Bioengineering Analysis of Orthodontic Mechanics

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Bioengineering

Analysis of Orthodontic Mechanics

Bloengi neering

Analysis of Orthodontic Mechanics ROBERT J. NIKOLAI, Ph.D. Professor, Department of Orthodontics St. Louis University Medical Center St. Louis, Missouri

Foreword by CHARLES J. BURSTONE

LEA & FEBIGER

PHILADELPHIA

1985

LEA & FEBIGER

600 South Washington Square Philadelphia, PA 191064198 U.S.A. (215)922-1330

Library of Congress Cataloging in Publication Data Nikolai, Robert J. Bioengineering analysis of orthodontic mechanics Bibliography: p. Includes index.

1. Orthodontic appliances. 2. Force and energy. 3. Bioengineering. I. Title. [DNLM: 1. Biomedical engineering. 2. Orthodontic appliances. 3. Orthodontics. WU 400 N693b] RK527.N54 1984 617.643 84-5712 ISBN 0-8121-0922-8

Copyright © 1985 by Lea & Febiger. Copyright under the Inter-

national Copyright Union. All Rights Reserved. This book is protected by copyright. No part of it may be reproduced in any manner or by any means without written permission of the Publisher. Prioted in the United States of America

Print Number:

5

4

3

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1

To My Parents

Foreword

The advancement of orthodontics and the quality of orthodontic treat-

ment are dependent upon a thorough knowledge of basic science. Although there are many orthodontic texts that cover the biologic basic sciences, such as growth and development, neuromuscular physiology, genetics, and anatomy, considerably less attention has been paid to bioengineering. Certainly no basic science could be more relevant to the orthodontic researcher and to the practicing orthodontist than bioengineering. Dr. Nikolai has written the first comprehensive text that can serve as a keystone to the understanding of bioengineering principles in general and their application to clinical orthodontics.

This text develops the fundamental concepts in an orderly way which will allow the orthodontist to understand and use his orthodontic appliances. Since most dentists in their predoctoral training receive little information relative to bioengineering, the development of basic concepts and their application is most important and necessary before moving on to more complicated clinical situations. The sequential presentation of material from the elementary principles to the more complicated force systems of multibanded appliances will allow both student and practitioner to gain a better understanding of how orthodontic appliances work and how to improve clinical therapy based on this understanding. Bioengineering has a broad application to orthodontics and, hence, the scope of the text is very wide. It includes a discussion of the force systems produced from orthodontic appliances, the behavior of the materials that are used, and correlates the relationship between the torce systems and the biologic changes that occur in the dentition and periodontium. The excellent use of clinical problems and examples through-

out the text insures that this is not just another engineering text, but truly a book on orthodontic bioengineering, which can serve both the student and the experienced practitioner. CHARLES J. BURSTONE, D.D.S., M.S. Professor and Head Department of Orthodontics School of Dental Medicine The University of Connecticut vii

Preface

As viewed by the observant bioengineer, the orthodontic appliance is a

unique assembly of small, structural elements. When activated, the appliance exerts a system of forces upon the human dentition. These forces cause, or at least serve to catalyze, biologic remodeling within the dentofacial complex toward achievement of finite, permanent displacements of teeth. From both structural and clinical perspectives, the principal component of the fixed, intraoral appliance is the arch wire. This wire is currently available to the clinician in at least five materials (metal alloys), round and rectangular cross-sections in various sizes, single- and mul-

tistrand formats, and in straight lengths or precontoured into arch forms. Prior to engagement and activation the practitioner may bend or twist the wire, using special tools to place or modify the arch form and detail the wire geometry to the specific patient. The appliance contacts the dentition through crown-affixed brackets and tubes, and the structural assembly may also include various kinds of metallic springs and polymeric elements to exert forces. Moreover, extraoral adjuncts may contribute to the total appliance.

The complexity of the orthodontic appliance as a structure is extended through its dynamic behavior and that of its foundation, the dentition; over time the appliance deactivates slowly in concert with movements within the biologic complex to which it is attached. The overall analysis requires an integrated examination of the appliance and the force system it delivers to the dentition. The characteristics of that delivered force system influence the format of the potential displacements of the teeth. The interested bioengineer now envisions a multifaceted analysis problem. The experienced practitioner, as well as the clinician-in-training, perhaps viewing structural analysis as "foreign territory," should gain some familiarity with the engineering of the orthodontic appliance, and this text has been prepared to assist in that education. Herein the subjects of force and structural analysis are discussed within the framework of practical, orthodontic treatment and at a level appropriate to the

intended audience. The presentation is seemingly theoretical for the most part, although much of the theory has been demonstrated to be ix

x

Preface

well founded through bench and clinical experimentation. The topic

coverage is largely a composite of analytic and graphical mechanics, materials science, and mechanics of deformable bodies (subject titles familiar to the engineering student), but presented with minimal mathematical involvement and with direct application to orthodontics. Substantial attention is given to the interactions between the activated appliance and the biologic systems and processes affected by the transmitted force.

This text is an outgrowth of class notes prepared and carefully revised, over a period in excess of ten years, for a bioengineering course presented to first-year students in an orthodontics graduate program. Not only intended for students, the book should serve the interested, experienced practitioner as well, whether or not previously exposed to formalized graduate or continuing-education coursework in orthodontic biomechanics. Because examples are used throughout, the text as-

sumes at least superficial familiarity with orthodontic therapy that would be gained, for example, through an introductory or survey exposure within the professional dental curriculum. Principles and procedures accumulate from the start in the book, but no preparation in mathematics and science, beyond traditional predental and dental curricula, are required. The text begins with a discussion of needed mathematical "tools." The subsequent three chapters introduce concepts of engineering mechanics in the orthodontic setting. Con-

sidered next is the response to applied force of the dentition and its supporting tissues. Structural analysis procedures are then introduced and applied to simple appliance elements. Chapters are devoted, individually, to bending and torsional analyses of arch wires and to extraoral appliances. Finally, force and structural analyses of entire representative orthodontic mechanics are undertaken.

Orthodontic bioengineering is becoming widely recognized as an important area of formal study for the aspiring clinician. The design and preparation of appliances have been traditionally learned from preceptors, and refined through actual clinical experience. Recent advances in appliances and techniques, however, have resulted from the joint contributions of practitioner expertise and engineering science. Although the latter has demonstrated its potential impact toward overall advance-

ment of orthodontic treatment, the organized application of bioengineering to orthodontics remains relatively obscure, apparently, to the majority of practitioners. Accordingly, the objective in preparing this text was to provide a means whereby a bioengineering approach to the analysis and design of orthodontic appliances and procedures might be outlined to the orthodontic community. Several individuals have intangibly, although importantly, contributed to the completion of this work. Vasil Vasileff, now retired from the School of Dental Medicine at Southern Illinois University, and formerly a faculty member at the St. Louis University Medical Center, presented

the author with his first opportunity to interact with graduate orthodontics students. Kenneth Marshall, former Chairman of the Depart-

xi Preface

ment of Orthodontics, hired and charged the author, trained in engi-

neering rather than in dentistry, to develop the bioengineering course within the graduate curriculum. Lysle Johnston, present Chairman of the Department, through his urging and encouragement, was instrumental in making this text, begun in earnest during a sabbatical leave, a reality. The author would be remiss without acknowledging the influence of

the students in the graduate orthodontics program at St. Louis University, particularly those whose thesis research projects he has supervised; their comments, criticisms, and suggestions were of substantial aid in this effort. Finally, to Charles Burstone, eminent researcher, clinician, and educator, who kindly consented to contribute the foreword to this text, the author expresses his sincere gratitude. ROBERT J. NIKOLAI, Ph.D. St. Louis, Missouri

Contents

1. Mathematical Topics Constants, variables, and functions 1 Frames of reference 3 Displacements of particles and solid bodies Trigonometry

8

10

An introduction to vector algebra 16 Dimensions and units 20 Measurements, computations, and numerical accuracy

21

2. Introduction to Analysis of Orthodontic Force

24

Mechanics 24 Force 25

Vector addition and decomposition of concentrated forces 29 The moment of a concentrated force 36 The couple 40 Resultants of force systems exerted on rigid bodies 46 Distributed forces and their resultants 50 Friction

53

Concepts of mechanical equilibrium and their applicability to orthodontic mechanics 56 Synopsis

69

3. Material Behavior of the Orthodontic Appliance

Internal structure of a solid material 73 Load-deformation behavior of a structural member 78 Mechanical stress 87 Mechanical and structural properties: Standardized testing Chemical and thermal influences 103 Selection of materials 110

71

91

xlii

xiv Contents

4. Energy Analyses in Orthodontics Concepts leading to the process laws Displacement 114

113 114

Mechanical work 115 Energy 117

Heat transfer and thermal energy 121 The conservation-of-energy law 122 Available energy 124 Activation and deactivation processes 125 Strain hardening and heat treatment of metals and alloys A work-energy analysis of the preparation of an arch wire Synopsis 144

5. Response of Dentition and Periodontium to Force

137 141

146

Mechanical response of the individual tooth to applied

force

148

Transverse crown force systems and tooth displacements Extrusion, intrusion, and long-axis rotation 163 Response of the periodontium to force 169 Displacements related to magnitude and duration of force Controlling the force-time pattern 178 Physiologically proper orthodontic forces 182 Dentofacial Orthopedics 189 Synopsis 191

151

177

6. Introduction to Structural Analysis of the Orthodontic

Appliance

194

orthodontic appliance: A structure or a machine? 196 Attachment of the orthodontic appliance to the dentition 197 A continuous-arch-wire appliance model 205 The

An overview of the structural analysis of an orthodontic appliance 208 The activation and deactivation characteristics of representative appliance elements 211 The action and response of tip-back bends: An illustration of the four-step procedure in orthodontic structural analysis 224 "Control" of the orthodontic appliance 228 Synopsis 231 7. Behavior of Orthodontic Wire in Bending

deformations, strains, and stresses 235 Force systems within the beam 241 Beam

Beam stiffnesses 248

233

xv Contents

Additional topics in elastic bending 252

Application of elastic beam theory to orthodontic arch-wire activation in bending 257 Inelastic behavior in bending 261 Orthodontic wire loops 264 Synopsis

269

8. Delivery of Torque by the Orthodontic Appliance Structural theory for the straight, circular shaft

272

273

Extension of theory to shafts having rectangular cross-sections 285

Application of shaft theory to the orthodontic arch wire 288 Isolation of the activating torque 288 Responses of the appliance and the dentition to torsional activation 289

Structural influences on active and responsive force systems 295 Anterior-segment torquing mechanics 299 Rectangular-wire torquing 299 Torquing spurs in the appliance 305 Inelastic behavior in third-order mechanics 310 Wire loops, springs, and torsion 314 Synopsis 318 9. Extraoral Appliances

322

The cervical-pull, face-bow appliance Occiusal-plane analysis 326

325

Buccal-view analysis of the cervical appliance

335

A coronal-plane view and comments on the asymmetric problem 343 Canine retraction with headgear 344 Extraoral force delivered to an anterior segment or an entire arch

348

Delivery of extraoral force to the mandibular arch 358 Dual-force headgear 361 The chin-cap assembly 362 Reverse-pull appliances 365 Synopsis 369

10. Force and Structural Analyses of Representative Orthodontic Mechanics Individual tooth malalignments Leveling displacements 374

374

372

xvi Contents

Rotational corrections

379

Bilateral action 385 Interarch mechanics 388 Intra-arch vertical positioning 392 Intra-arch retraction mechanics 402

Prepared, posterior-segment anchorage 412 Class III mechanics 417 Class II mechanics 424 Synopsis 435

Appendix I Glossary of Terms

437

Appendix II List of Symbols

458

Index

463

Mathematical Topics I Orthodontic bioengineering has emerged as an important subfield of

orthodontics, not only for research but for formal study by the practitioner-in-training as well as the experienced clinician. Bioengineering is appropriately termed an "applied science" and, as such, uses numerous mathematical concepts and procedures. This chapter introduces and explores the mathematical topics required in this text. These developments are not unduly rigorous and most will not be totally new to the reader. Familiarization with the necessary mathematics is undertaken initially in this text, rather than to introduce each topic individually as the need arises. As necessary, the reader may refer back to this chapter for a review of any specific concept. Moreover, a foundation is laid for uninterrupted and continuous study of the principles and applications of bioengineering to orthodontics.

Constants, Variables, and Functions Any object having physical attributes or characteristics may be described

qualitatively or quantitatively. For example, a specific orthodontic wire may be characterized as being shiny, made of stainless steel, round, and having a cross-sectional diameter equal to .018 in. The first three de-

scriptive words or phrases refer to qualitative attributes; the last is a quantitative characteristic. Descriptions, whether qualitative or quantitative, are helpful only insofar as their references or bases are familiar and understood. In describing the orthodontic wire, the adjectives were ordered from the least to the most definitive: "shiny" implies a comparison with some herein undefined luster or smoothness of surface; several types of stainless steel, differing from one another metallurgically, are

used in fabricating orthodontic appliances; and "round" is meant to describe the shape of the wire cross-section (but, alternatively, it might

refer to a curvature placed in the wire). Finally, the reference to the "diameter" of the cross-section with the measurement given in inches is

2

Bioengineering Analysis of Orthodontic Mechanics

the most definite because the dimension is well understood and the inch

is an established unit of length. This example suggests that descriptions in quantifiable terms are generally clearer because of familiarity with the bases of those terms. A particular stainless steel may be described in

terms of the constituent materials of the alloy, given in percent by weight or by volume, or implicitly, for example, by the American Iron and Steel Institute grade-and-type designation. (Given the AISI code for the alloy, its composition may be learned by consulting an appropriate metals handbook.) Because many objective descriptions undergo some form of change in the most general discussion, in practical considerations some limits or bounds on the extent of change are often defined or implied. The magnitude of a force, exerted by an appliance on a tooth, may lessen over a period of days as the tooth moves and the appliance partially or totally deactivates. In a structural analysis undertaken to be valid for only a short time, however, the force may be considered as unchanging. The temperature in the oral cavity may vary in a complex fashion and may be affected temporarily by ingesting hot or cold food or beverages. Often, however, it is sufficient to consider oral temperature as unchanging, but elevated some 30°F relative to normal room temperature. (In a study of the characteristics of a metallic arch-wire material, for example, even the fact that intraoral temperature is elevated with respect to the ambient air may be inconsequential.) A constant, then, is defined as a quantity that retains a fixed, unchanging value or level within the bounds of a particular discussion or investigation. A quantity that cannot be regarded as constant while under study, either because its variation is the subject of the study or the contribution of its changing is not negligible in overall considerations, may experience change associated with movement from one location to another, or directly or indirectly as time passes. Location or position may be related to time. The force in a helically-coiled spring may be altered by changing

the position of one end of the spring with respect to the other. The position or orientation of a malposed tooth under the influence of an activated orthodontic appliance may change with time. A variable, is, in general description, a definable characteristic or parameter under study that takes on more than one quantifiable value during the course of a discussion or investigation. Variables may be categorized as independent or dependent. Although this distinction may often depend on viewpoint, a few variables (e.g., time and temperature) are generally independent, and other parameters assume their values in some manner associated with the magnitude(s) of the independent variable(s). In the analysis of the helical spring, the force in the spring may be said to be dependent on the amount of deformation referenced from the passive (zero-appliedforce) configuration; alternatively, the extent of the deformation may be viewed as dependent on the size of the force applied to the spring. In

the example of tooth location versus time, however, the parameter "time" can never be realistically considered as dependent on the orientation or position of a malposed tooth.

3

Mathematical Topics

Whenever two or more variables are involved in a common discussion or investigation, and whenever these variables are interrelated such that one or more unique values of one variable are assumed upon the assignment of specific values to the remaining variables, the first is dependent on, or said to be a function of, the other (independent) variables. A functional relationship is usually expressed explicitly in the depend-

ent variable. For example, if the symbol, y represents a quantity that takes on a particular value upon the assignment of a value to x, the general relationship is written as y = function (x), or simply y = f(x). If the "data" relating y to x originate from an experiment, the function f(x) is subsequently sought toward the facility of determining all values of y within some range or bounds of the independent variable x. The variation in a parametric value might be expressed mathematically in a smooth, continuous manner, and an analytic relationship, an equation, may be written to represent it. This representation can always be set down in a graphical or tabular form, but there are times when a variable takes on individual, discrete, seemingly unrelated values and an equation cannot be derived easily, if at all. The graph or table is then necessary to visualize a pattern in and the range of the variable values. In this text, the typical equation is simple in form and often derivable from the graphical representation of the relationship among the variables.

Frames of Reference When expressing graphically the change in value of a variable, the values and change may be associated with the length of, or distance along, a reference line or axis. Many variables have the possibility of assuming

the value of zero, and this or another convenient reference point or origin is located on the axis. To associate length or distance with the variable values, a scale is needed. In Figure 1-1, an axis is established for

the parameter "time." The origin of the plot coincides with a relative time of zero (hours). The chosen scale assigned one hour to a specific interval, and twelve adjacent intervals "equals" one-half day. In this particular representation, amounts of time elapsed or time change are of interest and negative time readings have no real meaning. Hence, since time is not an algebraic quantity, there is no need to extend the reference

I

t

i

I

i

i

I

I

I

1

1 hr

FIGuRE 1-1. A one-dimensional plot relating time as a variable, in hours, to distance along an axis.

4

Bloengineering Analysis of Orthodontic Mechanics

line in both directions from the origin, which has, therefore, been placed

at the left end of the axis with time increasing to the right. The graphical relationship of one dependent variable to another, or possibly to two other independent variables, is usually established within a two-dimensional, rectangular-coordinates framework. A pair of reference axes are defined as the basis for the graph; the axes are perpendicular to one another with the origin of each the intersection point,

the now-defined origin of the framework as a whole. Ordinarily, one axis is positioned horizontally and the other vertically. When the variables may only assume positive values, the origin is placed at the lowerleft so that variable values associated with the horizontal axis will increase from left to right and values associated with the vertical axis will increase upward. As with the one-dimensional "plot" of Figure 1-1, a variable name (e.g., force) and units of measurement (e.g., ounces) are appended to each axis. A scale is implied by the distance associated with an individual unit division along each axis. When preparing a graph of the relationship between two variables,

symbolized, for example, by x and y (without specifying the units of measurement of either), points would be individually plotted in the x-y plane and each point would be designated by a number pair, or coordinates, (x,y). When plotting a two-dimensional relationship with respect to a set of horizontal and vertical axes, the x-coordinate of a point is often termed the abscissa and the y-coordinate the ordinate. When all points have been plotted, a curve is sketched as a "best fit" through all of the plotted points. An attempt may then be made to obtain an analytic

expression that relates x and y at or near to every point on the drawn curve. Figure 1-2 shows a straight-line relationship that is defined as to position and orientation by just two (x,y) coordinate pairs. A nonlinear relationship is also displayed that may necessitate the plotting of many points prior to sketching the curve to obtain an accurate representation. The scales chosen for the axes need not be identical, regardless of whether or not similar units of measurement are appropriate to the variables symbolized by x and y. The straight line in Figure 1-2 was drawn after the coordinates (1,2) and (2,4) were plotted. If the lower bound for x and y is zero, the plot appropriately begins at the origin of this x-y frame (0,0). The equation of the line is easily determined to be y = 2x. The curve sketched with respect to the same pair of coordinate axes suggests a functional relationship between x and y that is quite different from that of the straight line. Although the numerical relationship of x to y can be readily obtained for any specific point on the curve, the equation relating the two variables at every point on the curve is not easily derived from the plotted points. The two plots of Figure 1-2 may be unrelated to one another or may

be "members" of a family of curves. If the latter is the case, a third variable is involved and, for each value of interest of that third variable, an individual relationship exists between x and y. Plotting a family of curves in the plane is a method of visually describing the relationship among three variables within a two-dimensional reference frame. Gen-

(2,4)

(1,2)

0

1

2

3

4

x

URE 1-2. Relationships between pairs of variables displayed, referenced to an coordinate framework.

thy,

when plotting one curve or a family of curves with respect to a

me framework consisting of horizontal and vertical axes, the dependI variable, expressed graphically as a function of one or two independ-

I variables, is scaled on the vertical axis.

6


y

z

FIGURE 1-3. Relationships of an individual dependent variable to pairs of independent variables, sketched within an x-y-z-coordinate framework.

From a two-dimensional plot, in addition to enabling the determinalion of one or more values of the dependent variable for a given value of the independent variable(s), the slope of the curve at one or more specific

points is often useful. This slope represents the rate of change of the dependent variable with respect to a unit increase in the scaled dependent variable and is, geometrically, the inclination of the tangent to the curve at the specified point. The slope at a point may be positive or negative, depending upon whether the dependent variable is increasing or decreasing in value from that point. If the plot of the relationship between two variables is a straight line, the slope is the same at all points. Along the nonlinear curve in Figure 1-2, the slope decreases to x = 1, then increases with increasing values of x. The rectangular-coordinate framework in two dimensions may be extended to three with the addition of a third axis, mutually perpendicu-

lar to both axes in the plane and intersecting the origin of the plane frame. In the coordinate system in space, a functional relationship among three or possibly four variables may be displayed. Figure 1-3 shows a curve and a surface referenced to an x-y-z framework. Both the curve and the surface might represent functions in which z symbolizes the dependent variable and x and y the independent variables. The surface may serve to display a function for all values of x and y within some bounds; the curve, drawn in the surface, may graphically provide the functional relationship for only those values of the independent variables that are of particular interest, A function of three independent variables may be displayed in the spatial framework as a family of curves or

7

Mathematical Topics

The concept of slope can be extended to three dimensions. Along a surface the rate of change of z, for example, the dependent surfaces.

variable, may be considered at a point of the surface independently with respect to unit changes in x and y or with respect to a coordinate in the direction of a curve in the surface. A unique tangent plane is associated

with each point on a surface, and a slope may be computed for any specified direction in that plane. The interpretation of a functional relationship displayed in a threedimensional reference framework is difficult, particularly when sketched in two dimensions. Although the pictorial representation of analytic functions in three dimensions is generally beyond the needs of this text, coordinate systems have other uses, and an important one is the designation of reference directions in a plane or in space. In the study of dental anatomy, a coordinate frame is often placed in an individual tooth; one axis coincides with the long axis of the dental unit. Mutually perpendicular directions associated with this framework are occlusogingival (or occlusoapical), faciolingual (or labiolingual or buccolingual), and mesiodistal. Such a reference frame is shown in Figure 1-4. Note that these axes extend on both sides of the origin of the framework;

rather than adopting an algebraic scheme, a sense is specified (and the direction is implied) in discussions. For example, as indicated in Figure 1-4, "distal" indicates a specific sense in the mesiodistal direction. In a

l(ingual)

dOstal)

Origin

g(ingival) t(acial)

FIGURE 1-4. Mutually perpendicular coordinate directions used in dentistry.

m(esial)

$


framework associated with the entire dental arch or dentition, the refer-

ence directions are generally labeled anteroposterior, lateral or transverse, and vertical. A plane of reference is generally associated directly with the coordinate perpendicular to the plane, often with the additional indication of the "side" of the plane being viewed. For example, the occiusal plane is perpendicular to the occlusogingival direction with the view toward the tooth crowns. A buccal plane is perpendicular to a buccolingual direction; an identical lingual plane exists, but the implied view is toward the lingual surface(s). The faciolingual and mesiodistal planes are perpendicular to one another, references associated with an individual dental unit, and the line common to both planes is coincident with the long axis of the tooth. Note that the common terminology for the planes of reference for the dentition, or each arch, as a whole deviates from this pattern; the coronal, the occlusal (or transverse), and the sagittal planes are the mutually perpendicular references, and two of the associated, perpendicular axes have designations unlike their counterpart planes.

Displacements of Particles and Solid Bodies The distance between two points in space (e.g., A and B) is measured in

length units along a straight line connecting the two points. The displacement "path" of a particle moving from point A to point B, however, need not be along that line. The actual distance traveled by the particle in moving from A to B is measured along the path followed, and

may be sizable compared to the straight-line measurement. The displacement of a particle from point A is in magnitude the distance from A to a specific point on the path, and is dependent upon the elapsed time since its coincidence with point A. Displacements of orthodontic interest

are generally small in absolute measurement, but are comparable to characteristic intraoral dimensions. Such displacements may be difficult to obtain accurately because (1) often few, if any, potential references with the dentofacial complex remain stationary over time, (2) position measurements cannot be monitored continuously, and (3) total move-

ments may be an inseparable combination of orthodontic and orthopedic displacements and growth. Although full coverage of this subject is properly relegated to the study of cephalometrics, several portions of the topic fit into the discussions of this chapter. First, any displacement to be quantified, must be measured with respect to a reference frame. The displacement path of a particle must often be estimated by connecting straight-line distances between successive positions of the particle; the larger the number of position measurements, the more accurate the estimation. The displacement of a solid body as a whole is fundamentally described in terms of the displacement of one particle of the body and the angular movement of a line within the body passing through this particle.

9

Mathematical Topics

C, C

1800

01

I

I arc D"D'

C

FIGURE 1-5. Successive positions of a tooth undergoing planar orthodontic displacement, including a change in angular orientation.

Figure 1-5 depicts the displacement of a tooth as well as the displace-

ments of two particles (or points) of the tooth; in this example, the points and movements are planar. The displacement of point D may be expressed in terms of the displacement of point C and the relative movement of D with respect to C. The angular displacement of the tooth is the difference in orientations, 02 — 01, of the line through points C and D. In

fact, all lines of the tooth in the plane of the sketch, or parallel to it, experience the same angular displacement. Because the distance between points C and D in the tooth does not change during the movement, the displacement path of the particle at D with respect to, or as viewed from, point C is a circular arc. The radius of that circular arc is the line segment in the tooth from C to D, the change in angular position of the segment CD represents the angular displacement of the tooth as a whole, and the relative curvilinear displacement of the particle at D with respect to point C is along a circular path, the arc D'D', and equals the product of the length of the segment CD and the angular change, 02 01. In

this computation the angular change must be expressed in radi-

ans, where 22/7 radians is comparable to 180°. The units of the length CD

and the relative displacement DD' are then identical. The completion of the determination of the "absolute" displacement of the particle coinciding with point D must be deferred until later in this

10

Bioerigineering Analysis of Orthodontic Mechanics

chapter because this displacement is, mathematically, a vector quantity.

This section has, however, introduced or reviewed concepts associated with particle and angular whole-body displacements and recalled the relationship among the size of the angle, the radius, and the length of the circular arc for a circular segment.

Trgonometry As noted in the preceding section, an angle is the measurement of the

orientation of one line with respect to another in the plane containing the lines. The actual quantification implies a circle centered at the point of intersection of the lines; the curvilinear length of the circular arc between the lines divided by the radius of the circle yields the size of the angle in radians. The example of Figure 1-6 shows an angle of 22/21 radians, or 600. (Although the dimensionless units of degrees are more familiar than radians, many computations to obtain an angle as the result yield the answer in radians.) Angles are often categorized by size: acute angles are smaller than 90°, right angles are exactly 90°, obtuse angles are larger than 90° but less than 1800; and straight angles are exactly 1800. Although circles are described as encompassing 360°, the smaller interior angle between two radii never exceeds 180°. The sum of acute and adjacent obtuse angles within the intersection of two lines is 180°; these angles are termed "supplementary" to one another. A triangle is a plane, three-sided, closed figure made up of three line segments. The intersection of each pair of line segments defines an

QA = OB =

S

AB = s 600 = 22/21 radians s = (22/21)r

0 FIGURE 1-6. Angular measurement.

A

11

Mathematical Topics

angle. Each angle is measured interior to the figure and the three angular measurements always sum to 1800 or 22/7 ("pi") radians. Also, for

every triangle, the ratio of the length of a side to a function of the angle opposite that side is a constant. That function is defined as the "sine" of the angle; it is a dimensionless quantity that has bounds for angles between 0° and 180° of 0 and +1. Triangles are categorized according to the size of the largest interior angle. A right triangle contains, opposite to the longest side termed the "hypotenuse," a right angle. The other two angles, necessarily acute, must sum to 90°; such angles are said to be "complementary" to one another. A right triangle is pictured in Figure 1-7. The following trigonometric functions are defined for an acute angle within a right triangle: sine A =

opposite

=

a

hypotenuse c adjacent b cosine A = =— hypotenuse c opposite a tangent A = adjacent

(1-1)

b

"Opposite" and "adjacent" refer to the positions of sides of the triangle with respect to angle A and the ratios are of side lengths. Similar relationships may be written explicitly for angle B. The defined trigonometric functions are generally abbreviated "sin," "cos," and "tan." Table 1-1 contains trigonometric functional values and conversions between degrees and radians for angles between 0° and 90°. Three additional trigonometric functions—the cosecant (csc), the

C

a

b

FIGURE 1-7. A right triangle with angles and sides labeled symbolically.

12

Bioengineering Analysis of Orthodontic Mechanics TABLE 1-1. Degrees-radians equivalents and trigonometric functions Degrees 0 1

2 3

4 5 6 7 8

9

10

Radians

Sin

Cos

Tan

0.000

0.000

1.000

0.000

0.018 0.035 0.052 0.070 0.087 0.105 0.122 0.140 0.157 0.175

0.018 0.035 0.052 0.070 0.087 0.105 0.122 0.139 0.156 0.174

0.999 0.999 0.999 0.998 0.996 0.995 0.993 0.990 0.987 0.985

0.018 0.035 0.052 0.070 0.088 0.105 0.123

0.192 0.209 0.227 0.244 0.262 0.279 0.297 0.314 0.332 0.349

0.191

0.367 0.384

0358

22

0.375

0.934 0.927

23

0.401

0.391

0.921

24 25

0.419 0.436 0.454

0.407 0.423 0.438 0.454 0.470 0.485 0.500

0.914 0.906 0.899

0.515 0.530 0.545 0.559 0.574 0.588 0.602 0.617 0.629 0.643

11

12 13 14

15 16 17 18 19

20 21

26 27 28

0.471

30

0.489 0.506 0.524

31

0.541

32 33 34

0.559 0.576 0.593

35 36 37

0.611

38

0.663

29

0628 0646

39

0.681

40

0698

41

0.716 0.733

42 43 44 45

0.751

0768 0.785

0.208 0.225 0.242 0.259 0.276 0.292 0.309 0.326 0.342

0.982 0.978 0.974 0.970 0.966

0.141

Cot 1.571

57.3 28.6 19.9 14.3 11.4 9.51

8.14 7.12

0.158 0.176

6.31

0.194 0.213

5.14 4.70 4.33

0.231

5.67

59

1.15

0.855 0.838 0.820 0.803 0.785

49

Radians

Degrees

0.755 0.743 0.719 0.707

0.869 0.900 0.933 0.966 1.000

Cos

Sin

Cot

0.731

0.625 0.649 0.675 0.700 0.727 0.754 0.781

0.810 0.839

79 78

1.361

1.030 1.012 0.995 0.977 0.959 0.943 0.925 0.908 0.890 0.873

0.656 0.669 0.682 0.695 0.707

0.883 0.875 0.866

1.378

80

1.66 1.60 1.54 1.48 1.43 1.38 1.33 1.28 1.24 1.19

0.601

0.891

81

1.80 1.73

0.857 0 848 0.839 0.829 0.819 0.809 0.800 0.788 0.777 0.766

0.940

1.414 1.396

69 68 67

2.61

0946

1.431

89 88 87 86 85 84 83 82

1.204 1.187 1.169 1.152 1.135 1.127 1.100 1.082 1.065 1.047

0.384 0.404 0.425 0.445 0.466 0.488 0.510 0.532 0.554 0.577

0.951

1.484 1.466 1.448

90

77 76

4.01

0.956

1.501

-

1.344 1.327 1.309 1.292 1.274 1.256 1.239 1.222

0.249 0.268 0.287 0.306 0.325 0.344 0.364

0.961

1.553 1.536 1.518

3.73 3.49 3.27 3.08 2.90 2.75 2.48 2.36 2.25 2.14 2.05 1.96 1.88

1.11

1.07 1.04 1.00 Tan

75 74 73 72 71

70

66 65 64 63 62 61

60 58 57

56 55 54 53 52 51

50 48 47 46

45

13

Mathematical Topics

(sec), and the cotangent (cot or ctn)—are defined as the reciprocals of the sine, cosine, and tangent, respectively. The lengths of the secant

sides of a right triangle are interrelated through the Pythagorean theorem;

for the triangle of Figure 1-7 the relationship is = a2 + b2

(1-2)

or, in words, the length of the hypotenuse equals the square root of the

sum of the squares of the other two sides. Also, from the foregoing definitions and because A and B are mutually complementary: sin A = cos cot A = tan

tan A = cot csc A = sec

cos B = sin A sec A = csc B

B B

B

B

(1-3)

An oblique triangle, incorporating no right angle, is sketched in Figure 1-8. The law of sines for this triangle is

sinD

—

sinE

ci

e

—

sinF

(1-4)

f

Another relationship, valid for the oblique triangle, also relating sides and angles, is the law of cosines: d2 =

e2 =f2+

d2

2ef(cos D) —2fd(cos E)

+

e2

2de(cos F)

f2 =

e2

+ f2

(1-5)

Although the oblique triangle of Figure 1-8 is specifically an obtuse triangle, these two laws may also be used in analyzing an acute triangle.

d

e

FIGURE 1-8. An oblique triangle with angles and sides labeled symbolically.

14


y L

0

x

FIGURE 1-9. A line segment referred to an x-y-coordinate framework.

For the right triangle, the sine and cosine laws are replaced by the defini-

tions of the sine and cosine functions and the Pythagorean theorem. Figure 1-9 shows a line drawn in the plane of the page with the origin of an x-y-coordinate framework coinciding with one end of the line segment. Because the reference frame is rectangular, the line makes complementary angles with the x- and y-axes. These angles, labeled and are called the direction angles of the line OL. The "shadow" or projection of the line segment on the x-axis is the apparent length of the line seen from a vantage point far out on the y-axis; that length equals OL cos Similarly, the projection of OL on the y-axis is equal to OL cos The cosines of the direction angles are termed direction cosines. Enclosing the line OL in a rectangle, having adjacent sides coincident with the x- and

y-axes and making the segment OL a diagonal of the rectangle, and noting the two equal right triangles formed (which share a common hypotenuse), the following equations may be written: (x-projection)2 + (p-projection)2 = 0L2 + = 1

(1-6)

The development just completed using Figure 1-9 may be extended to three dimensions. In Figure 1-10 the coordinate framework is again

appended to one end of the line segment (although, in general, the segment may be remote with respect to the frame); the end points of the segment are (0,0,0) and (XN,YN,ZN). The direction angles are defined as in Figure 1-9, except there are three; for instance, in Figure 1-10 the angle 0,

15

Mathematical Topics

\

N

0 z

Q

\\ a y

ox

x

FIGURE 1-10. A line segment in space referred to an x-y-z-coordinate framework.

measured between the x-axis and the line segment and that measurement is made in the plane containing ON and the x-axis. The projections of the segment ON on the coordinate axes are is

y-projection = QN(cos x-projection = ON(cos z-projection = ON(cos

0!,)

respectively, and the three projections are also related to the coordinates of end points of the segment and its direction cosines: x-projection

0 XN y-projection = z-projection = ZN — 0

YN

—

0

(1-8)

An alternative two-step process may be used to obtain the projections. First, for example, project the line on the y-axis and into the x-z plane. The angle 4, measured between the segment ON and the x-z plane, is complementary to and the projection in the plane equals ON cos q5. Second, decompose the x-z plane projection, OQ, into x- and z-axis projections using the complementary angles a and y, measured in the x-z plane. The three projections may then be expressed as follows:

16


x-projection = ON(cos 4)cos a

y-projection = ON(cos Os,,) z-projection = ON(cos q5)cos y

(1-9)

The length of the line segment is related to the coordinate-axis projections through the three-dimensional form of the Pythagorean theorem: (x-projection)2 + (y-projection)2 + (z-projection)2

(1-10)

Dividing through by the square of the segment length and noting the resulting forms defining the direction cosines, the expression becomes +

+

=

1

(1-11)

The comparable relationship in the two-dimensional situation, previously examined, is the complementary nature of the two direction angles. (No simple relationship among the three direction angles exists spatially.) Before closing this section, it is important to note that a three-dimensional geometry problem may always be decomposed into two or three

two-dimensional problems. The original problem is, seemingly, expanded from one into two or three, but the gain is in terms of each of the

two or three "projections" being in its own plane setting.

An Introduction to Vector Algebra

physical quantities associated with orthodontic bioengineering possess characteristics in addition to magnitude. A most important entity, the point or concentrated force must be known in magnitude, in direction (and sense), and in point of application to be fully described. A multifaceted description is similarly associated with the displacement of a particle or a point of a solid body. Both the point force and displacement obey the laws of vector mathematics. A vector is a mathematical quantity possessing both magnitude and direction. The equations describing vectors are unique, and vector quantities combine with scalars and with other vectors according to specific mathematical laws and procedures. Vector quantities may be expressed in both graphic and analytic formats. The vector is pictured as a directed line segment. When drawn to scale, the length of the segment is proportional to the magnitude of the vector. The inclination or orientation of the segment, the angle between it and a specified reference line or axis, indicates the direction of the vector. The sense of the vector is denoted by an arrowhead affixed to one end of the segment. Given all other characteristics of a vector, with only two possibilities for sense, it may be specified algebraically. Sense is often linked with direction; two vectors having identical angles with a reference axis, but differing in sense, are said Several

17

Mathematical Topics

be opposite in direction to one another. Without being specific as to the physical quantity represented, a vector is shown in Figure 1-11 with an x-y-coordinate frame included in the sketch that serves as a reference to

for direction and sense. The displayed vector, indicated as having a magnitude of 100 units with the directed line segment subdivided into 5 equal length units, implies a linear scale factor of 100 to 5 or 20 to 1. Figure 1-11 is similar to Figure 1-9 except for the absence of the arrowhead in the latter sketch and the coordinate framework not attached to an end of the line segment in the former. The angles labeled in Figure 1-11 are those which the line of the vector, if extended, would make with the x- and y-axes; therefore, these angles are properly termed the direction angles of the vector. Individual multiplications of the vector length by the direction cosines gives the x- and y-projections of the vector. Converting these (scalar) projections to vector quantities by including the senses and multiplying each by the scale factor for the sketch yields the

x-, and y-components of the given vector. The lengths of the x- and y-projections are indicated as 4 and 3 units; with a 20-to-i scale factor, the x- and y-component magnitudes are 80 and 60, respectively. The rectangular components of a vector are, by definition, mutually perpendicular and individually parallel to a coordinate axis. Hence, the sum of the squares of the component magnitudes (or the lengths of their line-

segment representations) equals the square of the magnitude (or the length) of the original vector. The analytic expression for a vector is typically written in terms of its component magnitudes (projections) and unit vectors (dimensionless and of magnitude one). Vector quantities appear in boldface type in this

y

60 units

Scale: 1:20

80 units

0 FIGURE 1-11. A vector positioned in the x-y plane.

X

18


text to distinguish them from scalars. The unit vectors i and j, defined as

associated in direction and sense with the x- and y-directions, are shown in Figure 1-11; note the positive senses corresponding to increasing coordinate values. The equation for the vector displayed may be written as V = 80i + 60j

(1-12)

Shown in Figure 1-12 is a vector in space. The ends of the vector are coincident with points located with respect to the chosen x-y-z-coordinate framework; the "tail" of the vector coincides with point D(3, 1,2) and the vector symbol extends to point E(5,5,6). In this example, the vector represents a 60-g force. A parallelepiped, oriented to the coordinate frame, "surrounds" the vector, which lies on a main diagonal. The length DE of the force vector is found through the three-dimensional form of the Pythagorean theorem: DE = [(5

—

3)2

+ (5

—

1)2 + (6

—

2)21112 =

(1-13)

6

The scale factor for the sketch is, then, 60 g to 6 length units or 10 g to 1 length unit. The lengths of the sides DA (2 units), DB (4 units), and DC (also 4 units) of the parallelepiped correspond to the projections of the given vector, which are parallel to the x-, y-, and z-axes, respectively. The direction angles and of the force vector are the angles EDA, EDB, and EDC, respectively. Similarities in and differences between Figures 1-10 and 1-12 should now be apparent. Defining k as the unit vector y

/

/ /7 y,/

//

/'/I / /

4

1/

/11/

x

1 0(3,12) E(55,6)

Scale:

FIGURE 1-12. A vector referenced to a three-dimensional framework.

1

unit = log

19

Mathematical Topics

parallel to the z-axis and with sense corresponding to increasing z, the

vector equation for the 60-g force may now be written:

F=20i+40j+4Okg

(1-14)

The terms of Equation 1-14 are the x-, y-, and z-components of the force vector. By way of application, Figure 1-12 might be displaying a force exerted by a stretched elastic, pulling from a point (3 cm, 1 cm, 2 cm) in a lateral-anteroposterior-vertical reference frame. (Continuing, the point

(5 cm, 5 cm, 6 cm) is on the line of the force, but the "head" (arrowhead) of the vector located at that point in the figure is merely a result of the particular choice of scale factor for the sketch.) Equation 1-14 charac-

terizes the force as to magnitude, direction, and sense at a particular time, but does not indicate the point of application of the force. To complete the displacement discussion begun in an earlier section of this chapter, a vector addition must be carried out. The displacement of point D, the root apex in Figure 1-5, equals the sum of the displacement vector of point C in the crown and the relative displacement, D with respect to C. In Figure 1-11 the vector components having magnitudes of 80 and 60 units "add" to give the resultant vector V with magnitude of 100 units. (The sum of two vectors depends on both the individ-

ual magnitudes and their directions and senses.) In that example, the (rectangular) components were perpendicular to one another; the components of the apex displacement vector are not 900 apart. Note, however, that the parallelogram law of vector addition, when employed graphi-

cally, is easily undertaken regardless of the relative orientations of the vectors to be summed. The vector addition of the two displacement components is shown in Figure 1-13. The displacement of point C, in reality obtained, perhaps, from successive cephalograms, is drawn to a scale; in this example, the scale magnifies the actual displacement size. The magnitude of the relative displacement is the scalar product of the change in long-axis angulation (in radians) and the length of the axis segment CD. The directions and senses are from C to C' and D" to D', respectively, from Figure 1-5. To accomplish the summation, the two vectors are placed in series, with the tail of one meeting the arrowhead of the other, and they form adjaC (or D)

d(isplacement)D

d010

C, (or D")

FIGURE 1-13. Vector addition. The relationship among the displacement vectors of points in Figure 1-5.

20


cent sides of a parallelogram. The third and fourth sides of the parallelo-

gram are added (dashed) to complete the figure and the desired vector, the absolute displacement of point D, is coincident with a diagonal. The length of the resultant vector is measured and multiplied by the scale factor to obtain the magnitude of the displacement of point D. Because the two vectors summed in Figure 1-13 are nearly parallel and have comparable senses, the result in this example is almost as if the component magnitudes were added. Parallel components, however, are the exception and not the rule; the 80-unit and 60-unit components in Figure 1-11 sum to give a resultant magnitude of 100 units, not 140. Additional examples of vector addition are presented in Chapter 2.

Dimensions and Units The quantifiable parameters of orthodontic bioengineering are generally

dimensional and their magnitudes depend upon the units in which the parameters are expressed. Four dimensions are basic or fundamental: length, time, mass, and temperature. All other dimensions of interest may

be derived from these four; the most important is the dimension of force. The result of a dimensional analysis of an equation of classical physics yields force as the product of mass, length, and the inverse of time squared [mass(length)/(time2)}. Dimensions help to describe a quantity, and also to distinguish among apparently similar parameters. For example, pressure is dimensionally mass/length!(time2) or force! (length2) and, therefore, is not identical to, or a type of, force, although the two are related. One quantity already discussed, which is dimensionless (or nondimensional), is angular measurement; another, which will be introduced in Chapter 3, is strain. The numerical value assigned to a dimensional quantity (and some nondimensional quantities as well) is meaningful only when units are appended to that value. Each fundamental or derived dimension generally has several units or sets of units that may be associated with it. The numbers 1, 3, and 36 are different from each other, but when used as expressions of length in yards, feet, and inches, respectively, they are identical.

Because of biologic and materials-science influences on their field, dental practitioners are familiar with two systems of units. Metric units are common to the life sciences; length is measured in meters and mass in grams or fractions!multiples thereof (e.g., millimeters [mm] and kilograms [kg]). In this country the physical sciences have long employed the American engineering system of units; typically, lengths are in feet and forces in pounds-force. In both unit systems the basic unit of time is the second. The derived, composite unit of force in the metric system is

the "newton;" the gram-force is more common now in the United States, but is a distortion of the metric mass unit. In the American engi-

21

Mathematical Topics

neering scheme, force is considered a fundamental dimension and the

"slug" is the derived unit of mass (although pound-mass is also used in a manner comparable to gram-force). Conversions from one metric-units set to another is straightforward; "kilo" indicates 1000 and "centi" means "divide by 100," for example. Such conversions in the American engineering system are more difficult (e.g., 5280 ft/mile and 16 oz/lb);

this is probably a principal reason why the scheme has never been widely accepted outside of the United States. Orthodontists, until now purchasing arch wires having cross-seclions measured in inches, but themselves measuring arch lengths and discrepancies in millimeters, should be pleased about the pressures from abroad toward a worldwide move to the Système International d'Unités (French), or the SI units system. A modified metric scheme, the fundamental SI units of length, time, mass, and temperature are the meter, the second, the kilogram, and the degree Kelvin, respectively. Examples of SI units associated with derived dimensions include the newton (force), the pascal (stress or pressure; one newton per square meter), and the joule (energy; one newton-meter). Currently, the conversion to one international system of units remains incomplete. Hence, the clinician must possess some proficiency in converting from one unit

or set of units to another in the same system and in transposing or converting between the metric and American engineering schemes. Table 1-2 is provided to assist the reader with a partial listing of conversion factors within and between unit systems.

ants, Computations, and Numerical Accuracy Although not professing to be mathematicians, orthodontists are con-

fronted with, and make use of, a substantial amount of quantitative information. Treatment planning is influenced by computations performed after measurements are taken from casts and cephalograms. Today's vendors of wires, elastics, and the whole array of appliance ele-

ments are oriented in the materials and engineering sciences, and clinicians must be able to communicate appropriately with them. To understand the mechanical and structural functions expected of their appliances, and to match sets of appliances properly with the treatment plan, the practitioner must be able to undertake at least a cursory analysis of the force systems created by activation of those appliances. Physiologically-proper forces, to be exerted on the dentition, must be approximated. To follow the development and examples in this text, and more importantly to make clinical use of the knowledge to be gained in the study of the bioengineering topics to follow, the mathematical tools discussed in this chapter become indispensable. When possible, emphases will be placed on the appropriate graphic techniques to carry out an analysis, in

particular to help visualize relationships among variables and in the

22

Bioengineering Analysis of Orthodontic Mechanics TABLE 1-2. Conversions within and between units systems

Dimension(s)

Length (L)

Area (L2)

Multiply mils

0.001

centimeters meters inches

100 25.4

inches2 centimeters2

0.155

pounds pounds pounds ounces kilograms-force

Force (F)

by

10

645 16

454

4.45 0.278 2.2

to Obtain inches millimeters millimeters millimeters millimeters2 inches2

ounces grams-force newtons newtons pounds

Energy (F-L)

inch-ounces foot-pounds

720 1.36

gram-millimeters newton-meters (joules)

Stress (FIL2)

pounds/inch2 pounds/inch2 pounds/inch2

0.704 0. 07 04

grams/mm2 kilograms/cm2 newtons/meter2 (pascals)

Temperature (T)

degrees F—32

degrees C

6900 5/9 9/5

degrees C degrees F—32

Time (t)

seconds minutes

3600 1440

hours days

Angle

radians

57.3

degrees

1. Degrees F = °Fahrenheit Degrees C = 2. Angular measurement is nondimen-

sional but is quantified in degrees or radians.

3. Joules (J) and pascals (Pa) are derived units in the SI system. 4. Absolute temperature is measured in °Kelvin = °Celsius + 273 or kine = ¶ahrenheit + 460

manipulations of force and displacement parameters. When necessary, analytic procedures will be used and dimensional and units analyses included as a partial check on the correctness of terms and equations. In any analysis requiring mathematical manipulations, after an overview of the entire undertaking and coming to a decision regarding the appropriate procedure to be followed, the input or given data must be gathered or examined. The techniques of obtaining linear and angular measurements, and the accuracy of these data, are expected to be those ordinarily achieved in acceptable cephalometric exercises. Determinations of force magnitudes should generally be to the nearest ½ oz or 15 g. When input data for a problem to be solved originate from a previous statistical analysis, the accuracy of that data is to be determined; if

mean values of parameters are to be used, their standard deviations should be sought.

23

Mathematical Topics

Analyses within this text are to be undertaken and solutions pursued through principles of force and structural mechanics interfaced with

knowledge obtained from clinical experience and experimental research.

Quantifiable results will be sought by using mathematical procedures

introduced in this chapter. After any solution has been obtained it should be carefully checked. Two possible means of checking results are

(1) to proceed to the solution by an alternate method or approach, if available, or to change the order of steps in the solution procedure followed, or at least to substitute the numerical answers obtained into the mathematical relationships initially used to ensure satisfaction; and (2) to make sure that no mistakes exist in dimensions or units in the procedural steps. A dimensional analysis of an algebraic equation, for example, must show identical, net dimensions and units for each term. The solution to a problem involving mathematical operations is only as accurate as the data employed and the accuracy of the computations performed. Deficiencies in either can render a solution inadequate, even if the procedures are all correct. An accurate set of calculations cannot make up for inaccuracies in initial data. Hand-held, electronic calculators and digital computers will yield numerical solutions to six or more

figures, but no more digits should appear in the answer than can be justified. The evaluation of the accuracy of mathematical procedures is a complex matter and will not be investigated here. It is perhaps sufficient to indicate that most input data in orthodontic bioengineering problems are accurate to no more than three significant figures and, therefore, no more than three significant figures should be recorded in the solutions obtained.

Readings Davis, H.F., and Snider, A.D.: Introduction to Vector Analysis. 4th Ed. Boston,

Allyn & Bacon, 1979, Chapter 1.

Drooyan, I., Hadel, W., and Carico, C.C.: Trigonometry. 3rd Ed. New York, Macmillan, 1979, Chapters 1, 4, 5, and Appendix A. Greener, E.H., Harcourt, J.K., and Lautenschlager, E.P.: Materials Science in Dentistry. Baltimore, Williams & Wilkins, 1972, Chapter 1. Spiegel, M.R.: Mathematical Handbook of Formulas and Tables. Schaum's Outline Series. New York, McGraw-Hill, 1968.

Spiegel, M.R.: Theory and Problems of Vector Analysis. New York, Schaum, 1959, Chapter 1.

Thurow, R.C.: Atlas of Orthodontic Principles. St. Louis, C.V. Mosby, 1970, Chapter 5. Wilson, W.A., and Tracey, J.I.: Analytic Geometry. Boston, D.C. Heath, 1949, Chapters 1—3, 13.

Introduction to Analysis of Orthodontic Force 22 The activation of an orthodontic appliance creates a system of forces that

are transmitted through the members of that appliance to the dentition, there providing the potential to produce displacements of dental units. Relationships exist between force systems and potential displacements of bodies upon which the forces act. Hence, an understanding of the fundamental principles and procedures of force analysis is indispensable to the practitioner who fabricates, places, and activates the appliance toward the desired end: the controlled movement of teeth. A large portion of the analysis of forces to be discussed has its foundations in the subfield of classical physics known as particle mechanics. Several of the basic concepts may be familiar to the reader from an undergraduate survey course in general physics. Nevertheless, the discussion here commences with no assumptions of previous knowledge of the subject.

Mechanics as an area of study within the physical sciences, is concerned with the state of rest or motion of bodies subjected to forces. Mechanics is subdivided by the physicist according to the sizes of the bodies studied (from the smallest subatomic parts of matter—quantum mechanics— to the planets and galaxies—astronomy) and with respect to the degree of motion (from no movement whatsoever to bodies moving at rates of the order of the speed of light). Although evidence exists that the early Greek scientists understood and used some aspects of mechanics several hundred years nc., the fundamental principles of particle mechanics, as understood today, were formulated by Sir Isaac Newton in the seven-

Mechanics,

teenth century. Newtonian mechanics pertain to bodies of moderate size, if moving then not at the excessive speeds where mass and velocity

may become relatively, mutually indistinguishable, and is separable from relativity theory and celestial mechanics. 24

25

Introduction to Analysis of Orthodontic Force

Newtonian mechanics is divided into branches known as statics,

which treats bodies at rest and under the action of forces, and dynamics, which deals with moving bodies. Dynamics is further subdivided into kinematics, the study of motion itself, and kinetics, in which relationships between the force systems and the characteristics of body motion are explored. The subareas of study within mechanics also may be categorized according to interest in whole-body motions and displacements of solids, the deformations that solid bodies experience under loading, and the flow of fluids. The principles of Newtonian mechanics, together with their interactions with the biologic systems and processes involved, govern the study of orthodontic forces. The movements of individual teeth are examples of whole-body displacements. Because the activation of the orthodontic appliance requires deformation of one or more elements of the appliance, applicable principles of mechanics of deformable bodies are discussed in subsequent chapters.

This chapter first introduces the principles of Newtonian partide mechanics. The extensions of these principles, enabling the treatment of bodies that are large in comparison to the particle, are then discussed. Some modeling (idealizing from the actual) and approximations will be

found necessary or beneficial and, in each instance, the validity and limitations of the model or approximation must be assured and understood. Although introduced individually, note that in large measure the concepts and principles "fit" together or accumulate to ultimately form a packaged procedure to be employed in analyses wherein total descrip•tions are required of force systems induced in the orthodontic appliance

and in the dentition to which the appliance is attached.

Force is defined as an act upon a body that changes or tends to change the state of rest or the motion of that body. The categories of forces are numerous, and forces are described in many different ways. Two bodies are associated with every force; one body exerts the force and the other receives the force. A first manner of categorizing forces is according to the proximity of the two involved bodies with respect to each other. Contact Force

forces arise due to actual physical contact between the bodies; body forces exist between bodies that are some distance from one another. The most common, sizable body force is the attraction of the Earth for a body on or near the Earth's surface: the weight of the body. Newton's

gravitational law enables the determination of the weight of a body, which is directly proportional to the product of the masses of the Earth and the body and inversely proportional to the square of the distance between the centers of the Earth and the body. Other body forces include those existing in the presence of electric or magnetic fields, but these do not occur routinely in the study of orthodontic force systems and thus will receive little attention. Notably, the weights of teeth and

26 Bioengineering Analysis of Orthodontic Mechanics

appliance parts are small and may be neglected in comparison to the

contact forces present within the activated appliance. In analysis procedures, forces are associated with the areas or volumes of bodies with which they directly interact, and contact forces are associated with the surface areas over which the contact exists. If the contact area is small in comparison with the total surface area of the body, the force is modeled as exerted on a surface point or particle of the body and is termed a concentrated or point force. On the other hand, if the contact area is relatively large, the pattern of the force intensity over

the area may be an important consideration and, therefore, the force is explicitly termed distributed. For example, the force exerted by an elastic stretched against a hook, soldered to a bracket or to an arch wire, is reasonably concentrated, but the force arising between the periodontal ligament and the root, in response to appliance activation transmitted to the tooth, must be analyzed with regard to its distribution over the sizable surface area of contact. By definition, forces exerted on an individual particle are point forces that may actually or potentially only pull or push the particle. As previously mentioned, two bodies are necessary to the existence of every force. Moreover, whenever one force is created or present, a counterpart exists as well; when one body exerts a force on another, the second body also exerts a force on the first. Between the two bodies there is an action and a reaction, and which force is which depends solely

on point of view. In analysis procedures, however, just one of the two bodies will be examined at a time and that one body will be studied with respect to the forces exerted on it (not those forces exerted by it). Reactive

forces must not be confused with responsive forces; reactions exist as counterparts to both active and responsive forces. In the study of a body subjected to forces, the active forces are those purposely created or those that "load" the body; the responsive forces are also exerted on the body,

but are a direct result of the creation of active forces. The responsive forces are often undesired, but being present they must be included in the analysis. As a final categorization of forces for present purposes, forces are said to be external to a body under study when they are exerted by other bodies not part of the specific analysis. Internal forces arise within a body as a result of loading or activating the body under study. These internal forces "carry" or transmit the action through the body to the responsive locations, for example, to connections with other bodies.

In addition to the gravitational law, Sir Isaac Newton formulated a set of three laws governing the mechanics of a particle. Newton's First Law states that a particle subjected to a balanced system of concentrated forces will remain at rest, if originally at rest, or will move with constant speed in a straight line, if originally in motion. ("Balanced" force system: the net force is zero.) The Second Law indicates that, if the particle is subjected to an unbalanced system of forces, the particle will be accelerated in the direction of the net force exerted; in equation form, the net

27


force and the product of the particle mass and its acceleration, with the

three quantities expressed in consistent, related units, are identical in magnitude, direction, and sense. (Acceleration is the time rate of change of velocity and, like the concentrated force, is mathematically a vector quantity.) The Third Law states that paired active and reactive forces are

equal in magnitude, but are directly opposed to one another and are exerted on adjacent particles. A corollary to the Third Law is the existence of internal forces, as previously defined, in canceling pairs; the ramification of this corollary is demonstrated later in this chapter. Newton's First Law might be considered a special case of the Second Law, and in orthodontic force analysis a strong argument may be made to the point that the inertia of an appliance element or a tooth is negligibly small, in particular after the almost instantaneous response, in the form of soft-tissue deformations, immediately following the activation of an appliance. If the inertia, the product of mass and acceleration for a particle, may be neglected in comparison with the individual components of the force system created through activation, the orthodontic system may be said to exist in a kind of quasi-static state; the displacements that take place are small and occur over a relatively long period of

time and, at any instant, a force analysis may be carried out without erring appreciably, as if the system was at rest. Clearly, this approximation has profound consequences; for the purpose of most analyses involving the entire force system, a dynamic body or group of bodies is effectively replaced by a static body or assembly. In Chapter 1 the concentrated force was reported to fulfill the mathematical requisites of a vector quantity. In the complete description of a point force, then, its characteristics as a vector must be stated: magni-

tude, direction, and sense. Moreover, from a physical standpoint the location of the force on or in the body must be given. Figure 2-1 shows a point force exerted on a molar crown. From the sketch, the force is seen to have a magnitude of 200 g; in this two-dimensional idealization, the direction of the force (and that of the dashed line of action of the force) is 700 from the long axis of the tooth. The sense of the force may be said to be up and to the left with respect to the sketch as a whole, and the point of application is point B with the force pushing on that point. (If desired,

an equation in vector form may be written for this or any point force, following the establishment of a reference frame, and all of the necessary characteristics to describe the force, except point of application, would be contained in that equation.) Figure 2-1, then, provides a complete visual description of the 200-g force. The potential or actual effects of a force exerted on a body, whether active or responsive, are displacement and/or deformation. Although both effects are of interest in orthodontic structural analysis, only the association of displacement with force is discussed in this chapter. Disregard-

ing deformation resulting from force application is tantamount to as-

suming that the body, on which the force or force system acts, is perfectly rigid. Although this is a reasonable assumption for some bod-

28


200 g

FIGURE 2-1. A concentrated force exerted on a tooth crown.

ies, including individual teeth, it is not a valid assumption for others, for example, elastics. In considering effects of force application, however, bodies that are clearly nonrigid may be reasonably analyzed only within a framework that recognizes differences in passive and active configurations.

The rigid body, then, is a model of Newtonian mechanics used in analysis procedures when only completion of the description of the total force system or the whole-body displacement is sought. (In Chapter 1,

the displacement of point D of the tooth was obtained within the assumption that the tooth was rigid; otherwise, the relative displacement, point D with respect to point C in Figure 1-5, could not have been described as it was.) An important feature of rigid-body mechanics allows the "sliding" of a point force to any location on its line of action found most convenient for analysis purposes. This principle of transmissibility states that the mechanical, whole-body effect of a concentrated force, acting at a specific point, is unchanged by replacing the given force by another force having the same magnitude, direction, and sense as the original force, but acting at some different point on the line of action. Figure 2-2 shows an 80-g force in two different locations on its line of action (dashed); the potential for displacement of the tooth as a whole is the same, whether the force is pulling on the lingual side or pushing on the labial crown surface. Before proceeding further, the general dependence of orthodontic

force upon the independent variable "time" must be mentioned. The

29


—— 80 g

FIGURE 2-2. Two concentrated torces having identical displacement potentials. The principle of transmissibility.

concentrated orthodontic force often changes with time and that change

is typically reflected in the magnitude characteristic, although alteration of direction may also occur with time. Magnitudes of force generated by appliance activation decrease during the between-appointment periods and an accompanying change in displacement potential generally oc-

curs. The changes with time may be divided into short-term occurrences, primarily due to soft-tissue deformations immediately upon acti-

vation, and long-term changes associated with tissue remodeling. Magnitude increases and, perhaps, directional alterations take place when the orthodontist reactivates the appliance. Detailed discussions of these phenomena are undertaken later in this text; in the present chapter, force systems are analyzed as if all involved entities are "frozen" at a particular instant in time.

ition and Decomposition of Concentrated Forces Two

fundamental analysis manipulations with point forces are dis-

cussed in this section. First, toward enabling the prediction of the form and direction of a potential rigid-body displacement associated with a system of active forces, the forces are combined in such a way that the mechanical effect of the original system remains unchanged while the resultant is as simple as possible. Second, often in the process of obtaining the resultant of a force system, the replacement of a force by a set of force components with each component having a designated direction is worthwhile. Beginning the first procedure with the fundamental operation, a pair of concentrated forces having intersecting lines of action has, as its resultant, a single force. The two original forces define a plane; the resultant lies in that plane and its line of action intersects the point where the

30


lines of action of the original forces come together. The means of reduc-

ing the two concurrent forces to a single force, mechanically equivalent to the combination of the original forces, is known as the parallelogram law of vector addition. To illustrate, Figure 2-3 shows a pair of point forces: one originating from an interarch elastic (EL) and the other from an extraoral appliance (HG); both are exerted on a maxillary molar bracket or buccal tube. The view is from the buccal on the right side. To obtain the resultant graphi-

cally, the directions and senses are sketched exactly as they exist in a two-dimensional picture of the appliance action. The force magnitudes are represented in the lengths of the vectors according to the given scale; because it is twice the magnitude, the length of the headgear force vector is twice that of the elastic force. The given forces intersect at point 0, the center of the bracket in the buccal view, and their vector representations form adjacent sides of a parallelogram. The dashed lines are added to complete the geometric figure. The vector representing the resultant is drawn as the diagonal of the parallelogram; the "tail" of the resultant vector coincides with the point where the "tails" of the given vectors

meet. Its sense is then known, the direction of the resultant may be determined with respect to a chosen reference line with a protractor, and the length of the vector representation is converted to the force magnitude through the scale as indicated. Note that if one of the forces is seemingly pushing on the point and the other is pulling on the point of intersection of their lines of action, one or the other vector must be moved along its line of action until either both "tails" or both "arrowheads" or "tips" of the vectors meet at the point. Then the parallelogram may be completed and the sense of the resultant is determined; either all three "tails" or all three "tips" meet at the common point.

0

EL = 200 g

HG = 400 g 0=

+ 02

Scale: 1 in. = bOg lengths = 2.8 in. H = 280 g = 28°

FIGURE 2-3. Graphical determination of the resultant of frio concurrent forces. The parallelogram law.

31


With the aid of a rough sketch, the resultant of two concurrent point forces may be obtained analytically. Seen in the parallelogram construclion of Figure 2-3 is a pair of congruent triangles with a shared side, a

diagonal, coinciding with the resultant vector. Two sides and the included angle of each triangle are known through the characteristics of the given forces. The laws of cosines and sines may then be used to obtain the magnitude of the resultant force and the orientation of its line

of action with that of a given force or some other known reference. Either the vector lengths or the actual magnitudes of the given forces, because length and force size are proportional, may be used in the computations which would proceed as follows: R=

+ HG2 — 2(EL)HG(cos 0) [2002 + 4002 2(200)400(cos 400)]h/2

= 278 g or approximately 10 oz

sin0 EL

=

.

(2-1)

200

R

cb=27.5°

The rough sketch is necessary, accompanying the analytic solution, to determine and illustrate the sense of the resultant force and location of its line of action. If the two forces to be "summed" are perpendicular to one another, the graphical procedure remains essentially unchanged, but the parallelogram becomes a rectangle. The pair of congruent triangles become a set of adjacent right triangles, and the analytic solution is simplified somewhat with the cosine and sine laws replaced by the Pythagorean theorem and the definition of one trigonometric function. An example solulion is given in Figure 2-4. The vector summations of Figures 2-3 and 2-4 may be handled alternatively, using a corollary of the parallelogram law: the triangle law.

R

Q2

= 36 = 6.8 oz

P = 3 oz

I tan 6 =

Q = 6 oz

P

0 = 27°

FIGURE 2-4. The resultant of two concurrent, mutually-perpendicular forces.

=

3

= 0.5

32


0

JR

HG

R

P

0 Q

FIGURE 2-5. Resultants obtained for the given forces of Figure 2-3 (top) and Figure 2-4 (bottom) using the triangle law.

Graphical solutions for the examples of Figures 2-3 and 2-4, using the triangle law, are shown in Figure 2-5. Recall that the "tails" or "tips" of the given force vectors must coincide when using the parallelogram law; the solution by the triangle law requires that the "tip" of one given force and the "tail" of the other be coincident. The resultant is drawn from the "tail" of one given force to the "lip" of the other so that the three forces form the sides of a triangle. Note that in summing two concurrent forces by means of the triangle law, one given force may have to be displaced to an artificial location in order that the resultant emerges in the construction along its proper line of action; alternatively, keeping both given forces on their original lines of action in the graphical approach, the resultant obtained is correct in all characteristics except for its actual point of application. Hence, care must be taken in using the triangle law so that the proper line of action of the resultant is ultimately determined and designated correctly.

33


The resultant of any number of point forces having lines of action all intersecting at one common point is a single, concentrated force; the line of action of the resultant passes through that same point. To determine the resultant of a system of three or more concurrent forces, the parallel-

ogram law or triangle law may be used repeatedly. Two of the given forces may be combined into one, that one combined with a third given force, and so forth until all of the given forces have been included in the process; the resultant emerges from the final combination involving the

last given force. Because any two of the concurrent forces define a unique plane in which their resultant will lie, the two procedures may be used in a three-dimensional problem. If all of the concurrent forces act in the same plane, the triangle law may be extended to a polygonal law.

Graphically, the given forces are drawn to scale with the "tip" of one touching the "tail" of the next until all have been included, in any order desired. The magnitude, direction, and sense of the resultant are determined by closing the polygon with the vector resultant having its "tip" touching that of the last given force drawn (and its "tail" touching that of the force with which the sketch was begun). The number of sides of the polygon is one more than the number of given forces. In this construction all but two of the given forces must be moved from their actual lines of action. The line of action of the resultant is known, though, to pass through the point common to the lines of action of the given forces. An example of the procedure using the polygonal law is illustrated with three given forces in Figure 2-6. In this example the forces are "added"

in the order indicated by the equation in the figure. The dashed lines suggest the graphical procedure by two successive applications of the parallelogram law. In a force-system analysis, the decomposition of given forces into com-

ponents is often desirable. The procedure is the reverse of that of the parallelogram or triangle law of vector addition. In Figure 2-3 the 200and 400-g forces could be considered components of the 278-g force. In analyses of orthodontic force systems, however, the component directions often coincide with the mutually perpendicular, dental-coordinate directions. In Figure 2-4, with proper orientation, the 3- and 6-oz forces become occlusogingival and mesiodistal components of the 6.8-oz force. The number of mutually perpendicular, nonzero components equals, at

most, the number of coordinate dimensions in a given problem: up to two in the plane problem and three in the spatial problem. Although the determination of vector components perpendicular to one another was outlined in Chapter 1, by way of review consider the example of Figure 2-7 in which the occlusal and mesial components of the 300-g force against the molar are obtained. Graphically, the parallelogram (actually a rectangle because perpendicular components are desired) is constructed around the 300-g force as its diagonal, drawn

to scale. The components are adjacent sides of the rectangle; their "tails" touch that of the given vector. The component magnitudes are obtained from their lengths and the scale factor for the sketch. Analyti-

34


V A = 4 oz

7-7

'7

7

B = 6 oz

0 R = 7.5 oz

,, 7, C = 5 oz

7,

7

7 R=A+B+C

FIGURE 2-6. Three concurrent, coplanar forces and their resultant obtained through the polygonal law.

cally, each rectangular-component magnitude is the product of its direction cosine and the magnitude of the given force as indicated in the figure. The three-dimensional problem of decomposition into mutually per-

pendicular components cannot be easily handled graphically. The analytic solution, however, can be a direct extension of that used in the plane problem. A rough sketch, particularly helpful toward determining component senses, should be used. In proceeding, first establish the coordinate directions and, perhaps, sketch the corresponding rectangular parallelepiped around the given force. Second, obtain a set of angles, either (1) the direction angles (angles between the given force and the coordinate axes), or (2) the angles to project the given vector into one of the coordinate planes and onto the axis perpendicular to that plane and, subsequently, to project the component in the plane onto the coordinate axes defining the plane. Again, the procedure illustrated by Figure 2-8 is a review of the spatial relationship between the vector and its mutually

35


0

300 0

280 g

F0 = F cos 70° = 300(0.34) = 1000 Fm = F cos 20° = F sin 700 = 300(0.94) = 280 g FIGURE 2-7. Decomposition of a concentrated force into two rectangular components. The parallelogram law in reverse.

perpendicular components previously discussed in Chapter 1 (see Fig.

1-12). Note that only two of the three direction angles need to be meas-

ured or estimated; the third angle can then be determined from the geometric relationship requiring the sum of the squares of the directions cosines to be unity.

a

d

F

I, i/I

a: apical d: distal 0: occlusal I:

F=l2oz

lingual

Fd = 12 cos 30° = 10.4 oz F0 = 12 cos 75° = 3.1 oz

F, = 12 cos 65° = 5.1 oz

(cos 300)2 + (cos 750)2 + (cos 650)2 = 1

FIGURE 2-8. Three mutually perpendicular components of a concentrated force not located in any of the three coordinate planes.

36


The Moment of a Concentrated Force action of a concentrated force upon a particle tends to produce, simply, a push or pull displacement; the particle is small and the force seemingly covers the particle as a whole. The magnitude, direction, and sense of the force, together with characteristics of the particle and its supporting structure, are collectively a measurement of the potential of the force to displace the particle. A force, exerted on a small portion of the total surface area of a relatively large body, tends to produce a displacement of the body which is generally rotational. The moment of a force about, or with respect to, a specified point or line is a measure of the potential of that force to rotate the body, upon which the force acts, about the particular point or line. Consider a rigid body that is hanging from a cord attached to just one point of the body. Label that point 0. A force F is exerted on the body so that its line of action does not pass through point 0. Define the line OB so that point B is on the line of action of the concentrated force F and the two lines intersect at right angles to one another. As depicted in Figure 2-9, the line OB and the line of action of force F define a plane. If the distance from point 0 measured along OB is d to the line of action of F, the magnitude of the moment of force F about point 0 is the product of the force magnitude F and the distance d. Given the manner of support of the body in the figure, the application of force F would displace the body from the position shown and cause an angular change in the initially vertical line through point G (the center of gravity, through which the weight resultant acts). The moment of a force about a point is a vector quantity; the direction of the moment vector is perpendicular to the plane defined by the force vector and the point. The sense of the moment vector is determined by a rule associated with the apparent rotational tendency, clockwise or counterclockwise as viewed from above the plane of the force and the point about which the moment is determined: out of the plane if counterclockwise and into the plane if clockwise. The moment vector is located such that its line of action passes through the point about which the moment is computed, and may be specifically positioned wherever convenient along its line of action. The moment vector of magnitude Fd is shown in Figure 2-9. Force and moment vectors may not be directly combined in a force-system analysis and, because the two types of vectors may appear in the same sketch, the short slash is incorporated near the arrowhead in the symbol for the moment vector. To obtain the moment of a force about a point, three quantities must be completely known: (1) the force itself; (2) the point about which the moment is to be found or moment center (point 0 in Fig. 2-9); and (3) the The

length of the moment arm extending from the moment center and traversing the shortest distance to the line of action of the force (line OB in Fig. 2-9). From the determination of the moment of a force about a point, the

37

Introduction to Malysis of Orthodontic Force

Mzz = F(d) = F(d)(cos 0)

FIGURE 2-9. A concentrated torce and moment vectors with respect to lines (axes) through point 0.

38


potential of the force to produce rotation about a particular line through

the point may be obtained. The rotational potential with respect to a line, a moment axis, has the greater application in orthodontic mechanics. In Figure 2-9, the moment of F about point 0 is also the moment of the

given force with respect to the line Z-Z. To obtain the moment about another line through point 0, line L-L in Figure 2-9 for example, the magnitude of the moment about line Z-Z is multiplied by the cosine of the angle between L-L and Z-Z. The vector component is the directed projection of the moment about Z-Z on line L-L. Apparent now is the general, three-dimensional nature of the moment representation and computation. The moment axis typically does not lie in the plane of the force and the moment arm. In fact, if the moment axis is in the plane of the force and moment arm, either the axis intersects the line of action of the force and, therefore, the length of the moment arm is zero, or the axis is parallel to the force; in both cases the tendency for rotation about the specified axis is nonexistent, and, correspondingly, the moment with respect to the specific axis is zero. In the general problem, the force for which the rotational potential is sought

likely will not be perpendicular to the moment axis, and one of two approaches may be used: (1) the procedure followed in Figure 2-9 in which the moment vector with respect to point 0 was projected onto line L-L; or (2) the process of Figure 2-10. In the latter approach, the force R is decomposed into components parallel (P) and perpendicular (Q) to the moment axis S-S. Now, the moment of the resultant of several con-

current forces must be equal to the vector sum of the moments of the individual forces with respect to the same axis (a theorem established by the French mathematician Varignon in the late 1600s, before the advent

of vector algebra). Since the moment of P about axis S-S is zero, the magnitude of the moment of R equals that of Q: Qb; the sense is as shown in the figure according to the rule previously established. In problems in which the forces of interest all exist in the same plane, the moment concept may be put into a two-dimensional format. Rotational potentials are obtained with respect to axes that are perpendicular to the plane of the force. In sketches these axes will appear in end views, or as apparent points in the plane. Moment vectors with respect to these moment centers will all have a common direction: perpendicular to the plane of the forces. They may then be treated as algebraic quantities with the distinction between moments having clockwise or counterclockwise rotational potentials made merely by a sign convention. To determine the net rotational tendency with respect to a particular moment axis, owing to the action of several forces, either of two procedures may be followed. The moment vectors may be obtained for each individual force with respect to a particular point on the moment axis, these concurrent vectors may be combined into one by methods of vector addition already discussed, and the resultant moment vector may then be projected onto the moment axis to obtain the magnitude and sense of the rotational potential of the force system. Alternatively, the resultant of the system of forces may be sought and the moment of the resultant then found with respect to the specified moment axis.

