of elongation depend on soil ‘strength’ and water content (Taylor & Ratliff, 1969). The new volume of growing roots has to be accommodated by compressing adjacent soil both around the increasing circumference and also at the growing tip. There are four main types of root: (i) thin hair-like tap roots (e.g. ragwort) that grow vertically downwards; (ii) fleshy tap roots (e.g. carrots); (iii) fibrous roots that spread sideways as well as downwards (e.g. plantains); and (iv) tuberous roots (e.g. potatoes) that are fleshy and spread to the side. In the case of tap roots, the new volume is accommodated by horizontal displacement of the earth. With plants having horizontal root growth, the new volume is accommodated by displacements mainly in vertical planes so that, particularly in shallow-rooted plants, local displacements may manifest themselves as uplifted soil on the surface. Tree roots can lift paving stones and drains, etc., but the normal circumstance with most plants and crops is that there is little surface disturbance. Other types of root obtain support from other media (e.g. ivy on trees and walls). The properties of agricultural soils have been determined in much the same way as civil engineering soils, e.g. Greacen et al. (1968) employed penetrometers and recorded the changes in density around the penetrations. Critical state soil mechanics has been used for modelling plant root growth (e.g. Hettiaratchi, 1990). In modelling how roots grow, radial expansion of the root by temporary increase in plant turgor pressure must be important but that alone does not really explain how a root lengthens. Abdalla et al. (1969) considered that radial expansion produced a ‘weak zone’ of soil ahead of the tip of the root into which it would grow. Experiments showed that bladders expanding in glass ballotini (small glass spheres, available in diameters from 50 m upwards, from the Italian for ‘small balls’) produced tensile stresses at the ends (Richards & Greacen, 1986). Having lengthened, further expansion around the tip of the root weakens more soil ahead, and the cycle repeats itself. If we replace ‘weakening of soil’ by ‘separation of soil’, such ideas agree with the concepts in this book, and we should ask what work is involved. Furthermore, in the same way that cylindrical and spherical expansion models are employed to analyse civil engineering pile driving, there has to be a gap at the tip to permit forward motion: the elements of soil on the surface on one side of the point of a pile before driving, cannot be still attached to the elements on the other side of the point. Finite element methods (FEM) models of root growth (e.g. Kirby & Bengough, 2002) require a separation criterion just like models of any other ‘cutting’ process. The Greacen and Oh (1972) ‘turgor pressure balance equation’ has wide currency in the plant physiology literature. They argued that root growth occurs when the turgor pressure of the elongating cells is sufficient to overcome the constraints imposed by the viscoelastic cell walls and the soil matrix. They wrote a ‘pressure balance’ given by
turgor pressure inside plant cells wall stress external soil pressure 0 (14-5)
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Statics tells us that it should be forces (stresses areas) that must be equilibrated, not stresses. Two problems have to be solved and matched. The first is the radial expansion under turgor pressure of the thin cylinder representing the root. The second is the radial expansion of a cylindrical hole in a mass of soil. It is like the well-known problem in engineering of shrink-fitting one cylinder over another. Both radial expansions must match, and the solution to that problem determines the relationship between turgor pressure, soil pressure and cell wall stresses, not a ‘pressure balance’. Care is required in applying simple strength of materials analysis for a pressure vessel to the root growth problem, because new root material (meristem), in which chemical reactions are taking place, is being added to thicken the walls as the root system grows. It is not the original cell wall that is being stretched to ever-greater levels, as is the case in engineering pressure vessels. Chaplin (2008) has pointed out that if ‘wall stress’ in (14-5) does not mean what it seems (i.e. not the hoop stress pr/t in the expanding cylinder of radius r and wall thickness t) but rather the pressure p t/r causing the hoop stress, then (14-5) does make sense since the pressure difference producing cell expansion is given by (turgor pressure inside plant cells less the external soil pressure). The scale effect in fracture mechanics may be relevant to modelling root growth in that deformations take place at small scale, yet properties are determined in the large. How well root systems are anchored in the soil determines how cutting forces are reacted when harvesting crops. How well they are anchored also determines how easy or otherwise it to remove weeds from soil. The behaviour is rate and temperature dependent. Weeds pulled out slowly very often come away complete with root, but when pulled quickly simply snap off leaving the root in the earth. The response also depends on the state of the soil.
14.3 Earths, Sands and Burrows: Hole-making and Scraping by Animals Pigs are able to disturb the surface of the ground with their nose to an amazing degree, although as far as the author is aware no pigs have been instrumented for digging effort to see what is going on. Truffles (subterranean edible fungi) are found by pigs ‘rooting’ in this way: ‘to root’ in this sense comes from an old English word for snout. Clearly, the organ can withstand considerable force and abrasion. In contrast, dolphins are said to attach bits of sponge to their snouts as safety masks when hunting in sharp coral. The aardvark digs on the surface and opens out termite mounds with its claws (the Latin name Orycteropus afer means ‘African digger-foot’). Aardvark is Afrikaans for earth pig (but they are not pigs). The feet of rasorial creatures (poultry and other gallinaceous birds such as grouse, turkey and partridge) are designed to scratch the surface of the ground efficiently (same word as razor, to scrape). Digging (fossorial) creatures that make burrows have, in addition, to remove soil from the site by various means (a fossor was a minor clergyman in the early Christian church employed as a gravedigger). How are such animals equipped to do the job? The ability of fossorial creatures to excavate depends on tooth shape and size, ‘hands’ and ‘feet’, the mechanics of muscle and bone linkages, and how good they are at loosening and dislodging soil in front and passing it backwards as they tunnel. Like roots, many creatures make holes in the ground, either just passing through, or as homes or as places of refuge. For example, earthworms burrow by enlarging extant crevices and holes in the soil. The anterior body segments of such ‘hydrostatic skeletons’ are first elongated and extended into a crevice and then expanded radially to enlarge the hole (French, 1988, p. 188). The crawling kinematics of the earthworm and other peristaltic hydrostats are described in Quillin (1999), and Quillin (2000) measured the forces involved in the digging cycles of different-size worms, looking at scaling relationships with body mass.
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Earthworms feed primarily by pulling leaves into the burrows and secreting saliva to digest them externally (Darwin, 1881), thereby loosening and aerating the ground; they eat soil and defecate it only when food is scarce. Earthworms have no teeth but many other kinds of worms have mandible ‘teeth’ made of chitin (Chapter 13). Videler and his co-workers have studied radular teeth in limpets (gastropoda) and chitons (polyplacophora) and used them as models for the improvement of industrial cutting machines like dredgers (e.g. van der Wal et al., (2000) (Section 14.4). As explained in Chapter 13, the radula is the flexible ribbon in the mouth cavity on which are implanted several tens of transverse rows of teeth. It is used as a ‘rasp excavator’ during feeding on organisms such as algae living on and in rocky substrates. Echnidas (spiny anteaters) are able to submerge in soft soil to escape an enemy, burrowing straight down until only their spines are showing. The dynamics of burrowing in bivalves (molluscs that have two shells hinged together) was studied by Trueman (1966). The pressures during digging in sand, and the forces during probing and retraction of the foot, were measured during experiments. Initial penetration is by scraping and probing: progress by muscular contraction and generation of high pressures in the bodily fluids does not take place until part of the animal is beneath the surface of the sand so as to provide a ‘reaction buttress’ (French, 1988). Moles have forty-four teeth and five claws per foot and the naked mole rat (neither naked, nor mole nor rat, being a cousin of the porcupine and guinea pig) is a rodent some 75 mm long that has huge incisor teeth to carve out tunnels in hard desert sand in East Africa. While tooth wear must happen, as the incisors never stop growing this does not seem to be a problem. They have hairy toes to kick soil back to other sweeper moles in a relay behind in a tunnel and, eventually, into the open. To keep soil from the mouth while digging, they can close their lips behind the teeth. The platypus has large webbed front limbs. The webs fold away on land for the front claws to burrow and it rivals the mole for quick digging of tunnels on land (Mitchinson & Lloyd, 2007). Stein (2000) discusses the mechanics of digging by animals, which combines the design of their digging ‘implements’ and the properties of different soils – but not the toughness of soils. Gasc et al. (1986) studied the digging system of the Namib Desert golden mole, exploring these sorts of feature. Among subterranean rodents, three methods of breaking up soil were identified by Hildebrand (1985): (i) scratch digging; (ii) chisel-tooth digging; and (iii) head-lift digging. Scratch digging predominates among the geomyids (small rodents sometimes called ‘pocket gophers’) and is characterized by the alternate flexion and extension of the forelimbs to loosen soil with the claws, after which it is pushed or flung behind with the pads of the forefeet: in soft soils, the act of winning the soil and raking it under the body is combined into a single process. Chisel-tooth digging uses the procumbent incisors to break up the soil, which is subsequently removed from a burrow by head or feet. Powerful jaw and head muscles are required. Head-lift digging combines penetration by the incisors with head movement in a sort of ‘pick and shovel’ action. Figure 14-2 shows the primary components of rodents that dig in these three ways. Note the different widths, shapes and relative orientations of the self-sharpening incisors. Friction has to be overcome when digging and the non-smooth, yet low-friction, exterior surfaces of dung beetles have inspired widespread biomimetical research on low-friction surfaces (Ren et al., 2003; Tong et al., 2009). This sort of thing ties in with the way in which drag during swimming is reduced by the surface protrusions on the skin of sharks. In comparison with above-ground rodents, the smaller limbs of subterranean rodents and their spindle-like shape (fusiform) are acknowledged as adaptations to manoeuvring in narrow
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Scratch digging (Geomys)
Chisel-tooth digging (Cryptomys)
Head-lift digging (Nannospalax)
Figure 14-2 Primary elements of the digging apparatus in subterranean rodents performing (i) scratch digging; (ii) chisel-tooth digging; and (iii) head-lift digging, showing differences in skull shape, degree of incisor procumbency, placement of incisor roots and size of the claws of the forefoot (after Stein in Lacey et al., 2000).
and confined spaces. Regardless of digging mode, all subterranean creatures show some modification of teeth, head, neck and forelimbs to promote digging compared with their surface-dwelling equivalents. For example, the upper incisors of underground rodents are more chisel-shaped than those on the surface, enabling them not only to dig better but also to cut through roots. Other work relates to the proportions of limb bones, the leverage required of digging animals and the associated musculature that generate sufficient power to dig. The trade-off is that good diggers such as badgers cannot run very quickly on the surface.
14.4 Earthmoving and Ploughing Substantial prehistoric earthworks were dug by early man with tools of bone (or preferably antler, which is a tougher form of bone – antler is not horn). Silbury Hill at Avebury in Wiltshire, England, a 40 m (130 foot) high man-made chalk mound, is the largest man-made earthwork in Europe and dates from the Neolithic period. Later, bronze and iron tools appeared but the work was always done by hand until after the Industrial Revolution when steam-shovels appeared. Cuttings, embankments and tunnels on the early canals and railways were made by human ‘excavators’ (navvies) with picks, shovels, wheelbarrows and the occasional use of dynamite.
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A seed bed for planting requires that the soil be broken down into small units and that it be clear of the remains of previous crops in the same location and of newly grown weeds (it is then said to have a good tilth). Gardeners achieve this by digging, turning over and raking; the attainable result depends on a soil’s make-up, how well water drains away, and so on. Mechanized digging concerns powered devices that disturb the soil by continuous rotary or linear motion, turning it over, followed by harrowing as necessary to break it down even more finely. (A harrow is an implement with spikes, discs or chains hauled across a field.) It would be interesting to compare the changes in roughness of a ploughed field followed by harrowing: there is no reason why ‘centre line average’ (CLA) ideas employed for engineering surfaces cannot be applied at such a larger scale. Apart from the well-known use of ploughs in agriculture, cutting of ground materials is important in many engineering contexts such as dredging, tunnelling, bulldozing, ploughing trenches for cables and pipelines, and so on. In cultivation no soil is removed from the site. In contrast, when digging trenches, civil engineers have to remove spoil from the site at least temporarily. In addition, depths of trenches (approx. 1 m) are much greater than typical cultivation depths (that rarely exceed about 300 mm). In city streets, trenches for electricity, gas, water and telecommunications utilities are often dug by hand. Alternatively, excavators, back-loaders, etc., are employed. Such machines are intermittent in action, i.e. every bucketful has to lifted out of the trench and disposed of before a new cut is taken. It is possible to cut long trenches by excavators and draglines, again intermittently, but there are cases where trenching continuously by ploughing has advantages. Examples are where pipelines cross beaches or are entirely underwater. As explained by Palmer et al. (1979), drivers of draglines cannot see what they are doing on the seabed when operating from a barge, or from an ice sheet. Ships that dredge channels in waterways also operate blindly. Although all ploughing is a form of scratching, agricultural and trenching ploughs have different designs because, unlike an agricultural plough, a trench plough has no need to turn over the soil, and just needs to displace material in order to produce a groove or slot in the form of a narrow, stable trench (i.e. one that will not collapse into itself). Designs are also different because of the character of the soil in the two different cases (farmlands have been regularly cultivated for years, whereas land trenched or tunnelled or excavated for foundations may never have been disturbed in geological time). In addition, soil now on or near the surface may in times past have been far below the surface, and has been exposed by erosion, drying out or glacier scouring. In previous times such material experienced an overburden pressure that no longer exists or is far smaller. The properties of such an overconsolidated soil are different from a similar soil without such history. Normally consolidated soil has never been overloaded in history by pressures greater than the current overburden. Before the 1820s in the USA, there were no farms west of the Alleghenies. Some Native American tribes had ploughed the soil on the east coast before the first settlers arrived. As expansion took place it was found that the English-derived cast-iron ploughs commonly used on the eastern seaboard performed poorly in the prairie states, as they would not cut the virgin turf and became clogged with the dense soil. Sayenga (2007) discusses how the sharpness of such ploughs was improved by the use of crucible steel blades; he also relates how the term ‘plow steel grade’ became an indicator of the strength of wire ropes. This is connected with the introduction of steam ploughing in England in Victorian times, in which a plough having many blades was winched by traction engines back and forth across a field by wire ropes. (It could also be accomplished by a single ploughing engine with ropes in a triangle.) We say above that ploughing ‘cuts’ the soil: the first turnover is ‘fracture’ in the cutting sense. But note that ‘ploughing’ is used in tribology in a different sense, and rubbing on the
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underside or flank of a cutting tool is also sometimes called ploughing by metal cutters. In both these cases, material is disturbed on the surface and moved around, but no chips are detached, and the mechanism is the same as in scratching at low attack angles where material displaced from the groove forms ridges alongside the scratch (Chapter 6). Agricultural and engineering ‘ploughing’ is thus different from ‘ploughing’ in friction. We also note the use of the word drill in agriculture, where it means a shallow channel into which seeds are sown, rather than the making of a hole down into the ground. The seed drill is a machine to scratch such channels in already prepared ground, sow seed regularly and economically within them, and then cover them over with soil. The drill was Jethro Tull’s invention at the end of the seventeenth century and revolutionized the old-fashioned method of broadcasting by hand. Even so, Tannahill (1973) points out that the Sumerian ‘Farmers Almanac’ of 2500 BC described a combined plough and seed drill, and a three-bladed combined plough and sower was known during the Chinese Han dynasty.
