Cheese Rheology and Texture

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Cheese Rheology and Texture Sundaram Gunasekaran M. Mehmet Ak

CRC PR E S S Boca Raton London New York Washington, D.C. © 2003 by CRC Press LLC

Library of Congress Cataloging-in-Publication Data Gunasekaran, Sundaram, 1957Cheese rheology and texture / Sundaram Gunasekaran, M. Mehmet Ak. p. cm. Includes bibliographical references (p. ). ISBN 1-58716-021-8 (alk. paper) 1. Cheese—Texture. I. Ak, M. Mehmet. II. Title. TX382 .G86 2002 637′.3—dc21

2002034861

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2003 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 1-58716-021-8 Library of Congress Card Number 2002034861 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

Dedication To: My parents, Raga Palanisamy Sundaram and Kamala Sundaram, for inspiring me to always strive for excellence. My wife, Sujatha, and children, Suvai and Suman, for their love, support, and patience. — SG

My father, Haci Ak, and mother, Zeynep Ak, for giving me the opportunities they never had. My wife, Nese, who continuously supported my efforts and patiently endured the time I spent working on this book. My daughter, Asli, and my son, Efe, who cheered me up in times the situation looked hopeless. — MMA

© 2003 by CRC Press LLC

Foreword Two complex scientific areas, cheese and rheology, create an exponential increase in complexity when combined. This text makes a significant contribution to an understanding of this complexity. It underscores limitations and considerations in evaluating and conducting research on cheese rheology, points out some important gaps in our understanding of cheese rheology, and thoroughly reviews methods, theories, and applications of rheology in general and specifically for cheese. Rheologists will gain a better understanding of the physicochemical properties of cheese, and cheese researchers will be exposed to the wide range of rheological methods and the theoretical bases of those methods. Both groups should realize the need for collaborative research after exposure to the individual complexities of cheese and rheology. The diversity of observations, and seemingly contradictory observations, on the physical and chemical properties of cheese that appear in this text should not be surprising since many of the observations were made before instruments were improved and were specifically adapted to deal with unique properties of cheeses. Also, confusion resulted from: cheese scientists who used techniques inadequate to definitively measure physical properties of cheese; rheologists who chose test samples of cheese that did not possess comparable chemical properties except for the property to be measured; and inadequately defining the chemical properties of cheese. The authors have discussed unique characteristics of cheese that rheologists should be cognizant of in designing experiments. Comments on merits and deficiencies of wide range of rheological test methods as applied to cheese should assist cheese scientists in appropriately using the procedures. The chronology of cheese rheology research outlined in this book is encouraging as evidenced by the increase in collaborative research groups or research groups with better understanding of both research areas. The physicochemical properties of cheese have always been an important component in assessing cheese quality and value. The assessment was usually done by sensory evaluation, which was quite adequate because cheese was generally consumed in its “original” state. Development of heat-processed cheese products in the early 1900s prompted some research on the physical properties of cheese, primarily by modifying chemical properties, however, only limited research was done on rheological properties. The last several decades have greatly changed the forms and uses of cheese in the market place. Cheese has to be sliced or shredded by highspeed cutting devices; the melt and flow properties of cheeses have to be more carefully controlled; flavor intensities and flavor profiles have to be modified without adversely affecting physical properties of cheese; and cheese products must possess adequate stability, often under wide ranges of environments. This myriad of desired properties greatly increases the need for procedures to independently control specific properties and the need for adequate methods to measure the properties specifically © 2003 by CRC Press LLC

being controlled. The authors of this book have facilitated attainment of these goals by their thorough review of the present status of cheese rheology research and by providing guidance for further research efforts. Norman F. Olson Department of Food Science and Center for Dairy Research University of Wisconsin-Madison


Preface Rheology of cheese has been studied since the early 1950s. In fact, “Cheese Rheology” is the name of a chapter in the 1958 FAO Report*. Since then, many advances have taken place both in cheese technology and rheology. As cheese became an important part of the diet in many parts of world, the cheese industry responded by manufacturing new types of cheeses with varying textures to suit varied needs and to promote cheese use both as a table cheese and as an ingredient food. This flurry of new cheeses and applications and cheesemaking technologies has also brought about an acute need to characterize the rheological and textural attributes of cheeses to ensure their high quality. Thus, for food rheologists and food scientists, cheese is among the most popular subjects of study. In this book, we have attempted to summarize the vast literature available on cheese rheology and texture. Needless to say, the sheer volume of information available and the complexity of both cheese and rheology made this a particularly difficult task. Our goal was to bring together many of the dispersed published information on cheese rheology and texture in one book to serve as a comprehensive reference source. A unique aspect of this book is that it contains detailed descriptions of several methods to study rheology of foods in general and cheese in particular. This is to provide the interested readers the necessary basic information on many techniques reported in the literature which often do not have adequate explanation. Chapter 1 provides an overview of cheesemaking technology. Fundamental rheological test methods are described in much detail in Chapter 2. This chapter will facilitate the readers to gain a deeper understanding of the various rheological test methods. The uniaxial testing, one of the most widely used classes of rheological and texture testing methods, is the focus of Chapter 3. The fracture mechanics are an extension of the uniaxial test methods. These are discussed in Chapter 4. In Chapter 5, linear viscoelastic methods are described. This is now among the most popular rheological test performed on cheeses, and is also known as dynamic testing. Both the theory and applications are discussed in a manner benefiting those who are already familiar and those who are new to the subject. Chapter 6 focuses on nonlinear viscoelasticity of cheeses. This subject has not received much attention due to the lack of available instrumentation and the complexity of data analysis. This chapter will be more useful to those familiar with rheological analysis than to the casual reader. The discussion on cheese texture in Chapter 7 is limited to mechanical texture of cheese, as it is more in line with rheological measurements. Cheese meltability and stretchability, two of the most important properties of cheese used in prepared foods, are the topics of Chapters 8 and 9. The emphasis in these chapters is on measurement methods. The effects of various factors on cheese functional properties are addressed in Chapter 10. * Kosikowski, F.V. and G. Mocquot. 1958. Advances in Cheese Technology, FAO Studies No. 38. Food and Agriculture Organization of the United Nations. Rome, Italy. © 2003 by CRC Press LLC

Acknowledgments We would like to acknowledge many individuals who have contributed directly or indirectly toward making this book a reality. First and foremost, we would like to express gratitude to Professor Norman F. Olson, who was instrumental in helping us to initiate our first project on cheese rheology in 1989, when S.G. was a new assistant professor and M.M.A. was a graduate research assistant. Since then, with his expert knowledge and friendly personality, Professor Olson has been a source of great support. Thanks are also due to Dr. Mark Johnson, Dr. Rusty Bishop, John Jaeggi, and other past and current staff at the Wisconsin Center for Dairy Research. These people are invaluable resources for cheese research. This book draws from much of the research performed in S.G.’s laboratory. As such, the efforts of many graduate students and post-doctoral research associates are deeply appreciated. They include: Chyung Ay, James Colby, Kexiang Ding, Chang Hwang, Sun Young Kim, Sanghoon Ko, Gul Konuklar, Meng-I Kuo, Laura Marschoun, Kasiviswanathan Muthukumarappan, Hongxu Ni, Ramesh Subramanian, Salman Tariq, Deepa Venkatesan, Ya-Chun Wang, and Chenxu Yu. Thanks are also due to S.G.’s colleagues, Professors A. Jeffrey Giacomin and Daniel Klingenberg at the Rheology Research Center, University of Wisconsin-Madison, and Professor Karsten B. Qvist of KVL, Denmark. Thanks to Hallie Kirschner for typing parts of the manuscript. The financial support of Wisconsin Milk Marketing Board and Dairy Management Inc. for many of S.G.’s projects is also deeply appreciated. M.M.A. wishes to thank each member of his family for their full support and patience during the writing of this book. He expresses appreciation to the following: Suat Yasa and Murat Yasa of Aromsa Limited Company, for their interest in the book; friends Elsie and Warren Sveum, Sarah and Alvaro Quinones, Mar GarcimartinAkgul, and Arzu and Yann LeBellour for their constant encouragement; and former students Filiz Lokumcu and Metin Yavuz for their valuable assistance in gathering some of the publications. Sundaram Gunasekaran M. Mehmet Ak


Table of Contents Chapter 1

Cheesemaking — An Overview

Cheese Types Cheesemaking Milk Pretreatment Coagulation Syneresis Shaping and Salting Ripening Process Cheese References Chapter 2

Fundamental Rheological Methods

Definition of Rheology Basic Concepts Strain Stress Strain Rate Fundamental Methods Uniaxial Compression Uniaxial Tension Bending Test Specimen with a Rectangular Cross-Section Specimen with a Circular Cross-Section Torsion Test Vane Method Stress-Relaxation Test Analysis of Relaxation Behavior Creep Test Analysis of Creep Behavior Shear Rheometry Sliding-Plates Geometry Concentric-Cylinders Geometry Cone-and-Plate Geometry Parallel-Plate Geometry Capillary Rheometry Extensional Rheometry Lubricated Squeezing Flow Equations for Different Fluids in Lubricating Squeezing Flow References © 2003 by CRC Press LLC

Chapter 3

Uniaxial Testing of Cheese

Uniaxial Compression Measurements Structure and Composition Effects Stress-Relaxation Measurements Torsion Measurements Tension Measurements Creep Measurements Bending Measurements Vane Measurements Shear Measurements Lubricated Squeezing Flow Measurements References Chapter 4

Fracture Properties of Cheese

Fracture Mechanics Brittle Fracture Griffith Criterion Determination of KI Fracture Tests on Cheese Notch Tests Cutting, Slicing, and Shredding Cutting with Wire and Blade Eye/Slit Formation and Growth References Chapter 5

Linear Viscoelasticity of Cheese

Mathematical Relations in Linear Viscoelasticity Types of SAOS Measurements Strain (or Stress) Sweep Frequency Sweep Temperature Sweep Time Sweep Time–Temperature Superposition Application of SAOS in Cheese Rheology Linear Viscoelastic Region of Cheeses Cheddar Cheese Gouda Cheese Mozzarella Cheese Mozzarella: Time–Temperature Superposition Example Feta Cheese Imitation Cheese Quarg Cheese Processed Cheese Cox–Merz Rule References © 2003 by CRC Press LLC

Chapter 6

Nonlinear Viscoelasticity of Cheese

Pipkin Diagram Sliding Plate Rheometer Large Amplitude Oscillatory Shear Flow Spectral Analysis Discrete Fourier Transform Determining Material Properties Amplitude Spectrum Stress–Shear Rate Loops Effect of Wall Slip Constitutive Model for Cheese Relaxation Modulus Obtained from SAOS Relaxation Modulus Conforming to LAOS References Chapter 7

Cheese Texture

Texture Development in Cheese Cheese Manufacturing Factors that Affect Texture Textural Changes during Storage Measurement of Texture Texture Profile Analysis TPA Testing of Cheese Uniaxial Tests for Cheese Texture Measurement Compression Test Wedge Fracture Test Torsion Test and Vane Rheometry Texture Map Dynamic Tests Empirical Tests Crumbliness Cone Penetrometer Stringiness References Chapter 8

Measuring Cheese Melt and Flow Properties

Meltability Empirical Tests Objective Tests Steady Shear Viscometry Capillary Rheometry Squeeze-Flow Rheometry UW Meltmeter Viscoelasticity Index for Cheese Meltability Dynamic Shear Rheometry Helical Viscometry © 2003 by CRC Press LLC

Cheese Melt Profile Measurement UW Melt Profiler Determination of Melt Profile Parameters Graphical Method Modeling Melt Profile Constant Temperature Test Transient Temperature Test Conduction Heating References Chapter 9

Measuring Cheese Stretchability

Empirical Methods Instrumented Methods Vertical Elongation Horizontal Extension Compression Tests Helical Viscometry Fiber-Spinning Technique The Weissenberg Effect References Chapter 10 Factors Affecting Functional Properties of Cheese Properties of Milk Cheesemaking Procedures Addition of Starter Culture and Coagulants Curd Handling Cooking, Stretching, and Cooling Cheese Composition Moisture Content Fat Content Salt Content pH Post-Manufacturing Processes Aging/Ripening Freezing and Frozen Storage Heat Processing Other Factors References


1

Cheesemaking — An Overview

Cheese is one of the first and most popular manufactured food products. What perhaps started out as an accidental curdling of milk has been further refined into cheesemaking. Over several thousand years, cheesemaking has advanced from an art to near science. Cheese varieties have proliferated to suit varied conditions and requirements, especially during the last decade or so. It is estimated that more than 2000 varieties exist (Olson, 1995), and the list may still be growing. Cheese is now an important part of foods consumed in many countries (Table 1.1). In a recent survey, after spices, cheese was named the top ingredient that makes cooks feel more creative (Doeff, 1994). Several cheeses satisfy varied requirements in order to be used as suitable ingredients in various dishes from baby foods to baked products (Table 1.2). Battistotti et al. (1984) described the history of cheese and cheesemaking in much detail. This chapter provides a broad overview of cheesemaking. For further details, readers are referred to many recent books on the subject (Scott et al., 1998; Spreer, 1998; Law, 1999; Walstra et al., 1999; Fox et al., 2000).

CHEESE TYPES Today’s wide array of cheeses may be classified according to the country of origin, manufacturing process, or some end-use property. Classifying cheeses based on manufacturing and maturation processes by Olson (1979) produces a succinct list. A classification based on firmness and the maturation agent used produces a longer list but may be more relevant if textural and rheological properties are important (Figure 1.1). A classification based on the distinctive manufacturing process involved is also useful to understand the effect of the process on the cheese texture (Table 1.3). Other classifications of cheeses, e.g., according to milk source, overall appearance (color, size, shape), chemical analysis, etc., are also possible. Davis (1965) recognized the difficulty in classifying cheeses and attempted to group them based on the nature and extent of chemical breakdown during ripening or according to flavor. Such a classification is still not available. Fox (1993) proposed that the products of proteolysis could be most useful for classification. One of the main difficulties when using classification schemes is that differences exist in the moisture range allowed within various categories published in the literature (Banks, 1998). Davis (1965) assigned some empirical texture/rheological parameter values to the terms from very hard to soft (Table 1.4). The United States Code of Federal Regulations (CFR, 1998) stipulates certain standards of identity for cheeses classified according to their consistency, as listed in Table 1.5. The typical composition of milk and several cheese varieties is given in Table 1.6. In the United States, the cheese market is dominated (almost equally) by Cheddar and Mozzarella cheeses. They comprise about twothirds of the total cheese production over the past several years (Figure 1.2).


TABLE 1.1 Consumption of Cheese in Selected Countries Consumption per Capita (kg) Country North America Canada Mexico United States South America Argentina Brazil Venezuela Western Europe Denmark France Germany Ireland Italy Netherlands Spain Sweden Switzerland United Kingdom Central Europe Poland Balkans Romania Eastern Europe Russia Ukraine North Africa Egypt Southern Asia Japan South Korea Oceania Australia New Zealand a

1995

2000a

10.86 1.47 12.26

10.76 1.61 13.75

10.35 2.82 3.48

11.06 2.72 2.76

16.84 21.51 11.90 5.54 18.66 14.68 5.46 16.14 14.28 8.75

16.22 22.49 12.50 6.70 20.38 14.97 6.25 16.12 14.31 9.88

2.90

3.87

4.05

4.26

2.03 1.26

1.41 0.79

5.27

5.83

1.46 0.27

1.77 0.70

8.25 8.17

11.10 8.48

Preliminary

Source: After International Dairy Federation (www.dairyinfo.gc.ca/).


TABLE 1.2 Typical Requirements of Cheese as a Food Ingredient Requirement Crumbles when rubbed Sliceability

Shreddability

Flows freely when shaken

Examples of Food Applications Mixed salads Soup Filled cheese rolls (finger foods) Sandwiches (filled, open, toasted) Cheese slices in burgers Cheese slices on crackers Consumer packs of sliced cheese Pizza pie (frozen/fresh baked) Pasta dishes (lasagna, macaroni and cheese) Cheese sprinklings (on lasagna) Snack coating (e.g., popcorn) Dry soup/sauce mixes

Flowability when blended with other raw materials Ability to “cream” or to form a paste when sheared Nutritional value

Fresh cheese desserts

Meltability upon grilling or baking

All cooked dishes (including sauces, fondues, pizza pie)

Flowability upon grilling or baking

Most cooked dishes (e.g., pizza pie, cheese slices on burgers) Chicken cordon-bleu Deep-fried breaded cheese sticks Deep-fried burgers with cheese inserts Fried cheese dishes

Flow resistance upon deep-frying

Stretchability when baked or grilled Chewiness when baked or grilled Limited oiling-off when baked or grilled Limited browning when baked or grilled


Cheesecake Tiramisu Homemade desserts Baby foods

Examples of Cheese or Cheese-based Ingredient Feta, Cheshire, Stilton Stilton Swiss-type, Gouda, Edam Swiss-type, Cheddar, Mozzarella Cheddar Cheddar Swiss-type, Cheddar, Mozzarella Mozzarella, Provolone, Cheddar, analog pizza cheese, Monterey Cheddar, Romano, Provolone Grated Parmesan and Romano Cheese powders Cheese powders, enzyme-modified cheese Quark, Fomage frias, Cream cheese

Pizza pie

Cream cheese, Ricotta Mascarpone Cream cheese Dried cheeses, esp. rennet-curd varieties (high in calcium) Mozzarella, Cheddar, Raclette, Swiss, Romano, analog pizza cheese, PCPsa Mozzarella, Cheddar, Swiss, Romano, analog pizza cheese PCPs, Cream cheese PCPs, analog pizza cheese, custom-made Mozzarella or string cheese PCPs, analog pizza cheese Paneer, acid-coagulated Queso Blanco Mozzarella, Kashkaval, young Cheddar, analog pizza cheese Halloumi, Mozzarella, Provolone, Kashkaval, young Cheddar Mozzarella, Kashkaval

Macaroni and cheese Lasagna Pizza pie

Cheddar, Romano Cheddar, Romano, Parmesan Mozzarella, analog pizza cheese

Pizza pie Pizza pie

TABLE 1.2 (continued) Typical Requirements of Cheese as a Food Ingredient Requirement

Examples of Food Applications

Viscosity

Soups Sauces Cheesecake

Flavor

Most cheese dishes, soups Baked products Snack coatings Dressings Baby food Ready-made meals

a

Examples of Cheese or Cheese-based Ingredient Cheese powders, PCPs Cheese powders, Cheddar, Blue cheese, PCPs Cream cheese Cheddar, Romano, Swiss-type, Parmesan Cheese powders, enzyme-modified cheese Cheese powders Cheese powders Dried cheese Cheese powders

Pasteurized process cheese products.

Source: After Fox et al., 2000. With permission.

CHEESEMAKING Though there are numerous cheese varieties, the manufacturing processes of most of them share several common steps. Variations at one or more steps during manufacture produce cheeses of different textures and flavors. The essential steps in cheesemaking and some variations for a few types of cheeses are schematically illustrated in Figure 1.3. These steps are as follows.