39


0

0 R

7 I

I

/J

Mss 1

= Q(b)

FIGURE 2-10. A force in space, decomposed into components parallel and perpendicular to a given line, and its moment vector with respect to that line.

The former procedure is employed in Figure 2-11 to obtain the net moment of a pair of concurrent forces with respect to the z-axis of the reference framework. In this instance a two-dimensional solution may be carried out in the plane of the given forces; the z-axis would be perpendicular to the plane of the forces and in the sketch it is, in effect, point 0. The curved-arrow notation indicates the senses of the moments of the 120- and 90-g forces and of the net moment. Two moment computations are displayed with magnitudes and senses obtained with respect to moment axes perpendicular to the plane of the forces through points O and E. The net moment being equal to the moment of the resultant of the two forces enables the determinations of the actual moment arms of the 150-g force with respect to points 0 and E; the moment-arm lengths

are symbolized by o and e, respectively. Viewing the 120- and 90-g forces as x- and y-components of the l50-g force, whether or not to decompose

a force into rectangular components to obtain its moment will depend upon the ease of the actual decomposition process and the establishment of the moment arms of the components versus the determination of the actual moment arm of the given force. The latter can often be a

40


y

M0 = 90(3)1 + 120(2) 30 g-mm

= 420 g-mm J M0 30 o = -h-- = = 0.2 mm

d=

\ E(5,0)

ME

=

420

= 2.8 mm

x

0 FIGURE 2-11. Determination of the moment of a force about a point in two dimensions by using rectangular components of that force in the plane.

seemingly difficult geometry program while the moment arms of the

components become clear after the coordinates of the moment center and one point on the line of action of the force are known. The more complex problems of determining the net rotational effect of a system of forces, which may be two-dimensional but nonconcurrent

or three-dimensional in nature, have been outlined as to general approach. They are more easily discussed in detail, however, following the

introduction in the next section, "The Couple."

The Couple A pair of point forces having equal magnitudes and identical directions,

opposite in sense to one another and having noncoincident lines of action, when considered as one mechanical entity is termed a couple. Because the forces have the same magnitude, but are oppositely directed, the net potential of this special force system to translate the body upon which it acts is nil. The lines of action of the two forces are parallel, but are not coincident; as a result, this force system tends to rotate the body acted upon. Also, since the forces are nonconcurrent, they cannot

be combined and, therefore, the couple cannot be simplified further. The couple, occupying a specific plane as defined by the two forces, is illustrated in Figure 2-12. The rotational potential of the couple with respect to point 0, or about a moment axis perpendicular to the plane

through 0, is the sum of the moments of the individual forces with respect to the point. Using the procedure and notation of Figure 2-11,

41


y

M0 = 300(1) 300 g

+ 300(3)

= 600 g-mm) = 300(2)

= 600 g-mm)

2mm

= 300(1)) + 300(1))

•

=600g-mm) M0 = 300(3)) + 300(1)

C(2,2)

= 600 g-mm) 8(11) 300 g

D(4,0)

0

x

FIGURE 2-12. A couple and computations of its inherent moment.

noting that with respect to point 0 one moment is clockwise and the

other is counterclockwise, the net moment is found to be 600 g-mm counterclockwise. Using the points B, C, and D as individual moment centers, the same moment is obtained in the three computations—the same as that with respect to point 0. In fact, for all moment centers in or moment axes perpendicular to the plane of the couple, the same result emerges. A couple has, then, an inherent moment, and the magnitude of the couple itself is that of the inherent moment, the product of the size

of one of the forces and the perpendicular distance between the two forces. Because the moments of the individual forces are vector quantities, so also is that of the couple and it follows that the couple itself may be represented mathematically as a vector. The direction of the couple is

associated with the plane of the pair of forces and the sense with its rotational potential, either clockwise or counterclockwise, as viewed looking into the plane of the couple. Figure 2-13 displays a couple and its vector representation (and that of its moment). The direction of the vector C is perpendicular to the plane of the couple (and the "moment arm" b between the forces) and the sense is determined according to the convention noted earlier for moment vectors: out of the plane if viewed as counterclockwise, into the plane if clockwise. Note the slash through the symbol, again to differentiate between force and moment vectors. A couple is completely described when specified are its magnitude,

the plane of the two forces, its sense, and, if the body is nonrigid, the actual points of application of the two forces. The value of the couple as

42

Bloengineering Analysis of Orthodontic Mechanics C = Pb

FIGURE 2-13. A couple in its plane and its vector representation perpendicular to that plane.

an analysis tool, however, is in the determination of whole-body displacement potential and, when so used, the specific points of application are unimportant. Although its vector representation in Figure 2-13 is shown midway between the two forces, because the moment of the couple is the same with respect to every moment axis perpendicular to the plane of the forces, the vector has no particular line-of-action location and may be drawn through any point of the plane of the couple. This "freedom" associated with the couple vector has far-reaching implications with respect to certain force-analysis procedures. To review, a force acting on a rigid body has the potential, generally, to both translate and rotate the body. Sliding the force along its line of

action will not change this potential. The tendency to rotate the body about a specific axis is associated with the vector characteristics of the force and also with the location of the line of action of that force because the axis and the line define a moment arm. Moving the line of action of the force with respect to the moment axis alters the force direction or the moment arm and, therefore, changes the translational or rotational potential of the force.

The couple, however, has been shown to possess an inherent rotational potential and no tendency to produce translation. Apparently, then, as long as the characteristics of the couple are unchanged, the forces making up the couple may be altered without affecting wholebody response; it follows, though, that an alteration of one force of the pair necessitates a comparable change in the other. Indeed, the use of properties of and operations on forces already discussed, including the

43


parallelogram law, demonstrates that the following alterations may be

made without affecting the mechanical effect of the couple on the body as a whole: (1) the forces may be moved ("slid") along their individual lines of action; (2) the forces may be rotated in the plane of the couple, keeping their lines of action mutually parallel and the distance between

them unchanged; (3) the couple may be moved as a whole from the given plane to another parallel plane; and (4) the magnitudes of the forces may be altered, while preserving the equality of force magnitudes

and the moment of the couple, the latter by making a corresponding change in the distance between the lines of action of the forces. Figure 2-14 shows four mechanically-equivalent couples, one being a

curved-arrow representation which is convenient in two-dimensional analyses. These permissible movements of the couple, and the allowable correlated changes in the force magnitudes and the distances between

their lines of action, are embodied in the vector representation of the couple. The couple acting on a rigid body is, mathematically, a "free" vector, having no specific line-of-action location as noted previously. Frce and moment vectors may not be directly combined, and the slash through the moment-vector symbol is a reminder of this fact. Perhaps unfortunately, however, no symbolic distinction is made that would indicate the difference between the vector representing the moment of a single point force and that depicting a couple with its inherent

800 g 200 g .75 mm

-.

mm

200g

600 9-mm

800 g

FIGURE 2-14. Four mechanically-equivalent couples.

300 9

44 Rioengineering Analysis of Orthodontic Mechanics

moment. Note that the moment vector, representing the rotational po-

tential of the concentrated force with respect to a specific moment center or axis, is not a "free" vector. On the other hand, because a couple is a

free vector and through its two-force representation, a couple may be combined with a third force. The form of the resultant from this combination is particularly dependent on the direction of the third force with respect to the couple vector. Figure 2-15 shows the relative position of the force and couple vectors, enabling the placement of the two forces of the couple in a plane containing the third force, which leads to a simple resultant. Figure 2-16 illustrates the combination procedure. With the three forces in the same plane, the couple is represented as two forces having magnitudes equal to that of the single force; the distance between the forces making up the couple is then determined. Next, the couple is translated and rotated in the plane so that one force of the couple shares the line of action of single force and is opposed (in sense) to it. The now directly opposite forces cancel one another and one concentrated force from the original three remains, having the same vector characteristics as the original single force, but with a line of action some distance from it. The procedure of combining a force and a couple may also be carried out in reverse. Given a single force having specific characteristics, including location of line of action, a force and a couple may replace it without altering the whole-body displacement potential. The transla-

800 g-mm

H

FIGURE 2-15. Mutually-perpendicular force and couple vectors referenced to the plane of the force system.

45

Introduction to Analysis of Orthodontic Force 200 g

200g

200g

/

mm

/

/

200 g

I B

200 g

200g

/

FIGURE 2-16. Reducing the force and couple of Figure 2-15 to a single force.

tional potential remains unchanged if the force portion of the new sys-

tem has the same vector characteristics as the given force. To leave the rotational potential unaltered, the magnitude, direction, and sense of the couple must correlate with the position of the line of action of the force of the substituted system with respect to that of the given force. In Figure 2-17, in occlusal view the force exerted on the edgewise orthodontic bracket is "moved" by means of the addition of the couple to a parallel position through the long-axis of the tooth. Although seemingly adding to the complexity of a force system, the procedure provides a means to "free" the force vector from its specified location and permit lateral movement of its line of action. A couple is a special two-force system. In orthodontic application, active couples are often created upon bracket engagement of arch wires a

C = Qe

FIGURE 2-17. Moving the line of action of a force 0 through introduction of a couple C. The reverse of the process illustrated in Figure 2-16.

46


into which permanent bends or twists have been placed. The dimension

of the couple magnitude is the product of force and length. The couple tends to cause only rotation of the body on which it acts. In addition to its importance as a displacement-producing system, the couple is useful in analysis because it alone enables the transfer of a concentrated force, exerted on a rigid body, from its given location to a parallel, alternative line of action associated with the same rigid body.

Resultants of Force Systems Exerted on Rigid Bodies definition of force, presented earlier in this chapter, was stated in terms of its potential to influence the motion of the body acted upon. In

The

discussions of forces, moments, and couples thus far, references to translational and rotational displacements have been made. The word "resultant" was used previously in discussing the replacement of two forces by one through use of the parallelogram or triangle law. In the analysis of a class of particle- and rigid-body-mechanics problems, including certain applications to orthodontics, relationships are sought between an active system of forces and a displacement pattern to be achieved by, or to result from, force application. Because the force systems often are somewhat complicated in their actual formats, the replacement of the given system by another, which is mechanically equivalent but more easily related to the potential or actual displacement of the body, is usually desirable, if not necessary. Generally, this new or replacement system is simpler, and is termed a resultant of the original force system. It is possible to reduce any force system acting on a body, when relationships between the given system and whole-body displacement potentials are sought, to one force and one couple, and in some instances to just a single force or just one couple. Clearly, these are the most simple resultants.

In the process of determining the displacement format of a body subjected to a set of forces, the resultant of the entire, active force system is sought. (Recall that the active forces "load" the body and are distinguished from responsive forces, which come into being because of contact between the body and whatever supports it. Generally, the respon-

sive force system is present in some form, however abbreviated, even before the loading is applied to the body, simply because of the member weights and the physical contact of the body with its supporting structure. Loading the body, then, usually alters the responsive force system in some manner.) Although a more simple resultant of the active force system may exist, and often does, the resultant sought is usually that referenced to a point in the body known as its center of resistance. The location of the center of resistance depends upon the size and shape of the body, the distribution of the weight of the body, and the manner in which the body is supported. In general, the location of the center of resistance of a body must be found experimentally or approximately by

47


appropriate analytic analysis; simple force systems are applied and displacement responses are observed or determined and specific displacement formats are sought. an

Upon the application of loading to a body, which results in an activeplus-responsive force system, and if the entire force system exerted on the body is not completely self-balancing, one of three forms of displacement occurs. Translation is that displacement within which no line in the body experiences a change in angular position (orientation) with respect to a fixed reference line. If the body is rigid, during translation all points in the body follow parallel paths; although these paths may be straight lines, they may also be curvilinear without violating the requisites for translation. In order for a translational displacement to result from the application of force, the resultant of the applied, active force system mj.ist reduce to a single concentrated force, and the line of action of that force must pass through the center of resistance of the body. When the resultant of the active force system is a couple, the resulting displacement is termed pure rotation. All points in or on the body move in circular paths; the centers of those circles are all on one line or axis that pierces the center of resistance of the body. The vector representing the resultant couple is parallel to the axis of rotation. The most common form of displacement is a combination of translation and pure rotation and is known as generalized rotation. The resultant of the active force system is usually expressed as one force and one couple. The displacement may be difficult to describe except to note that the angulation of the body has changed. If the applied force system is two-dimensional and the resultant of the active force system reduces to a single force, its line of action does not pierce the center of resistance of the body, and for small displacements or if the resultant remains constant in all characteristics with respect to the body acted upon, the motion is circular about an axis perpendicular to the plane of the forces but not passing through the center of resistance. Except when the active force system consists entirely of couples, at least a portion of the resultant problem is the simplification of a concurrent force system. Although the parallelogram or triangle law may be used to reduce the concurrent system to a single force, an alternative procedure may be more simple if numerous forces are involved; its advantage in an analytic approach is organization. Useful in both two- and three-dimensional problems, the process follows a sequence of four steps: (1) a coordinate framework is established; (2) each given force is decomposed into its coordinate components; (3) the algebraic sum of each set of components is determined, thereby obtaining the resultant components; and (4) the resultant components may be combined into one single force, using the parallelogram law or the three-dimensional extension of it. A re-examination of a previous example serves to illustrate this procedure (see Fig. 2-3). Figure 2-18 shows the elastic and headgear forces referenced to a chosen framework in a buccal-plane view, the given forces decomposed into components along the reference axes, the resultant components,

48


0

0

m

d

192g

365 g

55 9

200 g

400 9

163 g

0

Rd = 400 sin 66° — 200 sin 74°

=365—192=173g = 400 cos 66°+ 200 cos 74°

Rd

d i

=163+55=218g

//1 I

R

I

= '11732 + 2182 = 278 g

FIGURE 2-18. The resultant of several concurrent forces obtained through the use of force components. A two-dimensional example.

and the single-force resultant together with the computations providing

the force characteristics sought. The resultant components as computed suggest that the displacement should have a strong, occlusally directed aspect, but some potential for distal movement also exists. The resultant force does not pass through the center of resistance of the tooth in this two-dimensional analysis. The procedure for determining the resultant of a general force system, consisting of both point forces and couples exerted on a rigid body, is as follows. First, when the entire active force system is under investigation, a reference point is chosen that often will be the center of resistance of the body if its location is known. A coordinate framework is

established, centered at the reference point, with coordinate axes o* ented according to the force-system layout. Second, each concentrated force is transferred from its given line of action to a parallel line passing through the reference point. Displacing the force in this manner requires

the addition of a couple to the force system. Following the approach shown in Figure 2-17, the magnitude of the couple added equals the product of the force magnitude and the perpendicular distance between the new and given lines of action of the force. The sense of the couple is that of the moment of the original force about the reference point. In a

three-dimensional problem, the couple will be represented in vector fashion with its direction perpendicular to the plane of the given and new force vectors. The given force system has now been transformed into a set of concurrent point forces at the chosen reference point and a set of couples (including those given and those created in "moving" forces).

49


Third, the forces are combined and the set reduced to a single concentrated force, using the parallelogram or triangle law or following the process outline with Figure 2-18 in which each force is decomposed into coordinate components. The couples, properly represented by free vectors, can all be gathered at any reference point and reduced to one couple. The choice of reduction procedures is the same as that for reducing a system of concurrent force vectors. The given force system now has been replaced by an equivalent system consisting of one point force, or

its two or three rectangular coordinate components, at the reference point and a single couple (or its two or three components). Fourth and finally, to the extent possible or desirable, the force and couple may be

combined; in this part of the process, the direct association with the ref eMnce point is lost (unless, as determined, no couple portion of the resultant exists as associated with the point). In the special circumstance when the force and couple vectors are perpendicular to one another, this pair may be reduced further to a single force. In the plane problem, if

both force and couple, as found with respect to the chosen reference point, are nonzero, these vectors are always mutually perpendicular and, therefore, may be combined to yield the single-force resultant. A two-dimensional resultant problem, using the procedure just outlined, is presented in Figure 2-19. The rectangular plate is viewed from above, initially resting on a frictionless surface. The three forces and the couple are then applied, and the resultant is sought toward description of the displacement produced by the loading. If the plate is homogeneous (its weight uniformly distributed throughout its volume), point 0 at

the geometric center is the center of resistance. By the choice of the 300 g

8mm

2mm

4—

8mm

—

lOOg

200g

L

X

= 300(4)) + 100(0)

= 300— 100 = 200 g

+ 200(4)) + 960)

= —200 g 01 200 g

R=V2002+2002=2800 200 1

°x =

=560g-mm) M0

—h—560 == 2mm

FIGURE 2-19. The resultant of a general two-dimensional force system acting on a solid body.

50


orientation of the x-y-coordinate framework, no decomposition of any of

the given forces is necessary in this example. Each force is then moved to point 0; the line of action of one of the given forces passes through 0, so the creation of the concurrent force system there requires the introduction of two additional couples. The resultant force components are determined, then the force portion of the resultant at point 0. The three couples may be combined in algebraic fashion, since they occupy a common plane. Note that the couple portion of the resultant has characteristics identical to those of the sum of the moments of the given forces with respect to point 0; this is always the case, whether the problem is twoor three-dimensional (with vector addition of moments necessary in the spatial situation). The motion resulting from the application of the force system may now be described: point 0 moves in the direction and sense of the 280-g force with accompanying counterclockwise rotation of the plate. The final sketch shows the single-force resultant following combination of the force and couple portions of the resultant (see the procedure shown in Fig. 2-16); the line of action of the single force is fixed and is totally unrelated to the initial choice of point 0 as the reference point for the solution. The typical resultant problems in orthodontic mechanics involve a

single tooth or a dental segment. Because the active force system is ordinarily applied to the labial or lingual crown surface(s), the problem is generally three-dimensional. Although seemingly a somewhat indirect and lengthy process, from the standpoint of ease of viewing, the elimination of the need to express couples in the vector format, and the definitions of orthodontic displacements, it is usually convenient to decom-

pose the actual spatial situation into two or three two-dimensional portions. Employing as the reference the center of resistance as seen in each dental coordinate plane needed, in each plane the force-system components are reduced to force and couple resultant components. If

then desired, as a final step in the solution, the components may be combined and the resultant expressed in the three-dimensional format.

Distributed Forces and Their Resultants Earlier, contact forces were noted as being associated with surface areas

of the bodies acted upon. If such forces directly affect only a small portion of the total surface area of the body, in modeling, that small area is

collapsed around a surface point and a concentrated force is exerted there. On the other hand, when a force system is spread over a relatively sizable portion of the surface area in a continuous manner, the system is termed a distributed force. A resultant, a point force and/or a couple, may

be determined to replace the distributed force system for purposes of examination of whole-body displacement tendencies. When considering associated deformations, however, it is necessary to differentiate among forces according to the sizes of contact areas involved.

51


Two familiar examples of distributed force are that exerted by a load of grain or coal against the floor and sides of a railroad car and the force of water pushing against a dam. In the analysis of any distributed force, the pattern of distribution must be established and, to this end, the total contact area is partitioned into small portions. The amount of the total

force associated with each small part of the subdivided area is determined; for each partial area the fraction of the total force divided by that

partial area is the intensity of force there (measured, for example, in pounds per square inch). Generally, the intensity of force varies from one location to another over the total area covered by the distribution. In the railroad-car example, the distributed force arises from the weight of the contents; the depth of the grain at any specific location measured vertically from the car floor is proportional to the local intensity of contact force against the floor. Water pressure increases directly with depth,

and the pressure at a point or level beneath the water surface is the intensity of force there. If the car floor and the dam-water interface are plane surfaces, the forces against each partial area are parallel to one another. In problems involving curved surfaces, the distributions may be decomposed into components yielding two or three systems of parallel, distributed forces. The number of parallel forces in a specific force distribution equals the number of parts into which the contact area is divided. The larger the number of partial areas, the better the approximation by a set of discrete forces of a continuous distribution. If the intensity of force at any location can be expressed mathematically in terms of the coordinates of the location within the contact area, an exact representation of the distribution may be available. Two examples of distributed force occurring with the activation of an orthodontic appliance are the "loading" of the periodontal ligament

by a tooth root and the force system generated within the adhesive when an engaged arch wire transfers contact force through a bonded bracket pad (or when a masticatory force is unintentionally exerted on the bracket). Distributed forces can be external or internal to a body under study; in an analysis of the railroad-car floor, the force of the grain

or coal is a distributed, external load, and every activated arch wire "contains" distributed, internal force. The dimensions of force intensity are force divided by length squared and the intensity of internal force is known as stress. Another vector quantity, stress is discussed in detail in Chapter 3. The systems of distributed force of orthodontic interest may be represented or modeled by one or more sets of parallel forces, with all forces in a set having the same sense. The resultant of a set of such forces is a single force having the direction and sense of the set, with its line of action passing through the center of the distribution (generally not the center of the associated contact area). Although the problem is inherently three-dimensional, many force distributions vary in only one direction parallel to the contact area. In such problems, the two-dimensional sketch of interest wifi contain directions referencing the magnitude of the force intensity from point to point and the associated

52


coordinate locating points of the contact area; the area itself will appear

only in edge view. In Figure 2-20, two particular distributions for which the determination of resultants is straightforward are displayed with their resultants. The distributions are assumed uniform in the direction perpendicular to the page and, for simplicity, the areas are assumed to be rectangular with the dimension not seen (perpendicular to the sketch) symbolized by w. The independent coordinate x, extending from zero to L, locates points along the edge of the area and, generally, the intensity (at any point represented by the height of the diagram) may be expressed mathematically as a function of the coordinate x. For any distribution of parallel forces, all having the same sense, the resultant magnitude equals the average intensity multiplied by the area covered by the distribution. For the uniformly distributed loading on the left in the figure, the intensity, which is the same everywhere, is clearly the average intensity. For the linearly increasing, triangular distribution, the average intensity is onehalf of the sum of the minimum (zero) and maximum intensities. Note that the resultant force may be computed as the area, within the boundary of the distribution seen in the figure, multiplied by the width w of the rectangular area covered by the distributed force system. In a dimensional analysis, force/(length2) multiplied by length gives force/length for the "area" of the sketched diagram, and this multiplied by another p

h

L

(L,p0)

(L,h0)

x

— — — — ——1

\ \\ P = h0Lw

4

.1

L

L

L

-

J

3

Lw

\\

\\

FIGURE 2-20. The resultants of uniform and linearly-varying, parallel, distributed forces.

53


length (perpendicular to the plane of the sketch) yields force. Also note

the locations of the resultant forces; their lines of action pierce the cen-

ters of the rectangular and triangular "areas" shown in the figure. Again, this development is restricted to distributions over rectangular areas where the intensifies vary only in one direction along one edge of an area. Do not confuse the two areas that have been mentioned: the rectangular contact area over which the force is distributed and the area within the distribution-of-force sketch.

Friction Two materials touching one another share a contact area. The resistance

to movement tangent to this area, of one material relative to the other, is known as friction. Friction may be generated between dry solid surfaces; the resistance to displacement may be lessened if a lubricant is placed between the contacting surfaces. Friction may exist between two solid surfaces, at a solid-fluid interface, or between fluid layers. Friction be-

tween solids is termed rolling or sliding, depending upon the form of relative movement attempted; rolling resistance is highly dependent on the amount of localized deformation where contact occurs. The resistance that precludes actual motion is termed static friction; that which exists during motion is called dynamic friction. Both the static and dynamics forms of sliding friction are of orthodontic interest. Sliding friction is

generated between arch wire and bracket when the wire "guides" the bracket during mesiodistal movement of an individual tooth or when the arch wire is slipped through posterior crown attachments in, for example, the retraction of an anterior dental segment. A classical model of frictional analysis is depicted in Figure 2-21. Before the active force P is applied, to produce intended horizontal movement to the left, the fixed surface on which the block rests responds only to the weight of the block with an upward force, perpendicular to the plane contact area. This force, often symbolized by N (representing "normal") is actually the resultant of a distributed force system

against the bottom, rectangular face of the block and, if the block is homogeneous, initially the distribution is uniform over the contact area. With the application of the force P at a relatively low magnitude, the

responsive distribution of the force against the bottom surface of the block becomes nonuniform and the magnitude, direction, and point of application of the resultant of the distribution are altered. With P sufficiently small, however, the block remains stationary and the changes in characteristics of the responsive force from the horizontal surface take place because the contacting surfaces are not perfectly smooth. (If absolutely no friction existed, the application of P, however small in magni-

tude, would result in motion with no change from the static (P = 0) configuration in the responsive surface force.) With the application of the force

the now-inclined, responsive, resultant force R may be de-

54


Wi

P

f

p ,-p 777'?

N

FIGURE 2-21. Active and responsive forces exerted on a block supported by a rough horizontal surface.

composed into components perpendicular (normal) and parallel (tan-

gential) to the contact area. Since no potential for vertical motion exists, the normal component is N and the tangential component, in the direction of but opposite in sense to the intended displacement, is the resistive force of friction. Increasing the magnitude of P results in a conesponding increase in the magnitude of the frictional force f, but motion does not occur until the force of friction reaches a maximum value corre-

sponding to a critical magnitude of the applied force P. A slight, additional increase in P beyond this critical size results in motion, and the force of friction generally decreases slightly from the "maximum 1' as the static situation becomes dynamic. A plot of the frictional force versus the applied force P for the model described is displayed in Figure 2-22. Controlled experiments in which frictional forces between plane surfaces were measured have determined that the maximum f and the level

of the frictional force following the initiation of motion are highly dependent on the relative roughness of the contacting surfaces and thtmagnitude of the normal force component N. In the classical model, fmax and the dynamic frictional force are individually assumed proportional to N;

the proportionality constants are the static and dynamic coefficients of friction, quantified empirically for pairs of contacting surfaces. Note that in the static model only the maximum frictional-force magnitude is equal

to the product of the static frictional coefficient and the normal force component N and the model does not consider the manner of distribution of contact force between the surfaces. The classical block-on-plane analysis gives a good fundamental basis

for examining the problem of friction between arch wire and bracket slot. The block and plane are analogous to the wire and slot, but while

55


Frictional Component

(Pent, fmax)

Static

Dynamic

Activating Force P FIGURE 2-22. Relationship between the magnitudes of active and frictional forces, P and f, for the block and surface of Figure 2-2 1.

only one contact area exists in the foregoing example (Fig. 2-21), the

orthodontic problem is three-dimensional in that the wire may contact the bracket slot and the accompanying ligation at as many as four locations simultaneously. The direction of the frictional force is bounded by the angulations of wire and bracket slot; the sense opposes the relative motion and depends upon whether force is exerted on the bracket or the wire (action-reaction). If the bracket is to move distally on the wire, the frictional force against the bracket is generally toward the mesial (and its action-reaction counterpart against the wire is directed distally). If the wire is to slide toward the posterior through a stationary bracket, the frictional force from the wire and against the bracket acts toward the distal (since the force of friction against the wire is directed mesially, opposing the intended motion). Oscillation between static and dynamic forms of friction occurs between the wire and slot in an activated retraction appliance as the relative movement takes place in small "steps." As viewed buccally, one or more of four separate forms of contact between bracket and arch wire may be observed during canine retraction. These four forms of contact are shown in Figure 2-23 and a pair of components, normal and frictional forces, would generally exist at each individual contact area. Influencing the normal component of contact force, and therefore the level of friction, are the angulation between arch wire and bracket and the tightness of the ligation. The wire material and that of the ligature, for a given bracket, also influence the frictional resistance. Beyond the classical model, the difference in contact areas that exists with round versus rectangular wire likely affects the magnitude of frictional resistance. For a more complete discussion of friction between bracket and wire, refer to Frank and Nikolai (1980).

56


Zero angulation, clearance

One-edge contact, clearance

f4

Two-edge contact, zero clearance

Snug wire/slot fit, zero clearance

FIGURE 2-23. Bucca! views of possible contact modes between bracket and orthodontic

arch wire.

Concepts of Mechanical Equilibrium and Their Applicability to Orthodontic Mechanics The definition of "mechanical equilibrium" of a solid body represents an extension of Newton's First Law for a particle. In effect, the practical definition says that the net tendencies for translation and rotation of the body are nil: if the magnitudes of the force and couple portions of the entire force system exerted on the body are both the body is in mechanical equilibrium. If the body was stationary before

this "balanced" force system was applied, the system adds zero net potential for whole-body motion. The balanced force system creates no

inertia, where inertia is proportional to the products of mass and the acceleration components of the body. When the total force system is unbalanced, one or the other or both portions of the resultant of that system are nonzero, an extension of Newton's Second Law governs,

57 Introduction to Analysis of Orthodontic Force

and the body undergoes accelerated motion and has been given some

inertia.

An important notion in this discussion is that of the "entire force system": all of the external forces and couples, both active and responsive,

exerted on the body under study. The definition of mechanical equilibrium may be extended to apply to groups of bodies or assemblies. Inter-

nal forces, by definition being within the body or assembly, exist in cancelling pairs and, therefore, collectively contribute nothing to a resultant computation. Internal forces, however, may become external, for

example, when an assembly is disassembled for purposes of analysis and one body or member is studied individually. Pairs of forces associated with the connection(s) of this body to others in the assembly are internal to the assembly, but one of each pair of forces is external to the one member. Under the influence of an activated orthodontic appliance, the dentilion should not be static. Over a finite period of time the teeth are to be moyed and, correspondingly, within the appliance itself, relative movements and changes in as-activated shapes may occur. The concept of the resultant of a system of forces has been discussed toward establishing a procedure by which forms of displacement caused by active forces might be predicted. Also created with the activation of the appliance, however,

are responsive force systems in need of description: the force system transmitted through the tooth and exerted by the root on the periodontal ligament, and that delivered by the appliance to the "anchorage," for example. Force analyses of the entire tooth, segment, and the appliance and its

elements or parts are to be carried out, but exact results need not be obtained. Note, then, that although appliance activation generally results in the creation of some translational and rotational inertia, in the dentition as well as in the appliance, these levels of inertia are not sizable. Both parametric parts of the inertia, the mass and the time rate of change of velocity (the acceleration), are small. Although the entire complex is more dynamic immediately following activation, during the short lime period of soft-tissue deformations, than during the much longer period of tissue remodeling, tooth displacement, and appliance deactivation, the contributions of inertia to the analysis may reasonably be neglected, even in comparison to the smallest of active forces. The pair

of governing vector equations of the orthodontic dynamics problem may, then, be written as Vector sum of all forces

inertia =

(Vector sum of all moments +jot&ti

liriertia)mass center =

0

With the inertia terms deleted, the relationships are those associated with the definition of mechanical equilibrium. The equations are necessary and sufficient conditions for a mechanically static system. Thus, at

58


any instant following activation, the dentition and appliance are said to

exist in a quasi-static state and the equations of mechanical equilibrium may be employed to obtain a close approximation to the relative magni-

tudes of the components of the active and responsive force systems. The rationale for the consideration of the mechanical equilibrium problem and its analysis is, therefore, established. The general problem may be stated as follows: Given a body or group of interconnected bodies subjected to a system of external forces and couples, such that the state produced by the system is mechanically static, determine the relationships that must hold among the characteristics of the external force system. Often the problem is one in which several of the characteristics of the force system are quantified and those characteristics that are not given explicitly are to be obtained. Nearly indispensable in proceeding toward the solution of all but the simplest of problems is the use of the free-body diagram, and several of

these diagrams may be needed to solve one problem. In the solution procedure, a body or assembly of bodies is earmarked for study and, as the name suggests, is "freed" from its surroundings (its supporting structure and all of its connections to other bodies); this is depicted in an isolated sketch. All of the external forces and couples exerted, both active and responsive, are shown in the sketch. As has been the convention in previous force diagrams, forces and couples exerted on (not by) the body are displayed. Not only the completed known portions of the force system, but also those forces and couples with unknown characteristics, must be properly depicted. Moreover, to aid in moment computa-

tions, distances and dimensions must be included. Omitting existing parts of the force system, including nonexistent forces or couples, and incorrectly identifying support structures of force-transmission mechanisms (connections) are typical mistakes made in these problems. Once the free-body diagram has been correctly completed, however, the interrelationships among the known and unknown force-system characteristics may be expressed in a rather straightforward manner. For each freebody diagram drawn, a set of scalar equations may be written, obtained from the pair of governing vector relationships written previously. In words, these equations say: All of the forces exerted on the body or group of bodies, pictured and under study, must balance, and the moments of all of the forces and coupies, with respect to any chosen moment center or axis, must also balance.

(This is an alternative way of stating that both the force and couple portions of the resultant of the entire force system must have zero mag-

nitudes.) The number of independent, scalar, algebraic relationships that may be written from each free-body diagram will depend upon the type of force system and whether the problem is one-, two-, or threedimensional.

In the quasi-static orthodontic problem, because body forces are nearly always negligible in comparison with contact forces, understand-

59


ing the manner of transfer of force through a connection is fundamen-

tally important. Connections of specific interest include those between the tooth crown and appliance element(s) and between individual parts of the orthodontic appliance. Before examining connections within the appliance, however, the displaying of responsive forces and couples having unknown characteristics and several general types of connections are considered. The complete description of a concentrated support or connection force generally requires the definition of several scalar quantities. The magnItude of a force or couple is the most common unknown in the equilibrium problem. To define the direction of the line of action of a force, one scalar quantity, the size of an angle relative to a reference line, is needed in the two-dimensional problem; two angles are required in the spatial problem. Although nearly always known in practical problems of orthodontic interest, the point of application of a concentrated force is described through the dimensions of the body or in terms of the coordinates of the point in relation to a specified reference framework. In approaching an equilibrium problem in which the vector characteristics of a force are sought, with the point of application of the force known the unknowns may be expressed as the magnitude and one or two angles or as the two or three components of the force referenced to an established set of coordinates. Figure 2-24 illustrates the two methods of displaying an unknown force at a point in two dimensions. On the left, the force is completely described by the magnitude, the one angle, and a convention to define sense of the vector; that convention could be an understanding that the force would appear to be pulling on the point

y

x

F

n

-__,,_,, , Scalar unknowns: F, 6

Scalar unknowns:

FIGURE 2-24. Two methods of illustrating a concentrated force in two dimensions, unknown except for point of application.

60


and the angle would be measured counterclockwise from the positive

extent of the x-axis, the angle having a value between 0 and 360°. On the right, the force is completely described in terms of its x- and y-components; two algebraic quantities provide the magnitude and sense of each component. Because the sign convention leading to the determination of sense is more straightforward in the component description of a force, expressing the unknowns as the components themselves is, seemingly, the more convenient procedure in an analytic approach to the equilibrium problem. To clarify, then, although sense often is noted as an individual force characteristic, sense is never a separate unknown in the equilibrium problem. A force at a point is totally described in terms of its components. In

picturing an unknown force in a free-body diagram, an assumption of sense may be made for each component; in solving an algebraic equation

for a component magnitude, the sign accompanying the answer will indicate whether the assumed sense was correct (plus sign) or incorrect (minus sign). Shown in Table 2-1 are several types of connections between bodies, the associated force systems exerted on each body (action and reaction), and the number of scalar quantities involved, exclusive of location of the connection. The first is, effectively, one-dimensional (although it may exist within a two- or three-dimensional problem), the next three are two-dimensional, and the final two are three-dimensional. The first two entries in the table indicate push-contact between two bodies that share a contact area. In the general two-dimensional analysis, the resultant of the distributed force system over the contact area would be displayed as two force components, one perpendicular to and the other tangent to the area at the center of the distribution. When it is reasonable to assume that friction between the bodies is negligible, the force of one body upon the other is perpendicular to the contact surface and, possibly, only one characteristic is an unknown: the force magnitude. The tangential component is the frictional force discussed previously, and it generally adds a second unknown at the contact site (except when proportional to the normal component, and then the sense of the frictional force must be correctly indicated). The third entry in the table, depicting a frictionless hinge connection (with the pin through the two members perpendicular to the plane of the sketch), generally transmits a force having unknown magnitude and direction. The two scalar quantities may be represented as the magnitudes of the components of the force; the components must be mutually

perpendicular, but the pair orientation as a whole may be arbitrarily selected (often most convenient with one component coinciding in direction with the long dimension of one of the interconnected members).

Shown as the fourth entry is the fixed support. Whereas the hinged connection possesses no inherent resistance to the angular movement of one body with respect to the other in the plane of the sketch and in the absence of friction, the fixed support or connection provides that resist-

.

61

Introduction to Analysis of Orthodontic Force TABLE 2-1. Connections between members and supports and associated force systems Force system

Description of contact or connection

\\

Action

Bar against fricfiorrless

from

surface

f

Hr

Reactive composaNs of bar against surface

Force components exerted by surface

Bar against rough

:::e

Action and reaction counterpart forces perpendicular to plane of contact. (One componeat magnitude is Unknown.)

Force of bar against surface

Notmal force

\ \ surtace

Remarks

Ft eaction

N2

t

are unknown.)

N2

Components exerted by pin

Bar restrained by frictiortless pin connection

/ p3

Force components from bar against P3I pin

Direction of resultant contact force generalty not longitudinal. (Two component magnitudes are unknown

Force system exerted by support on bar

Couple represents rasistance to rotation inherent in support (Action and reaction interchanged compared to toregoing examples. (Three component magnitudes are unknown.)

—

Action of bar against support

Fixed support of bar in plane setting

J

H4

3

/

I

C4

Frictronless support of shaft in

-

Fixed support of bar

1 C)

C) P)

C) C) P)

Indicated is potential for bar to slide to the right. Friction opposes this tendency. (Two component magnitudes

U

No resistance to longitudinat shaft displacement assumed. (Four component magnifudea are unknown)

Complete resistance to linear and angular dinplacements in all directions. (Six component magnitudes are unknown.)

ance, represented by the couple. In the plane problem only one unknown, its magnitude, is associated with such a couple; the sense of the couple (clockwise or counterclockwise) is obtained through a sign convention in the manner of that of an unknown force component. A number of three-dimensional connections are relatively common within structures and machines, encompassing from one to six potential unknowns, but just two of interest are displayed in Table 2-1. For the bearing-shaft connection, negligible resistance to shaft translation through and rotation within the bearing is assumed. This leaves a force vector and a couple vector, both perpendicular to the shaft axis, generally transmitted through the connection. Expressing each vector in com-

62


ponent form, a total of four potential unknowns is present. The force is

associated with the positional resistance of one• body relative to the other, and often equilibrium can exist in the absence of the couple; in such instances the couple is regarded as a secondary effect in primarily giving the connection added stability. The final entry in the table is the totally fixed, rigid connection. This connection, and a weld is a good example, prohibits both translation and rotation of one body with respect to the other in any direction. The force and couple are expressed as sets of three components each, giving a total of six potential unknowns. Before proceeding to a discussion of connections associated with the orthodontic appliance, a straightforward and applicable statics problem is examined. Consider an elastic that is stretched between two connection points in the orthodontic complex. The elastic at either end might be

activated against a hook attached to a bracket, a hook soldered or welded to the arch wire, or a loop bent into the wire, but assume that the elastic is stretched between just two connection locations. A reasonable assumption, based on the flexibility of the elastic, is that a single force is transmitted through the elastic and, therefore, through each connectioti point. Shown in Figure 2-25 are seemingly possible free-body diagrams of the elastic, the first with both forces of apparently unequal magnitudes drawn in arbitrary directions. If, however, the elastic is in mechanical equilibrium, and its weight is comparatively negligible, the moments of the two forces must balance. Choose first a moment center at point A;

with the line of action of P passing through A its moment arm, and therefore its moment, is zero. Hence, the moment of Q with respect to A

//

'4,

0

0

B

A

B

B

A

A

'P P

I

I,

I

t

FIGURE 2-25. Preparation of a free-body diagram for a static elastic stretched between two fixed points.

63


also be zero, and thus the line of action of Q must also pass through point A. The direction of Q is then determined as shown in the second diagram of Figure 2-25. Now, making a similar argument using point B as the moment center, the direction of the line of action of I? is determined; the two forces, accordingly, must share a common line of action as shown in the third sketch. Finally, because the entire system of forces must balance in any static situation, the forces P and Q must be equal in magnitude and opposite in sense. Their common line of action is determined by the positions of the two connection points, which both lie on the line. Hence, for any fwo-force member in equilibrium, the two forces must be equal in magnitude, opposite in sense, and share one line of action. Generally, the forces may be either pulling away from or pushmust

.ing toward one another, but an elastic has no resistance to pushing forces and, therefore, when activated it must be "in tension." Now, with the background of Table 2-1 and the force analysis of a stretched elastic, a number of connections associated with orthodontic mechanics are analyzed in Table 2-2. Although all are shown in a twodimensional format, with several a third dimension may be necessary to represent the connection force system completely. In the first entry just one potential unknown is present, the magnitude of P1. provided the orientation (angulation) of the stretched elastic has been or can be measured. Because the bracket in the second example has finite mesiodistal width, the couple, due to angulation of the wire within the bracket, as well as the force may be generated. This is somewhat analogous to the fixed support in two dimensions shown in Table 2-1. Friction, not included in this sketch but potentially present, would add a third scalar quantity. Examining this connection in an occlusal-plane view, assuming the presence of ligation, another force-couple pair must be present, and with friction neglected and round wire engaged, this connection is analogous to that of the bearing-and-shaft connection of Table 2-1. The third entry in Table 2-2 assumes that the rectangular wire will not rotate within the bracket slot; the couple shown is torsional in nature. Also generally present is an accompanying force transverse to the wire, and a total of three potential scalar unknowns in this connection not including friction. The fourth entry presents a somewhat simplified representation with friction included; analogous to the second entry in Table 2-1, this is just one of the possible configurations, in a view from the buccal perspective, already displayed. (See Fig. 2-23.) The fifth entry in Table 2-2 shows the resistance to rotation and translation provided by

a buccal tube. The component H5 may arise from a step bend in the engaged arch wire, a stop, and/or from friction. (If the arch wire was tied back, H5 would have the opposite sense.) Two additional components, a buccolingual force and another antirotation couple, would be seen in an occlusal view, reflecting a total of five potential unknowns associated with this connection. Finally, the connection of a J-hook to an arch wire has little; if any, inherent resistance to rotation. This is the three-dimensional extension of the pin connection with three associated potential unknown force components, two of them existing in the buccal-plane

64

Bioengineering Analysis of Orthodontic Mechanics TABLE 2-2. Orthodontic appliance interconnections and contact force systems exerted on the appliance members Force system

Connected members

I-look affixed to buccal tube and elastic

Remarks

Reaction

Action

Force trom elastic against hook

Hook force against elastic

Direct tension Lines of action along the stretched elastic. 1

Force system exerted Ofl

Bracket and arch

bracket

V2

wire

Bracket force system against wire

I

Second-order angulation and contact creates couple

V2

Bracket and rectangular arch

Force system from wire

Force system from bracket Third-order angulation and contact creates tingual root torque.

vof

wire C3

H3

Bracket and arch wire

Force components exerted by wire N,

Bracket-generated force components

Friction opposes relative displacement of wire and bracket, No second-order contact assumed.

f4

Stopped arch wire and buccal tube

Generated by stopped arch wire

Reaction of tube against wire

Slop contact with mesial extent of tube produces horizontal force

S

component. H5

v5

J-hook and hook soldered H6 H6

¶

Representative of highpull-headgear force delivery to maxillary anterior segment of arch wire

view. Note that the most general and complex connection, analogous to the final entry in Table 2-1, can be generated between the ligated bracket and a rectangular arch wire: all three force components (the mésiodistal component from a stop or friction) and all three couple components may exist for a total of six potential scalar unknown quantities. The two vector equations governing the equilibrium problem, which indicated that both the force and couple portions of the resultant of the entire force system must be zero, suggest that a maximum of six scalar relationships may be written among the characteristics of the force system properly displayed in the free-body diagram. Although the majority

of the quasi-static problems to be examined are three-dimensional,

65


many of these problems may be modeled into two dimensions. Also,

those problems that must be treated in three dimensions may be decom-

posed into several interrelated two-dimensional analyses and approached in parts, each in its own plane. As noted earlier, a relationship exists between the maximum number of independent scalar equations available, relating characteristics of the force system, and the dimensional level and format of that system. Several examples illustrating this relationship and the solution procedures follow. In Figure 2-3, two forces were shown acting on a bracket. Returning to that problem: the equilibrating force exerted on the bracket through its attachment to the band or its bond to the facial crown surface is to be determined. To be pictured in a free-body diagram, then, is the bracket itself and the forces exerted on it. The three existing forces are: the elastic force, the headgear force, and the unknown force. Ignoring the possibility that the two known forces may be slightly out of plane, relative to each other, and neglecting the bracket thickness, the problem then becomes two-dimensional. Moreover, the force system is concurrent with the lines of action of all three forces passing through the bracket center. The free-body diagram is shown in Figure 2-26. All characteristics of the elastic and headgear forces are known. The point of application of

Bg

EL = 200 g

= 400 g

+

Rm =

Rg = B8

97:O=_147O3O0S1fl 66° = 0

400 COS 66° — 200 COS 74° = 0

B8 = 163 = 55 = 218 g F8

+ 2182

278 g

FIGURE 2-26. Free-body diagram of an orthodontic bracket subjected to a twodimensional set of concurrent forces.

66

Bioengineeririg Analysis of Orthodontic Mechanics

the third force is known, but its magnitude and direction are not. There-

fore, two unknown scalar quantities are unknown and are pictured as mesially- and gingivally-directed components in the facial view. In the problem which is two-dimensional with the forces concurrent, only two independent scalar relationships may be written: the algebraic sum of force components in two mutually perpendicular directions in the plane must be zero to ensure a force balance. The solutions for the components of the bracket force and, subsequently, the resultant of those components are carried out in the figure. The solution yielding algebraically positive results indicates correct initial assumptions of the senses of the components. No moment-balance equation is written. (Clearly, a moment equation using the bracket center as the moment center is identically satisfied since the moment arms of all three forces are zero; hence,

no useful information is obtained from the moment-balance relationship. Another moment center might be chosen, and a moment balance written with respect to it, but the equation obtained would only be useful in replacing one of the two scalar force relationships.) The force just obtained is noted to be identical in all characteristics, except sense, to the

resultant of the elastic and headgear forces found in Figure 2-3; in all equilibrium problems involving a concurrent force system, the resultant of the given forces must be balanced by the force(s) that involve the unknowns. The lever is a second example that has applicability to orthodontic appliance analysis. Figure 2-27 shows a straight member that could represent an arch-wire segment balanced on a "fulcrum" by the forces P1 and P2 at the ends of the member. The fulcrum force and the relationship between the forces P1 and P2 to ensure equilibrium are sought. It is

P1

Pl

C

C

.

L1

L1

P0 x

= P0 — Pi — P2 = 0

P0 =

+

McP2L2)+P1L1)0

-

P2L2 = P1L1

FIGURE 2-27. A lever subjected to a two-dimensional parallel force system.

—

67


assumed that the applied forces exist in a common plane and are both

perpendicular to the long dimension of the member. Hence, the problem is two-dimensional and the free-body diagram reflects a parallel force system because neither of the applied forces has a component in the direction of the long dimension of the member. The responsive (fulcrum) force likewise can have no long-dimension component and reflect a balanced system. The magnitude of the fulcrum force is obtained by balancing the forces, algebraically summing the forces in the one direction of all three lines of action. The relationship between P1 and P2 is obtained through a moment balance using any convenient moment center; in this example, point C, which is immediately above the fulcrum, is chosen.

In general, for a two-dimensional parallel force system, two independent scalar equations of equilibrium may be written. One force and one moment equation were used to analyze the lever example. The general vector moment-balance equation previously described indicates that the center of mass of the body or assembly under study is to be used as the moment center. With the inertia effects absent or disregarded, however, it may be demonstrated that a force balance and a moment balance about the mass center demand that moments balance with respect to any selected moment center. It follows that the two scalar equilibrium equations for the problem of Figure 2-27 could have been two moment equations, and those equations would be independent of one another so long as the line through the two moment centers is not parallel to the

direction common to the three forces. Figure 2-28 shows a straight "beam" subjected to a transverse concentrated "load" of 300 g. The problem of determining the responses at the supports is an inversion of that of Figure 2-27; sketching the free-body diagram and using the equa-

300 g

I—

Pt'

6mm

II

P4

FIGURE 2-28. A structural member supported at its ends and subjected to a transverse, concentrated load. The correct free-body diagram shows a parallel force system in one plane.

68


lions of Figure 2-27 with P0 = 300 g, the magnitudes of the left and right

support may be shown to equal 100 g and 200 g, respectively. In the final example of this section, a cantilevered straight member is subjected to a loading consisting of forces at its right end, parallel and perpendicular to its long dimension, and a couple, all in a common plane. The left end of the member is engaged in a fixed support, and the components of the support force system are sought. The free-body diagram of the member includes the support resistances to any movement of its left end in the plane of the sketch. This is a general two-dimensional equilibrium problem and a total of three independent scalar equations may be written to relate characteristics of the total force system. An analysis is undertaken in Figure 2-29. A choice of sets of equations is

available: two force and one moment equations, one force and two 80g

I

I I I

lB

A

100 g

I 2100 g-mm

35mm

80 g CA

B

A

lOOg 2100 g-mm

x

— 100 = 0 = = 100 g

bOg M8 —

+ 2100

+ CA

=0

CA = 2100 —2800 = —700 g-mm 700 g-mm

FIGURE 2-29. A cantilevered structural member loaded at its "free" end. A twodimensional system of active and responsive forces and couples.

69


moment equations (but the line connecting the two moment centers may

not be perpendicular to the direction of the force-component balance), or three moment equations (but the three moment centers may not lie on one straight line). The free-body diagram in Figure 2-29 contains three unknown forcesystem characteristics: the magnitudes of the two force components and that of the couple. The reference frame adjacent to the free-body diagram defines the chosen directions and assumed positive senses for the force-balance relationships written below the sketches. Note that the assumed senses associated with the unknown magnitudes need not coincide with the sign conventions for the equations. In the solution, note that a negative result was obtained for the support couple; this indicates that the assumed sense designated in the free-body diagram was incorrect. Also, recalling that the moment of a couple is inherent, note that its contribution to a moment balance in a two-dimensional analysis is the same, regardless of the moment-center location. The couple portion of the load on the member was included in the moment-balance relationship written, even though the moment center chosen at the right end of the member eliminated contributions from the other applied forces (because their moment arms with respect to point B are zef o) and the couple and moment center appear to be coincident. A partial check of the solution obtained in the figure may be pursued by writing a moment-

balance equation using point A at the left end of the member as the moment center.

Synopsis This chapter introduced the concept of force. Various force manipula-

tions were discussed, including the determinations of the moment of a force and the resultant of a system of forces. The couple, a special force system, was defined and its use as a tool in force analysis, as well as its occurrence as a load and a mechanical response, was considered. Modeling of actual force systems and the bodies acted upon, toward simplification of analyses, were discussed. The resultant of an active force system is sought in order to describe the whole-body displacement that may be produced by the system. A rationale was developed for analyzing the quasi-static problem, applicable to the orthodontic appliance, slowly deactivating between appointments, using the governing equadons of mechanical equilibrium. Force systems produced by activated orthodontic appliances provide the catalyst for the processes that result in the displacements of teeth. Forces are also generated in orthopedic actions toward alterations of growth patterns and to cause movements of dentofacial bones. Hence, a first step in evaluating the potential of a given appliance, with regard to its tendency to produce desired displacements in a controlled manner,


or

in the design of a new appliance, is an examination of the forces

created upon activation that are associated with the actual or proposed appliance. This, however, is only the beginning of the total structural analysis. Thus far, deformations have not been considered, nor has the interface between the biologic and mechanical systems. To initiate the action of the orthodontic mechanics, the appliance must be activated; one or more appliance elements must be deformed from their passive configurations, thereby "loading" the appliance and inducing forces throughout it. Because only a part of the total appliance force system can be quantified through direct measurement and the use of equilibrium

equations, relationships between force and deformation must be sought. Such relationships involve the material characteristics of the appliance members as well as their sizes and shapes and will be discussed in the next chapter.

Reference Frank, C.A., and Nikolai, R.J.: A comparative study of frictional resistances be-

tween orthodontic bracket and arch wire. Am. J. Orthod., 78:593-609, 1980.

Suggested Readings F,P., and Johnston, E.R., Jr.: Vector Mechanics for Engineers. 3rd Ed. New York, McGraw-Hill, 1977, Chapters 1 to 4, 6, and 8.

Beer,

Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 1. McLean, W.G., and Nelson, E.W.: Engineering Mechanics: Statics and Dynamics. New York, Schaum Publishing, 1962, Chapters 2, 3, 5, and 8.

Mulligan, T.F.: Common Sense Mechanics. Phoenix, CSM, 1982, Chapters 2 and 3. Smith, R.J., and Burstone, C.J.: Mechanics of tooth movement. Am. J. Orthod., 85:294—307, 1984.

Thurow, R.C.: Atlas of Orthodontic Principles. St. Louis, C.V. Mosby, 1970, Chapter 4.

Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, CV. Mosby, 1982, Chapter 2.

Material Behavior of the Orthodontic Appliance

In planning the treatment of a malocclusion, the orthodontist selects a

series of appliances for sequential use in the correction. Each appliance is a kind of structure. Activated through a deformation from its passive, as-prepared configuration, the appliance attempts to return to that passive state, but is prevented, temporarily, by the resistance of the dentilion to which it is affixed. The appliance responds to the activation by exerting a system of forces against those teeth and portions of the dentofacial complex that hold it in an activated state. Creation of contact forces between the appliance and dentition is accompanied by force sys-

tems within the individual members of the appliance, transmitted through connections from one member to another. The responses of the

appliance members to orthodontic "loading" are fundamentally dependent on the materials of those members. The orthodontist uses a variety of materials: metallic-alloy arch wires, bands, brackets, and headgear components, polymeric "elastics" (bands, modules, 0-rings, chains, threads), adhesives, alginates, acrylics, plasters, among others. It is important, then, that the clinician be acquainted with the basic concepts of materials science as applicable to the specialty. When designing and preparing each arch wire, auxiliary, or retainer, the orthodontist must be able to predict the response of each appliance member to "in-service" conditions. The materials are subjected not only to mechanical actions associated with the induced force

systems, but the environment in which the materials must exist also influences their behavior. The basis for explaining or predicting the response of a structural or machine member under specified conditions of use is found in its properties. A property is, in a general sense, a descriptive quantity that gives a specific characteristic to the member. Properties may be categorized in many ways, but initially they should be divided into those associated directly with the material or materials of the member and those termed 71

72


properties, which are descriptive of its size, shape, and appearance. Material properties are further subdivided into characteristics that are independent of external influences, simply termed "material" properties, and those that are associated in some way with the conditions of use or the use environment: mechanical, chemical, thermal, and magnetic, for example. Consider, for instance, the decision of selecting a spring to produce a particular orthodontic displacement. Properties that enter into this design decision include the cross-sectional wire size, the coil diameter, and the overall passive length of the spring (physical characteristics), the stiffness and strength of the wire material (mechanical) and the spring itself (structural), the resistance of the material to corrosion (chemical), and the effects of a somewhat elevated and varying temperature (thermal). The materials of the members of the orthodontic appliance are the focus of this chapter. Behavior of ancillary materials, such as plasters, will not be discussed here. The responses of biologic materials during orthodontic treatment will not be overlooked, but are relegated to a subphysical

sequent chapter. Physical, mechanical, chemical, and thermal are the property categories of primary interest to the orthodontist; optical, electrical,

and magnetic properties, although not to be generally disregarded, are of little importance within the scope of this text and will not be discussed in this chapter. Properties may be interdependent and, as previously mentioned, are often influenced by conditions of use of the material and in-service environment. The characteristics of a stationary body, directly associated with its response to applied or induced forces, which include both material and physical contributions, are known as structural properties. The response to activation of an orthodontic wire loop will depend upon the wire material, the wire cross-section, the loop geometry, and the amounts and directions of the bends made in forming the loop. The behavior of an elastic module used intraorally will differ in some respects from the results of an evaluation of its mechanical characteristics on the bench in the laboratory under ordinary conditions of temperature and humidity or its response as part of an extraoral appliance. Material-property values are generally obtained in controlled experimentation. In-service conditions, as necessary, are created in the laboratory and the required measurements are taken. When widespread use of the results is expected, and to ensure reliability of comparisons, testing procedures must be standardized. Much of the standardization in the United States in testing, particularly in the determination of mechanical properties, has been and is established by the American Society for Testing Materials. Property quantification and certification of testing procedures in the dental fields is carried out under the auspices of the American Dental Association. In the sections to follow, the nature and structure of solid materials are surveyed and the interaction between load and deformation is examined and pertinent structural characteristics are described. The concepts of stress and strain are introduced, mechanical and structural properties

73


obtained through standardized testing are defined and discussed as needed for developments in subsequent chapters, and chemical and as

thermal characteristics of interest are considered. The chapter concludes with a set of guidelines to be used when selecting orthodontic-appliance materials.

Internal Structure of a Solid Material Any fluid or solid material is a conglomerate of particles. Forces, generally attractive, exist between pairs of particles, and their magnitudes are determined by particle densities and distances between particle centers. The forces of attraction in a fluid are relatively small and, as a result, the

material readily conforms to its container. The particles in a solid are dense and relatively closely packed; the associated cohesive forces are large enough that, generally, the solid retains shape in the absence of any containment. The smallest subdivided unit of matter, which is representative of a pure material (chemical element), is the atom. Submicroscopic in size, the atom is envisioned as a nucleus surrounded by electrons in a kind of planetary arrangement. The nucleus contains protons and neutrons; these subparticles have approximately identical masses, the proton possesses a small, positive, electrical charge, and the neutron is electrically neutral. The mass of an electron is approximately 1/2000th that of the proton, yet it possesses a negative charge equal in magnitude, but opposite in sense (sign) to that of the proton. The atomic number of an element equals the number of protons in its nucleus. The atomic weight is approxi-

mately that of the nucleus, neglecting the small contribution from the electrons, a figure that numerically is, in essence, the sum of the protons and neutrons and varies with element from slightly more than 1 to in excess of 250. In the element model, the electrons exist in concentric orbits, or "shells," with the nucleus at the center. Heavier atoms may possess more electron shells than light atoms, but independent of the material element each shell—its position designated radially with respect to the nucleus—has a specific electron capacity. Except for the outside shell, which may be incomplete, all shells are totally "filled." The outer shell contains, at most, 8 electrons; elements with like numbers of electrons in their outer shells have similar chemical behaviors and the number of outer-shell electrons is particularly significant in connection with mechanical, electrical, and magnetic properties. Atoms hold together because of the intra-atomic attraction between elecIrons and protons. Overall, an atom is said to be electrically neutral if the numbers of protons and electrons are equal, regardless of the number of

electron shells and the number of electrons in the outer shell. Although possessing a net charge of zero with numbers of protons and electrons equal, the stability of an atom with regard to its potential to combine with other atoms is largely tied to the outer shell and the

74

Bioengineering Analysis of Orthodonfic Mechanics

number of "valence" electrons in it. Certain materials are inert or chemi-

cally inactive because the pure chemical elements exhibit filled outer shells together with having no net electrical charge. Many elements do not occur in their pure forms in nature because their atomic instability has resulted in chemical reaction or combination with other elements. In short, the most stable form is that achieved in a configuration with the outer shell of electrons completely filled. With the exception of the noble

gases, to attain this stabilized state an element must either effectively gain or lose from one to four electrons. Filling the outer shell results in the atom assuming a net negative charge, and the atom that loses its valence electrons assumes a positive charge; in either instance the result is a transformation from atom to ion. Ionization yields the potential for the formation of primary, interatomic bonds, of which three forms exist. Ionic bonds are formed when the

valence electrons shed by one group of atoms are transferred to, and complete the outer shells of, other atoms. Table salt is the result of sodium losing its one valence electron to chlorine, electrically neutral with

seven electrons in its outer shell. Covalent bonding is the sharing of electrons. The simplest example is the combination of two hydrogen atoms, another is the sharing among four hydrogen atoms and a carbon atom to form methane, and a third is the formation of diamond solely from carbon atoms. Metallic bonds may be formed by atoms of elements having just a few (generally one or two) valence electrons which are shed easily; formed is a structure of positive ions and "free" electrons or an "electron cloud" and between them the bonding occurs. The type of primary bond is usually indicated by the number of valence electrons in one of the elements; in a few materials, more than one form of bonding can exist. Ionic and covalent bonding results in combinations of atoms known as molecules. Although the molecule is electrically neutral, or nearly so, due to the nonuniform distribution of electrical charges, intermolecular bonds (also termed secondary or van der Waals bonds) may form.

Several types of secondary bonds are defined, but for all forms their strengths are substantially less, often an order of magnitude lower than that of the typical primary bond. Interatomic and intermolecular forms of attraction pull the particles toward one another, but spaces also exist within the particle structures, primarily because mutually repulsive forces between their electron shells become dominant when two atoms become too close to each other. Figure 3-1 depicts the forces of attraction and repulsion, both shown in dashed plots, versus the distance between the centers of a pair

of atoms. The solid curve, the net force versus interatomic spacing, crosses the horizontal axis at the point of equal attractive and repulsive forces—the so-called "equilibrium distance." This equilibrium spacing or distance between atoms varies with the material elements because of atomic size and the number of adjacent atoms in the molecular structure. Also, temperature is an important, controlling parameter. As the temperature is raised (lowered) the mean spacing between atoms increases (decreases), accounting for the thermal expansion (contraction) of materials.

75


Interatomic Force

\ Force of attraction

N ed

0

Net

/

/

Atomic Spacing

/ Force of repulsion

/

ed: Equilibrium distance

I FIGURE 3-1. Plots of interatomic force versus distance between atoms.

Several material properties are related to the bonding characteristics of atoms and molecules. Density is determined by atomic weight and spacing. Overall strength of a material depends upon bond strengths. Material stiffness is related jointly to magnitudes of bond forces and particle spacings. Thermal and electrical conductivities are low in ionicand covalent-bonded materials and high in materials that are metalli-

cally bonded wherein the free electrons serve as carriers of energy. Prop-

erties are also dependent upon the arrangement of atoms within the material, and atomic arrangements are generally classified as molecular, crystalline, or amorphous in structure. The chemical composition of a limited number of atoms forms a molecule, having a net electrical charge of zero due to primary bonding. From the characteristics of the constituent atoms, the numbers of bonds, the

bond lengths, and possible relative positions of the atoms (molecular geometries) may be determined. Often, for a molecular composition, more than one atomic arrangement is possible; such structural variations are termed isomers. Molecular materials, solids in particular, are held

together through secondary bonding. Such solids have low melting points, indicative of the relative weak intermolecular bonding, but the larger molecules have greater strengths because of the increased num-

76


bers of such bonds. Basic to the understanding of molecular structures is

the study of hydrocarbons. In a so-called "unsaturated" hydrocarbon, multiple carbon-to-carbon bonds exist that enable the formation of a large molecule from several smaller molecules in a chemical reaction known as polymerization. One form of this reaction, termed "addition polymerization," beginning with like monomers of ethylene, is illustrated in Figure 3-2. The polymeric molecule, or polymer, obtained is made up of many repeating units, called mers. A number of polymeric materials are familiar to the orthodontist: waxes, alginates (impression materials), acrylics (for retainers), adhesives (for direct or indirect bracket bonding),

and the elastomeric (rubber or rubberlike) bands, modules, "chains," and "threads." Although polymerization can develop lengthy molecular "chains," and the formation of van der Waals bonds, or "cross-linking" (vulcanization of rubber, for example) can create two- and three-dimensional molecular networks having some strength, the majority of engineering materials are made up of thrde-dimensional, repeating patterns of atoms called crystals. The primary bonding is typically metallic, but secondary

bonding can result in similar highly ordered patterns such as in the molecular crystal of table salt. The pictorial conception of a crystal is a spatial structure of straight lines with the atomic nuclei (in table salt, the sodium-ion centers) at their intersections. The three-dimensional, repeating element of the crystalline structure is known as the unit cell. Some 14 different unit-cell geometries are defined and the most common cubic crystal representations are depicted in Figure 3-3. Various material properties may be correlated with crystalline structure, and often the unit cell is indicative of the material class of the solid. Some materials, however, can exist as a solid in more than one crystal structure (polymorphism or allotropism, analogous to isomerism in polymers). A prime example is iron, which may exhibit a body- or face-centered-cubic crystal structure, depending on its temperature or the presence of alloying elements. Materials that do not display the regular, repetitive patterns found in crystalline solids and many polymers, but instead reflect a disordered

A

H

I

I

H

H

H

H

H

H

I

I

I

I

I

I

C

C

C—C————C———-—C

I

I

I

I

I

I

I

I

H

H

H

H

H

H

H

H

me

Monomer

Polymer

H: Hydrogen C: Carbon FIGURE 3-2. The ethylene monomer and a portion of the long-chain molecule resulting from addition polymerization.

77

Material B.ehavior of the Orthodontic Appliance

Ii

I

Simple cubic

Body-centered cubic

Face-centered cubic

FIGURE 3-3. Representations of simple cubic, body-centered cubic, and face-centered cubic unit cells.

molecular structure, are termed amorphous. Gases and liquids are amor-

phous ("without form") as are a number of oxides including glasses. Although such materials are not generally of specific interest to the orthodontist, some aggregates of crystalline or molecular and amorphous materials may be. Plastics are often reinforced with various "fillers." The divisions between crystals (grain boundaries) contain amorphous matter. Polycrystalline materials are, by definition, aggregates of small crys-

tals. Those polymers having highly disordered molecular structures might be categorized as aggregate materials. The internal, submicroscopic structures of many materials are highly ordered, but imperfections exist. Although often cumulatively occupying a small portion or volume of the material, certain imperfections, particularly in crystalline solids, cannot be ignored owing to their substantial

impact upon properties. Within the crystals themselves atoms may be missing from individual cells, foreign atoms may have replaced some of those of the pure material, atoms may have become displaced to "in-between" (interstitial) locations within the cells, or foreign atoms may have entered the structure and assumed interstitial positions. An extra line or plane of atoms, termed a "dislocation," may be present within the crystal. Imperfections inherently exist at external surfaces where the exposure of surface atoms makes them more vulnerable to reactions with matter within the adjacent environment. Within crystalline materials at internal or grain boundaries, amorphous matter up to ten or more atomic distances in thickness is present which generally possesses at least some relevant property values inferior to those of the individual crystals.

Certain imperfections are created to enhance property values. For example, brass is a solid solution of copper and zinc, termed "substitutional" because zinc atoms are substituted for some copper atoms in the crystal structure to produce the alloy. Steel formation involves elevating the temperature of iron to a level such that the relatively small carbon atoms can fit into the middle of the face-centered-cubic cell structure of the iron. As indicated previously, some material properties are highly sensitive to imperfections. Although the weight fraction of carbon in steel is often less than 5%, the presence of the carbon greatly improves its strength properties over those of pure iron. On the other hand, den-

78


sity and material stiffness are termed "structure insensitive" properties

because they are affected only in proportion to the imperfection level or numbers. (For example, the density of stainless steel is equal to the sum of the products of the fractional amounts, by mass, and the densities of the constituent elements.) Atoms, molecules, and crystals within a solid are retained in particular positions because of the existence of an overall, internal, force balance. The potential for relative displacement is often present, however, either in the form of some external effect or simply because the particles are being maintained in a strained configuration. Actually, atoms are continually in motion, vibrating about their equilibrium positions with amplitudes dependent upon temperature; apparently, completely static states occur only at absolute zero. If temperatures become sufficiently elevated and the particles are initially in strained positions, the vibra-

tions can result in displacements to less constrained configurations. Mechanical action can cause particle displacements, often to more constrained configurations, as can an electric or magnetic field if the particle

characteristics are such as to respond to the field. A combination of mechanical and thermal inputs can cause a displacement that cannot be achieved individually at the same level as the combined action. The movements of atoms within a crystal often involve point imperfections; with relatively little mechanical thermal input, an atom can be displaced into a neighboring vacancy or can move a smaller foreign atom interstitially and take its position. Displacements of molecules usually require less input than average atomic movements because secondary bonds are weaker than primary bonds; however, bond density is sometimes an overriding factor. The displacement of entire crystals typically occurs with the grains sliding with respect to one another; secondary bonds with grain-boundary matter are broken and then re-established when the resistance to further movement exceeds what remains of the initial input. A somewhat superficial discussion of the internal structure of a solid material is contained in the preceding paragraphs. Much should be familiar from previous dental-materials coursework, and a more complete treatment of these topics may be found in the materials-science texts in the "Suggested Readings" at the end of the chapter. The objective of this section has been to examine internal material structure to the point of recognizing the existence of relationships with pertinent properties of orthodontic appliance materials. Those relationships will be recalled as the associated properties and material behavior are considered in subsequent sections of this chapter.

Load-Deformation Behavior of a Structural Member generation of forces within the members of an orthodontic appliance, in addition to those associated with interatomic and intermolecuThe

79


bonding, and the transferral of induced forces to and through the dentition and into the periodontium, results in the remodeling of the periodontal support system and the displacements of teeth toward corlar

rection of the malocclusion. Although the rigid-body model was used in the discussion of forces in Chapter 2, the rationale was a simplification in force analyses when deformations could reasonably be neglected in comparison to the dimensions of the body or member under study. In fact, for any stationary, supported body, loading and deformation cannot take place exclusive of one another. To the typical engineering struc-

ture, a "load" (an external force system) is applied which results in a pattern of deformation; although perhaps unwanted, that deformation must be included in design or analysis procedures. On the other hand, the clinician produces a deformation, in activating an orthodontic appliance, to induce the desired force system. Although the cause-and-effect roles seemingly are reversed in loading the appliance compared to the typical structure, the parts of the orthodontic appliance may reasonably be termed "structural members" because they carry and deliver force. Accordingly, certain appliance-member characteristics defined in this section will be referred to as structural properties. The practitioner generally produces and observes deformation before, during, and after activation of the orthodontic appliance. Each deformation, defined as any localized or overall change in shape of a mem-

ber, results in an alteration of the force system within the appliance. Most familiar are the deformations necessary to activate the appliance, producing the initial levels of force to be delivered to the dentition. Over a period of time, with movements of teeth, the initial appliance deformations are lessened with corresponding changes in force magnitudes. At a

succeeding appointment, if the appliance has not totally deactivated (lost all force) in the meantime, the practitioner may unload the appliance with configuration changes accompanying disengagement. An appliance member may be said to be structurally passive when it is subjected only to gravitational attraction and any forces responsive only to its small weight. Prior to appliance engagement and activation, the clinician may affect loading and unloading deformations that take an appliance element, typically a length of arch wire, from one passive configu-

ration to another. In the formation of arches or auxiliaries from as-received wire, the orthodontist must apply (often with pliers), and

then release, force systems in order to produce the desired shape changes. Several types of deformation are to be recognized. First, consider two particles within a structural member. A change in the distance between particles reflects a normal deformation. If the distance is increased, the deformation is termed tensile or extensional; if the distance is decreased, the deformation is compressive. A second fundamental deformation is

associated with a pair of lines of particles of the material. If the angle between the two lines changes, a shearing deformation has occurred. Normal deformations are depicted in Figure 3-4; note the changes in relative positions of particles at right angles to the normal deformation,

80


11 Passive

configuration

Compressive deformation

Extensional deformation

FIGURE 3-4. Normal deformations, extension and compression, illustrated on an atomic scale.

associated

with bonding forces and distances, the so-called "Poisson

effect." The schematic of a shearing deformation appears in Figure 3-5.

Both normal and shearing deformations can occur simultaneously within a member—indeed, even at the same location. These fundamental forms of deformation may or may not be visually recognizable, depending upon their sizes and the volumes of material affected; an obvious deformation is the extension associated with the activation of an "elastic." Two additional forms of deformation include both of the fundamental forms. Bending is identified externally by a longitudinal shape change in the member, often from a straight to a curved configuration, as depicted in Figure 3-6. In the analysis of bending, the member is conceptualized as an assembly of longitudinal fibers, all having the same shape.

Passive

configuration

Shearing deformation

FIGURE 3-5. An atomic representation of shear deformation.

81


I

-

I

Passive bar

Bar deformed in bending FIGURE 3-6. An illustration of bending detormation, together with two measures of this form of deformation.

all fibers are attached to one another in a solid member, bending results in the stretching of some fibers and the compressing of others. In Figure 3-6, if external forces hold the member in the deformed configuration the lower fibers are maintained in compression and the upper fibers in tension. Two external measures of bending deformation are defined; as shown in the figure, a transverse deflection 6 denotes the displacement of a cross-section of the member perpendicular to the longitudinal dimension and an angular coordinate U indicates the rotation of a cross-section. The final type of deformation to be introduced is twisting. Figure 3-7 illustrates that the external measure of a twisting deformation is an angular coordinate 4) which denotes the change in orientation of a transverse reference line in the cross-section. In a twisted member, because the amounts of rotation generally vary from one cross-section to another, and since cross-sections are interconnected, the characteristic internal deformation is a shearing deformation. (Shearing action often also exists within a member subjected to bending activation, but the accompanying deformation is negligible except, possibly, when the longitudinal dimension of the member is small.) Beginning with a structural member in a passive but supported configuration, the application of external force produces an associated deformation. To maintain the deformed configuration induced by the force system, the force system itself must be maintained and the member is said to be constrained. Releasing the load results in a second configuration change, and the member again becomes passive. By definition, the application of force causes an elastic deformation of the member if the passive configurations, before loading and following unloading, are geometrically identical. If, upon unloading, the member does not return Because

to its initial, passive configuration, the loading produces inelastic

behavior.

82


Passive bar

Fixed end

Twist

deformation

Bar deformed in torsion

FIGURE 3-7. A twist deformation and an angle 4' as a measure of it.

As a first example to study, totally from an external standpoint, the load-deformation behavior of a structural member, the bending of an arch-wire segment is examined. Consider an initially straight piece of wire between two fixed supports. The span, the distance between the supports, is sizable in comparison to the cross-sectional dimensions of the wire. The wire segment has been cut from a longer piece of as-

83


Applied force P

----Deflection ô

FIGURE 3-8. An arch-wire segment subjected to bending deformation produced by a transverse, midspan force.

received wire. A point force is applied midway between the supports to

generate a bending deformation; specifically, the external deformation of interest is the transverse deflection of the cross-section directly under the applied load. Shown schematically in Figure 3-8 are the undeformed shape (dashed) and a deformed configuration (solid), the latter generated by the applied force. The load-deformation curve, given in Figure 3-9, is the result of the plotting of corresponding values of applied force and deflected position of the midspan cross-section relative to its passive location. The load is slowly increased in small increments and the accu-

mulated deflection grows until the simple structure collapses (is no

Applied Force P 2

/ S: Stiffness p1: Proportional limit

el: Elastic limit

0

Deflection

FIGURE 3-9. A p'ot of applied force versus midspan deflection for the wire segment of Figure 3-8.