14.4.1 Agricultural ploughs A simple scratch plough is shown in Figure 14-3. It consists of a pole, the pointed tip of which is protected by an iron or steel cap. In ancient times it would have been drawn by humans or by animals (the Chinese invented the animal collar, but the idea did not reach Europe until the eleventh or twelfth century). Soil is lifted up the face of the stick where, depending on the brittleness or ductility of the ground, it breaks up under its own weight to fall back on the surface. Calculations for the forces and work required are similar to those for scratching in Chapter 6. Hansen (1969) describes experiments with a replica of the Dostrup ard, a primitive light plough dating from 350 BC, found in a bog in Denmark (see also Steensberg, 1976). A scratch plough fails to cut a furrow and turn over the soil, something that a mould board plough is able to do. According to Lester (1975), this device came from Central Europe and was the only significant invention in agriculture during the Dark Ages (400–1000 AD). It has three working parts: (i) a knife blade (coulter) that cuts vertically into the earth just ahead of (ii) a blade (the ‘ploughshare’) that cuts horizontally; and, behind these, (iii) a mould board that turns the resulting slices to one side (Figure 14-4a,b). Before the sixteenth century, singlefurrow ploughs in England were massive affairs requiring six to eight oxen to draw. Lighter two-horse ploughs were then developed in the Netherlands, and two-furrow ploughs later in the following century. The well-known prophesy that
Soil
Figure 14-3 Simple scratch plough (ard).
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Wheel adjustments
Stilts
Coulter
Frog
Beam
Hake
Brace
(A)
Mouldboard
Share
Furrow wheel
Land wheel
Turned by mouldboard Cut by coulter
(B) Cut by share
Figure 14-4 (A) Right- and left-hand views of a two-wheel horse-drawn general purpose plough showing the names of the main parts; (B) the work done by a single-furrow plough. The vertical furrow wall is cut by the coulter, the horizontal cut made by the share, the twisted furrow-slice is in the process of being turned by the mould-board, and the previously cut-and-turned slice is seen on the right lying at an angle (after Davies, 1949).
‘… He shall judge between the nations and shall arbitrate for many peoples; they shall beat their swords into ploughshares and their spears into pruning hooks; nation shall not lift up sword against nation, neither shall they learn war any more …’. comes from Isaiah, Chapter 2, verses 1–4, and is repeated in Micah, Chapter 4, verse 3. George Orwell (1938), in Homage to Catalonia (about his experiences in the Spanish Civil War), said that ‘… Ploughs, drawn by teams of mules, were wretched things, only stirring the soil, not cutting anything we should regard as a furrow. All the agricultural implements were pitifully antiquated being governed by the expensiveness of metal … There was a kind of harrow that took one straight back to the Stone Age. It was made of boards joined together, to about the size of a kitchen table; in the boards hundreds of holes were morticed, and into each hole was jammed a piece of flint which had been chipped into shape exactly as men used to chip them ten thousand years ago … It made me sick to think of the work that must go into the making of such a thing, and the poverty that was obliged to use flint instead of steel …’. Yet in Evelyn (1670), in a letter to the Royal Society, extols the virtues of the design of the Spanish sembrador (Figure 14-5) or
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Figure 14-5 Spanish Sembrador, combined plough, harrow and seed drill of 1670, as described by Evelyn.
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‘New Engin [sic] for Ploughing, and Equal Sowing all sorts of Grain, and Harrowing at once; By which a great quantity of Seed-corn is saved, and a rich increase yearly gained’. Even when all components are made of metal, a plough will still wear. Ransome’s invention of chilled cast iron and its application to ploughs in 1809 considerably speeded up cultivation, as mentioned in Chapter 9. Even so, there can still be rapid wear of present-day agricultural tools (Richardson, 1967). The tips of blade become blunt after only a few hundred hours of work, resulting in increased resistance to cutting (Thakur & Godwin, 1990). All ploughs must be able to cut to a controlled depth, and the draught forces should be a small as possible. In addition, when a plough is remotely hauled, it must be directionally stable. Originally, all ploughs simply rested under their own weight on the ground, so that when the friction angle rake angle , the weight of the plough caused it to dig in to greater depths than intended; conversely, when , the plough blade tended to be lifted out of the ground. Later, horse-drawn mouldboard ploughs were mounted on wheels to give better control of depth and, later again, depth could be set directly when ploughs came to be mounted on hydraulically controlled hitches on the rear of tractors. Not all soils are ductile: they can be brittle when frozen and ploughing ‘knocks clods out’ exactly as in the cutting of low (R/k) materials described in Chapter 3 (Aluko, 1988). In between these two extremes, coagulated lumps formed by ploughing may break under their own weight when lifted and turned over by the blade. Frost penetration during overwintering breaks up large clods from autumn ploughing. The whole range of types of chip described in Chapter 4 may be produced in soils, from continuous chips in the agricultural ploughing of wet clay, to brittle scallops (undeformed clods of earth broken in bending) in frozen clay. ‘Mole drains’ under fields are circular holes about 50 mm in diameter cut in rows a few feet below the surface. They carry water to bigger drains, positioned at right angles to the holes, that take it away from the site. Mole drains are made by hauling a torpedo-shaped device below the surface. Traction is provided on the surface through a stem down to the separating head. The turning moment caused by the different lines of action of the pulling force and the resistance of the soil is counteracted by a long skid ahead of the device on the surface. Considerable effort (200 hp tractors) is required for mole draining: steam ploughing engines were once employed for the job. Experiments to establish how implements deform and cut ‘ductile’ soil are performed (i) in the laboratory in glass-sided tanks filled with soil that enable flow fields to be established of scaled-down implements drawn along the glass face; (ii) in soil bins, which are pits in the floor in which larger scale experiments may be conducted; and (iii) in full-scale trials in the field. Forces are measured in the usual way with dynamometers. To permit easier interpretation of flow fields in the laboratory, markers such as grids of talc slurry, beads, etc., can be placed on the vertical soil surface before the glass is put in place or layers of different-coloured sand may be interspersed as the soil is prepared (e.g. Spoor & Godwin, 1978). Note that in many publications on how bulldozers, ploughs and tines behave, the rake angle is measured from the horizontal, not the vertical. Perusal of the soils literature shows that, as in metal cutting, slip line field solutions for the movement of ductile soils have been proposed for different friction conditions on the blade that agree quite well with experimental flow results. Godwin and O’Dogherty (2007) provide a valuable review. Recall that the pressure dependency of yielding in soils alters the make-up of slip line fields in comparison with those for metals and employed for deformation fields in cutting, as described in earlier chapters. For a Coulomb–Navier material (14-1) the fans become radial lines and logarithmic spirals (Section 14.1). Indeed, since the same pressure-dependent relation seems to apply to shear failure in brittle materials (Section 14.2), fracture lines along
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which shear cracks run in rocks also comprise radial lines and logarithmic spirals (Sikarskie & Cheatham, 1973) (Chapter 8). Cutting with a vertical blade is similar to the design of retaining walls in soil mechanics, where the sideways pressure required to cause soil to slip upwards (Rankine’s passive state) is of interest (Section 14.2). The solution is for incipient slip on a single plane caused by a small movement of the blade. Cutting concerns what happens at large displacements. Experiments show that after first slip, a series of slices is formed as in discontinuous chip formation in metals (Rowe & Barnes, 1961; Selig & Nelson, 1964). Similar behaviour occurs for forwardsloping blades (negative rake). Cutting with a positive rake results in forces upward from the cut surface, thus putting the putative chip into tension, so that the chip fractures into scallops (Chapter 4), unless the soil is ductile, i.e. its fracture toughness is high. Agricultural engineers have studied three types of blade characterized by depth of cut to width (t/w) values: (i) wide blades (t/w 0.5); (ii) narrow chisel tines (1 t/w 6); and (iii) very narrow knife tines (t/w 6). Type (i) may be analysed by two-dimensional theory particularly if, in model experiments, the blade overhangs the sample, otherwise there will be small end-effects; types (ii) and (iii) involve three-dimensional flow, and are more complicated (Figure 14-6). Narrow tines also involve the interesting feature of a critical depth, above and below which the modes of soil deformation are different. For depths smaller than critical, the f Rupture distance Crescent failure
Soil surface
Depth of crescent failure
dc d Depth of line
Lateral failure Soil wedge
Upper failure zone – crescent failure
r cos pʹ
Lower failure zone – lateral failure
r
pʹ
w α dc
dccotα
Figure 14-6 Soil failure patterns and critical depth concept (after Godwin & Spoor, 1977).
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soil is moved forwards, sideways and upwards, as in scratching (Chapter 6). When the implement tine goes to greater depths, soil below the critical depth moves forwards and sideways only, with no upward motion; the portion of soil above the critical depth behaves as before. Ahmed (1990) showed that that in cohesionless soils (where the shear yield stress is pressure independent and constant) the critical depth was small. If processes of separation take place both above and below the critical depth, it presumably is related not only to yielding but also to fracture toughness, i.e. is related to (R/k). It is interesting to speculate whether such an additional mode of deformation might exist for deep scratching of metals. The response is relevant to the behaviour of crampons, spiked running shoes, etc., discussed later. Separation in soils at the blade tip occurs, as with other materials, in boundary layers between the cut and uncut parts, so that the pattern of movement of soil is hardly affected by the presence of the thin layers in which separation takes place. As with other materials, however, the total work and forces required should include the work of separation. In practice, friction and work of displacing the spoil may predominate as found in civil engineering ploughing and excavating (see below). Attempts to reduce friction in ductile and sticky ‘heavy’ soils are shown by the use in Cornwall of spades having holes in the blade. Ren and co-workers (2003) have been inspired by the function of non-smooth knobs on dung beetles that reduce friction in burrowing, to produce low-friction agricultural implements and bulldozers. Beard (1994) describes an electrically lubricated plough that required 32 per cent less work in a loam soil having 12-17 per cent moisture. A simple field test for predicting the ease of cultivation was proposed by Chandler and Stafford. Clods of earth were split between a cone and a ball and the load-displacement response was recorded. The test is essentially determining (R/k) (see also Aluko & Chandler, 2006).
14.4.2 Civil engineering ploughs Many marine pipelines are not simply laid on the seabed, but placed in a trench. Pipes in trenches left open on the seabed experience reduced hydrodynamic forces generated by currents and by wave-induced bottom velocities, and resistance to lateral movement of the pipeline is improved. Danger of impact by fishing trawls and dragging anchors is also avoided. Choice of depth for a submarine pipeline trench is a delicate choice: pipes that are laid too shallow get bent or broken; trenches deeper than necessary involve considerable extra expense. Pipes in trenches that are backfilled have even better resistance to these factors, and the covering also restrains upwards buckling and enhances thermal insulation. Undersea and shore trenches do not usually use the displaced soil to cover the laid pipe; rather, new, heavy, material such as gravel is most often used for backfill to protect the pipeline from wave forces and to place it below possible changes of beach level after storms. Another reason for burying pipelines is that on Arctic beaches, onshore winds drive large ice masses over the surface to produce continuous grooves and gouged-out scars on the seabed and shoreline. Such grooves can be 5 m across by 100 m long. There is some similarity with how glaciers gouge out tracks (Croll, 2008). Control of depth is achieved by the long-beam plough shown schematically in Figure 14-7, rather as in an agricultural mole plough. For a sufficiently long beam, this type of plough is self-stabilizing and maintains the intended depth of cut (Palmer et al., 1979). Trench depth is altered by changing the distance between the pivot point of the beam and the skid/trimaran assembly. A plough of this design was employed below ice in the Canadian Arctic. Its length was some 20 m, breadth 10 m and height 5 m, and its weight was some 26 tonnes. Draft forces were between 90 and 140 kN for cuts between 0.6 and 1.5 m deep taken at about 0.15 m/s.
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Beam Skid e
ar
Sh
Spoil Trench
Heel (A)
(B)
Figure 14-7 Mechanics of the long-beam plough: (A) schematic diagram and nomenclature; (B) cross-section of trench (after Palmer et al., 1979).
Whether a soil responds to ploughing as if it were in a drained or undrained state depends upon microstructure: (i) cohesive and impermeable clay soils behave as undrained solids; but (ii) at the same rates of ploughing in sand and silt the properties are of the drained material. Van Os and van Leussen (1987) measured cutting forces in sand under water. The deformation rate was important, as may be expected, because dilatancy decreases the pore water pressure and increases the working stresses. Consequently, cutting forces were greater at faster ploughing speeds. The behaviour of multibladed ploughs in saturated sand was investigated by van Rhee and Steeghs (1991) using model cutting tests in a laboratory dredging flume. Palmer (1999) reported that increasing cutting forces with speed, when cutting saturated soils, has caused difficulties in large-scale projects. The forces might increase five-fold even at modest increases of speed, depending on the soil and its condition. Palmer develops the equation that governs pore pressure, which is diffusion in a moving medium. Scaling between model ploughing experiments and large-scale applications in the field is discussed. Because the force required to shear the soil depends upon the cross-sectional area of a trench (when toughness is omitted from the calculations), but the force required to lift the cut soil out of the trench is proportional to volume, cube-square scaling applies. Inertia effects are negligible at typical ploughing speeds up to 1000 m/h (300 mm/s). Palmer gives the appropriate nondimensional groups by which model tests and large-scale ploughing can be related.
14.4.3 Coal cutting machines, excavators The motions of many commercial devices for dredging shipping channels, trenching, tunnelling, coal cutting, excavating and the like, are the same as those of familiar machines in the workshop, but the scale is vastly different. For example, bucket-wheel trenching machines, some tunnelling machines and some rotary snowploughs act like circular saws. Other types of snowplough, full-face tunnelling machines, and shaft sinking and heading machines are, in essence, drills. So-called ‘continuous belt’ cutters such as coal saws, ladder dredgers and trenchers act like chain saws. Large-scale percussive chisels and drills are found in jack hammers (e.g. Karlsson et al., 1989; Lundberg & Okrouhlik, 2006). Figure 14-8 shows a Victorian coal cutter, acting as a mechanical pick driven by a bell crank; another sort described in the same article performed as a horizontal traversing slotting machine. Both were driven by compressed air from the surface which also cooled and ventilated the workings (Fernie, 1868). Modern machines are not dissimilar. A valuable series of reports by Mellor (1975–81) of the US Army Cold Regions Research & Engineering Laboratory (CRELL), discusses the kinematics and dynamics of such machines. Kinematics deals with the geometry and motion of a machine and its cutting tools, without reference to the forces and work required to cause the motion: dynamics deals with the forces required not only to cut, but to remove the spoil and, where relevant, to propel the device.
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Figure 14-8 Victorian coal cutting pick machine (Fernie, 1868).