MILK PRETREATMENT Milk used for cheesemaking is normally standardized and heat treated. In some cases, milk is homogenized. An acid-producing starter culture is then added. The standardization of milk has become necessary to ensure that milk obtained from several producers or dairies is of a “standard” composition and condition throughout the year. This is critical in cheesemaking because the legal standards of various cheeses specify certain fat-to-protein ratios. The fat-to-protein ratio is determined mainly by the fat-to-casein ratio in the milk (Fox et al., 2000) which can be modified by removing fat or by adding cream or skim milk or skim milk powder, etc. It is also common to add color (annatto or carotene) and calcium (in the form of CaCl2) to the milk and to adjust milk pH to a desired level, known as preacidification. Adding calcium speeds up coagulation or reduces the amount of rennet needed and produces a firmer gel. Heat treatment of milk is primarily intended to destroy the harmful microbial population and enzymes in raw milk to assure product safety and quality. Pasteurization is the most commonly used heat treatment (72°C with 15 s holding time). It © 2003 by CRC Press LLC

Acid Coagulated Soft Cottage Cream Quark Queso Blanco Baker’s Neufchatel Ricotta (Acid and heat coagulated from whey)

Surface Ripened

Semi-soft Brick Bel Paese Havarti Limburger Munster Oka Port du Salut St. Paulin Trappist Taleggio Tilsiter Soft Liderkranz

Concentrated (from Whey)

TYPES OF NATURAL CHEESES

Gjetost Myost Primost Enzyme Coagulated

Internal Bacteria Ripened Very Hard Asiago Grana Parmesan Parmigiano Romano Sabrinz Sardo Hard Caciocavallo Cheddar Cheshire Colby Graviera Ras Cheese with eyes Edam Emmental (Swiss) Gouda Gruyere Maasdam Samsoe

Mould Ripened

Internal Mould Semi-soft Blue Danablu Gorgonzola Roquefort Hard Stilton

Surface Mould Soft Brie Camembert Coulommiers Carre de l’Est

Semi-soft

Soft Salt-cured/ Pickled Domiati Feta

Caerphilly Mahon Monterey Jack Pasta filata Mozzarella Provolone Caciocavallo

FIGURE 1.1 Natural cheeses classified according to the maturation agent used and firmness. (After Vedamuthu and Washam, 1983; Fox et al., 2000.)

not only destroys most of the bacteria present, including lactic-acid bacteria, but also inactivates many enzymes. A gentle heat treatment, known as thermization (60 to 65°C with 15 to 30 s holding time) may also be used advantageously before or after pasteurization (Spreer, 1998). However, many cheeses are still produced from raw milk, especially in Europe (Fox et al., 2000). If the cheeses are made from unpasteurized milk, they must be cured for at least 60 days at not less than 1.7°C (35°F), and the label should indicate the manufacturing date or state “held for more than 60 days.” (NDC, 2000). In traditional cheesemaking, the acid produced by microorganisms present in raw milk lowers the milk pH to a level sufficient for subsequent coagulation. However, if the milk undergoes a heat treatment, selected cultures of lactic-acid bacteria


TABLE 1.3 Classification of Cheeses by the Distinctive Manufacturing Process Involved Distinctive Process Involved Curd particles matted together Curd particles kept separate Bacteria-ripened throughout interior Prolonged curing period Pasta filata Mold-ripened throughout interior Surface-ripened mainly by bacteria and yeasts Surface-ripened mainly by mold Curd coagulated mainly by acid

Characteristics

Example Cheeses

Close texture, firm body Slightly open texture Gas holes or eyes with eye formation throughout cheese Granular texture; brittle body Plastic curd; stringy texture Visible veins of mold (blue-green or white); piquant, spicy flavor Surface growth; soft, smooth, waxy body; mild to robust flavor Edible crust; soft, creamy interior; pungent flavor Delicate soft curd

Cheddar Colby, Monterey Jack Swiss (large eyes), Edam or Gouda (small eyes) Parmesan, Romano Mozzarella, Provolone Blue, Gorgonzola, Roquefort Brick, Limburger Brie, Camembert Cottage, Cream, Neufchatel

Source: After NDC, 2000. With permission.

TABLE 1.4 Empirical Texture/Rheological Parameter Values Used in Cheese Classification Logarithmic Scale Values Cheese Type Very Hard Hard Semihard Soft

Moisture (%)

Viscosity Factor

Elasticity Factor

Springiness Factor

< 25 25–36 36–40 > 40

>9 8–9 7.4–8 < 7.4

> 6.3 5.8–6.3 < 5.8 < 5.8

> 2.3 2–2.3 1.8–2 < 1.8

Source: After Davis, 1965.

TABLE 1.5 United States Federal Standards for the Maximum Moisture and Minimum Milk Fat for Classes of Cheese Designated by Consistency Consistency

Maximum moisture content (%)

Minimum milk fat in solids (%)

Hard grating Hard Semisoft Semisoft part skim Soft

34 39 50 (>39) 50 Not specified

32 50 50 45 ( 2). On the other hand, if L/D ratio is small (e.g., L/D < 1) the test results may be greatly affected by the specimen end conditions (e.g., friction effects). For instance, Chu and Peleg (1985) examined apparent deformability modulus, (determined as the engineering stress divided by engineering strain at 20% deformation) and failure conditions (i.e., failure stress and failure strain) of potato, bologna sausage, and process American cheese as a function of height-to-diameter ratio in the range 0.12 ≤ L/D ≤ 1.00. The flat (i.e., lower L/D ratio) specimens exhibited higher apparent deformability modulus (i.e., higher stiffness). This is particularly


ASTM Cutting Tool

DTOOL

DTOOL = 28.804 mm (+0.000/−0.025) DSPECIMEN = 28.6 ± 0.1 mm LSPECIMEN = 12.5 ± 0.5 mm

FIGURE 2.20 The ASTM cutting tool for cutting cylindrical compression test specimens. (Per ASTM D638.) (After ASTM, 1996. With permission.) 17 to 21 mm

14.7 mm

14-mm diameter Cheese sample Cheese borer

Cheese block

14 mm

FIGURE 2.21 Cork borer for cutting cylindrical cheese specimens. (After Luyten, 1988; van Vliet and Peleg, 1991.)

the case when there is considerable friction between the specimen ends and the machine plates (e.g., when plates are coated with emery cloth). For instance, the apparent deformability modulus of the cheese sample with L/D = 0.12 is about 3.5 times greater than that for the sample with L/D = 1.0. Culioli and Sherman (1976) examined the effect of contact surface conditions on force–compression behavior of Gouda cheese at crosshead speeds of 2.5, 10, and 50 cm/min and L/D ratio of 1.0. At any of three crosshead speeds the level of friction


TABLE 2.3 Recommendations for Cheese Sample Preparation for Uniaxial Compression Tests If it exists, remove hard rind of cheese Prepare specimens when they are cold (refrigeration temperature) Take sample from the same location in a cheese (e.g., center) and in the same direction Take samples from locations that are sufficiently away from each other since fracture may occur in the cheese loaf during previous sampling Cut specimens as slowly as possible, preferably using a motorized borer and cutting tool Lubricate all surfaces of the borer with mineral oil to minimize distortion during cutting out Use a tightly stretched, thin wire cutter, lubricated with mineral oil Measure actual dimensions of each specimen before testing Make sure that the size of specimen is large compared to the size of the heterogeneity Source: After van Vliet and Peleg, 1991.

at contact surfaces (i.e., with emery paper or with oil) exerted no influence on the force–compression behavior until 40% compression. At fracture point the force was lower when an emery paper was used at the interface than when the specimen ends were lubricated with oil. However, when the true stress is plotted against percent compression, the stress was greater when an emery paper was used at the interface than when the specimen ends were lubricated with oil. On the other hand, Luyten et al. (1992) found no clear effect of using emery paper or oil at the specimen–machine interface on the fracture stress of Gouda cheese. Moreover, they recalculated stress–strain curves from the data of Culioli and Sherman for emery paper and oil, and found no difference between them. The effect of aspect ratio on the fracture stress of young Gouda cheese is depicted in Figure 2.22. In this figure, data for different friction conditions (i.e., normal plates, lubricated, and emery paper) are pooled together since no clear effect of friction on the fracture stress is reported (Luyten et al., 1992). It is seen that the fracture stress of this cheese shows a tendency to become constant beyond the L/D ratio of 1.5. Although the uniaxial compression test appears simple in principle and practice (e.g., no need to grip the sample), the data analysis is complicated by the effect of friction between the specimen and the testing machine platens. Friction influences not only the magnitude of the force for compression but also the appearance of the compressed specimens (Culioli and Sherman, 1976). Barreling due to friction (Figure 2.12) is an indication of nonhomogeneous deformation. During compression the cheese specimen is to move relative to the platens and thus the force required to achieve a certain level of compression depends also on the friction. Therefore, stresses in the presence of friction (i.e., shear-plus compression) are always greater than in the absence of friction (i.e., shear-free compression). The common practice of reducing the friction is to lubricate the sample–platen interfaces with low-viscosity oil. It has been shown that type of lubricating oil can have a significant impact on the measurements (Charalambides et al., 1995). An alternative way of accounting for friction is to bond the sample to the platens using adhesives such as cyanoacrylate (Casiraghi et al., 1985). Charalambides et al. (2001) described a method based on the Cooke and Larke procedure to account for friction.


TABLE 2.4 Experimental Conditions Used in Uniaxial Compression Testing of Cheese Cheese Type Cheddar Leicester Brie UF-Feta Immature Cheddar, Cheshire, Leicester Mozzarella cheese analogs Camembert

Crosshead speed (mm/min)

Temperature (°C)

Sample Diameter, D (mm)

Sample Height, L (mm)

Aspect ratio: L/D

2.5, 6.4, 12.7, 25.4, 50.8, 127 25, 50, 250, 500, 1000 33.3

22

19

19, 29

1, 1.53

Room, 23–26, 30–37 5, 20

25

25

1

15

19.5

1.3

100, 200, 300, 400

13

15.3

15.3

1

Mineral oil; emery sheet Smooth hydrophobic paper Low-viscosity oil

5–1000

0–40

28.5–29.5

30

~1

Machine plates

20

20

23

20

0.87

Machine plates

10

20

13.8

10

0.72

Machine plates


Surface Condition Mineral oil

Reference Ak and Gunasekaran (1992) Vernon-Carter and Sherman (1978) Molander et al. (1990) Wium and Qvist (1997) Dickinson and Goulding (1980)

Yang and Taranto (1982) Mpagana and Hardy (1986)

Emmentaler

5, 20, 80

15

16.4

17.5

1.07

Gouda

5, 10, 50

20–21

7.5–35

0.3–3.5

Gouda Mild Cheddar, Sharp Cheddar, Monterey Jack Process American cheese Mozzarella, Cheddar, Processed cheese spread Gouda Gruyere, processed Mozzarella

0.1–500 10

20 4

10, 20, 25 (also cubes of 10 and 20 mm) 15 20

20–30 7, 10, 13, 20

1.33–2 0.35–1

5

Room

10–21

2.5–10

0.12–1

0.5, 5, 50

7, 22

57

20, 30, 40

0.35–0.70

10 3.6, 5.8, 7.9, 10.8, 14.4

20 room

20 20

20 5, 8, 11, 15, 20

1 0.25–1


Paraffin oil Mineral oil; emery paper Machine plates Machine plates

Machine plates; emery cloth (No.120) Paraffin oil; bonding with cyanoacrylate; machine plates Paraffin oil Machine plates; synthetic grease

Rohm and Lederer (1992) Culioli and Sherman (1976) Luyten et al. (1991a) Charalambides et al. (1995)

Chu and Peleg (1985)

Casiraghi et al. (1985)

Rohm et al. (1997) Charalambides et al. (2001)

Fracture stress (kPa)

200

150

100

50

0 0.0

0.5

1.0

1.5

2.0

2.5

Aspect ratio

FIGURE 2.22 Effect of aspect ratio (height-to-diameter ratio) on fracture stress of Gouda cheese in compression. (After Luyten et al., 1992.)

According to this procedure, measurements are made on specimens with a constant diameter and various heights. Then, results are plotted as true stress against 1/H for constant values of strain. The intercept of the resulting line (i.e., at 1/H = 0) gives the correction to be applied to the stress for that strain level. For uniaxial compression, Hencky strain can be written in terms of the deformation rate (or crosshead speed), Vz as below:  L (t )   Lo − ∆L   Lo − Vz t  ε H = ln   = ln   = ln    Lo   Lo   Lo 

(2.15)

where, L = current sample height, Lo = initial sample height, ∆L = deformation (= Vz t), and t = time. Since L ≤ Lo in uniaxial compression, εH will have a negative value, an appropriate sign for compressive strains. Most often the right-hand-side of Equation (2.15) is multiplied with –1 to make the resulting strain values positive for common use. In lubricated compression a cylindrical specimen of radius Ro and height Lo is deformed into a cylinder of radius R and height L. From the assumption of constantvolume deformation we can obtain radius at any time from the following relation: L  R = Ro  o   L

1/ 2

(2.16)

The true stress for lubricated compression is calculated from: σt =

[

F(t ) F(t ) L(t ) F(t ) Lo − Vz t = = A(t ) Ao Lo Ao Lo

]

(2.17)

where, F(t) = applied force at any time, A(t) = cross-sectional area at any time, Ao = initial cross-sectional area, Lo = initial length.


In bonded compression, the cross-sectional area in contact with the compression platens remains constant, and the stress in bonded compression is given as (Casiraghi et al., 1985): σB =

F (t ) π Ro2

(2.18)

The stress is then corrected, σBC, for the shape changes using the following equation: σ BC =

σB R2 (1 + o 2 ) 2L

(2.19)

This corrected stress equation is shown to be effective to bring results in bonded compression into agreement with those in lubricated compression up to a strain level of 0.37 for Cheddar, 1.4 for Mozzarella, and 0.8 for processed cheese spread, where strain is defined as ∆L/L. This definition of strain is used since it relates directly to the extent of bulging (i.e., δo). For the bonded sample the relation is given by the following equation (Christianson et al., 1985): δo =

3 ∆L R 4 o L

(2.20)

Kamyab et al. (1998) and Charalambides et al. (2001) analyzed the uniaxial compression test with friction between the sample and the compression platens, which is quantified by the coefficient of friction. They provided a scheme that enables calculation of true stress–Hencky strain curve from uniaxial compression data influenced by friction. The resulting analytical equation in its simpler form is given as: FL µL  = σt + σt  o  ∀o 3 L

3/ 2

 Do  L   o

(2.21)

where, ∀o = initial sample volume (i.e., πRo2Lo), µ = coefficient of friction, and Do = initial sample diameter. The results from compression tests are analyzed by plotting [FL/∀o] as a function of [Do/Lo] at fixed values of [Lo/L]. The intercept gives the true stress, σt, and the slope can be used to calculate the coefficient of friction, µ. Thus, using this procedure one can construct true stress–Hencky strain curves and determine variation of µ with strain. More parameters, such as the modulus of deformability and fracture energy per volume or toughness per volume, can be extracted from stress–strain curves from uniaxial compression. To facilitate parameter calculations, Ak and Gunasekaran (1992) suggested using polynomial expressions to describe the stress–strain curves as:


N

σ t = a1ε H + a2 ε 2H + a3ε 3H + ... =

∑a ε

i i H

(2.22)

i =1

where, i denotes an index for the coefficients (ai) and a power for the Hencky-strain term (εΗi ). The coefficients (ai) can readily be determined using a curve-fitting procedure. The modulus of deformability, ED, can be obtained from Equation 2.22 using the following definition:  dσ  ED =  t  = a1  dε H  ε H →0

(2.23)

The coefficient of the first term in Equation 2.22 becomes equal to the modulus of deformability, and this is why it is necessary to apply a constraint so that it has non-negative values in curve-fitting procedure. As an alternative approach, we shall mention that Wium et al. (1997) determined the deformability modulus of UF-Feta cheese as the maximum slope in the range 0 ≤ εΗ ≤ 0.05. The peak strain, εf, which may sometimes correspond to fracture strain, can be estimated by locating the strain at which the slope becomes zero. That is: dσ t =0= dε H

N

∑a ε

i −1 i H

(2.24)

i =1

The strain that makes the slope zero (or nearly zero with some tolerance) can be determined using a root-finding procedure with computation software. Once the peak strain is computed, then its value can be inserted back into Equation 2.22 to determine the corresponding peak stress, σf , which may sometimes correspond to fracture stress. Of course, in some cases, it may be easier to obtain the fracture stress and fracture strain values directly from the experimental data without using Equation 2.24. However, for cases where a distinct peak is not readily discernible, use of Equation 2.24 is a practical approach. It is important to restate that in compression tests the fracture usually starts in the interior of the specimen and often before the maximum stress is reached (Luyten et al., 1991a). One can obtain fracture work per unit volume (W) (or the total energy per unit volume, or modulus of toughness) from the area under the stress–strain curve (Figure 2.17): εf

W=

∫ σ dε t

H

(2.25)

0

The calculation of fracture work becomes simpler with the substitution of Equation 2.22 for stress in Equation 2.25. With this approach we can also easily


True stress (kPa)

70

A

60 50 40 30 20 10 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Hencky strain (-) 250 B Slope (kPa)

200 150 100 50 0 −50 0.00

0.50 Hencky strain (-)

1.00

1.50

FIGURE 2.23 Example showing extracting various parameters from stress–strain data (symbols) by fitting a polynomial equation ( σ = 214ε H − 695ε H2 + 1342ε H2 − 1158ε H4 + 355ε H5 ) to the true  dσ 

stress–Hencky strain curve (line) (A). From the plot of slope   vs. Hencky strain (B), the  dε H  modulus of deformability (= slope when Hencky strain approaches zero = 214 kPa) and fracture strain (εH,f = 0.95), and fracture stress (σf = 58 kPa) are calculated at zero slope. Integrating the true stress–Hencky strain polynomial equation between the limits εH = 0 and εH = 0.95, according to Equation (2.25), yields the value of fracture work or modulus of toughness as 35 kJ/m3. (After Ak and Gunasekaran, 1992.)

obtain the work up to any given strain or deformation by simply changing the upper limit of the integral. For instance, if we insert the proportional limit for the upper limit of integral the resulting area is termed the modulus of resilience and is the work done per unit volume to reach the proportional limit (Figure 2.17). The modulus of toughness is a measure of the ductility of a material. The larger the modulus of toughness relative to the modulus of resilience, the more ductile is the material (Fletcher, 1985). In Figure 2.23 we present a numerical example where the mechanical parameters mentioned above are extracted from the experimental data. As can be seen in Figure 2.23, there are more than one minimum and maximum points in the slope vs. strain curve. It is not clear if these points relate to some structural changes during the compression test.


F

∆L

L0 L

∆D/2

D0

∆D/2

D

FIGURE 2.24 When a material is compressed by ∆L (from original length Lo to final L), its diameter increases by ∆D (from original Do to final D). The associated axial and lateral strains are used to calculate the Poisson’s ratio. (Equations 26a and 26b.)

Another important rheological parameter is the Poisson’s ratio (Figure 2.24). When a specimen of length Lo is compressed to final length L it experiences a concomitant increase in diameter from original value of Do to final D. That is, the imposed axial strain brings about a lateral strain. The Poisson’s ratio is the ratio of lateral strain to axial strain as given below: ν=−

ε ( D − D0 ) / Do lateral strain = − lateral = − axial strain ε axial ( L − Lo ) / Lo

(2.26a)

or, in terms of Hencky strains, it is given as:

ν=−

ε H lateral ε H axial

ln  D D    O =−  ln  L L   0 

(2.26b)

The negative sign indicates that the lateral dimensions decrease as the axial dimensions increase. It also makes ν a positive number since the lateral and axial strains are of opposite sign. The Poisson’s ratio is a material property and is based on the observation that when a material is subjected to an axial force, let us say tension, it will not only elongate but it will also contract laterally (Riley et al., 1995).