84 Bloengineering Analysis of Orthodontic Mechanics

longer able to sustain the load). The overall shape of the curve is charac-

teristic of crystalline materials, regardless of the member geometry or the type of deformation induced. The initial portion of the plot is linear with the slope then decreasing as the loading (plotted as the ordinate on the graph) becomes relatively substantial. The end point of the linear portion of the plot is appropriately termed the proportional limit. The coordinates of the final point of the curve denote the largest values of deformation and force before failure. The application of a relatively light force to the member of Figure 3-8 results in a small deflection; the two values are the coordinates of a point on the curve of Figure 3-9, most likely below the proportional limit. If the load is held constant and the temperature is not elevated substantially, the deformation of the metallic member will generally remain unchanged, virtually independent of time. Upon release of the load the deformation will disappear and, in general, the original straight-line shape will be regained, provided the elastic limit has not been exceeded. The elastic limit is the extreme point on the force-deformation plot from which unloading will occur with no permanent deformation resulting from the load-unload cycle. Points 1 and 2 in Figure 3-9 are below and above the elastic limit, respectively. An unloading plot from point 1 traces the loading plot between the origin and point 1. The dashed unloading curve from point 2 is notably linear and parallel to the linear portion of the loading plot; the deformation coordinate of the point of intersection of this unloading curve with the abscissa (deformation axis) denotes the permanent deformation of the segment and a new passive configuration of the now bent wire. Internally, the loading to point 1 stretches some interatomic bonds and pushes other atoms toward one another, but no bonds are broken and, as the load is released, the atoms return to their prior equilibrium positions. Loading to point 2, however, severs some bonds and results in movements of crystals relative to one another; new bonds are established that are reflected upon unloading in a new, altered, passive configuration. The elastic limit for a crystalline member is found experimentally to be only slightly beyond the proportional limit and, to simplify computations, the elastic limit is often "placed" at the end of the linear portion of the plot, coincident with the proportional limit. The elastic limit divides the structural character of the wire segment and the load-deformation plot into regions of totally elastic behavior and inelastic response. The slope of the linear part of the curve is the bending stiffness at the "midspan" location of the arch-wire segment. In an orthodontic activation, the loading of a wire is intended to be to a point not beyond the proportional limit so that the totally deactivated configuration is known: generally identical to the preload, passive geometry. Hence, in an elastic activation, the stiffness when quantified enables the determination of (1) the magnitude of the load corresponding to the activating deformation—the load is the product of stiffness and - the amount of activation—and (2) the reduction in the as-activated level of

85


loading during deactivation—the product of stiffness and quantified

deactivation increment. The coordinates of the elastic-limit point on the plot are known as the elastic range and the elastic strength. In this example, the elastic range is the largest transverse bending deformation of the midspan cross-section

that the wire segment can experience and subsequently deactivate totally or be unloaded without sustaining a permanent bend. The elastic strength is the maximum magnitude of applied, midspan, bending force without inelastic material behavior occurring somewhere in the segment. A fourth notable elastic structural property is termed resilience. An energy-related quantity, and as such to be discussed in more detail in Chapter 4, resilience is, quantified, the area under the load-deformation diagram up to the elastic limit; for crystalline materials that area is triangular. Dependent collectively upon both elastic-limit coordinates, which themselves are intradependent with stiffness, a member exhibiting low stiffness but a high elastic range may possess a resilience similar to that of a member having greater stiffness but a lesser elastic range. Although the activation of an arch wire in bending is not intended to take the member beyond its elastic limit, the nonlinear portion of the load-deformation diagram is also important because inelastic bending is necessary by the practitioner to place permanent bends. Taking the example wire segment to point 2 on the load-deformation curve results in the appearance of a "gable bend" midway between the supports. How severe a permanent bend the wire can sustain without fracture depends upon the ductility of the wire in bending. Two measurements of ductility are the deformation to fracture, the horizontal coordinate in Figure 3-9 of the final point of the plot, compared to the elastic range, and the toughness, the total area under the load-deformation diagram, compared to the resilience. Both of these quantities just defined, compared to their elastic counterparts, are substantial for the very ductile ligature wire, for example. In contrast, both comparisons are 1:1 for a perfectly brittle member; the fracture point and elastic limit coincide on the load-deformation diagram. A second, pertinent, structural example is that of a stretched elastomeric module. Schematics showing the passive and an activated configuration are presented in Figure 3-10. One end of the module is fixed and the member is elongated by the application of a concentrated force at the other end. The overall deformation (e) is the extended length (L) minus the passive length (L0). As with the arch wire in bending, the larger the

load, the greater the deformation, but the module exhibits a two- or three-part load-deformation diagram, quite different from that of the arch wire. An elastomer is a rubber or rubberlike, often synthetic, polymeric material. The nature of the bonding of atoms and molecules, and the imperfections, make the material inherently more flexible than a crystalline solid of the same size and shape. A typical two-part diagram is presented in Figure 3-11; the solid curve represents loading to failure by rupture (separation). Along the lengthy, linear portion of the plot,


Stretching force

F

e

L

0

Passive

Activated

FIGURE 3-10. Activation of an elastomeric module in direct tension.

the elongation under load is reflected principally in the uncoiling of long molecular chains. (A typical three-part curve suggests, in the initial nonlinear portion, that secondary bonds must be broken before the major uncoiling of chain molecules can begin; that first part of the curve exhibits decreasing slope with increasing applied force.) As the uncoiling is

completed, material stiffness increases as primary bonds begin to be stretched; with continued increasing of the load the slope of the curve grows and rupture eventually occurs. Accurate determination of the coordinates of the elastic limit of the module is difficult because of the time-dependent nature of the mechanical behavior of the material (to be discussed subsequently). Most elastomers, however, are relatively brittle; their elastic limits are "high" on the load-deformation curves, beyond the linear initial or middle portions of these plots. Inelastic action,

therefore, is never expected in clinical use, as the name "elastomer" suggests.

The dashed curve in Figure 3-11 is representative of rather rapid unloading of the module. Unloaded from a point below its elastic limit, some deformation remains immediately after removal of the activating force, but over time, full recovery of the preload, passive configuration takes place as secondary bonds are re-established (dashed line). Strktly speaking, orthodontic deactivation cannot be shown in Figure 3-11 be-

cause the force in the module depends upon the time elapsed while

87


Stretching Force F

/ / / 0

/ / Extensional Deformation e

FIGURE 3-11. The force-deformation diagram for the module of Figure 3-10.

under load, in addition to the amount of elongating activation. Hence, stiffness, as the slope of the load-deformation curve, can only be approximated because it, too, is changing with time. As previously mentioned, time-dependent behavior is considered further in a subsequent section of this chapter.

Mechanical Stress concept of internal force was introduced in Chapter 2. The interatomic and intermolecular bonding forces discussed previously in this chapter are fundamentally part of any internal force system. Superimposed on the bonding forces, however, are internal force distributions The

arising from the external loading of the structure through the application of mechanical force or through the imposition of an activating deformation and maintenance of a constrained configuration. The loading of one member creates internal forces in the whole of the structure; these forces

seemingly travel through the loaded member, across the connections and into adjacent members, and ultimately into the supports or foundation of the structure. These internal forces, both from bonding and from external loading, not only exist throughout the lengths of the members, but are also distributed over the entirety of the member cross-sections.

88


In general, the intensity of internal force can vary in all directions within

a member. (In structural analyses, although the internal force system arising from external loading is superimposed on the bonding force system and, likely, also a residual internal-force distribution remaining from manufacturing or preactivation fabrication processes, the analyses often ignore all but the first on the basis of magnitude comparisons.) As a first example in this section, consider the activated elastomeric

module of Figure 3-10. Held in this constrained state, although stretched, the module is in mechanical equilibrium with its supports. The internal force system can be exposed in part by making an imaginary cut through the module and arguing that the resulting two parts are each in equilibrium as is the whole of the module. The specific loca-

tion of the "cut" along the module length is arbitrary in this example and is simply made perpendicular to the longitudinal direction. The two portions of the module are shown in Figure 3-12 together with the activating and responsive forces on the entire module. The external forces at the module ends are equal and opposite from a force balance; the exposed internal forces are equal and opposite because they are action-reaction counterparts. Assumed is a one-material, homogeneous module and, with the line of action of the external forces coinciding with the

longitudinal axis of geometric symmetry, the distribution of internal force, arising from the loading, is essentially uniform over the cross-section as illustrated. Also, with the resultant of the internal force system equal and opposite to the external force (from a longitudinal force balance on either module portion), the internal distribution is wholly pulling against the cross-section of the "cut." Finally, because the external loading is only at the ends of the module, and with its weight negligible, the pattern of internal forces on all cross-sections having identical orientations is the same from one end of the module to the other, regardless

of the longitudinal location of the "cut." The average intensity of the internal force in Fig. 3-12) is the resultant, internal-force magnitude (F) divided by the cross-sectional area (As) exposed by the "cut." In the determination of the initiation of inelastic material behavior or failure, intensity of internal force is the critical parameter. In short, but sometimes oversimplified, the larger the cross-section of a structural member,

the greater is the load that it can carry. Stress is defined as the intensity of internal force, per unit area, at a point within a machine or structural member. It is a vector quantity, obtained by dividing a vector (force) by a scalar (area). The dimensions of stress are force divided by length-squared (FIL2). A "cut" exposes an area internal to the member, and that area is divided into a number of subareas with a portion of the internal-force resultant exerted on each subarea. As the number of subareas increases, the size of each subarea becomes smaller and the pattern of internal-force intensity (stress) becomes more detailed. In the module example, the stress everywhere is normal to the area and is the same on every subarea of the exposed cross-section, but this is a special situation; in general, the stress may be oblique to the area and may vary over the whole of a cross-sectional area

89


"cut"

F=

Crosssectional area

F A,,

F

FIGURE 3-12. Activation and response at the ends of an elastomeric module, Internal force distribution over a cross-section of the module.

90


Stress

S5

Ss

S

Sn Sn

Side view

FIGURE 3-13. A general stress vector decomposed into normal- and shear-stress components.

of a member. Shown in Figure 3-13, within the total cross-section, is a typical subarea (As) (enlarged) with the stress s associated with it. Nearly always convenient is the decomposition of the stress vector into components normal (perpendicular) to and tangential to the area; these components are known as the normal and shear stresses, symbolized by a or s,, and r or s5, respectively. When the normal stress is apparently pulling against the area, as in Figure 3-13, it is called tensile stress. A normal stress that pushes against the area is a compressive stress, and is directly analogous to pressure in a fluid. Note the correspondence between the two categories of normal stress and the forms of deformation previously discussed. As an example illustrating the generation of shear stresses, consider an orthodontic bracket pad bonded to a facial crown surface and the bracket subjected to an external force parallel to the facial surface. Figure 3-14, on the left in perhaps a mesial or distal view, shows schematically the enamel crown surface, an exaggerated sketch of the deformed adhesive, and the loaded bracket. On the right, following a "cut" through the adhesive, which is made parallel to the load Q, a free-body diagram has

been drawn of the bracket, pad, and a portion of the adhesive as a whole. Neglecting weight and the faciolingual thickness, the distributed

internal force system exposed by the "cut" is entirely shear. A force balance indicates that the resultant internal force must equal Q. The average shear stress in the adhesive equals the resultant of the distribu-

tion divided by the area in shear k. In the examples of the module in tension and the adhesive in shear, the resultant internal force equals the product of average stress and the corresponding area. The computation in arch-wire bending, however, is somewhat more complex. Figure 3-15 depicts a wire segment with the "right-section cut" (perpendicular to the longitudinal dimension) exposing an internal cross-sectional area (seen only in edge view in the figure) and a typical internal force system including both forms of normal stress as well as shear stress, with none of the three uniformly distributed. In this segment, no inelastic behavior is present and the resultant of the tensile- and compressive-stress distributions is a pair of equal, opposite, and parallel forces—a "bending couple." The resultant of the shear-

91


Q

Average shear stress where A,.

=

total area exposed by the cut'

FIGURE 3-14. Shear loading of a bonded bracket and shear-stress distribution within the bonding adhesive.

distribution is an internal force tangent to the area exposed by the "cut" and parallel to the direction of external force causing the bending deformation. A complete discussion of bending is undertaken in Chapter 7. stress

Mechanical and Structural Properties: Standardized Testing The

response of a structural member to external loading depends

upon the size and shape, or geometry, of the member, the relationship of the loading pattern or characteristics to that geometry, and the mechanical properties of the material of the member. Problems of analysis

92

Bioengineering Analysis of Orthodontic Mechanics Compressive stresses

Wire segment Position and direction ot shear stresses

Distribution ot shear stress

FIGURE 3-15. Typical stress patterns internal to a structural member subjected to inplane bending.

or design of structures are often approached by taking experimentally

determined mechanical-property values, substituting them into formulas or equations that express the governing principles of mechanics and

include the contributions of geometric and loading parameters, and thereby obtain or check what might be termed structural potential or capacity. The differences, if any, in bending characteristics of two, .016-

in. -diameter, arch wires, for example, are founded in the mechanical properties of the wire materials. Hence, a familiarity with these material properties is important to the practitioner. Many of the fundamental mechanical properties of a solid material are commonly quantified through the performance of a static tension test. Each test specimen is carefully prepared to minimize experimental varia-

tion or "error" associated with the testing itself. The test volume (the middle portion, longitudinally) of the specimen is uniform in cross-sectional shape and size; the cross-section often exhibits at least one axis of

geometric symmetry. The straight specimen is to be held at its ends, some distance away from the test volume such that the manner of attachment of the loading system to the specimen will not influence the test volume and the data gathered. An example test specimen is shown in the "grips" of the testing apparatus in Figure 3-16. The grips transfer the loading, measured by the testing machine, to the specimen in the resultant form of a longitudinal force having a line of action that passes through the centers of the cross-sections of the test volume. With the weight of the specimen ne-

glected (compared to the load), the specimen is a two-force member, similar to the module of Figure 3-12 and as described in Chapter 2. The grips are attached to the "heads" of the testing machine; one head is movable and the other is stationary. The testing machine has the capability to apply a slowly increasing load (thus the term "static" test) and

Load P

-II

IV

Responsive

forceP

-

FIGURE 3-16. A tensile-test specimen in the grips of a standard test apparatus.

_____ 94 Bioengineering Analysis of Orthodontic Mechanics

to monitor continuously the magnitude of that load. A longitudinal or

axial dimension within the test volume is known as the "gauge length." Its initial value, before external loading is begun, is shown in Figure 3-16 as 4?,,. An "extensometer" is attached to the specimen at the extremes of

the gauge length, or an alternative procedure is employed, to enable instantaneous measurements of the gauge length which increases during the tension test. (The extensometer weight must also be negligible compared to the specimen load or counterbalanced so as not to create notable testing error or destroy the mechanical symmetry.) In the test itself, the loading is begun from zero and gradually increased until rupture of the specimen occurs. Readings of applied load (which equals the internal axial tensile force) and the corresponding extensional deformation (or extended length) of the gauge length are taken, either manually within equal loading increments or, by means of transducers and a stripchart recorder, recorded continuously by the testing machine and plotted as the test is proceeding. The extension of the gauge length during the tension test will depend upon its initial length; therefore, this extension or elongation is "normalized" with respect to the initial length. Axial deformation per unit length is a form of engineering normal strain: 4?

(3-1) e,,

The test under discussion produces longitudinal tensile strain; if the loading sense is reversed, the gauge length is lessened under pushing force and longitudinal compressive strain is induced. Corresponding to the normal strains are normal stresses against cross-sectional areas exposed by "cutting" the specimen perpendicular to its longitudinal axis. The uniformly distributed internal force system, as in the module example, has an intensity that depends on the size of the exposed area. The engineering tensile stress equals the resultant, internal force divided by the pas-

sive, cross-sectional area. Note that by converting force to stress and deformation to strain, the net effect is the dividing out of the volume of material in the test section and, thereby, eliminating the influences of specimen size from the test results. (As the test specimen elongates under load, the cross-sectional area generally reduces; recall the Poisson effect illustrated in Figure 3-4. Engineering stress and strain are obtaine& by dividing out the undefonned, as-prepared area and length of the test volume. True stress and strain are obtained by dividing the instantaneous, deformed area and length into the corresponding internal force and elon-

gallon, respectively. All stresses and strains discussed in this chapter and throughout the text are engineering stresses and strains.) From the paired readings taken during the tension test, converting force to stress and elongation or stretched length to strain, a tensile stress-strain diagram is prepared (either by hand or plotted by the testing machine). With modifications of coordinates of data only through division by constants, the stress-strain diagram in tension for a specific ma-

95


—'C C,) C,)

C

Stainless, nickel-chrome steel

Mild, structural steel E

x: tracture point

0

Tensile Strain

0

Tensile Strain

FIGURE 3-17. Stress-strain diagrams in tension, obtained from static testing, for mild, structural steel (left) and for a stainless steel (right).

terial has the same shape as the tensile, load-versus-deformation plot for

a structural member of the same material. Shown in Figure 3-17 are tensile stress-strain diagrams depicting the behavior of mild, structural steel, which exhibits a yielding phenomenon, and a stainless steel, which does not notably yield. Yielding is the straining of the material at virtually a constant stress well below the level at which rupture occurs. Illustrated in Figure 3-18 are diagrams for a ceramic material and twoand three-part diagrams for typical polymeric materials. The mechanical properties of interest are obtained from the stress-strain diagram for the material and from measurements taken directly from the specimen, and

are analogous to corresponding structural properties, many of which have been defined previously. A crystalline material obeys Hooke's Law, which states that the stress-

strain ratio is constant up to the proportional limit; the constant in this linear stress-strain relationship is the modulus of elasticity E (Young's modulus) in tension, the elastic stiffness of the material and the slope of the initial portion of the diagram. The ultimate strength of the material is the value of the maximum ordinate of the curve; as indicated in Figure 3-17 this may or may not be the stress level at fracture (but if the high point of the cprve is reached in loading, although data may be taken at greater strains, the imminent fracture generally cannot be prevented). The coordinates of the fracture/rupture point are ordinarily termed the fracture strain and fracture stress or strength. The proportional and elastic limits are close together on the curve; the coordinates of the elastic limit are termed the elastic-limit strain and elastic strength of the material. Accordingly, the simple relationship a = Ec, which is the expression of

96


Tensile Stress

A ceramic material

x: fracture point

Polymeric materials

0

Tensile Strain

FIGURE 3-18. Static stress-strain diagrams in tension for ceramic and polymeric materials.

Hooke's Law in direct tension, holds virtually up to the elastic limit. For

those crystalline materials that yield, the yield strength is the stress level at which this phenomenon is exhibited; because yielding immediately follows the initiation of inelastic action in tension, the values of elastic strength and yield strength are, for practical purposes, equal.

In the discussion of mechanical properties, the word "modulus" implies "per unit volume;" accordingly, the modulus of resilience of the material is quantitatively the area under the stress-strain diagram up to the elastic limit and the modulus of toughness is the total area under the diagram to the point of fracture. A material is relatively ductile or brittle, depending on the extent of the stress-strain diagram beyond the elastic limit. Clearly, the ceramic material of Figure 3-18 reflects no ductility at

all; the plot being linear to fracture indicates the nonexistence of an elastic limit and the inability of the material to take a permanent strain. Typical measures of ductility are, analogous to the content of an earlier discussion of structural characteristics, the fracture strain referenced to the elastic-limit strain and the modulus of toughness relative to the modulus of resilience. A third measure is obtained by taking the two por-

tions of the fracture specimen from the testing machine, mating the fractured surfaces, measuring the as-fractured gauge length, determining the corresponding strain value, and finally multiplying by 100 to obtain the percent elongation at fracture. A fourth measure of ductility,

97


similar to the third, is obtained from the pretest cross-sectional area and

the cross-sectional area of the fractured specimen at the fracture location: the percent reduction in area.

Noncrystalline materials, such as the polymers of Figure 3-18, seldom exhibit any useful level of ductility and often do not obey Hooke's Law. As a result, there is little distinction between elastic strength and ultimate strength. Although not exactly correct, the elastic limit may be taken as the point where the slope of the stress-strain curve begins to increase substantially toward rupture. The stiffness per unit volume is not a constant in those ranges where the curve is nonlinear; hence, the simple relation between stiffness modulus and the coordinates of the elastic limit, (cr/e)ej = E for crystalline materials, is not valid for those materials exhibiting a nonlinear first portion of their stress-strain diagrams. To determine the stress level, for a given activation in tension of an elastic for example, it is best to directly measure the force and divide that magnitude by the cross-sectional area. Mechanical properties of materials are obtained from simple experiments, tension tests for example, which generate the fundamental forms

of deformation cited previously. Because their values often depend upon the form of deformation generated in the test, these properties when quantified must be presented as "in tension," "in compression," or "in shear." In addition, the rate of loading during the test will influence mechanical-property values to some extent; for example, distinctions are made between static and impact tests. Furthermore, environmental conditions, particularly temperature in the test locality, may affect values, as may specimen-storage time and conditions prior to testing. Finally, the as-received internal material configuration will partially

determine the values of some mechanical properties, in particular for a metallic material the coordinates of the elastic limit and, therefore, its modulus of resilience. (The influence on properties of permanent deformations and accompanying "residual stresses" is considered in Chapter 4.) Material behavior in static compression is similar to that exhibited in a tension test, but some differences in test specimens and procedures, and in obtained mechanical-property values, do exist. In an axial compression test, with the sense of the loading reversed from that of the tension test, the as-prepared specimen often has a uniform cross-sectional geometry from one end to the other, and is short and thick to ensure symmetric loading throughout the test and to avoid the possible occurrence of lateral deflection or buckling. Load and deformation data are taken, converted to compressive stress and strain, and the diagram is constructed. For a crystalline material, the modulus of elasticity is generally the same in tension and compression, although the extension of the linear portion of the curve to a greater elastic-strength magnitude in compression than in tension is not uncommon. Noncrystalline materials may exhibit a substantially altered stress-strain diagram in compression compared to the tension plot. With molecular materials not extensively cross-linked, the chain molecules that uncoil at low to mod-

98


loads in tension clearly would not do so to any great extent in compression. Although some tensile strain is developed laterally from the Poisson effect accompanying the axial loading, with compressive loading, interatomic bond deformations begin almost immediately. Ultierate

mate strength in compression is defined and computed in the same manner as in tension, but the failure mode must be substantially different from the tensile rupture. The load-carrying capacity of a member in direct compression is lost when one of the following occurs: extensive lateral deformation (buckling); longitudinal splitting (also known as internal buckling); or a "slipping fracture," on a plane inclined with respect to the loading axis, due to the exceeding of the ultimate strength of the material in shear. (Although absent on right cross-sections in members undergoing direct tensile or compressive loading, shear stresses exist on all other planes exposed by "cuts" through the members and are largest on planes at 45° with the loading axis.) The nonlinear portion of the stress-strain curve in compression, seemingly related to ductility, is often reduced comparatively in materials that can sustain at least moderate amounts of inelastic action in tension; the opposite may be true with apparently brittle materials because right-section rupture cannot occur under compressive loading. Examples of actual or potential shear deformation, associated with orthodontic appliances, were cited previously; a schematic was shown in Figure 3-5 and the shear strain, generally symbolized by y or is defined as the tangent of the angle developed as two intersecting line segments at a point within the material, passively perpendicular, are inclined to one another with the member under load. In a test to determine mechanical properties in shear, the specimen and loading are typically similar to the bonded bracket assembly of Figure 3-14 except that

the specimen is quite small, is of one material, and in particular the moment distance of the applied force with respect to the specimen support is minimized. Readings of load and angular deformation obtained are converted to shear stress and strain, and the reduced data are plotted to give the stress-strain diagram. Two example diagrams are presented in Figure 3-19. The form of the plot for the crystalline material is, by now, familiar; mechanical properties as derived from the diagram are defined exactly as in the tension or compression test except each name ends with the phrase "in shear." The slope of the linear part of the shear-test curve, however, is generally often termed the modulus of rigidity G. From stiffness magnitudes, many metallic materials seem to be more flexible in shear; for arch-wire alloys, the ratio of elastic moduli in identical units,

shear to axial tension, is about 0.4. (Although measured differently, both normal and shear strains are nondimensional, so these stiffnesses are dimensionally alike.) Hooke's Law for pure shear loading is written T = G-y and is valid up to the proportional limit in shear. For ductile materials in general, the elastic and ultimate strengths in shear are about one-half of the comparable values in tension; for brittle materials, the ratio of ultimate strength in shear to that in tension approaches unity.

99


Shear Stress

A crystalline material

Shear Strain

0

FIGURE 3-19. Static stress-strain diagrams in shear for crystalline and polymeric materials.

The forms of the stress-strain diagram in shear and in compression are

similar for most molecular and amorphous materials; the initial nonlinear portion of the three-part curves in tension of some noncrystalline materials, as shown in Figure 3-18, is rarely seen under compressive or shear loading.

Previously depending upon mechanical-property data generated elsewhere, in 1977 the Council on Dental Materials and Devices of the American Dental Association announced the ADA Specification Number 32 for orthodontic wires not containing precious metals. The Specification includes a detailed description of the preparation and testing of arch-wire specimens toward the determination of the modulus of elasticity and "yield strength" for the arch-wire material in flexure (bending). A schematic of the cantilevered-beam test is given in Figure 3-20. A bending couple is applied to one end of the specimen where only rota-

tion is permitted; at the other end of the test span the wire is held against a fixed, knife-edge stop. The angular deformation measured is the rotation of the shaft, which is also the rotation of the end cross-section of the specimen engaging the shaft with respect to the passive orientation. A typical plot of the couple versus the rotational deformation, for a specimen of metallic material not exhibiting a well-defined yield point, is

100


Load couple C

FIGURE 3-20. An illustration of the fiexure-test arrangement dictated by ADA Specification No. 32.

presented in Figure 3-21. To obtain an equivalent yield point, approximating the elastic limit, the dashed line is drawn parallel to the initial, linear portion of the plot, intersecting the deformation axis at the specified offset (approximately 2.9° according to Specification No. 32), and the yield strength of the material is determined from the couple magnitude at the point. (The actual computations to quantify the elastic modu-

lus and yield strength in flexure are outlined in Chapter 7.) A third material property of orthodontic interest may then be computed; the ratio of yield strength to modulus of elasticity is termed the springback of the material and is a close approximation to the elastic-limit strain. Table 3-1 contains approximate values of these three properties in bending for five arch-wire materials. In addition to the static flexure test, an experimental procedure that may be terminated after the data confirms exceeding of the elastic limit, another is required which, in part, provides an indication of the ductility

of the wire material. In this second portion of the Specification, a 90° bend is placed in the wire, then removed, and then replaced at the same cross-section and in the same direction. The radius of the bend is specified. The 90-to-0-to-90° bending cycle is continued at a specific rate until fracture occurs. The number of cycles to failure is then compared with the Specification requirements, which vary somewhat with wire size. Before concluding this section, several additional comments associated with testing and mechanical properties are in order. Indicating that a material is relatively "hard" or "soft" does not pertain, strictly speaking, to any previously discussed property. Hardness is defined as the resistance of a material to localized, permanent, compressive deformation (indentation); it is, therefore, largely a surface phenomenon. Hardness numbers are determined experimentally by subjecting a material specimen to an indenter of a specific geometry exerting a designated load. Indices of hardness familiar to engineers are the Brinell and Rockwell hardness scales. Although hardness is very nearly, directly proportional to ultimate tensile strength for a number of metallic materials, this property must not be confused with material stiffness or resilience. Mechanical properties of some materials are difficult to quantify by testing because their stress-strain characteristics change with time under

101


Load Couple C

Equivalent

yield point

/ /

/

/ Stiffness

/

/ / 0

Angular Deformation 0

FIGURE 3-21. Couple versus angular deformation; typical plot of flexure-test data obtained according to ADA Specification No. 32.

load. Although not a factor at room or oral temperature with most metals, often it is with molecular materials, particularly certain polymers. Subjecting a typical, orthodontic, elastic module, thread, or "chain" to a moderate tensile load, and holding that load constant while monitoring deformation, a continuing elongation will be observed over a period of hours. The material is said to creep and, from data taken, a strain-versustime curve may be generated similar to the plot of Figure 3-22. If the temperature or time period is sufficiently high or long, the straight-line portion of the curve will give way to a nonlinear increase in strain fol-

lowed by rupture, even though the stress level may be substantially below the ultimate strength of the material as gained from a static, tension test at room temperature. A material that creeps will also "relax." Relaxation is the decrease with time of load carried under conditions of constant strain; a relaxation plot is presented in Chapter 4.

102


TABLE 3-1. Mechanical pmperties in bending for selected, orthodontic, arch-wire materials

Modulus of elasticity

Yield strength

Springback*

(x106 lb/in.2)

(x103 lb/in.2)

(x102 in/in.)

A gold alloy

15

150

0.94

Type 302 stainless steel

29

280

0.97

Elgiloy (Cr-Co alloy)

28.5

315

1.1

Material

Nitinol (Ni-Ti alloy)

4.8

195.

4.1

Beta titanium (Ti-Mo alloy)

9.5

170

1.8

upon unloading from the (equivalent) yield point: the yieldstrength-to-elastic-modulus ratio. Yield strength and springback values will vary with cold working or heat treatment. *The recovered, unit

The final topic considered in this section is fatigue, the progressive failure of a material undergoing loading that changes with time. Usually

such loading is cyclic, exhibiting a repeating pattern; the maximum stress generated may be substantially below the ultimate strength of the material obtained from a static test, and still result in an eventual fracture. Examples of cyclic loading pertinent to orthodontics include masti-

Strain

Stage 3

0

Time

FIGURE 3-22. A strain-versus-time plot illustrating the three stages of creep.

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Stress

Ultimate strength

Endurance limit

0

Number of Cycles to Fracture

FIGURE 3-23. A typical plot of the results of tests to fracture of a metallic member subjected to cyclic loading in bending.

(relatively high-frequency cycling) and the daily activation, continuous-force, deactivation pattern associated with an extraoral cation

appliance and interrupted loading (low-frequency cycling). A common fatigue failure is the progressive fracture of a bent-wire segment emerging from the acrylic which is part of a retainer; the removal and replacement of the appliance represents one cycle. Figure 3-23 shows the re-

suits of a series of fatigue tests in bending. The curve intersects the stress axis at the ultimate-strength level. The inclined portion of the plot is obtained from fractures occurring at maximum stress levels below the ultimate-strength value; the lower the stress, the greater the number of cycles to failure. Below a particular stress level, termed the endurance limit, fracture does not occur regardless of the number of load cycles. With the retainer in which the wire eventually fractures, the maximum

stress in the activated wire ioop is less than the ultimate strength but greater than the endurance limit of the wire material. Besides indicating

level of ductility, the previously described bending test to fracture within ADA Specification No. 32, owing to the cyclic-loading pattern required, also indirectly provides an approximate, relative measure of arch-wire resistance to fatigue.

Chemical and Thermal Influences With orthodontic application in mind, a discussion of the chemical and thermal behaviors of solids is undertaken. The chemical and thermal material properties themselves are not the principal focus of atten-

104


lion, but rather the potential for interactions that may result in deleterious effects upon the mechanical or physical characteristics of orthodontic materials. Some, and often all, of the members of the ortho-

dontic appliance must exist and "work" for substantial periods of time in the oral cavity. Materials used in the mouth must be biologically admissible and the potential for interference with acceptability, because of chemical reactions that may occur over time, either before or during intraoral engagement, must be realized and understood. The oral-cavity environment is inherently corrosive. The oral fluids are strong, potential reactants toward oxidation of metals. Saliva is known to contain salts and acids are often liberated during mastication; ingested food and drink vary widely in their levels of acidity or alkalinity (pH). Food debris may become lodged and remain for relatively long periods of time and provide the catalyst for initiation of corrosion; although generally a dental-care matter, the tendency for this occurrence is increased by the presence of appliances. Both metallic materials (e.g., restorations, wires, bands) and molecular solids (e.g., "elastics," cements, adhesives, acrylics) may be increasingly vulnerable to chemical degradation in the moist oral environment. The effects of chemistry and temperature upon the materials themselves, as well as the products of chemical reactions, must be of concern. The extraoral environment may also influence the structural capabilities of orthodontic appliance materials. The time period between manufacture and actual use can be substantial. Storage may occur with the manufacturer, with the vendor, and with the practitioner. Molecular materials, in particular, often have a finite "shelf-life," which may be reduced by high temperature and humidity. Parts of extraoral appliances come into contact with skin and hair, and the oils as well as perspiration, often together with particles suspended in the air, can and do have a degrading effect. Corrosion is defined as a deterioration because of a chemical reaction that results in apparent disappearance of the material attacked. Corrosion may occur by a chemical solution contacting the material; ionization ensues and the material is dissolved in the fluid. For example, silver will be corroded by a solution containing sulfides with silver sulfide formed in the solvent. A more common form of corrosion is known as electrochemical oxidation. Characteristic of metallic materials, the process begins with a reaction initiated by a fluid that removes electrons, thereby form-

ing positive material ions. In turn, chemical combination of these ions with electrically negative ions in the fluid may take place. Rusting starts with the stripping of three electrons from the iron element, forming the

ferric ion. The degree to which electrons are bound to the atoms in metals varies, and so also does their tendency to form compounds in the

presence of nonmetals having incomplete outer electron shells. Gold and platinum are relatively inactive and, in mining, are often found in the pure form; iron and aluminum, for example, as active metals, are continually oxidized and must be purified (reduced) chemically before metallurgical processing.

105


Generally, corrosion occurs as a combination of the two forms described. The facility for the ionization of a metal is dependent upon the nature of the fluid-solvent and the inherent activity level of the material; the latter is often termed the electrode potential of the metal and is a direct indicator of its corroding tendency. The determination of the ranking of metals by electrode potential is accomplished through use of the "standard cell" shown in Figure 3-24. Immersed in the solution is the test-metal

electrode on the left and a hydrogen electrode on the right, the two connected externally through a potentiometer. Ionization begins at both

electrodes; however, depending upon the electrode potential of the metal with respect to hydrogen, electrons will flow one way or the other, registering a potential difference on the meter. The experiment is

Potentiometer

H2

V TM ±

1-1k

TM

Electrolyte

Positive ion of test metal or alloy H2: Hydrogen gas Hydrogen ion FIGURE 3-24. A standard electrochemical cell.

106

Bioengineering Analysis of O,thodontic Mechanics

repeated, changing only the test metal. With all potential-difference val-

ues recorded, a ranking such as that given in Table 3-2 is obtained with hydrogen as the reference ion. Consider a cell such as that shown in Figure 3-25 with iron as the electrode on the left and water as the solution. From Table 3-2, iron has a

higher electrode potential than tin and, upon ionizing, electrons removed from the iron flow through the connector to the hydrogen electrode. The hydrogen gas is also ionized and water molecules are decom-

posed into hydrogen and hydroxyl (OH) ions. The electrons coming from the iron unite with hydrogen ions and hydrogen gas is liberated. 3-2. Electrode-potential rankings, with respect to hydrogen, of selected metals and alloys TABLE

The ranking to follow is in the order of most anodic, at the top of the list, to most cathodic at the bottom of the list. Magnesium Aluminum, active Titanium Cesium Vanadium Zinc Aluminum, passivated Chromium Iron

Stainless steel, active Cadmium Cobalt Nickel, active Lead-tin solder Tin

Lead

nconel*, active HYDROGEN Brass

Copper Bronze Monelt Nickel, passivated lnconel*, passivated Stainless steel, passivated Silver Palladium Mercury Platinum Gold lnconel is a nickel-chromium alloy. tMonel is a copper-nickel alloy.

107

Material Behavior of the Orthodontic Appliance Electron flow

Fe3t Iron ions OH—: Hydroxyl ion Fe(OH)3: Rust

FIGURE 3-25. A galvanic cell illustrating the electrochemical corrosion of iron.

ferric ions react with the water and oxygen therein to form ferric hydroxide: rust. The rusting of iron will occur in such a galvanic cell, in fact, whenever the iron is connected externally to an electrode having a lower potential (occupying a lower position in Table 3-2) and the elecThe

trodes are immersed in a solution (the electrolyte) wherein hydroxyl ions become available with ionization. In general, the electrode supplying the electrons is called the anode and the receiver of electrons is the cathode. The anode experiences this galvanic corrosion and the location of

the corrosion products will depend upon the relative ease with which the ions, forming those products, are able to diffuse through the electrolyte. In a galvanic cell containing iron and a less active metal, because ferric ions are smaller than hydroxyl ions and because the product requires three OH ions for every ferric ion, rust will usually be found at or near the cathode. The formation of a galvanic cell with the physical contact of an orthodontic-appliance member and an amalgam restoration, in the presence of saliva as the electrolyte, results in momentary ionization and electron flow and the dental "galvanic shock." This is an example of a composition cell established between two dissimilar metals. A tin roof, placed on a

108


building having steel siding, becomes cathodic and corrosion begins in

the presence of moisture and direct contact between roof and siding. Important in addition are the more subtle forms of electrochemical cells. A stress cell may form between two different locations on the same material, because of an imbalance in electrode potential. A vivid example of stress corrosion is the etching or corroding of grain boundaries; the

atoms at the boundaries are more loosely bound and, therefore, are potentially anodic with respect to those within the crystals. It follows, then, that a fine-grained metal is potentially more vulnerable to corrosive attack than is a coarse-grained metal. An imbalance in electrode potential at different surface locations of the same material results from localized inelastic action. The region of a permanent bend in an arch wire is anodic in comparison to the passively-straight portions of the wire. A concentration cell may occur involving one material (or two) when the electrolyte is nonhomogeneous. The surface area exposed to an electrolyte is potentially more active where the concentration is lower. In a one-material cell, the region exposed to a diluted electrolyte is anodic with respect to a region in contact with a concentrated electrolyte. The

significance here is the anodic nature of cracks, crevices, and areas under accumulations of debris or other surface contaminations. Moreover, often the deposits of corrosion products aggravate the situation and increase the rate of degradation; localized pitting of a material is a typical manifestation. Electrochemical corrosion requires the presence of two electrodes, physical contact between them for conduction of electrons, and an ionizing, ion-carrying fluid, the electrolyte. Retardation or elimination of corrosion tendencies necessitates minimization of the electrode potential or a break in the "circuit." In two-material systems, corrosion may be prevented by insulating one material from the other or one or both materials from the electrolyte. With one material, isolate it from the electrolyte, keep its surface smooth, clean, and free of sharp edges, and reduce its inherent electrode potential. Steel coated with zinc is "galvanized" to separate the steel from moisture. Because zinc is more active than steel, it is the potential anode should a crack or scratch expose the steel. Orthodontic stainless steel generally resists corrosion well because its surface is protected by passivation with chromium oxide, the nickel in the alloy allows a reduction in carbon content (and a lessening, then of the potential for chromium-carbide formation while retaining strength and ductility properties), and the result is a position in Table 3-2 just above silver.

The foregoing discussion has dealt primarily with "wet" corrosion in which a liquid electrolyte or at least moisture is necessary to the reaction. Dry corrosion is the result of a chemical reaction of the affected material with a gas, often air. With metals the specific gaseous reactant is oxygen and an oxide scale is formed, initially on the metal surface. In general, the rate of increase of the film thickness depends upon the existing thickness, the diffusion coefficients for the reactants, and the temperature. Continuation of the reaction is dependent upon the pene-

109


trabiity of the scale film by either the gas or the metal ions. An example of dry oxidation is the passivation of stainless steel; the chromium oxide film is very thin and highly impenetrable. Dry corrosion of molecular materials also occurs; of particular interest is the aging of elastomers. In this oxidation reaction the initial result is an increase in cross-linking (secondary bonding), which causes harden-

ing, stiffening, and embrittlement. Heat, light, stress, and ozone concentration influence aging, which takes place because of the existence of

free-end molecular chains. Antioxidants may be incorporated during polymerization to combine in a monofunctional manner with the chain ends to provide some resistance to aging. The practitioner must be aware of aging and its effects, particularly with regard to the time period and location of storage of polymeric, orthodontic-appliance elements. In a moist environment polymeric materials experience a combination of solution and aging corrosion. "Elastics" that have been stored may contain aging cracks that are catalysts for further and accelerated degradation following activation; intraoral placement of these elements tends to quicken deterioration compared to extraoral application. Another type of degradation occurring in the orthodontic setting involves extraoral appliances, in particular the neck pads, head caps, and

other nonmetallic elements that come into contact with the skin and hair. A form of wet corrosion occurs as skin and hair oils, and perspiration attack these molecular materials over time. Although causing some aging, of equal or perhaps more concern is the chemical reaction initi-

ated by the hydrocarbons that ultimately breaks down the material structure into small molecules, thereby drastically reducing mechanical strength. The progress of this deterioration is relatively slow, and material discolorations indicate the start of the process. Frequent cleansings of washable parts of the extraoral appliance, for example, retards this form of corrosion. The effects of temperature on a material are, to a great extent, manifested through influences upon properties already discussed. The fundamental material response to change in temperature is on the level of atomic activity which, in turn, affects volume. A typical unconstrained solid will expand upon experiencing an increase in temperature; such a material will contract as its temperature is lowered. (Some polymeric materials, however, react differently with temperature changes due to accompanying bonding alterations.) A structural member that is constrained against undergoing volumetric change will be subjected to induced thermal stresses if its temperature is altered; these stresses are superimposed upon those that exist due to mechanical loading. The increased vibrational motions of atoms and molecules, accompanying increased temperatures, generally affect mechanical behavior in that ductility characteristics are enhanced but strength properties are reduced. The rates of, or tendencies toward, creep and relaxation grow as the temperature rises. The greater the atomic motions, the more easily materials may be ionized; hence, increased temperatures generally raise the rates of corrosion and oxidation.

110


In general, the effect of an incremental change in temperature is greater upon nonmetals than upon metals. Pertinent to orthodontic application, the influence of the difference between room and oral temperature, or between that of ingested ice cream and coffee, is inconsequential upon arch-wire materials. The effects of such temperature differentials, however, are discernible in many of the elastomeric materials. Although the metallic appliance components do not experience large temperature ranges during actual orthodontic therapy, they may in soldering, welding, or heat-treatment procedures prior to engagement and activation. Note that when the temperature is raised to about 900°F and above, the carbon atoms in stainless steels tend to migrate toward the grain boundaries; they combine with chromium ions to form chromium carbide, and the passivating potential of the alloy is lessened. Hence, care must be taken in soldering or welding this material, and other metals and alloys, to concentrate the heat and hold it for only a very short time.

Selection of Materials In concluding this chapter, comments pertaining to the selection of ma-

terials are appropriate. Although materials and appliance members associated with orthodontic therapy are of specific interest, the same criteria are reviewed and examined in the intelligent selection of materials for any structural or machine application. The relationship of the relevant characteristics or properties of the material to its intended use must be the first and overriding considera-

tion in material selection. The orthodontic appliance is a structural mechanism and, with appliance members available in varieties of physical sizes, attention is focused primarily on mechanical properties. Archwire materials must be ductile if arch fabrication, including the place-

ments of permanent bends and twists, is necessary. Moreover, arch wires must be moderately flexible while possessing substantial elastic strength to provide desired activation and deactivation responses. All parts of the appliance must exhibit strength characteristics to sustain without failure the stresses induced in appliance activation, both at the moment of engagement and over time periods ranging from a few days to, perhaps, several years. Possible chemical reactions, which may affect mechanical behavior or result in unwanted product compounds, must be a factor of serious consideration, particularly in the selection of intra-

oral appliance materials. Ideally, the mechanical characteristics of a structural material should not be influenced by time or by temperature fluctuations that may occur in the service environment; however, with real materials, one or both of these parameters may exert significant impact on mechanical behavior.

111


Other factors to be evaluated are generally subordinate to those directly associated with in-service use and are given significant weight in the selection decision only when, from principal considerations, two or more materials seem equally suitable or when a compromise is warranted and can be tolerated. The ranking of these additional factors will vary, depending on the specific application and, often, on financial influences. Costs will clearly impact the selection, and they include the initial investment, expenses associated with subsequent fabrication, and costs related to storage of the material. A compromise directly related to initial cost has led to the virtual disappearance of gold alloys as archwire materials. Availability is another factor that may influence initial cost and may, at least temporarily, impact the selection procedure. In any business involving a product, inventories must be kept, and orthodontic practices are no exception. Accordingly, the effects of storage on the material and the needs for handling and working with the material, in transforming it from the condition "as received" to the state of readiness for immediate use, come into consideration. Chemical degradation limits the "shelf life" of a number of materials used by the orthodontist. Moreover, many cements, adhesives, alginates, and acrylics, for example, require storage in controlled environments or careful mixing to produce chemical reactions resulting in optimum material characteristics. Not to be ignored, but often rightfully near or at the end of the list of influencing factors, is appearance. Although advertising may attempt to inflate the importance of external attractiveness, the potential user must place this factor in its proper position in the list of selection criteria. Although often an apparently difficult decision, a sacrifice within the primary selection factor, that of relation of material properties to inservice use, should rarely be made solely to accommodate appearance.

Suggested Readings Burstone, C.J.: Application of bioengineering to clinical orthodontics. In Current

Orthodontic Concepts and Techniques. 2nd Ed. Edited by T. M. Graber and B. F. Swain. Philadelphia, W.B. Saunders, 1975, Chapter 3. Burstone, C.J., and Goldberg, A.J.: Beta titanium: A new orthodontic alloy. Am. J. Orthod., 77:121—132, 1980.

Greener, E.H., Harcourt, J.K., and Lautenschlager, E.P.: Materials Science in Dentistry. Baltimore, Williams & Wilkins, 1972, Chapters 3—6, 8, 11, and 13.

Hayden, H.W., Moffatt, W.G., and Wuiff, H.C.: The Structure and Properties of Materials. Vol. III. New York, John Wiley & Sons, 1965, Chapters 1—3, 6, 7, and 10. Jarabak, JR., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 3.

112


Kusy, R.P., and Greenberg, A.R.: Effects of composition and cross-section on

the elastic properties of orthodontic arch wires. Angle Orthod., 51:325—341, 1981.

Richards, C.W.: Engineering Materials Science. Belmont, CA, Wadsworth Publishing, 1961, Chapters 1—4, 8, 9, 12, and 13. Phillips, R.W.: Skinner's Science of Dental Materials. 7th Ed. Philadelphia, W.B. Saunders, 1973, Chapters 2, 3, 11, 15—19, and 32.

Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982, Chapters 3, 7, and 8. Van Vlack, L.H.: Elements of Materials Science. 2nd Ed. Reading, Massachusetts, Addison-Wesley Publishing, 1964, Chapters 1—3, 6, 7, and 12.

Energy Analyses in Orthodontics ]

The theoretical developments within classical mechanics have provided two analysis formulations. The four laws offered by Isaac Newton are

the foundation for one approach toward the solution of problems in mechanics, and two mathematicians—Jean Bernoulli, a Swiss, and J. L. Lagrange, a Frenchmen—are generally given credit for the more con-

temporary hypotheses and methods of work and energy. In Chapter 2 the procedures of Newtonian mechanics were introduced as based upon a set of equations of equilibrium or motion, depending upon the state of rest or movement of the mechanical system. The free-body diagram was introduced and employed as a principal tool in the solution of problems. Generally, the objective was the altering,

simplifying, or completing of the description of the set of forces and couples exerted at a particular instant of time upon or within the body or the elements of a structure or machine. In the analysis procedure employing work and energy, the mechanical system undergoes an actual or

contrived process that may be of short or long duration and results in changes in one or more of the characteristics of the system. To prepare for an analysis through use of the principles of work and energy, the body or set of elements or bodies included within the mechanical system under study must be defined. The initial state of the system is described at a particular time; physical, geometric, kinematic, thermal, chemical, and mechanical properties—any characteristics that may experience change during the ensuing process—are noted. The process that is to take place results from an interaction between the system and its surroundings. Therefore, the surroundings, which will influence

the changes in the mechanical system, must also be characterized. Of particular importance are (1) the set of forces and couples to be exerted by the surroundings upon the mechanical system under study and (2) any heat sources or heat "sinks" in the surroundings that may interact with the system. With the system and its surroundings adequately described to define the initial state, an examination of the process is undertaken. A process is the interaction between a defined system and its surroundings that results in a change of state (alteration of character, changes 113

114


in property values) of the system. The final state of the mechanical system is described in terms of the properties of the system following com-

pletion of the process. In summary, the analysis is, in general, toward the solution of the following problem: Given the initial state of a mechanical system to be studied and the description of the interaction be-

tween the system and its surroundings during the defined process, characterize the final state of the system. Several processes associated with orthodontic treatment may be analyzed more directly through energy methods than by Newtonian mechanics. Included, and to be discussed in this chapter, are the processes of activation and deactivation of the orthodontic appliance, the placements of permanent bends and twists in an arch wire prior to intraoral engagement, the various forms of heat treatment to which arch wire and some auxiliaries may be subjected, and the actual process of manufacturing arch wire. Because each process to be analyzed generally involves some displacement or motion of the system, and the interaction may include the transfer of heat toward affecting the thermal as well as the mechanical characteristics of the system, it is appropriate that the body or group of bodies or system elements to be studied be termed a "thermodynamic system."

Concepts Leading to the Process Laws In this section appropriate concepts are introduced or reviewed toward

stating and understanding the process laws that govern the work-andenergy approach to mechanics problems.

Displacement As described in Chapter 1, the displacement of a particle is the change in

its position with respect to a reference framework. A straight-line, particle displacement is shown in Figure 4-1. The initial position of the parti-

cle is defined by the vector r1, drawn from point 0 to point P. The particle then moves to point Q. Labeling the coordinate s along the displacement path, the distance covered may be symbolized as where prefix represents "change in" the parameter. Because the the delta displacement is along a straight line in this example, the distance traveled by the particle equals the magnitude of the displacement vector. The displacement vector may be expressed more generally as the difference between the final and initial position vectors, r2 — r1. The displacement of just one point or particle of the body cannot indicate whether or not the orientation or angulation of the body was concurrently changed. If the body is rigid and is moved, but without an angular change, the displacement of the body is a translation and the displacements of all particles of the body have identical vector characteristics. Alternatively, if the orientation of the reference line in the rigid

115

Energy Analyses in Orthodontics

y

Q .—

p

r2

0

x

FIGURE 4-1. The displacement of a particle along a straight-line path.

body is changed in the displacement, some form of rotation has occurred. If the body is nonrigid, linear displacements of particles or angular displacements of lines of the body may be the result of widespread or

localized deformations. Displacements accompanying deformations, both visible external changes in shape and alterations internal to the body, are of substantial interest in this chapter. Orthodontic displacements to be studied subsequently include that of one end of an "elastic" with respect to the other in an extensional activation, the displacements that occur in the elongation or compression from the passive state of a helical-coiled spring, the occiusogingival or faciolingual bending deactivation of an arch wire accompanying tooth movement, and the placement of a tip-back or toe-in bend in an arch wire. Mechanical Work

Displacements produced mechanically, whether or not the body experiences deformation, are associated with forces or force systems. Both

active and responsive forces may cause displacements; on the other hand, displacements can result in the creation of forces. Figure 4-2

116


y

F

P

0

x

FIGURE 4-2. A concentrated force exerted upon, and during its displacement doing work on, a particle.

shows a force exerted on the particle of Figure 4-1. As the particle under-

goes a displacement with the force acting on it, regardless of the relationship between the force and the displacement, the force is said to be doing mechanical work on the particle if the force has a nonzero component in the direction of the displacement. Although, in general, the magnitude or direction of the force may change during the particle displacement, if the vector characteristics of the force shown in Figure 4-2 are all constant, the work done by the force F on the particle as it moves from point P to point Q is equal to F(cos Although computed from two vectors, mechanical work is a scalar quantity. The component of the force involved is that having the direction of the displacement vector. If the force component and displacement vector have the same sense, the angle 4 is less than 900 and the work is algebraically positive; if the senses are opposite (q5 is between 90° and 180°), the work done is negative. If the force is continually perpendicular to the displacement vector, no work is done on the particle by the force. The force of Figure 4-2 would properly be termed an external force if

exerted on an isolated particle or a particle on the outside surface of a body; if the particle is internal to the body, the contact force would also be internal. Internal forces, occurring in canceling pairs, do zero net work on a rigid body; however, within a nonrigid body the net work of a pair of equal and opposite internal forces may be nonzero because the

117


particles on which they act may undergo unlike displacements. In the

absence of friction, the net work done by the force systems connecting one body to another is zero if the parts of the connection are rigid. A couple exerted on a body does work if the body experiences an associated rotation; the work is positive if, in the pertinent plane view, the couple and angular displacement have the same sense. Friction may also do work; if the force of friction is resistive to the displacement, the frictional force and displacement vectors will have opposite senses and the work of friction will be negative.

Work is done by the orthodontist in the placement of permanent bends or twists in arch wires; the forces are transmitted by the hands of the practitioner to and through the pliers and to the wires. In the placement of a tip-back bend, for example, the force system exerted by the pliers on the end of the wire, causing the rotational displacement beyond the elastic limit of the wire resulting in permanent deformation, is

a couple. The clinician does work on an elastic in stretching it, on a spring in compressing it, and on an arch wire in actively engaging it. Appliance elements do work on teeth as their force systems produce desired or unwanted displacements of dental units. In the typical bend or twist placement, activation, or deactivation, the force doing the work changes in magnitude with the displacement, requiring an averaging or a graphical procedure to quantify the work done; examples presented later in the chapter illustrate the methods of computing work in these instances.

The typical process in which mechanical work is done involves a change of state that is reflected partially in the displacements of particles of the mechanical system under study. A cause-and-effect relationship

exists between force and displacement in the process. The external forces that may do work on the system are exerted by elements of the surroundings of the mechanical system, indicating that this form of work is an interaction between system and surroundings. On the other hand, the work of internal forces occurs entirely within the system being

analyzed. In the overall analysis of a process, these two categories of work are often separated because of this difference and its ramifications.

Energy Newton's Second Law for a particle of mass in under the action of a

single, constant force F suggests that this force produces a proportional increase in the speed v of the particle with time t. For a one-dimensional motion, the relationship may be written as (4-1)

where Av and the displacement coordinate s have the same sense, identical to that of the force, as shown in Figure 4-3. The work done by F during the displacement of the particle from point P to point Q may be

118

Bioengineering Analysis of Orthodontic Mechanics m

P

F

Q

0

v+Av

v

As

At FIGURE

4-3. Changes in kinematic characteristics of a particle as work is done on it.

determined. Multiplying both sides of Equation 4-1 by As, noting that

speed is the time rate of displacement, and with appropriate mathematical operations, the following formula for mechanical work is obtained: Work = F(As)

+ Au)2 = + m[(v

—

v2}

(4-2)

The quantity ½mv2 is recognized as the kinetic energy of the particle; Equation 4-2 indicates that the work done on the particle results in a change in the kinetic energy of the particle. This relationship may be expanded to yield the law of kinetic energy: the work of all of the forces, internal as well as external, acting on a mechanical system, equals the change in kinetic energy of the system. The speeds used in the formula must be referenced to a stationary coordinate framework or one moving in a straight line with constant speed; such reference frames are termed Newtonian. Kinetic energy is energy of motion and the kinetic-energy law is a form of, and in one sense the basis for, the more general process law toward which the discussion of this section is directed. A rationale for partitioning the forms of work done in a process by the external and internal forces was suggested earlier. A parallel division separates the work done that is dependent on the displacement path(s) from that which is dependent only on the initial and final states of the mechanical system. An example of path-independent work is that which is performed by the force of gravity (weight). For processes during which the center of gravity of the system undergoes a change in elevation that is small compared to the radius of the Earth, the magnitude of the work done is the product of the weight and the elevation change. Because the force of gravity acts directly downward, gravitational work is negative when the elevation increases. A second example of path-independent work is that which is done on or by a helical-coiled spring in activating or deactivating it in direct tension or compression. In this example the magnitude of the force changes with the amount of spring deformation. Note that the force activating the spring and the resulting displacement always have the same sense; therefore, the work of activation on the spring is positive and by the spring (and on the attached body) is negative.

119

Energy Analyses in Orthodontics Passive state 0

, —

Initial state 1 ______x

Work12 =

Activated state A

Displaced state 2

x

Work done by gravity (left) and within a spring as it is stretched (right).

Figure 4-4 shows the forces and displacements associated with the work done by an external weight force (left) and that by the internal

forces in a spring (right). Because the amounts of work done are dependent only on the initial and final configurations of the system, and with

the form of the law of kinetic energy, such path-independent work is defined as energy change. Specifically, the work done by the weight force is, in magnitude, set equal to the change in gravitational potential energy, and the work done by the internal spring forces is, in absolute value, set equal to the change in strain energy of the spring. Moreover, in the examples of Figure 4-4, increases in energy levels of the bodies, increased elevation of the center of gravity and increased deformation of the spring, are seen to correspond algebraically to negative amounts of

work done on the bodies. To this point in the discussion, two forms of external energy, kinetic and gravitational potential, and one form of internal energy, strain energy, have been introduced. Energy is defined as the capacity of a body or system to produce an effect; thus far that effect has been, exclusively, mechanical work. Energy is a scalar quantity and generally is referenced

so as to be non-negative. Although kinetic energy has an apparently natural reference (zero speed), gravitational potential energy has no such convenient reference level. Although initially it might seem reasonable to set strain energy equal to zero for the passive state of the spring, this form of internal energy is added to the spring in the manufacturing process when it is given its coiled shape. In actual process analyses, only

changes in energy levels are determined. Zero-energy references are unnecessary, but for each individual process such references may be chosen in a reasonable manner, for each energy form, if so desired.

120


A work-energy analysis of standard testing procedures may be undertaken to determine certain mechanical-property values. Such tests were discussed initially in the previous chapter. The general process of interest is the loading of the test specimen. Because the loading is slow, the kinetic-energy change for the process is negligible. Thus, the law of kinetic energy reduces to "The total work of all forces, when individual contributions are algebraically summed, equals zero." With the work of

the weight force, the change in gravitational potential energy, also negligible, the process equation becomes the sum of two work contributions, that done by the external force system in loading the specimen (a positive quantity) and the work of the internal, responsive forces (a negative quantity), which must be zero. Substituting for the work of the internal forces, the final form of the process says that the work done in loading the specimen equals the increase in strain energy of the specimen. Figures 4-5 and 4-6 show the stress-strain and load-deflection diagrams associated with the standard uniaxial tension and ADA bending tests, respectively, which were described in Chapter 3. (In obtaining the diagram of Figure 4-5, the initial volume of the gauge-length portion of

the specimen has been divided out, converting force and deformed length into stress and strain.) With the loading force system increasing in magnitude with deformation, and the resulting need to sum quantities of the form F[AsI (in the tension test cr[A€J) to obtain the external work done, the total work in loading the specimen from its passive state to some deformed configuration is the area under the diagram up to that activated state. The process law says this area also equals the increase in strain energy. Therefore, in Figure 4-5, the process that takes the specimen from point 0 to point el, the elastic limit, requires positive, external

Stress

C

ci

area OABO: modulus of resilience area OCDO: modulus of toughness

0 0

B

Strain

FIGURE 4-5. A stress-strain diagram generated from the data of a standard tensile test.

121


p

Load Couple

a!

area OMNO: resilience area OPQO: toughness

Q

0

N

Angular Deformation

FIGURE 4-6. A load-deformation diagram plotted from the data of a bending test of a cantilevered beam.

work per unit volume, and increases the strain energy per unit volume an amount given by the triangular area under the stress-strain plot up to the dashed line. This area is also the magnitude of the mechanical property known as the modulus of resilience in tension ("modulus of" denoting

per-unit-volume). The amount of strain energy per unit volume that may be added to the specimen before fracture occurs is equal to the entire area under the stress-strain diagram, and is known as the modulus of toughness in tension. Analogous areas under the diagram of Figure 4-6 give the resilience and toughness of the arch-wire specimen in bending. Note that both the resilience and toughness magnitudes are dependent upon the "stress history" of the specimen. In their initial states the specimens contain unknown levels of internal energy; these levels are dependent upon the mechanical and/or thermal processes that occurred prior to the tests (including manufacturing operations). Just as a maximum load exists, which the specimen can sustain without fracture, the specimen also possesses a finite capacity for strain energy, and fracture will occur when this maximum level is exceeded. Heat Transfer and Thermal Energy

In general, a process may be initiated or accompanied by the transfer of

heat between the system and the surroundings. When heat is transferred to the system, a heat source must exist in the surroundings. If heat leaves the system, it is absorbed by a heat sink in the surroundings. In the

absence of perfect insulation between system and surroundings, and

122


with a temperature differential between them, heat will move from one to the other. Amounts of heat transfer are determinable; heat transfer, like mechanical work, is a scalar quantity, but it has sense, always moving from a region of higher temperature to one of lower temperature.

When heat flows to or from a system of finite size and capacity, the temperature of the system changes; the temperature increases when heat is transferred into the system and decreases when heat flows away from the system. Three forms of heat transfer are distinguishable: (1) conduction, by physical contact between system and surroundings; (2) convection, the transporting of heat between system and surroundings by a third medium, often a fluid; and (3) radiation, the emission and

absorption of rays traveling through space between system and surroundings. If the temperature is uniform within a system at a particular lime, then temperature is a property of the system and is state-related. A change in temperature of a body is reflected in the level of activity of the elemental particles that make up the body, sometimes termed the "particle kinetic energy" of the body. Appropriate, then, is the introduction of the concept of thermal energy, a form of internal energy dependent on the temperature of the body or system. Like other forms of energy, thermal energy is inherently a scalar, non-negative quantity. Its zero level might

be said to coincide with a system temperature of absolute zero, but because only changes in thermal-energy levels occur in process analyses, as with other forms of energy, a fixed reference level is unnecessary. Two types of internal energy now have been mentioned: strain and

thermal energies. Although individually they directly depend upon apparently separate, more fundamental properties, deformation and temperature, strain and thermal energies are not mutually exclusive. For example, the results of the tension tests of two specimens to fracture, initially identical in every characteristic except for a finite difference in the temperatures of the test environments, will generally yield significant differences in resilience and toughness. The lower the test-environ-

ment temperature, the more brittle the response of the specimen. Hence, although the strain and thermal forms of energy may be differentiated from each other, the more important consideration is their contribution to the total internal energy of the system under discussion. Existing experimental evidence suggests that the total internal energy of a body may depend upon other parameters besides deformation and

temperature, but only significant changes in the strain and thermal forms of internal energy occur in processes of orthodontic interest. The Conservation-of-Energy Law

A process occurs when an interaction between the system, defined for

study, and its surroundings takes place. Generally, but not always, a process results in a change of state of the system, usually reflected in an alteration in the total energy level of the system. The law of conservation

123


of energy says, in simple terms, that energy can neither be created nor

eliminated within the system. If, in a process, the total energy of the system increases, energy has been transferred from the surroundings into the system. If the total energy of the system decreases, an energy transfer to the surroundings from the system has taken place. If no net energy change of the system occurred during a process (but perhaps levels of individual forms of energy have changed), the net energy trans-

fer to or from the system is zero. Two forms of energy transfer have been defined: mechanical work of

the external forces exerted on the system and heat. Several types of energy, possessed by or contained within the system have been described: the two external forms of note are gravitational potential and kinetic energy and the two significant forms of internal energy are strain and thermal energy. In the general law to be stated, energy transfers are positive when proceeding from the surroundings to the system; energy changes of the system are positive when the final levels exceed the initial

levels. The work of the weight of the system equals the decrease in gravitational potential energy; one or the other, but not both, is included in the formulation. In orthodontic applications, changes in gravitational potential energy as well as electric and magnetic energy contributions rarely occur. With the foregoing definitions, terminology, assumptions, and sign conventions, a form of the law of conservation of energy, sufficiently general for orthodontic applications, may be narrated as follows: The mechanical work of external, contact forces and couples performed on the system plus the heat transfer into the system during a process equals

the increase in internal energy of the system during that process.

This "equation" is also known as the first law of thermodynamics. It is somewhat similar in form, but is actually independent of the law of kinetic energy in that the external work term may be eliminated between the two formulas, yielding an explicit expression for the work of the internal forces. A dimensional analysis of the first-law equation indicates that energy, energy transfer, and work are dimensionally identical. The dimension of work is the product of force and length, so the same is true for heat and energy change (although a variety of units are employed: pound-feet, newton-meters (joules), gram-millimeters, ounceinches, and British thermal units (Btus, for example). The law of conservation of energy is altered somewhat for application to orthodontic processes. First, changes in gravitational potential and kinetic energies may be reasonably neglected. Second, a third form of energy transfer may be defined, to be termed losses, energy escaping from or released by the system in an uncontrolled and usually undesired manner. Because the sense of losses is always from the system to the surroundings, with the sign convention previously assigned to energy transfers, losses are inherently negative quantities. With these alterations, the law of conservation of energy, modified for strict application to orthodontic processes, may be rewritten as follows:

124


The mechanical work of the external, contact forces on the system plus the

heat flow into the system minus the losses equals the net increase in the strain and thermal energies of the system.

A schematic of the orthodontic process law is given in Figure 4-7; therein SE, TE, and En represent strain, thermal, and total energy, respectively.

Available Energy A process generally begins with the system in total (mechanical, ther-

mal, and chemical) equilibrium. The final state as well is usually an equilibrium configuration. Any equilibrium state, however, may be constate (1)

En, = SE, + TE, System

WorkW

Losses L

Surroundings

Heat H

state (1)

Enf = SEf + TEf

FIGURE 4-7. A schematic depicting an orthodontically related process and the workenergy law that governs it.

125


strained such that, if released from its constraints, without any catalyst

an energy transfer to the surroundings occurs as the system proceeds toward a minimum-energy configuration. Orthodontic examples of these constraints are the maintenance of an elastic in a stretched state and an arch wire at an elevated temperature in a furnace. The available energy of a system is that portion of the internal energy that is released by removing all constraints and permitting the system to come to a completely passive equilibrium state. The total internal energy of a system may be expressed as the sum of its available and unavailable ("locked in") parts. The concept of available energy becomes more meaningful in the study of orthodontic processes within the following sections of this chapter.

Activation and Deactivation Processes When activating an orthodontic appliance, the practitioner applles exter-

nal force and does mechanical work on one or more elements of that appliance. Because negligible heat transfer occurs in such processes, this quantity of work equals the increase in internal energy; since little if any

change in temperature of the system takes place, the internal energy increase is reflected in the strain-energy form. The appliance is taken in the process from a passive state to a constrained, activated state. The visible evidence of the process and the added strain energy are the deformations of activation. The activation process occurs over a short time interval and negligible losses accompany it, but losses may occur over the relatively long time period to follow when the appliance is "working" on the dentition. If no losses, as defined previously, occur subsequent to activation, and the activation does not take the material(s) be-

yond their elastic limits anywhere in the appliance,

all

of the

strain-energy "input" is available to produce displacements of teeth or other movements within the dentofacial complex. As a first example, consider the activation of a linear, helical-coiled spring in compression. (This spring might be wound around an arch wire and positioned against the mesial extent of a canine bracket, for example; when activated the spring provides the distal driving force to do the work of retraction.) The design of the spring is such that, even when the compressive force pushes all of the coils into full contact with one another, the elastic limit of the spring-wire material is not exceeded. The laws of kinetic energy and conservation of energy say that the net contribution from the external and internal forces is zero and the work done by the external forces equals the increase in strain energy of the spring. The spring, compressed from the right end is shown in Figure 4-8; the initial configuration is drawn dashed, and the final, activated state is superimposed. The activated spring is "cut" to expose the internal force system and a free-body diagram of the right-hand portion is included in the figure. Note that the responsive force against the left end

126


Responsive force

d,

"cut"

force

FIGURE 4-8. The process of compressing an open-coil, helical spring.

of the spring does no work because no displacement occurs there. The relationship between the external force that compresses the spring and the spring deformation is illustrated in Figure 4-9. The spring is termed "linear" when the plot of this relationship is a straight line. The force is seen to increase with the spring deformation and the work done to produce the 8-mm compression of the spring from its passive configuration is the cross-hatched area under the plot:

Work =

=

(160)8 = 640

g-mm

(4-3)

The process law indicates that, with negligible losses during activation, 640 g-mm of strain energy have been gained by the spring. Figure 4-9 shows the loading or activation plot of the linear spring. Because the elastic limit of the spring has not been exceeded, and if the spring was formed from a hard-metal alloy such as stainless steel, no losses of energy will occur over time as an activated configuration of the spring is maintained. As a result, the unloading or deactivation plot will trace the loading curve, indicating that all of the added strain energy of activation is available energy. A second process may now be analyzed in

which the spring is an energy source in the surroundings and under study is the system on which the spring will do work (e.g., a tooth). The

work capacity of the spring in this second process, during which the spring is deactivated to some extent, equals the 640 g-mm of strain energy added to it in the activation process.

127 Energy Analyses in Orthodontics

200

(0

E .2?

160

F1)

0) C-)

0

LI-

-o

0)

a-

a

120

80

40

0

0

2

4

6

8

10

Compressive Deformation (millimeters)

FIGURE 4-9. The force-deformation plot for the linear spring of Figure 4-8.

The analyses of the activation and deactivation processes for an "elastic" are similar to those just discussed for the spring, but notable differences exist in the response of the elastic. Figure 4-10 shows the

activation plot for an elastic band or module, fixed at one end and stretched by an external activating force at the other end. As with the spring, the process laws indicate the equality of magnitudes of work done by the external and internal forces and the increase in internal

P1)

0 0

as

1.0

1

a (inches)

Passive state

FIGURE 4-10. The activation ot an elastic module and the associated torce-elongation diagram.

As noted in Chapter 3, because of the generally amorphous nature of elastic materials, the initial portion of the force-deformation plot is often nonlinear. To simplify the computation of the area under the loading curve, a straight line is sketched that enables equal, partial areas to be cut above and below the actual plot. The obtained triangular area is the mechanical work of the activating, external force,

energy.

(18)1 =

9

oz-in.

(4-4)

and is also the strain energy added to the elastic in the short-term activation process that stretches it, in this example, 1 in. Although springs and activated arch wires may be expected to deactivate partially, if not totally, during between-appointments periods, the activated elastic module often maintains much of its initial stretch for a

substantial length of time. Because of the relatively weak, secondary bonding within certain amorphous materials, when maintained in a stretched configuration many of these materials lose tension over time. The force-versus-time plot for the example polymeric element under constant strain conditions is illustrated in Figure. 4-11. The phenomenon exhibited, defined in Chapter 3, is "relaxation"—loss of force with time and without deactivation. An energy-analysis schematic of this process is provided in Figure 4-12. Like the spring, the activation of the elastic module provides it with available energy to do work on another system; however, although the activating process should not take the material of the module beyond its elastic limit, the amount of available energy may decrease with time due

129


24

U,

0 C

=

0 a)

16

0 0

LL

8

0

0

12

6

18

24

30

36

Time (hours)

FIGURE 4-1 1. A relaxation plot characteristic of some polymeric materials.

to relaxation. In the relatively long-term deactivation process, it is gener-

ally impossible to quantitatively separate the energy transferred from the elastic module into external-work and loss portions. To illustrate the effect, however, consider maintaining (constant extensional strain) the activation depicted graphically in Figure 4-10 for several hours. Figure 4-13 shows the force-deformation relationship upon subsequent total unloading of the module. Note that this plot is also nonlinear and the force at the initiation of unloading is somewhat reduced, due to relaxation, from the level immediately following the activation process. Again,

the straight line is sketched in order to aid in the computation of the area: (13)1

= 6.5 oz-in. = strain energy released

(4-5)

The difference between the initially induced strain-energy increment and the strain energy released are the losses of relaxation and the energy no longer available to do work. In other words, the 6.5 oz-in. was the available energy in the elastic at the time of total deactivation. In general, the magnitude of such losses depends on the material of the elastic, the magnitude of the initial force relative to the geometry of the elastic,

and the length of time the activation is maintained. Because the oral environment is potentially more harmful to the polymeric materials of "elastic" elements than the normal air environment, intraoral "elastics"

130


20 oz 4

d

-

£0+lin. Initial state

I I 4

I Losses

/ to + 1

I

p

in.

—I

15 oz

Subsequent state

FIGURE 4-12. An energy-analysis schematic of the relaxation process.

should be expected to experience somewhat more dramatic relaxation

and, therefore, greater losses than "elastics" that activate extraoral appliances.

Consider now the process of activating an arch-wire segment in bending. As shown dashed in Figure 4-14, the segment, supported at its ends, is initially (passively) straight. The clinician applies an occlusogingival force between the supports to engage the segment in a bracket. As with the spring and elastic, the force during the process increases from

zero to a maximum value at the point of bracket engagement. Once more the process law says that the mechanical work of the external

alyses in Orthodontics

(e1, F2)

I

to

0.5

e (inches) 13.

The unloading diagram for the elastio module of Figure 4-10.

equals the increase in internal (strain) energy of the segider study. (It is assumed that no displacements occur at the s during short-term activation, so no work is done by the reForces

forces (Qi'

and couples (C1,

there.) The force-deforma-

t for an activation to a state or configuration below the elastic hown in Figure 4-15. Negligible losses occur during and follow'ation. Hence, the work done in this activation of the segment is as in the previous examples:

+ (2)1.5 = 1.5 oz-mm

(4-6)

oz-mm, also the strain energy added, is the energy available to during deactivation. In the orthodontic deactivation process, in system under study remains the wire segment, the possibility at, in addition to the desired tooth movement, one or both of the

s might be displaced. If so, in total deactivation, the desired

132


Ge

P

C,

Ce

t: left r:

right

FIGURE 4-14. The activation of an arch-wire segment in bending.

energy transfer to the tooth would be less than the 1.5 oz-mm by the

amount of work done by the responsive forces in displacing the supports. (Similarly, in actual situations, the spring or elastic module could deactivate with both ends of the element undergoing movement.) If the tooth is somewhat farther out of alignment than was suggested in the foregoing discussion, the activation process could result in localized inelastic behavior of the segment. Point 1 on the loading curve in Figure 4-16 is the termination of the previous elastic activation; point 2 represents the final state of the present activation process. Once again using straight-line aids, dashed in the figure, the work done in activating the segment is quantified as the sum of three parts making up the total area under the plot:

+ (2)1.5 + 2(1) + + (1)1 = 4.0 oz-mm

(4-7)

The 4.0 oz-mm of work equals the strain energy input to the wire and, although no substantial losses as defined are expected if the wire material is relatively ductile (and the deformation at fracture is substantially in excess of 3 mm), the inelastic activation of the wire limits the energy available to do work. The deactivation process from point 2, depicting maximum movement of the tooth and assuming no work done at the supports of the wire segment, is shown as the solid line in Figure 4-17.

133


3 (I) Q)

0 C

0 0. ci)

0

0

U-

2

(1.5,2)

0 0

3

2

1

Deflection 8 (millimeters)

FIGURE 4-15. An elastic activation plot for the wire segment of Figure 4-14.