While machine tools can have problems of chatter (e.g. Welbourne & Smith, 1970), they are nevertheless very stiff in comparison with earthmoving devices, particularly those mounted on pneumatic tyres. We have seen above that skids are used on some ploughs to control depth of cut and similar considerations arise in the design of large excavating devices: how are the moments set up by the cutting edges reacted by the weight of the machine; what is the likelihood of the device ‘digging in’, ‘lifting off’ or otherwise overturning under load; and so on. These are interesting variations on the usual ‘load control’ and ‘displacement control’ constraints considered in cutting and materials testing. Chain- or rope-hauled coal cutting machines have a tendency to hop along the coal face when cutting, not at the stick–slip frequencies observed when off-load but at a frequency related to the rotational speed of the cutting head (Edwards, 1978). Pick speeds in mines are kept low because of the risk of dust and gas ignition, and the resulting chatter is, perhaps, inevitable. The properties of the materials being cut influence design: in soils, the forces and energy for cutting are often small compared with those required for acceleration and transport of the spoil, so that materials handling considerations dominate the behaviour. The minimum energy required to accelerate debris away is the kinetic energy imparted to the spoil, but since there is friction as material is transported to the outlet of the machine, much more power is required. For example, in deep drilling with an auger, in addition to the potential energy gained by the debris, by far the greatest work is in lifting the spoil against friction up the helical flight path to the surface (see Appendices in Mellor, 1975–81). If a machine operates under water, power is required to overcome fluid drag on all the rotor surfaces as well as to accelerate and eject the cuttings. In contrast, in rock, concrete or frozen soil the forces required for cutting are much higher than those for handling the broken material and the situation, perhaps, corresponds more with workshop cutting practice. Large machines have multiple cutting edges spaced around a wheel, or along a belt. With wide, drum-like, wheels single teeth could occupy the whole width of the cutting face, but it makes sense to stagger tools of smaller breadth across the width of the cutting face so as to smooth out the variation in cutting force for the complete machine. Figure 14-9 shows the cutter bar of a multitooth belt trenching machine, looking like a giant chain saw, taking a cut of depth t when the bar is forward-inclined at angle ϕ to the horizontal. (Force calculations for this sort of device are applicable to hand-held chain saws cutting timber). The
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ha
in
Burrowing in Soils, Digging and Ploughing
uc
u FC
V H
FT t
ϕ
Figure 14-9 Chain trenching machine showing forces on individual cutter and on whole machine (after Mellor, 1975).
machine is moving forward across the surface of the ground at a speed U. The chain moves with velocity uchain, and is shown operating in an up-cut milling mode where the depth of cut increases from zero at the bottom of the trench to a constant value along the straight sides of the inclined bar. The trajectory of a tooth is the combined motion of the chain and machine. The resultant force on a single tooth may be resolved into components normal and parallel to the direction of tool motion (i.e. to the tooth trajectory) from which the total force along (FC) and across (FT) the path at any instant may be determined by multiplying by the number of edges cutting. When the motion of the device into the workpiece is small compared with the speed of the chain or belt, i.e. when U/uchain is small, resolution along the tooth trajectory is approximately the same as resolution normal and parallel to the chain. The difference is important only when (U/uchain) approaches unity, as may happen when drive tracks slip on crawler vehicles. Note that in a similar fashion, the depth of cut should strictly be defined with respect to the tooth trajectory, but it suffices here to use the feed speed U of the machine. The total force in the chain is a maximum at the tension side of the drive sprocket and is made up of not only (i) the forces FC for cutting; and (ii) friction at tool tips; but also (iii) the movement of debris to the exit point; and (iv) the resistance to motion of the chain around the guides and top and bottom nose of the bar. The total power for cutting is given by the total chain force times the chain velocity. Mellor shows that forces for acceleration of debris in this sort of device require noticeable power demands only with very high-speed chains when cutting ‘weak’ materials (a sawmill bandsaw ripping lumber might be a similar example). Note that in the many worked examples given by Mellor, the forces for cutting are couched in terms of empirical relations of the sort sometimes used in wood cutting (Chapter 7), namely FC (constant1) tindex1 and FT (constant2) tindex2, where the constants depend upon
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2
1
H FC V FC
0
K=2
K=1
30°
60°
90°
ϕ
120°
150°
180°
–1 H/FC
K=1
V/FC –2
K=2
Figure 14-10 Horizontal and vertical forces on cutter bar of chain trenching machine in the upcutting mode at different angles of inclination ϕ to the horizontal. Greater K relates to greater cutting resistance (after Mellor, 1975).
tool geometry, material properties and operating speeds rather than fundamental properties of toughness and yield strength. The constants are supposedly linked to the uniaxial compressive strength of the workpiece. Mellor quotes 14 MPa for permafrost and 200 MPa for hard rocks, both of which are likely to fracture in a brittle fashion. It would be sounder to involve fracture toughness as well as ‘strength’, certainly when the energy for separation is significant, but the importance depends on the relative effort for excavation and for subsequent removal. In order to traverse the machine through the work it is necessary to exert a force H parallel to the direction of traverse and a normal reaction V perpendicular to it (Figure 14-9). Resolution of FC and FT gives
H FT sin ϕ FC cos ϕ
V FT cos ϕ FC sin ϕ
(14-6a)
and
(14-6b)
Figure 14-10 (cf. Figure 5-4b) shows how H/FC and V/FC change with the inclination ϕ of the bar. When the bar trails and cuts upward (as used in ditching machines) the propulsive force H increases initially as ϕ increases then passes through a maximum, when ϕ is about 60–70°, say, depending on conditions. As the bar gets steeper the propulsive force decreases and continues to fall as the bar swings back through the vertical position at ϕ 90°. At greater ϕ (say about 130–150°) H can go negative, meaning that the machine becomes self-propelling, or that ‘grabbing’ occurs (see Sections 5.2.4 and 5.3.1). Over the same range
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of ϕ, the vertical force is positive (i.e. upwards) initially and continually decreases to become negative (downwards) after ϕ exceeds about 45–60° so that the machine tries to pull itself down into the work, i.e. to dig in (load or displacement control). The downward pull reaches a maximum at ϕ about 130–140°. At greater ϕ the force decreases but remains negative. H is provided by the drawbar pull of the machine, which is often assumed to be proportional to the weight W of the device. When the machine is cutting, however, it is (W V) that applies, with V being either positive or negative depending on ϕ and whether upcutting or downcutting. It is clear therefore that there has to be compatibility between the vehicle weight, tyre or crawler traction, and the cutting forces in a propelled machine, i.e. between vehicle weight and power. In addition to forces, consideration has to be given to the moments of the weight and the cutting forces about the support system, as mentioned earlier. In machines such as excavators and back-loaders that remove material in buckets, the loading of an individual bucket is transient until the bucket is full, after which the bucket is removed to the point of discharge of the spoil. At any one time, multiple buckets on a chain (dredgers) have some buckets digging in the transient phase, some full and being lifted to the point of discharge, and the rest empty and descending to depth at which cutting takes place. Excavator buckets have teeth and, depending on the condition of the soil, fissures may run from the teeth to break open the soil, thus reducing working loads. The behaviour of teeth (not in terms of fracture) was investigated by O’Callaghan and McCullen (1964–65), who modelled the penetration in terms of a Prandtl-type punch analysis. Balovnev (1983) presents many slip line fields for cutting soils and applies them to the action of bucket excavators. The behaviour is non-steady, the soil entering the bucket and accumulating ahead of it in the form of a ‘drag prism’. Power requirements for digging and excavating depend mainly on separation from the main body of soil, followed by further movement of the dug soil inside the digging bucket or ahead of it. The picture is complicated since, on the one hand, hard, dry soil is broken off from main soil mass in lumps; on the other, moist, plastic soil is cut off in compact layers, later broken up inside the digging part. Balovnev also presents the more traditional (Terzaghi) approach for forces in this sort of problem in which formulae are given that sum terms relating to soil shearing, compression, and crushing; size of cut (width and height); tool geometry; friction against the digging tool; the volume of the drag prism; the weight of digging part; density of soil; and so on. Every term has a coefficient determined from experimental calibrations. Experience shows that these coefficients are not constant: a formula may be acceptable for a given design of tool, but not for a different one. The word bulldozer comes from the animal ‘bull’ (implying great strength) plus ‘doze’, which is a variant of dose (meaning to apportion). Bulldozers, graders and scrapers sometimes simply push around material already sitting on a surface such as a concrete base. While there is friction at the ground to be overcome, there is no separation work. At other times, they have to cut into soil to slide it to another location. Reece (1964–65) showed that most earthmoving machines cause a state of complete failure in the soil mass involved, and he analysed their action using classical static equilibrium methods that take into account soil cohesion, weight, adhesion and surcharge (but not toughness). Material being moved will slide up the blade of a bulldozer and, depending on the R/k value, may fall over its back in front of the machine unless the volume of the spoil is limited by controlling the depth of cut. In practice the self-weight of the ‘chip’ of soil may cause it to break apart and fall forward, so that a standing wave of excavated material is formed in front of the blade. How high soil is lifted upward along the blade is considered by Balovnev (1983) (Figure 14-11). When snow first falls its density is a minimum (about 100 kg/m3) and is comparatively easy to move. But should it pile up and compact, and form drifts (with densities perhaps
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2
1 (A) Direction of movement
1
3
(B)
Figure 14-11 Pattern of soil movement ahead of a bulldozer blade: (a) cohesive soil; (b) noncohesive soil (after Balovnev, 1983).
up to 600 kg/m3), it becomes more difficult to clear. (The volume of a compacted snow ball is much less than the snow it comes from.) Whether the snow is wet or dry has a big effect. There are two principal types of road and rail snowplough: push ploughs and rotary ploughs. The former behave like bulldozers and push snow to one side. It must be fun to drive a locomotive to charge into a drift at 50 mph! However, in long deep drifts and especially in railway cuttings, the brute force of the plough packs the snow so tightly that further progress is impossible without help from the pick and shovel brigade (Parkes, 1965). The first successful rotary snowplough was introduced on North American railroads at the turn of the nineteenth/twentieth century. The basic design comprises a large hooded wheel consisting of knives and scoops that revolve at speed, cutting into the snow to eject it from the top of the hood and away from the track. The biggest machines have something like a 4 m diameter wheel rotating at 150 rpm that can throw snow over 80 m. With wet and heavy snow, this type has difficulties and other types can be employed having horizontal augers (or separate cutting knives) with faster moving impellers that eject all snow. This design can also rake snow from high up in a drift down into the path of the plough.
14.5 Picks and Crampons Many devices use teeth or spikes for grip rather than to pierce or make holes (carpet edging; crampons on shoes for climbing telegraph poles or trees; running, golf and cricket shoe spikes; anchors). The hoofs of mountain goats have soft inner suction pads for ascent and their dewclaws act as crampons for descents (a dewclaw is a vestigial thumb that does not reach the
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ground in walking). Proper functioning crampon-like devices depend on the indented or cut medium reacting the imposed loading without giving way, in other words that a deep scratch does not form. Thus, when a picture is hung on a nail, the weight of the picture ought not to result in the plaster giving way beneath the nail, and the picture falling off the wall. When skiing, poles thrust into the snow are employed to push off with, and rely on the embedded sticks not moving much. On ice, rather than snow, it is difficult to get the pole to pierce in the first place. Similar things inform design of tread patterns on tyres, and the use of caterpillar tracks. The ‘going’ on racecourses concerns whether horses’ hooves can grip and is controlled by surface friction when the ground is hard. In softer ground, the action of the hoof of a galloping horse when landing is to indent, rotate and push the soil backwards in some sort of negative rake groove formation. In its way, it is no different from a picture on a wall as to whether the horse will slip. The going is sometimes decided by the experience of the clerk of the course on digging his heel into the ground. All the features of groove formation by prow/ridging or chip formation described in Chapter 6 can easily be demonstrated on a sandy beach with one’s tipped-forward shoe going forwards (positive rake/large attack angle that produces chips), or a tipped-backward shoe going backwards (negative rake/small attack angle that produces prows). The wetter the sand the more cutting with little side ridging. Inspection of impressions in the earth where running horses have skidded to a halt and the hoof has dug in, shows that the hoof acts as a positive rake tool. Presumably sums could be done that would estimate the speed at which the horse was travelling just before it drew up. The tusks of walruses are ever-growing canine teeth, which in a large male can be 1 m long. While they are mostly for show, they are used to haul the animal up on to ice floes. The mechanics of how much ‘purchase’ a tusk can have in ice (like an ice-axe), and what loads can be borne, is an interesting question that must be governed by the ideas in Chapter 8. In softer ground, how a spike responds to loading perpendicular to the surface will be rather like that of a narrow tine used for cultivation, except that the boundary conditions will be different: a single hand-held spike will be able to rotate through the ground (unless the wrist action is incredibly strong) rather than be kept perpendicular to the surface. Crampons on boots, and spikes on running shoes, will be more firmly orientated, being fixed to the sole of the shoe. Payne (1956) describes the design of a sprag (anchor) for use with tractors performing drainage work where haulage cables were wound on a winch mounted on the tractor. The usual design of anchor had to be disproportionately large to react the hauling forces, and even then it slowly ‘dragged’. The inverse problem to that investigated by Payne is how a pick functions: in contrast to wanting no movement, use of a pick relies on soil giving way when the tool is transversely loaded after insertion, so that spoil may be dislodged and dug out. In Hollywood swordplay, characters have been known to escape from on high by thrusting their sword through a thick curtain and dropping to the ground by a controlled cut of the fabric. Whether sufficient resistance could be generated by this means is a moot point. Not all embedded tools give way in such a controlled fashion. How is the number of spikes chosen, their length and their spacing, in an array on running shoes and the like? While the behaviour of an isolated spike may be described by the models in Chapter 8, there will be a critical distance between spikes at which the behaviour of each will interact. How does that affect the resistance to transverse traction? Godwin et al. (1984) investigate how tine arrangement affects harrowing forces where soil disturbance is required, rather than not. A similar question arises in the design of corkscrews as mentioned in Chapter 8: too shallow a helix may leave insufficient material to bear the load for pulling out so that the cork disintegrates; with hardly any twist of the spike, the corkscrew itself may just pull back out of the pierced channel.