FIGURE 2.25 Picture of compressed Mozzarella samples taken parallel and perpendicular to the fiber direction. (After Ak, 1993.)

Equations 26a and 26b are applicable only to homogeneous isotropic materials; that is, materials with the same properties in all directions. It assumes that the lateral expansion in uniaxial compression and lateral contraction in uniaxial tension is uniform in the radial direction. This may not be the case for some cheeses. For instance, Ak and Gunasekaran (1997) demonstrated that the kneading and stretching of the curd in hot water results in a Mozzarella cheese with anisotropic tensile and compressive properties. Anisotropy refers to the fact that material behavior is dependent on the direction in which stress is applied or on the direction in which sampling is done. As shown in Figure 2.25, the postdeformation appearances of compressed specimens taken parallel and perpendicular to the fiber orientation are clearly different. Thus, Poisson’s ratio calculations based on the lateral expansion of these two cases would certainly produce different values. For some cheeses, anisotropic mechanical properties may also be of commercial importance. For Gruyère de Comté, a Swiss-type hard cheese, it is reported that anisotropic rheological properties are important in the formation of eyes and slits (Grappin et al., 1993). The resistance to wire cutting of Cheddar cheese (4 months old) from different manufacturers is reported to vary with the cutting direction (Ney, 1985). Although allowable range of Poisson’s ratio is from –1 to 0.5, its value generally varies from 0 (totally compressible) to 0.5 (incompressible). For most metals it has a value between 0.25 and 0.35 (Riley et al., 1995). Rubber has a Poisson’s ratio about 0.5, making it nearly incompressible. On the other hand, cork has a Poisson’s ratio close to zero, which makes it a good bottle stopper. An axially loaded cork will not show a lateral expansion, thus making its insertion into a bottle easy. A negative value for Poisson’s ratio has been reported for polymeric and metallic foam structures (Lakes, 1987; Friis et al., 1988). Regarding cheese, experimental


results indicate that the Poisson’s ratios of Cheddar and Gouda cheeses vary between 0.40 and 0.45; that is 20% and 10% lower than the theoretical value of 0.5 for incompressible materials (Calzada and Peleg, 1978; Luyten et al., 1991b; Rohm et al., 1997). The Poisson’s ratio is related to the elastic constants such as Young’s modulus (E), shear modulus (G), and bulk modulus (K) by the following formulas (Findley et al., 1989): E = 2(1 + ν)G K=

E 3(1 − 2V )

(2.27)

Bulk modulus describes the change in volume in response to hydrostatic pressure (i.e., equal pressures in all directions). From Equation 2.27, for an incompressible (ν = 0.5) linear elastic solid we can compute E = 3G and K = ∞.

UNIAXIAL TENSION As far as the direction of applied stress is concerned, uniaxial tension is simply the opposite of uniaxial compression. However, a more fundamental difference between tension and compression tests is in the strain rate. When a specimen is deformed at a constant speed, the strain rate decreases in tension but increases in compression. Various features of different fundamental methods are listed in Table 2.5. Uniaxial tension tests are considered not suitable for routine measurements since they are more difficult to execute due to lengthy sample preparation and difficulty of gripping (Luyten et al., 1992). Specially designed grips are often necessary in order to eliminate slippage and breakage of sample in the grips. Grip surfaces can be scored or serrated to enable better holding. It is generally assumed that the grip assembly and the specimen ends are nearly rigid and all of the deformation is taking place in the gage section of the specimen. A large number of grips are commercially available for different purposes and materials (Figure 2.26). The existence of many sophisticated grips designed for particular materials is sufficient to show that the tensile test is difficult to perform even for engineering materials. Therefore, it is not a commonly performed test for cheese, particularly at temperatures above melting point of fat in the cheese. The test specimen used in tensile testing may have either a circular or a rectangular cross-section. The latter shape is more suitable for cheese. The ends of tensile specimens are generally enlarged to provide extra area for gripping and to prevent sample prematurely breaking at the grips. A typical tension specimen described in ASTM Standards for plastics is illustrated in Figure 2.27. The specimen must be aligned as perfectly as possible with the direction of stretching so that the long axis of the test specimen will coincide with the direction of the grip assembly. Strictly speaking, data from only those tests that produce failure in the gage length should be used to obtain material properties. In practice, specimens may fail near the grips where there is stress concentration. Therefore, a small notch can be made in the central part of the test piece to ensure the location where the fracture will start (Luyten, 1988) (Figure 2.28).


TABLE 2.5 Advantages and Drawbacks of Various Test Methods Used for Cheese Test Method I.

Test Type

Uniaxial Constant compression rate

Advantages

a) Difficult to obtain specimens with flat and perfectly parallel ends b) Friction affects the property calculations c) The strain rate increases during a test if a constant crosshead speed is applied (typical of Universal Testing Machines used in food studies) d) Rheological parameters depend on specimen size as a result of friction, varying strain rate, and the inhomogeneities in the cheese e) Start of fracture is often inside the test piece and does not correspond to the maximum stress in the stress–strain curve f) Assumptions of constant volume and perfect cylinder shape during deformation may not hold g) Test piece undergoes inhomogeneous straining Lubricated a) Biaxial extensional viscosity a) Difficult to separate elastic contribution squeeze may be determined relatively from viscous contribution to material’s flow easily when cheese response specimen is compressed b) Difficult to completely eliminate friction between lubricated plates c) Equations will not apply unless the b) Deformation rate can easily assumption of perfect cylinder shape holds be varied to obtain biaxial extensional viscosity as a function of strain rate Constant a) Constant force tests (creep a) Stress decreases during the test if the force tests) can be executed easily plate diameter is greater than specimen for long time scales provided diameter that necessary precautions are b) Strain rate varies during the test taken to avoid physical and c) Drawbacks mentioned above for the chemical changes constant rate case applies here as well b) The Young’s modulus and except that related to strain rate the compliance can be determined c) An apparent viscosity can be calculated from the so-called “secondary stage” where the strain rate is nearly constant d) Relevant to hole (eye) formation in cheese and sagging of cheese under its own mass


a) Easy to prepare samples and to perform the test b) Young’s modulus can be obtained c) Fracture stress and strain can be obtained d) Easy to vary deformation rate e) Small samples with less chance of containing undesired inhomogeneities

Drawbacks

TABLE 2.5 (continued) Advantages and Drawbacks of Various Test Methods Used for Cheese Test Method

Test Type

II. Uniaxial tension

Constant rate or force

III. Bending

Constant rate or force

IV. Cutting (wire or wedge)

Constant rate

Advantages

Drawbacks

a) Fundamental rheological and fracture properties can be determined, such as Young’s modulus, fracture stress, fracture strain, toughness b) Friction effect is not present c) Fracture initiation and propagation can be controlled using notched samples a) Easy to perform the test b) No need to fix the specimen to an apparatus c) Fracture can be observed mostly on the outside (tension side) of the specimen d) Test imitates closely the sensory evaluation of cheese by graders a) b) c) d) e)

a) It is often difficult to grip the sample, thus requires specially designed grips to hold samples b) Fracture may occur at the grips, which is avoided using special sample shapes (e.g., dog-bone shape) c) Test piece must be long compared to other dimensions for deformation to be homogeneous and for reliable stress–strain calculations. d) Strain rate decreases during a tension test a) Test can be used only for cheese of some rigidity and fairly short texture b) Large test pieces increase the possibility of containing an undesired inhomogeneity c) Length of samples must be much larger than the other dimensions, which is sometimes not practical d) Deformation is far from being homogeneous as it varies from a compressive strain to a no strain at neutral axis, and to a tensile strain at the outside Easy to execute and no need a) Only fracture energy is determined from to clamp sample this test, and other tests are to be carried Small test piece out to obtain other rheological and fracture Useful for determination of properties fracture energy of cheese b) Additional cracks may be formed due to Similar to biting food with inhomogeneities in the structure of brittle teeth materials Fracture is in tension c) Friction between the wedge and the specimen may contribute to the measured force, which can be reduced by lubricating the wedge

Source: After van Vliet, 1991a; Luyten et al., 1992.

Parameter calculations in uniaxial tension are similar to those given earlier for uniaxial compression. Strain and stress in uniaxial tension can be calculated using the following expressions, respectively:  Lo + Vz t   L( t )   L o + ∆L  ε H = ln   = ln   = ln   = ln(1 + ε)  Lo   Lo   Lo 


(2.28)

FIGURE 2.26 Various designs of commercial grips for tensile tests. (From www.itinscale.com/ grips.htm; www.dillon.fm/grip.htm#clevis; www.cscforce.com/gripping.htm. With permission.)

σt =

[

F(t ) F(t ) L(t ) F(t ) Lo + Vz t = = A(t ) Ao Lo Ao Lo

]

(2.29)

Here, Lo and L are initial and final gage lengths instead of total specimen lengths. In stress calculation it is assumed that the volume of specimen remains constant during extension (i.e., A(t) L(t) = Ao Lo). In strain calculation it is assumed that the deformation as a result of the crosshead movement is taking place in the gage length of the specimen. If this assumption is of suspect, one way to obtain strain values is


D G W

R

WO

L LO

Recommended tensile specimen dimensions (in mm) for samples of the thickness (mm) listed below (after Swallowe, 1999). Sample thickness

Dimension W L WO (min) LO (min) G D R

7 to 14 19 57 29 246 50 115 76

4 to 7 13 57 19 165 50 115 76

1.0; Z2/D > 0.5 (Nguyen and Boger, 1985). Here, DT is the diameter of the container, Z1 and Z2 are height of © 2003 by CRC Press LLC

material above and below the vane, respectively. Typically, the blades are made of stainless steel with the thickness less than 1 mm. Recently, a new instrument called “slotted-plate device” has been developed to directly measure static yield stresses of suspensions (Zhu et al., 2001). The slottedplate device is reported to be more reliable for evaluating smaller yield stresses and avoids possible secondary flows between the blades and nonuniform stress distribution along a virtual cylindrical surface — the key assumptions in the vane geometry. The success with the vane method has resulted in new applications and designs of this geometry such as oscillatory testing and texture analysis (Junus and Briggs, 2001) and hand-held versions of the vane instrument (Keener et al., 1999).

STRESS-RELAXATION TEST One of the fundamental tests to study viscoelastic response is stress relaxation. This test can be performed in (uniaxial) tension, compression, shear, bending, torsion, etc. When a constant strain is applied to a viscoelastic material isothermally, the stress necessary to maintain that strain is not constant but decreases with time. Hence, the decrease of stress at constant strain is called stress relaxation. Two kinds of relaxation experiments can be conducted: stress relaxation after a sudden step strain, which is often applied to solids; and stress relaxation following a cessation of steady flow, which is often applied to liquids (Figure 2.38) (Dealy, 1995; Ferry, 1980; Whorlow, 1980). Stress-relaxation response permits rapid characterization of material behavior as shown in Figure 2.39. When a step strain is applied to ideal elastic solid, a finite and constant stress will be reached. Ideal elastic solids store all the energy charged during the straining step and would expend this energy upon removal of stress to return to its original size and shape. In a way, ideal elastic solids possess a perfect memory of the initial state. Thus, the same stress should be kept on the specimen at

Strain

Cessation of flow

Step strain

Time

FIGURE 2.38 Two types of stress relaxation test: step strain for solids and cessation of steady flow for liquids.


Strain

Constant strain input

Time

Stress

Ideal elastic response

Stress

Time

Ideal viscous response

Time 2

Stress

Linear viscoelastic response

3

4

1

Time ε0

ε0

ε0

1

2

3

4

σ1=0

σ2

σ3

σ4

σ2 > σ3 > σ4

FIGURE 2.39 Step-strain input (A) and stress relaxation response of ideal elastic material (B), ideal viscous material (C), and viscoelastic material (D). The stress and strain on the viscoelastic material is schematically depicted at different times during the test at the bottom.


all times for strain to remain constant. On the other hand, for ideal viscous liquids the stress decays to zero immediately after the cessation of strain application. Ideal viscous liquids do not store any energy and have no memory of the initial state. However, various materials, including foods, are viscoelastic and exhibit an intermediate response where stress relaxes at a finite rate characterized by the relaxation time (Peleg, 1987). Viscoelastic materials can be considered as materials with “fading memory.” If a viscoelastic material relaxes to zero stress within a certain period (e.g., experimental time) it is further characterized as “viscoelastic liquid.” In contrast, a viscoelastic material is considered as “viscoelastic solid” if a finite stress remains unrelaxed (i.e., residual stress) after a sufficiently long time. With most foods, however, the “sufficiently long time” is on the order of few minutes due to chemical, enzymatic, and physical changes that foods normally experience. The residual stress after an arbitrary time for test duration (e.g., 10 min) is suggested as a quantitative measure of the degree of “solidity” of foods (Peleg, 1987). The relaxation experiment can be viewed as composed of two steps: the straining step and the relaxation step. Ideally, the straining step is instantaneous, but in reality it takes finite time. The time it takes to apply the step strain is called the rise time. The rise time depends upon the capability of the instrument used and the magnitude of strain. For instance, if the highest crosshead speed of a UTM machine were 1000 mm/min, then it would require 0.09 s to apply 10% deformation on a sample of 15 mm height. Since stress relaxation of a viscoelastic material is affected by the history of deformation, the time taken for the straining step is important (Meissner, 1978). Accurate stress-relaxation tests require the rise time of the applied strain to be short in comparison with the relaxation times to be measured. With the advanced rheometers it is possible to apply a step strain within few seconds or milliseconds (e.g., 20 ms to 1000 ms) (Lauger and Huck, 2002). Obviously, for a proper test the applied strain, and consequently the resulting stress, should be lower than the corresponding fracture value. Although in nonfood applications stress-relaxation tests can be continued for a long time, the test duration for foods is limited (on the order of minutes, e.g., 10 min or less) because degradation of sample may occur before the test is completed as a result of physical changes (e.g., moisture exchange with environment), microbial activity, and chemical and biochemical changes (e.g., enzymatic browning in fruits, oxidation in oil-containing foods) (Peleg, 1987). For a linear viscoelastic material subjected to an instantaneous constant strain (εo),* the initial stress will be proportional to the applied strain and will decrease with time (Figure 2.40). By linearity it is meant that when the applied strain is multiplied by any factor (e.g., doubling), the stress it produces also changes by the same factor (e.g., doubling). The rate of stress decay is quantified by a material characteristic time known as relaxation time, λ. The relaxation time in practice is defined as the time for stress to decay to about 37% of the initial level. However, single-relaxation time is often insufficient to fully describe the relaxation curve of most food materials. A better representation of relaxation curve is possible by using more than one relaxation time or, ideally, by using a continuous relaxation-time spectrum (Peleg and Normand, 1983). * The symbol ε hereafter denotes Hencky strain unless stated otherwise.


Stress

Straining stage Relaxation stage

σ0

0.37σ0

Rise time

Relaxation time, λ

Time

FIGURE 2.40 In actual relaxation test, the strain is applied over a finite time (rise time). The peak stress at the end of the straining stage is the inital stress (σo). Time taken for the stress to decay to 0.37σo during the relaxation stage is the relaxation time, λ.

For linear viscoelastic materials the stress decay with time t can be described, in tension or compression, by the following equation: σ(t ) = E (t ) ε o (2.44)

or E (t ) =

σ (t ) εo

where, the function E(t) is called the relaxation modulus. The relaxation modulus represents the change in stress per unit of applied strain and is a material property. For linear elastic solids the E(t) = E, the Young’s modulus. In shear configuration the corresponding equations are given as: τ( t ) = G( t ) γ o (2.45)

or G( t ) =

τ( t ) γo

where, τ(t) is the shear stress, G(t) the shear stress relaxation modulus, and γ the applied constant shear strain.


o

Analysis of Relaxation Behavior Stress relaxation is a basic test providing information on the viscoelastic character of materials rapidly. Although in principle relaxation test can be done in any configuration (i.e., compression, tension, shear, torsion, bending, etc) the most common one in cheese studies is compression. To determine linear viscoelastic region of a material in relaxation, a series of relaxation curves is obtained by sequentially increasing the applied strain. When the resulting data is replotted in terms of modulus vs. time, the curves within the linear viscoelastic region will overlap. The strain level at which the curve does not overlap indicates that the linear viscoelastic region is exceeded. Alternately, the linear range of the isochronal — the plot of stress against strain at a specific time — will indicate the extent of strain level over which the material response can be considered linear (Ak and Gunasekaran, 2001). The process of obtaining isochronal plots is illustrated in Figure 2.41. The data obtained at two constant strains (ε1 and ε2) are represented in Figures 2.41A and 2.41B. From these, data points (for, e.g., a, b, c, d in Figure 2.41A and 2.41B) are gathered at different times (e.g., t1 and t2). Then the corresponding σ(t) vs. ε plot is constructed for each of the times at which the stress response is measured (Figure 2.41C). The strain value at which the isochronal begins to deviate from linearity (indicated by dotted line in Figure 2.41C) is the upper limit of the liner viscoelastic region for the material. The hatched region in Figure 2.41C indicates the nonlinear range of the material studied. The mechanical model most suitable for quantification of relaxation behavior of foods and a variety of polymeric materials has traditionally been the generalized Maxwell model (Figure 2.42) with a discrete number of elements (Peleg and Normand, 1983):

σ (t ) = E(t ) = Eo + εo



n



t 

∑ E exp − λ   i

i =1

(2.46)

i

where, Eo is the modulus of the single spring (λ = ∞) in parallel to Maxwell elements in Figure 2.42, t the time, Ei the modulus of each Maxwell element, and λi the relaxation time of each Maxwell element. It must be mentioned that for a true viscoelastic liquid the first term (i.e., Eo) will be zero and the material will eventually relax completely. For linear viscoelastic behavior, the relaxation parameters are a function of time only. However, for nonlinear viscoelastic behavior, the relaxation parameters will be a function of time as well as imposed strain and strain history. An alternative model to describe relaxation and creep curves of viscoelastic solids is suggested by Peleg (1979, 1980): t = k1 + k2 t Y (t )


(2.47)

ε2 ε1 t

t

σ1(t)

σ2(t) c a

d

b

t1

t

t2

t1

t2

(a)

t

(b)

σ σ(t1)

Linear

c σ(t2)

a d Nonlinear b ε1

ε2

ε (c)

FIGURE 2.41 Plotting isochronals to determine linear viscoelastic range from stress relaxation data. (A) relaxation experiment at applied strain ε1; (B) relaxation experiment at applied strain ε2; and (C) isochronals plotted using data points a, b, c, and d from A and B at times t1 and t2 . (After Ak and Gunasekaran, 2001.)

and Y (t ) =

σ o − σ (t ) σo

(2.48)

where, σo is the initial stress and σ(t) the decaying stress. This linearization makes the calculation of model parameters easy as the slope gives k2 and the intercept gives the k1 . As it is also true for the parameters of the generalized Maxwell model,


E0

E1

h1

E2

h2

E3

h3

En

hn

FIGURE 2.42 Generalized Maxwell element for stress relaxation (E0, E2, E3, … En are spring stiffnesses; η1, η2, η3, … ηn are dashpot viscosities).

the dependency of the constants k1 and k2 on the applied strain is an indication of nonlinear viscoelasticity related to the structural modifications that occur during deformation. Peleg (1980) further stated that 1/k1 represents the initial decay rate, while 1/k2 represents asymptotic level of Y(t) when t→∞.