The cross-hatched area under the deactivation plot is

— (2.8)2.1 = 2.9 oz-mm 2

(4-8)

and is the energy available to do work on the tooth. The unavailable portion of the internal-energy input during activation is 4.0 — 2.9 = 1.1 oz-mm

(4-9)

7eering Analysis of Orthodontic Mechanics

/

(2.5,3) 2

e/

1

2

3

Deflection

(millimeters)

4-16. Loading of the wire segment of Figure 4-14 such that the elastic limit has 'ceeded.

is T

been "locked in" the wire at the activation site. This unavailable

differs conceptually and actually from the losses sustained by the

due to relaxation; although both phenomena result, in effect, in a

of potentially available energy to do work, energy was transother than through external work, from the elastic, but not from re, between the end of the activation process and the conclusion ctivation. (Relaxation of metals generally occurs only when they

135


3 a) C)

0 aa)

U

0

U-

2

1.1 oz-mm

/

/

/

/

/

/

/ /

(2.5, 2.8)

/\

0

0

1

3

2

Deflection

(millimeters)

FIGURE 4-17. The deactivation of, and the available and unavailable energies for, the wire segment activated beyond its elastic limit.

have been strained almost to their fracture point or when they are main-

tained under load at substantially elevated temperatures.) Relaxation and inelastic action have been noted in this section as counterproductive toward maximizing the available energy of the activation process. Friction is a third phenomenon that "detours" a portion of the strain energy available to a deactivation process. Recalling the linear

spring analyzed earlier in this section, now consider a canine as the system with the linear spring activated against it at the initiation of a process. A free-body diagram of the tooth during the process is shown

136


F' Holding couple C d

m

Spring force P

Frictiona' force

Cpenodontrun

a

FIGuRE 4-18. A buccal-view, free-body diagram of a canine during its retraction into a first-promo/ar extraction site.

in Figure 4-18. Ideally in the process no localized deformations should occur and, therefore, no change in the internal energy level of the tooth. In the absence of friction and tipping, the process law would say that the work done by the spring force (equal, in total deactivation, to the strain energy added in activating the spring) plus the work done by the periodontal force (sense of force to the right, sense of displacement to the left: negative work) equals zero. Considering bodily movement wherein no work is done against the couples, a portion of the energy transferred from the compressed spring during the displacement will be carried into the stationary arch wire by friction. If the displacement is indeed bodily movement, the division of available energy in the spring between the periodontium and the arch wire is in the same ratio as the periodontal and frictional forces, inasmuch as a mesiodistal force balance must exist throughout the process and the displacements of the particles, on which the three forces act, are all identical excepting locations. If tipping occurs, negative work is done

137


by the couples which further lessens the desired energy transfer from

the tooth to the periodontium. In this example, typical of processes in the presence of unwanted friction, considering the arch wire as the system under study for the moment, the energy transferred into the wire by the work of the frictional force is initially reflected in an increase in the thermal form of internal energy. If the arch wire is stationary (the frictional force having done work during the displacement of the bracket with respect to the wire), with the momentary, localized increase in wire temperature, all of that energy increment will be transferred immediately into the surroundings with an accompanying return of the wire to the ambient temperature. Hence, in such instances, in a very real sense, energy transfer in the form of the work of friction is a loss.

Strain Hardening and Heat Treatment of Metals and Alloys Mentioned earlier was the existence of a relationship between the level

of internal energy, with the system under study in an unconstrained state, and the mechanical properties of the materials of the system, the metals and alloys in particular. In addition, discussed previously was a sequence of processes in which energy was transferred into a material followed by restoration of the initial equilibrium state with its surround-

ings; the processes as a whole resulted in a net change in the level of internal energy because of inelasticity during loading. Apparent, then, is the ability to design processes which have as their objective to alter the internal-energy level and, thereby, change the mechanical properties of

the system materials. Also indicated in prior discussion was an upper limit to the internal-energy capacity, per unit volume, of a material; failure hypotheses known as the maximum-strain and maximum-internalenergy theories are often employed by practicing engineers in designing load-carrying members and structures. High levels of internal energy, principally in the form of strain energy, are accompanied by residual or "locked in" stresses. Superposition of loading on such stress states may lead to unexpected fractures; hence, a process, which will at least partially release residual stresses, may be carried out to advantage prior to the application of the load. Although the level of internal energy cannot be readily quantified, good indicators of high levels are substantial hardness and lack of ductility. Two categories of processes, carried out on metals or alloys before in-service use, alter the unconstrained-state internal energy in the material. Strain hardening is a process in which energy in the form of mechani-

cal work is transferred into the material, taking the material beyond its elastic limit. When strained elastically, the interatomic bonds are only stretched; upon release of the straining action, the deformation disappears. The loading of a material to a state beyond its elastic limit causes the breaking of interatomic bonds and the establishment of new bonds; permanent, relative displacements between grains occur in crystalline

138


materials with the result, upon unloading, of an altered external shape

and, generally, a different stored-energy level compared to that of the originally passive state. The strain-hardening process is undertaken toward an increase in internal energy; however, the process can, particularly in a high-temperature environment, result in a reduction in internal energy. When undertaken at normal room temperature or below, the process is called cold working. Strain hardening may be carried out at an elevated temperature to enhance the interaction between the strain and thermal forms of internal energy; this process is known as hot working. The strain-hardening process, when initiated from a relatively low internal-energy state with a moderate energy transfer, results in increases in the elastic strength and range, the resilience, and, perhaps, the ultimate strength; however, the process reduces the ductility. Because for each material an upper bound

exists to the amount of internal energy per unit volume it can retain without undergoing drastic structural change (including, perhaps, fracture), the degree to which the material can be cold- or hot-worked is limited. In general, when the point of maximum strain-energy capacity is approached, control of the process is lost and, subsequently, a sizable (and, perhaps, explosive) energy loss occurs accompanying fracture. Heat treatment is the general name given to those processes in which the catalyst toward changing the internal-energy level in the metallic material is the transfer of energy into the system entirely in the form of heat. The process is carried out in three steps:

1. The system temperature is elevated by placing it in a high-temperature environment (e.g., a furnace or a hot salt bath) or by electricresistance/induction heating; heat transfer is through convection in using the furnace or bath and by conduction with electrical procedures.

2. Upon reaching the desired temperature, the system is maintained there for a specific period of time. Because no losses are occurring with the system at the temperature of the furnace or bath, while energy is continually flowing into the surroundings with the direct electrical methods, the potential for temperature control is greater with the system enveloped in the high-temperature environment. 3. The system is returned to its initial-state temperature. Although the lime taken to elevate the temperature is not particularly critical, the step-3 time is if an objective of the heat treatment is a specific alteration of the atomic arrangement within the material. Note that sol-

dering and welding procedures also use heat to initiate processes. Although not discussed in detail in this text, a by-product of either process may be a heat treatment. Accordingly, these procedures must be carried out carefully with the realization of the potential effects upon localized internal-energy levels and mechanical properties. The difference between strain-hardening and heat-treatment processes is clear with regard to the forms of energy transfer initiating these

139


Losses (uncontrolled energy transfers from the system) are substantial in heat-treatment processes; losses are generally negligible with cold working as long as the energy input does not result in fracture. Ariother distinction between the process categories is in the internalenergy change achieved: typically an increase from strain hardening, but heat treatment may result in either raising or lowering the internalenergy level. Energy schematics showing strain-hardening and heattreatment processes are presented in Figure 4-19. Three types of heat-treatment processes should be familiar to the orthodontist. Following the placement of bends and/or twists in an arch wire, a cold-working procedure, a stress-relief heat treatment may be desirable, particularly if the bends are sharp and when complex loops have been fabricated. The purpose of a stress-relief heat treatment is the release of at least a portion of the residual stresses "locked in" the material, and recovery of some of the ductility lost through strain hardening. processes.

Depending on the material, the dimensions of the member, and the degree of strain hardening, the stress relief may also raise or lower the elastic limit and resilience and/or the ultimate strength. Step 2 in the stress-relief process in a furnace of stainless-steel orthodontic wire is generally carried out at 700 to 900°F for 3 to 5 mm; the time necessary at lower temperatures is excessive, from a practical standpoint, and higher temperatures will likely produce carbide precipitation and loss of corrosion resistance. An annealing heat treatment is carried out at a temperature substan1

Initial state

Work W

Heat

H

Losses L Strain Hardening

> En1

Heat Treatment

Enf

En,

FIGURE 4-19. Energy-analysis schematics of general strain-hardening (left) and heattreatment processes (right).

140


tially above that to stress-relieve. The temperature and resulting particle kinetic energy are high enough to dissolve at least a portion of the inter-

nal bonding and permit atomic reorganization. An anneal produces a substantial loss of internal energy and an accompanying reduction of all strength properties and resilience if the material was previously strain hardened. The process also increases ductility, making the material more malleable. The annealing of stainless-steel orthodontic wire requires only a few minutes at temperatures of 1800 to 2000°F. The mate-

rial is fully softened and any carbides are brought into solution. The cooling (third) step in the process is carried out rapidly by quenching in water or oil to prevent reformation of chromium carbides. The time at temperature is generally kept short to minimize the potential for grain growth and, therefore, maximize ductility and corrosion resistance. A hardening heat treatment is usually a long-term process with a step-

2 time of several hours, carried out at a temperature somewhat below that necessary to anneal and with the material often cooled rapidly by quenching. Hardening generally will improve strength properties, particularly with respect to their levels as annealed. Although the particular stainless steels from which orthodontic wire is manufactured will not take a hardening heat treatment, the strength and resilience of gold and chrome-cobalt-alloy wires are raised upon hardening (but at the "expense" of some reduction in ductility). With the effects on properties of strain hardening and heat treatments now outlined, an overview of the processes employed in the manufacture of orthodontic stainless-steel wire may now be undertaken. Initially a relatively large ingot of the material is heated to a temperature near that necessary to anneal, in order to facilitate the working operations to follow. At the elevated temperature the material is "hot rolled" (hot worked), which results in a reduction in cross-section and corresponding lengthening of the member. Now the ingot has been transformed into a straight bar having a round cross-section. To reduce the bar to the final cross-sectional size and shape, the bar is drawn through a series of dies. Each successive drawing is a strain-hardening process that decreases the ductility of the bar. Accordingly, heat treatments must be interspersed between drawings to keep the internal-energy level from becoming excessive. The formation of rectangular wire is accomplished either by drawing the initially round bar through a succession of dies that gradually change the cross-section from circular to rectangular or by performing a rolling operation on the round wire follow-

ing the drawing processes. At some point in the overall wire-formation procedure an annealing heat treatment must take place to ensure that the wire, as received by the practitioner, has the metallurgical format characterized by the carbon atoms in solid solution. The anneal should occur as near to the end of manufacturing operations as permissible, given the strength and resilience requirements of the final product. All of the orthodontic stainless-steel wires currently marketed contain the same principal constituents and virtually identical percentages of iron, chromium, nickel, and

141

Energy Malyses in Orthodontics

carbon. High- and moderate-resilience wires, and "standard" wires pos-

sess different strength, resilience, and ductility characteristics, notable in the hands of the clinician, because production detailing varies somewhat among wire manufacturers and according to desired mechanical properties. The as-received, high-resilience wires exhibit more strain hardening and, therefore, a higher internal-energy level; ligature wire, having a low elastic limit and substantial ductility, is fully annealed as received. With the marketing of the "beta titanium" (titanium-molybdenum alloy) wire, practitioners now have their choice among five arch-wire materials. Each material has specific time and temperature formulations for the different heat-treatment processes, and wires of the same material are often available with several different internal-energy levels. Accordingly, clinicians must be attentive to the vendor's instructional literature when placing permanent bends or twists or when considering or undertaking soldering, welding, or heat-treatment processes.

Work-Energy Analysis of the Preparation of an Arch Wire Orthodontists must be aware of the possibility of a wire fracturing in the

mouth during a between-appointments period. Care must be taken not to induce excessive force or energy into a wire during fabrication and activation. Mastication may load the appliance beyond that of activation and add to the internal energy present in the engaged wire. Because the amount of internal energy in a passive wire, even prior to bend or twist placements, cannot be quantified and may be substantial, the danger of raising the internal-energy level excessively is more subtle than that of overloading the wire with respect to force magnitude. Consider the preparation of an arch from a straight segment of asreceived, stainless-steel wire. The experienced clinician can, by the "feel" of the wire, gauge its relative stiffness in bending, but the level of ductility and the toughness cannot readily be estimated except from specifications often known only by the wire manufacturer. The wire segment is cut to an appropriate initial length and the arch form and offset bends are placed. The amount of additional internal energy localized at any cross-section is related directly to the size of the permanent deformation placed there. The strain energy induced in preparation of the overall arch form is distributed primarily over that which will be the anterior portion of the wire; the stored-energy increment per unit volume is nominal. The localized increases in internal energy will be greater at the sites of the offset bends. If, for example, this arch wire is to provide guidance and, with the buccal segments, anchorage for intraoral canine-retraction mechanics, second-order bends are placed including tip-back bends at the terminal molars; toe-in bends are also placed at the molar sites. Substantial amounts of additional internal, strain energy are induced

142


at the locations of the permanent bends, particularly just mesial to the buccal tubes where the lip-back and toe-in bends are superimposed on one another. Moreover, wire engagement likely activates the bends, and the work of the clinician in placing the appliance inputs additional strain energy at the molar sites. At no other cross-sections do both permanent deformations and induced orthodontic loading exist in this example, if fracture would occur, the most likely location is one of the "critical" cross-sections just mesial to a terminal molar. If the wire is ductile and possesses substantial toughness as received, there need be little fear of fracture subsequent to a moderate amount of inelastic bending followed by activation. The more highly resilient wires are generally more brittle, however, and the clinician may wish to increase the "effective toughness" of the wire as a safety measure before placement and activation of the appliance. This may be accomplished without the loss of resilience only through a stress-relief heat treatment, and only if the process is conducted properly so as to result in a net decrease in the internal energy of the passive wire (in particular, localized energy decreases at the critical cross-sections). Figures 4-20 and 4-21 depict the effects of bend placements, of activation, and the internal-energy capacities for a pair of otherwise identical

stainless-steel wires, without and with a stress-relief heat treatment prior to activation. The figures indicate conditions at the critical crosssections for the example under discussion. In Figure 4-20 curves 1 and 2

a)

0 0

U-

0

Deformation

FIGURE 4-20. The force-deformation diagram for an example arch wire, activated in bending tollowing permanent-bend placement.

143


a) C)

L2

0

HT

Deforniatton

FIGURE 4-21. The force-deformation diagram for an arch wire, activated in bending subsequent to permanent-bend placement and a stress-relief heat treatment

represent the loading and unloading of the wire during the placement of

the tip-back and toe-in bends; note the permanent set and the new positions of the elastic limit following bend placements. Curve 3, coincident with part of curve 2, illustrates an elastic activation process as the wire is

engaged in the brackets and buccal tubes. The triangular area under curve 3 represents the available strain energy of activation of the wire at the molar, revised through bend placements (strain hardening) from its as-received value. Curve 4 (dashed) is the remaining portion of the plot to fracture and the cross-hatched area represents the internal energy per unit volume that may be added to the as-activated level before fracture will occur. Mastication, which causes superposition of loading increments upon the activated state, would make the dot in the figure move along curve 4 and, perhaps, take the wire material at the critical section beyond the "new" elastic limit. The cyclic loading and unloading pattern of mastication may result in fatigue of the wire (defined in Chapter 3), which will effectively reduce the toughness, even when the elastic limit is not exceeded. As in Figure 4-20, curves 1 and 2 of Figure 4-21 indicate the place-

ment of permanent bends in the wire at the molars. To free at least a portion of the "locked in" (residual) stresses at the critical cross-sections,

144


stress-relief heat treatment is undertaken. This process has little effect on the as-bent geometry and no mechanical work is involved, so it may a

be represented simply as the dot marked "HT" on the plot of Figure 4-21. With some ductility remaining in the passive wire following bend placement, the stress relief results in a further raising of the elastic limit

and an accompanying increase in the resilience (beyond the levels achieved through bend placement only). Curve 3, here as in Figure 4-20,

represents the engagement of wire in the buccal tubes and brackets, activating the tip-back and toe-in bends. Because the heat treatment also

raises the ultimate strength of the wire with negligible effect on the deformation to fracture, both the maximum tolerable load and the internal-energy capacity are increased. As in Figure 4-20, curve 4 (dashed) and the cross-hatched area of Figure 4-21 are the remainder of the potential loading curve to fracture and the revised toughness, respectively. With identical bend placements in and activating deformations of the two wires, the gains in both load and additional internal-energy capacilies attributable to the stress-relief heat treatment are apparent from a visual comparison of the two figures.

Synopsis first four chapters contain the concepts of mechanics and materials science needed by the clinician to embark on an intensive, but not exceedingly complex, study of the orthodontic appliance in the manner of a structural bioengineer. To complement the use of Newtonian mechanics in appliance analysis and design, in this chapter the approach of work and energy has been discussed toward examining the mechanical and thermodynamic behavior of the appliance elements over a finite time interval. The concepts of mechanical work, heat transfer, external and internal energies, and system with its surroundings were introduced, followed by the statements of the law of kinetic energy and the principle of conservation of energy. Examples of orthodontic processes discussed were the activation and deactivation of a coiled spring, an elastic module, and an arch-wire segment. Strain hardening and heat treatments have been considered with emphasis on their applications to orthodontics. Prior to the pursuit of direct analyses of orthodontic-appliance members and assemblies, an examination of the kinematics and kinetics of tooth movement is undertaken in the upcoming chapter. Also appropriate is a discussion of the interface of the physiology and the mechanics, and the response of the periodontium and the dentofacial complex at large to the application of force. These and related topics are the subject of Chapter 5 and, with the tools of analytical mechanics, complete the basis from which a bioengineering study of orthodontic therapy and appliances may be approached.

These

145


Suggested Readings Allegheny Ludlum Steel Corporation: Stainless Steel Handbook. Pittsburgh, Al-

legheny Ludlum Steel, 1959. Beer, F.P., and Johnston, E.R.: Vector Mechanics for Engineers-Statics. 3rd Ed. New York, McGraw-Hill, 1977, Chapter 10. Fillmore, G.M., and Tomlinson, J.L.: Heat treatment of cobalt-chromium alloys

of various tempers. Angle Orthod., 49:126—130, 1979.

Fillmore, G.M., and Tomlinson, J.L.: Heat treatment of cobalt-chromium alloy wire. Angle Orthod., 46:187—195, 1976.

Greener, E.H., Harcourt, J.K., and Lautenschlager, E.P.: Materials Science in Dentistry. Baltimore, Williams & Wilkins, 1972, Chapter 7. Hayden, H.W., Moffatt, W.G., and Wulff, J.: The Structure and Properties of Materials. Vol. III. New York, John Wiley & Sons, 1965, Chapter 8.

Howe, G.L., Greener, E.H., and Crimmins, P.S.: Mechanical properties and stress relief of stainless steel orthodontic wire. Angle Orthod., 38:244—249, 1968.

Kohl, R.W.: Metallurgy in orthodontics. Angle Orthod., 24:37—52, 1964.

Marcotte, M.R.: Optimum time and temperature for stress relief heat treatment of stainless steel wire. J. Dent. Res., 52:1171—1175, 1973. McLean, W.G., and Nelson, E.W.: Engineering Mechanics, New York, Schaum Publishing, 1962, Chapter 18. Phillips, R.W.: Skinner's Science of Dental Materials. 7th Ed. Philadelphia, W.B. Saunders, 1973, Chapters 16 and 34.

Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982, Chapters 1, 7, and 8. Zemansky, M.W., and Van Ness, H.D.: Basic Engineering Thermodynamics. New York, McGraw-Hill, 1966, Chapters 3 and 4.

Response of Dentition and Periodontium to Force

activation of an appliance by the orthodontist creates a complex system of forces that is transmitted generally through the dentition and into the dentofacial complex at large. The system of forces is developed initially within the appliance due to the mechanical work of activation in which one or more appliance members are deformed from the passive configuration. The force system is carried throughout the appliance, The

transmitted from one member to another through their interconnections, and to individual tooth crowns. Force systems delivered to crowns are carried longitudinally through each tooth and are transmitted by the roots to and through the periodontal ligament into alveolar bone and to the maxilla or mandible and beyond. The alveolar process and the bones of the face respond to several categories of force. Physiologic forces exist continuously, but perhaps are most prominent during the eruption of teeth. Functional forces of mastication and deglutition periodically produce force against the dentition, as may the tongue and lips. Orthodontic forces are exerted through me-

chanical connections of activated appliance members to the tooth crowns and induce movements of individual teeth or segments. Forces delivered to the dentofacial complex that result in movements of the maxilla or mandible, or relative displacements of portions of these hard tissues (e.g., palate splitting), are generally termed orthopedic. An individual tooth receives crown force principally from the connection of arch wire to orthodontic bracket. Through incorporation of stops, loops, ligation, and use of round and rectangular wire, forces and couples may be transmitted in faciolingual, mesiodistal, and occlusogingival directions between arch wire crown through the "universal" bracket. Moreover, contact forces may reach the crown through other

than the bracket connections. Eyelets, staples, or buttons may be welded to the orthodontic band or bonded directly to the crown surface where desired and space permits, and appliance members (e.g., elastics, 146

147


ligatures) activated against them. Adjacent teeth within the same arch

may push against one another through direct crown contact; they also may "pull" on each other, interconnected through the transseptal-fiber system. Finally, teeth in the opposing arches may exert force upon one another through occlusal surface contact. The individual tooth is highly resistant to localized deformation and is positioned in a flexible supporting structure. The application of crown force displaces the tooth as a whole. The support response to short-term forces, such as those of mastication, is essentially elastic. Tooth displacements arising from forces existing over extended periods of time—orthodontic forces, for example—must b97'

analyzed in two parts: the initial displacement associated with the in* mediate response of the nonrigid alveolar process and the displacement occurring with the biologic remodeling of the tissues. The deformation of the periodontal ligament, accompanying initial tooth displacement, produces a responsive force system against the root. This system is the action-reaction counterpart of that delivered by the root to the ligament, and arises due to the resistance to deformation inherent in the ligament and the bone behind it. Because the stiffness of the ligament itself is relatively low, the deformation of it occurs through-

out the entire contact area between ligament and root; hence, the mechanical response to tooth displacement is a force system that is distributed over virtually the entire root-surface area. Initially, that response may reflect highly varying intensity from one location to another over the root-ligament interface; if the applied force remains for a period of time, remodeling of the alveolar process will alter that intensity pattern toward a more uniform distribution. The ligament is less than 0.5 mm in thickness everywhere, its weight and inertia are negligible, and the alveolar bone-ligament and root-ligament interface areas are virtually equal. Accordingly, the force distributions transmitted from root to ligament and from ligament to bone are nearly identical in all characteristics. Al-

though mature bone is much stiffer than the periodontal ligament, forces against the bone can result in slight deformations, particularly from pressure against the facial or lingual alveolar crestal regions. Upon

reaching the alveolar bone, ligament force against it resulting from crown loading travels into and through the cortical plate and into the bone proper. This force does not disappear; instead, it is distributed farther throughout the bone volume, decreasing in intensity with distance inward from the bone-ligament interface as the effective area of distribution grows. Several reasonable idealizations are proposed at the outset in this bioengineering analysis. The tooth is assumed to be a rigid body; it is displaced, but not deformed, under the action of crown loading and the accompanying response from the periodontium. The weight of the tooth is small compared to the contact forces that may be exerted on it; the weight of each tooth, as well as that of the periodontal ligament, may be neglected. Furthermore, even in the short period of initial displacement following creation of crown force, the inertias of all components of the

148


structural and biologic systems are never of significant magnitude to

warrant inclusion in the analysis. In structurally characterizing the penodontium, its response to loading is time-dependent and its biomatenial components must be separated due to differences in mechanical properties. As mentioned previously, the overall stiffness of the periodontal ligament is low compared to that of bone, and although the response mechanisms are different from one another, the stiffnesses of the periodontal ligament in tension and compression generally will be assumed equal in the light of present knowledge. The comparable stiffnesses of alveolar bone are at least one order of magnitude higher than those of

the ligament and the greater hardness and stiffnesses of the cortical plate compared to the spongy bone beneath it may be a factor in portions of the analysis. Mechanical strengths of the teeth and periodontal components need not be of concern; inherent (anatomic) controls against structural failures exist in response to most forms of possible crown loading. In general, replacing strength criteria in ordinary structural (engineering) analyses are pathologic considerations in the study of the periodontium and its influence on the appropriate characteristics of induced forces and the design of appliances.

The total analysis or design problem associated with orthodontic therapy is in two distinct and interesting parts. The "engineering" of appliances will be discussed in subsequent chapters. The biomechanics of the living system to which those appliances are attached is the subject of this chapter.

Mechanical Response of the Individual Tooth to Applied Force Under the action of an applied force system the single tooth experiences

a whole-body displacement due to the nonrigid nature of its supporting structure. To adequately describe such a displacement, a reference frame is established with respect to the initial position and normal orientation of the aligned tooth. This framework includes three mutually perpendicular axes. One axis is coincident with the long axis of the aligned tooth, the second is directed faciolingually and passes through the center of the proper bracket location, and the third is oriented mesiodistally. The cen-

ter of the framework, the point common to all three axes, is located approximately at the center of the crown and is labeled point CC. This reference system, described previously in Chapter 1, is shown within a typical tooth in Figure 5-1. Another useful reference point is the center of resistance of the tooth; the long axis passes through it and the point is designated the cre. Four additional reference points will be introduced subsequently; however, because the tooth is a rigid body, the displacements of these four points and the cre may be readily obtained from the absolute displacement of the long axis, a line perpendicular to the long axis through CC, and point CC itself. (Recall the two-dimensional displacement example in Chapter 1 and the associated Figure 1-5.)

149


a

e

a

f

d

m

0

FIGuRE 5-1. A localized reference frame fixed within the tooth and having its origin at the crown-center point CC.

Defined in Chapter 2 were three general forms of rigid-body displacement: pure translation from an active force system mechanically equivalent to a single force with its line of action passing through the center of resistance; pure rotation from an active force system having a single couple as its resultant; and a combination of translation and pure

rotation termed generalized rotation resulting from action equivalent to a force and couple referenced to the center of resistance. Orthodontic displacements as defined have clinical origins, but each may be categorized

under one of the three general forms just mentioned. Although multiple-rooted teeth do not possess the near-axial symmetry of the singlerooted teeth, no special differentiation in overall displacement designations is necessary; point-wise descriptions of root-apex movements, however, may be different for specific, whole-body displacements. The partitioning of the response to crown loading into immediate and longterm components was previously mentioned; nevertheless, a description of displacement ordinarily defines the position and orientation of the tooth, at a specified instant of time after loading, referenced to the preload configuration. The description of an orthodontic (individual-tooth) displacement is often a complex undertaking because the load is principally applied away from the long axis and through the bracket. For this reason, the displacement is decomposed into what is seen individually in the occlusal plane and in two mutually perpendicular vertical planes. Three forms of orthodontic translation are recognized: extrusion, intrusion, and bodily movement. The first two may be viewed from a vertical or transverse perspective, but ordinarily they are not discernible in an occlusal view because particle displacement vectors are parallel to the passive,

150


long-axis orientation. In bodily movement, particle displacement vectors

are generally parallel to the occlusal plane or perpendicular to the reference, long-axis angulation. Rotational movements, whether they are "pure" or "generalized," are typically described in terms of the initial and displaced orientations of the faciolingual axis or long axis and the intersection of the axis positions in the sequential configurations: the center of rotation. The two positions of the reference axis define a plane; the center of rotation is in

that plane and is often labeled cr0, and the rotation is about an axis perpendicular to the plane and pierces the cro. These reference lines and the cro are exemplified for a "transverse rotation" in Figure 5-2; note that

the cro may be located within or outside of the confines of the tooth itself. Transverse rotations are those tooth displacements during which the long-axis orientation (angulation) changes. During pure long-axis rotation, the angulation of the long axis is not altered. Transverse rotations

are known to the clinician as "tipping" or "torquing" displacements, depending on whether the center of rotation is apical or occlusal of the center of resistance. Simple tipping is a transverse rotational displacement produced by a single active force directed along the faciolingual or mesi-

odistal axis through point CC. Transverse rotations are called crown movements when the center of rotation is at or near the root apex and root movements when the cro is located within the crown, more specifically between bracket level and the occlusal surface (or incisal edge). During orthodontic therapy, the force system exerted on the crown is transmitted from the activated appliance, primarily through the bracket. The connections of arch wire, "elastics," and auxiliaries to brackets carry a force system to the crown having the general resultant consisting of a concentrated force and a couple. Individually, the force and couple vectors may have any direction; for analysis purposes each vector is decomposed into components in the occlusogingival, faciolingual, and mesio-

crc

cro

/ FIGURE 5-2. Several transverse rotations viewed from a facial perspective and, in each sketch, showing the location of the center of rotation, cro.

151

Response of Dentition and Periodontiurn to Force

distal directions. As previously mentioned, the crown may also be

subjected to additional forces during the period of activation; included

are mesiodistal contact from an adjacent tooth, force transmitted through an eyelet, staple, or button, the force of occlusion, and transseptal-fiber pull. With the exception of the fiber force, which is exerted at approximately the cementoenamel-j unction level, the applied force system is generally located at the occlusogingival position of point CC on the long axis, and any movement of the line of action of a force (to the cre, for example), requires the addition of a couple according to the pro-

cedure outlined in Chapter 2. To describe the responsive force system exerted on a tooth by the periodontal ligament surrounding the root, the composition of the ligament must be examined. Completely enveloping the root from cementoenamel junction to apex (apices), the ligament is a composite of fibers, cells, nerves, and blood vessels embedded in a soft-tissue matrix often termed the "ground substance." From a structural standpoint, two components, in essence, respond to crown loading: the fibers (in bundles) in tension and, in compression, principally the ground substance. The apparent shear resistance of the ligament is generated through redirection of the fibers from their primarily radial configurations with respect to the long axis. Accordingly, the response of the individual ligament is generally a distribution of tensile, compressive, and shear forces over the entire root-surface area while the action on the crown is a system of concentrated forces. The initial intensity of ligament force against a point of the root, produced by appliance activation, varies directly as the displacement of that point from the passive, preactivation state. In analyses of bodily movement and transverse rotations, a twodimensional model is used; for the other orthodontic displacements, two-dimensional modeling is unnecessary, but advantage is taken of the near-axial symmetry of the single-rooted tooth. Certain potential errors are recognized in the modeling of the distributed force transfer at the root-ligament interface. First, a limit exists to the amount of compressive ligament deformation; upon reaching this limit, the stiffness of the ped-

odontium increases substantially. Second, although most fibers are embedded in bone, some terminate in free gingiva; the latter result in a more flexible support structure occlusal to the alveolar crest than apical to it. Third, the remodeling of alveolar bone results in a redistribution of the responsive ligament force; hence, the distributed force changes, not only in resultant magnitude with gradual deactivation, but also in pattern with the long-term, biologic remodeling of the periodontium. The significance of each of these three modeling faults will be noted as they arise within discussions in the sections to follow.

Transverse Crown Force Systems and Tooth Displacements Attention is now restricted to the force-displacement analysis of a single tooth subjected to an active force system that is equivalent to a force and

152


couple at point CC; the vector representations of this action have no

occlusogingival components. Therefore, excluded from the discussion of this section are extrusion, intrusion, and long-axis rotation. Moreover, the active resultant at point CC and the problem itself may be decom-

posed, when necessary, into two plane-view (occlusogingival and faciolingual) analyses. Finally, changes in long-axis orientation are generally sufficiently small that the convenient configuration for instantaneous analysis can be the tooth position immediately following activation, and changes in angles between forces and the long axis, and in moment arms, with respect to the passive (preactivation) state, ordinarily, may reasonably be neglected. The orthodontic displacement defined as bodily movement is examined first to enable the introduction of terminology and procedures that will be used throughout this section. The resultant of the crown force system to achieve this translational displacement is mechanically equivalent to a single point force having a line of action passing through the center of resistance; hence, at point CC the equivalent active system is a force and

a related couple. Two typical sets of orthodontic mechanics that may seek to produce bodily movement are canine retraction, with the displacement directed distally, and incisal-segment retraction with the displacement directed lingually. If the active force system is transmitted to the crown at a bracket on the facial surface, the position of the line of action of the "driving force" may or may not be such as to exhibit a tendency only for a transverse displacement. In anterior-retraction mechanics, the line of action of the driving force, with correct positioning of the bracket, passes through the long axis, and therefore, this force has no moment about the long axis (and no potential to cause long-axis rotation). On the other hand, the driving force in canine retraction, when transmitted to the crown by a labial or lingual bracket, does have a moment arm with respect to the long axis of slightly more than one-half the faciolingual crown width: an accompa-

nying tendency for rotation about the long axis is thus exhibited. The two actions are shown in Figure 5-3, and throughout this section the existence of an occlusal-plane couple is assumed, when necessary, to eliminate long-axis rotational potential effectively from any analysis and permit plane-view study. (This couple counters the long-axis moment of the driving force. The effects of the eccentricity of the bracket with re-

spect to the long axis are discussed in Chapter 6.) Shown on the left in Figure 5-4 is the two-dimensional model of transverse crown loading referred to point CC; the plane view contains the long axis and either the faciolingual or the mesiodistal axis of the localized reference frame established previously. To "move" the loading apically the distance d to the center of resistance, a couple must be introduced that compensates for the change in rotational potential associated with the change in the line of action of F. This compensating couple C', with F now located at the cre, and the couple C from the original loading,

are shown in the center sketch of Figure 5-4; because the force F was moved a distance d perpendicular to its direction, the magnitude of C' is

153


Ffe

Fmd

FIGURE 5-3. Transverse crown forces applied faciolingually (left) and mesiodistally (right) through the bracket. (When the line of action does not pierce the long axis, the transverse force must be accompanied by an occlusal-plane couple of appropriate magnitude if long-axis rotation is to be prevented.)

the product of F and d. The couple C, having a whole-body, rotational-

displacement potential independent of its location on the tooth, now may simply be placed at the cre and, in order to yield a force-only resultant there, must be identical to C' in all characteristics except sense. Finally, on the right in Figure 5-4 is shown in most simple form the active force system referenced to the cre. Established completely, then, is the

in-plane relationship between the force and couple portions of the resultant action at point CC; for bodily movement the cou pie-force ratio must equal the CC-to-cre distance d, measured along the long axis (more

specifically, perpendicular to the lines of action of the applied force through CC and the cre). In general, the applied force system is first combined into a resultant at point CC to obtain the couple-force ratio, and that resultant is then transferred to the cre to obtain at least a qualitative picture of the tooth displacement to be anticipated from the given loading. The actual location of the center of resistance, generally dependent upon the size and

LI

C'Fd

F

(C = Fd)

FIGURE 5-4. The crown loading in bodily movement: the force and couple at point CC (left); the force moved to the cre with the compensating couple introduced (center); and the force system at the ore reduced to the single-force resultant (right).

154


and the manner of support of the body, must be obtained for the tooth by varying, in an experimental format, the couple-force ratio and, for each value, determining the displacement pattern of points on the root surface. Such studies have been carried out using both theoretical modeling (Nikolai, 1974) and laboratory experimentation (Burstone et al., 1982); the cre has been located close to midway between cementoenamel junction and root apex, making d approximately equal to one-half the total tooth length (if the bracket is positioned at midcrown, occlusoshape

gingivally).

Although a substantial simplification, discussion of the distributed responsive force system resulting from transverse crown loading within a two-dimensional model does encompass the essence of that response and, in particular, clearly differentiates among the individual forms of

associated, orthodontic displacements. Neglecting the inertia of the tooth, the resultant of the periodontal-ligament response to deformation by the root, referenced to the cre, must be equal and opposite to the equivalent crown loading at the point. For bodily movement that response is a single force F' as shown on the left in Figure 5-5, identical in all characteristics to the driving force F in Figure 5-4 except sense. Because the tooth experiences no change in angulation in this displacement, F' may be expressed as the sum of a tensile component F exerted on the trailing surface (in the plane view appearing as a curve) and a compressive component against the leading surface as shown in the center sketch of Figure 5-5. If the two halves of the total root surface as divided were identical reflections of one another, and the stiffnesses of the ligament were equal in tension and compression, the two compo-

nents would be equal and, herein, are modeled as such. (The slight angulations of the root surfaces and the deviations of some fiber directions from the normal with respect to the long axis are reasonably neglected in the modeling.) A sketch of the in-plane distributed force of the ligament upon the root is shown on the right in Figure 5-5. Although in bodily movement the displacements of all tooth points are identical, the tensile and compressive distributions are shown nonuniform to properly recognize the

F'

5-5. The plane view of the responsive force of the ligament against the root during bodily movement: the resultant force at the cre (left); the tensile and compressive components (center); and the modeled tensile and compressive distributed forces FIGURE

(right).

155


tapering of the root. Accordingly, the intensities are highest at the apex

where the root circumference is smallest and lowest where the root circumference is maximum at the cementoenamel junction. The difference in tensile stiffnesses of intra-alveolar and extra-alveolar fiber bundles is ignored as modeled here, based on the relatively small portion of the total root area in which the latter fibers are embedded. (The twodimensional modeling of the stress distribution, and use of the projected root areas, are tantamount to replacing the approximately round or oval root cross-section of the single-rooted tooth by a square or rectangular cross-section. Shown in Figure 5-6 are the as-modeled and typical, anatomically correct, root "slices" and the distributed force systems exerted on each periphery.) From the discussion of stress in Chapter 3, the maximum ligament tensile and compressive intensities, as suggested in the

model, are dependent on the magnitude of the driving force and the surface area and anatomy of the root. The transverse displacement known as simple tipping derives its name from the simple active force system—a single point force P with line of action perpendicular to the long axis and piercing point CC, which produces an alteration in long-axis angulation. The active force may have faciolingual and/or mesiodistal components; one component is shown on the left in Figure 5-7. The couple-force ratio for this loading is zero; no

applied crown couple exists in a plane containing the long axis. The transferral of the active force to the cre, however, requires the addition of a couple equal in magnitude to the product of P and d (where, again, d is the long-axis distance from the bracket to the cre). The active force

system referenced to the center of resistance is shown in the center sketch of Figure 5-7; the two components indicate that the potential displacement is a translation of the cre in the direction and sense of P and a rotation or tipping of the long axis with sense corresponding to that of C1,. A typical displaced configuration with respect to an initiallyinclined long axis is shown on the right in Figure 5-7. The intersection of the two long-axis positions is the center of rotation for the simple-tipping displacement, slightly apical of the cre. Note that simple tipping

includes displacements of both point CC and the root apex (apices). The resultant of the periodontal-ligament response to simple-tipping tension

tension

//

compression

a likeness of the actual root cross-section

the modeled root cross-section

FIGURE 5-6. 0cc/usa! views of root cross-sections and distributed force systems exerted thereon: the anatomically correct (left) and as-modeled (right) root slices for a single-rooted tooth.

156


/

/

P

= Pd

/

I

/

crc

FIGURE 5-7. A plane view of the simple-tipping displacement: the active crown force at point CC (left); the equivalent loading referred to the cre (center); and a typical displacement showing the crc location (right).

action, referred to the cre, is equal and opposite to the action referred

there (Fig. 5-7, center), and is shown on the left in Figure 5-8. Equivalent to this resultant of P' and is the set of four forces shown in the middle

sketch of Figure 5-8; the pair located more apically are individually smaller in magnitude than those positioned closer to the cementoenamel junction. Each of these four forces is the resultant of a quadrant of the total distributed ligament response sketched on the right in Figure 5-8; each quadrant is not exactly triangular due to the tapering of the root.

Both compression and tension exist on either "side" of the root. Proceeding occiusally from the apex, the stress (intensity) levels in the model decrease from relative maximum values at the apex, go to zero and change signs at approximately the cro level, and then increase toward relative maximum values at the cementoenamel junction. The absolute maximum stresses occur at the root apex, due to the minimum root circumference there. Throughout this section in the modeling of the

P,

c,

tension

compression

FIGURE 5-8. The responsive force system against the root during simple tipping: the resultant referred to the cre (left); an equivalent set of two tensile and two compressive forces (center); and the modeled distributed force system in four parts (right).

157

Response of Den tition and Periodontium to Force

distributed response, the tensile and compressive ligament-material

stiffnesses are assumed equal and unchanging over the entire root length as noted previously. Considered next is the response of a single-rooted tooth to the application of a second- or third-order crown couple. This example might be termed "academic" because, in the presence of friction and some existing faciolingual arch-wire resistance, the crown couple will always be accompanied by some amount of transverse force. Nevertheless, pure transverse rotation is worthy of brief discussion. Shown from left to right in Figure 5-9 are the activating couple referred to point CC, the couple moved to the cre, and the displacement format associated with the couple-only loading. The couple-force ratio is mathematically infinite and, with pure rotation occurring, the cre remains stationary and the cro coin-

cides with it. The resultant of the periodontal-ligament response, this couple expressed as four discrete (in this instance, equal) forces, and

the distribution of the response modeled in two dimensions, are depicted from left to right in Figure 5-10. Figure 5-10 is analogous to Figure

5-8; again, the nonlinearity and the maximum intensity at the apex are due to the root taper, but the stress changes sign between tensile and compressive farther from the apex than in simple tipping (associated with the difference in cro locations). Root movement is the term given to the orthodontic tipping-torquing

displacement in which the crown position remains essentially unchanged. To accomplish the displacement, a second- or third-order cou-

ple is applied to the crown in the presence of a transverse "holding" force. A typical root movement is the uprighting displacement, to change tooth angulation with the vertical toward proper orientation of the long axis with respect to the occlusal plane. The principal part and

C0

cro

5-9. A plane view of pure, transverse rotation: the couple loading at point CC (left); the couple moved to the cre (center); and the displacement format showing the coincidence of the cro and the cre (right). FIGURE

158


-

tension

compression

FIGURE 5-10. The response to transverse couple loading: the resultant at the cre (left); an equivalent set of four forces (center); and the modeled distributed force system of the ligament against the root (right).

motive action of the applied force system is the couple; the role of the

force is prevention of crown displacement which occurs when the couple alone is exerted on the crown (Fig. 5-9, right). The force H and the couple Ch are shown on the left in Figure 5-11 at the crown center point CC. In the center sketch of Figure 5-11 the displacement format is illustrated with the cro at bracket level; note the direction of cre movement and the sense of the change in long-axis angulation. Correspondingly, shown on the right in Figure 5-11 are the force and couple referred to the cre, which are needed to produce the displacement. The magnitude of this couple is C,, — Hd, where the product Hd arises from the movement of the force H from point CC to the cre. Accordingly, the couple-force ratio necessary to produce this displacement is greater than d that required for bodily movement.

Ch

H

C,, - Hd

FIGURE 5-11. A plane view of root movement: the components of the active force system referred to point CC (left); the displacement and cro location (center); and the equivalent loading at the cre (right).

159


In root movement the apex experiences the largest displacement of all points of the tooth; the maximum intensity of the ligament responses is located at the root apex. Shown on the left in Figure 5-12 is the resultant of the distributed force system exerted by the periodontal ligament,

referred to the cre. The force H' is equal and opposite to H and the couple Ch is in magnitude Ch Hd. This force and couple may be combined into a single force located somewhat apical of the cre, and then it may be divided into tensile and compressive components H; and on the trailing and leading root surfaces, respectively, and plane-view modeled as illustrated in the center sketch of Figure 5-12. These forces are the resultants of the tensile and compressive portions of the distributed re-

sponse shown on the right in Figure 5-12. If the center of rotation is occlusal of the cementoenamel junction, nowhere along either the trailing or leading surface of the root does the intensity of the distribution go to zero and change sense. In this way the distributed response is similar to that for bodily movement; however, during root movement the difference in maximum and minimum intensifies is substantially greater than during bodily movement, given the same magnitude of transverse force for the two displacements (Nikolai, 1975). The clinical objective of root movement is to locate the cro within the tooth crown; to achieve the desired result, the mechanics must be capable of developing a holding force of sufficient magnitude to obtain the needed couple-force ratio. The last of the recognized transverse orthodontic displacements, and perhaps the most difficult to achieve clinically, is crown movement. The objective is a transverse displacement of the tooth crown without moving the root apex; this is, as ideally performed, a tipping displacement with the center of rotation at the apex. The displacement is shown on the left in Figure 5-13. From the relationship between the point-wise displacement of the center of resistance and the sense of the long-axis change in angulation, the applied force system referred to the cre must be as shown in the center sketch of Figure 5-13, including both a force and a couple. Furthermore, with the cro at the root apex, the response of

H'

compression

tension

FIGURE 5-12. The periodontal response during root movement; the resultant located at the cre (left); an equivalent system of two forces (center); and the modeled tensile and compressive distributions of force (right).

160


C

Cd - Cq

crc

FIGURE 5-13. A plane view of crown movement: the displacement and the crc location (left); the applied force and couple referenced to the cre (center); and the crown force system at point CC (right).

the periodontal ligament against the root as modeled will divide into a

compressive distribution on the half-root surface corresponding to the sense of the active force Q and tension on the remaining surface area. The resulting implication is in combining Q and Cq into a single force; that force must be located between the cre and the cementoenamel junction. It then follows that the active crown force system referenced to point CC consists of a force and a couple as shown on the right in Figure 5-13, the magnitude of the couple is less than the product Qd as indicated, and the couple-force ratio is less than d. Figure 5-14 depicts the resultant components Q' and Cq' of the periodontal-ligament response referred to the cre on the left (components equal and opposite to those of

C'q

cre

tension

compression

FIGURE 5-14. The ligament response during crown movement: the components of the resultant of the response referenced to the cre (left); an equivalent, single-force resultant divided into tensile and compressive parts (center); and the modeled distributed response against the root (right).

161


the active force system in the center in Figure 5-13), the resultant liga-

ment response as a single force divided into compressive and tensile portions, and Qj, in the center, and the compressive and tensile distributed responses on the right. Crown movement is the only transverse orthodontic displacement for which the maximum intensities of responsive force occur at the cementoenamel junction. To summarize, all transverse orthodontic tooth movements as defined are produced by a combination of a force approximately perpendicular to the long axis (parallel to the occiusal plane) and a couple in a plane containing the long axis. Looking into that plane, when both components have nonzero magnitudes, the sense of the moment of the force with respect to the cre is opposite to that of the couple. The couple-force ratio, which controls the position of the cro, is given simply as a ratio of magnitudes and equals the occlusogingival distance from point CC to the location of an equivalent, single-force load. (Although the center of resistance is a fixed point in a tooth—its location is independent of the crown loading—the center of rotation is highly dependent on the characteristics of the crown force system.) In the model used in this section, the cro is always located on the long axis or its extension; in reality this may be somewhat (but not substantially) in error in root movement produced by a third-order couple and a sufficiently stiff holding-force potential that will place the axis of rotation coincident, or nearly so, with

the arch wire and, thus, the cro within the bracket slot as viewed mesially or distally. In the analyses of these movements, when necessary the loading may be decomposed into components contained in faciolingual and mesiodistal planes through the long axis; often the transverse action is totally in one or the other of these planes. In the

faciolingual plane, adjacent teeth exerting pushing forces through crown contact can control the cro location; in the absence of a stiff palatal

bar, no such strong control can exist exclusive of ordinary orthodontic mechanics in the mesiodistal plane. Shown in Figure 5-15 is a plot, theoretically derived, of occlusogingival center-of-rotation location versus couple-force ratio for the labiolingual, transverse displacements of an average-size, maxillary central incisor (Nikolai, 1974). The partitioning of the plot into two curves results from the couple-force ratio becoming infinitely large as the bodily-movement displacement is approached. Notable are the substantial effects on cro location as the ratio nears d from above or below. Given in Table 5-1

are results associated with five mesiodistal displacements of a typical canine. The root length and overall long-axis length of the tooth were 17 mm and 22 mm, respectively. The analysis was theoretical assuming linear relationships between intensities and displacements of rootligament interface points, equal ligament stiffnesses in tension and compression, and accounting for root taper. Notable are the substantial differences in extremum (maximum, minimum) interface force intensities among the four tooth displacements generated with the same magnitude of transverse force.

162


to 0.9

Distance to cro occliisally from apex, normalized by root length 1.6

1.2

d

0.8

to 0.45 ——

I

to 0.45

—0.8

1.2

—0.4

1.6

2.0

Couple-force ratio (cm)

—0.4

—0.8

—1.2

to 0.9 —1.6

FIGURE 5-15. Lab/olin qua! displacements of a typical, maxillary, central incisor: a plot of cro location along the long axis and its extension versus the ratio of couple to transverse force as the loading components at point CC.

Finally, the remarks of this section need not be restricted to singlerooted teeth. Although the analysis may divide the overall root-surface

area into multiple portions for individual teeth having two or three roots, this is not a significant, complicating factor; neither is the fact of root apices away from the long axis when restricting attention only to transverse displacements. From recent experimentation, the center of resistance of a multiple-rooted (molar) tooth is apparently slightly apical of the midpoint of the overall apices-to-cementoenamel-j unction length (Burstone et al., 1981). Hence, center-of-rotation locations for transverse displacements, in relation to total root length, may be insignificantly influenced by the number of roots.

163

Response of Dentition and Periodontium to Force TABLE 5-1. Couple-force ratios, center-of-rotation locations, and maximum stress values tar five transverse canine displacements

Displacement format

Simple tipping Bodily movement Crown movement Root movement

Mesiodistal force (g)

Couple-force

Center of

ratio (mm)

rotation (mm/mm)*

Maximum stresses Gingival Root margin apex (g/cm2) 230

60

0

60 60 60

14

0.35 Infinite

11

0

16

1.3

7

85 10 125

Infinite

0.5

125

245

135 20 50

Couple (g-mm)

Pure torquing

1020

The location ot the center of rotafion is given from the root apex and as a fraction of the root length. The model canine has a projected area (mesiodistal perspective) of 0.9 cm2, a root length of 17 mm, and a

length of

22 mm.

fModitied from Nikolai, R.J.: On optimum orthodontic force theory as applied to canine retraction. Am. J. Orthod., 68:290— 302, 1975.)

Extrusion, Intrusion, and Long-Axis Rotation remaining three distinct orthodontic displacements are characterized by several common features. The active force system includes no net force component perpendicular to the long axis of the tooth; therefore, no potential exists to alter the long-axis position or orientation. The resistance to any of these three displacements, as provided by the periodontal ligament, appears to be principally shear in nature. Because the inherent shear resistance of the matrix material of the ligament is low, however, the essence of the response originates with the fiber bundles and, as one or a combination of these displacements begins, fiber orientations are altered toward positions tangential to the root surface. Twodimensional modeling in analyses of these displacements is unnecessary and unwarranted. With single-rooted teeth the anatomy and force system approach a configuration of geometric and mechanical symmetry with respect to the long axis, and the modeling more appropriately takes advantage of this near-symmetry. The analysis is somewhat more involved, however, particularly with long-axis rotation, when the tooth under study is multirooted and the near-symmetry is absent. In pure extrusion the activating system is a concentrated force acting along the long axis and referred to point CC. In the clinical setting, however, the extrusive force is exerted at the crown surface. Moving that force first to point CC and then to the cre, as shown in the leftto-right sequence in Figure 5-16, the extrusive force originating on the facial crown surface is seen to be accompanied by a third-order couple The

164


Fe

FIGURE 5-16. An applied crown force with the combined potential for extrusion and lingual crown tipping: at the facial surface (left); an equivalent system at point CC (center); and the equivalent system at the cre (right).

and, with it, the potential to also tip the tooth; the crown would move

lingually and the root apex (apices) labially with the cro in the position indicated in Figure 5-9. Accordingly, the occlusally directed crown force must be augmented if the tipping potential is to be countered. (As mentioned earlier, the eccentricity of the typical appliance is discussed generally in Chapter 6.) In response to a purely extrusive potential, the periodontal ligament exerts a distributed force system having a resultant directed apically and positioned along the long axis. The distributed response arises almost exclusively from reorientation and stretching of the ligament fibers although, depending on actual anatomy, there may be some compressive resistance over a small area near the cementoenamel junction. The intraalveolar fibers will provide the majority of the total resistance because of their numbers and higher tensile stiffnesses compared to the supra alveolar fibers; otherwise, the distribution is apparently nearly uniform both circumferentially and longitudinally, and is modeled as such, for the single-rooted tooth. Other than to note the increased root-surface area inherent in the multirooted tooth, no substantial difference in extrusion analysis is necessary for multirooted versus single-rooted teeth. Of particular note, however, is the relatively weak periodontal resistance to extrusive displacements with little, if any, ligament reinforcement from bone compared to other tooth movements; control in this regard must be

provided in the decay characteristic of the activating force, perhaps through the arch wire, and by occlusion. Intrusion and extrusion are similar orthodontic displacements in that both activating loads are, ideally, axial forces, both pure movements are translational, and both resistances to displacement are associated with stretched ligament fibers. Differences between the movements include the sense of the load and of displacement potential and the creation of

165


apical pressure as a portion of the force system responsive to an intru-

sive loading. If the intrusive force is applied through the bracket at the facial surface, transferring it to point CC will introduce a third-order couple, opposite in sense to its counterpart in Figure 5-16, indicating that a tendency for labial crown tipping may accompany the intrusion. Shown in Figure 5-17 are the action and response associated with pure intrusive displacement potential. Of particular note is the presence of pressure at the root apex associated with localized ligament compression. With a multirooted tooth this pressure exists not only at each root

apex, but also in the bifurcation region. In the absence of substantial fiber stiffness, the pressures in the compression areas, which are small in comparison to the overall root-surface area, would be high even with a light, intrusive load. In their passive configurations, many of the intraalveolar fiber bundles are obliquely oriented with respect to the rootligament interface. This "hammock effect" protects the root apex or apices from excessive pressures resulting from masticatory, occlusal, and

intrusive orthodontic forces. If the intrusive force is substantial or is present continuously for a long time period, the pressure zones of the root-ligament interface grow and can envelope the entire root-surface area. Hence, because of the tapering of the root toward the apex, under intrusive force both fiber tensions and pressures from the ligament can exist at common points on the root surface; these pressures away from

C = F,e

fiber tension

apical compression

FIGURE 5-17. The force system associated with pure intrusion: the active and responsive resultants (left); and the active force referred to the ore together with the responsive distribution of tension and pressure (right).

166


the apex, relatively low compared to compressive intensifies at the apex,

are not shown in Figure 5-17 (and such pressures do not exist at all along the root in response to extrusive loading). A notable difficulty occurs in the attempt to quantify extrusive and

intrusive movements in comparison with similar analyses of other orthodontic displacements. The problem arises when tipping accompanies extrusion or intrusion, which occurs often in clinical practice, thereby altering the perpendicular-to-occlusal-plane distance between occlusal surface (incisal edge) and root apex. Shown in Figure 5-18 are two sets of initial (dashed) and final (solid) positions of a maxillary central incisor as viewed mesially or distally. The two initial positions are identical. The

amounts of extrusive or intrusive displacement must be determined in terms of two reference locations, one fixed within the tooth (e.g., the incisal edge) and a second outside the tooth (perhaps fixed within the maxilla or mandible, although "landmarks" within hard tissues may also move during the period of orthodontic treatment). The orthodontic displacement known as long-axis rotation is characterized by a change in the faciolingual- and mesiodistal-axis angulations as viewed from an occlusal perspective. To produce this displacement the active force system must exhibit a moment about the long axis, and such a moment is associated with a force, having a transverse component, applied tangentially to a facial, lingual, mesial, or distal crown surface. In pure, long-axis rotation no movement of the long axis occurs; hence, the center of rotation is on the long axis as shown in Figure 5-19. To obtain the pure displacement a first-order couple is required as illus-

\

Cant of the palatal plane

Dashed: initial position Sohd: Displaced position FIGURE 5-18. Combined orthodontic displacements of a maxillary central incisor as viewed distally: lingual crown tipping and intrusion (left); and labial root torquing and apparent extrusion (right).

167


I, (cN

/ FIGURE 5-19. An occiusal view of an applied first-order couple to generate long-axis rotation (left), and the initial (dashed) and displaced (solid) configurations together with cr0 location (right).

trated, and that couple might be made up of a mesiodistal force applied

through the bracket on the facial surface and a parallel force, but with opposite sense, exerted through crown contact with an adjacent tooth. If no provision is made or no potential exists for creation of the latter force,

the moment with respect to the long axis is present but the expected movement as seen in an occiusal view is a "rolling" displacement with the cro located away from (lingual of, in this example) the long axis. Figure 5-20 shows the sequence with the active force depicted on the left, the equivalent system at the cre (which coincides with point CC in an occiusal view) in the center sketch, and the displaced position (solid) of the crown on the right. The resultant of the periodontal-ligament force system, in response to a first-order crown couple, is a couple identical in all characteristics to the load except for sense. In the pure displacement, points on the surface of the root move circumferentially with respect to their counterparts on the alveolar-bone surface, lengthening and reorienting the fiber bundles toward configurations tangential to the root and perpendicular to the long axis. Fiber strains and stresses vary directly with the displacements of root-surface points, and the amounts of these displacements are directly proportional to the radial distances from the long axis to the points. Accordingly, the intensity of distributed responsive force is minimum at the root apex (of a single-rooted tooth) and increases in magnitude occiusally to a maximum near the alveolar crest. By reason of their attachment in the gingiva, the radial extra-alveolar fibers are more flexi-

168


//

F0

• cro

/ I' FIGURE 5-20. A rolling displacement in 0cc/usa! view: the applied force at the facial surface (left); the equivalent force system at point CC (center); and the displaced configuration and cr0 location (right).

ble and, as a result, the fiber tensions are less than those in neighboring,

intra-alveolar fibers. A schematic of the load and the distributed periodontal response is given in Figure 5-21; on the right is a qualitative plot of fiber stress against longitudinal position along the root surface. In the model, the single-rooted tooth is assumed conical in shape; therefore, at a specific occiusogingival level the fiber stresses do not vary in magnitude circumferentially. In reality, the variability in root-surface contour 0 cc

0 0

-J

Fiber

Fiber Stress

Co

FIGURE 5-21. Pure, long-axis rotation: a transverse view of the crown loading and the periodontal response (left); and the variation in fiber stress with longitudinal distance from the gingiva! margin to the root apex for a single-rooted tooth (right).

169

Response of Dentition and Periodontium to Force tension

compression tension

compression

tension

compression

FIGURE 5-22. Responsive periodontal distribution against the roots of a maxillary first molar undergoing long-axis rotation: apical view (left) and transverse view (right).

results in some localized compression of the ligament, but the percentage of the root surface experiencing pressure is relatively small. Of all the distinct orthodontic displacements, the multirooted tooth undergoing long-axis rotation presents the most substantial difference in response, compared to the single-rooted tooth. Apical of the bifurcation location, an assumption of axial symmetry is totally incorrect. The "long axis" of each root deviates from the long axis of the tooth as a whole with, generally, substantial distances between the long axis and the apices. With the cro on the long axis of the tooth, in this displacement the individual roots experience bodily movement in a curvilinear fashion, with pressures against the leading surfaces and fiber tensions exerted on the trailing surfaces catalyzing the bone remodeling necessary to achieve the movement. Apical and mesial views of the distributed periodontal responses to long-axis rotation of a maxillary first molar are depicted in Figure 5-22. also

Response of the Periodontium to Force applied to the tooth crown, originating in mastication, the activated orthodontic appliance, or from the lips, tongue, or facial tissues, results in the creation of a response within the supporting structure of the tooth. The individual tooth is positioned locally within a socket or Force

alveolus, lined with a soft tissue—the periodontal ligament—which separates the tooth root from the alveolar bone. The basal bone interconnects the units of the dentition; the alveolar processes of the upper arch blend

170


into the maxilla and those of the lower arch into the mandible. The

periodontal ligament completely surrounds the tooth root from cementoenamel junction to apex (apices). The ligament is composed of collagenous fibers in bundles, blood vessels, nerves, and cells embedded in an amorphous, viscous matrix (often termed "ground substance"). The ligament is a composite material capable of sustaining and transmitting

tensile loading primarily through the fibers; pressure is transmitted principally by the matrix material. The ligament passively exhibits nearuniform thickness of approximately 0.3 mm and low stiffness, at least an order of magnitude less than that of calcified bone. The primary function

of the ligament is to cushion the bone beneath it from the action of impulsive, masticatory loading of the dental units. The alveolar bone, also, is a heterogeneous material. Although the underlying portions of the bone are somewhat spongy, the layer adjacent to the periodontal ligament is dense and exhibits rather high surface hardness. This layer, the cortical plate (lamina dura), and the cementum anchor the radial fibers that run into and through the ligament. The alveolar bone is relatively stiff as a whole, although some deflection apparently occurs in the thin, crestal regions under the action of relatively high forces applied to the tooth crowns. Forces exerted on the dentofacial complex may be designated as orthodontic if their intent is the movement of teeth through remodeling of the alveolar process; during such remodeling, no significant alteration of basal bone occurs. On the other hand, forces may be exerted with the intention of displacing basal bone without producing relative displacements of teeth within the alveolus; such forces are rightfully termed orthopedic. The internal force system in the periodontium, arising from the application of a crown force, possesses an involved format, made so by the geometric and mechanical as well as the biologic complexities of the responding tissues. The problem, however, can be simplified somewhat through modeling, with a reasonable analysis of the distribution in the alveolus of this internal force pattern including three individual consid-

erations. First, because the periodontal ligament is thin compared to bone dimensions, the distributed force system carried to it by the tooth root is insignificantly altered while transmitted through it. Specifically, the tension-pressure system at the ligament and cortical-plate interface

may be considered identical to that between the root and ligament. Admittedly, a small reduction in intensities exists due to the slightly larger ligament surface area adjacent to the plate compared to the area against the root, but the difference is reasonably ignored. Second, the distribution of normal stress into alveolar bone, in response to loading at the plate surface by the ligament, may be envisioned through the aid of Figure 5-23. Shown on the left is a flat surface (in edge view) of a solid subjected to a compressive force and representations of resulting pressures at various distances from the point of application of the load. In a homogeneous material, such pressures have been shown to decrease in magnitude directly with the radial distance

from the location of the load. (Changing the sense of the load, an ap-

171


Compressive force

Pc

PA PB

Pa

p: compressive stress (pressure) FIGURE 5-23. A simplified representation of the alveolar-bone response to pressure loading by the periodontal ligament: the variation of compressive-stress magnitude with distance from the load (left); and the accompanying induced normal and shear stresses (right).

plied tensile force induces corresponding tensile stresses directed radially.) On the right in Figure 5-23 are shown stress-at-a-point sketches for

points A and B. Note the presence of circumferential tensile stresses at right angles to the radially directed pressures at points off the force line. (In general, for either sense of the normal surface load, the two mutually perpendicular normal stresses at each point will exhibit opposite senses. The shear stresses, present at all points like B and C not on the line of action of the concentrated load, indicate a tendency to distort the small

elements of bone material surrounding the points.) The two normal stresses being of the same order of magnitude, the circumferential stresses also decrease in size with increasing distance from the applica-

tion-location of the load. In the problem of analyzing the transfer of force from the root through the ligament and into alveolar bone, the root typically transfers a nonuniformly distributed load rather than a concentrated force. Figure 5-23 depicts, then, a substantial simplification of the response at the bone-ligament interface and within the alveolar bone to transverse crown loading; however, the model does reflect the essential features of that response. Third, a portion of the load applied to one tooth crown can be trans-

mitted directly to adjacent teeth through either crown contact or the transseptal fibers (to be discussed subsequently). Distributions of force by these means depend upon the characteristics of the load, the relative positions of the teeth, and the continuity of the fiber network. The effect of this distribution is twofold: a reduction from expected value of the force transmitted into the periodontal ligament surrounding the loaded tooth and the displacement(s) of adjacent teeth that were not subjected

172


to external loading. The phenomenon is most apparent when the crown

loading is mesiodistal in direction, although the effects of the transseptal-fiber system are also present in other orthodontic displacements. The preceding discussion provides a simplistic, qualitative description of the principal effects of the force system within the periodontium at any particular instant in response to crown loading. No general discussion of periodontal response is properly undertaken, however, without considering the influence of time upon the applied and responsive forces. An examination of the response to masticatory action serves well to introduce time as an analysis variable. Chewing forces are primarily intrusive and exist only infrequently during the average day. Moreover, when present, forces of mastication exhibit a periodic pattern, oscillating with jaw movement between zero and some finite value. The responsive forces are induced in the same pattern with the periodontal ligament tending to isolate the impact of mastication from the bone beyond it. The fiber bundles form a hammock-like structure around the root, stretching and relaxing in primarily an elastic manner in response to the occlusal impulses. Although the loading is transmitted into the bone, it is shortterm and "dulled" by the flexibility of the ligament; normally no lasting effects are produced in the ordinary, passive configuration of the dentition and its supporting system. Note that the individual tooth is inherently well protected against the effects of all short-term loading with the possible exception of extrusive force (although occlusion tends to prevent abnormal displacement). The overall response of the periodontium tends toward that of a viscoelastic material, with a rather substantial

time period under uninterrupted force required to produce lasting changes.

The response of the periodontium to the internal force system created by a sustained load, such as that produced by an orthodontic appliance, is fundamentally in two sequential parts. Immediately upon appli-

cation of the crown force system, the tooth undergoes an initial displacement owing to the flexibility of the periodontal ligament. As discussed previously, the form of the displacement depends upon the characteristics of the applied force system. With the exception of extrusion and, possibly, long-axis rotation, the extent of this initial displacement is limited by the complete compression of the ligament between root and alveolar bone. Transversely and intrusively, with allowance for possible small crestal deflections, the narrow periodontal ligament restricts an initial displacement to a maximum of approximately 0.5 mm reflected at the root apex (apices) or the alveolar crest. Any displacement is indicative of the appliance doing mechanical work upon the dentition which reduces the strain energy of activation and the magnitudes of the

components of the force system exerted on the dentition. The initial displacement ordinarily will not deactivate the appliance totally and the result is a "stand-off' between the appliance-teeth and periodontal systems, both in constrained configurations with respect to their previously passive states. If left in these constrained states, in a period of days the

173


long-term portion of the periodontal response would become evident

with the biologic remodeling of the periodontium allowing further dental-unit displacement accompanied by additional deactivation of the appliance. If allowed to continue indefinitely, the process might end only upon complete deactivation of the appliance.

The long-term response of the periodontal ligament to continued deformation involves two forms of biologic remodeling: the partial replacement of fibrous matter carrying tensile stress by new, stress-relieved material and the resorption of alveolar bone to relieve and redis-

tribute the pressure. In the portion of the ligament in which fiber stretching is developed and sustained, due to either direct tension or what would otherwise be shear, the tensile strains apparently heighten the activity of fibroblasts, builder cells within the ligament that initiate the formation of new collagenous fibers. The level of activity of these cells is

assumed to be at least partially dependent on the amount of localized pressure developed between the stretched fibers as well as the magnitude of tensile strain. Notable are the categories of fibers within the periodontal ligament. The intra-alveolar or principal fiber bundles radiate from the root and proceed into and through the ligament and into alveolar bone. These fibers are relatively short and differentiation among groups is generally made according to direction as noted in a facial view of the tooth and alveolus. Proceeding apically, included are the crestal, the horizontal, the oblique, the interradicular (attached to multirooted teeth), and the apical fibers. All

but the interradicular fibers are depicted passively in Figure 5-24. The extra-alveolar (supra-alveolar) or gin gival fibers are passively longer and are

embedded in the cementum and in the gingiva occlusal to the alveolar crest. In this group the cementoepithelial fibers radiate from the cementum near the cementoenamel junction, the circumferential bundles are totally

within the gingiva and surround each tooth, and the transseptal fibers weave their way mesiodistally from tooth to tooth, linking the dental units to one another. Given the displacement of a root point, the amount of accompanying fiber strain depends upon the passive length of the fiber. Of the radial groups, the cementoepithelial fibers are the longest and require the greatest length of time to remodel; to hasten readaptation and overall remodeling, and, in particular, toward retention of long-axis rotational displacements, these fibers are sometimes severed with subsequent reattachment occurring naturally in the displaced configuration. With nonextrusive tooth displacements the proximity of the root to

the alveolar-bone surface protects the ligament against failure due to excessive tensile strain. The real concern is sustained compressive deformation of the periodontal ligament. Where pressures exist substantially

in excess of normal blood pressure (25 to 35 g/cm2), occlusion of the vessels occurs and with it an interference in the supply of nutrition to the tissues. Pressures of sufficient magnitude not only interfere with normal blood flow, but drive the blood supply from the area of high

174


_— Crestal fibers

Horizontal fibers

Periodontal ligament

Oblique fibers

A

— Apical fibers FIGURE 5-24. The groups of intra-alveolar fiber bundles between root and cortical plate as seen in a transverse view of a single-rooted tooth.

175


pressure, reduce or prevent cellular activity, and even force the displace-

ment of the ligament matrix material to regions of lower stress. Sustained high pressure is accompanied by potential for regressive biologic alterations of the ligament (hyalinization, necrosis) and prevents remodeling of the alveolar bone at the cortical plate-ligament interface. Accordingly, not only to lessen the pain associated with tissue deformation, but also to guard against the possibility of ligament pathology associated with sustained high pressures, the magnitudes of the force-system com-

ponents transmitted by appliances to tooth crowns must be limited. The fibrous attachment of the periodontal ligament to the cortical plate enables the transmission of tensile as well as compressive distributed loading into the alveolar bone. A number of cells, in addition to the aforementioned fibroblasts, exist in the periodontal space and contribute to the remodeling process in the absence of excessive pressure. In regions of the bone-ligament interface where continuing fiber tension is present, the fibroblastic activity is accompanied by that of the osteoblasts. These cells play a role in the formation of osteoid, which develops along the outer surface of the cortical plate, adhering to both the surface and the stretched fibers emerging from the plate. In time this new material mineralizes to form bundle bone and eventually calcifies. Each small but distinct movement of the tooth root causes an incremental change in interface stress magnitudes and in osteoblastic action; as a result, the new bone appears in layers (lamellae). The overall effect on osteoblastic and fibroblastic activity, attributable to the initial magnitude of the fiber tension, is not well understood. Notable, however, is the apparent fact that removal of the fiber tension has little effect on the continuation of

bone maturation, once the osteoid has been laid down. Moreover, a reversal of the interface loading to create compression of the ligament against bone does not result in the immediate removal of osteoid; this new tissue exhibits an inherent resistance, significantly more so than mature bone, to such remodeling. Although sustained crown loading results in fiber remodeling and bone apposition in periodontal regions where the radial component (di-

rected out from the long axis) of induced stress is tensile, where the radial stress component is compressive, another form of biologic action occurs. If transverse movement of teeth is to take place, resulting from the application of orthodontic force, alveolar bone must be removed to accommodate the leading root surfaces while new bone is formed in the vacated space(s). The bone-resorption process is stimulated by yet another cell form, and the location(s) of the initiation of the process and the time required to complete it are apparently dependent upon the stress pattern within the periodontium which, in turn, depends upon the form

and magnitude of the applied crown force system. Osteoclasts in the periodontium seemingly are activated, beyond their normal levels, by moderate amounts of stress in certain configurations to break down the protein matrix of bone. When the bone resorption occurs at the bone surface adjacent to the periodontal ligament, the process is termed direct

or frontal resorption. Where the bone-ligament interface stresses are

176

Bioeng!neering Analysis of Orthodontic Mechanics

high, the process begins at some depth within the alveolar bone; this

internal bone remodeling is known as undermining or rear resorption. (Recall that the radial and circumferential normal stresses decrease with distance from a normal surface loading; see Fig. 5-23). Two hypotheses are offered in regard to the mechanical catalyst associated with the remodeling of alveolar bone. On the one hand, there is a strong tendency to relate the remodeling format to the radial stress— bone apposition to tension or the tensile stress component normal to the bone-ligament interface and resorption to the radial compressive stress or pressure. Alternatively, transverse remodeling of long bones is said to be keyed to the sense of the longitudinal stress—apposition by pressure and resorption by tension. Once again recall the discussion of Figure 5-23, particularly the fact that, at a point within the bone, the normal stresses at right angles are opposite in sense to one another. If the analogy is drawn between the longitudinal and transverse directions associated with the long bone and the circumferential and radial directions, respectively, within the alveolar bone, the so-called "piezoelectric effect" theory espoused by the orthopedist does not inherently conflict with the pressure theory of the orthodontist. Much remains to be learned and understood regarding the biologic response of the periodontium to mechanical stress. Moreover, the nahire of the orthodontic appliance and of therapy adds to the complexity of the matter. The effect of biologic remodeling "in isolation" would undoubtedly be a modification of the initially varying stress pattern toward a uniform distribution throughout the ligament and bone, as well as a reduction in the average pressure within the periodontium. Although the crown loading is lessened in overall magnitude as incremental displacements due to remodeling occur, it generally continues to exist and maintains a nonuniform pattern of stress against and within the alveolar bone. This matter is further complicated when several components are included in the crown force system and each is dependent for its magnitude on individual stiffnesses of appliance elements; the couple-force ratio, for example, may change as the overall displacement progresses. Finally, the clinician may interrupt the ongoing process and start another by first unloading, then modifying, and subsequently reactivating the appliance. In concluding this section, a comment on the resistance of cementum to resorption and the activity of cementoblasts is appropriate. These cells potentially sustain and repair the hard outer surface of the root. Both the cortical plate and the cementum are subjected essentially to the same pressures created through the application of crown force; however, a physiologic difference between the seemingly similar tissues is reflected

in a greater resistance to resorption inherent in the cementum. The cementoblast resides in the periodontal space and its routine activity is undoubtedly affected by sustained high pressure. Other factors have a role, but apparently the combination of conditions that sometimes result in root resorption during orthodontic treatment is not now well understood.