Chapter 15
Unintentional and Accidental Cutting Supermarket Plastic Bags, Falling Objects, Ships Hitting Rocks and Aeroplanes Hitting Buildings Contents 15.1 Introduction 15.2 Grounding and Collision of Ships: Diverging and Converging Tears 15.3 Progressive Dynamic Fracture 15.4 Accidents Involving Cables 15.5 The Twin Towers
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15.1 Introduction Unintended cutting may concern a deliberate cutting operation that goes wrong (a mistake when cutting a groove in carpentry, say, or when a blind hole becomes a through hole because the drill has penetrated too far), or may start out as something not remotely connected with cutting (e.g. holes in socks formed by overgrown toenails, when a ship hits a rock and is holed below the waterline, when a fork lift truck pierces the side of a van, the failure of a wind turbine at Aarhus in Denmark when a blade detached and cut through the mast). Soil cutting is involved when constrained buried pipelines buckle owing to thermal stresses (Palmer et al., 2003). Avalanches or mud slides may be triggered by earthquakes or other instabilities and result in damage by cutting. An example of where cutting allegedly triggered a ‘mud volcano’ occurred at Lusi in Eastern Java in 2006 (Dempsey, 2008). It continues to ooze enough mud to fill fifty Olympic-size swimming pools every day and some 30 000 people have been displaced. Various efforts to stem the flow (dams, channelling to the sea, plugging the crater with concrete balls) have failed (BBC, 2008). What produces a cut or scratch is not always obviously a ‘cutting tool’. An overloaded plastic bag from a supermarket can be stretched and pierced by the stiff corner of a cardboard box. When polythene was introduced for packaging, the carrot favoured by horticultural packers became short and with a blunt end so as not to pierce the plastic bag (Lester, 1975); traditional carrots had been big, long and with a pronounced point. The low bridge in the 1973 James Bond film Live and Let Die sheared off the top of a double-decker bus passing underneath (this has really happened in service with passengers). Is the bridge a cutting tool? Likewise with the iceberg that rent the Titanic. There is a presumption of cutting in the ‘high heels may tear the evacuation slide’ safety instruction on passenger aeroplanes. The fragments of rubber tyre that pierced the wing of Concorde at about 100 m/s at Paris in 2000 were effective as cutting tools (Mines et al., 2007). In the days of steam trains, water could be replenished at speed from water troughs placed within the tracks by means of a water scoop lowered for the purpose beneath the tender and raised afterwards (the mechanics of the process may be found in Inglis, 1951, p. 151). The Great Western Railway’s revolutionary Automatic Train Control (ATC) system that warned drivers whether signals were at danger whatever the Copyright © 2009 Elsevier Ltd. All rights reserved.
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weather, and would apply the brakes if required, was activated by a plunger moving over a 44 foot long ramp between the rails. In 1914, a water scoop came down of its own accord (the catch had not been replaced) and sliced off an ATC ramp that shot up through the back of the tender and through the leading coach to appear through the roof (Vaughan, 1985). The scoop certainly acted like a cutting tool. At Cambridge and Oxford the rivers are too narrow to allow side-by-side racing, so eights boats start in line and try to catch up and bump the boat ahead. The bows of Victorian boats were clad with metal and there had been the occasional injury at the bump to people in the boat ahead. In the Lent Bumps in 1888, a Clare College oarsman was killed, being impaled on the bow of a Trinity Hall boat, the steering gear of which had jammed (Durack et al., 2000). Thereafter, the Cambridge University Boat Club resolved that ‘an india rubber ball’ was to be attached to the prow of all boats and that remains in force to this day. (India rubber is an old-fashioned word for unvulcanized natural rubber or caoutchouc – a word of the Quechuan people of South America.) There is no foundation in the legend that St John’s College was involved in such an incident, was banned from the river in consequence, and changed its name to the Lady Margaret Boat Club in order to be permitted to row again (Hutchings, 2008). At the finishing line of the 1911 Madrid–Paris air race, the French Minister of War had an arm severed and was killed by an aeroplane banking too steeply near the crowd, when a blade of the propeller came off. Other gruesome accidents have concerned ships’ propellers cutting off water skiers’ limbs, and motorcyclists impaled on railings. A curious incident is related in Batchelor’s 1996 biography of Sir Geoffrey Taylor. A group including Taylor had ascended a chimney stack so as to study the aerodynamics of flechettes (small arrows, antipersonnel projectiles dropped from aircraft during World War I over enemy lines of defence and troop concentration). Some flechettes were left over having finished the launching experiments from the top of the chimney, and were brought down in a satchel by a member of the team. That person descended last, for some inexplicable reason, and the satchel caught on the rung of the ladder, spilling the surplus flechettes, but no one below was struck. Many types of defect cause limitations in traditional manufacturing processes (Johnson & Mammalis, 1977), and rings of metal can be cut off inadvertently during spinning operations. Other similar unintentional fractures in forming are described in Section 7.4.2. Accidental damage is unintended. In law, the intention of a person (mens rea) is paramount when deciding whether something is an accident or was deliberate. When a workman drops a hammer from high up on a building which hits and kills someone, was the result intended or not? Clearly some of the things discussed in this chapter as being accidental and unintentional do not fit that category when the perpetrators intend the consequence. The same sort of damage to the body work of a car may result from a driver accidentally scraping a wall, or from a vandal deliberately scratching the paintwork with a broken beer bottle. The consequences of accidentally dropping a screwdriver may be nasty, but screwdrivers have been used as substitute knives in attacks on the person. Several cases of this sort of thing have been described earlier, such as cuts produced benignly by the surgeon or malignly by an assailant (Chapter 11). In accidents and collisions involving people either (i) the surroundings stay still and the person moves (falling down stairs); or (ii) the surroundings move towards a stationary person (someone hit by a car). Many accidents, especially with colliding vehicles, can occur at high rates, and cutting during an impact can be surprisingly different from quasi-static experience. Strange things happen under extreme conditions: straw has been found embedded in trees after hurricanes have struck. How was it possible for aeroplanes to cut through the wire ropes of cablecars at ski resorts, and what happened during the Twin Towers incident? In crashes
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of vehicles, aircraft and so on, great play is made of energy-absorbing crumple zones and the like, to limit damage not only to the vehicle but also to the occupants. Nevertheless, seat-belted occupants may be thrown about within a vehicle, so ‘sharp edges’ are avoided in the design of vehicles. Safety glass in windscreens may be laminated or prestressed, not only to try and prevent fracture, but also to avoid sharp shards should it break. Motorcyclists are vulnerable in initial impact with other vehicles or safety barriers, and injuries may be produced by scraping the body along the road. Resistance to tearing by sharp objects is paramount in the design of safety clothing: it is a broad subject, just touched on in Chapter 8, but really beyond the scope of this book. Again, recent car designs do not incorporate bonnet mascots with sharp bits so as to minimize the likelihood of injury to pedestrians. Similarly, until 1975 British motorcycles had number plates mounted fore-and-aft on the front mudguard: newer machines have only rear plates as it was deemed that edge-on front plates could act as knives with the potential to cause severe laceration injuries to both riders and pedestrians. Not all accidental cutting occurs at high speed, but very often happens when control is lost and something slips. A device to absorb energy in steering columns of cars by tube inversion/ eversion involves a metal tube being completely rolled up inside or outside (like putting the rubber cover on the handle of a cricket bat) (Al Hassani et al., 1972; Stronge et al., 1984). The efficiency of the device is impaired should the tubes crack, and prediction of the number of cracks that forms shows how a transition from plastic flow alone, to plasticity and fracture, can take place even though one energy sink becomes two (Atkins, 1987c). The idea, but now with cracking as a trigger, has been applied successfully to the design of energy-absorbing train buffers. An understanding of the mechanics of fault and accident situations where cutting occurs enables the forces and energy of deformation and cutting to be estimated. This enables either a priori risk assessments to made, or a posteriori forensic investigations to be conducted. The topic is part of the general field of accident investigation where dents, bumps, scratches and other features on broken bodies can give information about the forces involved and the conditions under which an accident took place. In such investigations it is vital to answer the question of whether what is seen caused the accident or was the result of the accident (Felbeck & Atkins, 1996). Fault or accident damage may have immediate consequences so that an object can no longer properly perform its function (pierced plastic bags, sinking ships) and/or later hidden consequences, not at first obvious, e.g. in underground pipeline installations where bumps, dents and scratches, arising from improper trenching and laying, affect subsequent in-service performance (Jones, 1993). In unintentional cutting, (i) the ‘tool’ and the ‘workpiece’ may be solid objects; or (ii) one of the contacting bodies may be ‘rigid’, e.g. a massive wall hit by a car, or a floating log hit by a seaplane float; or (iii) in other cases both bodies may deform, e.g. a car hitting a central barrier on a motorway. In many situations one or both of the objects may be deformable hollow shell-like structures with various ribs, stiffeners and stringers. Initial contact is followed by denting, bending, stretching, detachment of welds releasing reinforcements, and possible fracture. As mutual contact progresses, the most highly deformed regions will attain the necessary combination of strains and hydrostatic stresses to initiate fracture according to ductile fracture mechanics, and when there is sliding between the bodies, one or more cracks may propagate with continued loading (Muscat-Fenech & Atkins, 1998). Thus the stiff bow of a ship accidentally (or, in war, deliberately) ramming the compliant side of another vessel can easily cut into the second ship; ancient and modern naval vessels have had specially designed bows for ramming. The special ram employed by police for battering down doors has a loose internal mass which spreads the load, otherwise just a hole is knocked in the door (Fenne, 2008).
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The mechanics of piercing plates and shells in Chapter 8 is relevant to all accidents involving liquid or gas storage tanks, the consequences of which depend on whether hazardous substances are under storage. Tankers on land, sea and in the air may be subjected to accidental penetration by projectiles and sharp-ended objects resulting from a collision, or deliberately in war. The degree of damage to hollow bodies (such as the leading edge of an aerofoil) impacted by smaller objects, such as hail and birds, will depend on the size and speed of the aircraft. (The bird strike problem of birds being ingested into engines is of great concern to aeroengine manufacturers.) When the striking object is small and supposedly undeformable, predictions can be made about damage and failure in the way calculations are performed about bullets striking targets. There will be ballistic limit velocities associated with different parts of the airframe below which fatal damage will not occur. Even when the striking object is deformable, estimates can still be made. However, when both the striking and stricken bodies are together deformable and large, and where the relative impact velocity is great, the problem becomes tricky and strange things can happen in collisions when a seemingly much harder, and stiffer, stationary object is struck by a seemingly soft body but which has high velocity and high kinetic energy. O’Neill (1967) relates the experiments of Bottone (1873) in which a rapidly revolving disc of soft iron cuts a hard steel plate, and a tallow candle fired from a gun is said to be able to penetrate a wooden board. There is a pub trick performed with a dry beermat that goes as follows: make a horizontal ring with the forefinger and thumb of the left hand and place the beermat over the hole. Stiffen the little finger of the right hand and drive it through the beermat at high speed and you have a clean hole through the beer mat. You get entirely different behaviour if you push slowly. It is important not to bend the little finger even slightly, or to lose one’s resolve, otherwise a very painful finger and no hole will result.
15.2 Grounding and Collision of Ships: Diverging and Converging Tears Glancing collisions of very stiff structures resemble the processes of scratching. Grooves may be made on the underbelly of a military tank when riding over very rough stony ground, for example, and using the ideas in Chapter 8 the forces and energy involved can be estimated from the size of the groove and the mechanical properties of the armour plate. A digger loading rocks into the back of a lorry will cause more than scratching when the rock is dropped, rather than placed, in the vehicle. Fractures may form in the dents. This is what happens in collisions of less stiff, hollow, structures where more extensive deformation (denting) occurs on which scratching is superimposed. Ultimately scratching itself becomes insignificant compared with the energy required for permanent deformation of the hollow body. It is not only the skin or plating of a hollow structure that is deformed but also the frames, ribs and stringers, and welds attaching all the latter fittings may break in the deformation, thus reducing the constraint to further plastic flow. One or more cracks may be initiated in the dented regions and tearing will be part of a ‘travelling distorted and holed region’ propagating in the direction of the glancing collision (e.g. along the hull of a ship). It is an example of elastoplastic fracture. The original shape of the damaged hull cannot be regained unless all the denting and scrunching is flattened out, after which the fractured plate can be welded back together (in practice, new plating is substituted to effect a repair). The forces involved and energy consumed by the travelling zone can be estimated from the plastic flow, fracture and friction (e.g. Wierzbicki & Thomas, 1993; Cerup-Simonson & Wierzbicki, 1997), and compared with the kinetic energy of the ship just before contact. Questions may then be answered regarding (i) whether the vehicle will be damaged; (ii) how long a tear may
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result; (iii) whether there will there be enough energy to ride over or along the pointed obstacle; (iv) whether the ship will sink; and so on. These sorts of calculations may also be applied to similar problems such as a ship stranded on a rock with a falling tide and ship grounding. Experiments have been performed in which instrumented ships have been deliberately run aground to see what happens, and the associated deformation and damage has been modelled with finite element methods (FEM) by Tornqvist (2003) (Figure 15-1), who has also modelled the structural results of explosions on-board ships. There are two types of framing in the construction of ships: the traditional sort with many horseshoe-shaped frames with a limited number of stringers running the length of the hull (cross-framing), and longitudinal framing, where there are many stringers and few frames. In grounding and collisions, cross-frames constrain deformation and fracture, whereas there is far less resistance to tearing with longitudinal framing. Figure 15-2 shows the author inspecting damage to a 46 000 tonne cruise ship that had struck rocks off the Scillies, south-west of
Figure 15-1 Numerical simulation of the full-size ship grounding experiments. (a, b) The finite element methods (FEM) model and the artificial conical rock; one side of the inner bottom plating is removed for illustrative purposes. (c, d, e, f) The FEM model after a deformation of 5 m (after Tornqvist, 2003).
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Figure 15-2 The author inspecting, in dry dock, damage to a 46 000 tonne cruise ship. Having struck rocks off the Scillies, the cross-framing of the ship reduced the severity of the damage, but a hole was made below the water line even so.
Cornwall, UK. The progressive arresting of damage at the cross-frames is seen, but eventually a hole appeared below the waterline. The Exxon Valdez oil tanker sailed over a rock in the Prince William Sound in Alaska in 1989 and leaked a great deal of oil. When she was dry-docked for repairs, it was discovered that instead of a single tear that might occur with a sharp obstacle, the extensive damage consisted of an unusual diverging tear, similar to what happens when a sheet of paper, held to a noticeboard by a drawing pin, is torn off; similar marks are found on the pages of newspapers where teeth have pulled them through the press, and they develop when a wall, under tension in a loaded plastic shopping bag, is punctured and the contents of the bag fall out. For a ship, this sort of damage is ecologically dangerous, since the longer the tear the wider the tear and the more oil is released into the sea. The mechanics of this sort of tear formation was investigated by Wierzbicki et al. (1998). In contrast to diverging tears, it is possible to produce converging tears in the same material by another method of loading (Muscat-Fenech & Atkins, 1994). A Boeing 737 landed at Hawaii in 1988 with the top of the fuselage having come off in mid-air. Converging tears are evident in the lower half of the fuselage. These tears are also familiar when opening packaging and when ripping wallpaper from walls; when sticky tape irritatingly tears to a point on unrolling from the dispenser, it is one-half of the ‘gothic window’ shape. Jacques Villeglé has illustrated a book with pointed tears. Converging and diverging tears are readily produced in baking foil. For the converging tear, cut two parallel slits inwards from an edge of the sheet, and pull the flap. For the diverging tear, make a hole with the tip of a ball-point pen somewhere in the middle of a sheet of foil; lay the punctured sheet on a flat surface, insert a fingernail, and drag the finger across the surface. Make sure that the nail does not lift off the end off the flap, otherwise a diverging tear may revert to a converging tear. Perhaps surprisingly, both diverging and converging tears are minimum energy deformations. This is easier to understand for the converging path where the work entails plastic
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bending of the tongue and fracture work along the sides: considering the parallel path as a reference, (i) were the path to diverge, the crack path is longer (third side of a triangle) so more incremental bending work and more incremental fracture work is necessary; but (ii) when the cracks converge, while more incremental fracture work is required, less bending work is required as a diminishing amount of material is being rolled up (Atkins, 1995a). The work for diverging tears also concerns plastic deformation of the flap and fracture work along the sides, but now the flap deforms non-uniformly as a concertina where the permanent deformation is restricted to discrete plastic hinges in the roofs and valleys. The wavelength of the concertina also increases with propagation (Wierzbicki et al., 1998). The difference between uniform bending in the one case, and discrete hinging with changing wavelength in a concertina in the other, is the cause of the different crack paths.