CREEP TEST As with stress relaxation, a creep test can be performed in different configurations (i.e., compression, tension, shear, torsion, etc). In an isothermal creep test, a constant stress is applied to the material, and the resultant strain is recorded as a function of time (Figure 2.43). In an actual test the stress application is not instantaneous but can be rapid such as by dropping the weight on the specimen. Analysis of Creep Behavior In a creep test a constant step-stress is applied to a material and the resulting deformation or strain is measured as a function of time. The distinction between constant stress and constant force is necessary, especially for highly deformable foods, because of the progressive change in the cross-sectional area of the specimen. Hence, a constant force (i.e., dead weight) results in a progressively increasing stress in uniaxial tension and decreasing stress in uniaxial compression (Purkayastha et al., 1985). Although in principle creep tests can be done in any configuration (i.e., compression, tension, shear, torsion, bending, etc.), the most common one in cheese studies is compression. For linear materials, the time-dependent compliance, D(t), is given by (Findley et al., 1989): D(t ) ≡ ε(t ) σ o

(2.49)

where, ε(t) is the tensile or compressive strain. Symbol J(t) is used to represent the shear creep compliance, that is, J(t) = γ (t)/τo, where γ (t) is the shear strain, and


Load removed Constant stress application

A

Stress

σo

0 B

t1

Time

t1

Time

Elastic solid response

Strain

εo quid

li cous

onse

resp

Vis

0

Recovery

Creep C 4

εi

Strain

3

5 2 Permanent strain

εi 1 t1

0 W

1

Time

Linear viscoelastic response

2

W

3

W W

4

5

FIGURE 2.43 Typical creep–recovery test. (A) application of instantaneous and constant stress (σo); (B) strain response of elastic solid and viscous liquid; (C) strain response of viscoelastic material. The application and removal of load (W) is shown at the bottom at various times along the creep–recovery curve for the viscoelastic material.


D(t) Creep Curve ∑Di

Delayed creep Steady state flow slope = 1/η

Do

Instantaneous compliance

Time

FIGURE 2.44 Typical creep compliance D(t) vs. time response of a viscoelastic material comprises of instantaneous compliance, delayed creep, and steady state flow (of viscosity η).

σ3

Strain

σ1 σ3 > σ2 > σ1

Time (a)

σ3 Compliance, D (t)

σ2

σ2 & σ1

Time (b)

FIGURE 2.45 Determining linear viscoelastic region by creep tests. A. Creep curves at different stress levels σ1, σ2, and σ3; B. Compliance D(t) vs. time for the corresponding creep curves overlap within the linear viscoelastic region (σ1 to σ2). Stress level σ3 is outside of linear viscoelastic region. (After Anon, 2002.)

τo the applied shear stress. The objective of creep tests is to determine material properties D(t) and J(t) from the experimental strain vs. time data (Figure 2.44). To determine linear viscoelastic region of a material in creep, a series of creep curves is obtained by sequentially increasing the applied stress (Figure 2.45). When the resulting data is replotted in terms of creep compliance vs. time the curves within the linear viscoelastic region will overlap. A typical creep compliance curve is shown schematically in Figure 2.44. Unless the applied force or stress is carefully selected the test terminates with the failure of the specimen, especially in tension. Quantification of creep behavior of foods and a variety of biological and polymeric materials has been traditionally based on the © 2003 by CRC Press LLC

E2

E3

En η1

E0

η2

η3

ηn

FIGURE 2.46 Generalized Voigt-Kelvin element for creep (E0, E2, E3, … En are spring stiffnesses; η1, η2, η3, … ηn are dashpot viscosities).

generalized Kelvin-Voigt model (Figure 2.46) with a discrete number of elements (Purkayastha et al., 1984): ε (t ) t = D(t ) = Do + + σo η1



n



t 

∑ D 1 − exp − τ   i

i =2

(2.50)

i

where, Do is the instantaneous compliance (= 1/Eo), t the time, η1 the Newtonian viscosity while 1/η1 being the slope of the linear portion of the creep curve after sufficiently long time, Di the delayed compliance of each Kelvin-Voigt element (= 1/Ei), and τi the retardation time of each Kelvin-Voigt element. The last term in Equation 2.50 is called the creep function and denoted by ψ(t) (Purkayastha et al., 1984). It must be mentioned that for a true viscoelastic solid material the second term (i.e., t/η1) will be zero, and the material will eventually reach an equilibrium creep compliance. Equation 2.50 also reveals that a typical creep curve is composed of three components, as illustrated in Figure 2.44. For linear viscoelastic behaviors the creep parameters are a function of time only. However, for nonlinear viscoelastic behaviors the creep parameters will be a function of time as well as imposed stress and stress history. As for the relaxation case, the Peleg model (Purkayastha et al., 1984) can be used to linearize and represent the creep behavior of foods. For creep data, the Peleg model is used to represent the creep function using constants k1″ and k2″ as: ψ (t ) = D(t ) − Do −

t t = " η1 k1 + k2" t

(2.51)

SHEAR RHEOMETRY Polymer and food-processing applications involve a wide range of shear rates as shown in Figure 2.47. Thus, various rheometry measurements based on different geometries are essential and complementary to each other.


Oscillatory/Vibrational Rotational Elongational Capillary Polymer Processing Food Processing 10−6

10−3

10−2

10−1

100

101

102

103

104

105

Shear rate (s−1)

FIGURE 2.47 Approximate shear rate ranges for different rheometry measurements and those involved in polymer processing and food processing applications. (After Riande et al., 2000.)

Rheological measurements based on shear flow are conveniently divided into two groups: (a) drag flows in which shear is generated between a moving and a fixed solid surface, and (b) pressure-driven flows in which shear is generated by a pressure difference over a closed channel (Macosko, 1994). Examples of shear-flow geometries belonging to the first group include sliding plates, concentric cylinders, parallel disks, and cone and plate. Examples of shear-flow geometries belonging to the second group include capillary or Poiseuille flow, slit flow, and axial annulus flow. The working equations for some of these measurement techniques are presented here. Interested readers are referred to other sources (Collyer and Clegg, 1988; Macosko, 1994; Steffe, 1996) for detailed discussions. The measurement systems described below can be used to conduct a variety of tests (e.g., steady shear, dynamic, relaxation, creep). In some tests, one system may be preferred over the others due to the shear-rate range or other advantages it offers. For instance, parallel-plate geometry is often preferred for measuring viscoelastic properties of solid cheese (e.g., relaxation modulus, creep compliance, dynamic moduli) since the sample handling is easier and the sensitivity to gap setting is less as compared to the cone-and-plate geometry. There are several companies manufacturing highly advanced rheometers and viscometers that will satisfy the measurement needs of researchers. Barnes et al. (1999) summarized the history of commercial viscometry and rheometry. Web sites of various companies offering rheological instruments are given in Table 2.7 for readers to have quick access.

SLIDING-PLATES GEOMETRY The schematic drawing of the sliding-plate geometry is shown in Figure 2.48. This relatively simple arrangement is generally used in defining shear viscosity. The


TABLE 2.7 Some Major Manufacturers of Rheometers and Viscometers Company

Web Site

Alpha Technologies ATS RheoSystems Bohlin Instruments Brookfield Camtel Dynisco Polymer Test GBC Scientific Equipment Pty Ltd. Goettfert Inc. Haake Infra Scientific Kaltec Scientific Paar Physica Porpoise Viscometers Pressure Profile Systems, Inc. Reologica Instruments AB Research Equipment (London) Ltd. Rheometric Scientific TA Instruments Vilastic Scientific Inc.

http://www.alpha-technologies.com/ http://www.atsrheosystems.com http://www.bohlin.com/ http://www.brookfieldengineering.com/ http://www.camtel.co.uk/ http://www.dynisco.com/ http://www.gbcsci.com/ http://www.goettfert.com http://www.thermo.com/ http://www.infra.uk.com/ http://www.kaltecsci.com/ http://www.physica.de/ http://www.porpoise.co.uk/ http://www.pressure-profile.com/ http://www.reologica.se/ http://www.research-equipment.com/ http://www.rheosci.com/ http://www.tainst.com/ http://www.vilastic.com/

Source: The Society of Rheology Web site, http://www.rheology.org/sor.

Moving plate ∆X

y

Fx,Vx sample

H

sample

x Stationary plate

L Width of plate = W; Area of plate, A = L.W

FIGURE 2.48 Description of sliding plate geometry. Application of shear force Fx moves the top plate by ∆X at a velocity Vx.

sliding-plate rheometer can be operated in either strain-controlled or stress-controlled mode. The shear strain (γ), shear rate (γ˙), and shear stress (τ) can be calculated from the following equations (Dealy and Wissbrun, 1989; Macosko, 1994): γ=

∆X Vx t = H H

(2.52)

Vx H

(2.53)

γ˙ =


τ=

Fx F = x A LW

(2.54)

where, L is the plate length, W the plate width, H the sample thickness, ∆X the sliding-plate displacement, Vx the velocity of sliding plate, Fx the shear force, and t the time. The shear viscosity is then computed from η = τ/ γ˙. The assumptions involved in the sliding-plates geometry include: (a) negligible inertial and edge effects to establish homogenous, simple shear flow, and (b) L and W are much greater than H, and H is as small as possible. The principal advantage of the sliding-plate geometry is that it is ideally suited for studying nonlinear viscoelasticity. The relative advantages and disadvantages of different rheometer geometries for studying nonlinear viscoelasticity are listed in Table 6.1. Further discussion on the theory and application of the sliding-plate rheometer for studying nonlinear viscoelasticity of cheese is presented in Chapter 6.

CONCENTRIC-CYLINDERS GEOMETRY The schematic views of different concentric-cylinder geometries are shown in Figure 2.49. The concentric-cylinder geometry has long been used in commercial viscometers. The concentric-cylinder system consists of an inner cylinder (called “bob”) positioned inside an outer cylinder (called “cup”). The sample is contained in the annular gap between the “infinitely” long bob and cup. In some cases the cup is rotated at a steady angular velocity while the bob is kept stationary, and in others, the bob is rotated at a constant angular velocity and the cup is fixed. The concentric-cylinder system is typically used with low-viscosity materials and mobile suspensions. The double-gap or double-Couette geometry (Figure 2.49) offers greater sensitivity than the other concentric-cylinder systems at low shear rates and viscosities as a result of its larger surface area. If sample drying (or skin formation) is likely to be an issue, which is a common problem in working with low-fat cheeses, it is better to use a solvent trap with the measuring system or alternatively a low-viscosity oil can be used as a barrier provided that the oil used does not interact with the sample to alter the sample properties. Working equations for shear strain γ, shear strain rate γ˙, and shear stress τ are given as (Macosko, 1994): Shear strain:

γ=

Ωi t R R (for narrow gaps; that is κ = i ≥ 0.99) Ro − Ri Ro (2.55)

and R=

( Ro + Ri )


2

Bob

Cup

Wi

Wi

Ri

L Ro

L z

r

a

Lb

Lb

sample

Standard concentric cylinder

Coaxial cylinder with conic base

Wi

Wi

trapped air

Mooney cell (recessed bottom cylinder)

Double gap (double Couette)

FIGURE 2.49 Different concentric-cylinder measurement systems.

Shear strain rate: γ˙ ( Ri ) ≅ γ˙ ( Ro ) =


Ωi R 2 Ωi = Ro − Ri 1 − κ 2

(for κ > 0.99)

and γ˙ ( Ri ) =

(

2 Ωi

n 1 − κ2/n

)

and γ˙ ( Ro ) =

(

−2 Ω i

n 1 − κ −2 / n

)

(for 0.5 < κ < 0.99) (2.56)

where n=

d ln Mi d ln Ω i

and

R=

Ro + Ri 2

Shear stress: τ( Ri ) =

Mi 2 π Ri2 L

(2.57)

where, Ri is the radius of bob, Ro the radius of cylinder, Mi the torque on bob and Ωi the angular velocity of bob, L the height of bob. Other alternative designs that are generally used to minimize end effects or make possible to account for the end effects are also depicted in Figure 2.49. The shear stress for the coaxial cylinder with conic base is given by: τ( Ri ) =

Mi 2 πR 2 L + 2 πR3  i 3 i  

(2.58)

With the Mooney cell or recessed bob the air is trapped underneath the bob and contributes practically no torque to the overall response. As can be seen from Equation (2.56), the shear rate changes across the gap for a wide-gap viscometer. This is a serious concern when using concentric-cylinder geometry with concentrated suspensions. Rotating the bob in a concentrated suspension causes particles to migrate away from higher shear-rate regions near the bob to lower shear-rate regions near to the cup (Abbott et al., 1991).

CONE-AND-PLATE GEOMETRY Sketches of different cone-and-plate (C&P) geometries are shown in Figure 2.50. The C&P system consists of a rotating (stationary) cone and a stationary (rotating) plate with a sample contained between them. The apex of the cone is essentially in contact with the plate. As seen in the following working equations the main advantage of using the C&P geometry is that the shear rate is approximately constant (i.e., independent of radial position) throughout the sample provided that the cone angle does not exceed a few degrees (e.g., ≤ 4°) (Macosko, 1994). This feature of constant shear rate is the reason why C&P geometry is particularly useful for studying nonNewtonian behavior.


Ω, φ

θ

α

sample

Rt r

R

R

Normal cone-and-plate system

Truncated cone-and-plate system

R hc Rd

Double cone system

Extended cone-and-plate system Cone-and-dish system

FIGURE 2.50 Different cone-and-plate measurement systems.

Shear strain: γ=

φ tan α

(2.59)

where φ is the angle of rotation and α the cone angle. It is true that for a small cone angle tan(α) ≅ α*, for which the shear strain becomes: γ=

φ α

(2.60)

It is seen that for a given cone angle the strain is homogeneous and independent of position in the sample. Shear rate: dγ dφ 1 Ω = γ˙ = = dt dt sin α sin α

(2.61)

For small cone angles the shear rate can be simplified to:

* Maclaurin series for tan(α) is given as: tan(α) = α + (α3/3) + (2α5/15) + … where α in radian unit (Thomas and Finney, 1988).


γ˙ =

Ω α

(2.62)

where, Ω = specified angular velocity. This equation shows that a uniform shear rate is realized experimentally with cone-and-plate system with small cone angles. According to the calculations of Adams and Lodge, reported in Lodge (1964), the errors involved in shear rate approximation by Equation 2.62 are 0.02, 0.18, 0.50, and 2% for cone angles of 1°, 3°, 5°, and 10°, respectively, for a material with a constant viscosity. Shear stress: τ=

3M 2 π R3

(2.63)

where, M = torque and R = radius of plate. This equation indicates that proper loading of the specimen is vital so that a full contact of the cone with the specimen is established since the torque measurement (and, consequently, the stress calculation) depends on R3. The C&P geometry is typically configured with cones having angles less than 4°. The cone angle shall be chosen with care since, for instance, for large cone angles the shear rate across the gap will vary; on the other hand, for the small cone angles there is higher chance for errors due to gap settings. For instance, for a cone angle of 10° the variation in shear rate across the gap is 3%, and the resulting error in calculated viscosity of Newtonian liquids is 2%. However, since the cone angles typically used is 4° or less the resulting error is quite small (Dealy, 1982). The C&P geometry cannot accommodate materials that contain particles since the particles can be subjected to grinding action near the tip of the cone. Therefore, quite often the tip of the cone is slightly truncated to allow measurements on particulate fluids. Cones are often slightly truncated, as shown in Figure 2.50, by removing the tip of the cone to make them more robust measurement tools. Errors due to cone truncation are generally negligible since the radius of truncation Rt is much smaller than R. The maximum error in torque associated with truncated cones can be calculated from the following equation:  R3 − Rt3  Maximum error = 1 −  100 R3  

(2.64)

For instance, if Rt = 0.2R then the maximum theoretical error in torque is 0.8%. Although C&P geometry is very simple and useful there are cases in which it shall not be preferred: (a) it is not recommended when conducting temperature sweeps unless the rheometer is equipped with an automatic system for thermal expansion compensation; and (b) it is not recommended for testing samples with particulate materials. The particles can jam the cone apex, giving erroneous data. Moreover, cone-and-plate system is not suitable for materials with a high concentration of solids, as the solids become expelled from the gap under high shear rates.


In the extended C&P geometry the apex of the cone does not touch the plate, instead there is a finite distance between the apex and the plate, denoted by hc in Figure 2.50. The shear rate for this case is determined from (Powell, 1988): γ˙ ec =

RΩ hc + R tan α

(2.65)

It is sometimes essential to replace the flat plate with a dish to contain liquid materials as seen in Figure 2.50. Truncated and extended cones can also be utilized with a dish in place of a flat plate. Furthermore, double cone (or biconical) sensors (Figure 2.50) have been developed and employed for measuring very low viscosity liquids with small sample volume. This geometry eliminates the free surface and variation of its shape with rotational speed, and minimizes the sample exposure to the environment. On the other hand, it introduces a new type of edge effect (Dealy, 1982).

PARALLEL-PLATE GEOMETRY A schematic drawing of the parallel-plate geometry is shown in Figure 2.51. The parallel-plate system consists of a rotating (stationary) upper plate and a stationary (rotating) lower plate separated by the sample to be tested. Although similar in many ways to C&P system, the major difference between the parallel-plate and C&P systems is that the shear rate in the parallel-plate system is not constant but varies across the sample. Thus, if the objective is to subject the entire sample to a uniform shear, then the parallel-plate geometry is not appropriate. On the other hand, parallelplate geometry is not as sensitive to the gap-setting errors as the cone-and-plate system (Macosko, 1994). The working equations for the parallel-plate geometry is given as follows: Shear strain: γ=

φr h

(2.66)

It is clear that shear strain is not homogeneous and varies with radial position r. Shear rate at the edge (at r = R): γ˙ R =

R Ω h

(2.67)

It is seen that the shear rate can be varied in two ways: (a) by changing the rotational speed, Ω, and (b) by changing the gap between plates or sample thickness, h. Shear stress: τR =


M 2 π R3

 d ln M  3 +  d ln γ˙ R  

(2.68)

Ω, φ

sample h R

Sample thickness or gap setting

R h

Rdish

FIGURE 2.51 Parallel-plate measurement systems.

where, M = torque on the rotating plate. The derivative term in the brackets makes the shear stress and viscosity calculations in the parallel-plate geometry more involved and difficult than those in cone-and-plate geometry. The accurate evaluation of the derivative term requires sufficient amount of torque versus edge shear rate data. For a Newtonian liquid the derivative term is equal to 1.0, and the equation reduces to: τR =

2M π R3

(2.69)

When the lower plate is replaced with a dish to contain the liquid sample there will be additional torque contribution caused by the increased frictional drag of the dish. Vrentas et al. (1991) presented an analysis of the dish effect (referred to as reservoir effect in their paper) for the flow of a Newtonian fluid in a parallel-plate rotational viscometer. According to their results, provided that the ratio of radius of


dish, Rdish, to radius of upper plate, R, is greater than 1.10, the following equation can be used to calculate the torque: M=

Mmeasured h 1 + 1.9 R

(2.70)

The torque, M, from this equation can be used in Equation 2.69 to calculate shear stress τR for a Newtonian fluid.