177


Displacements Related to Magnitude and Duration of Force force system applied to the tooth crown, toward the creation of a displacement, originates in the straining of elements of an appliance in the attachment of it to the dentition. The crown force system may consist of one or several components; they are individually concentrated forces or couples, each having a direction, sense, and an initial magniThe

tude dependent on the manner of appliance-to-crown contact, the amount of activation deformation, and the localized stiffness of the associated appliance element. The components of the applied force system generally lessen in magnitude as time proceeds from the instant of activation, and the significant portions of the force-versus-time pattern for the fixed appliance are (1) the initial magnitude, (2) the rapid decrease in magnitude associated with the deformation of the periodontal ligament, (3) the long-term decay resulting from remodeling of the alveolus (to be examined subsequently in more detail), and (4) any reactivation by the practitioner to begin a new cycle. Notable also is the possible creation of additions to the crown force system resulting from tooth displacement

caused by the principal activation. For example, an arch wire may be initially passive within a bracket, but mesiodistal crown tipping can eliminate the original second-order, wire-to-bracket clearance and gradually result in activation of a counter-tipping couple. In such a situation, although the original crown force system is decaying with time, a portion of the load is inactive initially, then grows from zero to some maximum magnitude dependent in a complex manner upon several parameters, and also, subsequently, decays in the absence of interference with the original activation. The intensity of internal force, or stress, at a particular instant and at a specific point within the periodontium, is directly related to the characteristics of the crown force system at that instant and the specific perio-

dontal location. The change in stress at a point with time follows the same temporal pattern as that of the crown load. The maximum stresses exist in the periodontal ligament and against root and the cortical plate at alveolar-crest or root-apex locations, depending upon crown loading, as discussed previously. Ideally, any tooth movement should be accomplished in a direct and reasonably rapid fashion while minimizing patient discomfort. Largely because of the differences in biologic responses

to stimuli found in virtually any sample of man, however, there currently exists no wide acceptance of one "best" force-time pattern to accomplish even the simplest single-tooth displacements. Nevertheless, the practitioner must be cognizant of several basic guidelines, whichever treatment philosophy is followed:

1. Continuous interruption of the blood supply to the periodontal ligament must be limited in time. 2. The amount of time under load, required to resorb sufficient alveolar

178


to permit a measurable tooth displacement, is apparently dependent upon the magnitude of pressure against the periodontal ligament. bone

3. Relatively light forces can generally produce the desired orthodontic displacements. Controlling the Force-Time Pattern The clinician, with the cooperation of the patient, can and must control

the magnitude-versus-time pattern of applied orthodontic force. Two displacements, together with appliance-element stiffnesses, are the key items in that control. The combination of appliance-member displacement(s) (activation) and stiffness(es) determine the initial level of applied crown force. The immediate, soft-tissue deformation and accompanying tooth displacement, and the same stiffness(es) determine the state at which biologic remodeling must begin and to what extent, if at all, the periodontal ligament has been hyalinized. In an over-simplified manner, orthodontic forces are sometimes categorized as "light" or "heavy," according to the initial magnitude of force generated within the appliance. In reality, the important consideration is not whether the initial appliance-element force is 50 g, 150 g, or 450 g, but rather (1) the maximum value and distribution of pressure in the individual periodontal ligament following soft-tissue deformation and (2) the stiffness of the appliance element generating the force. Accordingly, the classification

of appliances as generally "light-wire" or "heavy-wire" devices, with reference to stiffness is somewhat more appropriate. When periodontal-ligament pressures are sufficient to occlude blood vessels throughout a substantial portion of the tissue at the initiation of crown loading, as suggested in the aforementioned guidelines, the existence of a "rest and recovery" portion of the activation-deactivation cycle is imperative. Prior to the end of each cycle and reactivation, the maximum pressures must have dropped (as a result of remodeling and tooth displacement) to levels that permit the restoration of circulation and the

resumption of nutritional activity within the ligament. This requires almost complete deactivation of the appliance or the unloading of it. There need be little concern of losing the displacement achieved since whatever osteoid has been laid down is highly resistant to resorption compared to calcified bone. The intermittent orthodontic force exhibits a cyclic pattern with a period equal to the time between successive appointments. Applied by a heavy-wire appliance, the initial activation distance should be approximately twice, and generally never more than three times the expected, corresponding soft-tissue deformation. (Greater activations may produce some inelastic straining of the appliance as well as, perhaps, excessive force magnitudes.) Although the initial force may be rather high, so also is the appliance stiffness and immediately with the ligament deformation this force decays substantially. Bone remodeling then begins with undermining resorption likely in the higher pressure zones until

179


the

applied force drops off further with tooth movement. Again, this

force is termed intermittent because the combination of moderate activalion and high appliance stiffness results initially in ligament necrosis, but ultimately in pressure reduction to low levels long before the next

appointment to ensure the resumption of blood circulation within the soft tissue. A second type of acceptable orthodontic force is that typically applied by an extraoral or removable appliance. The interrupted force has a characteristic magnitude-lime pattern created by the patient who periodically unloads the appliance, thereby totally relieving all force for a

period of time. A typical interrupted, orthodontic force follows a 24hour cyclic pattern with the appliance in place and active for, perhaps, 10 to 14 hours a day. The force may be heavy and decay little; hence the inactive period each day must be sufficient to keep the periodontal liga-

ment healthy over the total period of use of the appliance. Continuous orthodontic forces do not, by their definition, decay to zero during the between-appointments period. Exerted by rather highly flexible appliance elements, activations must be to relatively low force levels so as to occlude no more than a small percentage of the vessels within the periodontal ligament (and not substantially interfere with nutritional activity). Bone remodeling in pressure zones is expected to occur primarily as direct resorption of the cortical-plate surface. Because the continuous-force cycle includes no "rest period," little interference with normal biologic functioning within the soft tissues can be tolerated. Figure 5-25 provides a qualitative comparison between the magnitudetime patterns of acceptable light and continuous, and relatively heavy, but intermittent, orthodontic forces. To produce the total desired displacement by a particular appliance configuration or set of mechanics often requires several months. A new activation-deactivation cycle begins each time the clinician reactivates

the appliance, and this reactivation may occur four or more times in the overall process, for example, in distally driving canines into firstpremolar extraction sites. If the teeth to be moved are receptive to ortho-

dontic force, a reactivation is necessary only when the appliance has exhausted all or most of the mechanical-work capability provided by a prior activation or reactivation. In the absence of some appliance failure, the total displacement is accomplished incrementally, and an examination of one activation-deactivation cycle provides the substance of the

entire tooth-movement process, even though perhaps only one-fourth or less of that total displacement is depicted. Figure 5-26 shows two time-versus-displacement plots for one acti-

vation-deactivation cycle and a between-appointments period of 3 weeks. The displacements are simple-tipping of canine teeth on opposite sides of the same arch. The solid plot represents an initial force magnitude of 60 g; the dashed plot was obtained following activation to 150 g. Each crown displacement takes place in 3 stages. Stage 1 is the almost immediate movement reflecting the deformation of the periodontal ligament by the tooth root; little difference in this first portion of the overall displacement is noted for the 2 teeth as, apparently, 60 g was

180


is

0 0

u

soft-tissue deformation

undermining resorption

Heavy, intermittent force

Light, continuous force

deactivation

Time

FIGURE 5-25. A one-cycle comparison of the magnitude-versus-time patterns of intermittent and continuous orthodontic force.

sufficient to completely compress the ligament at the root apex and alve-

olar-crest levels. During stage 2 the remodeling of fibers and bone begins and further displacement is virtually nonexistent. The length of this stage may vary from several days to several weeks or longer, depending on the level of pressure against the cortical plate. From the figure, stage 2 for the tooth initially receiving 60 g of load was completed in about 4 days; for the canine subjected to 150 g this second stage extended to nearly 2 weeks. As bone is resorbed in the pressure zones and the effect of resorption reaches the bone-ligament interface where complete softtissue compression had occurred upon loading, additional tooth movement signals the beginning of stage 3. When direct resorption takes place, stage 3 displacement is more gradual than when substantial undermining resorption is necessary before any further tooth movement can occur. This is reflected in Figure 5-26 in that, after stage 3 began for the tooth receiving 150% more initial force, the displacement progressed at a more rapid rate such that, when the patient appeared on day 22, the practitioner measured nearly the same accumulated canine crown displacement on both sides of the arch (Gianelly and Goldman, 1971). The overall results depicted in Figure 5-26 are indicative of the preference of some clinicians for heavy-force mechanics and others for lightforce appliances and techniques. Conceivably, however, had a third,

similar canine been subjected to an initial force of several hundred grams, stage 2 might not have been completed in 21 days; hence, no rest period for the ligament would have occurred between appointments, a

)flSO of Dentition and Periodontium to Force

I I I I I I

I

/ —U

/

/

/

/

/

/

/

/

Stage 3

—— Stage 2

Stage 1

5

10

15

20

Time (days)

E 5-26. The three-stage pattern of tooth movement. Plots of displacement versus Or a pair of canine teeth, one subjected to an initial distal driving force of 60 g and the other to 150 g (dashed).

182


-

reactivation of the appliance might endanger ligament vitality, and the

result of excessive initial force applied 3 weeks earlier is likely an exten-

sion of the total time to complete the full tooth movement. The graphics of Figure 5-26, in illustrating the three-stage nature of orthodontic tooth movement, are indicative of the first, betweenappointments period associated with a long-term displacement; the 1.5 mm is perhaps one-fifth of the total movement desired. A reactivation will produce another displacement-time pattern, similar to the first cycle if (1) nearly total deactivation of the appliance had taken place, (2) the periodontal ligament had been fully revitalized if hyalinization oc-

curred early in the cycle, and (3) the reactivation displacement of the appliance was the same as that at the start of the first cycle. In general, the lighter the force, the less significant may be stage 1 and the shorter is stage 2. Assuming that measurable movement does not take place during the first cycle, subsequent reactivations can be to somewhat larger

initial loads since cellular activity has been ongoing and the tooth or teeth are loosened with slightly wider periodontal space than before the orthodontic force was first applied. Also, the apparently insignificant difference in one-cycle displacements shown in Figure 5-26 may be the long-term result, indicating that an initial magnitude of force might be selected from a range of acceptable values. It is clear, however, that the range is bounded. If the force is too light, the rate of stage 3 displacement will be low and the between-appointments displacement small.

With the load magnitude too large, little more than stage 1 displacements will occur and, with subsequent reactivations and no relief for the tissues from the high pressures, accumulated damage to the periodontal ligament is likely.

Physiologically Proper Orthodontic Forces An orthodontic force or force system may be said to be "physiologically proper" when the force-time pattern produces a reasonable rate of tooth

movement without causing extensive interference with the normal biologic activity within the periodontium and, therefore, no irreversible damage to the tissues. Although very light forces fulfill this prescription, the efficient management of treatment demands that displacements be performed as rapidly as possible. Some discomfort is often associated with activation or reactivation of an appliance due to the relationship between the pressures developed and the pain threshold of the patient; again, efficient management to the end of gaining and maintaining good patient cooperation suggests minimization of this discomfort.The orthodontist, then, should strive to apply force that is controlled both with respect to initial value and, through astute selection of appliance elements, in terms of its decay pattern over time. From the foregoing discussion, the principal indicator of appropriate force is the pressure generated in the periodontium, more specifically, in the periodontal ligament. Because this pressure is actually compressive stress, and stress level is generally related to applied force through the area over which the stress is distributed, a first consideration is the num-

183

Response of Dentition and Periodontium to Force TABLE 5-2. Root-surface areas in centimeters squared for average normal permanent teeth

Maxillary arch

Mandibular arch

Tooth

Central Incisor Lateral Incisor Canine First Premolar Second Premolar First Molar Second Molar Third Molar

1.40 1.10 2.05 1 .50

1.40 3.35 2.70 1.95

1.05 1.25 1.60 1 .30

1.35

3.50 2.80 1.90

ber of teeth directly affected by the applied force system and the size of

the roots of those teeth. Table 5-2 provides the root surface areas of individual teeth of average size. Note that these areas vary from the smallest to the largest by a factor of 3.3. Given the value of an applied force and the tooth or segment to receive that force, the average stress in the periodontal ligament may be obtained by dividing the force magnitude by the affected root-surface area. Using this approach, then, nearly

twice the force magnitude to displace a mandibular central incisor should be applied to move a maxillary canine. Again, however, the key parameter is specifically compressive stress and three, additional, important factors must be considered in this analysis: (1) the direction of the applied load, (2) the couple-force ratio (or type of displacement desired), and (3) the load-time pattern. Refining the procedure of selecting an initial orthodontic loading magnitude to emphasize pressure and direction of load as criteria suggests replacing the overall root-surface area with projected areas. Table 5-3

gives these projected root-surface areas from three perspectives: occluTABLE 5-3. Projected root-surface areas in centimeters squared for average normal permanent teeth presented from three perspectives

Mandibular arch

Maxillary arch FL

MD

OG

Tooth

FL

MD

00

0.50 0.40 0.70 0.50 0.50

0.70 0.65 0.75 0.75 0.55 1.20 1.00 0.80

0.40 0.30 0.45 0.30 0.30 0.80 0.70 0.50

Central Incisor Lateral Incisor Canine First Premolar Second Premolar First Molar Second Molar Third Molar

0.25 0.25 0.70 0.60 0.60

0.45 0.50 0.75 0.60 0.60 1.10 0.95 0.65

0.20 0.20 0.35 0.30 0.30 0.85 0.75 0.60

1.35 0.95 0.60

FL: Faciolingual perspective MD: Mesiodistal perspective OG: Occiusogirgival perspective

1.05

0.95 0.65

184


sogingival, faciolingual, and mesiodistal. The only orthodontic displace-

ments I or which pressure does not cover such an area are extrusion and long-axis rotation, although only a partial (occlusogingival) area may be involved in an intrusive displacement. Recalling the differences in stress distributions produced in the various transverse tooth movements (Figs. 5-5, 5-8, 5-10, 5-12, and 5-14), Table 5-1 presented sets of theoretically determined couple-force ratios, center-of-rotation locations, and maxi-

mum stress values for five such displacements of an average canine. Noting particularly that the distal driving force is identical in four of the displacements, because of the sizable differences in maximum stress

values between simple tipping and bodily movement, for example, a substantially larger crown load may be used in attempting the latter displacement. The influence of root shape should also be brought into the determina-

tion of appropriate crown loading when warranted. Perhaps the best example of root-shape impact is in the comparison between long-axis rotation of the single- and multirooted tooth. The specific factor is the substantially greater amount of bone resorption required to bodily move the bifurcated-root apices that follow circular paths around the long axis.

In other instances, unusual root curvatures or cross-sectional shapes may provide significant input in determining load magnitude. A final, important factor is individual biology. Resistance to or ease of remodeling is affected by alveolar bone density and to an extent by cortical-plate and

periodontal-ligament thicknesses. A reasonable manner of including this parameter is to estimate by clinical observation and patient input, a nearly "optimum" orthodontic force magnitude for a specific tooth displacement early in treatment. Subsequently, by proportions according to projected root-surface area and type of movement primarily, appropriate load values may be established for succeeding displacements to be undertaken in the same dentition and within the same treatment philosophy. With all principal influencing factors now mentioned, a hypothetical procedure may be outlined which leads to the determination of the suggested, initial level of physiologically proper force with which to begin a specific orthodontic displacement in the average patient. The approach to be outlined arises from the impressions of clinicians, based on their experiences, together with theoretical and bench-experimental analyses. The fundamental bases are (1) the knowledge of normal blood pressure

and, therefore, a reasonable approximation of the ligament pressure necessary to interrupt blood flow at a point within that tissue, (2) the assumption that vessel occlusion over a sizable volume within the periodontal ligament for a period of several weeks is necessary to produce irreversible pathology, and (3) the understanding that, whenever direct pressure exists within a portion of the ligament (and at the interfaces), elsewhere in that tissue fiber tension is the direct stress that should not interfere substantially with nutritional activity in those areas. The procedure begins with the following hypothesis: an intermittent,

simple-tipping force should have an initial value such that the cone-

185

Response of Dentition and Periodonfium to Force

sponding average stress within the affected periodontal ligament is approximately 30 g/cm2. In other words, the initial magnitude of the trans-

verse-tipping force should be the product of 60 g/cm2 and the mesiodistal or faciolingual projection of the total root area transferring responsive tension and pressure. (In transverse movements, two projected areas are under normal stress—mesial and distal, for example. Hence, 60 g/cm2 is multiplied by one of these areas or 30 g/cm2 by the sum of the two areas.) If the applied force is to be continuous and no "rest period" will occur over a period of four or more weeks, the initial value should be approximately 0.4 of the corresponding intermittent magnitude (based on the assumption that such a force will decay, beyond stage 1, about 40% during one between-appointments period). For bodily movement, the stress distribution along the root length is substantially more nearly uniform than the response in simple tipping (see Figs. 5-5 and 5-8). Also, the ratio of maximum stresses to applied force is much lower in bodily movement (see Table 5-1). Accordingly, a larger initial force can be tolerated in bodily movement, and the suggested value is based on the average of the alveolar-crest and root-apex stresses taken from Table 5-1. The resulting formulas for the initial val-

ues of the force and couple portions of the intermittent, transverse crown load to produce bodily movement are Transverse force = 3.5(simple-tipping force) Couple = 0.5(transverse force)(total tooth length)

51

(Assumed is the crown force system effectively applied at point CC on the long axis, midway between incisal edge [occlusal surface] and the cementoenamel junction.) Again, examining Table 5-1 and Figure 5-11, the couple-force ratio for root movement is understandably somewhat greater than that for bodily movement. Once more employing as a portion of the basis the average of alveolar-crest and root-apex stresses equal for all transverse movements, the formulas to follow are offered for the torquing of roots through intermittent loading: Transverse (holding) force = 0.8(comparable bodilymovement force) Couple-force ratio = 0.6(total tooth length)

(5-2)

When several teeth are to undergo similar displacements as a single entity, for example, in the retraction of four incisors in either arch, the computations may be made for the segment as a whole using an average

tooth length and the sum of the projected root areas of the individual teeth. Proceeding as above, for transverse crown movement the following formulas are suggested: Transverse force = 1. 7(comparable bodily-movement force) Couple-force ratio = 0.4(total tooth length)

186


Although not occurring often in actual clinical practice, but appropriate

to include for completeness in the set of formulas for proposed, physiologically proper transverse force systems is this equation for pure torquing:

Couple = (comparable simple-tipping force)(root length)

(5-4)

The couple-force ratio is undefined because no separate crown force acts.

The three nontransverse orthodontic displacements—intrusion, extrusion, and long-axis rotation—might be characterized collectively by the dominance of fiber tensions within the responsive force system. Vir— tually no pressure occurs anywhere between root and periodontium in response to purely extrusive action; in long-axis rotation, the sizes and locations of pressure zones depend entirely on root anatomy. To complete the set of formulas for the force-system components, then, for the

remainder of the eight, previously defined displacements, in the absence of pressure as a principal influence, collective judgments of clinicians, based on their treatment observations, are the principal bases for the following suggestions: Extrusion: Force =

(80

g/cm2)(occlusoapical

projection of root-surface area in cm2) Intrusion: Force =

(60


projection of root-surface area in cm2) Long-Axis Rotation: Couple =

(30


projection of root-surface area in cm2)(average crown width)

Several addenda to these formulas are pertinent: 1.

If the tooth is against the cortical plate during an attempted intrusive displacement, the applied force computed from the formula should be doubled.

2.

If a multirooted tooth is intruded, the magnitude of force obtained from the formula should be increased 50%.

3.

If long-axis rotation of a multirooted tooth is envisioned, the couple value calculated from the formula is multiplied by 1.5 if the tooth has

two roots or by 2.5 if the tooth has three roots. It is worth repeating that the foregoing formulas are all presented for computations of initial, intermittent crown loadings. If the crown force system is continuous with only a moderate decrease in magnitude during the between-appointments period, beyond that associated with the rapid, soft-tissue deformation, the initial magnitudes should be 50 to 60% less, depending on the amount of long-term decay. On the other

187


hand, the interrupted force that is cycled daily may have a magnitude

larger than that of the corresponding intermittent force due to (1) the increased frequency of rest periods and (2) the generally longer accumu-

lated lime of total absence of the interrupted force between appointments. A reasonable procedure is to make the magnitude of interrupted, orthodontic force the product of the corresponding, intermittent force and the ratio of 24 to the number of hours the interrupted force is active during the day. Implicit in the considerations of responsive forces arising from the application of crown loading is the existence at any specific time of the quasi-static state defined in Chapter 2. Hence, the resultant of the distributed force system exerted on the root of a tooth under orthodontic load is identical, for practical purposes of analysis, in all characteristics except sense to the resultant of the force system applied by the appliance to the crown. The appliance as well as the dentition exhibit this quasistatic state. The reaction against the appliance accompanying the action of the appliance is transmitted to the anchorage location(s). Focusing on the appliance momentarily, the force system exerted by the anchorage is equal in magnitude but opposite in sense to that exerted by the crown(s) at the displacement site(s). In general, "anchorage" refers to stability, and in orthodontics the anchorage supports the appliance, receiving the

responsive force system transmitted through the appliance to it. Although in orthodontic therapy the entire force system developed by the appliance is sometimes intended to produce displacements, more often a portion of the appliance is attached to anchorage which, by definition, is to be displaced very little or not at all. The rationale for the use of the extraoral appliance is the location of anchorage external to the oral cavity, against the back of the neck or the cranium. Intraoral anchorage is often necessary or desirable, and the knowledge of the influences of the various force parameters on orthodontic displacement have led to the differential force concept.

The actual meaning of

differential force,

not unrelated to the

hypothesis of a physiologically proper force, is best explained by an example. Consider the distal movement of a mandibular canine into the space created by the extraction of the formerly adjacent first premolar. Using intraoral mechanics exclusively, the distal driving force is created by extending a helical-coiled spring or an elastic module between the

canine and the first molar. The forces against the canine and molar, holding the activating appliance member in a stretched configuration, are equal in magnitude (from a quasi-static analysis of that member) and tend to move both teeth. The force against the molar, however, may be distributed also to the second molar (if "tied" to the first molar) or the second premolar (through crown contact). If the anchorage includes all three posterior teeth mentioned, from the projected areas given in Table 5-3, the anchorage area is approximately 250% larger than that of the canine. Accordingly, although the forces at the anchorage and displace-

ment sites are equal in magnitude, the average periodontal-ligament stress is 3.5 times greater at the canine. The size of the active force (upon

188

Bioengineering Analysis of Orfhodontic Mechanics

the

canine crown) should be just large enough to produce primarily

direct resorption of bone and a reasonable rate of canine movement. The same force magnitude, then, should not be sufficiently great to create adequate pressure to produce appreciable displacement of the anchorage during the between-appointments period. Note that if the active force is relatively large, so also is the responsive force, and both units may move comparable distances over time. Furthermore, extreme force magnitudes can result in displacement of the anchorage while, in effect, producing ankylosis of the canine. Refer-

ring once more to Table 5-1, noting that the distal forces for four movements were equal, changes in the couple-force ratio are seen to alter the maximum-stress values and the overall stress pattern with respect to a uniform distribution as a reference. Apparently, then, a uniform stress distribution is ideal for the anchorage, although it is difficult to obtain in the clinical setting. More realistically, the creation of a crown couple with the appliance to counter in sense the tipping potential of a force against anchorage is a step toward reinforcing that anchorage, beyond that achieved by maximizing the root area within the anchorage. The point is made, then, that the phrase differential force is a misnomer; the differentials are actually in the average stresses and/or the stress-distribution patterns within the ligament at the displacement and anchorage sites.

The parameters that influence the determination of the initial magnitude of a physiologically proper orthodontic force have been indicated in the foregoing discussion, and a set of formulas was given. Through the literature, additional terms have been introduced to describe or categorize forces exerted on the tooth crown. The lightest of these actions have been termed threshold forces, just large enough to result in some remodeling which permits measurable movement over a lengthy, perhaps unrestricted, period of time. Differentiation between "light" and "heavy"

orthodontic forces was undertaken previously. Upper-bound or maximum orthodontic forces produce ligament necrosis for a time, are likely to be initially painful to the patient, but result in tooth movement with no significant, long-term pathology. Forces are termed excessive if they crush the periodontal ligament over a substantial portion of the area under pressure, probably result in little or no tooth movement during a between-appointments period, and if continuous beyond several weeks may cause irreversible tissue damage. During the 1950s and 1960s research was undertaken toward determination of optimum-force magnitudes associated with specific, orthodontic displacements. Values were sought that would produce desired movements most rapidly and cause only minimal interference with normal biologic activity and patient discomfort. A principal motivator in these investigations was the solidification, through in vivo experimentation, of the validity of the differential-force concept. In the late 1960s, however, papers began to appear in the literature exhibiting research results that tended to discount an optimum-force theory. The influence

of the magnitude-time pattern on displacement achieved has been

-

189


a) Ca

a)

E

a)

0

Light force,,,1!'

/I'

/ /

Threshold force

\HeavY force

\ \ Excessive

\

force

\

As-Activated Force Magnitude FIGURE 5-27. Qualitative plot of expected, be!ween-appointments displacement versus the initial level of orthodontic force.

shown to be a significant factor in addition to parameters rightfully con-

sidered in the determination of the initial force values. Collectively, from

these efforts, the acceptance of the hypothesis of an existing range of orthodontic force magnitudes has emerged for each specific displacement, which produces no long-term physiologic conflict. The plot of initial force magnitude versus displacement rate of Figure 5-27 provides a qualitative illustration of the present understanding of expected results with a typical patient. Dento facial Orthopedics Although the typical displacements performed during orthodontic treatment are of individual teeth or dental-unit segments, the total therapy in a specific case may, in part, call for dentofacial orthopedics. Examples of orthopedic forces are those induced by an appliance designed to widen the

maxillary arch through palate splitting, to tip the maxilla (altering the angulation as seen in a buccal view), and to exert influence on the pat-

tern of growth of the mandible. The appliance may be intraoral or extraoral, but the applied force must always be distributed over a substantial area and, correspondingly, the magnitude of the total load is large. The load may or may not be transmitted through the dentition, but the biologic remodeling is not intended to be within alveolar bone; in general, relative displacements of adjacent teeth in the same arch are unwanted. In palate splitting, the intention is to induce transverse tension along

190


the median suture. The appliance used is totally intraoral, producing bilateral action with force transmitted at least partially through the posterior teeth and often, in part, by direct appliance contact with the hard

palate. That portion of the load imparted through the lingual crown surfaces must originate from an appliance element sufficiently rigid such

that the individual teeth are not displaced relative to their alveoli. Because remodeling is to occur in tensile zones, the magnitude of physiologically proper force is not directly associated with the suture-bone in-

terface area, but with the force-time pattern and the faciolingual projected areas of the tooth roots through which the force is to be transmitted. Because patient discomfort is often used as an indicator of sufficient or excessive activation-reactivation of the appliance, the appliance should be designed so that the pressure generated in those periodontal

ligaments, through which the active force system is transmitted, approaches a uniform distribution. In orthopedic action designed to tip the maxilla, at least a portion of the anchorage may be extraoral; the mandibular arch may also be used in the anchorage. Depending on whether vertical displacements are desired only anteriorly, only posteriorly, or divided between anterior and posterior locations, the vertical component(s) of the load are applied

anteriorly or posteriorly as individual forces, as equal and opposite forces at both locations to form a couple, or as unequal and opposite forces at both locations to generate a force-and-couple resultant. Heavy, continuous arch wires are in place to prevent individual tooth displacements. The desired remodeling is primarily along the zygomatic process; the stress distribution at that interface is nonuniform and the center of rotation for the maxilla as a whole will be located, anteroposteriorly, approximately directly below the point in a buccal view where the stress distribution, or its extension, goes to zero (or changes sense from tension to compression or vice versa). Figure 5-28 illustrates two forms of —

/

—

N

-

Fan

Resultant

HG: Headgear Ill: Class Ill 00: Occlusion II:

Class II

FIGURE 5-28. Tipping the maxilla to open the bite anteriorly (left) and to close an open bite primarily through posterior vertical action (right).

191


0 o

Solid: Appliance activated Dashed: Appliance disengaged

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I I

I

I

I

I

I

I

I

I I

0

24

L 48

72

Time (hours)

FIGURE 5-29. The typical interrupted, force-time pattern of an extraoral appliance.

maxilla tipping. Note that horizontal (posteriorly directed) components

of force nearly always accompany the wanted vertical components, owing to the nature of the appliances. The magnitude of force applied should be gauged by imagining the maxilla as a single tooth with total length equal to the anteroposterior dimension. The tipping load is a force, a couple, or a force and couple, and the couple-force ratio for the desired displacement format may be estimated. Also note that the max-

illa is supported, in essence, only superiorly; as modeled as a large, individual tooth, then, it has only approximately one-half of the equiva-

lent periodontal support of a "tooth" with comparable support (root) surface.

To control mandibular growth downward and forward, orthopedic force may be applied by an extraoral appliance. The force is most often applied by a chin cap rather than through the mandibular dentition. The maximum levels of force used are governed by the acceptable pressures induced against the temporomandibular-joint members. In order that large forces may be employed, the force-time relationship is the typical interrupted pattern of extraoral appliances as depicted in Figure 5-29. This appliance is discussed in more detail in Chapter 9. -

Synopsis This chapter has endeavored to examine the biomechanical response of

the dentofacial complex, with emphasis on the dentition and its sup-

192


porting structures, to applied force during orthodontic treatment. The delivered force system, most often exerted through the facial crown surfaces, originating in the activation of an appliance, follows a discernible

path into and through the alveolar process and beyond. The presence of force within the tissues, augmenting the normal physiologic activity, catalyzes biologic processes that result in tissue remodeling, thus per-

mitting measurable displacements of teeth—individually or in segments—or of an entire arch. Eight distinct forms of orthodontic (individual-tooth) displacement have been defined. For each, the characteristics of the crown force system, the distributed responsive pattern of tension and pressure within

the periodontal ligament and against the root, and the displacement format, were discussed. Without becoming deeply involved in the physiology, the remodeling of the periodontal structures has been considered with emphasis on the role of induced internal-force intensity, in particular that of compressive stress (pressure). The movement of teeth by means of orthodontic therapy is known to be controlled by the forcetime pattern of the loading as well as the initial (as-activated) load characteristics; this pattern, in turn, is dependent on the structural properties of certain appliance elements and by the actual displacements that

occur. The seemingly closed loop notwithstanding, advancements through research are leading to the refinements of procedures to determine and apply physiologically proper ranges of orthodontic force. A "cookbook approach" to the quantification of initial levels of crown loading has been offered. The concepts of differential and optimum force were introduced and discussed briefly. Through the former, the term anchorage was defined. Some attention was given to the biomechamcal analysis of dentofacial orthopedic therapy. This chapter has provided descriptions of and guidelines for desirable characteristics of force systems delivered to the dentition in orthodontic treatment. These force systems are developed within and transmitted to the crowns by the orthodontic appliances. Beginning in the subsequent chapter and continuing throughout the remainder of this text, structural analyses of appliance elements are undertaken, individually and in relation to their roles in the appliance as a whole. The ultimate objective is the refinement and improvement of appliance designs toward devices that provide ideal magnitude-time patterns of orthodontic force at those locations where displacements are desired, while concurrently minimizing unwanted, secondary effects elsewhere.

References Burstone, C.J., Pryputniewicz, R.J., Bullock, C., and Hubert, M.: Centers of

rotation of the human maxillary central incisor. J. Dent. Res., 61 IADR Abstract 1095, 1982.

193


Burstone, C.J., Pryputniewicz, R.J., and Weeks, R.: Center of resistance of the

human mandibular molar. J. Dent. Res., 60 IADR Abstract 822, 1981. Gianelly, A.A., and Goldman, H.M.: Biologic Basis of Orthodontics. Philadelphia, Lea & Febiger, 1971, Chapter 4. Nikolai, R.J.: On optimum orthodontic force theory as applied to canine retraclion. Am. J. Orthod., 68:290—302, 1975. Nikolai, Periodontal ligament reaction and displacement of a maxillary central incisor subjected to transverse crown loading. J. Biomech., 7:93—99, 1974.

Readings C.H., and Johnston, L.E.: A clinical investigation of the concepts of differential and optimal force in canine retraction. Angle Orthod., 44:113—

Boester,

119, 1974.

Burstone, C.J.: Application of bioengineering to clinical orthodontics. in Current Orthodontic Concepts and Techniques. 2nd Ed. Edited by T.M. Graber and B.F. Swain. Philadelphia, W.B. Saunders, 1975, Chapter 3.

Burstone, C.J.: The biomechanics of tooth movement. in Vistas in Orthodontics. Edited by B.S. Kraus and R.A. Riedel. Philadelphia, Lea & Febiger, 1962, Chapter 5.

Burstone, C.J., Baldwin, J.J., and Lawless, D.T.: Application of continuous forces to orthodontics. Angle Orthod., 31:1—14,

1961.

Burstone, C.J., and Pryputniewicz, R.J.: Holographic determination of centers of rotation produced by orthodontic forces. Am. J. Orthod., 77:396—409, 1980. Graber, T.M.: Orthodontics, Principles and Practice, 3rd Ed. Philadelphia, W.B. Saunders, 1972, Chapter 10. Jarabak, J.R., and Fizzell, J.A.: Techniques and Treatment with Light-wire Edgewise Appliances, 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 7.

Reitan, K.: Some factors determining the evaluation of forces in orthodontics. Am. J. Orthod., 43:32—45, 1957.

Reitan, K.: Bone formation and resorption during reversed tooth movement, in Vistas in Orthodontics. Edited by B.S. Kraus and R.A. Riedel. Philadelphia, Lea & Febiger, 1962, Chapter 3. Ricketts, R.M., et al.: Bioprogressive Therapy, Book 1. Denver, Rocky Mountain Orthodontics, 1979, Section 1, Part 6.

Smith, R.J., and Burstone, C.J.: Mechanics of tooth movement. Am. J. Orthod., 85:294—307, 1984.

Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982, Chapters 10—12.

Thurow, R.C.: Atlas of Orthodontic Principles. 2nd Ed. St. Louis, C.V. Mosby, 1977, Chapters 2, 14.

Introduction to Structural Analysis of the Orthodontic Appliance

Beginning in this chapter and continuing throughout the remainder of

this text, force and structural analyses of orthodontic-appliance elements or members, and assemblies thereof, are undertaken. In the following discussions, the terminology and procedures used in structural analysis are introduced, and the orthodontic appliance is compared to the typical engineering structure. The manner of attachment of the appliance to the dentition is examined in detail, highlighting the associated force system and devoting particular attention to orthodontic brackets. The continuous-arch-wire appliance is then modeled to simplify a geo-

metrically and mechanically complex member for analysis purposes without altering its essential characteristics. A four-step analysis Focedure is introduced. The activation and deactivation behavior of several of the more common orthodontic-appliance elements is discussed. This chapter concludes with a section that examines the "control" of treatment progress and the roles therein of the appliance, the clinician, and the patient. A structure is an interconnected assembly of individual members; the assembly is designed to withstand without failure a specific loading and to transmit the force system induced by that loading into some foundation or supporting substructure. The load may be "dead" weight, some other static action, or it may be dynamic, varying characteristically with time. A "machine" may be distinguished from a structure. The typical structure is designed primarily to sustain a load and its principal components are usually stationary; a machine is designed primarily to transmit a force system from one site within the assembly—the input—to another— the output; moreover, the machine typically has movable parts. Buildings and bridges are familiar structures and a pair of pliers is an example 194

195


of a machine. Because both structures and machines include, when "in

service," members within which internal force systems exist, structuralanalysis procedures are as appropriate to machines as they are to structures. Each member of the assembly contributes toward the performance of

the desired function of the machine or structure. The loading of the structure or the input to the machine results in subjecting each element or member to a force system. The force system delivered to a specific member may be transmitted entirely through its connections to other members, or, it may result, in part, directly from external activation. Each member must be analyzed initially to determine the type of force system exerted on it and then the relationship of each component of that force system to the external load or input. After the external force system has been quantified, the internal force distribution throughout the volume of the member must be obtained toward consideration of possible failure of the member. Failure of a structural or machine member occurs when either stresses or deformations exceed their allowable values. Excessive stresses may result in outright fracture of the member or in drastic reduction of its ductility, for example. Excessive deformations may eliminate clearances that must exist or may result in structural instability or collapse of the structure. Stresses generally vary from point to point within the mem-

ber, and deformations ordinarily vary from one cross-section of the member to another; hence, the analysis is undertaken to determine not only the maximum values of stress and deformation, but also where within or along the member they occur. From the discussions within Chapter 3, the material and geometry (size and shape) of a structural or

machine member are known to influence, in addition to the applied force system, the induced stresses and deformations. The typical structural analysis proceeds according to one of two formats: 1.

-

Given the design or actual structure or machine and the failure criteria, to be obtained is the maximum allowable loading or input force system.

2. Given the design, the loading or input, and the possible modes of failure, computations are made to determine whether or not a failure might occur in the actual structure or machine. The analysis procedure follows a well-defined series of steps, regardless of the format. First, a force analysis is undertaken, for the structure or machine as a whole and for each member of the assembly, to obtain in terms of the loading or input the components of the force system exerted on the entire assembly and on each member at the connections. Newton's laws (Chap. 2) and the relationships governing continuity of displacements may be used to carry out this first step. Second, the failure criteria are established (fracture, excessive inelastic behavior, formation or extension of a crack, or deformation beyond established bounds) and

196


quantified for each member and for the structure as a whole. Each maxi-

mum permissible stress or deformation is modified in value by a "factor of safety" to account for inadequacies in the theory or model used in the

analysis. Third, "critical" cross-sections and locations of maximum stress therein are pinpointed within each member of the structure or machine. Fourth and finally, the stresses and deformations at the critical

locations are obtained in terms of the force system exerted, and are compared to the failure criteria as quantified to establish the structural integrity, or lack of it, of the assembly. A structural analysis of an existing structure or machine would be undertaken when a change in the loading or input force system is contemplated. Such an analysis would also be carried out as a primary facet in the preparation of a new design. The overriding design objective is always the creation of an assembly that is capable of sustaining, without

failure, the force system to be induced in it. Design constraints with regard to space, materials, and perhaps aesthetics may also require satisfaction. Several alternative designs may meet the principal design criteria, and one may have to be selected through secondary considerations

or judgmental factors. Often, after choosing one design from among those considered, further modification of it occurs that will not result in infringement on the failure criteria; excess material may be removed from regions of low stress, for example. In short, the preceding comments are intended to suggest that design and analysis are closely related.

The Orthodontic Appliance: A Structure or a Machine? The function of the orthodontic appliance is the application of specific

force-system components to selected intraoral sites to produce desired displacements of dental units. The appliance is similar to the typical structure or machine; forces exist at the connections of its elements and within its members when "in service," and it, therefore, may appropriately be subjected to structural analysis. The principal "loading" of the orthodontic appliance, however, is unlike that of a structure or machine; its load source is not an external force system (although the forces of occlusion or mastication may apply secondary, external loading). The "input" to the appliance is the deforming of it from its passive configuration to induce internal forces that are to be transmitted to the displace-

ment sites. The "output" is the force system delivered to the tooth crowns to produce selected movements of portions of the "foundation" to which the appliance is attached. In a strict sense, the action of the appliance is dynamic; most components of the internal force system decrease in magnitude with the passage of time in the activated state. This change is so gradual, however, and the masses of the appliance elements are so small, that analyses of the primary force system in the activated appliance may be undertaken

197


using static relationships with negligible error. Restricting attention only

to the generally quasi-static appliance system, failures might be assumed to occur immediately upon activation or not at all during the between-appointments period. Forces of occlusion, however, do provide real dynamic loading, superimposed on the appliance activation. Although this source of "load" is comparatively minor by itself, it can result in loss of structural integrity and failure with, for example, a frac-

ture within the bond between bracket pad and enamel surface. The "loading" of the removable appliance is dynamic when examined in the

context of a time frame of days or weeks, and progressive failure— resulting in the fracture of a wire component of a removable retainer—is not uncommon. Apparently required, then, is at least the direct incorporation in a static-appliance analysis of a factor of safety to account for the dynamic loads that may be intermittently applied to it.

The structural analysis of the ideal orthodontic appliance may be undertaken through a hypothetical design of the appliance. Principal design parameters include specific stiffnesses and elastic ranges of the members of the appliance to be activated. Generally, the activation is such that inelastic material behavior does not occur anywhere in the appliance; hence, maximum stresses may not exceed elastic-strength values. Also important are the appropriate member stiffnesses at anchorage locations, a somewhat subtle consideration to be taken up in later discussions of specific mechanics. The appliance design must con-

sider the possible paths of the force transmitted from the activation site(s) to the anchorage location(s), and the distribution of anchorage force. Pertinent in addition are the shape changes in the appliance and resulting alterations in the force system as deactivation due to tooth displacements takes place; not all of these changes in appliance configurations are mirror images of the activation deformations. And not to be overlooked, the materials of members must be chosen with full regard for the biologic environment in which they will be placed.

-

:achment of the Orthodontic Appliance to the Dentition intraoral orthodontic appliance is supported by the tooth crowns and its overall configuration is maintained through its connections to them. These connections also provide the final measure of continuity The

that permits the transfer of active force from the appliance to the teeth to generate the wanted displacements, regardless of the type of appliance. Although these connections may be made in a variety of ways, considering all phases, types, and philosophies of treatment, the most common is through the orthodontic bracket affixed to the facial or lingual surface of the tooth crown. The bracket as attached is meant to be integral with the crown; hence, it should exhibit high overall stiffness, although strength is more often the principal concern. Failures in bracket-to-crown attach-

ment may occur by fracture, either at or near the solid joint of the

198


bracket to its pad or to the band, or in the bond of the pad or band to the

crown enamel. The latter is more common. Attachment of the bracket assembly to the crown may be through cementation of the metal (stainless steel) band that encircles the crown or by bonding of the bracket pad to the facial or lingual surface. The merits or disadvantages of one scheme versus the other may be argued from several standpoints; however, of interest here is the structural comparison. The force system transferred from bracket to crown must travel through the cement or adhesive and the strength of the attachment depends on the size and shape of the attachment area, the characteristics of the force system transferred through it, and the inherent unit strength of the cement or adhesive itself (internally, between it and the band or pad, and its attachment to the enamel). The cement or adhesive may be stressed in many directions because of the various force-system components that may be present—individually and in combinations. In addition, tensile and sheat strengths differ for a specific product and between products. A detailed comparison would, therefore, be lengthy and inappropriate here; however, two general comments are pertinent. First, the bonding area of the band is at least double that of the typical bracket pad; hence, the inherent strengths of the bonding adhesives must be substantially greater than those of band cements. Second, the more nearly uniform the stress distribution within the adhesive, the stronger the bond; accordingly, failures of bonds of pads to enamel surfaces are more likely under loadings that tend to produce "peeling" (from first- or third-order couples or from occlusogingival or mesiodistal forces that have a moment arm with respect to the adhesive) or rotation (from second-order couples) than when the force system is direct tension, shear, or compression. The bracket-band and bracket-pad assemblies are compared pictorially in Figure 6-1 as to stress generation within the bond resulting from the application of a facially directed force. Although the uniform stress distribution is desirable, the average intensity is less within the band cement because of the greater area of attachment. Note also that a failure within the band cement will not result in complete disruption of the force-transmission path for the particular loading shown. Figure 6-2 illustrates several other simple force systems against the bonded bracketpad assembly and the resulting stress distributions within the adhesive.

F,

stress

tensile stress stress

FIGURE 6-1. Applied and responsive force systems exerted on bracket-pad (left) and bracket-band (right) assemblies.

199


Fa

tension

F,

compressive stress

compression

shear

FIGURE 6-2. Force systems exerted on the bonded-bracket-pad assembly as loaded by lingually directed compression (left), shear-peeling (center), and a pure couple (right).

Although brackets transmit the bulk of orthodontic force to tooth crowns, other means are also in common use. Buccal tubes are often

attached, in place of brackets, to terminal molar crowns. Buttons, eyelets, and staples are often affixed either to the band or bonded directly to the crown surface. They may be located either mesial to distal with respect

to the bracket on the facial surface, on the lingual crown surface, or, if space permits, on the mesial or distal crown surfaces. Most commonly, a ligature tie or an elastic is activated against the button or eyelet to create a moment about the long axis of the tooth as shown in Figure 6-3. Other orthodontic appliance elements may transfer force to the lingual crown surface. A transpalatal bar may be used to deliver buccolingual force or tipping (torquing) couples bilaterally, for example. Labiolingual treatment therapy divides the transmission of force from the appliance to the dentition between the facial and lingual crown surfaces in order that mesiodistal displacements might be accomplished without the tendency for long-axis rotation. Removable-appliance therapy may make no use of brackets at all. Lateral force generation must then be accomplished by Arch wire

/

I

P

Button

= P(e)

FIGURE 6-3. An elastic extended between a button and the arch wire to generate a moment about the long axis of the tooth (occiusal views).

200


(

0

cc

•

I )

C1=Q(e)

FIGURE 6-4. Occlusal views of a distal driving force exerted on the canine bracket (left) and the equivalent force system referred to point CC (right).

pushing, and both facial and lingual surfaces are contacted by elements

of the appliance. Accessibility contributes heavily to the rationale for placing the orthodontic bracket on the facial surface. Often resulting, however, is the eccentric transfer of at least a portion of the appliance force system, as noted in Chapter 5. Because the center of resistance is unreachable directly, the next most favorable reference point is on the long axis of the tooth and occiusoapically at bracket level (point CC in the local reference frame introduced in Chap. 5). Although through the bracket, labiolingual force and any couple may be effectively applied at point CC, mesiodistal and occiusogingival forces may not. These two components of force, commonly existing in the activated appliance, because their lines of action are in excess of one-half of the faciolingual crown width from point CC, embody sizable rotational potentials often undesired. Figure 6-4 (similar to Figure 2-17) presents occlusal views of a distal driving force delivered to the canine bracket and the equivalent force system at point CC. The couple (right) embodies the rotational tendency of the force Q applied to the bracket; if the rotation is unwanted, an

equal but opposite (in sense) first-order couple must be generated within the arch-wire-to-bracket force system or by means of some other appliance-to-crown contact (through an eyelet- or button-to-arch-wire ligation, for example). On the other hand, distolingual rotation of the

canine can be accomplished by using the distal driving force and a mesially directed force applied through point CC, the latter created through crown contact with the adjacent premolar; this force system is presented in Figure 6-5. An occlusogingival force at the bracket tends to 0

d

m

FIGURE 6-5. Occlusal view of a pair of forces acting on the canine crown as a couple capable of producing distolingual rotation.

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extrude and tip the crown lingually or intrude and produce facial crown

Upping, depending on the sense of the force. Countering an unwanted rotational potential is often attempted through placement of an active, third-order twist in an engaged, rectangular wire. Figure 6-6 illustrates the problem in attempting pure extrusion and the ideal elimination of the accompanying Upping potential. (Alternatively, if lingual crown Upping without extrusion is desired, it may be obtained through the sole application of a third-order torque having the appropriate sense or with a lingually directed force applied through point CC. The slight difference in the two displacements, in terms of the locations of the centers of rotation, was discussed in Chapter 5.) The force system carried to the orthodontic bracket from the appliance activation site(s) is transmitted primarily by the arch wire. The bracket is designed principally to accommodate the arch wire and the ligation completes the engaged confinement of that wire. Figure 6-7 (left) presents a view of the ordinary edgewise bracket from a mesial or distal

perspective. The faciolingual and occlusogingival dimensions of the bracket slot admit a range of cross-sectional sizes of round and rectangular wire. The "wings" occlusal and gingival of the slot provide support

F0

F0

C3 = Fje)

Initial position: dashed

FIGURE 6-6. Extrusive displacements: with tipping, achieved with a force located oft of the long axis (left) and with a force and couple having their resultant along the long axis of the incisor (right).

202


/Pad d

Base

FIGURE 6-7. The typical edgewise bracket: mesiodistal view (left) and facial view (right).

for the ligature tie or elastic "0-ring" which, when in place, closes the

slot. (Pins, clasps, cams, and "locks" provide resistance to arch-wire disengagement within other types of bracket assemblies.) The faciolingual dimension from the lingual surface of the slot to the lingual extent of the bracket itself is as small as strength and stiffness demands will allow to minimize the distance from arch wire to the long axis (although this "concept" is exchanged in straight-wire-therapy appliances for offsets built into the brackets). Figure 6-7 (right) illustrates a facial view of the typical bracket. Edgewise brackets are classified by occiusogingival slot dimension, by mesiodistal width, and by the absence or presence of a vertical slot ("single" versus "twin"). Irrespective of slot size, the four common bracket designations are: narrow-single, wide-single, narrowtwin, and wide-twin (in order of increasing mesiodistal bracket width). Bracket materials must be strong, stiff, and, true of all intraoral appliance elements, chemically enert in the oral environment. The so-called "universal" bracket design enables the transfer from the arch wire of all six possible components of the force system carried by the wire to it. The occlusogingival and faciolingual components of concentrated force are transferred by direct contact of wire and slot or

ligation as viewed from the mesial or distal. Note that the facially directed force from the wire pushes against the ligation, which may be less

stiff and have less strength than the bracket slot itself. Ordinarily, the arch wire may be able to slide mesiodistally through the bracket slot, depending on the wire-to-slot clearance and the type and tightness of ligation. The transfer of the mesiodistal force component generally does not rely upon friction; instead, it depends more positively on a stop affixed to the arch wire or a ioop placed in the wire. The stop or loop may make mesiodistal contact with a bracket, creating an action-reaction pair of "push" forces, or force may be transferred between the loop and the bracket by a simple auxiliary element capable of carrying tension (a fled length of ligature wire, for example). Typical transfers of the three force components between arch wire and bracket are illustrated in Figure 6-8. The capability of the arch wire to effectively transfer first- and sec-

ond-order couples through the bracket and into the tooth crown is related directly to the mesiodistal bracket width. Each couple consists of a pair of forces generated through wire-to-bracket contact at the mesial

203


FIGURE 6-8. Transfers of concentrated force components from arch wire to bracket: occiusogingival (left), faciolingual and lingually directed (center), and mesiodistal (right).

and distal extents of the bracket. As illustrated in Figure 6-9, and re-

called from discussions in Chapter 2, the couple magnitude depends on force size and the shortest distance between the lines of action of the forces. Accordingly, the potential couple size increases with the mesiodistal bracket width. The clearance between arch wire and slot is also a factor, which is discussed further in Chapter 7. In the generation of a first-order couple, one of the forces arises from contact between wire and ligation; hence, if the ligation has low stiffness, the capacity to transfer this couple through the connection of arch wire to bracket is substantially impaired. Accordingly, several alternative means of applying a first-order couple or moment to the crown have been developed and three are examined in Chapter 10. A third-order couple may be transferred by a straight, round wire to the bracket only by friction, but two more positive mechanisms are available to apply the third-order couple to the tooth crown. A rectangular arch wire, having a diagonal cross-sectional dimension greater than the

occiusogingival width of the bracket slot, may transfer this couple through two-edge contact within the slot. (The bracket slot must be stiff enough not to "open" under this loading to the extent that the wire may spin within it.) The distance between the two forces forming the couple, then, is smaller than the diagonal dimension of the wire cross-section. Arch wire

Bracket pad

Wing

Ligation

FIGURE 6-9. Bending couples transferred from arch wire to bracket-ligation system: first-order couple (left), and second-order couple (right).

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An alternative procedure is to loop the round arch wire passing through the bracket such that, as activated, two-point contact is made against (1) the ligation closing the bracket slot and (2) the crown occiusal or gingival

to the slot (depending on the desired sense of the couple). The perpendicular distance between the forces is substantially greater than that of the rectangular-wire procedure, but one of the forces pushes against the ligation and, in the absence of an accompanying lingual driving or holding force, demands at least a moderately stiff and strong closure of the bracket slot. Figure 6-10 depicts the transfer of a third-order couple from arch wire to crown using the two mechanisms just described. More detailed discussions of arch-wire torsion are contained in Chapter 8. Orthodontic force may be transmitted to the bracket other than from the arch wire. Typically, coiled springs or "elastics" are used to produce mesiodistal driving forces within one arch against individual teeth. Because of their small cross-sectional sizes and relatively high flexibilities, "elastics" are most commonly used to transfer interarch force. Various auxiliaries may be engaged and activated against individual brackets; generally, such devices are anchored in some fashion adjacent to the tooth that is to receive the active force. These auxiliaries are discussed later in this chapter and in Chapter 10.

The partial connection of the appliance to the dentition, and the transfer of some orthodontic force, may be other than through the orthodontic bracket. Although similar to the bracket and, like it, transferring force directly to a crown through a bonded base or band, the buccal tube

Q

II I

P

I

Torquing loop

I

Facial surface

I

I I

I

I

I

FIGURE 6-10. Transfers of third-order couples from arch wire to crown: by a rectangular wire to the bracket (left) and by a ligated looped round wire (right).

205


special characteristics. Generally affixed to the terminal molars in one or both arches, the buccal tube is open only at its mesial and distal ends and, therefore, transmits buccally directed force from a wire just as well as lingually directed or occlusogingival force. Its mesiodistal dimension is greater than that of the widest brackets; hence, the tube may more effectively transfer first- and second-order couples. Buccal tubes are also made to accommodate the "inner arms" of the face bow, a principal element of a class of extraoral appliances (discussed in Chap. 9). Connections of appliance elements may avoid the bracket or buccal tube entirely and transfer force to the crown directly. Examples already mentioned include the button, eyelet, and staple against which elastics or ligature-wire ties may be activated, and the attachments for lingually affixed appliance elements. Forces may also be delivered first to the arch possesses

wire, and then carried by it to the brackets, buccal tubes, and to the tooth crowns. Because such forces nearly always have mesiodistal com-

ponents, delivery demands rigid attachment of spurs or hooks to, or loops bent into, the arch wire. Typical carriers of such force include the interarch elastic and the J-hook, the latter an extraoral-appliance element.

A

Continuous-Arch-Wire Appliance Model continuous arch wire is a curved structural member, typically extending from terminal molar to terminal molar, and as engaged and activated it generally contacts through brackets and buccal tubes six or more—often at least ten—tooth crowns. At each primary connection site, where arch wire and bracket or tube make physical contact, as The

many as six independent components of a transmitted force system may exist. Although an essentially static analysis may be undertaken, there M no externally applied loading and all components of force are generated through deformations of appliance members occurring in the engagement-activation process. In short, the analysis of a highly indeterminate, geometrically complex structure is required if the force system and the structural characteristics of the wire are to be examined in detail. The task is formidable for the highly skilled structural analyst having sophisticated computer assistance, and is not a problem to be undertaken by the orthodontist. The clinician should be capable, however, of pursuing an approximate analysis that includes the principal features of the activated arch wire or appliance. To this end a model of the continuous arch wire is now proposed and developed. The ideal arch is geometrically symmetric with respect to the midsagittal plane. Often, this symmetry exists in the arch exhibiting a malocclusion, and thereby demands bilateral treatment mechanics. The concur-

rent presence of symmetry in both the dentition and the appliance as activated allows an analysis involving just one-half (the left or right half)

of the orthodontic mechanism. Such an analysis requires a figurative

206


s(uperior)

C

F,

Ft

Arch wire— i(riferior)

L FIGURE 6-1 1. Components of the force system in the arch wire transferred across the midsagittal plane in the presence of symmetry: occkisal view (left) and coronal view (right). Either sense of each component can exist.

"cut" through all portions of the appliance that span the midsagittal plane. In many instances only the arch wire must be "cut," and the symmetry demands that, of the six force-system components that could, in general, be transmitted across the plane, only the three shown in the sketches of Figure 6-11 may possibly be nonzero. Any analysis model should take advantage of midsagittal-plane symmetry when the geometry and force system permit it. Various phases of orthodontic therapy often treat groups of teeth as if each group is an individual entity, either in the application of force to produce wanted displacement (in the retraction of an anterior segment, for instance) or in arranging anchorage (in intraoral, canine-retraction mechanics, for example). In portions of orthodontic treatment, the adjacent central and lateral incisors may be considered as a single unit, and so also may the second premolar, the first molar, and possibly the first premolar (if not extracted) and the second molar (if fully erupted and aligned). Such segmentation can substantially simplify analyses, particularly in the presence of midsagittal-plane symmetry. Two distinct curvatures generally exist in the dental arch—that of the anterior portion as observed from an occlusal perspective and the anteroposterior curvature as seen in a buccal view, the latter often termed the "curve of Spee." In the analyses of many orthodontic procedures, the effects of the curvature of the anterior segment may be suppressed without significantly violating the overall structural integrity of the appliance; such suppression serygs to simplify the geometric aspects of the problem. With the exception of analyses of anteroposterior leveling mechanics, the effects of the curve of Spee often may also be neglected. Furthermore, localized first- or second-order bends, although their ef-

207


S

p

-1--I

a

/

Actual arch form: dashed

FIGURE 6-12. 0cc/usa! and bucca/ views of the continuous arch-wire model.

if active or stabilizing may be prominent and, therefore, must generally be included, need not be shown explicitly in sketches of the arch wire to be analyzed; the same may be said of third-order twists and of the details of arch-wire loops. Often, then, the overall shape of the contirtuous arch wire may be modeled as shown in Figure 6-12, consisting of three solidly connected, straight-section segments. The central and lateral incisors would always engage the middle (anterior) segment. First premolars (if present) and dental units distal to them would contact the posterior segments of the arch-wire model. The canines would generally fects

be located at the anterior extremities of the posterior segments, but, more appropriately, in nonextraction therapy, they may be part of the anterior or middle segment of the model. Although each arch-wire or appliance analysis is, in reality, threedimensional in general character, the best approach is often through the use of several plane views that are commensurate with the component forces and couples already mentioned. Perhaps the most useful view, particularly with midsagittal-plane symmetry, is that from a buccal perspective. Explicit in this plane are anteroposterior (labiolingual in the anterior segment, mesiodistal in the posterior segments) and occlusogingival components of force, third-order couples in the anterior segment, and second-order couples in the posterior segments. Practically

any analysis of mechanics intending to produce anteroposterior displacements will necessitate a buccal-view evaluation. Anteroposterior movement may also be examined in an occlusal-plane view, but the particular value in this plane analysis is in evaluating mechanics designed to produce mesiodistal displacements in the anterior segment and buccolingual movements in the posterior segments. The force-system components between arch wire and crowns apand faciolingual pearing in the occiusal view include components of force and first-order Anteroposterior or lateral appliance, differences in the action of an asymmetric left side versus right side, may be viewed from the occlusal perspective. An analysis in the coronal plane, from an anterior or posterior view, is

208


used less frequently. The coronal-plane perspective is helpful, however,

in studying anterior-segment consolidation, third-order torque in the buccal segments, and the transpalatal bar, for example, or in examining occlusogingival actions, left versus right side, in asymmetric situations. Occlusogingival forces, mesiodistal forces, and second-order couples in the anterior segment, and buccolingual forces and third-order couples in the posterior segment, appear in the transverse-plane view.

An Overview of the Structural Analysis of an Orthodontic Appliance The complexity of an orthodontic-appliance analysis increases with the

number of teeth to which the appliance is attached, when those teeth are directly involved in attempting the desired displacement pattern. The movements of an individual tooth, with the appliance activated to extrude or intrude it, tip or torque it, or correct a rotation, all may make use of practically an entire arch for anchorage and often are not strongly

dependent on the overall geometry of the continuous arch wire and dentition. These analyses are not initiated here, but will be discussed subsequently. The more immediate objective is to introduce and outline the structural-analysis procedure for those mechanics that involve the whole of the appliance attached to a dental arch or make use of appliances in both arches andlor an extraoral appliance. Indeed, the problem requires a careful examination of the actions, responses, and displacement potentials throughout the dentition as a whole and of the structure affixed to it. To begin the analysis, a peripheral examination of the active configu-

ration versus the passive shape of the appliance as a whole must be made. Note specifically the numbers and positions of the individual teeth to which the appliance is engaged. At each bracket or buccal tube where contact exists, the active or passive nature of the connection, with respect to each of the six individual components of the most general force system, is examined. Often, several components are zero at a contact location and in many instances only one component is nonzero, at least until a displacement occurs that may activate one or more additional components. In all but the simplest analyses, some modeling is generally warranted in accordance with the guidelines of the previous section. If the individual activation displacements and force systems associated with adjacent teeth are similar, the dental segment so identified may be treated in the analysis as a single entity. If one or more of the arch-wire curvatures are not relevant to the analysis, they may be neglected in formulating the model. Note that after the modeling to simplify the analysis procedures has been undertaken, the model must be carefully scrutinized regarding its propriety, not only at the instant of activation, but also as to its continued validity throughout the days or weeks during which displacements of teeth and corresponding changes in the activated-appliance configuration are occurring. After the model

209


formulated and found acceptable, the next steps are to check for the presence or absence of midsagittal-plane symmetry and to determine the number of necessary plane views together with the specific views to be employed. The entire appliance assembly, in general, is a combination of the intraoral appliances contacting the individual (maxillary and mandibular) dental arches, any interarch attachments, and, if used, the extraoral appliance. Each of the existing subassemblies, considering prior modeling, is separated from the others and the dentition, and is isolated; using the necessary plane views, the force system exerted on each subassembly is sketched. Included are all components arising from appliancecrown contacts, intraoral forces associated with interarch interconneclions, and force systems created through the activation of "headgear." The directions of concentrated forces and dimensions or distances that may be needed as moment arms are also shown in these sketches. Next, is

the transmission of force through the appliance members is qualitatively traced from the location(s) of principal activation(s) to (1) the sites of wanted displacements and (2), if existing, the appliance-crown conneclions and any other contact locations intended to experience little or no

movement during the between-appointments period. If no anchorage exists, the activation sites and the force system against the appliance will be divided (halved) on either side of a plane of symmetry; in the majority of such instances the activation is bilateral with symmetry with respect to the midsagittal plane. The transmission of force is then followed from the activation locations on either side of the plane to the plane of

symmetry itself, where action-reaction matching of the force-system components is carried out. The total number of force and couple components, exerted on the complete appliance assembly, may now be deter-

mined. From discussions in Chapter 2, in each plane view a total of three, scalar, force- and moment-balance relationships are available from Newton's laws and the concept of the quasi-static state of the appliance. The number of force and couple components to be quantified will exceed the number of quasi-static, balance equations by the number of needed "load" versus deformation relationships associated with the activation of the appliance. The complete analysis will likely require a disassembly of the appliances "on paper," separating the auxiliaries from the arch wires, and an examination of each appliance member in a manner similar to that previously described for the principal appliance subassemblies. This detailing will "expose" the force systems that exist at the sites of interconnections between appliance members, and these systems are interrelated in pairs through Newton's law of action and reaction. Each member of the appliance assembly may be placed in one of two categories: (1) those members that are physically deformed from their passive states to achieve the activation of the assembly as a whole; and (2) those members in which force systems arise in response to the activation. For the members in the first category, if the entire force system external to and within the appli-

ance assembly is to be quantified, the amounts of activating displace-

210


ment-deformation must be measurable and the load-deformation rela-

tionships must be known. The structural analysis of an orthodontic appliante is accomplished in two principal, sequential parts. First, all of the forces and couples acting on the appliance as a whole and on each appliance member at the interconnections are determined. This procedure, described in a general manner previously, requires the use of force- and moment-balance equations and load-deformation relationships. If the purpose is only to obtain the force systems delivered to anchorage sites, to decide whether or not anchorage reinforcement is advisable, for example, then the analysis may be terminated following the completion of the first portion. The second part of the analysis examines each appliance member internally.

For each, the critical sections and points thereon are located and, as potential sites for initiation of structural failure, are checked for stress levels and amounts of deformation. These stresses and deformations are determined from geometry and material characteristics of the individual

member and the force system exerted on it, and may then be related back to the parameters of the activation itself. This second portion of the analysis may be undertaken for one of several reasons:

1. A failure may have occurred and the need exists to make some modification of the appliance or in the activation of it. 2. Under consideration is the possibility of replacing one or more mem-

bers of the appliance or altering the character of the activation to modify the response, and the structural integrity of the appliance under the new conditions is to be examined. 3. An entirely new design is contemplated, including the shapes and sizes of the appliance members and their materials, with selections to be based in large part on allowable stresses and deformations as related to those that will be generated in the new appliance.

The complete structural analysis of the orthodontic appliance incorporating a continuous arch wire can be a complex undertaking as previously noted. The foregoing, general approach to the process as carefully outlined partially attests to the magnitude of the task. With appropriate modeling, however, and with the knowledge to be gained in the remainder of this chapter and those to follow, the clinician will be able to incorporate the principal features of an appliance in a reasonably straightforward analysis procedure. To this end, the foregoing discussion might be condensed into the organized, four-step process suggested here:

1. Knowing the specific displacement(s) desired, both in form and extent, the individual components of the total force system acting on the dentition, which should produce those displacements, are identi-

fied together with any components present to control the movements.

2. One or more free-body diagrams of the orthodontic appliance as a

whole, and subassemblies and individual elements thereof as

211


are sketched. These diagrams include the action-reaction counterparts of the components identified in step 1. The entire force system exhibited in each diagram is investigated through Newton's laws (action-reaction and the force and moment balances). needed,

3. With the aid of the results of step 2 and, once more, using the actionreaction law, the force system(s) transmitted to the responsive locations (anchorage) are evaluated to determine what, if any, unwanted displacements might occur. Stress and deformation computations for individual appliance members may also be undertaken if a possible, structural failure is of concern.

4. As warranted from the results of step 3, appropriate precautions or alterations should be considered and undertaken to minimize any potential, undesired "side effects" of the appliance objective. Such action could involve, for example, reductions in induced force magni-

tudes, making member substitutions in or additions to the existing appliance, or altering parameters intended to control the orthodontic displacements.

The Activation and Deactivation Characteristics of Representative Appliance Elements The completion of the first portion of the structural analysis of an ortho-

dontic appliance initially requires the quantification of the forces and couples induced in those auxiliary elements or at the individual, specific locations along the arch wire where mechanical work is done by the clinician on the appliance to achieve activation. Examples of this work are the compressing of a spring, the stretching of an elastic, and the deflecting of an arch wire to gain bracket engagement. The work of activation is usually concentrated at only a few locations, properly termed activation sites, and at each site knowledge of the amount of localized deformation and the associated element or wire stiffness enable the determination of the induced force system. A subsequent force-system

analysis, assuming that all of the dentition to which the appliance is affixed behaves like a rigid foundation or support-base, is sufficient to approximate the initial magnitudes of the responsive forces throughout the dentition. Although somewhat idealized, such an analysis of the appliance is appropriate because a state of maximum, overall internal loading and deformation exists under the assumption of a rigid dentition and periodontium. The soft-tissue deformations, which in reality permit some immediate tooth movement following activation, serve to partially "relieve" the appliance, releasing a portion of the strain energy induced by the mechanical work of activation and, thereby, reducing the stresses and deformations from their maximum values. As time proceeds following acti-

212


vation and stage 1 of tooth movement (defined in Chap. 5), additional

deactivation will likely occur during the between-appointments period with stage 3 displacements. Although some of these movements may serve to generate force systems at locations that were passive when the appliance was initially engaged (the guiding arch wire attempting to keep a canine upright during retraction, for example), this work is accompanied by a transfer of strain energy from one location to another within the appliance. Overall, stage 3 displacements result in a net, continuing decrease in strain-energy level, stresses, and deformations in and of the appliance with respect to the as-initially-activated state. The orthodontist may disengage the appliance before complete deactivation occurs as a result of tooth movement or possible relaxation of some appliance elements. If the unloaded configuration of the appliance is not identical to the original passive configuration, signaled are energy losses, energy transfers other than to the dentition in producing displacements or to the clinician in the unloading process, or increases in unavailable, stored energy in the appliance resulting from exceeding the elastic limit somewhere during activation. Inelastic behavior may be reflected in the metallic appliance members and relaxation in the nonmetallic members. Although the configuration change following a load-unload cycle is more often the rule than the exception, of at least equal and probably of more importance is the fact that the induced increment of strain energy of activation generally cannot be completely controlled with respect to the subsequent work done by the appliance in moving teeth. With the influence of the clinician removed as soon as activation is complete, the appliance will seek to take on a minimum-energy configuration. The added strain energy, transferred to the appliance at the activa-

tion site(s), is immediately distributed throughout the appliance such that the assembly as a whole assumes the deformed configuration characteristic of the lowest energy level possible, consistent with the constraints to which it is subjected. Moreover, with every small tooth dis-

placement that takes place, the appliance adjusts itself to a new, minimum-energy configuration. The ideal appliance, then, is one that most nearly permits the orthodontist complete control over the manner in which the strain-energy input of activation will be distributed and dissipated over time. The first individual orthodontic element to be discussed is not part of an appliance at all, and is used before brackets are affixed and only when and where interproximal space is to be created to accommodate the placement of bands. These "spacers" are small polymeric rings or discs, placed interproximally adjacent to gingival margins, and are "loaded" or activated, in their placements, in direct compression. Figure 6-13 shows a free-body diagram of the activated spacer and the actionreaction, counterpart forces exerted upon the adjacent teeth. Compliance with Newton's laws requires that all four forces shown be equal in magnitude. This magnitude at the time of activation is determined by the amount of deformation of the spacer required to place it and the

213


Passive

fl

F4

F4

spacer (representation)

F3

F3

(F3=F4) FIGURE 6-13. The spacer unloaded and activated in direct compression (center) and the adjacent teeth subjected to spacer action (facial views).

load-deformation relationship for the element. Typically, the average

stiffness of a spacer in compression is moderate with the cross-section in the mesiodistal plane (as activated) sizable compared to its dimension in the direction of loading. The width of the ordinary orthodontic band is, however, only a fraction of the periodontal-ligament width at the gingival margin and, as a result, the ligament need not be fully compressed by this loading to create the needed space. Because the forces exerted by

the spacer on the two teeth will be of equal magnitude throughout the active period, the absolute displacements of the individual dental units will be controlled by the resistance of each tooth to movement, either inherent (projected root surface area, crown contacts along the dental arch) or created artificially through an anchoring of one tooth or the other in an attempt to limit the movement to just one unit. Helical-coiled springs may be used as auxiliary elements of both intraoral and extraoral orthodontic appliances. Typically activated by a force applied along the axis of the spring at one end and supported at the other, the member may be loaded in either direct tension or direct compression. Regardless of the sense of the applied force, neglecting its weight compared to contact forces generally makes the spring, like the spacer, a two-force member. Helical springs, to be activated in tension, are usually "wound" (fabricated) to assume a closed-coil, passive configuration while compression springs must exhibit an open-coil (space-between-coils), passive state. Tension springs may be activated (stretched) almost indefinitely and are inherently stable when loaded; however, an elastic limit does exist as the activation actually produces a rather uni-

form bending deformation of the coiled wire. On the other hand, the compression spring can be deformed only until all coils contact one another; this spring is ordinarily designed such that the closed-coil state is achieved with a force producing maximum stresses less than the elasticlimit stress of the spring material. Of some concern must be the stability

of an activated compression spring in the absence of lateral support potential and, to guard against the possibility of lateral buckling (uncon-

214


a,

0

0

Deformation

FC

FIGURE 6-14. Helical tension and compression springs (left and center); typical loaddeformation diagram for the linear spring (right).

trolled, large deformation), this spring as an appliance element is usu-

ally wound around the arch wire. The majority of metallic, helical springs are termed "linear" in that they exhibit straight-line load-deformation behavior beneath the elastic limit. The structural properties of a helical spring are primarily dependent on the spring material, the wire cross-section, the coil diameter, and the passive length. Helical springs in tension and compression are shown schematically in Figure 6-14. A typical load-versus-deformation plot for a linear spring is also presented (similar to Fig. 4-9). Given the plot for a specific spring, a measurement of its axial deformation from the passive state provides the magnitude of force at its ends. Figure 6-15 shows a free-body diagram of a compression spring wound around an 0

a,

P

FIGURE 6-15. A helical compression spring making lateral contact with the arch wire about which it is wound, midway between the spring ends.

215


arch wire, as activated and making lateral contact with the wire midway

between the ends of the spring. A force balance indicates that the end forces have vertical components that must sum to the force of the arch wire against the spring coils. The lateral forces arise if the spring tends to

buckle laterally or if the section of arch wire, over which the spring is wound, is curved. (An available spring assembly of interest is activated by a pulling force against its "jacket" and an axial wire transferring force from an end coil; the axial wire passes through the spring body. This element exhibits favorable characteristics of both standard tension and compression springs; it is loaded in tension, but activation actually compresses the spring. Accordingly, the assembly incorporates the closedcoil "guard" against exceeding the elastic limit and the axial wire and jacket prevent buckling.) To activate a pulling force in a specific direction within the orthodontic appliance, or to transmit a tensile force from one site to another, an elastic may be used in place of the tension spring. The popularity of the elastic, generally descriptive of the array of bands, 0-rings, modules, chains, and threads of rubber or rubberlike polymeric materials, is associated with its favorable characteristics of high flexibility, small crosssectional dimensions, and low cost. Free-body diagrams of the elastic band (or ring or the looped and tied thread) and the module, loaded in direct tension, are shown in Figure 6-16. A typical load-deformation plot, similar to a portion of Figure 3-11, is also presented. An undesirable feature of the elastic, associated with many polymeric materials, is the tendency toward relaxation. Defined in Chapter 4, the result of reP

Q

a)

0

Elastic module

Elastic band

0 P

Elongation

FIGURE 6-16. An elastic band or looped thread (left) and an elastic module or chain (center) subjected to two-point tension, and an example load-deformation diagram (right).

216


laxation is the loss of force magnitude with lime under condilions of constant stretch (strain). Figure 6-17 is a four-part plot for an elaslic showing (1) inilial activalion, (2) maintenance of constant strain for a finite lime period, (3) rather rapid unloading to a stress-free state, and (4) recovery in the absence of

reacivalion. Notable in parlicular is the fourth segment of the plot which indicates that, for some polymers and given sufficient lime, the original passive configuration may be nearly, if not totally, regained if the elaslic limit of the material was not exceeded during activation. (As noted in Chapter 4, these elements are termed elastics, partly because of their sizable elaslic ranges. Strictly speaking, though, the materials are

quite brittle with measurable elongations to failure by rupture not sizably beyond the elastic-limit, extensional deformalions.) Although several comparable characteristics already menlioned suggest preference for the elaslic over the tension spring, a noteworthy advantage of the metallic spring is the virtual absence of lime-dependence in its structural properties. Convenient to transmit tensile force, when moderately high, extensional sliffness is desired, is the looped and lied (twisted, "pig—tailed") length of ligature wire. The ductility of this wire is substantial and the elaslic strength of the "dead soft" material is quite low; this combinalion permits the procedure. Figure 6-18 qualitalively compares the tensile stress-strain behavior of the annealed (as-received), stainlesssteel ligature wire with the same wire material having average resilience.

a)

C

0

U-

I loading

0

relaxation

Elongation

FIGURE 6-17. A four-part hysteresis diagram for an 'elastic" appliance element

217


C))

C')

C)

C') C)

0)

C

Strain-hardened wire

Annealed wire

0

Tensile Strain

FIGURE 6-18. Tensile stress-strain diagrams tor annealed and resilient stainless-steel wires having identical cross-sections.

Note that the tying process inelastically deforms the wire, adding to the

strain energy stored locally in the wire; each twist reduces the ductility and the effective deformation to fracture. If the number of twists are excessive and this cold-working process itself does not cause fracture, a small additional load superimposed upon the activated ligation, in overall appliance activation or later from mastication, for example, might cause the ligature to rupture. The longitudinal flexibility of the ligature tie is practically nonexistent, despite its small cross-section; the elastic modulus of its material, generally stainless steel, is high and the relatively low stiffness site is that of the twist-tie. In summary, ligature wire should be expected to carry responsive force; it will not perform as an activating element like the elastic module or the coiled spring. In its intended use (to close the bracket slot) and to generate secondary activation (holding force), this wire serves well as long as the localized embrittlement at the twist-tie is not so extensive that fracture occurs with the superposition of masticatory force upon appliance activation. The rubber-band elastic may be stretched between two points, creating the action-response force pair shown in Figure 6-16, or it may be activated in three- or four-point tension. In the "up-and-down elastic" application (between the maxillary and mandibular arches) the element may be looped around three or perhaps four brackets or spurs (affixed to arch wire). Unless constrained locally at several points, the activated

218


elastic will assume a configuration with the longitudinal strain the same

throughout its stretched length. If the load-deformation behavior of an elastic band has been or can be obtained in two-point tension, an analysis of the three- or four-point, equalized-tension problem can be accomplished. First, the magnitude of the tensile force existing throughout the band is obtained from its effective elongation, which is one-half the difference between the passive and activated circumferential lengths, and the load-deformation plot. Second, with the force magnitude the same on either side of the bracket or spur (because the strains are the same), as is shown in Figure 6-19, the direction of the delivered force (to the bracket) bisects the angle formed by the elastic-band segments. The magnitude of the delivered force is obtained from a force-balance on

Q

/ I

— — — - I. — — -

elastic-band segment

P

0 = 2(P cos

0)

P

FIGURE 6-19. A free-body diagram of the portion of an elastic band contacting a bracket; the analysis yields the direction of the force delivered to the bracket.

219


the segment free-body diagram shown, with the equation given in the

figure.