15.3 Progressive Dynamic Fracture Structural dynamics concerns the vibrations of assemblies of elements in structures, such as frameworks, gives the natural frequency of vibration, and what may happen under forced periodic loading and so on. An interesting question is what happens when an element breaks for some reason while the framework is under load. Fracture could be caused by the member being defective in some way and fracturing at a load smaller than the design load; fracture could also occur in a bridge, say, were the member accidentally cut by rail or road traffic passing over, or from outside by being crashed into by an aircraft, or by being shot away in war. Central to the philosophy of fail-safe, or damage-tolerant, design is that a damaged structure should still safely carry the design loads. This aim is achieved by employing structures with many redundancies where, even with load shedding from a number of damaged or broken elements, the damaged structure can still bear the design loads. Damage takes various forms: plastic hinging, fracture, buckling, etc., and perhaps combinations of all. In structural integrity calculations of the load-carrying capacity of a structure with one or more failed members, it is customary to perform static analyses of the structure with the elements removed. Clearly it is possible to get a quasi-static cascade of failures once one member has failed when the remaining damaged structure cannot sustain the applied loads statically (e.g. Gross & Maguire, 1983). McConnell and Kelly (1983) made a study of the progressive failure of warehouse racking initiated by collision with a moving fork-lift truck. In this sense, dislodgement of elements of a framework is a form of cutting. But an equilibrium analysis that predicts a damaged structure should be safe will not be conservative and may not provide a correct picture of events. For, when a premature failure of an element takes place with the structure under load, energy stored in the structure is released and this inaugurates a state of transient vibration about the new equilibrium position. The members of the structure will therefore experience transient loads and displacements greater than the values given by the static analysis, in the manner of a mass vibrating on a spring. Clearly there is the possibility that these dynamic stresses and strains may produce failure in another member of the structural system even though the new steady-state stresses and strains in the damaged system may be below failure levels. Failure of a second member will excite further vibration and either more failures will occur or the process may arrest. Casciati and Faravelli (1984) discuss this sort of thing in connexion with seismic reliability analyses of buildings, and the problem is of importance in the damage survival assessment of tethered offshore platforms (Lotveit & Wangenstein, 1985). Pretlove et al. (1991) demonstrated that there are structures that appear to be safe from progressive fracture on the basis of static calculations, but which are unsafe when transient
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Safe statically Safe
I 0.6
Unsafe
Unsafe dynamically
II 0.7
M
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Probability of survival
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III L
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1.0
Figure 15-3 Safety boundaries for the spoked-wheel structure taking into account the statistics of member failure load. Computer simulation: probability of static survival. Computer simulation: probability of dynamic survival. 3 Experimental: probability of static survival. () Experimental: probability of dynamic survival. – – Theoretical boundaries for zero spread of member failure load. — Theoretical boundaries for 6.1 per cent standard deviation of member failure load (after Pretlove et al., 1991).
motions are taken into account. A multiply guyed mooring post associated with an offshore installation had allegedly failed when, with the mooring ropes under tension, a failure in one was said to have led to progressive failure in several others, even though according to static calculations, the system was supposedly fail-safe. They analysed, and performed experiments on, a twelve-member radially tied structure with two degrees of freedom to model the offshore failure (a sort of bicycle wheel arrangement). The nature of cascaded failures was predicted by means of Monte Carlo type computer simulation and compared with actual experiment with good agreement. Safety boundaries were established between the three regions of (i) always statically safe; (ii) statically safe but dynamically unsafe; and (iii) always unsafe, as shown in Figure 15-3. The study provides satisfactory evidence that the dynamic effect is important and easily seen to occur in practice. Although progressive dynamic failures of the sort discussed here involve vibrations, wave effects are not part of the solution. The following impact problems do require consideration of stress and displacement waves within structures.
15.4 Accidents Involving Cables A barrage balloon defence consists of a number of thin wire ropes held aloft by balloons and located around some important installation on the ground that has to be protected against dive bombing and low-level aerial attack. Attacking aircraft would want to fly low in order to pinpoint the target, and the idea was that aircraft would be deterred from so doing as they would be cut through by the cables. Barrage balloon defences were employed up to 10 000 feet (3000 m). In 1943, the US Army Air Forces (USAAF) made a low-level bombing attack on the Ploesti oil refinery in Roumania with B-24 Liberators. The installation was protected by barrage balloons flown at about 700 m (about 2000 feet) tethered to the ground by cables of
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diameter 2.5 mm. Seven planes happened to fly through the cables, yet none was severely damaged. Instead the cables broke. In France in 1961 and 1989, and in Italy in 1998, aeroplanes collided with cableways resulting in terrible accidents when gondolas full of people fell to the valley below (Lombard, 1998a; Marchelli, 1998; Nejez, 1998). What is surprising about these incidents was that while the ropes broke, the planes were hardly damaged (apart from some cutting into the wings). The pilots felt nothing and managed to continue flying and to land. Why did the ropes break in the aircraft impacts and not the wings? It seems contrary to common experience: a violin string slowly deflected to the side by a screwdriver will eventually snap; but if pushed by a block of butter, it is the latter that is cut through. The ‘minimum length of tow rope’ problem is well known in dynamics. When the slack in a tow rope is taken up by one car pulling another, or by a tug hauling a ship, or by a motor boat and water skier, energy seems to disappear. That is, just after a tug of mass m having velocity vtug, jerks into motion a stationary ship of mass M, the velocity of the two together is [m/(M m)]vtug, from conservation of linear momentum. The kinetic energy of the system before was mvtug2/2; afterwards it is m[m/(M m)]vtug2/2. Consequently, energy of magnitude [M/(M m)] mvtug2/2 has disappeared. What has happened is that the towrope has been stretched and the missing energy is stored as elastic strain energy in the rope. Now the maximum allowable strain energy per unit volume that can be stored in the rope before it breaks is given by f2/2E where f is the breaking stress of the rope (assumed to be in the elastic range) and E is the Young’s modulus. The volume of the rope is LA, where L is length and A the cross-sectional area, so the total amount of energy that can be stored in the rope is (f2/2E)(LA). For a given rope with particular f and A, the minimum length of rope required to absorb all the energy is Lmin [M/(M m)] E mvtug2/f2A. What happens if we apply these simple ideas to barrage balloons? There is uncertainty, perhaps, whether we can because the circumstances are not the same, the rope being struck transversely, not longitudinally, and uncertainty about what length of rope ‘knows’ about the impact. Nevertheless, assuming that the rope is stretched either side of the impact point, the tug is the aircraft and the ship is the rope. Either Lmin could be predicted and would be the shortest length of rope that would not break when struck by an aircraft travelling at vtug or, for given length of rope, vtug could be evaluated as the maximum speed of aeroplane that would be brought down by the rope barrage. Note that the solution does not say where the rope will break. Since the mass of the aeroplane is most likely to be much greater than that of the rope, a simple calculation would say that unless all the kinetic energy of the aircraft can be stored in the rope, the rope will fracture. A wire rope of 2.5 mm diameter has a breaking load of some 2.5 kN, meaning that f 500 MPa. For the Ploesti 700 m long rope, calculations show these limiting speeds are 89 m/s (500 kg aircraft), 63 m/s (1000 kg) and 20 m/s (10 000 kg) assuming a full 210 GPa for E for steel. Wire ropes are not solid steel and E is less. For E 70 GPa, the corresponding aircraft velocities are 155, 110 and 35 m/s. A better analysis goes as follows (Pretlove, 2008). The collision between aircraft and rope is an ‘impact’, where rate effects and inertia come into play. This sort of thing has been discussed in Chapter 10 on the cutting of unrestrained bodies and the behaviour of devices like strimmers. When a wire is deflected and let go so as to vibrate, simple vibration theory assumes that the stress waves that communicate the changes to the ends of the wire from the point of excitement will have run up and down the wire many times, been reflected and damped out, all within the time that a new deflected shape is taken up. In reality, the deflected transverse shape itself is produced by a disturbance wave that moves out from the point where the wire was deflected. The two waves (one across and one along the wire) have
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different velocities and both types produce variations in stress up and down the length of wire, which turn out to be important in impact situations. The velocity of (i) the stress wave along the rope is (E/), where E is the elastic modulus of the rope and is the density; and (ii) the deflexion wave transverse to the rope is (T/A), where T is the tension in the rope and A its cross-sectional area. The velocity of a longitudinal wave in a solid steel rope is (210 109/7850) 5000 m/s. For f 500 MPa in a barrage balloon rope of diameter 2.5 mm, the greatest tension T in the wire is 500 106 3 0.00252/4 2500 N. The working load in normal operation is less than this by a factor of safety, say 3, which results in a working stress of 167 MPa. The velocity of the transverse wave is thus (167 106/7850) 146 m/s. The transverse wave is thus considerably slower than the longitudinal wave. In an aerial cableway, too, E is perhaps 25 times bigger than the stress in the rope given by (T/A) so that the longitudinal stress wave speed is about five times faster than the transverse deflexion curve (Oplatka & Volmer, 1998). The quasi-static assumption of simple vibration theory, that the deflected shape and resulting changes in axial tension are produced instantaneously, fails when the perturbation (‘plucking’) occurs under high-rate impact conditions. Understanding what happens has to take waves into account. Furthermore, the behaviour of wire rope is affected by high loading rates and that has to be incorporated in analysis. A steel rope is a complicated structure made up of extremely hard (patented) wires but whose overall quasi-static behaviour is nevertheless relatively flexible. When loaded at a sufficiently high rate, however, a steel rope breaks relatively easily. It is common experience that some materials fracture more easily when subjected to a ‘jerk’. The speed of the B-24 Liberators was, say, 225 mph (100 m/s). Consider Figure 15-4, where the centre of the wire is supposedly deflected by 1 m by the aeroplane, the time taken for which is 1/100 10 ms. In that time the longitudinal wave will have travelled 5000 0.01 50 m, and the transverse wave 146 0.01 1.46 m, above and below the point of impact: the ends of the rope will have no idea of what is happening. For purposes of calculation let us assume that the shape of the deflected rope after 10 ms is two quadrants of a circle of radius 1.46 m, from which the extension of the rope is 2 1.46 [(/2) 1] 1.67 m. This extension stretches the 2 50 100 m length of rope affected after 10 ms by the longitudinal wave, resulting in a tensile strain of 1.67/100 0.017. Consequently, the extra stress is 210 109 0.017 3.5 GPa using the full E, or 1.2 GPa using E 70 GPa. In either case the stresses are far above the failure stress of 500 MPa, so the rope has broken. The cross-sectional area of the rope is (0.0025)2/4 5 106 m2, so the extra load is 18 kN or 6 kN. The horizontal force on the leading edge of the wing is given by resolution which, for the quarter-circles in Figure 15-4, is double these loads. The time taken for the longitudinal wave to reach the top and bottom of the 700 m wire is 700/5000 140 ms. The top of the wire, even with the balloon attached, is more like a free end where a tensile wave is reflected as a compression wave, but the bottom of the wire is built in, so that the tensile wave is reflected as another tensile wave to double the stress in the rope. Fractures at the bottom of the rope are possible when collisions occur sufficiently far below the mid-point of the long rope, so that the longitudinal wave can reach the anchoring point before the transverse wave has produced sufficient sideways deflexion to produce the axial fracture strain. These calculations suggest that there is a critical impact speed below which wave effects will not matter, when the assumptions of simple vibration theory are acceptable. Below this speed, the wire or the colliding object (or both) may or may not suffer damage depending upon their
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1.46 m
Figure 15-4 Assumed transient geometry of barrage balloon cable after impact by aeroplane travelling at 225 mph (100 m/s). In the time taken to produce a central deflexion of 1 m, the longitudinal wave will have travelled 50 m, and the transverse wave 1.46 m above and below the point of impact: the ends of the rope will have no idea of what is happening. For purposes of calculation the shape of the deflected rope is assumed to consist of two quadrants of a circle of radius 1.46 m.
geometry and properties. In war, one side may string wire between trees across a road or track to take enemy riders off motorcycles. In contrast, motor vehicles may be able to crash through boundary wire fences. Housner (quoted in Irvine, 1981) determined the critical speed assuming that the limiting velocity of the plane occurs when the tensile failure strain of the rope is attained. For impact against high-strength cold-drawn steel wires having a failure strain of 0.007, it is about 150 m/s (say 300 mph). The idea that wires held aloft by barrage balloons over some installation would cut through aeroplane fuselages and wings is therefore correct when plane speeds are relatively slow. Modern planes (especially military) have far higher speeds than the critical, of the order of 350 m/s, so barrage balloon defence would appear to be no longer effective. The incident at Cavalese in the Dolomites, northern Italy, in 1998 involved a US military jet (a Prowler having dead weight 14 tonnes, maximum laden weight 29 tonnes) that hit the ropes at 240 m/s (540 mph). Figure 15-5 shows the position of the plane relative to the ropes as seen from the direction of travel. The upper rope is the track rope (51 mm diameter, weighing 15 kg/m; breaking load 2500 kN); the lower is the haulage rope (20 mm diameter, weighing 1.4 kg/m; breaking load 2000 kN) (Oplatka & Volmer, 1998). Both ropes had been in place for about eighteen months. The track rope produced a tear about 1 m long in the wings, and the haulage rope tears in the wing were about 0.5 m long. The working tension in the track rope was about one-third of its static breaking strength; it was about one-fifth for the haulage rope. The axial breaking strength is reduced by transverse forces (they ‘interact’) and it takes a transverse force of about two-thirds of wire breaking
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Figure 15-5 Ski cablecar accident at Cavalese in the Dolomites, northern Italy, in 1998. The wing of the aircraft collided with the upper track rope at one point. The lower haulage rope was hit at two points approximately 1.5 m apart, one by the wing and the other by the tailplane (after Oplatka & Volmer, 1998).
strength to cause a first wire fracture in the case of the track rope; and about four-fifths in the case of the haulage rope. Thereafter, under the same axial load, the tension in the remaining wires increases and lower transverse forces can cause successive wires to fracture. The question therefore is whether the aircraft was able to exert the required transverse force on the wires and break them, or whether the inertia of the ropes was sufficient to generate forces capable of slicing through the wing before the two ropes themselves broke. Impact calculations (Oplatka & Volmer, 1998) show that the transverse forces exerted on the rope increased so rapidly that rope fracture occurred after just a few milliseconds, which is confirmed by the limited length of the tears in the aircraft panels: if the ropes had not fractured, they would have sliced through the wing. These calculations show that a cableway impacted at a sufficiently high speed is incapable of stretching quickly enough to accommodate the displacement due to a moving aeroplane and failure occurs at the point of impact in a very short time. Lombard (1998b) gives a detailed analysis of the problem and shows that fracture occurs when the longitudinal speed of propagation of the transverse disturbance provoked by the impact is greater than the speed of propagation of the transverse movement of the rope assumed to carry a tension equal to the breaking load. The force exerted by the rope is applied for the very short time interval of milliseconds. The plane receives an impulse (given by the product of mean force by the time of application) that causes a small loss in the normal speed of the plane. The corresponding loss of kinetic energy is that required to deform and fracture the rope. The reduction in speed of fast aeroplanes is usually minimal.