CAPILLARY RHEOMETRY Capillary tube rheometry is a well-established technique for studying shear properties of materials. It has been applied to study viscosity of cheese (Smith et al., 1980), butter (Shukla and Rizvi, 1995), and many other food materials (Halliday and Smith, 1995; Sharma et al., 1993; White et al., 1993). Capillary viscometers are often used in laboratories and as an on-line instrument in process industries to measure viscosity (Roberts, 2001). The capillary rheometer consists of a small tube through which an incompressible fluid is forced to undergo steady axial laminar flow, either by means of an imposed pressure or a piston moving at a constant speed (Figure 2.52). The capillary rheometers can also be designed to have several capillary sections of different diameters in series so that non-Newtonian fluids can be characterized in a single pass of fluid (White et al., 1993). The quantities normally measured are the volumetric flow rate, Q, and the driving pressure, Pdriving. When a moving piston generates the flow, the driving pressure is related to the piston force (Fpiston) and reservoir radius (Rreservoir) as follows (Dealy and Wissbrun, 1989): Pdriving =

F piston 2 π Rreservoir

(2.71)

The important assumptions made in the analysis of capillary flow are (Macosko, 1994): (a) fully developed, steady, isothermal, laminar flow; and (b) fluid velocity is zero at wall — that is no slip at the wall. The total pressure drop (∆P) for flow of fluid from a reservoir, through a capillary and out to the ambient pressure consists of two components (Dealy and Wissbrun, 1989)*: Pdriving − Pambient = ∆P = ∆Pend + ∆Pcapillary

(2.72)

where, ∆Pend = excess pressure loss due to the entrance and exit flow (i.e., ∆Pend = ∆Pentrance + ∆Pexit), and Pambient = ambient pressure. The components of total pressure drop are schematically illustrated in Figure 2.53. * Although ∆ means “final-initial,” which makes ∆P a negative value; for convenience we consider the term ∆P as “Phigher – Plower” to make it a positive quantity.


W (Dead weight) or Vz (Constant velocity) z Plunger r Rr

Sample Reservoir or barrel section

R

Capillary section

L

Q (Volumetric flow rate)

FIGURE 2.52 Schematic drawing of a piston-driven capillary rheometer.


∆P

∆Pentrance

∆Pcapillary

∆Pexit In

Out

Reservoir section

Capillary section

FIGURE 2.53 The pressure profile in different sections of a capillary rheometer.

The working equations for capillary rheometry are given as (Macosko, 1994): Wall shear stress:

τw =

1 ∆Pcapillary 1 ∆P 1 ∆P − ∆Pend = = 2 ( L / R) 2 [( L / R) + CB ] 2 ( L / R)

(2.73)

where, CB (= ∆Pend /(2τw) is the Bagley correction, which takes into account the pressure losses in the entrance and exit of the capillary. The Bagley correction is either applied to capillary length-to-radius (L/R) term or to the pressure term, as written in Equation 2.73 The Bagley correction procedure involves measuring the pressure drop for a number of capillaries having different lengths (thus, different L/R ratios) at selected values of apparent wall shear rate. It is common practice to use at least three tubes of the same diameter but different lengths. The magnitudes of end corrections are determined from Bagley plots as shown schematically in Figure 2.54. It may be necessary to apply corrections to the measured volumetric flow rates (Qm ) if there is wall slippage. Wall slip is to be suspected when plots of τw vs. apparent shear rate at wall γ˙aw , (see below) for capillaries with the same L/R ratio but different diameters do not fall on a single curve. In accounting for slip, the slope of [Qm/(πR3τw)] versus [1/R2 ] is taken as the corrected slip coefficient, βc, and this parameter is used in the following equation to calculate the corrected volumetric flow rate (Qc): Qc = Qm − β c π R τ w


(2.74)

∆P

. γaw3

. γaw2

. γaw1

∆Pend3 ∆Pend2 ∆Pend1

e3

e2

e1

L /R

FIGURE 2.54 Bagley plot for pressure corrections for the end effects. The measured pressure drop (∆P) for different length-to-radius ratio (L/R) of the pipe is obtained for different wall shear rates (γ˙ aw1, γ˙ aw2, γ˙aw3). The correction factors (∆Pend1, ∆Pend2, ∆P end3) are obtained from the intercepts.

Apparent or Newtonian shear rate at the wall is given by: γ˙ aw =

4 Qc π R3

(2.75)

For a non-Newtonian liquid the shear stress at the wall τw is unchanged while the shear rate at the wall is calculated from Weissenberg-Rabinowitsch-Mooney equation (Macosko, 1994):  3 1 d ln Qc  γ˙ w = γ˙ aw  +   4 4 d ln Pc 

(2.76)

The term in square brackets is called the Rabinowitsch correction. The slope (dlnQc/dlnPc) is equal to 1.0 for Newtonian fluids, and to (1/n) for power-law fluids with n being the flow-behavior index. Once shear rate and shear stress are known at the same location we can then calculate shear viscosity (η = τw /γ˙w) and construct either the flow curve (τw vs. γ˙w) or viscosity curve (η vs. γ˙w). The capillary rheometer can be operated in two modes: (a) controlled volume or displacement mode, where Q is controlled and ∆P is measured, and (b) controlled pressure mode, where ∆P is controlled and volumetric flow rate (Q) is measured. The first mode can be realized using a Universal Testing Machine with constant crosshead speeds. The second mode can be realized either by applying a dead weight on the plunger or by using gas pressure to move the plunger.


EXTENSIONAL RHEOMETRY Extensional or elongational rheometry is a relatively new area of active research when compared to shear rheometry, and began to receive increased attention around 1970. (Macosko, 1994; Doraiswamy, 2002; Barnes et al., 1989). Its development stems from the observation that several industrially important polymer processing operations such as extrusion, molding, fiber spinning, calendaring, blowing, coating, and foam production involve significant extensional deformation in addition to shear deformation (Cogswell, 1981; Macosko, 1994; Baird, 1999). Currently, commercial elongational rheometers are available to measure extensional properties of polymer melts (Schulze et al., 2001; Meissner and Hostettler, 1994). Moreover, the filamentstretching rheometer has been developed to measure extensional properties of mobile polymer solutions (Tirtaatmadja and Sridhar, 1993; Sridhar, 2000). Extensional flows are more sensitive to variations in molecular structure of a polymeric material, and thus offer a powerful means of polymer characterization (Münstedt et al., 1998). It is possible for polymers to have identical shear flow properties while exhibiting extremely different extensional flow properties. Only for deformations that are either very small or very slow, the theory of linear viscoelasticity provides relationships between material functions determined using various kinds of deformations (Dealy and Wissbrun, 1989). For instance, the following limiting relation between extensional and shear properties is established (Barnes et al., 1989; Dealy, 1995): ηE (ε˙ ) ε˙ →0 = 3η( γ˙ ) γ˙ →0

(2.77)

where, ηE is the tensile or extensional viscosity and η is the shear viscosity. For Newtonian fluids ηE = 3η for all values of strain rates. This relationship is named as Trouton ratio, TR, defined as (Jones et al., 1987): TR =

η E (ε˙ ) η γ˙ = 3ε˙

(

)

(2.78)

For calculating TR, the shear viscosity should be evaluated at a shear rate numerically equal to 3ε˙ . Trouton ratio is exactly 3 for inelastic flows, and any departure from the value of 3 is associated with viscoelastic effects (Jones et al., 1987; Barnes et al., 1989). It is clear that when the relation given in Equation 2.77 is valid there is no need to make extensional tests since the extensional viscosity can be calculated from the shear viscosity function determined at small and slow shearing experiment (Dealy and Wissbrun, 1989). However, for large and rapid deformations the relation given in Equation 2.77 is not valid, except at small strains, and therefore it is essential to make extensional measurements. There are several kinds of extensional-flow geometries such as uniaxial extension, squeezing flow, sheet stretching, fiber spinning, bubble collapse, stagnation flows, and entrance flows (Macosko, 1994). Direct extensional-flow measurements have an advantage over shear measurements in that the measurement does not involve


a sample–instrument interface, and thus there is no slip problem to consider (Cogswell, 1981). Some of these extensional methods are already applied to cheese with some success, as described in Chapter 9. Here we limit our attention to lubricated squeezing flow (LSF) technique, which is the most popular and promising method of studying extensional properties of (melted) cheese.

LUBRICATED SQUEEZING FLOW Squeeze flow or squeezing flow is often used to determine flow properties of highly viscous materials such as polymer melts and semiliquid or semisolid food products (e.g., cream cheese, peanut butter, melted cheese, tomato paste, butter, dough, etc). A schematic diagram of the squeezing-flow geometry is shown in Figure 2.55. Although it is a simple test to perform, the analysis of squeezing flow may not be straightforward, particularly if there is friction between the specimen ends and compression plates (i.e., unlubricated squeezing flow). In unlubricated squeezing flow, if a significant shearing component is present it alters the pattern of outward flow of material between the plates. Thus, it is necessary to lubricate the sample–platen contact surfaces in order to eliminate the shear in the sample and obtain purely shear-free or biaxial extensional flow. Lubricated or unlubricated squeezing flow can be conducted in constant volume or constant area configurations depicted in Figure 2.55. The material is squeezed out between two parallel plates at either controlled force (or stress) or controlled speed (or strain rate). Quite often the upper plate is moving at a constant speed while the lower plate is stationary. The squeezing-flow configuration represents one of the few cases where specimen loading and cleaning of equipment are fairly easy. The simple geometry of the lubricated squeezing flow (LSF) makes it also convenient for performing stress relaxation or creep experiments. Replacing the lower plate with a shallow dish results in a new configuration named as “imperfect squeezing flow” (Lee and Peleg, 1992) (Figure 2.55). LSF has been developed and used first by Chatraei and Macosko (1981) to measure biaxial extensional viscosity of polydimethyl siloxane and polyisobutylene melts under constant stress. The LSF technique was introduced to food rheology in the mid-1980s (Casiraghi et al., 1985) and has since been applied to various kinds of food products (Campanella et al., 1987; Campanella and Peleg, 1987b; Hoffner et al., 1997; Shukla et al., 1995; Huang and Kokini, 1993; Bagley et al., 1990; Corradini et al., 2000; Suwonsichon and Peleg, 1999; Wang et al., 1998; Ak and Gunasekaran, 1995). Recently, Campanella and Peleg (2002) reviewed LSF applications to semiliquid foods. A good example of application of LSF for cheese is the UW Meltmeter, the cheese meltability measurement device developed by Wang et al. (1998). In this, the fat melting from cheese at high temperature self-lubricates the compression plates, making it an ideal test method. The UW Meltmeter is discussed in detail in Chapter 8. The LSF geometry and velocity profile in the specimen and the lubricant layer are depicted in Figure 2.56. In an ideal situation, the lubricant undergoes shear deformation, and the sample undergoes biaxial extension. The working equations for LSF method are given as follows (Chatraei and Macosko, 1981; Macosko, 1994):


R

R

Sample

H

r(t) Unlubricated constant volume squeezing flow

Unlubricated constant area squeezing flow

R

R

Sample

H

r(t) Lubricated constant volume squeezing flow

Lubricated constant area squeezing flow

R Sample Imperfect squeezing flow = Constant Load, W or = Constant Velocity, Vz

FIGURE 2.55 Configurations of unlubricated squeezing flow, lubricated squeezing flow, and imperfect squeezing flow tests.

Axial Hencky strain:  H ε H = ln   Ho 

(2.79)

Axial strain rate: ε˙ H =

1 dH Vz = H dt H

(2.80)

where Vz is the velocity in the vertical direction (e.g., crosshead speed of a universal testing machine).


Upper plate (moving) δ/2

Lubricant layer

The ideal velocity profile

H

Specimen

Lubricant layer

δ/2

*Shear in the lubricant *Extension in the sample

Lower plate (fixed)

FIGURE 2.56 Lubricant and sample velocity profiles in ideal lubricated squeezing flow (the sample thickness H is much greater than the total lubricant layer thickness δ). (After Papanastasiou et al., 1986.)

Radial or biaxial strain:  R 1  H ε B = ln  = − ln  2  Ho   Ro 

(2.81)

where, the constant volume assumption, (R/R0) = (H0/H)1/2, is applied. Radial or biaxial strain rate: ε˙ B = −

1 dH 2 H dt

(2.82)

It is assumed that at any moment the lubricant film thickness, δ, is smaller than the specimen thickness, H, and therefore, H + δ ≅ H. This is justified at the start of the test, but may be questionable at later stages when the specimen thickness becomes small at large strains (e.g., when εΒ = 2.0, H/Ho = 0.018). Biaxial (compressive) stress when the gap is fully filled with sample: σB =

F π R2

(2.83)

In this case Rspeciman ≥ Rplate = R. Biaxial (compressive) stress when the gap is partially filled with sample: σB =

F π r(t) 2

In this case r(t) represents the instantaneous radius of the specimen.


(2.84)

Biaxial stress growth coefficient (Dealy, 1995): η+B (t, ε˙ B ) ≡

σB ε˙ B

(2.85)

Biaxial extensional or elongational viscosity:

[

]

η B (ε˙ B ) = lim η+B (t, ε˙ B )

t →∞

(2.86)

For Newtonian fluids the biaxial elongational viscosity is six times the shear viscosity, ηB = 6η. One important consideration in LSF, which is often not addressed in food-related studies, is the loss of effective lubrication, which limits the maximum achievable strain. Macosko (1994) mentions that the strain in LSF is limited to 1.0–1.5 because of the loss of effective lubrication. It is experimentally demonstrated that lubrication is maintained up to a higher total strain if the lubricant has a higher zero-shear viscosity (Chatraei and Macosko, 1981). The optimum ratio of the zero-shear viscosities of the sample to the lubricant is reported to range from 500 Pa.s to 1000 Pa.s (Papanastasiou et al., 1986; Soskey and Winter, 1985). The criterion for good lubrication is given as (Macosko, 1994): 2δ η L R 2 < < 20 H ηS δ 2

(2.87)

where ηS is the sample viscosity, δ the lubricant thickness, H the sample thickness, ηL the lubricant viscosity, and R the plate radius.

EQUATIONS

FOR

DIFFERENT FLUIDS

IN

LUBRICATING SQUEEZING FLOW

Analytical solutions for lubricated-squeezing flow of Newtonian and non-Newtonian fluids are given in various publications in the rheology literature. We present below these equations describing the specimen thickness as a function of time under constant load, or the load as a function of time under constant velocity for Newtonian and non-Newtonian fluids. (1) Newtonian fluids (a) Flow under constant load–constant volume (Lee and Peleg, 1989): 1 1 W t = + H (t ) Ho 3ηΛ

(2.88)

where, W is the constant load, η the Newtonian viscosity, Λ the specimen volume, and t the time.


(b) Flow under constant load–constant area:  Wt  H (t ) = Ho exp − π 3 R 2 η  

(2.89)

where, R is the radius of the plate (= radius of the sample). (c) Flow under constant velocity–constant volume: 3ηΛ V [H(t)]2 z

F (t ) =

(2.90)

where, Vz is the squeezing speed. (d) Flow under constant velocity–constant area: F (t ) =

3π η R 2 Vz H(t)

(2.91)

(2) Power-law fluids (a) Flow under constant load–constant volume (Campanella and Peleg, 1987b): 1/ n     t  W   1  1 = +    H (t )  Ho1 / n n  n2+1 K 3 Λ    

n

(2.92)

where, K is the consistency index, and n the flow behavior index. (b) Flow under constant velocity–constant volume: F (t ) = 3

n +1 2

ΛK

(Vz )n

(2.93)

[ H (t )]n+1

(c) Flow under constant load–constant area: 1/ n    W  t  H (t ) = Ho exp  − n+1   2 2  3 π R K  

(2.94)

(d) Flow under constant velocity–constant area: F (t ) = 3


n +1 2

 V  πKR  z   H (t )  2

n

(2.95)

(3) Herschel-Bulkley fluids (a) Flow under constant load–constant volume (Ak and Gunasekaran, 2000):   1 1 −n 1 3 τo Λ  ,1 + , +  1 / n  Hypg2 F1 , n H (t ) W  n n  ( H (t ) W )  1/ n   1 1   −n 1 3 τo Λ  1   ,1 + , t  = − ( n +1)  1 / n Hypg2 F1 , n Ho W   ( Ho W )  K Λ 3  n n  

(2.96)

where, Hypg2F1[a,b,c,z] is the hypergeometric function 2F1[a,b;c;z] (Mathematica 3.0, Wolfram Research; Andrews, 1992). The validity of this hypergeometric solution, Equation 2.96, is verified by the fact that it reduces correctly to Equation 2.88 for n = 1 and τo = 0, and to Equation 2.92 for τo = 0 (Ak and Gunasekaran, 2000). (b) Flow under constant velocity–constant volume: V   Λ  F (t ) = 3 τ o + K  3 z   H(t)  H (t ) 

n

  

(2.97)

(c) Flow under constant load–constant area:   W − πR2 3 τ  1 / n  o H (t ) = Ho exp − n +1  t 2   πR 3 K  

(2.98)

(d) Flow under constant velocity–constant area: n  V    F (t ) = 3 π R 2  τ o + K  3 z    H (t )   

(2.99)

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3

Uniaxial Testing of Cheese

Uniaxial testing is the most popular configuration for evaluating mechanical and rheological properties of cheeses. In this chapter, we attempt to summarize the extensive literature on different cheeses. The available data are diverse and are often collected by various testing modes, which can be grouped under the word “uniaxial.” Uniaxial testing is also the most popular method for instrumental evaluation of cheese texture (see Chapter 7). One way to handle such diverse literature on properties of different cheeses would be to present the available data according to the type of cheese. However, we organized the information according to the specific test method (e.g., uniaxial compression, tension, relaxation, etc.), since effects of many experimental factors (e.g., deformation rate, cheese age, etc.) on rheological properties of different cheeses bear some similarities. Rigorous analysis of the literature data is seriously hampered by the lack of standardization in terms of sample preparation, measurement conditions, parameter evaluations, and reporting style. It may not be feasible to specify particular requirements for each of these issues since the objectives of measurements can be totally different (e.g., quality control, correlations with sensory results, etc.), and the prevailing conditions under which the tests are made may also vary. However, minimum requirements in data reporting may be (and perhaps should be) universally agreed upon, which would greatly facilitate comparison of results from different sources. Masi (1987) reported a major attempt to improve the comparability of results from different laboratories and to identify the most suitable measurement conditions for generating reproducible mechanical data on cheese. Even in this collaborative work the participating laboratories used different experimental conditions (i.e., sample size and shape, test temperature, sample handling, sample age at testing, number of samples tested, etc.). The recommendations based on the results of this extensive study are given in Table 3.1 for cheese, as well as general recommendations for reporting compression results. Some of these suggestions are no longer relevant due to advances in the instrumentation. For instance, the recommended crosshead speeds are naturally based on the characteristics of uniaxial testing machines available at that time and the capabilities of the strip-chart recorders, which are practically obsolete now. A group at the International Dairy Federation (IDF) also formed a working team (IDF E703) to develop standards for testing and reporting uniaxial test results on cheese. Though a draft was prepared, it was never officially published.* Nevertheless, the special bulletin of IDF (1991) includes expert opinions on important issues related to cheese rheology and texture measurements, and recommendations on test methods. * Philip Watkinson, Fonterra Research Center, New Zealand, personal communication.