Although elastics are always activated in direct tension and helical springs are generally loaded uniaxially, the arch-wire segment spanning several brackets may be subjected to one loading pattern or several patterns concurrently. The most common arch-wire activation is in bending, and when activated in this manner, the wire behaves structurally as

a beam. Detailed discussions of arch-wire bending are contained in Chapter 7, but some general comments are appropriate here. First, a distinction is drawn between occlusogingival and faciolingual bending, simply in terms of the direction of activation and the plane containing the external force system. Second, the anchorage may be totally mesial or distal of the activation site, or it may be divided between locations mesial and distal to the activated section of the wire. For example, a symmetric arch might use the posterior segments for anchorage in mechanics designed to intrude or extrude the incisors; the model of such an arch is shown in buccal view on the left in Figure 6-20 and resembles the cantilevered beam of Figure 2-29. A lingually malposed second premolar might be tipped into alignment by a facially directed force, activated by tying the adjacent arch-wire section into the premolar bracket. In this

example, the anchorage is split between the dental units mesial and distal to the premolar, and an occlusal-view, free-body diagram of the arch-wire segment, showing the concentrated faciolingual forces, appears on the right in Figure 6-20. Third, innumerable load-deformation plots exist for a single arch wire in bending. Although each plot is approximately a straight line for metallic beams, if the deformations are not large and the elastic limit is not exceeded, the slope (stiffness) is dependent not only on the material and the size and shape of the cross-section, but also upon the characteristics of the loading, including its position with respect to the anchorage and, for rectangular cross-sections, the direction of the loading with respect to the cross-sectional geometry. When activating an arch wire to produce a crown force, the displacement must be measurable so that the wire "works" during the betweenappointments period with the force induced having a physiologically proper magnitude. Although such activations may readily be obtained through displacements perpendicular (normal, transverse) to the long I

I

I

I

cp pp

/

buccal view

/

\

occlusal view

FIGURE 6-20. Buccal and occiusal views of arch wires activated in bending by transverse forces: the wire as a cantilevered beam (left) and with divided anchorage (right).

220


dimension of the wire, no matter how small the cross-sectional area the

longitudinal (mesiodistal) wire stiffness is much too great. Two procedures are used to effectively reduce the mesiodistal stiffness of the arch wire: (1) More flexible elements are incorporated, nearly always bilaterally, in series with the arch wire, generally stretched between the ends of the wire and posterior anchorage. An example of this arrangement is shown in the buccal view on the left in Figure 6-21. Often this is done in anterior-retraction mechanics incorporating stops, springs, and, perhaps, ligature wire or elastic thread. (2) A loop is bent into the arch wire such as shown in the facial view on the right in Figure 6-21. The more wire incorporated in the loop, the lower the resulting mesiodistal stiffness. Because they are formed through inelastic bending procedures, and most often activated in bending, loops are examined in some detail in Chapter 7. Among other uses, loops are used in space-closing procedures and to "unravel" crowded incisors. Couples, as well as concentrated forces, may be transmitted between arch wire and bracket or buccal tube. Two of the three couples or couple components load the arch wire in bending; the third is associated with activation in torsion. To produce an active, second-order bending couple, in the passive configuration an angular malalignment must exist, as viewed from a facial perspective, between bracket slot or buccal tube and the section of arch wire to be engaged. Accordingly, clinicians place second-order bends in the wire or use preangulated brackets and tubes. The mesiodistal bracket width can greatly influence the magnitude of the second-order couple. Besides defining the distance between two contact forces (generated between wire and diagonally opposed edges of the bracket slot or buccal tube) forming the couple, for a given arch wire increased width (1) reduces any second-order, angular clearance between wire and bracket slot or buccal tube, (2) decreases the adjacent interbracket distances, and (3) tends to increase the as-activated, con-

tact-force magnitudes. In short, the wider the bracket or tube, the greater the potential, second-order bending stiffness. Many characteristics of the first-order couple are similar to those of its second-order counterpart. Located in an occlusal plane, the first-

Coiled

Hooks

Buccal tube Arch wire

FIGURE 6-21. Methods of increasing the longitudinal flexibility of the arch wire: a spring stretched between hooks affixed to the wire and a terminal-molar buccal tube (left) and a loop bent into the arch wire (right).

221


order couple arises from a difference in passive angulation between the

arch-wire section to be engaged and the axis of buccal tube or the lingual surface of the bracket slot. Again, the action is bending, and stiffness is substantially affected by an alteration in the mesiodistal bracket or tube

width. Notable with the first-order couple generated specifically between bracket and arch wire, however, is the dependence of stiffness on

ligation, inasmuch as one-half of the force pair exists due to contact between wire and the ligation. Moreover, the ligation may affect the bending stiffness from two aspects: (1) through its own stiffness, and (2) in determining the effective mesiodistal width of the bracket. Because

the ligation can weaken the connection of arch-wire to bracket and lessen efficiency in transferring first-order couples, other means are available to produce long-axis rotations (eyelet-to-arch-wire ties, wedges, and levers, for example); procedures are discussed in Chapter 6-22 shows a typical second-order couple transfer on the left and first-order couple transfers suggesting conventional ligature ties on the right. As with the transfer of a transverse force, anchorage for an active first- or second-order couple may be totally mesial or completely distal to the activation site, or the anchorage may be split (divided) with it partly mesial and partly distal to the activation location. Appropriately noted at this juncture is a comparison of similar parameters appearing in a load-deformation plot in bending for an arch wire, depending on the form of activation. If the wire is activated by a transverse force, the magnitudes of the force and the transverse bending deflection at the point of activation along the wire are plotted. At least the initial portion of the plot is linear and the slope—the stiffness—has dimensions of force divided by length. If the bending activation is by a first- or second-order couple, the deformation is angular; the stiffness in this mode has dimensions of the product of force and length per unit of rotation. Example plots for small-deformation, elastic behavior, together with sample units for stiffness, are given in Figure 6-23. An area under any load-deformation plot represents energy or energy transferred, always having dimensions of force times length [FL]. Couples created in planes containing the longitudinal "axis" of the arch wire induce bending; the couple applied to the wire such that its 10. Figure

plane is perpendicular to the mesiodistal direction creates a twisting

Cd Cm P

C,

FIGURE 6-22. Arch wires activated in bending by couples transferred to them by brackets: an active second-order couple exerted on a wire and transmitted to divided anchorage (left) and bilateral, first-order couples delivered by a wire segment to correct central incisor rotations (right).

222


C)

0

IJ_

ci 0

C)

Stiffness

(g-mm/deg) Stiffness (g/mm)

0

Deflection (mm)

0

Rotation (deg)

FIGURE 6-23. Arch-wire load-deformation diagrams in bending; the active force system is a transverse force (left) and a couple (right).

deformation. The twisting or torsional couple can arise from a rectangu-

lar wire and an edgewise bracket, the wire and bracket slot passively angulated, one with respect to the other. The rectangular wire may experience third-order activation following permanent twist placements by the clinician or through the use of "pretorqued" brackets. Torquing spurs bent into round wire are also used to generate third-order action. As with other forms of arch-wire "loading," third-order activation may be sustained by anchorage entirely on one side or the other of the activation site, or the anchorage may be divided between locations mesial and distal to the location of the activation. Free-body diagrams of an archwire segment subjected to a third-order couple with the anchorage split, and of a cantilevered wire loaded by a third-order couple, are shown on

the left and right, respectively, in Figure 6-24. Note that the bracket exerts less influence on the third- than on the first- or second-order stiffness of the arch wire. The mesiodistal bracket width is a factor in

9 Cload

C.,

C action

Cleft

FIGURE 6-24. Free-body diagrams of an arch-wire segment loaded in torsion; the segment on the left reflects divided anchorage, but only one anchorage site exists for the segment on the right.

223


torsional wire stiffness only insofar as it affects interbracket distances.

(As noted previously, a third-order couple in a rectangular wire will be transmitted to a tooth crown through a bracket only if the occlusogingival slot width is less than the diagonal dimension of the wire cross-section.) The form of the load-deformation plot for third-order arch-wire behavior is the same as that for first- or second-order-couple loading; for twisting as for bending by couples the loads, deformations, and stiffnesses individually have identical dimensions. In short, the load-deformation plot on the right in Figure 6-23 could qualitatively be representative of the arch wire subjected to active torsion or bending. Chapter 8 is entirely devoted to the third-order behavior of arch wires and associated topics.

For all of the fundamental activations previously discussed, uniaxial tension and compression, the cantilevered or divided-anchorage archwire beam subjected to a concentrated transverse force or couple, and the wire activated by a twisting (third-order) couple, force- and momentbalance analyses made of the member at any specific time following activation results in several quasi-static relationships that are less than the number of unknown characteristics necessary to complete the description of the entire force system. With the exception of the arch wire subjected to a couple loading and having divided anchorage, one additional relationship is needed; for the exception noted, two are needed. For each of these problems, the load-deformation plot provides the equivalent of one equation; with the deformation quantified with respect to the passive state, the plot yields the magnitude of the induced load. In those problems of couple loading and partitioned anchorage, one additional relationship is obtained, dependent on the relative distances between the load and the anchorage sites. Thus, the first portion of the structural analysis is handled, completing the description in all characteristics of the external force system exerted on the arch-wire segment. Detailed discussions of such analyses are taken up in subsequent chapters. The load-deformation behavior for a helical spring in tension or compression and for the arch wire in bending or torsion is virtually independent of time, at least within the elastic limit, because of its material properties. Moreover, provided no inelastic action has occurred, the deactivation or unloading plot will retrace the activation curve. Furthermore, should the elastic limit be exceeded in activation of a metallic appliance element, the deactivation (unloading) plot will be nearly linear with the same slope as that of the linear portion of the activation curve. (This was first discussed in Chapter 3.) Accordingly, for metallic appliance members, the relationship between load and deformation during deactivation is, in essence, known from the plot of these parameters obtained from activation data. Such an extrapolation, however, cannot be made for appliance members of polymeric materials because, generally, their structural behavior is nonlinear and time-dependent. A typical activation-deactivation pattern for a polymer was described previously in this section (see Fig. 6-17).

224


The Action and Response of Tip-Back Bends: An Illustration of the Four-Step Procedure in Orthodontic Structural Analysis Until in-depth studies of bending and torsional behavior of individual

members are undertaken, structural evaluations of orthodontic appliances cannot be completed. Nevertheless, in view of the concepts and procedures introduced in this chapter and the rather general discussions herein, with particular consideration of the analysis initiated, a look at an arch-wire example is appropriate. Chosen for examination is a continuous arch wire that fully engages the mandibular dentition from first molar to first molar. As often occurs,

this example appliance is geometrically and mechanically symmetric with respect to the midsagittal plane. All permanent teeth are fully erupted mesial to the first molars and have been aligned occlusogingivally and faciolingually. Because of the symmetry and alignments, together with the nature of the activations, the analysis is undertaken through a right-side buccal view of one-half of the arch wire and the involved dental units. (The additional potential for faciolingual tooth movements is recognized, owing to the development of force systems not within the "plane" containing the long axes, but actions and responses appearing in coronal-plane views are not investigated here.) The intended activations in this example are those produced by bilateral tip-back bends in the wire at the terminal molars; the angular, passive malalignments between wire segments and buccal-tube slots give rise, upon full engagement, to second-order couples exerted by the wire on the molar crowns (through the buccal tubes with the action-reaction counterpart couples against the wire). Three variations in the overall "loading" are to be examined, associated with the bend geometry and the occlusogingival position of the buccal tubes with respect to the "line of bracket slots": (1) the couple alone exerted on the molar; (2) the couple accompanied by an extrusive force; and (3) an intrusive force as well as the couple exerted on the molar crown. Before beginning the force analysis, some modeling is appropriate, as mentioned earlier in this chapter. Proposed is the consideration of the right-side half-arch as "L"-shaped as viewed from an occlusal perspective; the corner of the "L" marks the intersection of anterior and posterior portions of the arch model. The central and lateral incisors engage the anterior section and the canine, premolars, and first molar the posterior section of the wire. Because the intended action is against the first molar, it is separated in the model from the middle segment consisting of the canine and premolars. Shown on the left in Figure 6-25 is the right-side, first molar sub-

jected to the couple created by the tip-back bend. The displacement tendency is that of distal crown tipping with a center of rotation coinciding with the center of resistance in the approximate location indicated. On the right in Figure 6-25 is a free-body diagram of the half arch wire.

225


1P Ce

DC6

FIGURE 6-25. Buccal views of a terminal molar subjected to tip-back-bend action in the absence of net occlusogingival force (left), and a free-body diagram of one-half of the arch-wire model showing fhe responsive, contact force system exerted by the dental units (right).

Newton's law of action and reaction yields the couple at the first-molar

site. Because the half-arch is quasi-static, like the dentition, the entire force system acting on it must be balanced in this view. In the absence of substantial second-order bends mesial to the tip-back and, then, assuming second-order clearance between premolar and canine brackets and the wire, together with third-order clearance in the anterior section (or round wire engaged and containing no torquing spurs), the response to the couple loading must be in the form of occlusogingival forces mesial

to the activation site. The pair of equal, parallel vertical forces shown form another couple, opposite in sense to the couple against the lip-back bend; the product of one of the forces and the distance between the pair of forces must equal the magnitude of the tip-back-bend couple in order to yield a balanced force system exerted on the half arch wire. The specific locations of the vertical responsive forces depend on the bending stiffness and the passive geometry of the arch wire relative to the bracket placements and their orientations, but most likely both will act within the middle segment—perhaps against the second premolar and canine. The net effect of the response tends to rotate the line of middle-segment brackets clockwise, as viewed in Figure 6-25. If the arch wire has high bending stiffness, the response to the couple loading may travel to the incisal segment with a tendency there toward intrusion. The sketch on the left in Figure 6-26 shows the combined effect of the lip-back bend and an occlusogingival malalignment of buccal tube and wire, the latter creating a tendency for extrusion. The action-reaction counterparts of the force and couple against the molar are shown against the posterior end of the half arch wire on the right in the figure. Again, assuming ample second-order clearance throughout the middle segment and no third-order activity in the incisal region, the response mesial to the molar is again in the form of occlusogingival force. In fact, the total force system exerted on the half arch wire by the engaged dentition may be balanced, with the given action at the terminal molar, by a single, resultant gingivally directed force as shown. The location of this balancing force is probably at the first-premolar site if the wire exhibits moder-

226


P6

C6

0 (= P6)

FIGURE 6-26. Buccal views of a terminal molar subjected to tip-back-bend action in combination with a net extrusive force (left) and a free-body diagram of one-half of the arch-wire model showing the response of the dentition (right).

ate bending

stiffness, but all three teeth in the middle segment are likely subjected to intrusive action. If the wire is very flexible, the second premolar will receive the greatest portion of the distributed, intrusive potential.

The third variation is the most common, found in mechanics designed to establish posterior anchorage; the action against the terminal molars is a combination of distal crown tipping and intrusion. Shown on the left in Figure 6-27 is the right-side molar subjected to the counterclockwise couple and the gingivally directed force, The action-reaction counterparts act against the posterior extent of the half arch wire on the right in the figure. For a third time, assume that the response mesial to the terminal molar is void of couples. Note that, although the addition of one responsive, occlusogingival force could provide, in the free-body diagram of the half arch wire, a vertical force balance, a moment balance would be impossible. Accordingly, two vertical forces of mutually opposing senses are necessary to enable both force and moment balances. Moreover, because the intermediate vertical force must have a magnitude equal to the sum of the other two forces and a sizable counterclockwise moment is necessary for the overall moment balance, the more

C6

-)

FIGURE 6-27. Buccal views of a terminal molar subjected to tip-back-bend action together with a net intrusive force (left) and a free-body diagram of one-half of the arch-wire model showing the response of the dentition (right).

227


mesial of the two parts of the vertical, responsive, force system is likely

located in the incisal region as shown. Hence, the intermediate, vertical force is exerted on the middle segment. For each of the three active force systems against the terminal molars, three of the four steps in the previously outlined analysis procedure have been discussed. Figures 6-25 through 6-27 show the desired (or obtained) actions at the activation site, the terminal molar (step 1), and the action-reaction transition to the appliance (the arch wire—with symmetry, the half arch) and the subsequent balancing of the force system on the wire (step 2). The effects of the response elsewhere in the dental arch, gained from a second action-reaction transition, were then noted;

the three variations in response in the order considered are pictured from left to right in Figure 6-28 (step 3). The displacements that occur in

these mechanics are to be restricted to the terminal molar. Undesired side effects, then, are movements within the middle and incisal segments produced by the responsive force systems. The ideal may not be realized, however, inasmuch as the molar demands a sizable active force system commensurate with its root-surface area, and this area is larger

than that of any of the teeth mesial to the molar. Beyond keeping the active force system against the molar as small as reasonably possible, considering the smaller root-surface areas of teeth receiving responsive force, little further discussion of step 4 is reasona-

ble here. The distribution of responsive force is associated with the structural behavior of the appliance, and in this instance knowledge of arch-wire bending (flexure) is required—the topic of Chapter 7. It is notably necessary, in particular when the tip-back couple is accompanied by the intrusive force (see Fig. 6-27), to unitize the middle segment rather than permit an individual tooth to take a majority of the vertical

response. Of concern, in addition, must be the intrusive responsive force against the incisal segment seen in the third variation (Fig. 6-27 and Fig. 6-28, right). The potential for intrusive displacement is not only strong because of the relatively small root-surface area of the half incisal segment, but the line of action misses the center of resistance by some

distance if these teeth have typical angulations and, as a result, the tendency for flaring is also present. One suggestion, assuming the intrusion and flaring are unwanted, is to place up-and-down elastics to transfer a portion of the gingivally directed force to the maxillary arch. Alter-

natively, the patient could wear, for a portion of each day, an FmId

t

F1

I

FIGURE 6-28. The transfers of responsive force systems to the dentition resulting from the terminal-molar activations of Figures 6-25 (left), 6-26 (center), and 6-27 (right).

228


appropriate extraoral appliance engaging the incisal segment of the

mandibular arch wire. Clearly, these "remedies" draw on concepts and discussions yet ahead in this text and, as already mentioned, the distribution of the responsive force system is partially dependent on bending characteristics of the arch wire that receive in-depth attention in the

chapter to follow. In short, the example discussed here is typical in format of the whole of Chapter 10; as noted at the beginning of this section, the purpose was to provide a glimpse of a portion of the structural analysis procedure, the foundation for which has been laid in the foregoing sections of the present chapter.

"Control" of the Orthodontic Apptiance of its relation to the structural behavior of the mechanism over time, before closing this chapter several comments pertaining to the "control of the appliance," to use the clinician's terminology, are in Because

order. A difficult concept to define directly, "control" may seem to make reference to displacements of teeth, but a broader interpretation should

be understood. In anticipation of activation or reactivation within the oral cavity, the appliance is prepared with regard to the existing configu-

ration of the dentition and the realignment desired, expecting certain tooth movements to occur in the interval between appointments. Because the displacements of dental units are accompanied by geometric and mechanical alterations of the appliance from the as-activated state, and the orthodontist has no "hands on" influence during the betweenappointments period, control must be built into the appliance design and the activation of that appliance. After the patient has left the office or clinic, three factors govern: (1) the biologic response of the tissues to the induced force system; (2) the minimum-energy principle; and (3) the actions of the patient. The first factor was discussed in Chapter 5. The second factor has been mentioned, but to repeat the principle indicates that, given the locations and formats of connections of the appliance to the dentition and anchorage, the appliance will assume a configuration, within the constraints imposed by those connections and the dentition, which makes the total

strain-energy increment associated with activation quantitatively as small as possible. This strain-energy increment, directly related to the appliance deformation from its original passive state and the accompanying induced forces, is distributed throughout the mechanism. Any crown displacement causes a change in the appliance configuration (due to the connection of mechanism to dentition), a reduction in the strain

energy (due to an energy transfer in the form of work done by the appliance in moving the tooth or teeth), and a redistribution of what remains of the increment to define a new, minimum-energy state. The third factor is well understood as to its potential adverse impact. If any

229


portion of the appliance is removable (elastics, an extraoral appliance, or

a retainer, for example), the patient may infringe upon the control estab-

lished by the clinician. Patient cooperation and control go together; a patient who is 100% cooperative leaves the maximum amount of appliance control to the orthodontist. The ideal orthodontic appliance would permit the operator to predict, in extent to within the limits imposed by the incompletely known biologic response factor, the displacements of all points of the appliance where force is transmitted to the dentition and, in effect, to completely but remotely be in control of the deactivation process at all times. Of obvious importance is control of displacements at the anchorage sites. In this regard the extraoral appliance presents a paradox: the rationale for development and use of headgear is the location of the anchorage away from the dentition, and yet, with this appliance, the patient may interfere with operator control to a substantial and, perhaps, critical extent. The process of activating and engaging the orthodontic appliance produces a system of forces within the mechanism that reaches the dentition through the appliance-to-dentition connection sites. Although permitting some initial deactivation with primarily soft-tissue displacements, the resistance of the periodontium that prevents large initial tooth movements generally keeps the appliance under load (activated) for an extended period of lime. Accordingly, during active treatment two sets of forces exist as action-reaction counterparts at the connection sites. One set is primarily and continually related to the deformed configuration of the appliance with respect to its passive state; the other determines in large measure the course of dental-unit displacements over the finite time period. Control of the overall force system, then, must account for this system's influence on both the appliance and the dentition, and all characteristics of each component of the system (force or couple location, magnitude, and direction and sense), together with variations in these characteristics with time, must be considered. With respect to the dentition, force and couple levels should be large enough initially so that, after the soft-tissue movements, sufficient magnitudes remain to produce measurable orthodontic displacements during the between-appointments period. On the other hand, force magni-

tudes cannot be so great as to inflict undue pain and/or result in ankylosis. Concurrently, the force system delivered to intraoral anchorage sites, if any, must be small enough (through distribution as well as by limiting the size of the resultant) not to cause unwanted movements there. Furthermore, the initial force levels chosen will be affected by their expected decay patterns over lime (discussed in Chapter 5). Displacement paths of the dental units will be directly influenced by directional characteristics of the force system; displacements early in the be-

tween-appointments period may alter initial force directions with respect to teeth and, thereby, affect subsequent movements within the same period. Particularly difficult to control are couple-force ratios over time, inasmuch as the individual decay patterns of couple and force are

230


related to separate stiffness and deformation parameters. The biologic

system reacts to the local force system as a whole, and not to the individual components. Ideally, force control relative to the appliance should be built into the design of the mechanism. The force system created with the activation

of the appliance must not permit any type of failure at the instant of activation when the stresses and localized appliance displacements are generally greatest. To the point mentioned previously, the allowable "load" on the appliance must account for the expectation of superposition of masticatory forces. Particularly with regard to the possibility of fatigue failure, special precautions must be taken with appliances or appliance elements that are removable and, as a result, are subjected to interrupted force systems. Displacement control pertains primarily to the orthodontic tooth movements, but not to the complete exclusion of the appliance. Allow-

ances must be made, and again largely in the design, such that the mechanism, as it deactivates, will not make unwanted contact with the tissues, thereby giving rise to the potential for trauma or spurious displacements. Displacements, like forces, are vector quantities having direction as well as magnitude. As discussed in Chapter 5, the directions of tooth movements are influenced by the characteristics of the delivered

force systems and, to some extent, by alveolar-bone biology. The bounds on dental displacement magnitudes are directly related to the extent of appliance activation and the between-appointments time period; even in unlimited time, orthodontic tooth movements will not continue beyond total deactivation of the appliance. With metallic appliance elements the orthodontist uses the passive configuration of the appliance and the elastic limits of appliance elements

as guides in activation. Ideally, if the activation does not exceed the elastic limit, the appliance will deactivate to its former, passive state. This does not mean, however, that the movements will be, at their greatest extents, exactly as "read" from the activation, because this "reading" assumes immovable anchorage. Although the majority of metallic, activated appliance elements are expected to produce intermittent force as defined in Chapter 5, displacement control of elements producing continuous force depends upon unloading at the appropriate time by the clinician or the patient, or on intraoral abutments. Because interrupted forces are continuous during their "on" periods, the same is true for this third class of loading. When the continuous force is produced by a polymeric element, the deterioration-aging and relaxation of the material may hinder the maximizing of controlled displacements,

but in an uncontrolled situation (the patient missing one or more appointments, for example), a drop in force magnitude or discontinuance of the force due to failure of the element may be quite helpful. Note that the orthodontist can use the clearances that may exist between arch wire and bracket or buccal tube, particularly with regard to controlling rotations. The wire may be used as an active appliance element, it may be a guide to control the direction of a displacement with a

231


driving force produced by a separate appliance element, or it may be the "trigger" of an abutment system to limit the extent of a specific displace-

ment component. In canine retraction on a continuous arch wire, some distal crown tipping may be permitted; however, when second-order clearance disappears, a couple countering the tipping moment of the driving force arises with bracket-wire contact. In preparing for Class IT mechanics, posterior-segment crowns may be tipped distally; secondorder bends are placed in the posterior portion of the mandibular arch wire of the Class II appliance, but may become active only when force against the segments begins to upright these teeth and "burn" (eliminate) the anchorage. These two sets of mechanics, and others, are examined in detail in Chapter 10, and means of control of forces and displacements are discussed concurrently.

Synopsis objective of this chapter was to provide the reader with a foundation, based in part on the concepts and procedures of structural analysis, The

for the understanding of the topics of arch-wire bending and torsion, extraoral-appliance force systems, and a variety of orthodontic treatment mechanics, discussed from a bioengineering standpoint in the final four chapters of this text. The uniqueness of the orthodontic appliance, neither a structure nor a machine by their engineering definitions, has been noted. The sequence of procedures in a general structural analysis was outlined, the entire process was seen as divisible into two main portions (related to the external and internal force systems associated with individual appliance members), and a four-step approach to the analysis of the orthodontic appliance was introduced. The complexity of the appliance was noted as to its geometry, material characteristics, and the force systems transmitted between appliance

and dentition. When evaluating the continuous arch wire or an appliance as a whole, the primary actions and responses may be examined in

a less complicated environment through reasonable modeling procedures; preparation of an arch-wire model has been undertaken with the importance of validity evaluation emphasized. Substantial attention was given to the connection sites of appliance to dentition, in particular to the orthodontic bracket and its characteristics with influences on both the properties of the transmitted force system and the structural behavior of the arch wire. The orthodontic appliance was discussed as a quasi-static and indeterminate structure, and in this light the force-deformation characteristics of a number of the more basic auxiliary elements were examined. Mention was also made of the various formats and components of arch-

wire activation, the many stiffnesses that may enter into an analysis, and the associated anchorage configurations; however, it is only an overview with the two chapters to follow devoted to these considera-

232


tions. An examination of the effects of activated, tip-back bends served

as a first experience involving an arch wire with the four-step analysis approach. Finally, the clinician's concept of "control" was considered with note of its applicability both to the desired displacements within the dentition and the structural behavior of the orthodontic appliance. This text is not intended to convert orthodontic students or practicing clinicians into structural analysts. The practitioner does, however, contribute significantly to the advancements in design of orthodontic appliances and, as important, is potentially in the best position to knowledgeably critique existing designs from the standpoint of their "in-service" performances. Accordingly, the orthodontist needs at minimum a superficial understanding of the contributions of the structural parameters of the appliance elements to the total behavior of the mechanism to properly employ it in therapy and to evaluate the results achieved with respect to treatment objectives.

Suggested Readings Hsieh, Y.Y.: Elementary Theory of Structures. Englewood Cliffs, N.J., Prentice-

Hall, 1970, Chapters 1, 2. Jarabak, JR., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapters 3, 8.

Laursen, H.I.: Structural Analysis. 2nd Ed. New York, McGraw-Hill, 1978, Chapters 1, 2.

Behavior of Orthodontic Wire in Bending

The orthodontic arch wire, more than any other form of structural mem-

ber, resembles a beam. The beam is characterized by its geometry, its loading, and its response to the force systems exerted upon it. The longitudinal dimension of this structural member is large in comparison with its cross-sectional measurements; the interbracket distance, a characteristic length for the arch-wire segment, is ordinarily at least ten times the typical cross-sectional dimension. Beam loading, as well as its responsive force system, generally consists of concentrated and distributed forces directed perpendicular to the longitudinal dimension; couples may be included in the force system. The applied forces and responses may exist in more than one plane; the force system exerted on an archwire beam may be directed occlusogingivally and/or faciolingually. The activation of a beam is reflected in characteristic displacements with respect to a passive configuration. The primary reference for beam deformations is the passive, longitudinal beam axis that pierces the centers of all of the beam cross-sections. Also associated with beam deformations are the right cross-sections which, in the passive state, by definition, are everywhere perpendicular to the beam axis. The two forms of beam deformation of principal importance are the transverse deflection, the displacement of a beam-axis point perpendicular to the passive orientation of the axis at the point, and the cross-sectional rotation, not a twisting but

rather a tipping, angular displacement of the right cross-section. In keeping with the model established in Chapter 6, the typical archwire segment, to be activated in bending, is envisioned as passively flat and straight. Depending on the plane(s) in which the bending occurs, facial or occlusal views of the arch-wire beam are used in analysis. Most often, the bending force systems are exerted by ligated brackets and

buccal tubes, although forces may be transmitted to the arch wire through hooks or loops—by elastics stretched against them, for example. When an auxiliary force acts on a wire in a direction oblique to it, that force is decomposed into longitudinal and transverse components; only the transverse component contributes directly to bending action. Occlusogingival and faciolingual bracket and buccal-tube force systems 233

234


(but not mesiodistal forces from, for example, stops, ties to anchorage,

or friction) are associated with the arch-wire bending problem. First- and second-order couples, originating from angulations and two-point contacts between wire and bracket or buccal tube, also enter into the bending analysis. The arch-wire segment, activated in bending through en-

gagement in brackets and buccal tubes, may test the analyst to distinguish between the "load" and the responsive portion of the force system; moreover, the arch-wire beam problem is generally quasi-statically indeterminate, necessitating the use of force-deformation relationships in the solution in addition to force and moment balances. The responsive force system, exerted by the "support(s)" of the arch-wire beam, is the action-reaction counterpart of the force system against the anchorage. Clearances between bracket or tube and wire often play a role in the bending problem. Occlusogingival, dimensional differences

may be important to the determination of a transverse deflection. Changes in shape from the passive beam (arch) configuration are generally consequences of bending activation; depending on the amount of second-order clearance, for example, a faciolingual-plane couple may or may not be developed at a support location as part of the bending force system.

The many facets of the arch-wire bending problem already mentioned suggest a complex analysis procedure. Reasonable simplifications, however, beginning with the straight-segmented, passive, arch-wire model, enable a manageable examination of the significant contributions to the analysis. A first bending example appears to be an inversion of the lever of Figure 2-27. The member of Figure 7-1, however, is subjected to a transverse load located between two anchorage sites. Characterizing the beam by its longitudinal axis, it is shown in the

5)

I-

0 0

2L

U-

Midspan, transverse stiffness

P

Midspan Deflection

FIGURE 7-1. A simple example of a split-anchorage beam: passive (dashed) and activated beam-axis configurations (upper left); a free-body diagram of the beam (lower left); and the force-deflection pattern of activation at midspan (right).

235


sketch in the upper left in the figure, passively as a dashed, straight line

and having a solid, curvilinear shape following activation. Such an arch-

wire beam might be maintained in its deformed configuration by the relative occiusogingival positions of three brackets. Assuming that the initial position of the intermediate bracket represents a malalignment to be corrected, the brackets at either end of the wire beam are affixed to teeth that are not to be displaced; the anchorage might be termed "divided" or "split." To make this first example straightforward, distances between adjacent brackets are equal and sufficient second-order clearances are assumed to exist such that no couples arise with activation. The free-body diagram of the activated beam, shown in Figure 7-1 (lower left), is identical to that for a lever having the fulcrum at its midpoint. Force and moment balances indicate that the support forces Qi and QT are equal in magnitude and each is half that of the force P. but none of the three can be quantified by a quasi-static analysis alone. To approach the solution from this point, the configuration of the active state, with respect to the passive beam-axis shape, must be examined. Consider the wire as first engaged in only the support brackets, with activation then achieved by depressing the midsection of the wire to reach the slot of the intermediate bracket. The greater the vertical malalignment, the larger the value of P upon full engagement. From foregoing discussions of material behavior (Chap. 3), the expectation is a linear force-deflection relationship for the metallic wire as shown on the right in Figure 7-1. The slope of the plot is a bending stiffness, the quantity needed to complete the solution; the product of this stiffness and the activating deflection yields the magnitude of P. With the foregoing as an introduction, this chapter develops with a discussion in the following four sections of the geometric and mechanical parameters involved in elastic beam analyses. Next, direct application to activation and deactivation of arch wires in bending is under-

taken. Subsequently considered is the inelastic bending of metallic beams toward an understanding of the influence of the placement of permanent bends in an arch wire which is then to be activated. The chapter closes with a structural analysis of orthodontic wire loops—their characteristics and the influences of loop geometry and wire properties upon activation and deactivation in bending.

am Deformations, Strains, and Stresses reference frame established for determining bending deformations is affixed to the longitudinal axis of the passive, straight beam. The beam axes for arch wires are easily located due to the symmetry of circular and rectangular cross-sections. The general beam problem is three-dimensional with two coordinates of a rectangular framework locating a point The

in the cross-section with respect to its geometric center and the third coordinate locating the cross-section. More specifically in the develop-

236


ment to follow, the origin of the frame coincides with the geometric

center of the reference cross-section of the passive beam. The longitudinal coordinate x is measured along the undeformed (straight) beam axis. The y and z axes, completing the framework, are coincident with perpendicular axes of symmetry within the reference, right cross-section of the beam before loading. Although the force systems associated with the activation of arch-wire beam may be located in either the x-z or y-z plane

or in both, when the force system is three-dimensional in character, separate "in-plane" analyses may be undertaken and the results subsequently superimposed. Accordingly, a plane of bending is defined as that

plane containing both the undeformed and the deformed longitudinal axes of a beam, as well as the external load and responsive force system. Arch-wire bending may be reasonably analyzed in parts, individually in the faciolingual and occlusogingival planes as necessary, followed if de-

sired by superposition of the results. In discussions to follow the x-z plane is the plane of bending and an angle 6 is introduced as a cross-sectional, rotational coordinate.

As many as five displacement parameters may be associated with in-plane bending. They are described here in their usual order of importance. First is the linear displacement of a beam-axis point perpendicular to the passive beam axis, quantified by a z-coordinate measurement. Second is the rotation of a cross-section, symbolically indicated by the angle 6. These are the principal displacement parameters and both may be expressed mathematically as functions of the longitudinal coordinate x. Shown in the upper sketch of Figure 7-2 is a cantilevered beam sub-

jected to a concentrated load at its free end; in a separate sketch, the undeformed (dashed) and deformed configurations of the beam axis are

depicted. A reasonable location for the origin of the x-z frame is the center of the fixed-end cross-section, and indicated in the figure are the transverse-deflection and cross-sectional-rotation parameters for an arbitrary value of the longitudinal coordinate and for the "free" end of the beam. The collection of transverse displacements of all beam-axis points yields the deformed beam axis. Because, for all values of x associated with the activated beam, continuity demands that the longitudinal axis and rotated right cross-section be perpendicular to one another, 6 represents the slope of the beam axis, generally varying from one cross-section to another. A third displacement parameter designates the longitudinal shortening of the beam. With axial (longitudinal) force systems excluded in the bending analysis, the curvilinear length of the beam axis is unchanged

by activation, but the x-coordinate distance between the ends of the loaded beam is less than the length. Nearly always in the elastic bending of engaged, activated arch wires, this form of displacement may be neglected in its influence on other beam parameters, as is implied in the symbols z(L) and 6(L) in the example of Figure 7-2. (In leveling procedures within orthodontic treatment, however, because of high mesiodistal wire stiffness and, often, the real presence of longitudinal loading, this shortening can have a sizable influence on displacement potential,

237


F

L

0(x)

z(L)

z

FIGURE 7-2. Sketches of a cantllevered beam serving as an example to diagrammatically define the bending displacement z, the transverse deflection, and 0, the rotational displacement of a cross-section.

particularly of incisors.) The fourth and fifth displacement parameters

pertain to deformations of the cross-section itself. In the activation of metallic beams, the induced transverse force systems will rarely alter the passive, in-plane, depth dimension of the cross-section; this potential deformation, which might be significant with soft materials or very short beams, is routinely neglected here. Finally, experimental research has

shown that, under bending action, the right cross-sections of some beams will warp, but no warping occurs when the end view of the plane of loading coincides with an axis of symmetry of the beam cross-section. In summary, in all beam theories the z and 0 deformations are recognized as sizable. Small-deformation theories of bending neglect all forms of deformation except these two. When transverse and angular displacements of cross-sections are substantial, however, the effects of longitudinal shortening must be considered; such analyses are highly involved and, because of their complexity, they are not discussed theoretically in this text.

238


In beginning an examination of structural bending, the solid beam is considered to be composed of a large number of longitudinal fibers, each having a very small cross-sectional area in comparison with that of the beam as a whole. One fiber coincides with the longitudinal beam axis. All fibers of the beam as modeled are passively straight and, in the plane of bending, all deformed fibers have identical curvatures. Because these fibers are interconnected, however, reflecting the structural integrity of the beam as a unit, the majority of these fibers experience longitudinal

strain upon activation. Depending on the location of the fiber, this "bending strain" may be tensile, compressive, or zero. Figure 7-3 shows an exaggerated, activated shape of a beam segment in in-plane bending. With the outline of the passive segment superimposed (dashed) and the segment subdivided to simulate the fibrous idealization, the fibers on the side of the beam axis nearer to its center of curvature are seen to have been contracted; the fibers on the other side of the axis are stretched. Note also that the amount of fiber (bending) strain varies with the distance from the beam axis. As shown in Figure 7-3, a local coordinate u is used to measure this distance with the posi-

Neutral surface

FIGURE 7-3. A beam segment in passive (dashed) and activated configurations. Illustrated are the neutral surface, bending strains of the beam fibers," and the coordinate u that locates specific fibers with respect to the neutral surface.

239


five sense taken toward the stretched fibers. An in-plane bending activa-

tion results in all fibers having the same u-coordinate value experiencing

equal bending strains and, considering only circular and rectangular cross-sections, the group of fibers undergoing no longitudinal strain coincides with the cross-sectional axis of symmetry perpendicular to the plane of loading. In Figure 7-3 the apparent beam-axis fiber is actually an edge view of the collection of unstrained fibers known as the neutral surface of the beam. Because the cross-sections of arch-wire beams ordinarily do not warp upon activation of the member, the fiber-strain pattern is associated solely with cross-sectional rotational deformation and the strain values are, therefore, proportional to the u-coordinate locations of the fibers. The overall beam shape in plane bending is represented by the curvature of the beam axis, within and coincident with the edge view of the neutral surface. In general, each point on the bending axis is associated with a beam cross-section and has its own center and radius of curva-

ture. If all bending-axis points in a beam segment possess the same center and radius of curvature, that segment is said to be experiencing "pure bending," terminology that will be further explained in a subsequent section of this chapter. Figure 7-4 shows a very short segment of a beam with a typical, relative rotation of one neighboring cross-section

with respect to another. The portion of beam axis within the segment takes on, approximately, the shape of a circular arc with activation, and the center of curvature of that arc is point C. The associated radius of curvature is r and its inverse will be symbolized in this chapter by k. The beam-axis segment is of curvilinear length 3x, where 5 generally indicates "small amount of' or "small change in" the variable that follows it. The angle 4 denotes the relative cross-sectional rotation. Because & is a very short arc length and approaches in magnitude, and because the longitudinal (bending) strain is the deformation per unit length of a fiber having a passive length 8x, the relationship or

(7-1)

is obtained. The coordinate u locates the beam fibers experiencing bending strain e,, and k is the average curvature of the short beam-axis segment. On any specific beam cross-section located by the coordinate x, then, the rate of increase of bending strain from the neutral surface, a constant for the cross-sectional location, is directly dependent on only the beam-axis curvature at that location. On the typical cross-section, in-plane bending deformation of an initially straight beam produces compressive strains in those beam fibers

on one side of the neutral surface and tensile strains in the remaining fibers. The magnitude and sense of any fiber strain at a point in the beam depend on location, defined by the two coordinates x and u for in-plane bending, the geometry and material of the beam, and the force system producing the bending deformation. On any beam cross-section,

240


C

=

5x =

GH= Also,

GH

=

4

So,

= k(8x) = e0(ôx) U

U

FIGURE 7-4. The geometry from which may be derived, for an activated beam, the relationship between the intensity of bending strains on a specific cross-section and the beam-axis curvature there.

241


(en)max

Neutral axis

(So)rnax

FIGURE 7-5. Fiber-strain and elastic fiber-stress patterns on a typical circular or rectangular cross-section of a beam subjected to in-plane bending.

the fiber strains are proportional to their distances from the neutral sur-

faces and, for beams with circular or rectangular cross-sections, the max-

imum tensile and compressive strains are equal in magnitude. A typical fiber-strain pattern is shown on the left in Figure 7-5 with, for cross-sections of interest here, the neutral surface (seen in edge view) midway between the pictured longitudinal boundaries of the beam. The corresponding bending-stress pattern is obtained from the stress-strain

law for the beam material and the established strains. Because archwire-beam materials are reasonably homogeneous, the stress-strain relationship for any one fiber is that for the material as a whole. Since such materials are metaffic, they behave within their elastic limits according to Hooke's law. The theoretical development and results to follow reflect equal material stiffnesses in tension and compression, a reasonable as-

sumption (verified by experimental research) well beyond the elastic limit for beam material of interest. Shown on the right in Figure 7-5, then, is the elastic, fiber- (bending-) stress distribution associated with the adjacent strain pattern. At any specific point of the cross-section, for totally elastic material behavior, the ratio of stress to strain equals Young's modulus for the beam material. Increasing the beam loading to produce inelastic action will not destroy the antisymmetry seen in the figure, but it will change the relationship between the bending (fiber) stress s,. and u from the linear pattern shown to a nonlinear pattern, corresponding to the general plot of stress versus strain for a typical, ductile, crystalline material (see Fig. 3-17, right).

orce Systems Within the Beam The beam of Figure 7-1 may be described as having a "span" of magnihide 2L, simply supported (because the responses at the beam ends are "simply" concentrated forces, and activated by a transverse, point load

242


"midspan." The responsive forces obtained earlier seem to suggest that the load is divided and its halves are transmitted left and right from the midspan site to the beam ends where they are transferred from the member to the supports. This is an oversimplification, but correctly implied is the generation of a force system within the beam, created with application of the load. The internal force system may be "exposed" by figuratively cutting the beam perpendicular to the beam axis at a desired longitudinal location, revealing two adjacent right cross-sections. One at

or the other beam portion is isolated for study; either may be chosen since the force system exerted on one cross-section is the action-reaction counterpart of that exerted on the other exposed cross-section. As an example, the beam of Figure 7-1 has been "cut" half way be-

tween the left support and the load, and a free-body diagram of the shorter, left-hand portion is shown in Figure 7-6. (Even though activation produces deformation, for purposes of force analysis the passiveconfiguration sketch may be used when longitudinal shortening is minimal.) In the activated state, any part of the beam is static (or quasi-static) as is the entire member. Although the force system exerted by one portion of the beam upon the other at the "cut" site is distributed throughout the cross-section, resultant components perpendicular and tangent to the cross-section are initally considered. First, the presence of the support response at the end of the segment and the need for a transverse force balance suggest the existence of a tangential or shear component V. (The symbol V is used because the majority of beams are ori-

ented with the passive longitudinal axis horizontal; the shear force, then, is directed vertically.) Second, because no axial loading is present, a horizontal force balance indicates that the net longitudinal force com-

ponent at the "cut" is zero. (When present in the practical structural problem, axial forces and their effects may be handled separately and superimposed upon those associated with bending.) Third, since the

L

P

2

7-6. A free-body diagram of the left quarter segment of the beam of Figure 7-1, exposing components of the resultant of the internal force system at a specific crosssection and permitting the determination ot their characteristics. FIGURE

243


support response and the shear component in the example are not col-

linear, a moment balance cannot exist without the presence of a couple M at the "cut" site. (The symbol M is used because the couple is known as the bending "moment" to the structural analyst.) The magnitudes and correct senses of V and M from quasi-static analysis are given in the figure for the example, simply-supported, split-anchorage beam. In obtaining these components a force analysis of the entire beam is necessary prior to making the "cut" in order that V and M may be expressed in terms of the load and longitudinal dimensions. In the beam problems of interest here, an in-plane bending analysis will involve external loadings consisting of concentrated forces and/or couples located in the x-z plane. The internal components V and M will

be partially dependent on the load and will generally vary with the longitudinal coordinate x from one cross-section to another along the beam length. Using the example analysis as a guide, V may be seen to be unchanged along a beam segment upon which no external force is ap-

plied while M, computed from force and distance, varies in a linear fashion (increasing or decreasing with distance from the force to the "cut" site). Proceeding from one cross-section to another longitudinally, an encounter with an applied, concentrated force makes V "jump" to a new value; M is also affected, but to a less dramatic extent. In the same

analysis procedure, encountering an applied, concentrated couple makes the internal bending moment "jump," but the couple has no effect on V. In detailed beam analyses, plots of V and M versus the longitudinal coordinate x are often sketched. Such diagrams, showing typical varia-

tional patterns of these resultant components of the internal force system, are presented for the example problem below the free-body diagram in Figure 7-7. The determination of the critical cross-section, the location of the largest M value is important in beam analysis; the figure provides that information for the example problem. Although not exhibited in the example, a segment of finite length may exist in a beam in which the bending couple M remains constant. Correspondingly, the shear force V must be zero throughout the segment and the segment is said to be in pure bending. Within the segment the deformed beam axis has the shape of a circular arc and, therefore, every cross-section within the segment has the same center and radius of curvature and, as mentioned previously, the same bending strain pattern. The foregoing comments suggest the existence of relationships between the couple M and the bending stress and deformation patterns for the general beam. Appropriate first, however, is a look at the influence of the shear component V upon bending. Although V is related to M (reflected in Figure 7-7 in that the slope of the M-diagram is everywhere equal to the corresponding shear force), its direct effect on important bending parameters is minimal. Experimental results have shown that transverse deformations attributable to V are generally very small. The component V is the resultant of the transverse shearing-stress pattern, and the average and maximum shearing stresses are typically less than

244


P

P

P

2

2

V

P 2

L L

x

2

M

x

FIGURE 7-7. Shear and bending-moment diagrams below a free-body diagram of the beam of Figure 7-1.

10% of their counterparts in bending. Only when the beam is extremely short are the shear effects significant, and such beams are nonexistent in the orthodontic appliance. Hence, the remarks to follow focus solely on one component of the internal-force system in bending.

245


Noted previously was the presence of a fiber-stress pattern perpendicular to a right cross-section within the activated beam (Fig. 7-5). For

the two cross-sectional shapes of interest and considering in-plane bending, that pattern is divided in half by the neutral surface. Directly viewing the cross-section, the edge of the neutral surface seen is termed the neutral axis and it coincides with the y-coordinate axis defined earlier. Bending stresses are zero on the neutral axis that divides the distribution into tensile and compressive parts; the resultants of the two parts of the

bending-stress pattern are equal, parallel forces, and their resultant is the bending couple M. In general, then, like M, the magnitudes of these forces and the sr-to-u ratio may vary from one cross-section to another, mathematically dependent on the longitudinal coordinate x. A diagrammatic representation of the relationship among the bending stresses, the pair of force resultants, and M is sketched in Figure 7-8. The bending(fiber-) stress value at a point depends upon the coordinates x and u to specify the location within the beam, upon the loading that enables the determination of M, and finally upon the cross-sectional geometry of the beam. To fill several needs an analytic expression in these parameters is now developed. Figure 7-9 depicts a typical beam cross-section in edge view, and a line of fibers parallel to the y-axis and the associated internal force upon

which attention is focused. The desired, explicit expression for M is obtained by summing the moments of all such "fiber-line" forces about the neutral axis, also seen only in edge view in the figure. Because each fiber has a very small cross-section and is activated in direct tension (or compression), the 0-notation is again used and the fiber-line force equals the product of stress (unchanging in the y-direction) and the fiber-line area: OP =

(7-2)

F, =

F,

M=

FIGURE 7-8. The internal bending couple as the resultant of the fiber-stress distribution.

246


Neutral axis

= 3M = (SP)u

U

a "tiber"

r

N

Outline ot tensile tiber-stress distribution

FIGURE 7-9. The contribution of the force in one beam fiber to the internal bending couple M.

The moment arm for each fiber-line force 8P is the coordinate value u,

and the sum of the moments is M: M = sum of [u(sn)6A1

(7-3)

Again referring to Figure 7-5 and recalling the beam-material model, if the in-plane bending does not take any fiber beyond its elastic limit, the value of is proportional to the distance between the neutral axis and the fiber-line (which is, again, the coordinate value u). Using this proportion, Equation 7-3 may be rewritten: Sfl

M = —[sum of (u 2SAfl

(7-4)

The quantity in brackets is a geometric parameter of the beam cross-section; it is termed a second momenf of the cross-sectional area and is often symbolized by the letter I. The ratio of Ito c, where c is one-half of the beam depth measured in the plane of bending, is known as the secf ion modulus for the cross-section. Values for the second moment of area and the section modulus are given in Figure 7-10 for circular and rectangular cross-sections; note that, with the loading plane vertical with respect to the sketches, the beam depths 2c are the diameter and the dimension h for the circular and rectangular cross-sections, respectively. Returning to Equation 7-3, the ratio of bending stress to coordinate u may be written as Sn: u

=

(sn)max : c

(75)

247

Behavior of Orthodontic Wire in Bending For the circular cross-section:

For the rectangular cross-section: b(h3)

64

and

and

c

c

32 d

6 h

where c = —

h

FIGURE 7-10. Second moments of area I arid section moduli I/c for circular and rectangular cross-sections.

The

desired equation may now be expressed in either of two forms: M=

= (Sn)max1

(76)

This equation relates M, the bending stress in a particular line of fibers (perpendicular to the plane of bending), and a geometric parameter of the cross-section (generally a constant throughout the beam length). The arch-wire-beam activation ordinarily should be fully elastic in order that the totally deactivated and pre-load passive configurations are identical. Accordingly, the maximum magnitude of loading to be exerted on the beam, without anywhere exceeding the elastic limit of the beam material, is of interest. This quantity is known as the elastic strength in bending. From the form of Equation 7-6 and Figure 7-8, the maximum

fiber stresses on a cross-section are located at the points farthest from the neutral axis, and these maximum stresses are proportional to the value of M. Considering the entire beam, the elastic strength depends on the maximum value of M, the internal bending couple at the "critical cross-section," which is dependent on the loading and manner of support of the beam and its longitudinal geometry. Accordingly, the elastic strength may be expressed analytically by setting the maximum bending stress in Equation 7-6 equal to the elastic limit value for the beam material (in tension or compression), and specifying the critical cross-section:

248


= (Sn)erL

(74)

The critical bending couple is, therefore, computed from mechanical and geometric property values for the beam. The relationship between and the loading parameters is obtained from a free-body analysis of one of the two beam segments isolated by "cutting" the whole member at its critical section. Finally, then, the elastic strength of a given beam subjected to a particular loading configuration may be determined as maximum or limiting values of parameters associated with the active force systems.

Beam Stiffnesses The theory of elastic bending has its origins in experimental observation

and research proceeding to an empirical development. The initially straight member takes on a curved form when activated or loaded, with the associated deformations largely expressible in terms of the shape of the longitudinal beam axis. Passively plane right cross-sections have

been found to remain plane with activation for problems of interest, even when the deformations are substantial, leading to a simple, linear fiber-strain pattern (see Fig. 7-5). Hooke's law then provides the bending-stress distribution on a cross-section; the resultant is the internal bending couple. This couple is related to the external force system exerted on the beam through force and moment balances. To this point in

the discussion, three governing relationships have been mentioned which contain variables dependent on the longitudinal coordinate x: (1) the curvature of the beam axis and the u-coordinate provide the fiber strains; (2) fiber stresses are everywhere proportional to fiber strains; and (3) the internal bending couple is obtained from the bending-stress distribution and the cross-sectional geometry. An explicit expression, or a set of expressions, for the deformations

of the activated beam, in terms of the beam material, geometry, and loading, is now required. An intermediate result is obtained from eliminating and u from the aforementioned three relationships; or

M=(EI)k

(7-8)

In essence, this equation interrelates deformation (k), force (M), and stiffness (El) parameters; the first two are generally dependent on x, but the third is a constant and is known as the unit bending stiffness. The next step in the derivation eliminates k in favor of its differential expression in terms of z and x and, although the subsequent mathematical operations

are beyond the scope of this discussion, several associated comments are appropriate. First, relatively simple equations are obtained only when the slope of the beam axis is, for all x values, small compared to

249


unity (0 less than 45°), and when longitudinal-dimension changes with

activation are negligible; therefore, in invoking these restrictions, the result is a small-deformation theory. Second, the mathematical procedures that must be undertaken yield equations for both the cross-section-rotation parameter 0 and the transverse deflection z. In these derivations use is made of physical constraints to the actual deformations that exist along the beam; such constraints are integral in the supports of the beam and, for the arch-wire beam, exist where the wire engages the brackets or buccal tubes. Third, a number of assumptions involving geometry and material have been made and it is well to recall the list here: cross-sectional symmetry; in-plane bending; elastic activation; Hookean material; and values of the elastic modulus F equal in tension and compression. The split-anchorage, arch-wire beam of Figure 7-1 again serves well as an illustrative example. The span is 2L and the bracket-wire clearances are assumed sufficiently large that transverse forces alone are developed upon activation. With midspan loading, the responsive forces at the anchorage locations have magnitudes that are each one-half that of the active point load. Given is the magnitude of the midspan deflection necessary to activate the member; in perspective, the occlusogingival or faciolingual crown malalignment of a canine or premolar has been measured. If the midspan, transverse stiffness of the wire beam can be determined, and the activation is totally elastic, the product of this stiff-

ness and the activating displacement is the magnitude of the midspan load.

With the external force analysis completed for the entire beam in terms of the applied force F, the next step is the determination of the bending couple on an arbitrary cross-section as a function of P and x. An x-z-coordinate frame is established with its origin taken, for example, at the left-support position with the x-axis coincident with the passive con-

figuration of the beam axis. The undeformed (dashed) and deformed (solid) beam configurations and the reference frame are shown in Figure 7-11. Because of the geometric and mechanical symmetry in the example problem with respect to the midspan cross-section, the M-diagram (Fig. 7-7) and the transverse-deflection pattern exhibit the same symmetry.

z

FIGURE 7-11. An x-z-coordinate frame superimposed upon the passive (dashed) and activated (solid) configurations of the longitudinal beam axis for the Figure 7-1 example.

250


With the slope of the beam axis horizontal at midspan and the symme-

try, in effect the solution may be pursued from this point using only one-half of the beam. The equation for the couple M is written from free-body analyses of segments of the beam, or obtained from the M-diagram, and substituted in Equation 7-8. The curvature k is then expressed implicitly in terms of z and x as noted previously, and subsequent mathematical operations yield equations for 0 and z. In the example given,

the equation of interest is PL3

or

P

LEJ

(7-9)

obtained by substituting x = L into the more general expression. A linear relationship between P and z is noted and the ratio of P to z is the midspan, transverse-stiffness equation desired. This stiffness is seen to be directly proportional to the unit bending stiffness El and inversely proportional to a power of the characteristic, longitudinal dimension. With these quantities determined for the wire beam selected and the malalignment (z at x = L) measured, Equation 7-9 provides the value of P and the forces transmitted to anchorage also become known. From the results of this example problem the parametric influences on bending stiffnesses may be discussed in general. A first effect is that of beam material with stiffness proportional to Young's modulus E. A second influence is that of the size and shape of the cross-section. Because the parameter I is a fourth-power function of the cross-sectional dimensions, small changes in cross-section can substantially affect the bending stiffness. For instance, due to the differences in values of I, replacement of a round wire by a square wire having an edge length equal to the round-wire diameter increases the bending stiffness by 70%. Third, a characteristic beam length or span has sizable impact on bending stiffness. In arch-wire bending, this influence is largely that of inter-

bracket distance. In the point-load example, the third power of the length appears in the transverse-stiffness formula; in general, as will be

seen, this L-influence will be determined by the type of loading and which stiffness (transverse or rotational) is being computed. Two other considerations related to the example problem are of interest. First is the influence of the position of the load between the supports upon the transverse stiffness at the beam-axis point under that load. For purposes of comparison, shown in Figure 7-12 is the example beam with the load moved from midspan, left to the one-third-span location. The free-body diagram indicates that the mechanical symmetry is no longer present; the desired stiffness is given in the sketch. A comparison of the expression with Equation 7-9 shows that a lateral shift in the load position a distance equal to 17% of the span (214 results in an increase of

about 27% in the stiffness. Second is the influence of the presence or absence of second-order clearances at anchorage in occlusogingival bending (or first-order clearances in faciolingual bending). Figure 7-13 depicts the original example problem once more, except

251


At x = P

P

2L

(El

= 7.6k

L3

I

I

3

3

3

FIGURE 7-12. The split-anchorage example beam with the applied, transverse force moved to the one-third-span location: a free-body diagram of the entire beam and the expression for the transverse stiffness at the load site (above, right).

At x = L: P

x

4c' P

4

FIGURE 7-13. The split-anchorage example beam with no rotational deformation permitted at its ends: a free-body diagram of the entire beam and the expression for the transverse stiffness at the load site (above, right).

_________ 252


that the supports permit no beam rotations at those locations (indicative

of zero second-order clearance if the figure shows a facial view). The free-body diagram reflects the presence of couples as part of the support responses; in effect, these couples oppose the rotational tendencies. With the additional constraints on the shape of the deformed beam axis—

its slope forced to be zero at the supports—the stiffness should be expected to rise compared to that expressed in Equation 7-8. Indeed, the transverse, midspan stiffness given on the right in Figure 7-13 for the beam having "fixed ends" is four times that for the same beam material, length, and cross-section, but with ample clearance permitting end rotations.

Additional Topics in Elastic Bending With elastic strength and stiffness defined, a third structural parameter

of interest in bending is elastic range. A counterpart to elastic strength,

range is the maximum measure of a flexural deformation which the beam can experience without an inelastic response occurring somewhere in the member. Hence, elastic strength is directly associated with load and elastic range with the corresponding deformation. In the splitanchorage problem of Figure 7-1, the range of interest is the maximum, elastic, transverse deflection at midspan. The M-diagram indicates the largest bending stresses in the beam to be at midspan, the location of the is equal to PLI2 (Fig. 7-7). The elastic critical cross-section, and range is obtained by first substituting for PLI2 in Equation 7-9 and then eliminating Mcrjt from the resulting expression using Equation 7-7. The largest, elastic midspan deflection may then be expressed as (Sn)eiL2

3Ec

or

(7-10)

In general, elastic range is influenced by the characteristic length dimension, the beam depth in the plane of bending, and elastic-limit strain for the beam material. The power to which the length dimension is raised in the range expression is always one less than that appearing in the corresponding stiffness equation. Instead of the elastic-limit strain, the ratio of elastic-limit stress to modulus of elasticity is often explicit in the range equation; these two mechanical properties are typically more readily available, from reference materials or experimental data, than is the strain value. The existence or amount of second-order clearance at the supports affects the range as well as the stiffness. If the beam is fixed or "built-in" at both ends such that no second-order clearance is present at either site, the value of is lessened to PL/4 and the

253


elastic range becomes (Sn)eiL2

6Ec

or

(eu)eiL2

6c

(7-11)

For the same reason that the stiffness was increased, the elastic range is reduced when rotational beam deformation is constrained. In this example, requiring the beam-axis curvature to be zero and exhibit horizontal slope at its ends results in a 50% reduction in range. Elastic range and strength are significant parameters in beam analysis and design because they signify the onset of inelastic material behavior. In the discussion of orthodontic bending, because the actual loading is often induced by a transverse or rotational deformation, range is perhaps the more important of the two. However, because arch-wire activation should normally not produce inelastic bending, the pertinent of the

two parameters (depending upon the loading) should be known or quantified before engagement of the wire. Because of the interrelationship among range, stiffness, and strength, with second- (or first-) order clearance already seen as influencing the first two, it affects strength as well. Just noted was the halving of in the example problem, as it relates to the load on the beam, by eliminating that clearance at both supports. Returning to Figure 7-7 and recalling Equation 7-7, making no changes in the example problem other than eliminating clearances at the supports results in a twofold increase in the critical value of the load P (the elastic strength). Typically, alterations in bending parameters that raise (lower) beam stiffness will also raise (lower) the corresponding elastic strength and concurrently lower (raise) the elastic range. Thus far the discussion of bending behavior has concentrated on the beam having split anchorage and subjected to a transverse, point load applied between the supports. Two additional problems having orthodontic application are the split-anchorage beam under in-plane, couple loading and the cantilevered beam activated by either a transverse, concentrated load or a couple at its "free" end. As with the example problem, of interest are the anchorage responses, the locations of the critical cross-sections and the associated, internal bending couples, the stiffnesses, ranges, and strengths, and appropriate comparisons among results. Shown in the upper sketch in Figure 7-14 is the free-body diagram of a beam of span 2L activated by an external bending couple of magnitude M0. (The loading may either be an applied couple, or the couple induced as a result of a clockwise, midspan, rotational deformation On.) Rotational clearances are ample at the supports and the antisymmetry of the deformed member helps in determining the responsive forces at the ends of the beam. The internal bending couple is zero at each beam end, increases linearly toward the middle cross-section, and at midspan it "jumps" an amount equal to M0; the value of is M012. The stiffness of interest is the ratio of the applied-induced couple to the cross-sec-

254


M0

2L

M0

M0

2L

x

L

z

3M0

3M0

4L

L

z

FIGURE 7-14. Free-body diagrams of split-anchorage beams subjected to couple loading at mids pan. No rotational resistance at the anchorage sites exists for the beam in the upper sketch; the anchorage prevents rotation of the ends of the beam in the lower sketch.

tional rotation at midspan and the relationship is M0

6E1

00

L

or

00=

M0L (6E1)

(7-12)

The cross-sectional, rotational range for this beam is (Sn)eiL

(en)eiL

or

3Ec

3c

(7-13)

and the elastic strength may be expressed as M0 =

I 2(Sn)e1

(7-14)

255


Shown in the lower sketch in Figure 7-14 is the same beam except that no cross-sectional, rotational clearance exists at either end support, and as a result the anchorage response includes a couple in addition to the transverse force at each end of the beam. Computations of the values of the responsive components require the use of indeterminate-beamanalysis methods and the constraints of zero slope of the beam axis at its ends. The relationship between couple and cross-sectional rotation at midspan for this beam is M0

8E1

M0L

or

=

00

=

(7-15)

8E1

and the rotational range is (en)eiL

or

(7-16)

Pertinent here is a comparison of stiffnesses and ranges for the beams that are identical except for clearances at the anchorage sites; constraining the cross-sectional rotations at the ends of the member results in a 33% increase in stiffness and the same percentage decrease in elastic range. Also of particular note is that the elimination of rotational clearances at the beam ends has no effect on the critical bending couple and, therefore, the elastic strengths are equal for the two beams. Free-body diagrams of cantilevered beams are shown in Figure 7-15. Because of the existence of just one anchorage site (support), prevention

of both rotational and translational displacements must be present there. Although some small amount of cross-sectional, rotational clearance could exist in reality, the equations to follow assume a complete constraint and, therefore, zero slope of the beam axis at its left end. The support response against the beam on the left in the figure includes the expected, transverse force and couple, and the internal bending couple

P

= PL

=M0

L

z

7-15. Free-body diagrams of beams cantilevered from their left ends. The free-end loads are a transverse force exerted on the beam at left and an in-plane couple on the beam at right. FIGURE

_________ 256


varies with x in a linear manner from zero at the right end of the member

to the maximum, value of PL at the built-in, left-end support. On the other hand, the same beam on the right is loaded by a concentrated couple at its free end and, with no net potential in the load to translate the member, the response at the support is just an opposing couple. The internal bending couple is unchanged from one cross-section to another along the beam length (pure bending throughout); hence, even if the applied couple M0 loading one beam was equal in magnitude to the product PL for the other beam, the differences in M-diagrams indicate unlike responses of the two, perhaps otherwise identical, members to their loadings. For the beam on the left in Figure 7-15, the stiffness of interest is the

ratio of the load to the transverse deflection of the free end and the relationship is P

3E1

or

PL3

(7-17)

The cross-sectional rotation of the free end of the same beam is (7-18)

Of

therefore, at least this amount of rotational clearance must be present within whatever maintains the activation (e.g., a bracket) if the loading is not to also include a couple. The range of interest in this example is the largest, transverse, free-end deflection that will induce maximum fiber (bending) stresses at the left end not exceeding the elastic limit of the beam material, and that range is or

3Ec

(7-19)

With Mcrit equal to PL at the built-in end of the cantilever, the elastic strength of the beam is P = (sn)ei4

(7-20)

For the cantilevered beam subjected to pure bending on the right in Figure 7-15, the rotational stiffness of its free end is M0

El

or

M0L

(7-21)

and the corresponding transverse deflection is = (7-22)

257


(Unless

the transverse position of the right end of the beam is main-

tained by some partial support that would exert a vertical force upward and, by its presence, change the problem entirely, a transverse deflection downward is induced by the given loading.) The range for crosssectional rotations of the free end of this beam is (Sn)eiL

Ec

and, with

or

(en)eiL

(7-23)

C

equal to M0, its elastic strength is given by = (Sn)e11

(7-24)

of Elastic Beam Theory to Orthodontic Arch-Wire in Bending I

The orthodontic appliance is activated by imposing on a passive configu-

ration the deformation of one or more elements of that appliance. The deformations of metallic elements are characteristically bending or twisting; the former predominates. Examples of bending activations are those of arch wires in occlusogingival and faciolingual leveling procedures, utility arches used to intrude or extrude incisal segments, and cantilevered arch-wire segments used to displace individual teeth. Responsive bending deformations also occur in, for example, a face bow in the activation of a cervical-pull headgear and in the buccal-segment portions of an arch wire into which anterior torque has been induced. The majority of orthodontic wire loops are activated in bending. Note that under consideration here is the deformation imposed following whatever forming of the as-received member is undertaken by the practitioner. Although such forming procedures take the material beyond its elastic limit and result in geometric alterations from the as-received configuration, the activating deformation is intended to be elastic in nature. Inelastic bending is discussed in a subsequent section of this chapter. The question arises regarding the validity of applying an engineering bending theory to orthodontic arch wires, inasmuch as these members have very small cross-sectional dimensions compared to the typical structural member. Note that, within the theory, the format of the elastic bending-stress distribution, exhibiting a linear variation from the neutral surface, is as simple as the resultant, internal-force analysis and the continuity of strains will permit. To be sure, the imposition of residual stresses in the manufacturing process and in the placements of bends and twists, prior to appliance engagement, will have some effect on the subsequent structural behavior of the activated wire, but such influences of substance are usually localized and reflected directly in the mechanical properties of the wire material. Arch-wire-bending experimen-

258


tation with straight segments has resulted in reasonable agreement between theoretical predictions and actual measurements, provided the deformations are small and assuming Young's modulus in bending is equal to the magnitude of the material stiffness derived from a tensile

test.

The ADA specification for orthodontic wire, discussed briefly in Chapter 3, includes an elastic-bending (flexure) evaluation and the theory wherein the pattern of couple loading versus cross-sectional rotation is examined and structural properties are determined. Although the theory has apparently been accepted by the orthodontic community, care must be taken to avoid exceeding the limits of its applicability. The theory will yield erroneous results when the activating deformations are substantial; its accuracy is suspect, for example, when a transverse deflection of a continuous arch wire exceeds in magnitude that of an adjacent interbracket distance or one-half of the distance between supports in a split-anchorage arrangement. In addition, the theory as discussed here applies, in a strict sense, only to passively straight beams. To this point, the arch-form curvatures of continuous wires across the incisal segment and throughout the posterior segments are not locally severe and are present only in the occlusal plane. The arch-wire model introduced in Chapter 6 suggests that only in the vicinity of the canines is the curvature sufficient to void practical use of the theory. The presence

within the beam span of a permanent bend however, may affect the accuracy of a theoretical analysis in the plane of that bend and, in particular, at the specific location of the bend. In an overall analysis, the inf usion of error originates not so much with the residual stresses, which are usually highly localized, but rather with the change in passive geometry associated with the bend placement. The selection of an arch wire for use in an appliance, for which the Th activation will result in bending deformation of that wire, must be based primarily upon its structural characteristics in bending. The most important parameter is stiffness, which is not only the key to the solution of the indeterminate problem but also provides the elastic activation and

deactivation rates (load per unit of deformation) in bending. Elastic range is significant because prevention of inelasticity during activation is desirable. If the load to be applied is measurable, the strength parameter

provides an upper bound for an elastic activation. Appearing in the bending formulas of this chapter are the modulus of elasticity E and the stress and strain values at the elastic limit. Young's modulus is affected

only slightly, but the elastic-limit coordinates may be substantially changed, by strain-hardening or heat treatment of the arch wire prior to in-service use. The recent additions of nickel-titanium and titanium-molybdenum alloys to the list of available orthodontic-wire materials has broadened the spectrum of values of all three cited mechanical properties. The three structural characteristics are also dependent on wire geometry, more specifically on the cross-sectional dimensions, the shape of the cross-section, and length (span) dimensions. The dependence of stiffness, range, and strength in elastic bending upon arch-wire geomet-

259

Behavior of Orthodontic Wire in Bending TABLE 7-1. Dependence of stiffness, range, and strength in bending upon material and geometric properties of round and rectangular wires

Cross-section Material

L3

tr*

Stiffness

rot

E

tr

Range

Strength

ro

Length

L1 L2

(Sn)ei(E1)

tr

L

Round

Rectangular

d4

b(h3)

d1

h

d3

b(h2)

L (Sn)ei

ro

0

*tr; Deformaton is transverse deflection. tro: Deformation is cross-section rotation.

nc and material properties, for both transverse-deflection and cross-sec-

tional-rotational formats, but restricted to solid (single-strand) wires, is given in Table 7-1.

Additional comments regarding cross-sectional geometry are in order here. Note was previously made of the influence of shape and the fact that, because of the additional material present, the bending stiffness of a square wire is potentially greater than that of the round wire,

the two cross-sections having equal depth dimensions. Rectangular wires also exhibit greater strength; however, since only the beam depth is the cross-sectional influence in the range formulas, shape does not impact elastic range. In addition, the differences in the two dimensions of a rectangular (nonsquare) wire produce, for a given orientation of the wire in the brackets, differences in the structural bending properties for occlusogingival versus faciolingual bending. Still another influencing factor is the physical configuration of the wire, be it solid, layered, or stranded. Because in layered or stranded wire the longitudinal elements may slide somewhat with respect to one another upon activation (and deactivation), for a particular overall cross-sectional size and shape, these wires exhibit less strength, greater range, and, most significantly, lower stiffness than their solid counterparts. Braided stainless-steel wire is available in both round and rectangular cross-sections and will take

some permanent bending, exhibiting ductility, without substantially compromising the braiding. Without the use of loops placed in the wire, the practitioner is generally unable to vary the longitudinal dimension(s) of the arch-wire beam to take advantage of the influence of this geometric parameter upon the structural properties under consideration. Except when using a specialized cantilever or when leaving a particular tooth or segment out of the activation, the mesiodistal geometry of the dentition controls the archwire beam spans. From a practical standpoint, then, a quantitative ex-

260


TABLE 7-2. Effect of material of a solid arch wire, activated in bending by a transverse force, upon stiffness, range, and strength

Stiffness*

Range*

Strength*

0.52

0.97

0.54

steel

1

1

1

Elgiloy (Cr-Co alloy)

0.98

1.13

1.11

Nitinol (Ni-Ti alloy)

0.17

4.2

0.70

Beta titanium (Ti-Mo alloy)

0.33

1 .86

0.61

Material

A gold alloy

Type 302 stainless

'Given figures are not actual values, but, in each column, are compared to a typical value for the stainless steel.

amination of the effects of material and cross-sectional size and shape

upon stiffness, range, strength in bending is appropriate. Table 7-2 presents the comparative influences of five wire materials, considering wire beams having all other characteristics identical and taking stainless steel as the reference material (setting its three property values each equal to unity). Table 7-3 examines the impact of cross-sectional geometry comparing solid round and rectangular beams having all other characteristics identical. The smaller cross-sectional dimension of each nonsquare, rectangular wire is assumed oriented occiusogingivally. Values of the

parameters are, in each column, normalized with respect to a noted reference wire and, where pertinent, the plane of bending. The fact that the second moment of area (I) varies with the fourth power of the crosssectional dimensions causes the substantial differences in stiffness and

strength between the smallest and largest wires listed, even though their depth dimensions differ by a factor not greater than two. One additional influencing factor, referred to earlier in the stiffness discussion, is the location of the load between specific, split-anchorage positions. An illustrative orthodontic example is associated with the distal displacement of a canine into a first-premolar extraction site, using an arch wire for guidance. The wire beam is supported anteriorly by the incisors and also on each side by a posterior segment. Considering the

half arch (one side), the primary load on the wire is a second-order couple exerted by the canine bracket, created as the tooth tends to tip under the action of the distal driving force. With no anchorage loss, the

261

Behavior of Orthodontic Wire in Bending TABLE 7-3. Influence of cross-sectional dimensions of the arch wire upon stiffness, range, and strength in occiusogingiva! and faciolingual bending by a transverse force

Stiffness

Range

Strength

1*

1.5

1*

1.7

1.3

1.5

2.7 4,2

1.2

2.1

1.1

6.1

1.0

2.9 3.9

.016 x .016

2.9

1.3

2.5

.016 x .022 ogt .016 x .022

4.0

1.3

7.6

1.0

3.5 4.8

.017 x .025 og .017 x .025 fI

5.4 11.8

0.9

.019 x .026 og .019 x .026 fI

7.9

1.1

14.8

0.8

Wire size (in.) .014 .016 .018 .020 .022

round round round round round

.0215 x .0275 og .0215 x .0275 Ii

1.3

12.1

1*

19.8

0.8

4.5 6.6 5.8 7.9 7.9 10.0

*Given figures are not actual values, but, in each column, are compared to the value for the wire indicated by the asterisk (*)

tog: Occiusogingival bending. 4f1: Faciolingual bending.

tooth, and with it the couple, move an average distance of 7 mm in the

retraction process, assuming approximately 20 mm (mesiodistally) between lateral-incisor and second-premolar bracket centers. Although a detailed consideration of such a moving-load problem is too complex to include here, the closer the load is to either anchorage site, the higher

are the relevant bending stiffness and strength and the lower is the range. In the canine retraction example, none of the three parameters changes with canine position as much as 30%; therefore, a reasonable, approximate analysis may be undertaken with the canine located mid-

way through its intended displacement: at the midspan location between the anchorage sites.

elastic Behavior in Bending fundamental, characteristic bending deformation is the rotation of one beam cross-section relative to a neighboring cross-section. The beam

The

fibers that are farthest from the neutral surface of the member experience the largest bending strains and, if the relative rotation becomes sufficiently great, the extreme fibers under tensile stress can rupture and a progressive failure of the entire member by fracture may be initiated. If

262


the beam material is metallic and ductile, however, as is true of many

orthodontic, arch-wire materials, the stretched fibers can experience substantial strain beyond their elastic limit without rupture, and thereby enable the placement of permanent bends. In plane bending of a straight beam the fiber strains have been noted to increase linearly with distance from the neutral surface (see Fig. 7-5). If the beam material obeys Hooke's law, then the fiber-stress pattern is the same as that for the bending strains with the limit of Hookean be-

havior approximately coinciding with the generation of elastic-limit stresses in the extreme fibers. Whenever the beam loading exceeds the elastic-strength level, however, some extreme fibers are strained beyond their elastic limit and, upon unloading, a portion of the relative rotation remains and the member exhibits permanent deformation (with respect to the previous, passive shape). The beam has been permanently bent. In the fabrication by the practitioner of an arch wire from an as-received, straight piece of wire, permanent bending deformation is widely distrib-

uted over at least the anterior portion of the wire as the arch form is prepared. At any cross-section in the anterior region, the permanent bend may be far from severe. On the other hand, the placement of a loop in that wire requires localized bends that take the wire material substantially beyond its elastic limit and, therefore, demands considerable ductility.