15.5 The Twin Towers The exterior columns of the World Trade Center (WTC) were thin-walled box beams made of steel. A Boeing 767 moving with a cruising speed of 240 m/s (500 mph) hit the exterior wall, cut through it, and disappeared in the smoky cavity, on 11 September 2001. As remarked by Teng and Wierzbicki (2005), it appeared to the casual observer that the façade of the Twin Towers did not offer any resistance at all, and that the plane’s wings and fuselage sliced through the
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exterior columns as if they were made of cardboard. The silhouette of fuselage, two engines and two symmetric narrow slots cut by the wings is clear in photographs. A slightly larger opening was caused by the falling floor, which dragged the sections of the exterior columns inside the building. Fragments of the airframe ploughed through the truss-like floor of the WTC Towers and hit the core structure. How was it possible that the relatively weak, light and airy airframe damaged the apparently heavy lattice of steel columns? Damage caused by Japanese kamikaze pilots in World War II deliberately crashing into ships is similar, as are mid-air collisions of aircraft. The problem is one of mutual indentation of crossed hollow beams of comparable strength that impact against one another at high velocity. They mutually deform with a complex interactive failure pattern. The general problem is formidable. Even so, depending on the geometry and properties of the two colliding bodies, certain simplifications are possible. For example, all the different structural members of the wing of an aeroplane may be lumped together and ‘smeared’ into a box beam of the same span as the aeroplane and having the same mass. In the case of a Boeing 767, the equivalent box beam has a thickness of over 100 mm. In comparison, the thickness of the hollow external columns of the Twin Towers is 9.5 mm. The 10:1 ratio thus permits the segment of the wing, which collided with the building, to be treated as a rigid undeformable mass to a first approximation. The problem then becomes the analysis of a thinwalled box beam subjected to high-velocity impact by a rigid mass (Teng & Wierzbicki, 2005). Quan and Birnbaum (2002), among others, have made FEM simulations of the attack on the Twin Towers and Teng & Wierzbicki (2005) emphasised the role of fracture in their analysis. They divided into three phases: (i) instantaneous cutting by the wing through the front flange; (ii) tearing of the side webs; and finally (iii) tensile fracture of the rear flange. The impact problem is dominated by the local inertia of the box column so that plastic deformation and fracture are concentrated to the immediate vicinity of the impacted parts of the column and wing. It was found that the fracture process started immediately, and continued as plate tearing on the side webs, soon leading to tensile/shear fracture on the rear flange. In each stage, the resisting forces arising from plastic deformation and fracture were calculated and the time history of the velocity of the impacting wing section was determined. The minimum impact velocity to cause fracture of the exterior column was found to be 155 m/s. The energy lost by the wings cutting through the exterior column dissipated about 1.1 MJ of energy, which was only about 7 per cent of the initial kinetic energy of the wing, so the wing had little difficulty in cutting through the outer column. The remaining 93 per cent of the kinetic energy was then transferred into the interior of the building causing fatal damage to the floors and core structure. Teng and Wierzbicki’s analysis also suggests that the exterior column would have been able to stop the aeroplane wing – or at least prevent a local shear failure – if the average yield stress of the steel had been double its value of 350 MPa. Thus, had the plane hit the base of the Towers which were made of higher strength steel having a yield stress of 700 MPa, the aircraft might have been deflected by the exterior walls. The analysis also helps to explain subsequent stages of the impact in which parts of the airframe continued onwards through the truss-like floor of the WTC Towers to hit the core structure. Teng and Wierzbicki’s analysis does not consider the effect of the fuel inside the tanks within the wing boxes. The fuel will greatly increase the mass/unit length of the wings and add to their destructive ability. High-velocity impact of fuel-filled tanks into deformable structures is a challenging problem in itself.
Appendix 1
Friction Forces on a Wedge-shaped Tool Cutting Orthogonally An inclined wedge of included angle represents a sharp cutting tool. In Figure 3-1, the wedge is tipped up slightly through a clearance angle so as not to rub the cut surface. The offcut therefore contacts, and slides up, only one surface of the wedge (the rake face). The inclination of the wedge to the vertical is called the rake angle. When the wedge is tipped right over can become negative. Different fields of study define the inclination of a cutting tool in different ways. Sometimes a so-called ‘cutting angle’ is referred to without saying what is meant. In metal cutting, the rake angle measured from the normal to the cut surface is invariably used. The wedge angle is thus (90 )°. In ploughs and other earth-moving equipment, the angle of inclination of implement blades from the cut surface is often used and is confusingly called the rake angle (Chapter 14). With bevelled knife-like tools, as in microtomes, where the included angle of the blade is small, the wedge (knife) angle itself is often all that is quoted, but the clearance angle must then be known to give the inclination of the rake face. When clearance angles are small ( 5°, say), it is often permissible to think of the wedge angle as the complement of . For simplicity, we shall refer only to and , so that the clearance angle is implicitly included in . Friction forces may be expressed algebraically in terms of either or ; one form is sometimes more convenient than the other depending upon the problem. Clearly, substituting (90 )° or vice versa transforms the expressions. A free-body diagram of the offcut (Figure A-1A) has the contact force N from the rake face pushing the offcut upwards and the friction force F (which opposes the motion of the offcut up the rake face) pushing down towards the cutting edge. A free-body diagram of the tool (Figure A-1B) has N pushing down on the wedge, and the friction force F between offcut and tool trying to move the wedge up the sloping face. F and N on the offcut are reacted by how the workpiece is gripped, and F and N on the tool are reacted by the resultant applied force on
Tool
� F
Chip
Fres N
N
FC
Fc FT
Fres
FT
F
A
B
Figure A-1 Free-body diagrams of the offcut and of the tool. Copyright © 2009 Elsevier Ltd. All rights reserved.
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the wedge that has components FC parallel to the cut surface and acting towards the cutting edge, and FT perpendicular to the cut edge and acting downwards into the cut surface. This is the sign convention often employed in metal cutting; in other fields, positive FT is considered to act upwards away from the cut surface (Section 3.2). Resolving horizontally and vertically
FC Fsinα Ncosα
FT Fcosα Nsinα
FC Fcosθ Nsinθ
FT Fsinθ Ncosθ
(A1.1a)
and
(A1.1b)
or
(A1.1c)
and
(A1.1d)
We have assumed that all forces meet in a point (the resultant forces on the offcut and tool are in line), so that moment equilibrium is automatically satisfied. Equivalently, resolving along and perpendicular to the rake face we obtain
F FTcosα FCsinα
N FCcosα FTsinα
(A1.2a)
and
(A1.2b)
or
F FTsinθ FCcosθ
(A1.2c)
and
N FCsinθ FTcosθ
(A1.2d)
For Amontons/Coulomb friction F N. Then
FC N(µsinα cosα)
FT N(µcosα sinα)
(A1.3a)
and
(A1.3b)
or
FC N(µcosθ sinθ)
(A1.3c)
369
Appendix 1 and FT N(µsinθ cosθ)
(A1.3d)
Again, by division of the above pairs of equations,
FC FT (µtanα 1)/(µ tanα) Hα FT
(A1.3e)
and
FC FT (µ tanθ)/(µtanθ 1) H θ FT
(A1.3f)
These equations, but with different symbols and sign convention, may be found in Williams (1998). Positive FT means that the tool is pushed into the surface, so when tan, FT will be negative and the tool lifted away from the surface. The offcut in Figure A-1 is shown contacting the blade only on one face, and there is supposedly no contact with the cut surface under the blade owing to the clearance angle. When cutting through the middle of a block of material with a knife, there will be contact on both faces of the blade and no clearance angle. In fact, most knives will have contact with both offcut and cut surface even when cutting thin slices, unless the blade is angled to the cut surface. When there is contact on both faces of a symmetrical wedge, it is readily shown that FT is zero because the forces balance automatically, and then FC is doubled. In books on metal cutting, the force equilibrium is done slightly differently. Figure A-2 shows the well-known force-circle construction (but omitting the forces on the shear plane for reasons explained below). (i) Instead of FC, FT and N, the resultant force Fres acting on the face of the wedge in contact with the chip/offcut is used where Fres (FC2 FT2) (F2 N2) (1 µ2)N for Coulomb friction. (ii) Instead of , the angle of friction is employed, where tan , whence Fres N sec.
FC α
β−α FT
Fres
F
β N
Figure A-2 The force-circle construction of metal cutting but omitting the forces FS and FN along and across the shear plane because it is not known a priori how much of the external work done goes to separation work (which depends on an area term) and how much goes to plasticity (which depends on volume). To obtain the shear force per unit width (FS/w), a ‘reduced’ cutting force/ width given by [(FC/w) R] must be resolved along the shear plane.
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There results
FC Frescos(β α)
FT Fressin(β α)
FT FCtan(β α)
(A1.4a) (A1.4b)
and
(A1.4c)
Despite looking quite different, Eqs (4) are the same as Eqs (3). If tan is substituted for in Eq. (3e), standard trigonometric functions will give Eq. (4c), and it is easily shown that the force-circle relations of metal-cutting theory such as
F Fres sinβ FCsec(β α)sinβ
N Frescosβ FCsec(β α)cosβ
(A1.5a)
(A1.5b)
are identical with these basic force equilibrium relations, taking care to observe the sign conventions employed in different presentations, e.g. Alexander & Brewer (1963). All the above relations for equilibrium of forces on the cutting tool will apply irrespective of the manner in which the offcut/chip is formed (in bending or by shear). However, unlike traditional force circle diagrams, the force circle in Figure A-2 does not include the forces along and across the shear plane that relate to the ‘internal’ work of plastic shearing. This is because when internal work has two components (one for plasticity that can be written in terms of a force, and another for separation work that cannot be expressed as a force), a direct equilibrium approach for forces relating to internal deformations cannot be employed. This is much clearer if a work analysis is employed, as in Chapter 3 and elsewhere in the book. That is, it is not known how much of the work rate performed by the tool (given by FCV, where V is velocity) is employed in supplying work of separation (toughness that depends on an area term) and how much goes to chip formation (plasticity that depends on volume). Traditional theories have only plasticity, so resolution of external forces along the internal shear plane, to give a stress to coincide with the yield stress in shear, is correct, but it is not when separation work is involved in addition to yielding work. To obtain the shear force FS along the slip plane in these circumstances, a ‘reduced’ cutting force/width given by [(FC/w) R] is resolved along the shear plane to give
FS [(FC /w) R]sec(β α)cos(φ β α)
(A1.6)
This shows that an intercept is expected in plots of FS vs AS, where AS ( wt/sinφ) is the area of the shear plane, exactly as found in practice (Section 3.6.6). Were R 0 in Eq. (6), the traditional force-circle resolution for FS would be recovered. Note that these remarks apply only to consideration of the internal work components; the force-circle construction is correct for the relations between (F,N) and (FC,FT).
Appendix 2
Friction in Cutting As explained in Section 2.7, friction has been modelled in this book mostly in terms of Amontons/Coulomb friction where F N. The friction force F and normal force N on the rake face of the cutting tool are usually determined by resolution of dynamometer forces FC and FT as explained in Appendix 1. F and N are rarely measured directly and there are surprisingly few plots of F vs N in the cutting literature. Wallace and Boothroyd (1964), Hsu (1966), Childs and Maekawa (1990) on cutting metals; Klamecki (1976) and Kobayashi and Hayashi (1989) on wood; Aluko (1988) on soils; Williams et al. (2009) and Patel et al. (2009a,b) on polymers, are papers that present friction plots. In many cases, particularly when cutting engineering materials, it is found that the coefficient of friction given by tan F/N is not constant and systematically varies with depth of cut and rake angle. In the absence of any explanation for the experimental departure from the assumption of constant in modelling, some average value over the whole range of t for given may be employed to check cutting forces and power. Alternatively, another model for friction in cutting postulates that the average friction stress on the rake face qF ( F/Lw, where L is the contact length between chip and tool) is some proportion m of the shear yield stress k of the workpiece, i.e. qF mk (m 1; sticking at m 1). Here again, however, m is not constant. The forces F and N are the integration of the distribution of shear tractions and pressure p over the contact area between chip and tool. Experiments have been performed to try and discover the distributions of and p. Four types of measurement have been made: (i) using photoelastic tools to cut soft metals such as lead (Andreev, 1958; Usui & Takayama, 1970); (ii) using split tools where the forces on different lengths of the rake face are measured (Kato et al., 1972, on aluminium, copper, lead and zinc; Shirakashi & Usui, 1973, on brass and steel; Barrow et al., 1982, on nickel–chrome steel; Childs & Maekawa, 1990, on steels); (iii) using ingenious transparent tools that enable chip motion over the rake face to be viewed directly (Nakayama, 1957, on lead; Doyle et al., 1979, on soft copper, aluminium, lead and indium; Bagchi & Wright, 1987, on steel and brass); and using deformable tools (Rowe & Smart, 1972). Photoelastic tools were employed as early as the 1920s to determine the stresses within tools themselves (e.g. Coker & Chatto, 1922) as part of the Institution of Mechanical Engineers (IMechE) Cutting Research Committee investigations. A paper by Toropov and Ko (2003) has a number of references to papers in the Russian literature on instrumented cutting tools. In broad terms it is found that the contact pressure is greatest near the tool edge and tails off towards the end of the contact region with the tool where the tool curls away, Childs (1980). The shear (friction) stress is constant for some distance from the cutting edge, and then it too tails off to the point of loss of contact. The constant frictional stress is at about the level of the shear yield stress of the workpiece and the region it encompasses is the sticking region. The conditions under which large portions of the interface between the contacting bodies can be welded together are often found in commercial metal cutting, and relate to surface cleanliness and surface finish, environment (lubricants and coolants), high shear rates producing high temperatures that may even produce a molten surface layer, and mutual solubilities of the bodies in contact (e.g. Keller, 1963; Rowe & Smart, 1972). Copyright © 2009 Elsevier Ltd. All rights reserved.