TABLE 3.1A Recommendations for Compression Testing of Cheese Measurement Conditions

Data to Be Reported

a. Shape: vertical cylinder or prism b. Size: aspect ratio > 1 (aspect ratio = sample height/sample width or diameter) c. Crosshead speed: 5 cm/min and two other speeds in the range 0–20 cm/min d. Boundaries: Both bonded and lubricated platens

a. Rupturea (or yield) stress b. Rupture (or yield) strain c. Elastic modulus d. Rupture (or yield) work per unit of volume e. Failure mode

a

If the cheese does not show rupture or yield, then the stress and the work per unit of volume should correspond to 80% deformation. Source: After Masi, 1987. With permission.

TABLE 3.1B Information to be Included in Compression-Test Reports Contextual Information

Measurement Conditions

Results

a. Detailed description of sample b. Sample dimensions c. Sample history (including preparation and conditioning)

a. Temperature of environment b. Relative humidity of environment c. Sample temperature d. Interface between sample and compression surfaces I. Material II. Roughness III. Dimensions e. Compression rate f. Machine details Accuracy of (1) compression rate, and (2) force measurement g. Initial position of crosshead in relation to the sample h. Response time

a. All original and derived results b. Complete force–deformation curve

Source: After McKenna, 1987. With permission.

The theory of fracture mechanics and the methods specially designed to study fracture properties of materials are discussed in Chapter 4. Fracture properties reported in this chapter are those that are often routinely determined even though the primary objective of the research is not to study fracture behavior. For instance, work-to-fracture (or area under stress–strain curve) values are given in this chapter, whereas the specific fracture energy data are reported in Chapter 4.

UNIAXIAL COMPRESSION MEASUREMENTS Mechanical properties commonly determined from uniaxial compression tests on cheese include modulus of deformability ED, fracture stress σf , fracture strain εf , and work to fracture Wf . All these variables are defined in Chapter 2 (see Figure 2.23). © 2003 by CRC Press LLC

The numerical results for these properties extracted from many publications on different cheeses are listed in Tables 3.2 to 3.5. In the following sections, we will frequently refer to these compilations. For viscoelastic materials, Mohsenin and Mittal (1977) suggested the term “modulus of deformability” instead of Young’s modulus, which is reserved for engineering materials obeying the Hooke’s law. The modulus of deformability is the slope of the “initial linear” part of the stress–strain curve. There are several methods for getting a representative value of the slope. One of these methods is to fit the stress–strain data to a polynomial equation (see Chapter 2, Equation 2.22) and to determine properties from the resulting fit equation (Ak and Gunasekaran, 1992). The other approach is to take the maximum slope within the strain range from 0 to 0.05 (Wium et al., 1997). Another method is to take the slope at a particular strain level (e.g., 5%) as the modulus of deformability (i.e., secant modulus described in Chapter 2). Linear regression on the data pertaining to the initial part of the stress–strain curve is yet another option to calculate the modulus of deformability. Watkinson and Jackson (1999) suggested a new procedure to calculate the modulus of deformability using the gradient of the inflection at the lowest strain in a stress vs. strain curve. This new procedure was used to calculate modulus of deformability ED for three cheeses and compared with the results from three alternative procedures, including the simple polynomial fitting suggested by Ak and Gunasekaran (1992). As shown in Table 3.6, the ranking of ED for each cheese was the same for each procedure, and the relative magnitude of ED for each cheese along with the coefficient of variation was similar for each procedure. Watkinson and Jackson (1999) discussed some special features of their procedure. The usual practice in uniaxial compression of foods is to run the crosshead at constant speed, since Universal Testing Machines (UTMs) machines are normally designed to do that. It is quite unusual, but perhaps highly necessary, to see compression tests made on cheese at constant true strain rate. The true strain rate continuously increases in uniaxial compression at a constant crosshead speed as the specimen height decreases, and this has a great effect in uniaxial compression of foods (Peleg, 1977a; 1977b). We are not aware of any such studies on cheese except that of Jaros and Rohm (1994). A simple device has been described earlier by Luton et al. (1974) for use with Instron testing machines to produce a constant true strain rate in compression or tension tests. Jaros and Rohm (1994) described a method to conduct uniaxial compression tests at constant strain rate using an Instron testing machine. Based on the analysis of rheological data on 136 Swiss cheese samples, it was shown that stress and strain at fracture are significantly lower in constant strain rate compression than in constant speed compression, due to the differences in strain history of the two modes. The modulus of deformability, obtained at 0.04 strain level, is, however, not affected by the test setup. The authors reported the following equations relating fracture stress σf and fracture strain εf from the two modes of deformation: σ f (constant strain rate) = 1.27 σ f (constant speed)0.917

(3.1)

ε f (constant strain rate) = 1.01 ε f (constant speed)0.867

(3.2)


TABLE 3.2 Values of Modulus of Deformability from Uniaxial Testing of Several Cheesesa

Cheese Variety Apericube processed Bel, France Appenzell Arzua-Ulloa Spanish soft cheese — type I Arzua-Ulloa Spanish soft cheese — type II Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Blue Brick UF-Feta 8–10 weeks Caciocavallo Caerphilly Camembert 1-d before brining Camembert 1-d after brining Camembert 8-d-old Camembert 15-d-old Camembert 22-d-old Camembert 29-d-old Cheddar Cheddar Cheddar 20 days Cheddar 8 weeks old Cheddar 64 weeks old Cracker Barrel Tasty: Kraft, USA Danbo cheese with 45% fat Danish Feta Double Gloucester Double Gloucester Edam Edam Emmental Emmental Emmentaler 4 months Emmentaler 4 months Emmentaler 4 months Emmentaler 4 months English Mature Cheddar Galbanino Galbanino Galbanino Galbanino Garrotxa-type goat milk cheese Gouda Gouda


Initial Strain Rate s–1b

Modulus kPa

— — 46 46 110 220 330 440 4.2 — 5.56 5.56 5.56 5.56 5.56 5.56 — — 1.4–110 32 32 — 833 — — — — — — — 4.76 19 76.2 19 — 0.556 11 28 56 500 — —

70 22 84 44 229 221 225 176 687 8 357 429 749 1070 251 0 48 180 242 640 290 580 189 470 1000 850 ~500 290 18 470 139 182 234 182 890 600 1000 1400 1800 343 405 390

Ref. Agrawal et al., 1997 Prentice et al., 1993 Almena et al., 1998 Almena et al., 1998 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Wium and Qvist, 1997 Masi and Addeo, 1986 Prentice et al., 1993 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Schlesser et al., 1992 Prentice et al., 1993 Prentice et al., 1993 Ak and Gunasekaran, 1992 Hort and Le Grys, 2001 Hort and Le Grys, 2001 Agrawal et al., 1997 Madsen and Ardö, 2001 Agrawal et al., 1997 Prentice et al., 1993 Agrawal et al., 1997 Prentice et al., 1993 Agrawal et al., 1997 Prentice et al., 1993 Agrawal et al., 1997 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm and Lederer, 1992 Rohm et al., 1992 Agrawal et al., 1997 Masi, 1989 Masi, 1989 Masi, 1989 Masi, 1989 Saldo et al., 2000 Prentice et al., 1993 Prentice et al., 1993

TABLE 3.2 (continued) Values of Modulus of Deformability from Uniaxial Testing of Several Cheesesa

Cheese Variety Gouda Gruyère Gruyère Gruyère-type strong cohesion Gruyère-type strong cohesion — Tension Gruyère-type weak cohesion Gruyère-type weak cohesion — Tension Jarlsberg La Serena 2 days old, with lactic starter La Serena 2 days old, without lactic starter La Serena 60 days old, with lactic starter La Serena 60 days old, without lactic starter Lancashire Lancashire Leicester Mahon >150 days Mahon water > pH > fat. However, in another investigation, the sequence obtained was different (Rüegg, 1985): water > fat > pH > Ca > (Nsol-NPN), where Nsol means soluble nitrogen and NPN nonprotein nitrogen. Regardless of its position in such ranking, it is well established that a strong relation exists between water content and mechanical and textural properties for several cheeses (Amantea et al., 1986; Luyten, 1988; Tunick et al., 1993; Prentice, 1992; Visser, 1991; Taranto et al., 1979; Rohm et al., 1992; Tunick et al., 1991). As the moisture content increases, the resistance of cheese to deformation, and hence its modulus of deformability, decreases. The stress at


Firmness (empirical units)

250 200 150 100 50 0 0.20

0.30

0.40

0.50

0.60

Volume fraction of protein

FIGURE 3.6 Strong contribution of protein content to the overall firmness of cheese. (After de Jong, 1978. With permission.)

fracture also decreases with increasing water content. The strain at fracture, however, either remains unaffected or increases with the moisture content, depending on the age of cheese (Visser, 1991; Rohm et al., 1992). The effect of water content on the modulus is usually explained by following reasons: (a) a high water content means a low protein content, which is the stress-carrying component; (b) water is a low-viscosity liquid that occupies the space between the fat and protein, and acts as a good lubricant; and (c) more swollen protein particles due to high water content offers less resistance to deformation (Luyten, 1988; Prentice, 1992). There can be significant variation in rheological properties within a single cheese block as a result of various reasons: (a) presence or absence of a rind; (b) frequency of turning during ripening; (c) development of moisture gradient; and (d) nonuniform proteolytic activity (Prentice et al., 1993). It is likely that not only the magnitudes but also the trends for a particular property (e.g., deformability modulus) can vary with the position in a cheese block, from surface (where moisture loss may dominate) to center (where proteolytic activities may dominate). Based on the data of Steffen (1976) reported by Prentice et al. (1993), there can be a variation of about 50% in firmness with distance from the surface in an otherwise uniform cheese. Recently, few studies have been conducted where hydrocolloid-based edible coatings were applied to semihard and brined cheeses for different purposes (e.g., moisture regulation, appearance, protection against microbial contamination) while still maintaining their desirable textural properties (Kampf and Nussinovitch, 2000). In terms of mechanical properties, the outcome of such coatings was promising, especially for semihard cheese (Kampf and Nussinovitch, 2000). Fat is one of the primary constituents contributing greatly to rheology, texture, and organoleptic characteristics (e.g., imparting a desirable mouthfeel to cheese) of cheeses (Marshall, 1990). It is also considered important in transport and packaging (Green et al., 1990). The contribution of fat to cheese quality is better appreciated when it is removed or reduced in manufacturing low-fat varieties. A reduction in fat content often adversely affects texture and flavor of cheese (Olson and Johnson,


1990; Jameson, 1990). The level of intact casein in 225-day-old Cheddar cheese was twice higher in the reduced-fat cheese than that in the full-fat cheese. But the rate of decrease in the levels of intact casein was practically independent of the fat content (Fenelon and Guinee, 2000). The technological strategies for coping with the challenges in making of reduced-fat cheese and low-fat cheese and some regulatory issues have been discussed in recent reviews (Olson and Johnson, 1990; Drake and Swanson, 1995; Mistry, 2001). Reduction in fat content of hard cheeses (e.g., Cheddar) affects not only the flavor development but also the texture. It is well known that the texture of a reduced fat cheese is firmer and more elastic than that with full-fat content. Marshall (1990) conducted a thorough study to determine effects of changing fat and moisture [as moisture in nonfat solids (MNFS)] contents on the sensory, rheological, and structural properties of the processed cheese analogs. Numerical results from this work are presented in Tables 3.2 to 3.5. As can be seen from the data in these tables, the rheological parameters tended to decrease with an increase in MNFS, probably because of moisture acting as a plasticizer in the protein network. The fat content affects the microstructure of cheese. Full-fat cheeses of all varieties (i.e., Cheddar, Mozzarella, processed, and Swiss cheeses) are characterized by a protein matrix interspersed with fat globules existing in various sizes and shapes. Low-fat cheeses, however, have fewer and smaller fat globules within the dense protein network. The consequence of the protein-dominated structure of lowfat cheeses, is a firm and rubbery body and texture (Mistry and Anderson, 1993). In an earlier work, Green et al. (1981) examined, by different microscopic techniques, the structure development in Cheddar cheese from concentrated milks throughout cheese making and maturation. As the milk used in cheesemaking became more concentrated, the protein network became progressively coarser and this, in addition to large loss of fat into whey, led to higher resistance to compression (i.e., higher firmness). The role of fat in rheology of cheese was occasionally treated in terms of composite material behavior (Prentice, 1992; Luyten and van Vliet, 1990). The cheese is viewed as a composite material with casein-water forming the matrix and fat globules acting like fillers or inclusions. The mechanical properties of a composite material therefore depend on the properties of the matrix, the volume fraction and properties of the filler, and the mechanical interaction between the filler and matrix. According to Prentice (1992) the only interaction between fat and casein is friction. Green et al. (1990) studied the composite behavior of cheese analogs containing fat globules in a protein matrix. They varied fat concentration and fat hardness, and they either emulsified the fat with a neutral detergent (minimizing interaction between fat globules and the matrix) or emulsified with sodium caseinate (increasing interaction between fat globules and the matrix). The higher fat content at room temperature (70 to 80% liquid form) increased the lubrication effect of fat and reduced the fracture properties in compression and in wire cutting. Using different amounts of sunflower oil (completely liquid at room temperature) instead of butterfat decreased the maximum stress from 304 kPa at 6.2% oil to 165 kPa at 20.7% oil, and Young’s modulus from 115 kPa at 6.2% oil to 77 kPa at 20.7% oil. On the other


hand, substituting partially hydrogenated oil (50% solid) for butterfat decreased the maximum stress from 349 kPa at 6.8% fat to 282 at 21.5% fat, but increased enormously the Young’s modulus from 152 kPa at 6.8% fat to 2061 kPa at 21.5% fat. Moreover, butterfat contributed more to the composite behavior when it was emulsified with sodium caseinate (i.e., more interaction with the matrix). It seems that simple substitution of fat by protein, water, and fat replacers results in a low-fat cheese that is inferior in many respects to its full-fat counterpart (Aryana and Haque, 2001). Several characteristics of fat or lack of it in low-fat cheeses can contribute to this result. Mechanical properties of fats, and their contribution to cheese and probably to perception of texture by consumers, are influenced by their chemical composition and temperature. The solid fraction of fat changes sharply in a relatively narrow temperature range (e.g., 10 to 35°C). However, water, for instance, within this temperature range, will have more or less constant properties. The interaction between fat and protein is hard (if not impossible) to mimic by the interaction between water and protein, or between fat replacers and protein. Commercially available fat replacers, of which none can yet fully duplicate functional and sensory attributes of fat, are one of three types: lipid based, carbohydrate based and protein based (Akoh, 1998). Interaction between water and fat replacers, in contrast to water binding of casein (Geurts et al., 1974), can also have a large impact on the properties of the final cheese. Another point worth mentioning is that during ripening of traditional cheeses, the fat globules can aggregate (and possibly coalesce) to increase in size and decrease in number, and this has an impact on the sensory properties of the cheese. The clumping of fat globules was observed in Cheddar cheese (Kimber et al., 1974) and Mozzarella cheese of varying fat contents and aged for 6 weeks (Tunick et al., 1993) and 50 days (Kiely et al., 1993), as well as in Cheddar-type cheese (Guinee et al., 2000). Kiely et al. (1993) suggested that fat aggregates are formed due to proteolytic destruction of the casein network. They also suggested that aggregation of fat globules could be the reason for the age-related increase in freeoil formation observed in low-moisture, part-skim Mozzarella cheese by Kiely et al. (1991). Tunick (1994) also reported that free-oil formation in Mozzarella cheese increased with the percentage of fat and protein breakdown. However, homogenization of cheese milk and cream greatly reduced free-oil due to the reduction of fat droplet size, while homogenization of skim milk had no effect (Tunick, 1994; Oommen, et al., 2000). Agglomeration of fat globules into larger particles was microscopically observed in both stirred-curd and stretched-curd Mozzarella cheeses after baking in a conventional oven (Paquet and Kalab, 1988). The agglomeration of fat globules is difficult to occur in low-fat cheese where the massive protein matrix keeps the small fat droplets well dispersed. This contributes to the rubbery texture and mouthfeel of the low-fat cheeses. Lastly, the contribution of free fatty acids and fat-soluble flavors to the taste of traditional cheese must be mentioned. Even after aging for 12 months, sensory panelists noted a flat flavor and lack of Cheddar flavor in reduced fat (50%) cheese (Olson and Johnson, 1990). The physical state of the fat globules (i.e., proportion of solid fat) determines its relative rigidity or stiffness in comparison to the casein matrix. Thus, the relative contribution of milk fat to the overall cheese properties is highly temperature


Modulus of elasticity (kPa)

40

y = -34.4 x + 57.9 R2 = 0.96

30

20

10 0.7

0.8

0.9

1.0

1.1

1.2

Fat/SNF

Modulus of elasticity (kPa)

40 y = -21.5 x + 66.9 R2 = 0.90 30

20

10 1.5

1.7

1.9

2.1

2.3

Water/SNF

FIGURE 3.7 Effects of fat and moisture contents on the elasticity modulus of different cheese varieties. (Masi and Addeo, 1986.)

dependent. Melting of milk-fat globules occurs over a large temperature range from –30 to 40°C due to varying melting points of triglycerides (Dufour et al., 2000). It is entirely liquid above 40°C and completely solid below –30°C. Between these extremes it is a mixture of crystals and oil, where the latter is a continuous phase. Masi and Addeo (1986) reported a clear relation between the fat content (expressed as the ratio of fat to solids-nonfat, SNF) and the modulus of elasticity for Mozzarella cheese (Figure 3.7). However, it may be incorrect to attribute the decrease in the modulus entirely to the increase in fat content since moisture contents of these experimental cheeses also changed. The size of variation in the fat content was 4.7 units, whereas that in the water content was 3.2 units; that is, rather comparable numbers. As shown in Figure 3.7, we can also plot the modulus against the water content (as water/SNF ratio) for the same cheeses and still obtain a good correlation. A recent study by Madsen and Ardö (2001) on Danbo cheese with three fat contents provides further results regarding effects of fat and water on rheological properties (Table 3.11). On one hand, when the fat content of Danbo cheese is


TABLE 3.11 Uniaxial Compression Properties of Low-Fat, Reduced-Fat, and Normal-Fat Danbo Cheese Parameter

Low-Fat

Reduced-Fat

Normal-Fat

Deformability modulus (kPa) Fracture stress (kPa) Fracture strain (–) Work to fracture (kJ/m3)

208 194 1.3 127

143 92 1.1 55

189 94 1.1 61

Fat (%,w/w) Moisture (%,w/w)

13.6 52.9

16.7 53.3

25.0 47.4

Source: After Madsen and Ardö, 2001. With permission.