Experimental investigations of the bending behavior of structural members have shown that often beam cross-sections remain plane (do not warp) under load, even when the load produces a moderate amount of inelastic action. In other words, the strain pattern remains linear well beyond the elastic limit of the beam material and through substantial cross-sectional rotations. For metals, however, the slope of the typical stress-strain diagram decreases after the elastic limit is exceeded: hence, taking a metallic beam beyond its elastic limit results in a nonlinear bending-stress pattern such as that shown on the left in Figure 7-16. Because of the linear relationship between fiber strain and the coordinate u, inelastic behavior and stress nonlinearity begin in the extreme fibers and proceed toward the neutral surface as the load is increased beyond the elastic strength. The portion of the cross-section incorporating the fibers not strained to the elastic limit is known as the elastic core, as indicated in the figure. Note that even with inelastic behavior the antisymmetry of the bending-stress pattern remains for the round and rectangular beam cross-sections. The sketches on the left in Figure 7-16 depict the rotational deformation and bending-stress patterns of the loaded (activated) beam. As verified by bench experimentation, deactivation from this state is tanta-

mount to superimposing a reversed, totally elastic loading on the activated configuration. In other words, the resulting bending-stress pattern upon partial unloading is obtained by, in effect, subtracting the stresses of Figure 7-5 from those shown on the left in Figure 7-16. If unloading is continued to its completion, the extreme fibers reach the zero stress state before the load is totally withdrawn and subsequently

263


Elastic

core

N (5n)max

FIGURE 7-16. Bending deformation and stress patterns associated with inelastic bending: cross-sectional rotation and bending-stress pattern of the activated state (left) and permanent rotation and remaining residual bending stresses following total deactivation (right).

undergo a reversal of stress. A residual-stress pattern of the form shown

on the right in Figure 7-16 ultimately accompanies the permanent bend

induced. The pattern indicates a reversal in sense of the stresses (as load-induced) in fibers distant to the neutral surface and the incomplete unloading of fibers within the elastic core near the neutral surface. The resultant of this residual-stress distribution may be expressed as four longitudinal forces (two couples) that must exactly cancel one another in

the final reduction to yield a net internal bending couple M of zero magnitude. Typically in orthodontic therapy, the placement of first- and secondorder bends in the arch wire is followed by engagement of that wire. An activation, then, intended to be elastic, is superimposed on prior inelastic action. Because the bend placements produce the residual-bending-

stress patterns just discussed and result in additional strain energy "locked" into the wire, some alterations in structural properties have occurred and the response to activation of the bent wire will be changed somewhat in comparison to that of an otherwise identical, straight wire.

In general, the described inelastic action, upon unloading, results in increased elastic range and strength and reduced ductility. The severity of the permanent deformation, how localized or widely distributed it is, and the longitudinal position and extent of that deformation with respect to the activating load and anchorage, are significant. Of substantial importance also is the directional nature of the effects of bend placements and, therefore, the direction-sense relationships of the bends to the activation. For example, if the permanent bend is second-order, the responses during activation will differ, depending on whether the activating, superimposed bending action is in the same direction and sense, in the same direction but opposite sense, or in another (e.g., the first-

264


order) direction. Quantification of these property changes appears to be

virtually impossible, but further consideration of the phenomena is contained in the section that follows.

Orthodontic Wire Loops Although the orthodontic arch wire is often activated in bending to

create faciolingual or occiusogingival tooth movement, routine mesiodistal displacements through arch-wire deformation cannot be accomplished directly because of the extremely high longitudinal stiffness of wire. To reduce this stiffness at a specific location, a wire loop may be incorporated in the arch, effectively providing a spring of manageable flexibility between segments of the arch wire. In general, the orthodontic wire loop may be bent directly into the continuous arch wire, or a loop or spring may be fabricated from a piece of wire and activated between a pair of neighboring teeth. The "legs" of a wire loop extend occiusogingivally from the arch-form plane into the "body" of the loop, the shape of which serves to primarily characterize the loop. The loop having the simplest geometry is shown in Figure 7-17. Generally, this ioop is wholly contained in one plane, its body gingival to the line of bracket slots, and its position usually within the interbracket space between adjacent teeth. The loop shape is attained by the placement of localized bends in a straight piece of wire; hence, the wire material must exhibit substantial ductility to sustain such inelastic strains without fracture. Again, the basic rationale for incorporating loops in appliances is to gain mesiodistal flexibility, although some loops simply serve as stops or as hooks along the arch wire. A typical wire ioop is characterized by the size, shape, and material

F

FIGURE

(right).

F

7-17. A simple wire loop: passive (left) and activated by mesiodistal pulling

_____ 265


'Teardrop"

FIGURE 7-18. Orthodontic wire loops: 'teardrop," '7," "L, ' and "box," from left to right.

from which it is fabricated, its overall occiusogingival dimension, its

as-formed geometry, and whether it "opens" or "closes" when activated mesiodistally. Several loop geometries are shown in Figure 7-18, and the association of names such as "teardrop," "T," "L," and "box" is usually made by viewing the individual sketches. The loop in Figure

7-17 and three of the four loops in Figure 7-18 are termed "opening loops;" the area within the loop form tends to enlarge and many (or all) of the bends tend toward reductions in curvature as the loop is activated mesiodistally by pulling action. On the other hand, the same activation of the box loop in Figure 7-18 results in a reduction of the area within the rectangular geometry and two of the four bends tend toward increased curvature; this loop is described as a "closing loop." Several properties influence the mesiodistal stiffness of a wire loop. Figure 7-19 serves to define this stiffness in terms of load and deformation parameters and, as indicated by the diagram, below the elastic limit

a

Activation a

FIGURE 7-19. The mesiodistal-pulling activation of a loop (left) and the associated force-deformation pattern including the graphic definition of the pertinent stiffness (right).

266


the loop reflects the Hookean behavior of its material. Because the load-

ing of the ioop produces bending deformation, this stiffness is affected by the wire material (through its modulus of elasticity E) and by the size and shape of the cross-section (through the parameter I). The curvilinear length of wire within the ioop is also highly influential: the more wire present in the ioop, the lower the stiffness. Also impacting stiffness

are the number and severity of bend placements, the procedures in forming the bends and the overall loop geometry, and the relationships between direction and sense of activation and those of the bend placements in fabrication. Strictly speaking, two mesiodistal stiffnesses must be noted, inasmuch as mesiodistal activation may be pulling or pushing, the latter depicted on the left in Figure 7-20; these stiffnesses are generally unequal due to the presence of residual-sfress patterns left by fabrication of the loop. Unfortunately, the influences on stiffness are too complex to permit an analytic determination; desired quantifications must be obtained experimentally. Although the foregoing discussion pertains to mesiodistal loop stiffnesses, noteworthy in addition is the potential, multifaceted nature of loop activation and the possible involvement of as many stiffnesses as there are modes of activation. As can be seen on the right in Figure 7-17, upon mesiodistal pulling activation the permanent bends at the base of the loop tend to remain generally as bent (because the internal bending couples there from mesiodistal activation are small) and a second-order phenomenon appears. If appropriate compensation is not provided in loop formation, second-order couples arise in the presence of inadequate wire-bracket clearances. The magnitudes of these tipping couples depend heavily on the passive angulations of the loop legs with respect to one another, the associated stiffness, the amount of mesiodistal activation, and the second-order clearances. Although not inherent in the typical loop, first-order angulation can be incorporated in the leg fabrication to enable the generation of first-order-couple action upon engagement.

Q

p

P

C, C

FIGURE 7-20. Loop activations other than by mesiodistal pulling: the opening of a simple reverse-closing loop by mesiodistal pushing (left) and an occiusogingival activation of a stepped "L' loop (right).

267


Another mode of activation is introduced by fabricating a passive "step," either occiusogingivally as shown on the right in Figure 7-20 or faciolingually. The magnitude of the occiusogingival forces shown as

part of the activation force systems will depend upon the size of the step, the related stiffness, and the amount of occlusogingival activation. The loop containing a faciolingual step, or the loop formed in rectangular wire and passively exhibiting a third-order, relative angulation between leg ends, may undergo torsional as well as bending deformations

upon activation, and will be discussed further in Chapter 9. In summary, all of the individual stiffnesses are generally influenced by the wire and loop properties already mentioned. The previous section of this chapter dealt with inelastic bending and the residual-stress pattern induced with the placement of a permanent bend was noted. The typical orthodontic loop is a composite of straight and bent portions of an initially straight segment of wire. An activation of a ioop may tend to increase or decrease the curvature of a specific permanent bend, and the sense of that activation apparently has an influence on the associated stiffness related to the residual-stress pattern left by fabrication (see Fig. 7-16). In fact, in identical mesiodistal activa-

tions, the closing-loop counterpart of the opening loop of Figure 7-17 has been shown to exhibit a greater mesiodistal pulling stiffness, even though the closing ioop contains slightly more wire (Lane and Nikolai, 1980). Furthermore, the severity of the permanent bend, in terms of the as-bent curvature of the beam axis at the bend, interacting with the beam depth, may also impact stiffnesses, including those akin to firstand second-order effects. Upon activation, much of the deformation seems to occur within the straight sections of the loop while the permanent bends tend to retain their bent shapes. To enhance flexibility, at locations of bends through angles of 90° or greater within the loop, helices may be incorporated. In the typical loop the localized mesiodistal and second-order stiffnesses may be reduced by the presence of helices; apparently most influential is

the increase in curvilinear length of wire necessarily accompanying a helix. Helices placed where the ioop legs blend into the arch form can reduce the potential of a loop to tip teeth adjacent to it. Furthermore, depending on loop geometry, the presence of a helix may affect other stiffnesses as well. For example, the loop of Figure 7-17 depends somewhat for its activation on the opening of the angle between its legs. Comparing it with the loop of Figure 7-21, the incorporation of the helix in the loop body increases the mesiodistal flexibility while the helices at the base of the loop in particular reduce second-order stiffnesses and the tipping potential just previously noted. The mesiodistal elastic ranges of a ioop are influenced, like the associated stiffnesses, by properties of the wire and the loop geometry; analogous to the straight-wire bending discussion, the impact upon a range of the wire depth dimension in the plane of bending is not as great as it is upon stiffness. A particularly important influence on loQp range is the severity of the bend(s) in the body of the ioop. The free-body diagram of Figure 7-22 shows the location and relative size of the critical, internal

268


F

F

FIGURE 7-21. An opening loop containing he/ices: passive (left) and activated by mesiodistal pulling (right).

bending couple within a teardrop ioop. The maximum bending strains

induced on activation are found within a permanent bend where residual bending stresses also exist. Of concern is not only an activation possibly taking the wire material within this bend beyond the elastic limit,

Mcrut

P

FIGURE 7-22. A free-body diagram of one-ha/f of the loop of Figure 7-17 showing the location of and an expression for the critical internal bending couple.

269

Behavior of Orthodontic Wire in Bending TABLE 7-4. Typical values of stiffness and elastic range for stainless-steel wire loops subjected to mesiodistal-pulling activation*

Loop description

Wire Size (mils)

Wire

in loop

Stiffness (gm/mm)

Range (mm)

(mm)

17 x 25 21 x 25

Teardrop

14

325 525

1.7 1.8

17 x 25 21 x 25

Bulbous

15

275 430

2.0

17 x 25 21 x 25

Reverseclosing

17,5

355 610

2.3 2.5

17 x 25 21 x 25

Reverse-closing with Helix

25

245 455

2.5 2.9

1.9

From Engel, EL. (1977). Notes:

The occiusogingival height of all loops was approximately 6.5 mm. None of the loops was stress-relieved following fabrication. The above stiffness and range magnitudes are averages involving several vendors' wires.

but also that the bend placement has locally embrittled the wire mate-

rial, substantially reducing ductility within the bend. Accordingly, care should be taken both to limit the severity of bends within the body of the loop and to activate the loop to a state somewhat below its predetermined elastic range. Table 7-4 contains approximate magnitudes of stiffness and range associated with mesiodistal-pulling activation for several common loops, but it is important to remember that both properties may be influenced rather significantly by fabrication procedures and, therefore, variations in obtained property values for seemingly identical loops prepared by different practitioners should be expected.

Synopsis The discussions of this chapter have examined the concepts of structural

bending behavior and the application of the theory to orthodontic appliance members—the arch wire in particular. Although the bending problem for the actual, continuous arch wire is almost hopelessly complex, reasonable modeling can result in a manageable problem and analysis. Substantial care was taken in the development of the theory, but with some voids necessary because of mathematical procedures beyond the scope of the text; the thrust was toward understanding the parametric influences on this most common form of appliance activation occurring in orthodontic therapy. The structural response of an arch-wire beam is dependent on sev-

eral geometric and mechanical properties, most of which may be selected and controlled by the practitioner. The ke.y parameter is stiffness and, following its definition with respect to the specific bending prob-

270


under analysis, the orthodontist must make a decision as to wire material, cross-sectional size and shape, and a solid versus a twisted, layered, or braided longitudinal format. Material ductility is an additional consideration if permanent bends are to be placed prior to activation and, in this regard, a superficial examination of inelastic bending was undertaken. Consequences of superposition of elastic activation on prior inelastic behavior received attention and led to a discussion of the functions, characteristics, and behavior in bending of orthodontic wire lem

loops.

It is important to note once again that an American Dental Association specification standardizes the evaluation of the orthodontic arch wire in both elastic and inelastic bending. Property values obtained from the results of structural tests are quantified through use of engineering

bending theory. Perhaps this fact gives support to the developments undertaken in Chapter 7 that are not only pertinent but also necessary to the structural considerations of the example orthodontic mechanics discussed in Chapter 10.

References E.L.: A force-activation comparison of retraction loops used with the "018" and "022" bracket assemblies. Master's thesis, Saint Louis University,

Engel,

1977.

Lane, D.F., and Nikolai, R.J.: Effects of stress relief on the mechanical properties of orthodontic wire loops. Angle Orthod., 50:139—145, 1980.

Suggested Readings American Dental Association Specificaton No. 32 for orthodontic wires not con-

taining precious metals. J. Am. Dent. Assoc., 95:1169—1171, 1977.

Andreasen, G.F., and Barrett, RD.: An evaluation of cobalt-substituted nitinol wire in orthodontics. Am. J. Orthod., 63:462—470, 1973.

Andreasen, G.F., and Hilleman, T.B.: An evaluation of 55 cobalt substituted nitinol wire for use in orthodontics. J. Am. Dent. Assoc., 82:1373—1375, 1971.

Andreasen, G.F., and Morrow, R.E.: Laboratory and clinical analysis of nitinol wire. Am. J. Orthod., 73:142—151, 1978. Brantley, W.A.: Comments on stiffness measurements for orthodontic wires. J. Dent. Res., 55:705, 1976. Brantley, W.A., Augat, W.S., Myers, C.L., and Winders, R.V.: Bending deformation studies of orthodontic wires. J. Dent. Res., 57:609—615, 1978.

Brantley, W.A., and Myers, C.L.: Measurement of bending deformation for small diameter orthodontic wires. J. Dent. Res., 58:1696—1700, 1979.

211


Burstone, C.J.: Variable modulus orthodontics. Am. J. Orthod., 80:1—16, 1981.

Burstone, C.J.: Application of bioengineering to clinical orthodontics. in Current Orthodontic Concepts and Techniques. 2nd Ed. Edited by T.M. Graber and B.F. Swain. Philadelphia, W.B. Saunders, 1975, Chapter 3. Burstone, C.J., and Goldberg, A.J.: Maximum forces and deflections from orthodontic appliances. Am. J. Orthod., 84:95—103, 1983.

Burstone, C.J., and Goldberg, A.J.: Beta titanium: a new orthodontic alloy. Am. J. Orthod., 77:121—132, 1980.

Creekmore, T.D.: The importance of interbracket width in orthodontic tooth movement. J. Clin. Orthod., 10:530—534, 1976.

Goldberg, 1.' and Burstone, C.J.: An evaluation of beta-titanium alloys for use in orthodontic appliances. J. Dent. Res., 58:593—599, 1979.

Goldberg, A.J., Vanderby, R., and Burstone, C.J.: Reduction in the modulus of elasticity in orthodontic wires. J. Dent. Res., 56:1227—1231, 1977.

Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C. V. Mosby, 1972, Chapters 2, 3. Kusy, R.P.: On the use of nomograms to determine the elastic property ratios of orthodontic arch wires. Am. J. Orthod., 83:374—381, 1983. Kusy, R.P.: Comparison of nickel-titanium and beta titanium wire sizes to conventional orthodontic arch wire materials. Am. J. Orthod., 79:625—629, 1981.

Kusy, R.P., and Greenberg, A.R.: Effects of composition and cross-section on the elastic properties of orthodontic wires. Angle Orthod., 51 :325—342, 1981.

Kusy, R.P., and Greenberg, AR.: Comparison of elastic properties of nickel-titanium and beta titanium arch wires. Am. J. Orthod., 82:199—205, 1982.

Lopez, I., Goldberg, J., and Burstone, C.J.: Bending characteristics of nitinol wire. Am. J. Orthod., 75:569—575, 1979.

Nikolai, R.J.: On optimum orthodontic force theory as applied to canine retraction. Am. J. Orthod., 68:290—302, 1975. Popov, E.P.: Introduction to Mechanics of Solids. Englewood Cliffs, N.J., Prentice-Hall, 1968, Chapters 6, 11.

Smith, JO., and Sidebottom, 0. M.: Elementary Mechanics of Deformable Bodies. London, Macmillan, 1969, Chapters 7, 8.

Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C. V. Mosby, 1982, Chapter 4, 5, 9. Yoshikawa, D.K., Burstone, C.J., Goldberg, A.J., and Morton, J.: Flexure modulus of orthodontic stainless steel wires. J. Dent. Res.: 60:139—145, 1981.

Waters, N.E., Houston, W.J.B., and Stephen, C.D.: The characterization of arch wires for the initial alignment of irregular teeth. Am. J. Orthod., 79:373—389, 1981.

Delivery of Torque by the Orthodontic Appliance 0

0 The

force system transmitted by and through a structural or machine

member, capable of producing pure rotational displacement about a lon-

gitudinal axis, is known as torque. The resultant of a torsional force system is a couple; therefore, the terms "torque" and "couple" are sometimes used interchangeably. In a strict sense, and in the typical engineering setting, however, torque is the more correct term to describe a distributed force system carried by a "shaft" from one location to another within a structure or machine. In orthodontic terminology, torque is often associated with the angulations of long axes of teeth and pertains to the positioning of root apices with respect to crowns. Second-order torque was described in Chapter 5 as a force system used to alter a long-axis orientation in a faciolingual plane that results in mesiodistal displacement of the root tip. Activated uprighting springs exert a second-order torque, and these auxiliaries are

examined later in this chapter. Receiving principal attention herein, however, are the structural considerations of third-order torque which, with respect to the dentition, pertain to faciolingual root movement and control. Although third-order torque may be delivered to an individual dental unit to produce a wanted displacement, perhaps the most common occurrence is in the mechanics of retraction of the maxillary anterior or incisal segment. To reduce overjet, to upright procumbent anteriors, and to angulate incisors with respect to vertical requires that the appliance deliver third-order torque. Third-order torque is often part of

the force system exerted on buccal segments to alter posterior arch width, to cite another example. In typical edgewise orthodontic mechanics, third-order torque is generated when, in order to achieve bracket engagement, a rectangular wire must be twisted about its longitudinal axis during the activation

process. The third-order, passive "mismatch" between wire and bracket, as seen in a mesiodistal view, is created or possibly augmented by a permanent deformation of the wire, by the use of pretorqued brack272

273

Delivery of Torque by the Orthodontic Appliance

ets,

or by a combination of the two. To generate third-order action

within a round-wire appliance, torquing spurs or loops are bent into or affixed to the round arch wire and, as seen in a mesial or distal view, they are passively angulated with respect to the long axes of the teeth to be displaced. A round-wire torquing auxiliary, overlaying the main arch wire from canine to canine, is used when appropriate in the incisal-segment retraction mechanics of Begg therapy (Begg and Kesling, 1977). Such torquing spurs may be incorporated in a continuous arch wire as well. The generation of torque at an activation site results in the delivery of

a torsional force system to the dentition. The action-reaction counterpart of the desired torque is that exerted by the dentition on the appliance, and the wire must carry it to the anchorage site(s). The quantification of torque delivered, then, involves an analysis of the wire and, as with the

activated arch-wire beam, presents the analyst with an indeterminate problem. Regardless of the number of anchorage locations, the number of unknown torque resultants (couples) exceeds the number of available, moment-balance relationships of quasi-statics. The analysis must make use of the known or measurable activating twist(s) that are necessarily performed to gain bracket engagement. Torque as the force system

generated and twist as the form of activating deformation are interrelated in the structural response of the loaded member. The complete description of the third-order force system, paralleling the solution of the bending problem, requires the augmentation of the equations of quasi-statics by relationships incorporating torsional stiffness. Torsional action, specifically in the anterior segment, may be generated by an extraoral appliance as well as through intraoral mechanics. Intraoral torsional mechanics are covered in this chapter, but all extraoral-appliance discussions are deferred to Chapter 9. In the sections that follow, torsional theory is developed for the straight, circular shaft and the extensions and modifications of that theory to accommodate the rectangular cross-section. Next, the theory is discussed with respect to its applicability to the orthodontic arch wire. Orthodontic mechanics of anterior-segment retraction are then detailed with emphasis on the role of third-order behavior; both the edgewise and round-wire approaches are examined. Because permanent bends or twists are often placed prior to third-order activation, inelastic behavior of metallic materials in torsion is explored and, subsequently, the superposition of an elastic activation on the residual-stress patterns of clinical fabrication. Finally, loops and springs used in torquing mechanics as well as other auxiliaries having torsional aspects to their activation-deactivation behavior are considered.

Structural Theory for the Straight, Circular Shaft In engineering terminology a "shaft" is a slender member providing

mechanical means by which, through rotational motion about an axis,

274


power may be transmitted from one location to another. Power is work

done per unit time and work requires a force system acting through an accompanying displacement (see Chap. 4). The shaft displacement is a rotation about its longitudinal axis and the resultant of the force system producing or reacting to the displacement is a couple. Power is transferred to, carried along, and taken from the shaft; the principal force system existing within the member is known as torque. Transient responses occur during start-up and shut-down of the mechanical system, but while the shaft is turning at constant rotational speed, the input and output torques (couples) balance one another and the shaft experiences an unchanging, overall twist deformation. Early designs of shafts were based on observations within simple experimentation. The typical shaft had a circular cross-section associated with the relative ease of fabrication, the facility of attachment of pulleys and gears to the member, and the character of the shaft motion. A theoretical basis for the design of the circular shaft has been firmly

established through research. Force systems may be interrelated through Newton's laws. Torque is the resultant of an internal distribution of shearing stresses on a shaft cross-section and is transmitted longitudinally within the member, unchanged between locations of forcesystem input or output. The torque-twist relationship is associated with

the stress-strain behavior of the shaft material in shear and with the shaft dimensions, and is independent of the shaft speed under steadystate (constant rotational motion) conditions. Torque may also be transmitted to, from, and through a stationary shaft, and the arch wire may perform the mechanical function of such a shaft. Although the character of shaft loading differs substantially from that of a beam, analogies may be drawn between bending and torsional responses. Influential in their

structural responses are cross-sectional size and shape, longitudinal dimensions, and the mechanical properties of the beam and shaft materials. Accordingly, of interest and principal subjects of the developments to follow are torsional stiffness, range, and strength.

The essential features of the torsional behavior of a shaft may be obtained from a rather simple example. Figure 8-1 illustrates a solid, circular member of radius R and deformable length L. The member is fixed against all movement at its left end and any bending deformations

R

FIGURE 8-1. Side view (left) and end view (right) of a straight, solid, circular shaft.

275


and stresses (caused by shaft weight, for example) are assumed negligi-

ble (or such effects can be superimposed on the torsion analysis). The line MN is on the outer surface and, in the passive state, is parallel to the axis of circular symmetry of the member; point 0 is on the axis of geometric symmetry. The end cross-section is a right cross-section; the angle MNO is a right (900) angle. Figure 8-2 depicts the deformation of the shaft, externally visable, following the application of torsional loading to its "free" end. The resultant of the load is a couple T, which represents torque and distinguishes this force system from an applied bending couple. Because the shaft is not rigid and only its fixed end is restrained from movement, a rotational deformation is distributed along the length of the member as characterized externally by (1) the distorted shape of the generator (labeled MN' as activated); (2) the unaltered, straight-line configuration of the shaft axis; and (3) the rotation of the reference, free-end radius to the as-activated position ON'. The radii remaining straight lines and the shaft axis undeformed by the loading are a combined result of the characteristics of the load and the symmetry of the member; as the magnitude of the applied couple is increased, indefinitely until fracture begins,

the pattern of deformation remains that shown in the figure. The amount of "twist" of the free end (with respect to the fixed end) of the shaft, the angle NON' in Figure 8-2, is, however, directly related to the magnitude of the couple loading. The evidence of longitudinal shear resulting from torsional activation is obtained by imagining the unwrapping of the outer surface, "skin," of the deformed shaft. The flattened skin is rectangular, having dimensions L by the shaft circumference, and is pictured in Figure 8-3. Corresponding to the change in inclination of the generator, any small area of the skin that is passively rectangular deforms into the shape of a parallelogram, as shown, upon activation of

M

Torsional couple T

L

FIGURE 8-2. The shaft of Figure 8-1 subjected to a torsional couple applied at its end.

free"

276


Shaft circumfererice

—

L

FIGURE 8-3. The outer surface of the shaft of Figure 8-2; the shaft "skin" has been unwrapped and flattened into a plane, but not otherwise further deformed.

the shaft; the demise of the right angles of the rectangle is the indication

of shearing strain. In the example under discussion the load is the torque input at the right end of the shaft; the torque is transmitted the length of the member and into the left-end support. A free-body diagram of the entire shaft is shown on the left in Figure 8-4; only the single couple loads the member and a moment balance for the static shaft yields a resultant, responsive couple of the same magnitude T at the fixed end. An analysis of any Responsive torque =T

Internal torque

=T

I

I

I—

Applied torque

T

I

T e

FIGURE 8-4. Free-body diagrams of the shaft of Figure 8-2; the entire shaft of length L (left) and a portion of the shaft of arbitrary length t (right) measured from the free end.

277


portion of the shaft, such as that shown on the right in Figure 8-4,

indicates that the resultant of the internal force system on every right section is a torsional couple equal to T. Further indication that the internal torque is the same on all right cross-sections between the shaft ends is the constant slope of the deformed generator MN' in Figure 8-3. (The internal bending couple in a beam changes from one right cross-section to another, when the loading includes transverse forces, due to the moment-arm changes; the shaft, however, is directly analogous to the beam in pure bending.) The deformed generator has an apparent curvilinear shape as viewed in Figure 8-2, but, in the general torsion problem,

it is made up of a series of straight lines in the "plane skin" view with sites of slope changes corresponding to locations of external torsional couples.

In between torque input-output locations, relative rotational displacements of cross-sections exist; the amount of relative twist is directly proportional to the longitudinal distance between cross-sections. In the example shaft, the absolute rotation of the left-end cross-section is zero as constrained; the relative rotations are simply additive along the shaft length (because no intermediate external couples are present) and sum to the rotation of the free-end cross-section. This free-end displacement, indicated by the angle NON' in Figure 8-2, is the twist produced by the input couple T; it is symbolized by the Greek letter 4 (phi) and is analogous to the bending-deformation angle 0. Note that torsional loading of the straight, circular shaft warps no cross-section of the member. The character of unit deformation (strain) on the surface of the shaft is determined by comparing the active and passive configurations of a skin element. Consider a small rectangle, with its sides parallel and per-

pendicular to the shaft axis, as the passive element. Torsional loading displaces the element circumferentially and distorts it with negligible changes in the side lengths. As deformed, the element takes the shape of a parallelogram, indicative of the presence of shearing strain. The passive and activated elements are shown in Figure 8-3 and, in an enlarged view, in Figure 8-5. As noted in Figure 8-5, the angular distortion of the element (the angle between the passive and active configurations of the generator at a specific, cross-sectional location) is the localized, surface shearing strain. Shearing strain was initially defined in Chapter 3; Figures 8-5 and 3-14 are similar, although exhibiting rotational and straight (linear) shear, respectively. For a shaft experiencing only torsional loading, all deformed generators are similar, indicating axial symmetry in the surface shearing-strain pattern. For the example problem under study, the surface shearing strains are everywhere equal and are by the magnitude (in radians) of the angle NMN' in Figures 8-3 and 8-5.

For any shaft cross-section, the axial symmetry results in internal shearing strains varying only with the radial coordinate. The circumferential shearing strains for any specific radial coordinate r are all equal

and the maximum shearing strains exist at the shaft surface (where

278


I

I

I

I

shearing strain (es)max

FIGURE 8-5. Small element of the outer surface of a straight, circular shaft shown enlarged (from Figure 8-3) and exaggerated in passive (dashed) and activated (solid) positions and shapes.

r = R). Because torsional loading does not deform the shaft axis, where r = 0 no shearing strains exist. Moreover, for the circular shaft, since

radii from the shaft axis remain straight with activation (ON becomes ON' in Figure 8-2), no radial shearing strains exist and the magnitude of circumferential shearing strain at a point is directly proportional to the coordinate r. Shearing strains at a point generally vary with the passive orientation (edge directions) of the rectangular element there. Shearing strains are always accompanied by shearing stresses. Shearing stresses of equal magnitude exist at a point at right angles to one another in pairs, and the stresses of interest in the arch-wire torsion problem are longitudinal and circumferential; note the element orientation in Figure 8-5. For a crystalline material, paired shearing stress and strain are proportional to one another up to the elastic limit in shear. A stress-strain diagram in shear for a ductile, metallic material is depicted in Figure 8-6; this is a repetition of one plot in Figure 3-19. The material stiffness in shear, also known as the modulus of rigidity, is the slope of the linear portion of the curve and is usually symbolized by G. The stiffness G and the modulus of elasticity in tension or compression are interrelated; for metals and alloys G is typically about O.4E. Also for ductile metallics, the elasticlimit stress in shear is approximately one-half of its counterpart in tension.

279


(1)

0)

0,

a)

U)

0) C

0) ci)

U)

Elastic limit in shear

C

Shearing Strain

FIGURE 8-6. Characteristic plot of stress versus strain in shear for a metallic material.

The sketch in Figure 8-7 is of one side of a three-dimensional, internal, material element of the shaft showing shearing stresses acting on areas appearing in edge view from this perspective, which is radial and the same as that of Figure 8-5. (The reference line shown is parallel to the shaft axis.) Because no normal stresses exist on the element, it is said to be subjected to pure shear. (Tensile and compressive stresses do exist in

the shaft, but on elements having orientations differing from that of Figure 8-7.) When the torsional loading of the shaft nowhere causes stresses exceeding the elastic limit, the shearing strains and correspond-

ing shearing stresses are proportional to one another at every point within the shaft. Linear shearing action may be compared to that of a pair of scissors. The shearing strength of a shaft tends to resist the attempt of torsional loading to rotate each cross-section with respect to its neighbor. Another example of rotational shearing-stress development is within the adhesive attaching the bonded bracket-pad to the tooth when a second-order couple is transmitted from arch wire to the bracket and into the crown. Figure 8-8 is a view along the shaft axis into an internal cross-section

through which a torque T is transmitted. Circumferential shearing stresses are shown on the exposed areas of typical, diametrically opposed shaft elements. (A corresponding elemental area appears in edge view in Figure 8-7; the edge labeled a-a is positioned perpendicular to

280


parallel to the shaft axis

ss

*

/*

a

FIGURE 8-7. Equal circumferential and longitudinal shearing stresses exerted on an internal element of the activated shaft.

the shaft axis.) Each elemental area may be termed representing a very small portion of the total cross-sectional area A. The resultant shearing force on the small area M has the direction and sense of the shearing stress and is equal to the product (M); this elemental force may be symbolized as 6F5. The elemental forces exerted on the two elemental areas shown are of the same magnitude and direction, but are of opposite sense, due to area location with respect to point 0 (the end view of the shaft axis). These two forces form a couple of magnitude The sum of all such couples over the entire cross-section is the resultant internal torque T.

281

Deli ve,y of Torque by the Orthodontic Appliance

FIGURE 8-8. An end view into an internal cross-section of the shaft of Figure 8-2 showing circumferential shearing stresses on two elemental areas.

The torque transmitted through the right cross-section of Figure 8-8 may be initially expressed as T

sum of

(8-1)

Refinement of this formula requires knowledge of the manner in which the shearing stress varies through the area A. Figure 8-6 shows a singlevalued relationship between stress and strain and, because this shearing strain varies only with radial location on a specific cross-section, shearing stresses are likewise independent of circumferential position of the elemental area. For totally elastic behavior, stress and strain are proportional to one another and, therefore, the variation with r of the stress as well as the strain exhibits the pattern of Figure 8-9. Mathematically expressed, the pattern indicates = (ss)max:R

or

=

r

(8-2)

282


(Ss)max

Resultant torque T

FIGuRE 8-9. Linear relationship between radial position and magnitude of oircumferential shearing stress on the cross-seofion of Figure 8-8 for totally elastic behavior.

Substituting Equation 8-2 into Equation 8-1 yields T=

(5s)max

[sum of (r2&A)]

(8-3)

resulting in the summation now associated solely with the cross-sectional area. The sum is the second moment of area with respect to the shaft axis and, for the solid circular cross-section, is equal to (8-4)

This geometric parameter is known as the polar second moment of area, is often symbolized by the letter I, and is analogous to the second mo-

ment of area I, which appears in the bending analyses of Chapter 7. 21.) In terms of the maximum shearing stress, occurring at the outside surface of the

(Referring to Figure 7-10, for circular cross-sections, J =

shaft, and for totally elastic material behavior, Equation 8-1 can now be written as T=

or

T=

(ss)m;xITR3

(8-5)

283


An analogous expression, explicit in the internal bending couple M, was

generated in Chapter 7; the two equations are similar in form. The structural response of a shaft in torsion is characterized by plots of internal torque versus the relative rotations of cross-sections at the ends of shaft segments through which torque is transmitted unchanged. For the simple shaft example of Figure 8-2 the internal torque is constant

throughout the entire length L (and equals the applied torque) and the left-end cross-section cannot move or deform. Accordingly, the appropriate plot, given in Figure 8-10, depicts the applied torque in terms of the free-end twist 4). The slope of the linear portion of the graph is a torsional stiffness and a formula for this parameter may be readily derived. Needed are the stress-strain relationship for the shaft material in shear below its elastic limit from Figure 8-6 = G(e5)

(8-6)

and Equations 8-2 and 8-5 previously developed. Referring to Figures 8-2, 8-3, and 8-5, and noting that the arc = length NN' is small in comparison to L, NN' can be equated not only to R4), but with negligible error also to the product of L and the angle NMN; this yields the relationship R4) = L =

(es)max

(8-7)

ci,

-C) C)

a a

Tei)

Torsional stiffness

Angle of Twist 4'

FIGURE 8-10. Plot of activating couple versus free-end twist angle for the shaft of Figure 8-2.

284


The formula sought, the ratio of torque to twist, obtained through elimi-

nation of shearing stress and strain from the equations noted, is

TirR4GJG 2L

L

88 (-)

Torsional stiffness is seen to depend directly upon the shaft radius and

the modulus of rigidity of the material and inversely upon a characteristic length. Again, note the similarity between this formula and its counterpart in the bending developments of the previous chapter. The coordinates of the elastic-limit point of the plot in Figure 8-10 are critical twist angle and the critical torque. In general, the critical crosssection of an activated shaft is that where the internal torque is a maximum and, typically, a critical shaft segment exists. In the example shaft of Figure 8-2 all cross-sections are critical and the critical segment is the entire shaft. If a continuous arch wire is delivering third-order action (torque) to the incisors, the critical "shaft" segments are generally between the canines and the lateral incisors. Under critical conditions, the maximum shearing strains in the shaft are the elastic-limit values for the shaft material. The critical twist as described, then, is the elastic range

for the shaft and, from Equations 8-6 and 8-7, its formula may be derived: —

(es)eiL

—

(55)eiL

1?

—

RG

(89)

The maximum elastic twist of the critical segment is seen to depend on

mechanical properties of the shaft material, the shaft radius, and the length of the critical segment. Once again, the analogy may be drawn between the bending and torsion analyses; similar parameters of the structural member exert like influences on the elastic range. The critical torque is that magnitude of internal torque corresponding to the generation of elastic-limit stresses in shear at the shaft surface of the critical segment. Through a moment balance, this level of internal torque may be related to the system of torsional couples applied to the shaft. The critical torque may be directly expressed, then, by substituting the elastic-limit value for the maximum shearing stress in Equation 8-5: —

(ss)eil

—

R

Tcrit

(5s)eIITR3 —

2

(8-10)

In the example problem, because the entire shaft is the critical segment and the internal torque is everywhere equal to the applied couple at the free end of the shaft, the value obtained from Equation 8-10 is the elastic strength of the shaft in torsion.

285


Extension of Theory to Shafts Having Rectangular Cross-Sections Although perhaps not of direct interest to many edgewise practitioners,

the foregoing development of torsional theory for circular shafts was undertaken for three reasons:

1. The axial symmetry, not only in geometry but also in loading and structural response, leads to the least complex formulas for torsional stiffness, range, and strength. 2. This theory provides a basis from which to generate corresponding relationships for the shaft having a rectangular cross-section.

3. Third-order force systems cannot be directly transmitted between round wire and bracket, but can be delivered from round wire to crown by means of torquing spurs and, therefore, circular-shaft theory has application to specific orthodontic appliances. In extending the theory to admit rectangular cross-sections, contributions from the principal influencing parameters—shaft material, length dimension, and cross-sectional size—might be expected to be similar to those exhibited in the circular-shaft formulas. Material properties (elastic-limit strain and stress and modulus of rigidity), when appearing in the equations, each did so to the first power (direct or indirect proportionality); the shaft radius or diameter, however, contributed in a nonlinear manner in two of the three formulas. In this chapter the dimensions of the rectangular cross-section are symbolized by b and c where, by definition, c is less than or at most equal to b. With bending deformation, the response of the rectangular wire depends on the direction of activation; in torsion no such interdependence exists, but one of the two cross-sectional dimensions of the rectangular shaft will be seen to be a more dominant influence than the other. The first analyses of rectangular shafts in torsion were carried out experimentally and by analogy, and were deemed reasonably accurate only when the ratio of b to c was large. In time an exact mathematical solution was obtained, general in terms of b-to-c ratio but highly complex in form. A third approach was that of empirical adaptation, be-

ginning with the form taken from the circular-shaft derivations and using experimentation to complete the formulas. The results of the ex-

perimental studies enabled the clarification of how the individual dimensions b and c should appear in the equations and the quantifications of dimensionless coefficients included in the formulas to generalize them.

The empirically determined formulas for stiffness, range, and strength for the straight shaft having a rectangular cross-section, analogous to Equations 8-8 through 8-10, are as follows:

286


T

=

f3bc3G Tcrit

aL(es)ei —

abc2(Ss)ei

aL(ss)ei /3cG

The coefficients alpha (a) and beta (/3) have values that are dependent on the ratio of b to c; the relationships are presented graphically in Figure

8-11 and representative coefficient magnitudes are given in Table 8-LBoth coefficients are seen to increase monotonically, and, as the b-to-c ratio becomes large, they individually approach as an asymptote the value of one-third. A cursory examination of Equations 8-11 suggests that square (b = c) shafts are stronger and exhibit higher torsional stiffness than "ribbon" shafts (b >>c) having the same cross-sectional area. The cross-sectional stiffness parameter, analogous to I for the circular shaft, is seen to be proportional to b(c3). In general, the smaller of the two cross-sectional dimensions, c, seems to be the more influential, but the contribution of the b-to-c ratio also has some impact. The materialand length-parameter influences are assumed, by the chosen forms of the empirical expressions, to be the same as for the circular shaft; the determination that these parameters do not impact the values of a or /3, for a specific ratio of b to c, ensures the validity of that assumption. Also note that the individual influences of the two cross-sectional dimensions depend upon the plane of bending when the member is subjected to transverse (flexural) loading; in torsion, the only load "direction" is longitudinal and, therefore, the cross-sectional dimensions of the shaft affect a specific dependent parameter (e.g., elastic range) in a singular manner. /3

0.33

0.30

a

lic

0.20 13

b

0.10

0

0

1.0

2.0

3.0

6.0

Ratio of b to c FIGURE 8-11. Plots of the coefficients and /3 as functions of the ratio of crosssectional dimensions of the rectangular shaft.

287

Delivery of Torque by the Orthodontic Appliance TABLE 8-1. Coefficient values for specific ratios of cross-sectional dimensions for the rectangular shaft

b/c

a

f3

1.0 1.5

0.208

0.141

0.231

2.0 3.0 6.0

0.246 0.267 0.299

0.196 0.229 0.263 0.299

The complete mathematical solution to the rectangular shaft problem

provides, in addition to expressions for stiffness, range, and strength within the elastic limit of the shaft material, the strain and stress patterns throughout the member. Several points of interest arise from this theoretical development and from experimental observations. Associated with the absence of axial symmetry in the geometry are the appear-

ance of warping of the passively plane cross-section under torsional loading and a complex shearing-strain pattern on the noncircular crosssection. A portion of the strain pattern is exhibited in Figure 8-12; for

linear elastic behavior the shearing-stress pattern is identical to

it.

b

Resultant torque

FIGURE 8-12. Typical cross-section of a rectangular shaft. Characteristic distributions of shearing strains along major and minor axes of geometric symmetry and a diagonal associated with torsional loading are antis ymmetric.

288


Strains are seen to be zero at the center of the cross-section and at each

of the four corners; the corners remain right angles when the shaft is loaded. The shearing strains change in magnitude nonlinearly from the center (axis) point outward; the patterns are antisymmetric along any line passing through the center point and the cross-section boundary. Shown in the figure are the strain patterns along the axes of symmetry in

the plane of the cross-section and a diagonal through a corner. The maximum values of shearing strain on the cross-section occur at the outer-surface points that are along one axis of symmetry; these are the closest surface points to the shaft axis.

App'ication of Shaft Theory to the Orthodontic Arch Wire analysis of the behavior and effects of an orthodontic arch wire in the presence of third-order activation can be a formidable bioengineering problem. To the foregoing theoretical developments and applicable quasi-static relationships must often be added complications imposed by the existence of several input and output torques and the curvature of the arch-wire "shaft." When the entire, continuous arch wire is involved in third-order mechanics, use of the model developed in Chapter 6 can be both reasonable and prudent. Combining the dental units into segments reduces the number of force-system components to the principal contributions. In general, the pure torque input to a curved shaft, as it is transmitted longitudinally through the member, is modified gradually and a bending component appears. Considering the block-U-shaped wire model, a third-order activation in the anterior section, for example, provokes a bending response in the posterior sections, arising from the 900 change in longitudinal orientation between anterior and posterior portions of the model arch. With the assistance of the arch-wire model, the following problems The

of interest may be examined: (1) the torquing of one tooth with a straight-shaft transmission of load to anchorage; (2) unilateral or bilat-

eral third-order activation of the terminal molar(s) or posterior segment(s) with, possibly, some response located in the anterior segment; and (3) perhaps of greatest interest, the torquing of four incisors or six anterior teeth as a segment, with midsagittal-plane symmetry present and responses in the posterior segments. Isolation of the Activating Torque A third-order couple is almost never exerted on a tooth in the total absence of other force-system components. This torque can be effectively transmitted to the crown by a facial or lingual arch or, in some instances, by a transpalatal bar. Pure third-order loading produces a center of rotation coincident with the center of resistance and, therefore, a crown-displacement potential toward the facial or lingual, depending on the sense

289


of the torque. If a faciolingual driving force is not part of the initial

activation, then the combination of appliance-crown contact and bending stiffness of the appliance will result in the appearance of a faciolingual force tending to restrain the crown. Furthermore, an occlusogingival force may also be present, resulting from a "secondary response" to be described in the following subsection. For purposes of analysis, however, it is both possible and practical to examine the effects of thirdorder, active torque upon the arch wire and any existing anchorage, in the absence of other contributions to the force system at the activation site(s), and subsequently superimpose the influences of the remaining components of the total force system. The couple transmitted from the appliance, reflected in two-point crown contact and a resulting pair of forces in the mesiodistal plane of the tooth, is associated with structural characteristics of the appliance and the amount of third-order activation with respect to passive configu-

rations. In conventional edgewise mechanics, permanent localized twists of the rectangular wire, placed prior to activation, create the potential for third-order action. Recently, pretorqued brackets have become widely available. Third-order, anterior torque may also be activated extraorally, as noted in Chapter 9. Interbracket twists or brackets pretorqued at different angles may be used to influence the distribution of torque within a dental segment. As noted earlier, Begg therapy uses a somewhat different torquing format to be contrasted, subsequently,

with that of the twisted, rectangular wire; once again, however, the couple loading is produced through an angular deformation. To the point, regardless of the approach or treatment philosophy, the creation of third-order torque occurs as a result of a particular form or component of activating deformation; hence, it is possible, and also contributory to

the relative ease of overall analysis, to isolate torsional action and its effects from the total appliance load.

Responses of the Appliance and the Dentition to Torsional Activation The

third-order activation of an orthodontic appliance results in

action-reaction counterpart couples exerted on the dentition and appliance at the site(s) of the activation. The load on the appliance is transmitted through it to the support (anchorage) location(s). The anchorage responses are described in format through application of the equations of quasi-statics to the appliance. A single moment balance is initially employed to relate load and response, but individual forces as well as couples may appear with couple loading, due to wire curvature, and necessitate the added use of force balances. In this regard, separate considerations of primary and secondary responses are undertaken here, with primary responses examined first. An active, third-order couple may be generated against an individual tooth within the dental arch. The reaction is a single torsional couple exerted on the appliance, and this torque is transmitted to anchorage, ordinarily through an arch wire. If the activation site is a terminal molar,

____ 290


T=T1+T2

Responsive torque

(

Applied

1torque T

Responsive torque

—

T1

€1

€2

FIGURE 8-13. A straight shaft restrained against rotation at both ends and subjected to torsional loading at an intermediate cross-section.

or if a segmented arch wire is engaged, a single anchorage site is possible (although several adjacent teeth may contribute). The basic, quasistatic analysis through a moment balance simply requires that the net activation and anchorage torques be equal in magnitude but of opposite

sense (assuming negligible wire curvature). Figure 8-2 is applicable here;

the fixed end represents the anchorage. Alternatively, the anchorage may be split or divided with the torsional load partitioned and carried by the wire to support sites on either side of the activation location. Such a

straight shaft is depicted in Figure 8-13 and the given equation is the result of the moment balance. If the active torque is located midway between the anchorage sites, the responsive torques are equal if no variation in geometric or material characteristics exists from one end of the shaft to the other. Otherwise, twist-deformation equations must be used to interrelate the responsive torques. The primary response to an incisal or anterior segment activated in torsion is divided between the posterior segments when a continuous arch wire carries the reactive load distally. Often the entire force system as well as the geometry exhibits symmetry with respect to the midsagittal plane. The sketch on the left in Figure 8-14 shows the right half of the activated arch-wire model in buccal view and the essence of the force

F6

C6 F3

F3 = F5 + F6

FIGURE 8-14. Free-body diagrams of the buccal view of an arch-wire model subjected to anterior, torsional loading. Posterior-segment response shown in its resultant form (left) and as distributed among dental units (right).

291

Deliver,' of Torque by the Orthodontic Appliance

system exerted on it: a third-order couple against the half anterior seg-

ment and an equal, but opposite, second-order couple from the posterior segment. On the right in Figure 8-14, the posterior response is shown in some detail, including occlusogingival forces, which must obey an oc-

clusogingival force balance, and diminishing second-order responses from individual dental units within the posterior segment. The distribution of torque within the anterior segment is considered subsequently. Another example of third-order mechanics exhibiting midsagittalplane symmetry is that of bilateral torquing of the terminal molars or the buccal segments. Figure 8-15 illustrates transverse views of two activated appliances from a posterior perspective. On the left is the archwire model, sketched as if the anterior teeth are not engaged. The bilateral torsional activation results in bending deformation of the anterior portion of the arch, exaggerated in the absence of anchorage response there. On the right is shown a transpalatal bar engaging the molars, contacting the lingual crown surfaces. If the bar is very stiff, under bilateral torque no response of consequence will exist between it and the palatal vault. In both sketches the moment balance is inherent in the activation; the couples on the left and right sides of the arch are equal in magnitude, but opposite in sense to one another. As noted in Chapter 7, when the responsive force system in bending is distributed over several individual dental units, that distribution is nonuniform with the tooth or teeth nearest to the activation site providing the largest portion(s) of the response. The same is true in torsion. For example, if the first molar on the left side is torqued, the second premolar and, if present, the second molar are the principal anchorage units. If the buccal segment on the left side is alone subject to third-order

activation, and a continuous wire engages all three segments, the response is distributed principally in the anterior segment, and the intensity of the distribution decreases with curvilinear distance along the wire from the activation site. The free-body diagram of the model arch wire from a posterior perspective is shown in Figure 8-16. Torque in the buc-

FIGURE 8-15. Bilateral torsional activation of the buccal segments posteriorly viewed; reactions against an arch wire (left) and a transpalatal bar (right).

292


F1

F1,

c2(

FIGURE 8-16. Free-body diagram from a posterior perspective of a continuous arch wire subjected to unilateral torquing associated with activation of the left buccal segment.

segment is transformed into bending potential in the anterior region, resulting in occlusogingival force responses there and, in the absence of substantial bracket-wire clearances, second-order couples. If any internal force is carried beyond the anterior segment, the remaining bending potential becomes torsional, with further change in longitudinal direction, and is delivered to the right-side posterior segment. On the left in Figure 8-17 is a free-body diagram of the transpalatal bar with right-side, unilateral activation. With this sense of the applied torque, the response may be partly adjacent to the activation site and exerted by the palate in opposing potential rotation of the bar. On the other hand, because only pressure (no tension but, perhaps, some small amount of frictional shearing force) can be generated between bar and cal

=

FIGURE 8-17. Unilateral, right-side couple loading of a transpalatal bar and the response as viewed from the posterior: clockwise activation (left) and counterclockwise activation (right).

293


palate, reversing the sense of the activating couple results in a different

response format, shown on the right in Figure 847. Two forms of secondary response exist as the result of third-order activation of an incisal or anterior segment. From a buccal perspective the arch-wire model resembles a beam, cantilevered from the posterior segments. Figure 8-18 (left) depicts the activated configuration of such a

beam, loaded by a couple at its "free end." The sketch represents the torqued arch wire well except the deformed shape suggests that no constraint is present to prevent occlusogingival displacement of the anterior segment. Such constraint is, in fact, provided by the anterior brackets. Accordingly, consider the arch wire engaged in the posterior segments and, before undergoing third-order activation, the anterior segment is

aligned occlusogingivally with the bracket slots. Accompanying the torque from the twist to gain bracket engagement is an anterior intrusive potential. The complete free-body diagram from a buccal perspective is

shown in the center of Figure 8-18, together with the action-reaction counterpart force system against the maxillary anterior crowns in the application of lingual root torque on the right. Note that changing the sense of the anterior torque reverses all aspects of the response shown in the figure.

Activation of the anterior torque provokes another secondary response which must be viewed from an occlusal perspective. Suggested on the left in Figure 8-19, resulting from the combination of loading and wire curvature, is a tendency for the posterior sections of the arch wire to assume buccally displaced positions. The associated, free-body diagram from an occlusal perspective is shown on the right in the figure; the action-reaction counterparts of the transverse forces induced tend to widen the dental arch in the posterior in the absence of arch-wire constriction before activation. The same arch-widening potential is present for either sense of the third-order action in the anterior portion of the wire. Secondary responses exist in another example previously examined, that of bilateral activation of torsional couples in the posterior segments, if the arch wire also engages the anterior segment. The twist activations produce bending potential in the anterior portion of the wire as noted previously in Figure 8-15, left. Resistance to immediate achievement of that anterior deformation is provided by bracket-wire engagement. In this example, two activation sites must exist and the internal force sys-

segment support

cp

Ta

FIGURE 8-18. Buccal-view diagrams of the arch wire and anterior teeth with third-order, anterior activation. Vertical displacement potential associated with the torquing action (left) and a free-body diagram (center) together with crown loading (right).

294


Fpe

Tar Ta. Tar

FIGURE 8-19. Occlusal-view diagrams of the arch wire associated with anterior, torsional loading: the posterior, buccal-displacement potential (left) and the free-body diagram assuming no initial, transverse, arch constriction (right).

tems resulting from third-order loading on both sides of the arch are carried mesiafly by the arch wire toward the midline. A free-body diagram of the wire, viewed from the posterior, is illustrated on the left in Figure 8-20. The canines in this example are assumed to be within the posterior segments. Occiusogingival, responsive force systems are exerted by the incisal pairs and, to balance the system, forces must also be induced in the buccal segments. The force system exerted against one

maxillary incisal pair is shown on the right in Figure 8-20. Note that changing the senses of the posterior-segment torques reverses the pattern of the bending potential in the anterior section of the wire and the senses of the secondary responses.

Tpr

FIGURE 8-20. Posterior views of bilateral, buccal-segment torquing with anteriorsegment appliance engagement: free-body diagram of the arch wire (left) and the force system transmitted to an incisor pair (right).

295

Deliveiy of Torque by the Orthodontic Appliance

Structural influences on Active and Responsive Force Systems Shaft stiffness is dependent on material, cross-sectional shape and size, and longitudinal dimension. Examined first in this subsection are cantilevered wire "shafts" within the context of the previously introduced arch-wire model. Figure 8-21 shows schematically a rectangularwire segment subjected to an activating torque near one end, anchored

at the other end, and having an effective length L. In conventional, edgewise mechanics, the potential for torsional action is achieved by placing permanent, third-order twists adjacent to the anchorage sites and/or through use of "pretorqued" brackets (having slots angulated with respect to the passive wire) at the activation sites. Superficially, perhaps the simplest problem is the torquing of a terminal molar with anchorage provided principally by the remainder of the posterior segment. The initial value of the molar couple is the product of the activating twist (within the elastic range) and the torsional stiffness of the wire segment. In estimating this stiffness, the nonrigidity of the anchorage unit must be considered; in effect, the real characteristic length L of the straight segment is greater than the interbracket distance between the molar and the adjacent tooth. Note in addition that distribution of the responsive couple becomes an important consideration when the active and resultant responsive torques are equal in magnitude, and the tooth adjacent to the molar is smaller in root-surface area. To approach a uniform distribution of the response among the sev-

eral, available, posterior teeth (in the absence of discrete, permanent twists between pairs), the potential for the wire between the individual

Anchorage

L

Applied torque T

FIGURE 8-21. The cantilevered, rectangular-wire shaft modeL

296


units of the anchorage to twist must be minimized. Therefore, within

the anchoring segment the torsional stiffness of the engaged segment of wire must be as large as possible. Furthermore, an attempt to isolate or contain the anchorage, such as not to involve the anterior teeth, requires that a portion of wire between the posterior and anterior teeth exhibit high torsional and bending flexibility. (Termination of the activated wire

segment between the anterior and posterior dental segments is one obvious way of isolating the anterior teeth from any effects of molar torquing.) In the presence of midsagittal-plane symmetry, the analysis of incisal- or anterior-segment torquing may be accomplished by considering

either side of the arch. When the wire is "cut" at the midline, it becomes, in effect, a pair of mirror-image cantilevers. Assuming for this discussion that the four incisors are torqued as a unit, each cantilever is supported by a posterior segment with a canine at the mesial extent of the anchorage. The permanent twist in the cantilevered wire segment (in the absence of pretorqued brackets) is placed immediately mesial to the canine bracket. The product of torsional stiffness of the cantilever and the activating twist yields the third-order couple delivered initially to the incisal pair. In essence, the torque is transmitted mesially from anchorage; hence, a portion of the torque is first taken by the lateral incisor and the remainder is delivered to the central incisor. Unless an appropriate twist is placed in the wire between the two incisors, or an equivalent arrangement is attained through use of pretorqued brackets, the lateral incisor will generally receive the larger portion of the resultant couple at activation. (Likewise, in the distribution of the responsive force system, the canine will receive the greatest portion of that resultant at activation.) In fact, because of the flexibility of the periodontal ligament, the first-phase movement of the lateral incisor prevents its acceptance of the entire activation and permits some third-order action to reach the central incisor. As further third-order displacement of the lateral incisor occurs with time, the apportionment of the resultant couple within the pair changes and the central incisor receives an increasing percentage of the total torque. Because the root-surface area of the canine is generally greater than that of the lateral incisor, the distribution of the responsive force system within the posterior segment does not raise the level of concern indicated in the previous example. As a result, the decision as to choices of wire and arch configuration may focus more directly on the requirements at the activation site. Although providing an interesting analysis problem, the effective application of third-order action to a posterior segment is clinically feasible only under special conditions. A first difficulty is that of properly distributing the active torque over a segment including three to five teeth, which demands locally high appliance stiffness or differential twists between the teeth of the segment. Second, transmission of the resultant couple through a continuous arch wire results in the anterior segment becoming the first line of support (anchorage). This couple

297


must have a sizable magnitude if wanted displacements are to occur,

and the root-surface area of the anterior segment will be substantially deficient to prevent generally undesired movement there. If bilateral action is wanted, only the dental units to undergo displacement should engage the appliance; no anchorage potential is needed and, ideally, the anterior segment should not be involved. If unilateral action is needed, then progressive torquing (one tooth at a time) should be the approach to keep the resultant-couple magnitude small throughout the mechanics and some, if not all, of the responsive force system should be transmitted other than to the anterior segment (e.g., to the palatal vault if in the maxillary arch or by some means directly to the other posterior segment in the same arch). In the preceding discussion involving cantilevered shafts, the potential problem of insufficient anchorage was noted. The availability of support, both mesial and distal to the activation site, lessens the tendency

for undesired anchorage displacement. (Two response couples, their magnitudes summing to that of the activation-site couple, are ordinarily associated with a larger, total root-surface area than that of the single

anchorage site.) Consider, for example, the torquing of a first molar using a rectangular-wire segment engaging the first and second premo-

lars and the first and second molars. To accomplish the task, in the absence of pretorqued brackets, localized permanent twists are placed immediately mesial to the second molar and distal to the second premolar brackets. These twists are generally made equal in magnitude, such that the wire engaging only the divided anchorage sites would be passive (Fig. 8-22, above). If the twists are unequal, engagement only into

anchorage produces equal but opposite torques against the second molar and the premolar pair (Fig. 8-22, below). Activation at the molar, through elastic twisting to permit bracket engagement there, superimposes a couple load on the molar and response couples at the anchorage sites.

If a uniform, straight, wire segment is the principal appliance element, the partitioning of the molar load for transmission to the two anchorage sites depends on the characteristic, shaft-segment lengths. Because of the division of the mesial response between the premolars, resulting from the flexibility of the periodontal ligament, the effective distance from activation site to mesial support is greater than that to the distal support. Increasing the characteristic length lowers the stiffness; hence, the second molar absorbs more of the activating couple than does the premolar pair. (Since the second premolar, being adjacent to the first molar and having a smaller root surface area compared to it, receives

more than half of the total mesial response torque, the second molar taking the larger portion of the total response may be advantageous.) The division of torque between mesial and distal anchorage sites for this example problem is depicted on the left in Figure 8-23 and corresponds to the pretwisting of Figure 8-22, above. To illustrate the potential influence available to the clinician, as to the partitioning of torque in a di-

298


Equal

permanent twists

Unequal permanent twists

FIGURE 8-22. Permanent twists placed in a rectangular-wire segment prior to third-order activation in the presence of split anchorage: equal twists (above) and unequal twists (below).

vided-anchorage arrangement, the superposition shown on the right in Figure 8-23 reflects the pretwist geometry of Figure 8-22, below, and the activation itself. The facility is present to effectively modify the split of the responsive torque (as apparently dictated by interbracket distances) through pretwisting to take advantage of the relative sizes of the anchorage units that must absorb that torque. Pretorqued brackets that are appropriately chosen can accomplish the same end.

299

Dellveiy of Torque by the Orthodontic Appliance

perm

i-I

.+ T7

T45

I FIGURE 8-23. The partitioning of anchorage in the example of the torqued first molar with equal (left) and unequal (right) permanent pretwists.

Anterior-Segment Torquing Mechanics

Rectangular- Wire Torquing The conventional, edgewise practitioner uses a rectangular arch wire in anterior torquing or retraction mechanics. Third-order action potential is achieved through the presence, in the passive state, of an angular differential between the wire cross-section and the bracket slot at each anterior location where torque transfer is desired. The longitudinal dimension of the wire shaft seems to be controlled by the mesiodistal, dental-arch dimensions in the anterior segment. The wire must "fill" the brackets; that is, the diagonal dimension of the wire cross-section must exceed the occlusogingival width of the bracket slot in order to maintain a twist activation. The wire material must be reasonably resilient, exhibiting a substantial elastic range in torsion, to accommodate sizable twist

activations. Generally, one of two orthodontic displacements is attempted in these mechanics: bodily movement or root movement. In the absence of pretorqued brackets, the clinician typically prepares

the arch wire for third-order activation by placing permanent twists immediately mesial to the intra-arch anchorage sites. The often-bilateral twist placements may be quantified as shown schematically in Figure 8-24: the angle between the arch-form plane and the axis of symmetry of the pliers. The amount of torsional activation achieved upon complete

300


30° of incisal torque

FIGURE 8-24. A buccal view of a rectangular arch wire held at midarch with pliers, indicating that 30° of torque has been placed.

appliance engagement is, however, actually less than this permanent-

twist angle; the principal portion of the difference is the third-order clearance between wire and bracket slot. This clearance may be computed from the actual cross-sectional dimensions of the arch wire and the occiusogingival bracket-slot width. The clearance is depicted on the left in Figure 8-25 and calculation of the clearance for a .019- by .026-in. wire in a .022-in, slot is presented on the right in the figure. Given in Table 8-2 are the third-order clearances for four combinations of wire and slot sizes; nominal dimensions were employed in the computations. True third-order clearances depend upon the actual wire and slot dimensions and, in effect, upon a stiffness of the bracket slot. In response to

Third-order clearance = Angle VWX — Angle VWY

For a .019- by 0.26-in, wire in a 0.22-in. (slot) bracket: slot size

Third-order clearance

19

Tangent of LVWY

LVWY = 36.2° 22

Sine of LVWX =

V'192 + 262 LVWX = 43.1°

Clearance

43.1 — 36.2

= 6.9°

FIGURE 8-25. Third-order clearance between rectangular wire and edgewise bracket: schematic with geometry (left) and an example quantification (right).

301

Deliveiy of Torque by the Orthodontic Appliance TABLE 8-2. Third-order clearances between rectangular arch wires and bracket slots

Wire size (inches)

.016 x .017 x .019 x .021 x

.016 .025 .026 .027

Slot size (inches)

Clearance (degrees)

.018 .018 .022 .022

7.7 2.3

6.9 2.2

the third-order activation, the slot will "open" slightly as the wire attempts to deactivate, torsionally, within it. Although stainless-steel brackets apparently exhibit adequate "slot stiffnesses," vendors have had to reinforce the slots of some plastic brackets and they continue to seek reduced slot flexibility through improved designs and materials. Many clinicians conveniently refer to "torque," quantitatively, in

terms of the size of the permanent-twist placement(s) in degrees. Pretorqued brackets are sized, in part, according to the third-order angulations of their slots with respect to a faciolingual reference. Just as the units of orthodontic force are, properly, grams or ounces, however, ac-

tual torque is determined in gram-millimeters or ounce-inches, for example. To quantify torque correctly as a couple, the combination of shaft stiffness and twist activation are necessary. Earlier in this chapter primary influences were seen to be the cross-sectional dimensions of a solid, rectangular wire (shaft) in determining structural parameters in torsion. Table 8-3 presents relative theoretical values of elastic stiffness,

range, and strength in torsion, per unit shaft length and a common material, for four rectangular wires: typical working- and stabilizingwire sizes associated with the .018- and the .022-in, bracket systems. The torsional stiffness of the largest wire is seen to be nearly five times that of the smallest wire in the table. Also noteworthy is the inverse relationship between size and range. (Nominal dimensions, Equations 8-11, and Figure 8-11 were used in the preparation of Table 8-3.) Additional influ-

ences on all three dependent, structural parameters are the wire material, longitudinal dimension, and the physical character of the wire shaft. Pertinent to physical character, rectangular wire is now marketed 8-3. Comparisons of structural properties in torsion of rectangular arch wires influenced solely by cross-sectional dimensions TABLE

Wire size (inches)

Stiffness

Range

Strength

.016x .016 fl17 x .025

1*

1.5 1.2

1*

zo

1.1 1*

2.5 3.1

.019 x .026 .021 x .027

a6 3.5

4.7

*Given figures not actual values, but, in each column, compared to value for wire adiacent to asterisk (*)

302


in several multistranded configurations. Because the strands may move

with respect to one another in activation or deactivation processes, the stiffness in torsion of stranded wire is less than that of the solid wire of the same overall cross-section and material. Moreover, the stiffness is

dependent on the number and size of the individual strands and the

"braiding" pattern within the wire as a whole. Elastic range and strength are similarly affected. Torque is generated in the appliance as the operator produces thirdorder deformation of the rectangular wire to achieve engagement of the anterior brackets; the larger the amount of necessary activating twist, the greater the initial level of torque in the wire. Figure 8-26 illustrates a torque-versus-twist plot for a full-size (for .022-in, bracket slots), stainless-steel, maxillary arch wire. The "torque" is actually the sum of the four couples transmitted to the four maxillary incisors; the twist is that measured at the midsagittal plane, relative to the before-engagement orientation there. The nonlinear portion near the origin of the plot is associated with bracket-wire clearance, bracket-slot flexibility, and manner of ligation; beyond this portion the relationship is, essentially, linear to the elastic limit, reflecting the overall pattern of Figure 8-10. The thirdorder range of this arch is beyond 35° and the slope of the linear portion of the curve, the "working stiffness," was determined to be approximately 217 g-mm/deg. A relatively simple relationship among the relevant parameters for the entire segment is the following: Whole-segment torque = (Working stiffness) )< x (Induced twist — Clearance)

8-12

The nominal bracket-wire clearance for the arch of Figure 8-26 is approximately 3°; the horizontal-axis intercept (dotted extension of the linear portion of the plot) is slightly greater than 3°, reflecting the flexibiities of the bracket-slot and ligation. If the elastic range is sufficiently great to disallow inelastic behavior

during activation, and not minimizing the importance of bracket-wire clearance, the dominant, indispensable parameter to convert the activating twist into torque induced is the third-order stiffness. Shaft theory, examined earlier in this chapter, indicates that, in addition to material, stiffness of a solid, rectangular arch wire depends on cross-sectional size

and longitudinal dimension. In the bilateral delivery of torque to the incisal segment, "longitudinal dimension" refers, in essence, to the amount of wire between the canine and the lateral incisor. Table 8-4 presents third-order stiffnesses obtained through bench experimentation (Wagner, 1981) for 16 maxillary incisal-segment torquing arches. The material was moderately resilient stainless steel, and the occlusalview arch form coincided with the Bonwill-Hawley diagram. Particularly notable are the decreases in stiffness related to the localized, inelastic wire deformation: the permanent-twist placements and the "V" bends. The values in Table 8-4 should be looked upon as indicating trends because of differences in arch lengths among patients; further reductions

Torque by the Orthodontic Appliance

217

10

20

30

40

Angle of Twist (degrees)

6. A typical third-order activation plot for a full-size, rectangular-wire, incisalrquing arch.

ss may

be attained through use of additional wire in loops more

than the simple opening loops used in the experimentation. torquing mechanics generally involves one of two attempted orthodontic displacement: bodily or root movement. is the more difficult; separate activations of the torque and ng force are necessary, and the two must be maintained over specific couple-to-force ratio. If the angulations (in a sagittal

304

Bioengineering Analysis of Orthodontic Mechanics TABLE 8-4. Torsional stiffnesses for representative, stainless-steel, rectangular-wire, incisal-segment torquing arches (in gram-millimeters per degree)*

Bend, twist placements

None (flat arch) V-bends, no twists V-bends and twists Teardrop loops and twists

.018

.018

.022

.022

Working

Stabilizing

Working

Stabilizing

180

190

160 110

170 145 110

270 230 160 125

285 265 235 180

80

'From Wagner, WA. (1981).

view) are not to change during the displacement, the initial torque mag-

nitude will remain constant (assuming no relaxation of the wire). Accordingly, the driving force should also experience no decrease in size with time and segment movement. In reality, this is an impossible set of mandates to the appliance and, therefore, some compromise must be accepted and undertaken in actual therapy. Alternatively, the torque is the principal activation in root movement. The accompanying faciolingual force is primarily responsive; as a secondary activation it arises against the tendency of the torque to move the crowns as well as the roots. The objective is to produce a displacement characterized by a center of rotation located occlusoapically near the bracket level; the ligation and anchorage must be sufficiently strong and stiff to permit rapid

development of a holding force and adequate to prevent sizable dis-

placements of crown points while the torque deactivates with the wanted movements of the roots. The discussions of transverse, orthodontic displacements in Chapter 5 revealed that the couple-to-force ratios for bodily and root movement are quantitatively similar. Using the physiologically proper force theory proposed in Chapter 5, for average-sized, maxillary, incisal segments the total intermittent torque requirement is between 3000 and 3500 g-mm at activation. The accompanying faciolingual force is 200 to 300 g and is split between the two sides of the arch. All four rectangular wires of Table 8-4 can generate in excess of 3500 g-mm of torque within their

elastic ranges. For a chosen rectangular arch-wire size and configuration, together with bracket selection to enable determination of clearance, substitution of the torque requirement and the working stiffness into Equation 8-12 enables estimation of the appropriate amount of twist activation.

The combination of arch curvature and anterior torsional activation produces several secondary effects that were mentioned previously. Of these effects, perhaps the one of most interest and concern pertains to control of the vertical position of the segment while faciolingual bodily or root movement is progressing between appointments. Any changes in the level of third-order activation in the anterior segment is accompanied by a tendency for occlusogingival movement of the anterior section of the wire. Consider, for example, the anterior portion of the wire at

305


bracket level with posterior engagement complete, but immediately be-

fore torsional activation and engagement into the anterior brackets. Figure 8-18 shows the result following activation of lingual root-torque in the maxillary anterior segment. In the absence of vertical constraint, the twist activation causes downward movement of the anterior portion of the arch wire; in the clinical situation, the existence of localized (bending) stiffness results in the development of an extrusive force exerted by the wire upon the anterior brackets. If the torque deactivates somewhat, subsequently, the accompanying tendency for extrusion is also reduced. Depending on the specific arch, each 10 to 15° change in elastic twist may be accompanied by as much as a millimeter of vertical movement of

the anterior section of the arch wire if unrestrained. This deflection, together with measured vertical stiffnesses between 100 and 200 g/mm for the arches of Table 8-4 (Wagner, 1981), exemplify the very real potential for extrusive or intrusive movements of the anterior teeth during torquing mechanics. Compensation for the vertical-displacement tendency can be created; the effect can also be used to produce vertical movement in concert with the torquing mechanics. Necessary at the outset is consideration of the desired vertical positions of the anterior teeth with respect to their locations prior to the torquing mechanics. Also note that both activation and deactivation of the twist affect the vertical-displacement potential. As an example, consider the attempt at bodily-movement, anterior retraction while holding the vertical positions of the maxillary anterior teeth. As indicated previously, change in long-axis angulations and reduction in torque magnitude as activated are cause and effect. To approach maintenance of the initial couple-to-force ratio over time and bodily retraction, the driving force must be generated by a low-stiffness appliance element. Ideally, the lingual root torque should not lessen with displacement and, correspondingly, the vertical-displacement potential should also remain unchanged with time. Accordingly, the position of the anterior section of the arch wire, prior to third-order activation, should be somewhat apical of the line of brackets; the amount of this vertical "malalignment" is to be proportional to the magnitude of the forthcoming twist activation. Again, the rule of thumb is 1 mm for every 10 to 15° of actual twist activation. If intrusion is wanted with these torquing mechanics, the passive, vertical "malalignment" of the wire and brackets must be exaggerated. Torquing Spurs in the Appliance As

an alternative to the use of rectangular wire, torquirig spurs

formed in round wire may be used to deliver third-order couples (with or without accompanying driving or holding forces) to the anterior teeth (or, for that matter, to any tooth). The wire must be ductile to accept the permanent bends of spur fabrication without fracture and, concurrently, resilient to sustain third-order activations approaching (and often exceeding) 90° without becoming inelastic. The individual spur may be

________________ 306


FIGURE 8-27. Facial views of torquing spurs: a single, narrow spur engaging an individual crown (left) and a T-loop contacting a pair of teeth (right).

affixed to an arch wire or an integral part of the wire. The spurs may be

attached to or part of a continuous arch wire or incorporated in an auxiliary (as in the Stage III appliance of pure Begg therapy). A single spur may contact an individual crown as shown on the left in Figure 8-27, engaged between the wings of a siamese or in a Begg bracket, or one spur may act on several teeth as suggested by the T-loop on the right in Figure 8-27. The typical lingual root torque generated by a torquing spur is de-

picted in the series of sketches in Figure 8-28. On the left, the spur is shown passively; it is often fabricated to lie near to or in the arch-form plane. In a view from the mesial or distal perspective, the center sketch shows the spur (dashed) following activation, an incisor with a bracket on the facial surface, and the forces exerted by the spur tip and by the arch wire on the bracket. On the right, the lingual holding- or drivingforce component, generally present and exerted by the arch wire, has been added. A vertical (occlusogingival) force component may also exist, as described previously. The labially and lingually directed forces

Passive spur

Arch wire

Lingual force

I

LingUalroot

torque

FIGURE 8-28. Sagittal views of a torquir,g spur and its effects: typical passive angulation (left), tip and in-bracket force pair acting on the crown (center), and with the holding or driving force from the arch wire added (right).