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Zorev (1963) proposed (, p) contact stress distributions between chip and tool based on Kattwinkel’s 1957 study where a single cut, t 0.5 mm deep, was taken on lead with a PMMA photoelastic tool having a rake angle of 10°. Figure A-3(a) shows that the normal stress p rises steeply towards the tool tip at the left hand end, from zero at the right hand end where the chip loses contact with the rake face, to give a concave-up distribution. The interfacial shear stress attains the value of the shear yield stress k over a region of length s near the tool tip, where sticking/adhesive friction prevails with transfer of workpiece metal on to the rake face. Elsewhere in the contact region where the normal stresses are lower, sliding friction is assumed and p, where is some coefficient of sliding friction. This coefficient of friction is called ‘real’ in order to distinguish it from the value of ‘F/N apparent’ which is often called the coefficient of friction in cutting, and which we have shown varies with depth of cut and rake angle. For simplicity, Figure A-3 shows k as constant and ignores workhardening. Zorev (1963) employed a power law for the pressure on the rake face in Figure A-3 given by p po (x/L)n
(A2.1)
where x is measured towards the tip of the tool from an origin at the end of the contact region, and po is the maximum pressure at the cutting edge which will vary with t and . For the normal pressure distribution shown in Figure A-3(A), n 1; Zorev’s results for cutting steel in air suggested 3 n 4. In an experiment using a 5° rake angle tool with t 0.15 mm, he found metal transfer over a length of about 0.53 mm when the contact length was 1.26 mm, thus giving s/L 0.4.
n1
p
k
p
k �p
�p
s
s
x
l
l A
x
B
Figure A-3 Based on Kattewinkel’s experiment, Zorev assumed stress distributions on the rake face of a tool given by p po (x/L)n. (A) The usual contact stress distribution where n 1; (B) the contact stress distribution where n 1, which seems more appropriate when the sticking zone occupies a large proportion of the rake face.
373
Appendix 2
The relationships for the variation of F and N, or the variation of the average normal stress qN ( N/Lw) on the rake face with the average friction stress qF ( F/Lw), that are expected from Zorev’s stress distributions may be determined (Atkins, 2009). We have
F wks µ real [1 (s/L)]n1N
(A2.2a)
and
q F k(s/L) µ real [1 (s/L)]n1q N
(A2.2b)
Equations (A2.2) predict plots that pass through the origin with an initial slope of real when there is no sticking (s 0), followed by a diminishing local slope given by real [1 (s/L)]n1. When s L (complete sticking) the slope of the diagrams become zero. Data from restricted contact ‘Klopstock’ (1925) tools, where high values of (s/L) may be achieved, follow the predictions of Eq. (A2.2); see Hsu (1966) and Wallace and Boothroyd (1964) for HE-10-WP aluminium alloy; Sharma et al. (1971) on steel; and Childs and Maekawa (1990) for BS970 708M40 steel, in the as-received and MAXIM version of the same alloy. Atkins (2009) demonstrates how po and n may be determined from these data, and shows that n 1 at biggest (s/L) (Figure A-3b). However, most data from experiments in which tools having rake faces at least as long as the ‘normal’ contact length are used, follow ‘F Fo N’ linear relations rather than Eq. (A2.2a). This behaviour is found in materials as diverse as metals, polymers (even bulk PTFE), wood, graphite/clay mixtures and so on. Williams et al. (2009) and Patel et al. (2009a,b) have modelled cutting (including toughness) employing F Fo N directly. Atkins (2009) attempts to rationalize the different results. It is clear that both non-linear F vs N and linear-with-intercept F vs N are able to explain the systematic reduction in app and at greater depths of cut, but the reason why app systematically alters is not explained. Sticking over length s along part of the rake face limits the frictional force because k p near the tip. Could it be that the sticking length s, the pressure distribution (po and n), and contact length L between chip and rake face of the tool, all adjust themselves at a given depth of cut so as to minimize the frictional traction F, as suggested by Oxley (1989)? Alternatively, friction work in terms of s, po, n and L should be included in the whole minimization exercise for that includes plasticity and separation work to find absolute minima. It should be noted that in finite element method simulations of cutting, prechosen friction conditions are inputs and not part of the solution as would be ideal (Felice et al., 2007); the same is true for slip line field solutions for plastic flow fields in cutting.
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INDEX [Note in some cases the symbol used for the quantity id listed after the name, e.g. clearance angle ; coefficient of friction ; coefficient of restitution e; and so on]
A Abrasion 143 Abrasive papers 150ff Acceleration of spoil 346 Accidental cutting 353ff Accidents involving cables 360ff Acupuncture 195, 277 Adaptive remeshing in FEM 67 Adhesive strength of coatings 161 Adiabatic discontinuous chip formation 77, 95 Aerodynamic lift on arrows 206 Aeroplanes crashing into buildings 364 Agglomeration 143 Air hardening 225 Al dente 294, 297 Allometry (scaling in biology) 323 Amontons friction 23 Amputation 272 Anchors 350 Angle grinders 172 Angle of friction 23 Anaerobic fuel reserve (oxygen debt in muscles) 27 Anatomical dissection 272 Anisotropy 6, 42, 160, 172 Apparent ‘scale effect’ in cutting 66, 154 Apple corer 139 Archaeometallurgy xi, 149, 202ff Ard 339 Arms 1, 202 Armour 1, 202 Arrows 202, 210ff Aspect ratio of spalls 47
Aspherical lenses 144 Attack (by animals) 322ff Augers 134, 346 Autopsy 272 Awl 189 Axe 6, 174, 273
B Back stroke 176, 181 Backward-pointing teeth 308 Ballistic impact 95 Ballistic limit 210, 356 Barbed wire 268 Barbs 196, 267 Bare-knuckle fighting 202 Battleaxe 174, 203 Bayonet 203 Beaks 268 Bearded execution axe 31 Bee sting 3 Bending (of nails, staples, arrowheads) 197, 212 Berkovitch indenter 146 Biaxial loading 15 Bimetallic blades 46, 230 Biological design and function xii Biological materials 259, 285 Biomechanics xi, 260 Biomedical engineering xi, 274 Biomimetics xi Biopsy 272 Bird strike 356 Bite forces 25, 292, 310ff Bite force quotient (BFQ) 311 Bites 263 Blanking 70, 190 Blunt dissection 274 Bluntness, compensation for 242ff
Blunt tools 98, 221ff Bolsters 173 Bolus 296, 312 Boring 3, 9 Bows 5, 202 Boundary layers of separation 19, 32, 64, 68 Brain operations 275 Bread Slicer 3, 248 Brickbat 248 Brittle chipping 33 Brittle offcuts 43 Brushes 5 Buckling (of nails, staples, arrowheads) 197, 212 Buffing 151 Built-up edge (BUE) 24, 92ff, 146, 223, 238 Bulb planter 189 Bulldozer 5, 349 Bullets 3, 202 Bump races (rowing) 354 Burin 166 Burnished area on cropped face 72 Burrowing in soils 327, 335 Burrs 4, 73, 166, 228 Butcher’s ‘steel’ 229
C Cabochon 170 Calkins 200 Caltrop 196 Cambridge critical state model (soils) 331, 334 Can opener 5, 133 Canine teeth 308 Cannula 3, 276 Cans 4
407
408 Carbide tools 226 Carnassial teeth 308 Carpentry 1, 272 Carving 1, 142, 169 Carving meat 286 Castigliano theorems 19 Cast iron 15 Castors 146 Catapults 202 Catheter 276 CATRA goniometer (measurement of sharpness) 232 Cementite 9 Centre of percussion 31 Centrifuge modelling 330 Ceramic tools 226 Cereal crops 246 Chain mail 203, 214ff Chainsaw 7, 172, 345 Chasing 143, 166 Chatter 79, 346 Checks in veneer peeling 101 Chewing 1, 285 Chip flow angle C 120 Chips, classification of 96ff Chips having sawtooth profile 95 Chips of varying thickness 78, 95 Chips, types of 75ff, 342 Chipboard 107 Chisel 8, 171 Chisel tines 343 Chisel-tooth digging 336 Chopping blocks 284 Circular saw 176, 178 Civil engineering ploughs 344ff Classification of chips 96ff Claws 264 Clearance angle 36, 176 Clearance in cropping and guillotining 72, 131 Cleavage 24 Coal-cutting machines 345 Coatings 160 Coefficient of friction 23, 165, 371 Coefficient of restitution e 29 Cold chisel 8, 173 Cold sectioning 76 Collision of ships, vehicles 355 Comb cutters 246, 250 Comminution 22, 76, 143 Complementary strain energy 18 Compliance 16 Compliance calibration equation 18
Index Conkers 202 Contact mechanics of teeth 312ff Continuous band wire 302 Continuous chips 33 Continuous offcuts by elastic bending 47 Continuous offcuts by elastoplastic and plastic bending 51 Continuous offcuts by shear 52 Continuum mechanics and microstructure 78 Control of depth in ploughing 337, 344 Controlled separation of parts 1, 6 Converging tears 279, 358 Cooking food 285, 293 Core samples 139 Corinthian helmet 203 Cork borer 139, 189 Corkscrew 196, 351 Corundum 9 Coulomb friction 23 Coulter 135, 339 Crack existence 33 Crack initiation in ductile solids 23 Crack opening displacement 23 Crack resistance curve JR 21, 52 Crack stability 17, 30, 33, 48, 56, 241 Cracks 16 Cracks at the tool tip 53 Cramp 27 Crampons 350 Crater wear 233 Creasing 6 Creep 13 Crispness 296, 314 Critical attack angle crit 145 Critical crack tip opening displacement C 23 Critical crack tip opening displacement C, connexion with edge sharpness 240, 321 Critical depth in cultivation 343 Critical depth cr at transition in cropping 72, 200 Critical impact speeds for cutting by cable 362 Critical stress intensity factor KC 20 Cropping 70 Crossbow 213 Crunchiness 296, 314
Crushing 143 Cube-square scaling xii, 21, 192, 206, 345, 345 Curved blades 111ff, 127 Curved blade, optimum shapes 250 Cuttability 233ff Cutting as a branch of elastoplastic fracture mechanics xiii Cutting by symmetrical wedge 43 Cutting by wire 300ff, 332, 360 Cutting distance (tool wear) 234 Cutting edge geometry 230ff Cutting forces from FEM 70 Cutting fabrics ‘on the bias’ 6, 224 Cutting fabrics ‘on the cross’ 6, 224 Cutting force FC 37 Cutting force fluctuations 46, 86 Cutting force waveforms 91 Cutting pre-tensioned sheet 40, 274, 279 Cutting ‘stress’ 8 Cutting threads 142 Cutting through the thickness 70 Cutting with more than one edge 141ff Cutting with tools of different edge radii 239 Cylinder lawnmowers 246, 253
D Damage-tolerant design 359 Damascus steel 230 Dartboards 197 Dead metal zone 24, 146, 238 Deck of cards analogy 52 Deciduous (milk) teeth 311 Decoration of pottery 142 Dentistry 1, 272 Defence 1, 269, 267, 322 Deformable hollow structures 355 Deformable tools 371 Deformation transitions 218 Deglutition 308 Delamination 45 Delicatessen slicer 297ff Depth of cut t 24 Depth of bite 285, 315 Determination of R from cr 74 Dew-claw 350 Diamond tools 226 Dibber 189
409
Index Dicing 2 Diet 3 Diffraction gratings 154 Digging 7, 327, 336 Dimensional analysis 47, 51 Disc harrow 118 Disc slicer 116 Discing in ballistics 199, 209 Discontinuous chips 33, 41 Discontinuous-complex chips in cutting polymers 98 Dishing in ballistics 191 Disposable tools 3 Displacement control 36, 317 Diverging tears 279, 358 Dopstick 169 Double wedge test 291 Down milling 107, 178 Draw filing 185 Dredging 336, 345 Dressmaking 6, 224 Drill string 140 Drilling 1, 3, 134ff, 262 Drilling in agriculture 339 Drilling muds 140 Drilling of bone 275 Drills in civil engineering 139 Droop in FC vs t plots 61 Dropweight testing machines 208 Dry machining 235 Drypoint 166 Ductile offcuts 49ff, 119ff, 127 Dullness (of tools) 221ff Dynamic cutting 245 Dynamic equilibrium states in cutting 96
E
(ER/k2) ratio for brittleness and ductility 22, 314 Earthmoving 337 Earths (burrowing animals) 335 Eel spears 196 Effective fracture toughness R* 129 Efficiency of lawnmowers 255 Elastic fracture mechanics 13ff Elastic recovery 13 Elastic strain energy 43 Elasticity 13 Elastoplastic fracture mechanics 49 Electronic speckle interferometry 242 Empirical relations 51, 181, 347 Enamel 270
Endoscope 273 Energy (work) 7, 12, 29 Energy imparted to weapons 206ff Energy methods xiv Energy scaling 79ff Engineering strain 14 Engineering stress 14 Engraving 142, 166ff Environmental dependence 58 Erosion 2, 163ff Erroneously high shear yield sttrength 65 Essential work of fracture method 61 Etching 167 Excavators 345 Excavating rock 192 Explosively-reactive armour 204 Extensible materials 6, 12 External work done 16
Force control 36, 317 Force fluctuations 86 Forensic investigations 205, 355 Fork tines 295 Forward stroke 176 Fossorial creatures 335 Fracture mechanics 11ff Fracture toughness R 16, 59 Franz classification for wood offcuts 86, 100, 102 Fresco 168 Fret saw 176 Friction 13, 73, 115, 336, 367 Friction drilling 138 Friction force F on rake face 38, 63, 367, 373 Friction in cutting 23, 173, 301, 371 Fullering 201 Fuzzy grain (wood) 223
G F Factor of safety 16 Fail-safe design 359 Fault conditions 3 Feathers 5 Feather armour 204 Feeding 269 Fid 190 File hardness 187 Files 8, 151, 171, 181ff Files, resharpening 187 Fingernails 265 Finite element methods (FEM) xiv, 45, 58, 67ff, 91, 105, 209, 301, 334, 357, 373 Firearms 202 First cut (files) 182 Fish hooks 196 Flak jackets 205 Flensing knives 277 Flesh simulants/substitutes 262, 278 Flint knapping 47 Floppy offcuts 37, 114 Flow drilling 139 Flycutter 136, 152 Food-cutting devices 283ff Food, effect of cooking 283 Food, effect of freezing 283 Food fracture toughness 292 Food grater 35, 181 Food mixers 248 Food shelf life 284 Foodstuffs xii, 283ff Force circle plots 66, 369