reduced from 25 to 16.7% (a change of 33%), there was no change in the rheological parameters except the modulus. On the other hand, when the fat content is further reduced from 16.7 to 13.6% (a change of 19%), all parameters increased dramatically. It was concluded that water is not an adequate substitute for fat in order to obtain good quality, low-fat Danbo cheese (Madsen and Ardö, 2001). Considering Gouda cheese as a composite material, Luyten (1988) and Luyten and van Vliet (1990) determined the effect of temperature on the compression properties of the cheese. They pointed out that at 10°C the fat is stiffer than the matrix, and therefore the E for high-fat (60%) cheese is greater than the E for lowfat (10%) cheese. By the same token, the opposite is true at high temperature (26°C) where fat is nearly entirely liquid and contributes little to the modulus of the cheese. At the middle temperature (20°C), there was no influence of volume fraction of fat, indicating that the modulus of the fat is equal to the modulus of the protein matrix and is about 100 kPa (Luyten and van Vliet, 1990). The authors estimated that the modulus of the fat particles at 14°C to be 460 to 880 kPa. This is comparable to the modulus of deformability of a variety of cheeses listed in Table 3.2. It is also important to note that part of the temperature effect is due to the changes in rheological properties of the protein matrix. The theory of composite materials predicts a decrease in fracture strain with an increase in volume fraction of the filler. Since this was not observed in Gouda cheese, Luyten and van Vliet (1990) concluded that the milk-fat globules are relatively small (0.1–10 µm, Mulder and Walstra, 1974) and do not create stress concentrations to cause crack initiation. The fracture stress of Gouda cheese is, however, affected by the fat content and test temperature. The fracture stress decreased with the increasing fat content, and the temperature effect was stronger in high-fat cheese than in lowfat cheese. The importance of pH for a variety of cheese has been discussed from the cheesemanufacturing point of view (e.g., Lawrence et al., 1983; Lawrence et al., 1984; Chapter 1). The apparent effect of pH is more striking for Mozzarella and string cheeses. It is known that the kneading and stretching process of Mozzarella cheese curd is best performed at about pH 5.2 to 5.4 (Kosikowski, 1982). Kimura et al. (1992)


reported on the structure and properties of string cheese at different pH and demineralization levels. Their results indicated that stringiness is limited by the calcium/phosphate (Ca/P) ratio in curd and not specifically by the curd pH. However, curd pH is an operational parameter that is associated with the Ca/P ratio. It is known that cheese has a “short” consistency (i.e., small fracture strain) if it has a low pH or a high salt content (Luyten et al., 1982). The effect of salt content on rheological properties of unripened Camembert (Mpagana and Hardy, 1986) and Gouda (Luyten, 1988) cheeses has been reported. Mpagana and Hardy (1986) adjusted the salt content of Camembert cheese by first keeping small, cylindrical pieces (2.4 cm diameter, 3.0 cm height) in saturated NaCl solution (pH = 5.0) at 15°C and then storing the brined pieces at 15°C and relative humidity of 95% for three days to assure homogeneous salt distribution within the samples. The test pieces (1.38 cm diameter, 1.0 cm height) were compressed to 20% of their original height (i.e., 80% deformation) at an initial strain rate 0.017 s–1. Although only the salt content was intentionally altered by applying different brining times, the water content of cheese would have decreased as well due to brining. Nevertheless, the modulus of deformability and fracture stress of Camembert increased exponentially, while the fracture strain decreased linearly with the salt content of the cheese (Figure 3.8). These findings are consistent with the results on other cheeses, such as Mozzarella cheese (Cervantes et al., 1983), Gouda (Luyten, 1988), and Feta cheese (Katsiari et al., 1997). Watkinson et al. (2001) preferred to work with a model cheese system, obtained by direct acidification, in order to study effects of pH on rheological properties during ripening with minimal confounding from other variables. It is observed that at a given ripening time, the fracture strain increased with pH in the range of 5.2 to 6.2, except there was a distinct maximum at pH 5.8 for the 87-day data. At a given pH value, the fracture strain increased with ripening time, with the exception of a constant value after seven days for the pH 6.2 cheeses. It shall be recalled that the local maximum in fracture strain of young Gouda cheese (1-week-old) was positioned at a lower pH 5.2 (Luyten et al., 1982). Further results from Watkinson et al. (2001) demonstrate that, in general, fracture stress increases with pH (e.g., at day 7 from about 160 kPa at pH 5.2 to about 270 kPa at pH 6.2) and modulus of deformability decreases with pH (e.g., at day 2 from about 1200 kPa at pH 5.2 to about 825 kPa at pH 6.2). Changes in rheological properties of cheese curd during the initial stages of ripening were studied at 20°C as a function of pH (5.45–6.05) and storage time (2–14 days) using a specially developed extrusion-flow technique (Ramkumar et al., 1998). The maximum force exerted on the grated curd during extrusion testing tended to increase with pH, reaching the maximum at pH 5.90. Increasing solid-like behavior with pH was also observed in oscillatory shear results for the cheese curd, indicating that the effect of pH is persistent in both small and large deformation regimes. Moreover, the tendency to exhibit more solid-like response is in accordance with the pH effect on the fracture stress (Watkinson et al., 2001). Perhaps the most important factor affecting rheological and other properties of cheese is proteolysis during maturation. The proteolytic activity that contributes


Modulus of deformability (kPa)

1000

100

y = 109.3e0.09x R2 = 0.91 10 0

5

10

15

20

25

Salt content (g NaCl/100 H2O)

Engineering fracture stress (kPa)

1000

100

y = 37.6e0.10x R2 = 0.91 10 0

5

10

15

20

25

Salt content (g NaCl/100 g H2O) 1.2

Fracture strain (-)

1.0 0.8 0.6 0.4 y = -0.016 x + 0.85 R2 = 0.56

0.2 0.0 0

5

10

15

20

25

Salt content (g NaCl/100 g H2O)

FIGURE 3.8 Effect of salt content on the mechanical properties of Camembert cheese. (After Mpagana and Hardy, 1986.)


greatly to the rheological and textural properties of cheeses basically means a reduction in the levels of intact casein (i.e., αs1-casein and β-casein). It is usually observed that αs1-casein fraction is hydrolyzed to a greater extent than β-casein, although the extent may depend on the cheese variety and the type of proteolytic enzyme (Fox, 1989; Trujillo et al., 1999; Dervisoglu and Yazici, 2001; Katsiari et al., 2000). Many ripening agents from different sources are responsible at different stages for the hydrolysis of caseins into small peptides and eventually to free amino acids, thus contributing to changes in both the texture and flavor during ripening (Fox et al., 1993; Farkye and Fox, 1990). It is of great importance to the cheese industry to be able to predict and control, and particularly accelerate, the ripening process (Fox et al., 1996; Fedrick, 1987; van den Berg and Exterkate, 1993; Folkertsma et al., 1996; Law, 1987; Saldo et al., 2000). Degradation of αs1-casein and β-casein during ripening of various cheeses is shown in Figure 3.9. We must note that only the trends should be considered and no comparison shall be made between different cheeses since the percent values are based on different reference quantities (e.g., expressed as % of levels in milk, or % of levels in fresh curd, % of levels in 1-day-old cheese, etc). An interesting example for effect of hydrolysis of para-casein on properties of cheese was reported for Camembert cheese (Schlesser et al., 1992). During early stages of ripening (up to 15 days) there was an increase in deformability modulus (called elasticity in the original paper) of Camembert cheese, which later on decreased with further aging until zero. After aging for more than 29 days the samples became semifluid (Schlesser et al., 1992). Consistent with the expectations based on previous findings, mechanical properties of Tybo Argentino cheese (semihard variety) decreased with ripening at 10°C and 60% relative humidity (Bertola et al., 1992). Although there was much scatter in the results, the deformability modulus of this cheese decreased from about 65 to 15 kPa during 120 days of ripening. Assuming a linear dependency, this means a decrease of 0.4 kPa/day. Ak and Gunasekaran (1995) reported a similar change for Mozzarella cheese over one month of aging. The strain at fracture did not vary with ripening time, and the average value was reported as 70% or εf = 1.20 (Bertola et al., 1992). Noël et al. (1996) determined rheological properties of Parmigiano Reggiano aged for up to 28 months and established relationships with the sensory properties. These authors reported not only standard rheological parameters, such as apparent elastic modulus and fracture parameters, but also the proportional limit and the modulus of resilience (see Chapter 2). The strain at apparent elastic limit of Parmigiano Reggiano cheese was about 7.6%, regardless of its age, while the stress limit increased from 117 kPa at 12 months to 191 kPa at 28 months. The apparent elastic modulus of the cheese increased with the age almost linearly with a rate of 60 kPa/month during maturation from 12 months to 28 months. The fracture strain and fracture stress values decreased linearly with the age of the cheese at a rate 0.006 units/month and 1.5 kPa/month, respectively. Raphaelides and Antoniou (1996) reported that for both traditional and UF-Teleme cheeses, the main changes in cheese structure, as reflected in mechanical properties, took place within the first month of ripening (Figure 3.10). Wium and Qvist (1997)


Mild Cheddar

100 Alpha (s1)-casein (%)

Monterey Jack Sharp Cheddar

80

Cheddar

60

La Serena Kulek

40

Feta

20 0 0

10

20

30

40

50

60

Maturation time (week)

Mild Cheddar

120

Monterey Jack Sharp Cheddar

Beta-casein (%)

100

Cheddar

80

La Serena Kulek

60

Feta

40 20 0 0

10

20 30 40 Maturation time (week)

50

60

FIGURE 3.9 Proteolysis in numerous cheeses as measured by the decrease in intact αs1casein and β-casein. (Data from different sources referenced in text.)

reported on the fracture properties of three types of UF-Feta determined by uniaxial compression and dynamic tests. As can be seen in Table 3.4 the stress at fracture for these cheeses varied between 20 to 46 kPa, with the Tin Feta cheese always having higher values than the Red Brick Feta cheese and the Blue Brick Feta cheese. Moreover, when these cheeses were evaluated by sensory means (i.e., nonoral firmness: the resistance of a cube of cheese to moderate squeezing between thumb and forefinger; and oral firmness: the resistance of a cube of cheese during normal mastication), the Tin Feta cheese was selected as the firmest cheese, followed by the Red Brick Feta cheese and the Blue Brick Feta cheese (Wium and Qvist, 1997). On the other hand, the Tin Feta cheese also had the highest n value (Table 3.7), as well as tan δ, both signifying the more viscous character of this cheese. It seems that the parameters n and tan δ do not play a significant role in sensory evaluation of cheese. The modulus of deformability (ED) values (Table 3.2) in uniaxial compression of three UF-Feta cheeses ranged from 176 to 465 kPa (Wium and Qvist, 1997). It can © 2003 by CRC Press LLC

Deformability modulus (kPa)

200 160 120 80 Traditional

40

UF-unheated UF-heated

0 0

1

2

3

4

5

Maturation time (month)

FIGURE 3.10 Effect of maturation time on the deformability modulus of different kinds of Feta cheese. (After Raphaelides and Antoniou, 1996. With permission.)

be seen that the Tin Feta* had higher ED values at all rates than the Red Brick Feta cheese and the Blue Brick Feta cheese, indicating that the Tin Feta cheese is the stiffest. Wium and Qvist (1998) and Wium et al. (1998) reported on changes in rheological properties of UF Feta cheese made by acidification using glucono-δ-lactone (GDL). In these studies, the gross chemical composition (e.g., moisture, salt, pH, total N, and fat) of the cheeses was kept essentially constant while studying the effects of rennet concentration, coagulation method, and storage time. Measurement of proteolysis by capillary electrophoresis showed that αs1-casein was degraded more than β-casein due to the relatively faster action of chymosin on the former casein fraction, and also, the greater inhibitory influence of NaCl on the hydrolysis of β-casein by enzymes (Fox and Walley, 1971). Similarly, the ripening of Meshanger cheese was characterized by a very rapid and complete hydrolysis of αs1-casein, while the β-casein remaining essentially unattacked (de Jong, 1976). Substantial amounts of αs1-casein and β-casein were hydrolyzed by the high level of residual rennet in Feta cheese having salt-in-moisture concentrations from 3.99 to 4.32 and pH from 4.65 to 4.71 (Samal et al., 1993). The compression variables (ED, σf , εf , Wf) for UF-Feta cheese generally decreased with storage time, and that was ascribed purely to the proteolysis since the gross chemical composition remained essentially constant. It was further observed that stress at fracture, modulus of deformability, and work to fracture all increased significantly with rennet concentration, more so in the young cheese, with some showing a maximum at some intermediate levels (Figure 3.11). This was associated with the formation of a coarser network at higher rennet concentrations and viewed as a potential way of making soft variants of UF-Feta cheese with a smooth texture by using less rennet. On the other hand, strain at fracture decreased with increasing rennet concentration, again consistent with coarser * The Tin Feta is packed in tins containing brine and the Red Brick Feta and Blue Brick Feta are packed in Tetra-Brick packages without brine.


80

0.5

0.3 40 0.2 20

Fracture strain (-)


0.4 60

0.1

stress strain 0 0.001

0 0.01

0.1

1

10

Rennet concentration (g/kg)

FIGURE 3.11 Effect of rennet concentration on fracture stress and fracture strain of Feta cheese made from ultrafiltered milk. (After Wium and Qvist, 1998.)

structure explanation. Luyten and van Vliet (1996) stated that the strain at fracture is rather low for very young Gouda cheeses (e.g., 1-day- or 2-day-old) due to the large inherent defects (e.g., incomplete fusing of curd particles) present in such cheeses (Luyten, 1988). A recent interesting work on the fracture stress of fused curd grains (Lodaite et al., 2002) indicated that the applied compressive stress, the degree of syneresis prior to fusion, and the fusion time exerted significant positive effects on the quality of fusion and the magnitude of fracture stress. A low fracture strain implies a short texture and possibly a crumbly behavior (Luyten et al., 1982). In general, the shortness of cheese increases with maturation. Watkinson et al. (1997) has shown that the strain at fracture of New Zealand Cheddar cheese initially increased in the first 28 days, and thereafter it decreased with further aging. Their results, combined with those from Creamer and Olson (1982), are plotted in Figure 3.12, which shows a large decrease in fracture strain during maturation of Cheddar cheese. Similar trends have been reported for several other cheeses (Luyten, 1988; Ak et al., 1993; Luyten and van Vliet, 1996). The initial increase in εf is attributed to the process of fusion until the proteolysis dominates and causes a decrease in εf or an increase in shortness. However, we must note that the relation between fracture strain and ripening time depends on other factors, such as pH. A fresh but acid cheese can have a fracture-strain value similar to that of a mature cheese (Luyten et al., 1982; Luyten and van Vliet, 1996; Rohm et al., 1992). Moreover, for 7-week-old Gouda cheese, a maximum in the strain at fracture was found at around pH 5.2 to 5.25 (Luyten et al., 1982). It is common to see that effects of one compositional variable are confounded with another compositional variable and with ripening time. Rohm et al. (1996) determined the impact of seasonal variations in raw-milk quality on the composition and fracture properties of Emmental cheeses. Regarding the composition, the iodine value (IV), which is an indicator of the softness of the © 2003 by CRC Press LLC

1.6 Strain at fracture (-)

Watkinson et al. (1997) Creamer and Olson (1982)

1.2

0.8

0.4

0.0 0

100

200

300

400

500

Maturation time (day)

FIGURE 3.12 Variation of strain at fracture for Cheddar cheese as a function of maturation time. (After Creamer and Olson, 1982; Watkinson et al., 1997.)

Strain at fracture (-)

1.8

1.6

1.4

1.2

1.0 0

20

40

60

80

100

120

Age of cheese (day)

FIGURE 3.13 Fracture strain of Emmental cheese as affected by maturation. (After Rohm et al., 1996.)

milk fat, and thus cheese, increased, as expected, from 34.3 in winter cheeses to 41.5 in summer cheeses. Moreover, Emmental cheeses produced during winter showed accelerated lactic-acid degradation and propionic-acid generation, and reduced secondary proteolysis than those produced during summer. It is observed that the fracture strain of Emmental cheese is not affected by the seasonal variations, but decreased with the maturation time (Figure 3.13). Emmental cheese made in different seasons exhibited considerably different fracture stress (Figure 3.14), most probably due to the seasonal variations in fattyacid composition of the milk fat (Rohm et al., 1996). The mean values of fracture stress, which correlates well with sensory firmness, are about 165, 126, and 91 kPa for mature cheeses (16 weeks) made in winter, spring, and summer, respectively. This is in line with the increase in iodine value of milk fat as the season changes from winter to summer. The iodine value is associated with the number of double


350


summer cheese 300

spring cheese

250

winter cheese

200 150 100 50 0 0

20

40

60

80

100

120

Age of cheese (day)

FIGURE 3.14 Fracture stress of Emmental cheese as affected by maturation time and production season. (After Rohm et al., 1996.)

Stress at fracture (kPa)

160

120

80

40 0

1

2

3

4 5 6 7 Production month

8

9

10

11

12

FIGURE 3.15 Seasonal variations in fracture stress of Vorarlberger Bergkäse. (After Jaros and Rohm, 1997.)

bonds in fat. As the number of double bonds increases, the solid fraction of milk fat at a given temperature decreases (Jaros et al. 2001). Therefore, the iodine value of the milk fat serves as an indicator of the firmness of fat, and in turn, of cheese. Jaros and Rohm (1997) reported on the seasonal variations in mechanical properties of Vorarlberger Bergkäse (a smear-ripened, Gruyère-type, raw-milk hard cheese). Figure 3.15 shows apparent fracture stress for Bergkäse as a function of the production month. The mean values of fracture stress for summer and winter cheeses are reported to be 107 and 133 kPa, respectively. The corresponding fracture strain values for summer and winter cheeses are found to be 0.48 and 0.61, respectively (Jaros and Rohm, 1997).


Without altering the gross chemical composition of the cheese (Swiss Appenzeller), Jaros et al. (2001) increased the amount of unsaturated fatty acids in milk fat by adding shredded rapeseed to the cattle diet. Those cheeses made with milk from the modified diet exhibited significantly lower values for the modulus of deformability and the fracture strain. The property values of the cheese with and without the rapeseed are given in Tables 3.2 to 3.5. It is shown that the rapeseed feeding can be utilized advantageously to eliminate the firmness differences in cheeses of summer and winter periods, and therefore provide the consumer with more consistent products (Jaros et al., 2001). Ohashi et al. (1982) demonstrated by wire-cutting tests the effect of seasonal variations on mechanical properties of milk-rennet curd in connection with the composition of milk. In agreement with results of the other studies cited above, the elastic moduli of the curd were as follows for the spring, summer, autumn, and winter seasons: 55.3, 37.1, 44.4, and 50.1 kPa, respectively.

STRESS-RELAXATION MEASUREMENTS Stress-relaxation properties are very important in many industrial operations, as they control the materials’ response to external mechanical forces. Generally speaking, stress relaxation can be due to physical events or chemical processes (Ferry, 1980; Grosberg and Khokhlov, 1997). In any case, the relaxation process is regulated by the ratio of two variables: the characteristic time of a material (λ) to the characteristic time of observation (t), hence the Deborah number, De = λ/t. A high De (De>>1) corresponds to solid-like behavior (i.e., no relaxation), whereas a low De (De1/3 of the melting temperature) 4. Fatigue failure — failure due to repetitious loading either above or below yield stress; may be further classified into low-cycle, high-cycle, and gigacycle fatigue damage It must be emphasized that any fracture or crack mechanism is closely related to the material’s microstructure. In polymers, these mechanisms are dominated by the long and flexible macromolecules (Schirrer, 2001). Macromolecules are long series of monomers whose backbones are composed of linked carbon atoms. The cone angle of carbon atoms is fixed at about 70º (Figure 4.1). Therefore, the relative position of the linked carbon atom chain, i.e., the macromolecular backbone, is limited to some extent. The stiffness of the monomer and the space it occupies dictate the stiffness of the macromolecule. A large condensed assembly of macromolecules is the polymer. It may exist in either amorphous or crystalline structure, depending on its temperature. At material temperatures below its glass transition temperature, Tg, the macromolecules assume a glassy or amorphous disordered structure in which the smallest elementary volume of the material is about the size of the monomer. At temperatures above Tg, the material is said to be in the “rubbery” state. Figure 4.2 shows a typical modulus vs. temperature relationship for a polymer. In the glassy state, interactions between nonlinked atoms are strong, and any applied load is distributed atom to atom. When a small elastic load is applied, all carbon–carbon bonds are stressed, and their cone angles are strained. Larger loads lead to nonrecoverable plastic deformations. In the rubbery state, molecular interaction at the atomic level does not exist. Under applied loads, the entanglements deform about each other, and the atoms are free to twist on the carbon–carbon cone. The elastic properties are due primarily to the entropy variations of the entanglement positions, which are nearly proportional to the macroscopic strain (Schirrer, 2001). True rubbery materials may exhibit linearity between applied stress and strain up to strain levels of 10.