Gang saw 172 Gas-filled foams (foods) 285 Gauge blocks 5 Geomechanics xii, 327ff Glancing collisions 355 Glass tools 36 Glass transition temperature Tg 54 ‘Going’ on racecourses 351 Gouges 174 ‘Grabbing’ by tool 116, 348 Graders 349 Grain orientation in wood 102, 102ff Gramophone needles 155 Grape shot 206 Grating 1 Graver 166 Green (wet) wood 100 Grinding 3, 76, 143, 150ff, 221 Grindstones 150, 225, 228 Grit blasting 163 Grooves 2, 42, 141, 351 Grounding of ships 356ff Growth and coalescence of voids 23 Gullets 172, 235 Guidewires (drilling) 276 Guillotining 1, 7, 70, 123ff, 174, 279
H Hachoir 2 Hack axe 144 Hack saw 176
410 Hadfield’s manganese steel 154 Hair clippers 246 Halberd 210 Hammer 7, 28 Hanging chads 199 Hardness Harpoons 196 Harvesting crops 1, 245ff Head-lift digging 336 Healing of wounds 280 Healing in diabetic animals 281 Hedgecutter 7, 246 Helical offcuts 113, 131, 171 Herbert’s file and cutting tool testing machine 184 Herbert’s inverted pendulum 165 Hertzian contact 143 Hertz-Roesler cone crack 192, 205, 355 Hierarchical composites xiv High speed steel (HSS) 226, 230 Histological sections 2, 39 Hodograph 52 Hole flanging 191 Holes 2, 70, 335 Hollow drills 139, 276 Hollow-ground knives 228 Hollow punches 189, 201 Hollywood 281, 351 Holystone 6 Honing 221, 229 Hopkinson bar 9, 58, 105, 106, 211 Horizontal auger snow plough 350 Horner effect in veneer peeling 102 Hot chisel 200 Hydrostatic stress 23 Hygenists 5 Hypodermic needles 3, 278 Hysteresis 13
I Iaido 209 Ice 6 Ice axe 351 Ice cream scoop 5 Ice gouging 144 Ice skating 224 Ice scour 6 Impact mechanics 28 Incisor teeth 296, 308 Incremental enery balance 18 Indentation hardness 150, 155 Indenting 2 Indexable tool inserts 3, 226
Index Inertia cutting 246 Ingestion 308 Injections 200 Intaglio printing 167 Interactive light techniques 169 Intercepts in FC vs t plots 64, 237 In vivo fertilization (IVF) 276 Iron-clad ships 204 Irreversible deformation 15 Isometric scaling (allometry) 324
J J-shaped stress-strain curve 12, 261, 270 Jabs 8 Javelins 202 Jaws 310 Johansson blocks 5
K Kerf 171 Keyhole surgery 272 Klopstock limited contact tools 88, 373 Knife ‘run through’ 29 Knife tines 343 Knives 142, 173 Knives, instrumented 207, 277
L Lance 204 Lapping 8, 169, 229 Lavatory paper 6, 199 Lawnmowers 7, 253ff Layered materials 48 Least damage in cutting 78, 299 Length scale 23, 79, 301 Ligaments 27 Linocuts 167 Liquid-filled foams (foods) 285 Lithography 167 Load control 36, 317 Lobotomy 275 Lodging of cereal crops 246 Log splitters 7 Loppers 7, 246
M Mace 204 Machinability 173, 233ff Machine 7 Machining economics 182 Machining maps 96, 104, 157 Magnus effect 206 Mallet 7
Mandible 307 Mandolin 2 Mannesmann process 190 Manufacture of tools 227ff Mastication 308 Martensite 158, 225 Matching 30 Maxwell effect 331 McKenzie classification for wood offcuts 103 Mean stress 23 Mechanical impedance 30 Mechanical microscope 39, 272 Mechanical properties from scratch tests 149 Medical engineering 274 Medicine xii, 1, 259 Meristem 335 Mesh size in FEM 68 Meshless methods in FEM 67 Mezzotint 167 Microcracks 12 Microhardness 36 Micromachining 144, 223, 239 Microstructure and continuum mechanics 78 Microtome 2, 35, 272, 280 Milling 3, 8 Minimal access surgery 272 Minimum cutting forces 304 Minimum cutting speeds 8, 248 Minimum quantity lubrication 235 Mixed mode of cracking 21, 44 Mixity in cutting toughness 63, 242 Model (scaling) 21 Modes of cracking 21 Molar teeth 308 Mole drains 342 Molecular dynamics simulation of cutting 67 Momentum 29 Mould board plough 339 Mouth feel 286, 296ff Muscles 7, 24ff, 271, 286, 310, 324, 335 Mushet steel 226 Mutual cutting 3, 227, 365 Mutual scratching 159
N Nails 264 Nailing 2, 196 Nano machining 144, 150, 239 Napier’s rotary cutting tool 123 Necking 209
411
Index Nicking 7 Nominal strain 14 Nominal stress 14 Non-proportional scaling 85, 324 Normal force N on rake face 38, 371 Nose bar in veneer peeling 101 Notch sensitivity 22, 294 Nut crackers 3 Nut splitters 315
O Oblique cutting 111ff, 224 Obliquity angle i 120 Occlusion of teeth 312 Offcuts having least damage 49 Oil stone 229 Optimum blade angle 49 Optimum clearance in cropping 73 Orbital cutters 300 Orthodontics 312 Orthogonal cutting 35ff Oscillating cut paths 242
P Packaging 1 Paint films 48 Palaeontology xii, 259, 269ff Palette knife 6 Paper guillotine 124 Papier-mâché armour 204 Pargetting 142 ‘Partially-discontinuous’ chips 95 Particleboard 107 Pastry cutter 189 Pebbles, shapes of 165 Ped 327 Peeling 1 Pencil hardness 160ff Pencil sharpener 1, 134 Pendulum testing 152, 236 Penetrate 2, 190 Penetration of armour 209ff Penetrometer tests 261, 296, 331 Percussive tools 192 Perforate 2, 166, 190 Perforation of armour 209ff Phlebotomist 277 Photoelastic tools 58, 78, 226, 371 Picks 350 Piercing 2, 192ff, 273 Piling 196, 332, 334 Pinking shears 127, 190
Pizza cutter 118 Plane strain 21 Plane stress 21 Planing 1, 3, 6, 35 Plaque 5 Plastic bags 3, 355 Plastic surgery 273 Plastic work 49 Plough press in book binding 124 Ploughing 1, 225, 327, 337ff Ploughing frozen soil 47 Ploughshare 339 Plugging (ballistics) 218ff Plywood 107 Poleaxe 174 Polishing 8, 142, 151, 221 Porous materials 86ff Postage stamps 6 Post mortem 272 Potato peeler 283 Power 7, 25 Precision 6 Preening 268 Pre-molar teeth 308 Pressure bar in veneer peeling 101 Pressure-dependent yielding 328 Pressure flaking 47 Pre-tensioned sheet 40 Primary shear angle φ 53, 65 Primary shear in cutting 50 Primary shear strain 53 Pritchel 200 Progressive dynamic fracture 359 Progressive failure of warehouse racking 359 Prongs 2 Prototype (scaling) 21 Prow formation 50, 144, 238, 245, 349 Pruning shears 127 Pterygoid walk 310 Pull stroke 176, 230 Punching holes 189, 199 Push-slice xiv, 111ff, 204, 224, 229, 232, 242, 279, 301, 319 Puncturing 189
Q Quartz tools 35 Quatrain 205
R (R/k) ratio 22, 237, 344 Rabbit’s tooth principle (sharpening) 230
Rachises (of feathers) 5, 204 Radial cracking 209 Radula 307 Raised grain (wood) 223 Rake angle 24 Rake angle in oblique cutting 121 Rake face of tool 36 Rapping 262 Rasorial creatures 335 Rasp 182 Rate of cutting 9, 54, 301, 312 Rate dependence 13 Razor sharpness 232 Razor wire 268 Reaction to cutting forces 248, 335 Reciprocating blades 119 Red muscle (dark meat) 25 Refiners for wood pulp 108 Relaxation 13 Residual stresses 41, 143, 223, 240 Relief angle 36, 176 Relief in design 169 Remotely-operated submersible vehicle (RoV) 248 Restrained workpiece 36 Reversible 13 Ridge heights in scratching 147, 166 Ridged teeth 316 Rigid-plastic fracture mechanics 49 Rigid-plasticity 13 Ring-pull cans tops 4 Road traffic accidents 278 Roots 333 Rough surface of cropped edge 72 Rotary cutting tool 123 Rotary lawnmowers 224, 255 Roulette 166, 199 Routers 180 Ruling of diffraction gratings 154 Run through (of knives) 207 Ruts 146
S Saliva 296, 313 Sandblasting 2, 163 Sanding 143 Sandstorms 163 Saw teeth 174 Saw teeth, bevelled-pointing 177 Saw teeth, intertooth spacing (pitch) 176
412 Sawability 173 Sawing 1, 6, 171, 276 Sawtooth profile chips 94 Sash gang saw 178 Scale effects 16, 79ff, 164, 220, 330, 335 Scaling of teeth and body size 323 Scaling factor 22, 79 Scaling in fracture mechanics 21, 192, 335 Scaling parameter Z (R/kt) 84 Scalloped knives 172, 230 Scallops 45 Scalping dies 188, 202 Scissors 7, 125ff Score lines 42 Scorper 166 Scouring 6 Scraffiti 168 Scraping 1, 5, 335, 349 Scraping of journal bearings 6 Scratch digging 336 Scratch hardness 150, 155, 157ff Scratch plough 8, 339 Scratching 2, 5, 142 Sculptures 142, 169 Scythes 8, 202, 246 Seasoned (dried) wood 100 Secateurs 127 Second cut (files) 182 Secondary shear in cutting 50 Seed bed 338 Segmented chips 95 Self-propelled cutting machines 348 Self-sharpening blades 229 Self-sharpening teeth 321 Semiconductor wafers 2 Separation criterion in FEM 67 Separation of materials 1 Separation at the cutting edge 55 Separation at punch strokes smaller than plate thickness 74 Serrated chips 95 Serrated knives 172, 230 Setting of guillotine and scisor blades 222 Setting of saw teeth 3 Shape factor Y(a/W) 20 Shaping 3, 142 Sharp dissection 274 Sharpening of tools 227ff Sharpness 3, 98, 221ff, 276 Sharpness, connexion with C 240, 321 Sharpness, measurement of 231 Shaving 1, 247
Index Shear box test 329 Shear flow angle S 120 Shear plane angle φ 53, 56 Shear strain 15 Shear stress 15, 371 Shear yield strength k 15 Shearing through the thickness 70 Sheep shearing 1, 246 Shells 3 Shields 203 Shrapnel 3, 203, 206, 278 Shredding 1, 133 Sickle 7, 245 Sinews 27 Single point incremental forming (SPIF) 188 Silicon discs 141 Similarities between processes 187 Single crystals 239 Skiing 224 Skin 261 Skin simulants/substitutes 262, 278 Skip-tooth blades 172 Slack quenching 204 Slice-push ratio xiv, 111ff, 204, 224, 229, 232, 242, 279, 301, 319 Slingshots 202 Slip line field fracture mechanics 58 Slip line fields 58, 91, 94, 238, 315, 332, 342, 349, 373 Slitting sheets 133 Slivers on cropped edge 73 Smooth surface of cropped edge 72 Snow 349 Snowboarding 224 Snow ploughs 345, 350 Softening stress-strain curves 13 Soil mechanics xii, 327ff Sol-gel coatings 161 Solvents 13 Spade 142 Spade drill 135 Spears 210ff Spikes 323, 350 Split tool dynamometers 371 Splitting of wood during nailing 197ff Sports injury and protection 205, 278 Stabler’s rule 121 Stagnation point in flow field 238
Staining 39 Standing wave formation 50, 144, 238, 245, 349 Sticking friction 24, 367, 371 Softening stress-strain curves 13 Soil mechanics xii Solvents 13 Spacing between shear bands 96 Spalls 41, 45, 141 Speeds of wood working machinery 180 Spears Spear throwers 7 Specific cutting pressure 122, 154 Specific work of cracking R or JC 16 Specific work of separation 16 Spherical-ended tools 146 Spines 267 Split tool dynamometers 58 Spreading 1, 5 Spreadability 2 Springback 13 Standing wave in front of tool 5, 187, 202 Stapling 198 Stellite tools 226 Sticking friction 78 Sticking zone at the cutting edge 58 Stiffening stress-strain curves 13, 50 Stiffness 11, 16 Stings 263 Stimulation of muscles 25 Stonemason 169 Stoping 192 Strain 14 Strain rate 54 Strength of materials 16 Stress 14 Stress concentration factor Kt 19 Stress intensity factor K 19 Stress waves 31 Stretch ratio 14 Strigil 5 Strimmer 8, 224, 255 Stropping 151, 229 Subgrain cell structure at separation point 59 Subsurface deformation 8, 64, 223, 237 Suit of armour 203 Surface free energy 15, 31 Surgery 272, 274 Synchrotron X-ray diffraction 240 Swords 3, 204
413
Index T Tapered cuts 107, 131 Taste (organoleptic) panels 296 Tattooing 274 Taylor equation for tool life 234 Tear chip 96 Teeth as cutting tools 307ff Teeth, ideal shapes 320 Teeth, mechanical properties 313 Teeth, sharpness of 320 Teeth, sharpness and penetration force 321 Teeth, structure 270, 313, 322 Teeth, types of 317 Teeth, wear of 320 Temperature dependence 9 Temperature fields in cutting 54, 94 Temperature rise in cutting 54, 235 Tempering 225 Tenacity 8 Tendons 25, 261 Texture (foods) 286, 296 Theories of strength 16 Thermal shock of tools 226 Thermomechanical treatments 225 Thin films 33, 45, 160 Thorns 267 Three-body wear 164 Thrust force FT 37 Time dependence 13 Tines 196 Tin opener 5, 133 Tins 4 Toboggan runs 146 Toenails 3, 265 Tool materials 224ff Tool signature 142 Tool travel 39 Tools running the temper 225 Touchstone 158 Toughness 8 Toughness variation with thickness in cropping 133 Toughness/strength (R/k) ratio 62, 237, 344
Tranquillizer darts 206, 224, 277 Transient beginning of cutting 68, 71 Transient end of cutting 70 Transitions in cutting 73, 146 Transitions in types of chip 76, 90 Transition temperature 34 Transparent tools 371 Transportation of spoil 345 Trauma 277 Trenches 338, 345 Trepaning saw 134 Trephine 273 Triaxial compression test 329 Tridents 196 Trocar 273 Trochoidal path 178, 333 Trowel 142 Tunnelling 335, 345 Tup 28 Turgor pressure 1, 334 Twin Towers 364 Twist drill 134 Twisting of offcut in guillotining 131
U Uncontrolled separation of parts 1 Uncut chip thickness t 24 Unintentional cutting 353ff Unit power 122 Unrestrained workpiece 36, 245ff Up milling 107, 178
V Velocity discontinuity V* 53 Veneer peeling 100ff, 223, 239 Viscoelasticity 13
W Warm sectioning 76 Water mass transfer 285 Wave effects 31, 361
Wave speeds in cables 362 Wax 6, 77 Wear 151 Wear lands 239 Wear of cutting tools 65, 227 Wear of teeth 270 Wear rate 155 Wear resistance 155 Wedge equilibrium equations 38, 367 Wedge hardness 216 Wedge included angle 36 Wedge indentation 314 Wedge loading of inclined plate 211 Wedge-opening loading specimen 21, 44 Wedge test for foodstuffs 288 Whalebone (baleen) 312 Whetstone 225 White muscle (white meat) 24 Whorls of butter 5 Wiggly cut paths 242 Wire cutting 78, 300ff, 332, 360 Wisdom teeth 308 Wood 99 Woodpeckers 263 Wood pulp 108 Wood veneer 2, 100 Woodcuts 167 Work (energy) 7, 12 Work done per volume 15, 53 Work of formation of new surfaces 31 Workhardening index n 52, 241 Wounds 277ff Woven fabrics 223
Y Yaw of projectiles 207 Yield strength y 15 Young’s modulus E 15
Z Z parameter (R/kt) 55, 84 factor 19