70° Cone Linked carbon atoms

FIGURE 4.1 Schematic representation of a macromolecule as a string of monomers whose backbone is comprised of linked carbon atoms with limited movement (within the cone angle of 70°) about each other. (After Schirrer, 2001.)

Glassy

Iog (Modulus)

Leathery

Rubbery

Viscous Flow

Tg Temperature

FIGURE 4.2 Modulus vs. temperature relation for a typical polymer. Glass transition temperature, Tg, is identified at the transition from glassy to leathery state.

Sometimes a third state, semicrystalline, can be defined when the material temperature is close to Tg. The semicrystalline state is characterized by a more or less regular small rigid lamellar arrangement of macromolecules with flexible amorphous macromolecules connecting the crystalline states.


Material

Environment

Loading Rate

Fatigue

Material Resistance

Crack Driving Force

Applied Stress

Crack Size

Crack Geometry

Loading Rate/Cycles

FIGURE 4.3 Various factors contributing to material resistance to fracture and crack driving force.

When strained, a network of entanglements in both glassy and rubbery states deform with increasing (in tension) or decreasing (in compression) distance between the entanglements. In the glassy state, the elongation is irreversible due to the atomic interactions, and the energy input is converted to heat. In some cases, a few molecules may break, creating a crack or cavity. Such an event entirely changes the material microstructure.

FRACTURE MECHANICS A fracture in a material is a failure mechanism that involves stable or unstable propagation of a flaw (e.g., a crack) within the material structure. Often the purpose of fracture-mechanics analysis is to prevent fracture (or propagation of an existing flaw). This criterion also applies to undesirable fracture in cheeses that will lower the overall quality and consumer appeal. However, the eye (slit) formation in some cheeses is not only desirable but is facilitated. In such cases, the focus is to limit the extent of crack growth. Regardless, it is useful to consider various factors that contribute to crack growth and those factors that tend to resist it. These factors are (Figure 4.3): (1) crack driving force – applied stress, crack size, crack geometry, and loading rate/cycles; and (2) material resistance factors — type of material, environment (temperature, chemical/physicochemical factors), loading rate, and fatigue. Naturally, when the driving force exceeds the material resistance, the crack will propagate. Under a specific set of conditions, the crack size that balances the driving force and resistance is known as the critical flaw size. These forces can be considered to act in one of three basic modes schematically illustrated in Figure 4.4. They are: Mode I — opening; Mode II — sliding; and Mode III — tearing. In the case of cheese and many other materials, Mode I, the opening mode, is most relevant. It represents the crack pulling open due to forces


Mode I (Opening)

Mode II (Sliding)

Mode III (Tearing)

FIGURE 4.4 Typical failure modes in engineering materials.

acting perpendicular to the crack. Mode III is relevant when forces are applied perpendicular to a crack, causing the material to tear and slide along itself, and thus move out of its original plane. Separating strings from string cheese (Izutsu et al., 1991) is as an extreme example of Mode III behavior. In Mode II, the forces are acting parallel to the crack, causing an in-plane shear. There are three broad regimes to analyze the fracture mechanics of materials: linear elastic, elastic–plastic, and limit load. These regimes are illustrated in Figure 4.5. In linear elastic fracture mechanics (LEFM), only localized yielding around the crack tip is considered. A stress intensity factor, KI , represents the crack driving force. This is defined as:

K I = Yσ πa

(4.1)

where σ = applied stress, a = crack size, and Y = a dimensionless constant depending on material geometry and loading mode (more on this later under Determination of KI). The subscript I refers to Mode I described above (Figure 4.4). Since Mode I is the most common, the subscript I is sometimes omitted. The material resistance is measured by fracture toughness KIc. Fracture occurs when KI = KIc. Fracture toughness is a material property, i.e., it is independent of material geometry and test procedure. It is a measure of the energy per unit area necessary to give a new crack surface (Williams, 1984). In elastic–plastic fracture mechanics (EPFM) analysis, a large section around the crack tip is considered yielding. Depending of the extent of yielding, it may be termed “contained” or “full.” The crack driving force represents the work done under applied stress in the area around the crack tip. It is a function of crack and material geometry, applied stress, and elastic–plastic stress–strain relationship of the material. The limit load analysis or diffuse dissipation assumes that the entire cross-section of the material becomes fully plastic before it begins to fail. This is appropriate for highly ductile materials. It is possible to have this regime in conjunction with one of the other regimes (Williams, 1984).


σ

σ

1

2

w a Crack

Crack

td

σ

σ

LEFM

EPFM (contained yielding)

σ

σ

3

4

Crack

Crack

σ

σ

EPFM (fully yielded)

Limit Load (diffuse dissipation)

FIGURE 4.5 Failure mechanisms: 1. LEFM — Linear elastic fracture mechanism, td>b) the stress concentration increases to a large value and can be approximated as:

σm =2 aρ σ

(4.4)

Therefore, even at an acceptably low applied stress the maximum stress can exceed that of the material fracture stress. If the material cannot relieve this stress concentration by plastic flow around the crack tip, the crack will grow, thereby lowering the total energy of the system. Bui and Ehrlacher (1981) described the damaged zone around a crack propagating in a brittle material. If the material was elastic and fails when the maximum


σ

y σm

σyy σ

σxx

2b

x 2a

σ

FIGURE 4.8 Distribution of local stresses around an elliptical crack inside an elastic plane subjected to stress σ (After Riande et al., 2000.)

principal stress reaches a critical value σc, the thickness of this damaged zone td is given by:

⎛ k ⎞ td = ⎜ ⎟ ⎝ σc ⎠

2

(4.5)

where k = initial stiffness. The theoretical failure stress is the stress needed to break atomic bonds. This is on the order of E/10, where E = Young’s modulus of the material. However, the actual fracture stresses are several orders of magnitude lower. This is due to the heterogeneous distribution of the stresses in the material. For example, in a material containing several microcracks scattered within the entire volume, stress concentrations easily approach the fracture stress. In materials without such microcracks, inclusions or impurities contribute to the lowering of fracture stress. Given the high stress concentration at local flaws in a material, fracture is always considered to originate from such locations (Gordon, 1968). Various inhomogeneities and flaws in cheese may range from 10–5 mm to 10 mm or larger (Table 4.1). The practical problem with the stress-concentration approach is that σm approaches infinity as ρ approaches zero, as would be the case for an infinitely sharp © 2003 by CRC Press LLC

TABLE 4.1 Estimated Size of Various Inhomogeneities and Flaws in Cheese Type of Inhomogeneity/Flaw

Estimated Size (mm)

Casein sub-micelle Paracasein micelle Protein network strands Fat globule Unevenness of network Precipitates of salts, amino acids Curd grains Acid spots Holes Curd pieces (in Cheddar cheese) Difference rind-center

10–5 10–4 10–3 10–3 to 10–2 0.01 to 0.1 1 1 to 10 10 10 20 10 to 100

Source: After Walstra and van Vliet, 1982; Luyten, 1988.

crack. Since all materials are considered to contain infinitely small flaws, they all should fail upon application of an infinitesimal stress.

GRIFFITH CRITERION To overcome problems due to the local stress concentration approach, Griffith (1920) proposed an energy balance approach. According to this, an existing crack can grow or a new crack can form if and only if such a process would result in net decrease or, at best, no change in total energy of the system. The energy needed for the process is given in terms of specific surface energy (γs, J/m2) of the two new surfaces created (Williams, 1984):

γs =

σ f 2 πa 2E

(4.6)

where σf = failure stress; E = Young’s modulus; and a = flaw size. For a typical elastic system where applied energy is absorbed locally around the tip of a sharp crack the energy balance may be written as (Williams, 1984):

∆U1 = ∆U2 + ∆U3 + ∆U 4

(4.7)

where U1 = energy input; U2 = stored energy; U3 = energy dissipated (around the crack tip); and U4 = kinetic energy. If we further consider that the crack is growing such that increase in crack area is given as ∆a, then we can write:

⎛ ∆U1 ∆U2 ⎞ ∆U3 ∆U 4 − + ⎜ ⎟= ∆a ⎠ ∆a ∆a ⎝ ∆a © 2003 by CRC Press LLC

(4.8)

It is important to note that the changes in energy occur not due to crack displacement in the material, but due to increase in area. The left hand side of Equation 4.8 ∆U3 is known as the energy release rate (G) and on the right hand side, is the ∆a fracture resistance (R). The fracture resistance represents the work required to fracture a material. For crack initiation, U4 = 0. Therefore, we can write the following differential form for incremental values of the variables:

⎡ dU1 dU2 ⎤ dU3 ⎢⎣ da − da ⎥⎦ = da

or

G=R

(4.9)

Incidentally, the symbol G is used in fracture mechanics after Griffith, who introduced the energy-balance approach. Williams (1984) provides a detailed account of evaluating G for different test configurations. It can be assumed that the energy required to produce a crack is the same for each increment (i.e., R = a constant). Therefore, G = GIc where GIc is known as the critical energy release rate. It can be shown that:

⎛ 1 − ν2 ⎞ 2 for plane (biaxial) strain GIc = ⎜ ⎟K ⎝ E ⎠ I GIc =

K12 for plane (biaxial) stress E

(4.10)

(4.11)

where ν = Poisson’s ratio and E = Young’s modulus. When Equation 4.9 is first satisfied, the crack propagates in a stable manner. However, uncontrolled fracture occurs when (Williams, 1984):

dG dR ≥ da da

(4.12)

The material’s resistance to fracture R can be determined by knowing σf , fracture stress of a sample with a crack of size a in a fracture test (Broek, 2001). Typically, a residual strength vs. crack size diagram can be drawn. Such a plot can be used advantageously when combined with a crack growth curve (Figure 4.9) to determine the permissible crack size ap corresponding to permissible residual strength pp and the time taken for the crack to grow to that limit (tp). The surface-energy approach helps to understand fracture mechanics without having to assume infinitely sharp cracks and infinitely high stress concentrations within the material. It should be emphasized, however, that in LEFM only the elastic deformation (stored) energy is available for fracture. The energy dissipated due to material flow does not contribute to creating new surfaces, and hence is not available for crack propagation. The EPFM considers plastic flow around the crack


Residual Strength, Pres

Crack Growth Time, t

tp

Pp

ap

a=0

Crack size, a

FIGURE 4.9 Residual strength and crack growth curve. (After Broek, 2001.)

tip (see Figure 4.5). For the EPFM the resistance to crack propagation R is GIc plus the energy dissipation in the flow region beyond the vicinity of crack tip. Thus, R is denoted by J (or JR) to account for this additional component. We should note that for the limiting elastic case J + R = GIc. It is worth pointing out that G is the energy release rate of the entire system and can include energy stored, whereas, R and J describe material behavior. Williams (1984) presents additional discussion on fracture mechanics of polymeric systems that exhibit such energy dissipation. Cheese being a viscoelastic material, strictly speaking, the LEFM is not directly applicable. However, unfortunately, theories suitable for evaluating combined fracture and flow in cheese are not available, and reasonable predictions can be made using the theory of LEFM. This is especially true if the flow region around the crack tip is small compared to the size of the crack (case 1 in Figure 4.5). In engineering materials, cracks usually occur rapidly. For cheese, crack propagation is a slow and prolonged process. For example, formation of eyes or holes in Gouda cheese takes about one week (Luyten, 1988). Based on the LEPM concept, the slow crack growth rate a˙ is measured as a function of KI or GI. That is,

a˙ =

da = A KI n dt

(4.13)

where A = material constant and n = exponent. As a˙ tends to zero, KI reduces towards a low threshold value Kth, and as a˙ tends to infinity, KI tends to approach KIc, the fracture toughness. Near Kth the crack growth is slow, and at Kc the crack growth is rapid. The material constant A varies, depending on the material and environmental conditions (e.g., temperature).


DETERMINATION OF KI The stress-intensity factor KI is normally determined experimentally by “v-notch” test or other similar tests. As shown earlier for the case of a through-the-thickness crack of length 2a in an infinite plate subjected to a tensile stress σ, the stressintensity factor is given by

K I = σ πa

(4.14)

The Equation 4.14 is the same as Equation 4.1 given Y = 1. However, for finite specimens Y¦1 and alternate expressions have been developed (Williams, 1984; Anderson, 1995):

πa ⎞ ⎤ ⎡ 2w K I = σ πa ⎢ tan⎛ ⎝ 2 w ⎠ ⎥⎦ ⎣ πa

1

2

(4.15)

where w = width of the plate. Note that as a/w approaches zero (i.e., for a large plate), KI approaches that of infinite plate value. Therefore, for all geometries, KI is written as:

KI =

P φ( a w) B w

(4.16)

where, P = applied force, B = thickness of the plate and φ(a/w) is a calibration factor. Some common test specimen and notch geometries and the corresponding solutions for KI based on finite element analysis are presented in Table 4.2. The American Society for Testing and Materials (ASTM) has defined a certain specimen size to obtain valid results for KIc in metals. Recommendations are also available for plastics. Such recommendations do not exist for food and biological materials. Therefore, even if the experiments were performed carefully, the test results of cheese should be treated with this fact in mind — i.e., test geometry does affect the validity of the LEFM theories. Moreover, for viscoelastic materials, even though the testing and data analysis procedure are the same, the validity of K and J are not guaranteed (Anderson, 1995).

FRACTURE TESTS ON CHEESE Fracture properties of cheese can be determined by any of the fundamental materials testing methods discussed in Chapters 2 and 3. These methods include compression, tension, shear, torsion, and bending. Details of these methods and accompanying results have been presented in previous chapters. Here we will briefly discuss some results that are directly relevant to determining fracture of cheese using notchedtension and notched-bending tests. In general, tension or bending is best suited for determining fracture properties because it is easy to observe the crack initiation and propagation. However, tension tests are difficult to perform with soft materials such


TABLE 4.2 Stress Intensity (KI) Factors (φ(a/w) in Equation [4.16]) for Different Test Geometries φ(a/w)1

Crack Geometry

[

2 tan m 0.752 + 2.02n + 0.37(1 − sin m)3 cos m

]

S n w 1.99 − n(1 − n)(2.15 − 3.93n + 2.7n 2 ) 2(1 + 2n)(1 − n)1.5 3

[

m (1.122 − 0.561n − 0.205n 2 + 0.471n 3 + 0.190n 4 ) 1− n

1

m = πa/2w; n = a/w

Source: After Anderson, 1995.


]

30

Stress (kPa)

50 37

20

25 10

12.5 0

0 0

0.1

0.2

0.3 0.4 Strain (-) A

0.5

0.6

0

0.2

0.4

0.6 0.8

1.0

Strain (-) B

FIGURE 4.10 Stress–strain curves of A: six-week-old and B: 10-week-old Gouda cheese in compression (____), tension (— — -), and bending (-.-.-.-). (After Luyten, 1988. With permission.)

as cheese and other food and biological materials. Therefore, bending is the preferred fracture-test mode. The fracture properties can be determined using compression tests, and a good set of data may be obtained due to the small sample size used and small effect of sample inhomogeneities (Luyten, 1988). We shall note that compression tests are often carried out until sample failure even when the object of the test is not to determine the fracture properties. Luyten (1988) evaluated fracture properties of Gouda cheese in tension, compression, and bending. The stress–strain curves in all three modes agree well for small strains (Figure 4.10). However, large differences are noticed in fracture stress and strain values (normally estimated at the peak of stress–strain curve). As presented in Table 4.3, the results indicate the modulus values are fairly similar because they correspond to linear (initial) part of the curve (measured as the initial slope). The fracture stress and strain values in tension, bending, and compression are low, intermediate, and high, respectively. With some corrections (Luyten, 1988), the fracture stress values from three testing modes may agree better than the fracture strain values.

NOTCH TESTS Testing of specimens in tension and bending with known crack size by means of a notch is perhaps the most popular fracture test for engineering materials. In fact, the tension test results presented in Figure 4.10 and Table 4.3 for Gouda cheese are for notched specimens. For a proper notch test, the initial crack should be sharp (so that the stress concentration is high). Therefore, the notch is made by pressing a razor blade into the material to a measured distance (the crack length). Luyten (1988) performed notched tension tests on Gouda cheese of different ages with different notch lengths (0 mm to 5 mm) to determine the notch sensitivity for crack initiation and propagation. The aged Gouda cheese is less notch-sensitive than the young cheese for crack initiation, and once initiated the crack propagates more rapidly in mature cheese than in the young cheese (Figure 4.11). The perceived “brittleness” of mature cheese is thought to be due to its propensity for higher crack propagation rate compared to the younger cheese. © 2003 by CRC Press LLC

TABLE 4.3 Comparison of Modulus (E) Fracture Stress (σf) and Strain (εf) of 6-week-old Gouda Cheese at 21°C Measured in Compression, Tension, and Bending Tests

Test Mode

Applied Strain Rate (1/s)

E (kPa)

σf (kPa)

εf

Tension

0.0139

189 201 221 230

30.5 26.4 28.9 22.4

0.33 0.31 0.31 0.28

Bending

0.0181

188 173 169 188 165

41.3 40.8 37.6 43.6 44.1

0.48 0.43 0.47 0.62 0.54

Compression

0.0167

202 203 173 154 244

49.7 45.7 45.8 45.3 53.3

0.87 0.80 0.89 0.96 0.82

Source: After Luyten, 1988.

FIGURE 4.11 Results of notch sensitivity tests on young (A) and mature (B) Gouda cheese samples in tension. The dotted lines connecting solid circles (•) and open circles (o) indicate decreasing stress with increase in notch length for crack initiation and crack propagation, respectively. Some points (x) are calculated values. (After van Vliet et al., 1991. With permission.)


FIGURE 4.12 Force–deflection curves from single-edge notched bending tests at a loading rate of 10 mm/min. (After Charalambides et al. 1995. With permission.)

Charalambides et al. (1995) and Kamyab et al. (1998) performed single-edge notched bending (SENB) tests on cheese. They followed the LEFM test protocol of the European Structural Integrity Society for testing of polymers (Williams and Cawood, 1990) using the test geometry for SENB as given in Table 4.2. This protocol has also been used for a model food system (Langley et al., 1994). For sufficiently large samples, the bending test specimen is prepared such that width (w) is twice the thickness (B), and the ratio of span (S = distance between supports) to B is four (Kamyab et al., 1998). Accordingly, samples of length, L = 88 mm, w = 18.5 mm, B = 9.3 mm, S = 74 mm, were used. Five-millimeter- to 6-millimeter-long notches were made using a razor blade at the center such that 0.45

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