307


at the bracket could be combined into a single labiolingual force; how-

ever, the labially directed force in the figure is the counterpart of the spur-tip force, the two forming the couple, and combining the forces at the bracket results in loss of identity of the couple-and-force loading. Of interest is a comparison of the third-order couples generated by rectangular wire and a torquing spur. The differences in the directions of the pairs of forces and the moment arms are shown in Figure 8-29. Those forces exerted by the rectangular wire are directed nearly occlusogingivally while the force pair from the spur acts labiolingually. Because the net translational effect of the force pair is zero (with the individual forces parallel and equal in magnitude), this difference in directions is meaningless with respecf to the action delivered to the dentition. The inherent moment arms for the rectangular-wire and spur couples are the labi-

olingual dimension of the wire cross-section and the occlusogingival height of the spur, respectively; they differ by approximately one order of magnitude (a factor of ten). Hence, if the size of each force generated

by the spur is about one-tenth that of each force of the pair generated within the bracket, the results are approximately equal third-order couples applied by the two appliances. Two other comparisons are noteworthy. First, one force of the pair exerted by the spur, if the wire is engaged in an edgewise bracket, acts against the ligation (which tends to be the weakest, most flexible part of the bracket assembly); alternatively, the Begg bracket incorporates a vertical slot. Second, although a torque having either sense (clockwise or counterclockwise as viewed from a mesial or distal perspective) may easily be generated by a rectangular

wire, the spur tip can only push against the facial surface; hence, to apply a labial-root (lingual-crown) torque with a labial-appliance spur, the tip must contact the crown occlusally to the bracket. Torquing the maxillary incisal segment with spurs is conventional in

Moment arm for torquing spur

Moment arm for rectangular wire

FIGURE 8-29. Sagittal views of maxillary incisors loaded by rectangular-wire torque (left) and spur torque (right) showing differences in force directions and inherent moment arms.

308


FIGURE 8-30. The typical passive geometry of a Begg torquing auxiliary.

III of Begg therapy. Figure 8-30 shows a passive Begg torquing auxiliary in an occlusal view. Spurs, approximately 3 mm in height, are bent into the wire; one spur contacts each incisor facial surface with the auxiliary also engaging the canines. The auxiliary is usually formed in .012-, .014-, or .016-in, wire and is typically employed in conjunction with a .020-in, main arch wire. One purpose of the main arch wire is to help to counter the extrusive potential associated with the activation of the auxiliary. Third-order stiffnesses of Begg torquing auxiliaries are substantially less than those of the edgewise wires; correspondingly, the activations of the spurs are greater with the passive spur configurations generally in or near to the arch-form plane. Torsional- and vertical-stiffness data for Stage III appliances are presented in Table 8-5 (Leaver and Stage

TABLE 8-5. Torsional and vertical incisal-segment stiffnesses for six Begg, Stage lii maxillary-arch appliancest

Auxiliary*

Wire size (inches)

Spur angle (degrees)

.012 .014

0 0 0 45 45 45

.016

.012 .014

.016

Overlaying a .020-in. main arch wire. Leaver, SR., and Nikolai, R.J. (1978).

Torsional stiffness

(g-mm/deg)

Vertical stiffness (g-mm)

33 40

41

51

42

70 45 46

61

65

31

39

309


Nikolai, 1978); the passive angle of the spurs with the arch-form plane

has, apparently, little influence on either stiffness. Figure 8-31 illustrates a torquing arch incorporating spurs; however, this design is quite different from the Begg auxiliary. First, it is a continuous, maxillary arch wire engaging all teeth through the terminal molars. Second, spurs will contact only the central incisors; the torque is transferred bilaterally along the wire from posterior anchorage directly to the central incisors, with third-order action then apparently carried to the lateral incisors by the transseptal-fiber system within the alveolus. The passive geometry of the arch reflects the attempt to negate unwanted secondary effects; a curve of Spee is placed to counter the potential for extrusion that accompanies third-order activation and the posterior sections are curved toward the lingual to offset the tendency for torsional activation to enlarge the posterior arch width. The appliance may be formed in highly resilient stainless-steel wire with the passive configuration of the spurs in the arch-form plane. The spur height is a clinical variable with its upper bound determined by the distance from bracket line to gingival margin and, for heights of 5 to 6 mm, the typical torsional stiffness of the arch fabricated in .016-in, stainless steel wire is approximately 45 g-mm/deg (Mellion and Nikolai, 1982). Comparing this value to the Begg counterpart in Table 8-5, the difference is principally due to the greater "shaft" length (canine to central incisor) in the arch under discussion. Note that the torquing spur itself is, in effect, a cantilever beam; the greater height reduces the bending stiffness of the spur, but it also enlarges the moment arm of the torsional couple. Be-

FIGURE

wire.

8-31. An occiusal view of the passive configuration of a hybrid, torquing arch

310


cause

the overall activation of the anterior section of the arch is primarily

twisting, with only a small contribution from spur bending, for otherwise identical, round-wire torquing arches, the actual torque generated (in g-mm, for instance) is virtually proportional to the height of the spurs. To summarize, regardless of the type of anterior-segment torquing appliance used, the key load parameter is the third-order couple to be transferred from the wire to the tooth crown. This couple is determined principally by a third-order stiffness and the amount of rotational activation. Rectangular-wire stiffnesses are relatively high and the amounts of third-order activation should be correspondingly small; torsional stiffnesses of spur-incorporated round wires are less, thus creating the need for greater angular activations.

Inelastic Behavior in Third-Order Mechanics Prior to twist activation of an arch wire the clinician may alter its config-

uration, through bending and/or torsional deformation, taking the wire beyond its elastic limit to achieve a desired, pre-engagement shape. Such a process changes the external geometry of the wire and induces residual stresses; consequently, mechanical and structural parameters

are affected. Substantial ductility is required of the wire so that the working of it, which strains the material beyond its elastic limit, will not cause immediate—or, while the appliance remains activated, progressive—fracture. Inelastic twisting is examined first in this section. Recall that thirdorder loading of a straight shaft produces a circumferential strain pattern on a cross-section characterized by zero magnitude at its geometric center (through which the longitudinal shaft axis passes) with, generally, the strain increasing with radial distance from the center. On a circular cross-section, the circumferential strain at a point is directly proportional to the radial coordinate of the point. On the rectangular cross-section, these strains vary radially in a nonlinear manner. The maximum strains occur, then, at points on the external boundary of the cross-section. For circular shafts with their axisymmetric circumferential strain patterns, all points on the outer perimeter of the cross-section are subjected to maxi-

mum values. The largest circumferential strains on the rectangular cross-section occur at the two outer-surface points nearest the center (and are on one of the axes of symmetry). The patterns of shearing strains on the two cross-sections of interest are maintained, with increasing load, throughout the elastic range and well into the inelastic region of material behavior. Below the elastic limit, stress and strain are proportional; the patterns of both circumferential stress and strain are

shown in Figure 8-9 for the circular shaft and in Figure 8-12 for the rectangular shaft. The formula for elastic range of a shaft indicates direct dependence of

311


critical twist angle upon the characteristic longitudinal dimension (Equa-

tions 8-9 and 8-11). The practitioner typically localizes the permanenttwist placement. This establishes a sizable twist-set within a relatively small portion of shaft length and, as a result, a concentration of residual shearing stress is created. Shown on the left in Figure 8-32 is the loadunload plot for the short section of shaft (wire) in which a permanent twist is placed. On the right in the figure is a curve describing in graphic fashion the circumferential stress-strain relation during the twist placement for the element on the shaft outer surface experiencing the greatest shearing strain during the process; the final point of the plot reflects the residual shearing strain and stress. For a round wire, a direct analogy can be drawn between inelastic bending and twisting when discussing stress patterns. In bending, the variation in normal stress with distance from the neutral axis was considered and, in torsion, the shearing-stress dependence on radial distance from the shaft center is examined. The sketch on the left in Figure 8-33 shows the pattern of increasing circumferential shearing stress with the radial coordinate r for the critical section of the circular shaft that has undergone torsional loading beyond the elastic range for the member as a whole. Within the elastic core of radius re., the relationship between the stress and r is linear; beyond the core, shaft elements have been inelastically strained and the relationship is nonlinear. The center sketch depicts the change-in-stress pattern assodated with the unloading of the shaft section. As noted previously in Chapters 3 and 7, unloading of a structural member always has elastic characteristics, thus the linear relationship between stress-change and the coordinate r for the metallic shaft material. Although the resultants of the load and unload stress distributions shown must be equal in magnitude (because the couple returns to zero upon unloading), the differences in distribution yield, upon superposition of one upon the other,

0) 0) 0)

C)

C)

:3

(I) (5 a)

-c

(I)

Shear Strain e0

Angle of Twist 0

(stress reversal)

FIGURE 8-32. Load-deformation plot for a shaft section in placement of a permanent twist (left) and a stress-strain plot for an element in the section at the shaft outer surface (right).

312


Load

Unload

Residual

FIGURE 8-33. The load (left), unload (center), and net residual (right), circumferential, shearing-stress patterns for a metallic shaft strained beyond its elastic limit.

net residual-stress pattern shown on the right in Figure 8-33. The difference between the load and unload third-order deformations is the permanent twist (Fig. 8-32, left). Upon overall unloading, elements near the shaft axis do not completely lose the stress induced with loading. At a specific radial coordinate beyond the elastic core, the loading stress and unloading stress-change exactly cancel one another, and near the outer surface of the shaft the elements experience a stress-reversal upon unloading. For the circular shaft this residual-stress pattern is the same along all radial-coordinate lines, commensurate with the axial symmetry. In the rectangular shaft the patterns are similar to that on the right in Figure 8-33, but they are more nonlinear and relative maximum values vary with radial-coordinate direction. Some changes in pattern occur as the diagonals of the rectangular cross-section are approached; however, the stress magnitudes are not critical in these regions. If the wire material can adequately take a permanent twist without failure, the subsequent concern is the superimposition of an elastic activation. From a practical standpoint, considerations are restricted to rectangular wires. (Round wires that easily fit into brackets will spin within them under torsional activation, with or without placement of permanent twists; torquing spurs incorporated in round wire to enable transfer of third-order actions are bent into the wire.) Two points of practical note are to be made. First, the activating (elastic) twist to gain bracket engagement and create the torque is undertaken in a sense opposite to that of the preceding, permanent-set twist. Thus, the largest shearing stresses of the activation itself, occurring near and at the outer surface of the wire, will be, at the sites of permanent-twist placement, superimposed on the largest residual stresses having the same sense. Accordingly, the maximum net shearing stresses will be at points on the outer surface of the cross-section and will be greater than if no permanent twists had been

the

placed. Distant from the outer surface the activating and residual stresses are of opposite senses and will tend to cancel one another. In short, placements of permanent twists followed by opposite-sense,

third-order activation results in concentrations of sizable shearing stresses at outer-surface locations of the shaft. Localized inelastic action

313

Deliveiy of Torque by the Orthodontic Appliance

may even occur, which will tend to reduce the concentration of stresses

and distribute them inwardly. A wire material with substantial ductility

is, therefore, absolutely necessary to avoid approach of the fracture point with activation. Second, bench-experimentation has shown that permanent-twist placements tend to lower the third-order stiffness of the arch wire (Table 8-4). The greater the amount of the permanent set, the larger the decrease in the stiffness. For a given magnitude of twist placement, the percentage reduction in stiffness seems to be essentially independent of cross-sectional size. In appliances incorporating torquing spurs, third-order elastic action is superimposed upon a residual-stress pattern remaining from the formation of those spurs. The analysis is complicated from several standpoints, and only a superficial examination is warranted here. Whether formed directly in the arch wire or prepared separately, the spurs are

bent into an initially straight segment of wire. Localized bends are placed to form the spur tip and where the spur joins the arch form. If the spur is formed directly in the arch wire, the three bends are located in a

common plane; if the spur is attached to the wire, the attachment is partially accomplished through inelastic bending that wraps the legs of the spur around the arch wire, and this bending is in a plane that is perpendicular to that of the spur-tip preparation. The activation of the torquing arch of Figure 8-31, for example, is in two parts: the wire segments between the central incisors and the canines experience the bulk of the elastic deformation as they are loaded in torsion while the spurs experience a small amount of bending deformation. Each spur, behaving much like a cantilevered beam, undergoes its maximum bending strains at the junctions with the arch form and experiences virtually zero strain of activation at the tip (apex). Whatever bending activation does occur at the junctions is in a plane at right angles to that of the permanent bends. Torsional couples are transformed into bending couples at these junctions and, although the stress analysis is extremely difficult

there, it is likely that these are critical locations (of maximum total stress). If the spurs are formed separately, then wound around the main

arch wire, the bending activation is directly superimposed upon the residual-stress patterns left by the winding. From the discussion of inelastic bending in Chapter 7, it is preferable to orient the windings such that the activation of the spurs tends to further wind, rather than partially unwind, the bent wire at the junction. The maximum net bending stress will be less with the former procedure. As noted earlier, third-order activation in the anterior portion of the arch wire produces second-order response in the posterior regions due to the curvature of the arch form. In edgewise, anterior-retraction mechanics, the clinician may place loops just mesial to the lead teeth of the

posterior segments. These loops, incorporated in the arch wire, are formed by inelastic bending in the faciolingual plane. Potential wire fail-

ure associated with the superimposition of the activating and residual bending-stress patterns within the loops need not be of great concern if the wire material is ductile, but the influence of these loops on stiff-

314


is notable. First, the presence of the loops effectively increases the longitudinal dimension of the shaft portion of the arch wire and, thereby, lowers the third-order stiffness as indicated in Table 8-4. Second, viewing the arch wire from a buccal perspective, the anterior portion is cantilevered from the posterior segments, the loops also effectively increase the beam length and, as a result, the vertical stiffness is reduced in the anterior section to possibly lessen the impact of the generally unwanted, secondary effect discussed previously. The more wire incorporated in these loops, the greater are the third-order and vertical flexibilities in the anterior sections of the arch. Note, however, that the loops oriented gingivally must be passively open to be effective if the

nesses

third-order activation is to produce lingual-root torque. Loops may also have a more direct association with torsion, either in terms of their own internal force systems or their effects upon the teeth; such auxiliaries are discussed in the following section.

Wire Loops, Springs, and Torsion Structural aspects of orthodontic wire loops were first considered in

Chapter 7, and that discussion was restricted to loops that provide mesiodistal flexibility to the continuous arch wire and were activated in flexure within the plane of their bends. Appropriate here in this chapter is the extension of that discussion to consider loops bent directly into the wire or auxiliaries that either experience internal torsional loading when

activated or are intended to transmit torquing couples to individual teeth in order to impart root-movement potential or control. These loops and auxiliaries are exemplified by the "half-box" loop and the uprighting spring. The usual objective in activating a half-box loop is to exert a faciolingual force on an individual tooth. The loop, as prepared from a straight segment of wire through placement of four, right-angle bends, is shown on the left in Figure 8-34. The vertical legs are typically 5 to 6 mm in length and the horizontal section spans one or more interbracket dis-

0 Q

Ce

Passive, buccal view

Occiusal view

FIGURE 8-34. A half-box loop shown passively in a faciolirigual view (left), and as activated and subjected to its typical force system from an occiusal perspective (right).

315


tances,

depending on the specific application. The loop is activated in

the direction perpendicular to the plane of the loop as formed. The sketch on the right in Figure 8-34 shows an occlusal view of the ioop as

ordinarily activated. The active and responsive force systems are included in the figure; capability must be present for generating the firstorder couple(s) to disallow contact between the loop and the gingiva. The faciolingual forces at the activation and anchorage sites are equal and opposite to one another, and the resultant couple is equal in magnihide to that formed by the two forces. (Additional actions may be imparted at the activation site, provided the equilibrating responses can be developed. The loop could also actively transmit a first-order couple, an occlusogingival force, a second-order couple, and, if the ioop is bent in

rectangular wire, a third-order couple to an appropriate malaligned tooth.) In Figure 8-35, the three sections of the half-box loop are separated, making the force systems at the "corners" external. Following activation, the reaction of the tooth crown at the activation site loads one leg in bending; the mesiodistal view of the vertical leg on the left shows the force system of a cantilever (plus torsion if C1 is present). The bending couple in the left leg becomes a twisting couple (Qv) in the midsection of the loop with the right-angle turn in the wire. The horizontal section,

as shown from an occlusal perspective in the center sketch of Figure 8-35, is subjected to a combination of twisting and bending; action-reaclion analysis gives the force system at its left end, and force and moment balances would yield the responsive components at the right end of the section. A mesiodistal view of the right leg of the loop is shown on the

right in Figure 8-35. The 90° turn converts the twisting and bending couples at the right end of the midsection into bending and twisting couples, respectively, at the gingival end of the right leg of the half-box

Qv

0 I

v

QV4I

FIGURE 8-35. Free-body diagrams of the three straight-segment sections of the activated halt-box loop of Figure 8-34.

V

316


ioop, and this is the critical section of the ioop (where the internal bend-

ing and twisting moments are concurrently maximum). The right leg of

the loop, then, is also subjected to combined bending and twisting. (With additional components from activation at the occiusal end of the left vertical leg, the analysis becomes more complex. The numerous variations will not be examined here, but in most instances the critical section wifi be located at the bend diagonally opposite to the activation site.)

Of the various structural properties of the loop, interest centers on the faciolingual stiffness since this characteristic, together with the activating displacement, yields the initial crown load. Generally, this stiffness is dependent on wire material, cross-sectional size and shape, and curvilinear length (the amount of wire in the loop). From the foregoing bending and torsional analyses, the mesiodistal wire dimension and the occlusogingival lengths of the vertical legs will be the most influential geometric parameters. All other properties held constant, the greater the

separation between activation and anchorage sites, the greater the length of the horizontal section and the larger the overall flexibility of the loop. Should the action of the loop include an occlusogingival loading component, the midsection length in particular will influence the occlusogingival stiffness.

The uprighting spring is an auxiliary designed to exert a secondorder couple on a tooth crown, adding to an appliance the capability of controlling root-tip position (long-axis angulation) in the faciolingual plane. The spring is formed, usually in round wire, by placing several

permanent bends to obtain the desired shape. One leg of the spring engages the bracket slot (which may be either horizontal or vertical) at the activation site and the other leg is generally hooked over the main arch wire. The passive (dashed) and active (solid) configurations of the spring are shown in Figure 8-36. As indicated, a helix is often incorporated in the spring at its apex, both to lessen stiffness and to increase elastic range.

Shown on the left in Figure 8-37 is the free-body diagram in a faciolingual plane of an activated spring. The second-order couple is the action-reaction counterpart of the desired component of the force system, but the pair of forces—one at the bracket and the other exerted on the hook by the contacting arch wire—arises in response to balance the system. The analysis parallels that for the half-box loop with C1 absent except for differences in activation planes and in transposition of active and responsive force components. For the spring, the size of the couple is determined by the associated stiffness and the magnitude of the displacement necessary to engage the hook with the main arch wire. The

action-reaction counterparts exerted by the legs of the spring on the tooth and the arch wire are illustrated on the right in Figure 8-37. Note that the arch wire, through its own bending stiffness in the faciolingual plane, can sustain at least a portion of the usually unwanted responsive

force that tends to extrude the tooth. The force exerted by the leg hooked over the arch wire is divided and transmitted by the wire to the

317

Delivery of Torque by the Orthodontic Appliance Line of bracket slots

Activated

—__

—

Passive

-7

FIGURE 8-36. Passive (dashed) and activated (solid) configurations of an uprighting spring in a facial view.

teeth on either side of the interbracket contact. Shown in Figure 8-38 is a

pair of free-body diagrams of segments of the spring, formed by a "cut" immediately to the right of the helix. The left-hand portion of the spring is subjected to a combination of bending and compression. The moment balance shows that the couple at the "cut" is slightly smaller in magnitude than the couple delivered to the bracket. The right-hand portion of the spring behaves as a cantilevered beam. The maximum internal bend-

ing couple exists, then, continuously throughout that portion of the spring between the bracket and the helix. e

P

FIGURE 8-37. Free-body diagram of the uprighting spring (left) and the force system exerted by the spring (right).

318


P

=( - b)

I

P(t-b)

P

FIGURE 8-38. Free-body diagrams of two portions of the uprighting spring exposing an internal bending couple adjacent to the helix.

Unless

extrusion is a portion of the desired tooth movement, the

spring should be designed to generate an adequate couple at the bracket while suppressing the vertical forces that respond to the activation. Examining the moment balance with the sketch on the left in Figure 8-37, the proper procedure is to make the distance between the bracket and the "hook-on" site (a moment arm) as large as possible. Alterations in spring-wire material, cross-section, occlusogingival height, or number of helices all influence the second-order stiffness of the spring and, as a

result, the magnitude of the as-activated couple for a given, passive geometry. These adjustments also affect the range and strength of the spring, but will not impact the ratio of second-order couple to extrusive force at the bracket. On the other hand, changing the characteristic mesiodistal dimension of the spring alters both the ratio of couple to force and the relevant stiffness.

Synopsis This

and the previous chapter have dealt with the force-system and

structural analyses of the orthodontic arch wire, subjected through clini-

cal activation to bending and torsional loadings. Engineering theory, experimental research, and clinical observations have been integrated toward presentation of the current development of the subject matter and the means to undertake evaluations of existing intraoral appliances

319


and, perhaps, to proceed toward new designs. Study of the fundamen-

tal bending problem, with its activations perpendicular to the longitudinal wire dimension, has proved to be more straightforward than that of the third-order problem. The clinician, however, must become familiar with actual torque and its relationship to torsional deformation and the third-order, structural characteristics of the arch wire and bracket systems if an integrated understanding of the appliance, in all of its aspects and capabilities, is to be achieved. Helpful to the reader in following the developments of this chapter is the parallelism between bending and torsional theoretical concepts. Stiffness links activating deformation to induced force system. Stiffness, range, and strength have the same general definitions, and are influenced by the same wire parameters and to similar extents, whether considering bending or torsion. The curvature of the arch form provides difficulties in the clinical arena as it causes transformation of torsional couples into bending couples, and vice versa, but the continuous-archwire model aids in understanding this phenomenon and the associated secondary effects that have been described. The emphasis in this chapter, not explicitly stated previously, has focused on the instantaneous, structural aspects of the third-order problem. Little attention is given to the time-influencing, deactivation process because the present level of knowledge has apparently been gained primarily from clinical experimentation; the actual results have often been less satisfactory than those envisioned. The mechanics of anterior retraction, for example, are in need of substantial, controlled study. Often in actual therapy, the achieved angulations are not those desired and an additional step in treatment becomes necessary, demanding revised and lengthened third-order mechanics. The foregoing discussions have attempted to enlighten the orthodontic student and practitioner on a perplexing subject and a difficult portion of therapy. Hopefully, some gain has been achieved in the direc-

tion of further understanding of torsion and third-order mechanics.

References Begg, P.R., and Kesling, P.C.: Begg Orthodontic Theory and Technique. 3rd Ed.

Philadelphia, W.B. Saunders, 1977, Chapters 6 and 11. Leaver, S.R., and Nikolai, R.J.: Mechanical analysis of the Begg torquing auxiliary. Aust. Orthod. J., 5:133—141, 1978. Mellion, N.P., and Nikolai, R.J.: Torquing maxillary incisors with a continuous, round-wire appliance. J. Dent. Res. (Special Ed.), Abst. 1107, March, 1982. Wagner, J.A.: Mechanical Analysis of Anterior Torque in Rectangular Orthodontic Wire. Master's thesis, Saint Louis University, 1981.

320


Suggested Readings Andreasen, G.F.: Selection of the square and rectangular wires in clinical prac-

lice. Angle Orthod., 42:81—84, 1972.

Arbuckle, G.R., and Sondhi, A.: Canine root movement: an evaluation of root springs. Am. J. Orthod., 77:626—635, 1977.

Blodgett, G.B., and Andreasen, CF.: Comparison of two methods of applying lingual root torque to maxillary incisors. Angle Orthod., 38:216—224, 1968. Boman, V.R.: A radiographic study of response to torquing spring action. Angle Orthod., 32:54—58, 1962.

Burstone, C.J.: The biomechanics of tooth movement. In Vistas in Orthodontics. Edited by B.S. Kraus and R.A. Riedel. Philadelphia, Lea & Febiger, 1962, Chapter 5. Gianelly, A.A.: Bodily retraction of maxillary incisors with round wires. Am. J. Orthod., 66:1—8, 1974.

Holdaway, R.A.: Changes in relationship of points A and B. Am. J. Orthod., 42:176—193, 1956.

Hurd, J.J., and Nikolai, R.J.: Maxillary control in Class II, Division 1 Begg treatment. Am, J. Orthod., 72:641—652, 1977.

Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapters 2, 3.

Kusy, R.P.: Comparison of nickel-titanium and beta titanium wire sizes to conventional orthodontic wire materials. Am J. Orthod., 79:625—629, 1981. Kusy, R.P.: On the use of nomograms to determine the elastic property ratios of orthodontic arch wires. Am. J. Orthod., 83:374—381, 1983.

Kusy, R.P., and Greenberg, A.R.: Effects of composition and cross-section on the elastic properties of orthodontic wires. Angle Orthod,. 51 :325—342, 1981.

Kusy, R.P., and Greenberg, A.R.: Comparison of elastic properties of nickel-titanium and beta titanium arch wires. Am, J. Orthod., 82:199—205, 1982.

Mitchell, D.L., and Kinder, J.D.: A comparison of two torquing techniques on the maxillary central incisors. Am. 1. Orthod., 63:407—413, 1973.

Neuger, R.L.: The measurement and analysis of moments applied by a lightwire torquing auxiliary and how these moments change magnitude with respect to various changes in configuration and application. Am. J. Orthod., 53:492—513, 1967.

Newman, G.V.: A biomechanical analysis of the Begg light arch wire technique. Am. J. Orthod., 49:721—739, 1963.

Popov, E.P.: Introduction to Mechanics of Solids. Englewood Cliffs, NJ, Prentice-Hall, 1968, Chapter 5.

321


Rausch, ED.: Torque 45:817—830, 1959.

and its application to orthodontics. Am. 1. Orthod.,

Schrody, D.W.: A mechanical evaluation of buccal segment reaction to edgewise torque. Angle Orthod., 44:120-126, 1974.

Smith, J. 0., and Sidebottom, 0. M.: Elementary Mechanics of Deformable Bodies. London, Macmillan, 1969, Chapter 9. Steyn, C.L.: Measurement of edgewise torque in vitro. Am. J. Orthod., 71 :565— 573, 1977.

Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982, Chapters 6 and 9.

Ext rao ral

Appliances )

A class of appliances familiar to the practitioner is characterized by the

extraoral positions of the activating elements and supporting structure. These appliances, which individually or collectively are often termed "headgear," receive widespread use, primarily because they have the advantage of remotely located responsive force (anchorage). The task of force-displacement analysis is eased because the effects of the responsive force system, broadly distributed, need be of little concern; therefore, the practitioner may focus attention on the active force system and its potential to produce or restrain orthodontic or orthopedic displacements. The principal components of the typical extraoral orthodontic appliance are:

1. A face bow or a pair of "J-hooks" that delivers the active force to intraoral locations where, either directly or indirectly, the force is transferred to the arch wire or through brackets or buccal tubes to the tooth crowns.

2. The active force generated bilaterally by one or a pair of traction bands or by one or more pairs of elastics or springs. 3. The responsive force distributed to cervical, occipital, or parietal anchorage areas by a neck pad or head cap.

The headgear apparently most popular with practitioners may be described as those that provide active force primarily directed toward the posterior, have intraoral mechanical connections to the maxillary arch wire or to teeth of the maxillary arch, and possess geometric and mechanical symmetry with respect to the midsagittal plane. Recent innovations, however, have improved the potential for controlled, unilateral displacement; delivery of extraoral force to the mandibular arch is

receiving increased acceptance, and extraoral appliances from

which anteriorly directed forces are supported have been designed and marketed. Extraoral appliances may be classified from several standpoints. Distinctions between types may be drawn according to the origin or direc322

323

Extracral Appliances

tion of the active force. This force may be generated against a neck pad

or head cap. Clinicians often refer to headgear as cervical-, straight-, or high-pull, indicating the buccal-view angulation and line-of-action position of the active force. The appliance having posterior anchorage and the active force on one or both sides divided into two separate, mechani-

cal actions is termed a "dual-force" headgear. Those appliances that generate anteriorly directed active force are becoming known as "reverse-pull" mechanisms. Headgear may also be categorized according to the intraoral locations of active-force delivery: the maxillary and/or mandibular arch, the first or terminal molars, the canines, or the incisal or anterior segment(s). The chin-cap assembly might be placed in a separate category because it delivers active force to an extraoral location: the

chin. Although it is often employed to produce orthopedic action or restraint, because of similarities to other types of headgear (in terms of anchorage and activating elements) the chin-cap assembly is included in the discussions to follow. The development of this chapter proceeds with the mechanical eval-

uation of headgear, generally subdivided according to the location of active-force delivery as seen from an occlusal or buccal view. Discussions are primarily focused on the relationships among active force systems, alone or in concert with intraoral mechanics, and the corresponding, potential, orthodontic and/or orthopedic displacements. There is little attention to structural considerations of the extraoral appliance itself, beyond that which follows in the introductory section of this chapter. The analysis problem is three-dimensional; the individual, active forces generally have three nonzero components with respect to a sagittal-coronal-transverse reference framework. The spatial problem is examined first in an occlusal view in which anteroposterior and faciolingual aspects may be considered, and then from a buccal view where vertical and occlusogingival aspects, together with the anteroposterior contributions, may be analyzed. All of the needed geometric parameters will be assumed known or the facility to take required anatomic and other measurements is available. The characteristics of the activating force system (from traction band[s], elastics, or springs) are also assumed given or measurable. Sought in the analyses are descriptions of potential displacements that are desired or, if the clinical objective is to negate unwanted but likely movements arising from intraoral mechanics, the degree to which the objective can be met. Before beginning the individual analyses, several overall, pertinent considerations should be kept in mind. First, the analysis of the force system must be undertaken with the extraoral appliance in its activated configuration. Strictly speaking, this is always the rule with any structure; however, often the difference between passive and activated formats are small enough to produce negligible variance in the directions or moment arms of forces. These differences are generally not substantially

influential, for example, in the bending or torsional analyses of arch wires. The point is well made here, however, because the deformations associated with the activations of face bows and J-hooks may be sizable.

324


In fact, with the face bow in particular, several adjustments in the sagit-

tal and transverse planes made "at the chair" are undertaken in anticipation of the deformations to accompany activation. Second, with the exception of some mixed-dentition treatment, the headgear rarely acts in the absence of intraoral-appliance interaction. When present, the arch

wire is nearly always interconnected with the delivering element of extraoral force. Hence, the tracing of the transmission of force is always

necessary, even though originating extraorally, throughout whatever intraoral appliance is engaged, whether activated independently or not. Third, a substantial portion of the prediction of displacement or displacement restraint originates from a buccal-view analysis. In effect, a complete division of the entire dentofacial complex and appliance(s) is made at the midsagittal plane, and the analysis is carried out on the right half separately. With complete symmetry with respect to the midsagittal

plane, the force system transmitted from one side to the other (e.g., through the center of a face bow) has no mechanical influence in the buccal view. With an asymmetric situation associated with appliance geometry and/or loading, however, such is not the case and appropriate notations are made when applicable. Fourth, the size of the force to be generated in the activating element and transferred to a bow or J-hook end is determined by the factors discussed in Chapter 5 and an estima-

tion of the amount of generated, active force to be, in effect, lost in transit to the intraoral site of force application. This force drop is small with face-bow transmission, but friction and, perhaps, a portion of the force not delivered in the desired direction may result in sizable losses in or diversions of force carried by J-hooks. Also noteworthy is the pattern of active force versus time: continuous when the headgear is in place and activated, but overall often a cyclic, interrupted function, as illustrated in Figure 5-29. Often when used in permanent-dentition treatment, the extraoral appliance is worn for just 10 to 14 hr each day. This interrupted force pattern must influence the choice of magnitude of the extraoral traction when the headgear complements intraoral mechanics; the sizes of intraoral and extraoral force must be selected to account for their interaction. Fifth, details of structural analysis are suppressed in this chapter.

The emphasis here is not with designs of face bows or head caps or potential structural failures. In short, the components of the extraoral appliance must be capable of generating, carrying, and transmitting forces without excessive deformation, fracture, or loss of stability. Because face-bow and J-hook materials are ductile, exceeding an elastic limit, whether to place a permanent bend or upon activation, generally should not be of concern. The practitioner must remember, however, that elastics and traction bands relax (the former more quickly than the latter) and, therefore, must be replaced periodically. Routinely, activation should be accompanied by the actual movements of forces generated in and by the traction bands, elastics, and springs, and activating elements should not be chosen solely by the distances over which they must be elongated.

325

Extraoral Appliances

The Cervical-Pull, Face-Bow Appliance cervical-pull appliance consists of a face bow, a neck pad, and the element(s) that, when engaged, induce the activating force. The face bow incorporates an inner bow, which fits intraorally, and the outer bow. The two parts of the face bow are interconnected, either solidly (e.g., welded) or by means of a hinge. When the face bow is in place, that connection is to be slightly anterior of and between the lips and, The

when geometric symmetry is present, intersecting the midsagittal plane. The inner-bow arm ends engage buccal tubes that are affixed to the facial crown surfaces of the maxillary first or second molars. The active force is delivered bilaterally to the outer-bow arm ends, transmitted anteriorly through the outer-bow arms, then through the bows connection into the inner bow, posteriorly through the inner-bow arms and through stops into the buccal tubes. The active force is generated by the elongation of

an elastic traction band, one or more pairs of elastics, or a pair of springs, attached to the outer-bow arm ends. The accompanying responsive force(s) are transmitted to the neck pad which, in turn, distributes that force over the back of the neck. The forces delivered to the outer-bow arm ends generally have com-

ponents in all three dentofacial-coordinate directions. Although the principal components (left and right sides) are directed posteriorly, small lateral components usually exist due to the transverse distance between the outer-bow arm ends that exceeds the neck width. Vertical components are often present, due, in part, to the anatomic positions of the maxillary molar crowns with respect to the cervical region. The elastic(s) or spring, stretched between an outer-bow arm end and a fastener on the neck pad, is a two-force element under axial load. Figure 9-1 shows a free-body diagram of an elastic traction band and the neck pad as a unit. The responsive force of the neck against the pad is primarily pressure distributed over a curved, rectangular area. Depending on the actual direction of pull, a minor component of the distributed response

may exist tangential to the neck pad and perpendicular to the plane view. Initially and, perhaps, irregularly over time, a frictional distribution may also exist. The essential features, however, are given in the two-dimensional sketch. The plane of the diagram generally makes an angle that is between 15 and 200 with the occlusal plane. The pressure distribution is apparently somewhat nonuniform, but not to the point of concern. The tensile strain in the elastic band is essentially constant from one end to the other (if not immediately upon activation, then shortly there-

after when movements of the head have occurred and the band has sought a minimum-energy configuration); accordingly, the magnitudes of the contact forces with the outer-bow arm ends are equal. The resultant of the pressure distribution, then, must dissect the angle between the lines of action of the forces exerted by the outer-bow arm ends. The

326


Distributed cervical response

P0

Pc,

FIGURE 9-1. Responsive force system exerted against cervical anchorage.

maximum pressure will be along the line of action of the resultant of those two forces which, in the symmetric situation, will be in the midsagittal plane as shown in the sketch. Note that the substitution for the traction band of elastics or springs, fastened directly to the neck pad, results in no substantial change in Figure 9-1 if the individual activating elements generate forces of the same magnitude. If one elastic or spring is stiffer than its counterpart on the other side of the head, the neck pad will tend to migrate toward the side of the stiffer element and, although the forces to the outer-bow arms may be unequal as activated, with time and the neck-pad migration, these forces will approach one another in value.

0cc/usa/-Plane Analysis Figure 9-2 depicts a free-body diagram in an occlusal- or transverseplane view of an activated face bow that is geometrically and mechanically symmetric with respect to the midsagittal plane. The symmetry requires equal lateral force magnitudes, each P0 sin offsetting one another. An anteroposterior force balance, with the symmetry, yields = P0 cos 00, existing due to buccal-tube contact against the stopped, inner-bow arm end. Also generally appearing in the occlusal view, as

327


L0

Co

CO

P0

P0

FIGURE 9-2. A free-body diagram in the occiusal plane of an activated, symmetric face bow.

part of the responsive force systems of the buccal tubes against the

inner-bow arm ends, are first-order couples and lateral forces. Because they are not directly related to the forces activating the appliance or to

each other, midsagittal-plane symmetry and equilibrium conditions demand that, separately, the couples (C0) and lateral forces (L0) be equal

in magnitude but opposite in sense to one another. An occiusal view of the force systems exerted on the molar crowns is presented in Figure 9-3; the posteriorly and lingually directed force components and couples are action-reaction with their counterparts in Fig-

ure 9-2. If the face bow produces molar movement, with the driving forces at the buccal surfaces and not through the centers of resistance, the occlusal-view displacements tend to be distolingual-rolling. The length of the buccal tube and the bending stiffness of the inner-bow arm

together can produce first-order couples, however. With the senses shown in the figure(s), the couples C0 may exist immediately upon acti-

vation, due to placements of toe-out bends in the inner-bow arms, or they may arise in time with initial molar movements, to counter the

328


F0

00

Qo

FIGURE 9-3. 0cc/usa! view of the force system transmitted to molar crowns by the face bow of Figure 9-2.

distolingual-rotational component of displacement. (The generation of

each couple is through two-point buccolingual contact of an inner-bow arm end within the buccal tube. The difference in magnitudes of the two contact forces, if any, is the lateral-force component L0. The action-reaction within the buccal tube is shown in Figure 9-4. The potential size of the couple is enhanced by the length of the buccal tube, which is also the magnitude of the inherent moment arm.) The pair of lateral force components, with the senses indicated, reflect a tendency to decrease the posterior width of the maxillary arch. They would be created upon activation when the passive distance between the inner-bow arm ends in less than that between the buccal tubes. Another more subtle source is in response to the potential for labial tipping of the molar crowns, present in occlusally directed components of force delivered by the innerbow arm ends. This latter phenomenon is more easily seen in a view from a posterior perspective to be presented subsequently. Generally, as noted, the lateral forces and first-order couples may exist in response to bends placed in the inner-bow arms prior to activation, or in response to initial molar displacements. Their senses may be those of Figures 9-2 and 9-3, or the senses of the couples and/or those of the lateral forces may be opposite to those shown. For example, opposite senses for the couples would result from placements of toe-in bends. Lateral forces created in the activation of an appliance exhibiting a passive distance between inner-bow arm ends greater than the posterior arch width measured between the buccal tubes would be opposite in

sense to those shown. The lateral components may also exist in response to a potential in the force system delivered to the molars to tip the crowns lingually. In Figure 9-3 the anteriorly directed forces F0 repre-

329


n I

j

I

I

nner-bow arm

Tooth

crown

Together,

these forces form C0

-J

FIGuRE 9-4. Two-point contact in an occiusal view, between inner-bow arm end and buccal tube, which generates C0.

sent buccal contact between the molars and intraoral mechanics (a tiedback, continuous arch wire or activated Class III elastics, for example). These forces are included in the sketch because they, and the intraoral mechanics, may exist in conjunction with the headgear. If the function of the extraoral appliance is to transfer forces, responsive to intraoralmechanics activation, from the molars to the cervical region, the poten-

330


tial displacements associated with the forces F0 are to be negated by the forces First-order couples, opposite in sense to those shown and

generated by toe-in bends in the inner-bow arms (and/or in, if present, the continuous arch wire), would help to stabilize the molars. The cervical-pull appliance may be modified to produce asymmetric action against the maxillary terminal molars; desired is the creation of

either a greater distal movement of one molar than the other or the displacement of one molar while maintaining the position of the other. Several face-bow designs have been offered, and mechanical asymmetry must be present to achieve the wanted displacement pattern. Figure 9-5 shows a free-body diagram of a face bow possessing geometric, midsag-

B

P1

Q1

Cl

C2

Q1

+ C) — P2c2}

Q2

+ C)— P1c1}

FIGURE 9-5. A free-body diagram in 0cc/usa! view of a face bow activated by unequal elastic or spring forces.

331


ittal-plane symmetry in the passive state and activated by concentrated

loads against the outer-bow arm ends, which differ in magnitude. As mentioned previously, such a differential in delivered forces cannot be maintained by a stretched traction band or by unlike elastics or springs activated against a neck pad. Recommended instead is the activation of a pair of elastics or springs, with one of the pair differing from the other in stiffness and/or amount of activating deformation, against a well-fitting head cap. Below the figure are the equations for the magnitudes of the distal driving forces, action-reaction counterparts of Qi and Q2. The formulas are obtained from an anteroposterior balance of forces exerted on the face bow and a moment balance about a convenient point such as B. Note that P1 and P2 are occlusal-plane components of the total elastic or spring forces. If the distances c1 and c2 are small compared to the lateral dimension c between the buccal-tube slots, with the outer-bow arms as activated close to but not touching the cheeks, a reasonable approximation is and Q2 individually approaching P1 and respectively.

A disadvantage in the asymmetric design just described is the dependence upon stability of the head cap. Another procedure to obtain unequal driving forces, which will accommodate neck-pad anchorage, is to engage an asymmetric face bow. These face bows are available with the inner and outer bows hinged or solidly connected to one another. The hinged face bow, with separate free-body diagrams of the inner and outer bows, is sketched in occlusal views in Figure 9-6. The axis of the hinge is vertical and any frictional couple that might be present at the hinge has been neglected. To maximize the potential differential in sizes between Qi and Q2, the hinge is located as distant from the midsagittal plane as is functionally permissible. Because the magnitudes of the forces delivered to the outer-bow arm ends (from a traction band) are equal (P0) and given the hinge location, to prevent the outer bow from rotating about the hinge (from a moment imbalance, clockwise with the hinge to the left of the midsagittal plane as shown in the figure), resulting in one outer-bow arm contacting the cheek and interfering with the wanted force system, one outer-bow arm must be longer than the other. The longer arm is on the same side of the midsagittal plane as the hinge; the objective is a balance of moments of the two traction-band forces about the hinge axis, point H. Since these two forces are of equal magnitude, the direction of the force between bows, transmitted through the hinge, may be obtained through the graphical construction shown in Figure 9-6; the line of action of Ph bisects the angle between the lines of action of the forces exerted on the outer-bow arm ends, and the magnitude Ph is somewhat less than twice one outer-bow force as indicated. The hinge force delivered to the inner bow is noted to include a lateral component, a generally unwanted side effect of the asymmetric face bow. Because of the hinge position and since the angle between Ph and Pa is not large, the intersection of the line of action of Ph and the transverse line between the buccal-tube centers, point E, is some distance away from point B. The larger this distance, the greater the mechanical asymmetry. Force and moment balances from the free-body diagram of

P0 Pb

\

PU

I

'

\

Pd

= E

——=

'-4 B

1

02

4— b1

where b = b1 + b2

b2

FIGURE 9-6. 0cc/usa/-view, free-body diagrams of inner- and outer-bow portions of an asymmetric, hinged face bow. 332

333


the inner bow yield the formulas given in Figure 9-6 for Qi and Q2; in an approximate analysis it is often reasonable to equate cos 0d to unity (one)

since this angle is usually small. Although the hinged face bow should be able to deliver a substantial differential in the distal driving forces delivered to the molars, inherent with the hinge connection between bows is a lack of stability; the outer bow may rotate rather easily and an arm contact the cheek. A patient, wearing the appliance at night, for example, will amend the delivered

force system when sleeping with either side of the face against the pillow.

Several asymmetric face-bow designs exist that exhibit a solid connection between inner and outer bows. One such design has the connection between bows off the midsagittal plane and the passive posterior extents of the outer-bow arms the same. This face bow, with a traction band and neck pad has been shown to produce only a small differential in the forces against the molars (Drenker, 1959); the mechanical asym-

metry is only slight and, in essence, is due to the bending-flexibility differential in outer-bow arms associated with their unequal lengths. A larger differential has been obtained with a design displaying the rigid inner-to-outer-bow connection, on or off the midsagittal plane, and outer-bow-arm ends exhibiting different posterior extents. This type of face bow is shown in Figure 9-7; again, the activation is delivered by a

P0

Q1

Q2

Pd

1

b2

01 = =

where b = b1 + b2 FIGURE 9-7. 0cc/usa/-view force diagrams for an asymmetric face bow exhibiting a solid, midline connection between inner and outer bows.

334


stretched

elastic band. The force diagram on the left suggests the

achievement of unequal actions delivered to the molars; the outer-bow arm lengths and the patient anatomy must result in the line of action of the resultant of the traction forces (each of magnitude P0), passing to one side of point B. The free-body diagram on the right in Figure 9-7 shows only the anteroposterior force components; because the lateral force components are colinear, their analysis may be undertaken independent of the anteroposterior force system. A moment balance indicates that here also, as with the hinged face bow, the magnitudes of the forces Qi and Q2 are in the same ratio as the distances b2 and b1. The size of the differential in delivered distal driving forces, using the asymmetric face bow and traction-band activation, depends on the eccentricity with respect to point B of the lines of action of the resultant active force. If this eccentricity is greater than one-half of the intermolar width measured between the buccal-tube centers, the mesiodistal forces to the molars will be opposite in sense to one another; although improbable in practice, "on-paper" analysis indicates this is a possibility. If the intent of the differential is to move just one molar, the achievement of a near-zero force against one and a sizable distal driving force against the other molar requires careful adjustments by the practitioner, particularly with respect to outer-bow arm lengths and as-activated configurations. Some assistance can be gained through concurrent use of intraoral mechanics delivering a balancing, mesial force to the molar receiving the smaller distal headgear force. As previously indicated, all asymmetric face bows exhibiting unequal distal driving forces also exert lateral action. The net active lateral force has been symbolized in Figures 9-6 and 9-7 as Pi. but its division between the molars cannot be quantified by force and moment balances alone. In an overview of the problem, it is first necessary to realize that a posteriorly directed force transmitted to the inner bow, whether positioned in the midsagittal plane or not, tends to widen the passive distance between the arm ends as indicated on the left in Figure 9-8 in the

FIGURE 9-6. Occlusal-view diagrams displaying lateral-force action delivered by an inner bow to the molars.

335


presence of lingually directed forces against the arm ends (action-reac-

lion counterparts of the buccally directed forces on the molars). To be superimposed on this effect at the molars is the response of P1. which produces forces at the molars that are identical in sense (although perhaps not equal in magnitude) as is indicated in the center sketch of the figure. The superposition, arguing only on the basis of senses of forces, results in an excess of lateral force on the short-arm side and a relatively small, net lateral load on the molar adjacent to the long arm of the outer bow. Assuming that the lateral forces from the posterior component of action are somewhat larger than those associated with the asymmetry, and that P1 is approximately halved between the molars, the net lateral response at the molars will generally be as shown in the diagram on the right in Figure 9-8. A third contributor to the lateral action exists if the inner-bow arm ends do not fit passively into the buccal tubes, due for example, to expansion or constriction of the inner bow with the intention of altering

the intermolar width. Activation of this effect alone creates forces against the molars that are equal in magnitude, but opposite in sense to one another. This facet of the problem is also quasi-statically indeterminate and a complete analysis of the total problem is beyond the level of the present discussion. If the face bow reflects geometric and tractionforce symmetry with respect to the midsagittal plane, this symmetry will extend to the delivered lateral force components; with asymmetric geometry and action, the lateral forces transmitted to the molars will likely differ substantially from one another in magnitude, and may be of the same or opposite sense depending on the relative influences of the three possible contributions as discussed. In concluding the occlusal-view analysis of asymmetric face bows, note that these appliance elements are "reversible." The longer outerbow arm can be positioned on either side of the face by simply flip-flopping the face bow. Two configurations of the bow then are available, and the geometric and mechanical configurations individually are mirror plane. images of one another relative to the

Buccal-View Analysis of the Cervical Appliance An analysis from a buccal perspective yields substantive component relationships between the active forces delivered by the extraoral appliance and potential intraoral displacements. If the molars alone receive the action of a face-bow appliance, should tipping, bodily, or torquing displacements be expected and in which direction(s)? If a stiff arch wire carries the loading anteriorly from the maxillary molars, might the maxila be rotated and, if so, might the anterior bite be opened or deepened?

One-half of the face bow is seen in the typical buccal view, and an analysis involving just that half bow and the interacting half-arch would

be convenient. In general, "cutting" the face bow at the midsagittal plane exposes an internal force system there with resultant force and couple vectors having, individually, any conceivable directions in space.

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In the presence of midsagittal-plane symmetry, however, the internal

force system transferred across the plane may be shown to include only a mesiodistal force and a first-order couple, and neither of these components appears in a buccal view. Midsagittal-plane symmetry is often present in extraoral mechanics, and the buccal-view analyses to follow will assume that symmetry unless otherwise noted. Figure 9-9 shows a free-body diagram of the right half of a symmetric face bow as viewed from a buccal perspective. One outer-bow arm and one inner-bow arm appear in the sketch. The weight of the half face bow is neglected and contact force systems exist at three locations (although anteriorly where the left portion of the face bow connects to the right, no force-system components are present in this view as noted previously). The elastic, spring, or traction band contacts the outer-bow arm end and the force Pb is the buccal-plane component of the total force transmitted through this connection. (From a practical standpoint, the total activating force generally makes a small angle with the associated sagittal plane and, accordingly, in magnitude, Pb often differs negligibly from it.) The force system exerted by the buccal tube on the inner-bow arm end in this view is shown in its components; Q0, was seen earlier in Figure 9-2, is perpendicular to the occiusal plane, and Cb is a second-order couple. Measurement in this plane of the magnitude and direction of the force delivered to the outer-bow arm end, the anteroposterior distances i and o, and the angle 0b between the bow arms in this view, with force and moment balances, enable the determinations of the magnitudes of the three components at the inner-bow arm end. Although the figure shows the half face bow in its activated state,

FIGURE 9-9. A free-body diagram in a bucca! view of one-half of a symmetric face bow.

337


note that the sense of the couple Cb is directly related to the vertical

movement of the anterior tip of the face bow during activation. When activated, the face bow in Figure 9-9 rotated clockwise (from a buccal perspective); its anterior tip moved down, generating the counterclockwise, responsive couple Cb. The sense of the couple corresponds to the location of the line of action of Pb with respect to the inner-bow arm end. Had that line of action been positioned above the inner-bow arm end, the sense of the couple would have been clockwise and the displacements of the face bow generally and its anterior tip would have been counterclockwise and up, respectively, during activation. This knowledge is useful in the preparation of the face bow before engagement; as activated, the anterior tip must be positioned in the embrasure between the lips. Figure 9-10 shows a free-body diagram in the buccal view of the right maxillary molar; the buccal tube (which receives the force, originating extraorally, from the inner-bow arm end) is in effect, integral with the tooth. The three components of the force system exerted on the crown through the tube are action-reaction counterparts of the components of Figure 9-9. For now, any effects of intraoral mechanics upon the molar

are not included. The distributed periodontal response to the molar crown loading, as it would appear in this view, is shown as three resultant components referenced to the center of resistance of the molar. Estimating the long-axis distance a between the tube and the center of resist-

ance as one-half the overall tooth length, with the analysis of the free-body diagram of the half face bow, (Fig. 9-9), force and moment

R,

= R, = = Cb + Q0(a) FIGURE 9-10. A free-body diagram in a buccal view of a right maxillai'y molar subjected to face-bow action.

338


and C, as indicated. Since permit the determinations of this force system is action-reaction with the resultant components of the tooth action on the periodontium, the cervical-pull headgear exernplitied displays the following orthodontic-displacement tendencies associated with the buccal view: overall distal movement, distal crown tipping, and extrusion. A more direct procedure to determine the displacement potential in the buccal view would avoid the buccal-tube force analysis and proceed directly to the center of resistance. Such a process can be validly undertaken provided that the molar alone receives the head-gear force and is not in contact with any other appliance elements. Shown first in Figure 9-11 is a sketch showing the half face bow of Figure 9-8 and the molar of Figure 9-10. The force-system transmission through the inner-bow-armend and buccal-tube contact is now internal. Needed here, besides the magnitude and direction of Ph, is a line parallel to Pb through the center of resistance of the molar; the dimension e and the direction of Pb could be taken directly from a headplate radiograph, for example. The force Pb is then "moved" to the new location at the center of resistance, decomposed into mesiodistal and occlusogingival components, and the couple inserted to compensate for the change in lines of action of Ph (see Fig. 2-17). The transformed force system, now at the center of resistance of the molar, is displayed in Figure 9-12 and is action-reaction with the system of Figure 9-10; each component reflects an orthodontic-displacement tendency as previously described. The graphical force analyses of Figures 9-11 and 9-12, together with the associated displacement-potential interpretation, indicate that the direction of Ph and the proximity of its line of action to the center of resistance of the molar form the basis for the displacement-format prediction in the sagittal plane. The direction of the activating force is that balances

cre

FIGURE 9-11. A buccal view of the molar, the half face bow, and the activating force on the right side.

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=

= Pb(e)

e

FIGURE 9-12. A force system at the center of resistance of the molar, mechanically equivalent to Pb of Figure 9-10.

of the traction band, elastic, or spring as it connects to the end of the

outer-bow arm. Restricting attention to neck-pad anchorage, in the buccal view the line of action of Pb must pass through points at the outerbow end and at the back of the neck. Figure 9-13 shows an activated, cervical-pull face bow oriented such that the line of action of Pb passes through the center of resistance of the molar. Although no potential for second-order tipping results from this configuration, a sizable extrusive component of force is present with Pb directed toward the cervical region. The method of eliminating the extrusive potential, generally present in the cervical-pull appliance, is shown in Figure 9-14; the outer-bow arm bent down allows Pb to approach a direction parallel to the occiusal

plane, but also produced is a large distal-crown-tipping couple. The physical parameters available in this extraoral appliance with which to

ore

Pb

FIGURE 9-13. A buccal view of the face-bow configuration to minimize the potential for second-order tipping with the appliance subjected to cervical-pull action.

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C0 = Pb(e)

=

FIGURE 9-14. Outer-bow arms bend down to eliminate the potential for extrusion in cervical-pull action.

vary the location and direction of the line of action of Pb are the length of

the outer-bow arm and the angulation of the outer-bow arm, as activated, with respect to the occlusal plane. Specific combinations of arm length and angulation can result in coincident lines of action of for example, referring to Figure 9-13, an identical action can be obtained with a shorter outer-bow arm and a larger angle 6h or a longer arm and a

smaller angle. It is possible to create a mesial-crown-tipping couple, opposite in sense to the couples in Figures 9-12 and 9-14, but a long, outer-bow arm is needed that must be bent severely upward with respect to the occlusal plane. A line of action is obtained that passes superior to the center of resistance of the molar, but a sizable potential for extrusion is also created (again, because the line of action of passes through the cervical region). The buccal-view analyses discussed thus far have assumed that the maxillary molars are the sole recipients of the extraorally generated force

system. Although primarily providing distal-driving potential, and often also producing a tendency toward extrusion, the face-bow cervical-pull headgear may as well cause second-order rotations. If an arch wire is engaged into and stopped and tied against the buccal tubes, the force system delivered to the buccal tubes may be transmitted anteriorly and, therefore, distributed throughout the dentition and into the maxilla at large. To distribute all components of the delivered force in a nearuniform fashion to the entire arch—particularly the occlusogingival (vertical) and second-order rotational components—the arch wire must possess high bending stiffness. With such a wire engaged, the arch approaches a rigid entity and the procedure of Figures 9-11 and 9-12 may be used to predict displacement of the half arch as a unit, but a needed

reference is the center of resistance for the half arch. Although not

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known precisely, for an arch with all spaces closed the anteroposterior coordinate of the center of resistance will approximately divide the total

root-surface area of the teeth engaged in half; this places the reference point slightly anterior of the second premolar in a nonextraction case with engagement terminating with the first molars. With the known

geometry of the maxilla, the vertical coordinate may be reasonably placed near the apex level of the premolars. (Note that reducing the bending stiffness of the arch wire will effectively move the center of resistance posteriorly.) Figure 9-15 depicts an appliance assembly in which the cervical-pull force is, in effect, delivered to the entire half arch. (The magnitude of Pb here must be much larger than that delivered to and absorbed by the molar alone; assuming similar displacement formats, the delivered force

to the half arch should be magnified by a factor equal to the ratio of root-surface areas affected.) With the angle of the activated outer bow approximately the same as the inclination of the applied extraoral force, in addition to the distal action that would tend to reduce an overjet, the rotational potential is clockwise with the line of action of Pb occlusal of the center of resistance. The resulting tendency is to close down the bite in the anterior region. To create the opposite potential, toward opening the bite anteriorly, the line of action of must be located superior to the center of resistance. Hence, the outer-bow arms must be long and must be bent upward. Moreover, because of the sizable force required, the face bow must be very stiff since the application of force to its outer-bow arm ends will tend to deflect them toward the occlusal plane. Also seen in this arrangement, sketched in Figure 9-16, is a strong tendency for extrusion; this must be countered by posterior occlusion if opening of the bite (in the anterior region) is to be part of the overall result. A somewhat unusual application of the cervical-pull face-bow headgear is examined in closing this subsection. The observed, vertical move-

e

FIGURE 9-15. Cervical-pull, face-bow delivery of extraoral force to the entire maxillary arch to reduce overjet and an anterior open bite.

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= Pb(e) (Pb)

_

0

FIGURE 9-16. Face-bow geometry necessary for reduction of an overbite.

ment of the anterior tip of the face bow during activation has suggested

a means of intruding maxillary incisors or up to six anterior teeth. Spurs are affixed to the face bow to contact the incisal edges. Upon activation, the force applied to the outer-bow arm ends, in the conventional appliance transmitted through the outer and into and through the inner bow in an uninterrupted manner to the molars, is transferred instead in part to the anterior teeth by means of the spurs. The molars provide intraoral

anchorage and a fulcrum. A free-body diagram in the buccal view is shown in Figure 9-17. Again, symmetry with respect to the midsagittal plane is assumed and one half of the face bow is sketched. To obtain the desired action, the force must be located so as to produce a counter-

clockwise moment about the end of the inner-bow arm. Force- and moment-balance relationships are written in the figure. The three components at the inner-bow arm end plus the force Pa against one half of

the involved anterior teeth yield a total of four unknowns, one more

Fa

e

H'

ov

00 = 0, = F8 + = F8e) —

FIGURE 9-17. Modification of the face bow to gain the potential for anterior intrusion.

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than the number of available quasi-static equations. To complete the

solution, a reading of 2Fa would have to be taken with a force gauge following activation of the appliance. A

Coronal-Plane View and Comments on the Asymmetric Problem The delivery of force by the symmetric face-bow, cervical-pull headgear, viewed from the posterior into a coronal plane, is shown in Figure 9-18. This third plane view of the activated appliance indicates the potential, noted earlier, for faciolingual displacement of the molars arising from the vertical components transferred from the inner-bow arm ends and the locations of their lines of action with respect to the centers of resistance. Because the bow arms have circular cross-sections, with the delivery of force to the molars at the buccal surfaces and not through the centers of resistance, and if the senses of the vertical forces are as

L

Pv

L

L

Qv

Qv

+

L

FIGURE 9-18. Posterior views of the face bow (above) and the molars (below) showing buccolingual responses to vertical force components.

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shown, a substantial tendency exists for lingual crown tipping. If the

inner bow is initially passive buccolingually, the initiation of lingual crown tipping will generate the lateral forces (L) as indicated in the figure. Due to their characteristic lengths, however, the inner-bow arms are rather therefore, the magnitudes of these lateral components will build slowly and sizable lingual displacements could occur before the forces would become significantly resistive. Alternatively, the inner bow may be activated buccolingually such that the lateral forces

exist immediately upon engagement of the appliance, either with the senses shown or their opposites. In the latter configuration the lateral forces would enhance the lingual-tipping potential unless the vertical forces have opposite senses to those shown, and as such provide the potential for facial crown tipping. In summary, without the tendency for development of third-order resistive couples, the existing buccolingual stiffness of the inner bow is alone the available inherent deterrent to the tipping tendency associated with the vertical force components delivered to the molars. Throughout this section, in viewing force diagrams from a buccal perspective, the half face bow has been used and, with midsagittalplane symmetry present, the force system transmitted between the left and right halves of the face bow has no components in the midsagittal plane. In the asymmetric situation when, for example, the component in the buccal plane of force to the right-side outer-bow arm end is larger than its counterpart on the left side, perhaps all three midsagittal-plane components (anteroposterior force, vertical force, and third-order couple) will exist in the right-side buccal view. These components are not large compared to the individual, corresponding components transmitted from the inner-bow arm end to the buccal tube, but they render the problem quasi-statically indeterminate. Even the sense of the midsagittal-plane components may be difficult to determine; however, if the face bow is geometrically symmetric in the passive state, the observation of the displacement of the joint of the inner bow to outer bow upon activation can provide the sense(s) of one or more of the components. The full

analysis of the asymmetric headgear, to extend that portion that was undertaken earlier (see Figs. 9-5 through 9-7), is beyond the general level of presentation of this text and, perhaps, is of questionable clinical relevance.

Canine Retraction with Headgear movement of the maxillary or mandibular canine teeth into spaces left by the extractions of first premolars requires the controlled action of distally directed force. That force may be activated against posterior-segment anchorage, but to avoid the potential of unwanted mesial movement of those segments, associated with the response to the activation, extraoral anchorage may be used. Driving forces may be created through The

345


the extensional activation of a traction band, elastics, or springs, and

transmitted to the dentition through J-hooks. The elastics or springs are also attached to a head cap or neck pad; the traction band transmits the responsive force through a neck pad to the cervical region. The extraoral

ends of the J-hooks connect to the activating element(s) in a hinged fashion such that only a force—no couple—may be transmitted through the connection. Hence, the ends of the 1-hook tend to line up with the direction of the activating force. The end of the curved portion of the J-hook is formed around the arch wire and makes contact with the mesial extent of the canine bracket (in the absence of the need for an interconnecting, sliding jig); again, only a force can be transmitted through the connection. The principal intent is to deliver a distally directed force, without mechanical interruption, to the canine. The arch-wire system provides overall guidance—attempting, initially, to keep the canine in faciolingual alignment as the movement begins, and thereafter to influence the displacement format. Whether these mechanics are symmetric or not with respect to the midsagittal plane, sufficient in a transverse perspective is an occlusal

view and an analysis of just one side of the appliance because the Jhooks do not contact one another. The noteworthy features of Figure 9-19 are the occlusal-plane components of the right-side force of activation P0. and two responsive force components. If the elastic or spring is fastened to a head cap, P0 is posteriorly directed; with a cervical-pull strap, as indicated previously, the force has a small lingual component.

The J-hook connects to other elements only at its ends. Similar to a two-force member, the J-hook attempts to align its ends on the line of action of P0 upon activation. Because one end is intraoral and the other is extraoral, the alignment cannot be accomplished and L0 is created, exerted by whatever guides the J-hook. The third force on the J-hook, K3, is the collective response to the activation in this view of the bracket and

arch wire. These three forces must balance as indicated by the closed force triangle. The action-reaction counterpart of R0 is shown decomposed into distal and lingual components in the occlusal view of the canine that completes the figure. The distal component actually acts against the bracket and is the driving force: two comments concerning that component are pertinent: 1. Because its line of action does not pierce the long axis of the canine, antirotation mechanics are necessary (dashed in the figure) if distolingual rotation is to be prevented.

2. A portion of this component may not reach the canine, but due to friction it is taken into the arch wire and carried to the posterior segment.

The lingual component is generally undesired; acting through the arch wire, its tendency, at least during the initial portion of the canine movement, is to constrict the intercanine arch width. As the displacement proceeds, though, relative to an unchanging direction of R0, the distal

346


J-hook

R0 F0

P0

Force triangle

P0

(RO)d

Canine

— '(Counterrotation couple)

FIGURE 9-19. 0cc/usa/-view diagrams of the force systems exerted on the J-hook, canine crown, and arch wire during retraction of the right maxillary canine.

component increases and the lingual component decreases in magni-

tude due to arch curvature. Figure 9-20 shows a right-side buccal perspective of the maxillary canine together with the guiding arch wire, the J-hook, a sliding jig, and the active force component Ph depicted as posteriorly directed, characteristic of a straight-pull headgear. With the canine receiving the entire force in the absence of friction, the displacement analysis may be carried

347


Arch wire

jig

Pb

J-hook

FIGURE 9-20. A right-side buccal view of a maxillary canine under retraction through use of a straight-pull headgear.

out in somewhat the manner of Figures 9-11 and 9-12. Noting the posi-

tion of the center of resistance of the canine in this view relative to Pb, without the arch wire engaged the distal displacement would be largely uncontrolled crown tipping. To the extent that this tipping is unwanted, the arch wire through its bending stiffness is the available source of a counter-tipping couple, either present immediately upon arch-wire engagement through an active, second-order bend, or allowed to initiate and grow (in magnitude) following initial canine tipping that eliminates any second-order clearance and establishes two-point contact between bracket slot and arch wire. If the applied force includes a vertical component, that component will be transferred from the J-hook to the arch wire at their contact location. In the absence of the sliding jig, that vertical component will be largely carried to the canine but, depending on the bending stiffness of the arch wire, it may be distributed anteriorly and posteriorly from the contact location. With cervical anchorage, an occlusally directed component will be part of and add the tendency to extrude the canine and enhance anteroposterior, whole-arch curvature. On the other hand, activation of high-pull action from a head cap produces an opposite, vertical, displacement potential. Figure 9-21 shows the activation of a high-pull headgear to the mandibular canine in a buccal view. The location of the line of action of passing occlusal to the center of resistance of the tooth indicates a tendency for distal crown tipping which the arch wire may resist. The direction of reflects the existence, in addition to the distal driving component, of a vertical component with potential to extrude the canine and, with an engaged arch wire possessing moderate stiffness, to reduce toward reversal of any curve of Spee. Generally, the delivery of extraoral force to mandibular locations demands the consideration of several fac-

348


Pb

FIGURE 9-21. A right-side buccal view of mandibular canine retraction with a high-pull headgear.

tors beyond the direction of applied force with respect to the arch-form plane and the line-of-action location relative to a center of resistance. This discussion is undertaken in a separate, subsequent section of this chapter.

Extraoral Force Delivered to an Anterior Segment or an Entire Arch

section considers the bilateral delivery of force, originating extraorally, to various locations along the arch wire with the force carried into the oral cavity by J-hooks. The extraoral forces are generated by elastics or springs fastened posteriorly to a head cap or a neck pad, or possibly by a traction band transferring responsive force through a neck pad to cervical anchorage. The activating element(s) are connected to the J-hooks that carry the force to the arch wire. Contact with the arch wire on each side is made between the central and lateral incisors, distal to the lateral incisor, or distal to the canines. Through incorporation of This

sliding jigs, with the J-hooks contacting their mesial extents, the extraoral force may be transmitted directly to the buccal segments. Generally, force-transfer points on the arch wire are chosen accord-

349


ing to the intended distribution of the force delivered from the extraoral

source. The extent of that distribution depends largely on characteristics of the arch wire. Again, the connections allow transfer of force in any direction, but 1-hooks will not transfer couples. If the incisal segment alone is to receive the force, the transfer points are between the central and lateral incisors; if the six anterior teeth are to be displaced as a unit, the 1-hooks will likely deliver the force on each side between the lateral incisor and the canine. In these two instances, either the arch wire is not stopped posteriorly or loops in the wire are activated to aid in the displacement. If the intent is to affect the entire arch (and produce orthope-

dic movement), the transfer points are distal to the canines and the buccal segments are appropriately stopped and ligated together to ensure both anterior and posterior distribution of the active force. Shown in Figures 9-22 and 9-23 are occiusal-plane views of applied and responsive force systems with delivery (action-reaction) to the maxillary incisal segment and to the maxilla as a whole, respectively. Both free-body diagrams reflect geometric and mechanical symmetry relative to the midsagittal plane. In Figure 9-22 the distribution is shown uni-

Incisal

response

F0

PG

P0

FIGURE 9-22. A free-body diagram in an occlusa( view depicting extraoral-force delivery to the maxillary incisal segment.

350


response F0

Co

p0

p0

FIGURE 9-23. A free-body diagram in an 0cc/usa! view depicting extraoral-force delivery to the maxillary arch as a unit.

form over the four teeth, a reasonable expectation in view of the contactpoint locations and the fact that each force essentially reaches just two dental units. In the whole-arch distribution of Figure 9-23, the occiusalplane component of the contact force on one side divides with one por-

tion transmitted toward the anterior and the remainder carried first to the terminal molar and then, by means of ligation, anteriorly throughout the buccal segment. To attempt to quantify the distributions of force is to undertake the solution of a highly quasi-statically indeterminate problem, particularly in the latter instance. With a stabilizing arch wire engaged, however, those distributions probably appear somewhat as they

are sketched, and a rough approximation for one side is obtained by dividing P0 by the number of teeth affected in the half arch. Several additional points of consideration are pertinent to this occlusal-view discussion. Again, due to the direction of P(, and the location of

351


its line of action with respect to the anterior end of the J-hook, the lateral

forces appear as first noted in the canine-retraction procedure with headgear. The magnitudes P0. the design of the head cap, and the softtissue facial anatomy of the patient relative to the intraoral J-hook connection points all influence the size of the lateral force F0. For example, if

cervical pull is appropriate, the narrower width of the neck results in relatively small lateral forces when elastics or springs are activated against a neck pad. In contrast to the canine-retraction process, because in the present mechanics the force from the J-hook is transmitted directly to the arch wire, the entire force delivered is effective. Continuing

in the occlusal view, the intent is to transfer—wherever the delivery points along the arch wire are—posteriorly directed forces; the applied forces have that direction. Friction may, however, divert some of the force generated in the activating element. Frictional resistance will accompany the presence of the lateral force and essentially in direct proportion to the size of that force. The frictional force is transmitted by direct contact from the J-hook to its guide, into the anterior extent of the head cap, and to the side of the face or head. Also, in the arrangement of Figure 9-22, friction between arch wire and brackets may take a portion of the delivered force, intended exclusively for the incisal segment, posteriorly into the buccal segments. Precautions taken to keep the delivered force in the anterior segment include reducing the cross-sectional

wire size in the buccal segments and using light ligations—generally avoiding bracket-wire binding and angulations. A last point to be considered is midsagittal-plane symmetry. Although with the face-bow and canine-retraction headgear, asymmetric or unilateral action is now and then required, not so with the extraoral appliance presently being discussed. A symmetric activation is a possibility, using unlike elastics or springs, but with force delivery to the arch wire the inability to sufficiently control the distribution of that force seemingly renders the attempt useless. As in previous analyses within this chapter, the view of the extraoral appliance from a buccal perspective leads to the substance of the potential displacement format. Of particular importance are the direction and location of the line of action of the activating force with respect to the dental units to be affected. As in earlier discussions of symmetric headgear, attention is focused on one elastic or spring or traction-band force and the right half of the arch (to which the J-hook is attached). In the buccal view the J-hook will align with the direction of the force generated in the activating element, so both the angulation of the force and its line-of-action location are determined by two points: the connection of

the activating element to the head cap or neck pad and the point of contact of the anterior end of the J-hook with the arch wire (or spur affixed to it). The relative magnitude and the sense of the vertical component of the active force are controlled largely by the anchorage arrangement (high-, straight-, or cervical-pull). The direction of the buccalview component of active force may range from 45° or more above the occlusal plane to, perhaps, 15° below it. This angulation is also affected

352


somewhat by the point of delivery to the arch wire; for example, for a

given head cap the high-pull angulation is greater with delivery distal to the canine than when the J-hook attachment is between the central and lateral incisors. Although these headgear are often employed to reinforce buccal anchorage intraorally, they may be used to produce wanted displacements and the function of the arch wire is solely to control the distribution of the extraoral force. The analysis procedure, first illustrated in Figures 9-11 and 9-12, may be used. (This procedure may also be used in sepa-

rate analyses of the effects on the same portion of the dentition of extraoral-force application and those of intraoral mechanics, the two to be superimposed.) The process requires initial identification of the portion of the half arch to receive the headgear force. Next, the location of the center of resistance of the segment is approximated in the manner discussed previously (influenced by the distribution of root-surface area and by the distribution of force determined by the extent and the bending stiffness of the arch wire). The characteristics of the delivered force are then evaluated in terms of the dental segment receiving it and the force is "moved" to the center of resistance. Figures 9-24 and 9-25 depict

buccal views of straight- and high-pull-headgear action to the entire maxillary arch. To distribute the force as uniformly as possible, the attachment is to a location near the center of the half arch; a stabilizing arch wire having a large cross-section should be used. The extraoral force is transferred to the center of resistance of the half arch, the necessary couple is added in association with the line-of-action displacement in Figure 9-24, and in Figure 9-25 the force is decomposed into anteroposterior and vertical components. The proper size of extraoral force is

determined by the procedures discussed in Chapter 5; the magnitude

FIGURE 9-24. A buccal view of straight-pull, extraora! force delivered to the maxillary arch.

353


FIGURE 9-25. A buccal view of high-pull extraoral force delivered to the maxillaiy arch.

must be relatively large to produce an orthopedic displacement of the maxilla.

In Figure 9-25 the potential whole-arch displacement components are

intrusive and toward the posterior; if the line of action passes through the center of resistance, bodily movement (translation—no rotation) should occur. Any overjet should be reduced and any intrusion of a previously leveled arch would diminish an existing anterior deep bite. With the straight-pull action of Figure 9-24, the potential for reducing overjet is again present, but vertical displacements are associated with the overall rotational potential (clockwise) suggested by the couple. Although no whole-arch vertical movement should take place, clockwise rotation of the maxilla, as viewed from the right side, results in closing or deepening of the bite anteriorly. If the mechanics are changed only by using a flexible rather than a stiff arch wire, the center of rotation will be more anteriorly located and the vertical action would be more concentrated in the locality of the connection of the J-hook to the arch wire. The high-pull headgear would tend to intrude the middle portion of the half arch, reducing any curve of Spee toward reversing that curvature. Addi-

tional effects of using the more flexible arch wire would be a minor shifting of the force distribution posteriorly and possibly reducing the stability of the mechanics. (In Figure 9-24 this reduction in stability would become a concern only after some rotation had occurred, creating

an angle between the occlusal-plane and J-hook directions. Action would then, in effect, be transferred toward the anterior, resulting in more pronounced anterior bite closure and a reversed curve of Spee.) Extraoral force is delivered to the anterior or incisal segment to pro-

354


duce localized displacements there or to offset potential, unwanted dis-

placements associated with intraoral mechanics. To concentrate action to the incisal segment, the engaged arch wire may be flexible and left rela-

lively unrestricted in the posterior segments to slide through brackets and buccal tubes. Figure 9-26 shows the action on an incisal segment from a straight-pull headgear. Replacing the I-hook force with an equivalent force system at the center of resistance of the segment reveals, in addition to the potential for lingual driving, a rotational tendency toward increasing the long-axis angulation with respect to the occlusal plane. In the absence of crown contacts with canines and without resistance from the arch wire from the posterior, a simple-tipping displacement is produced. Substitution of cervical-pull force has little effect on the moment arm (and the potential for rotation) but adds the tendency for extrusion. Replacing the straight-pull with high-pull action provides an intrusive potential and reduces the length of the moment arm. The delivery of extraoral force to the incisal or anterior segment may also generate a third-order torque in the arch wire. Figure 9-27 shows the delivery of high-pull extraoral force to a hook, affixed to the arch wire and extending gingivally from it, conceivably in an effort to direct the line of action of the delivered force through the center of resistance of the segment. Because the active force has a moment arm with respect to the arch wire, a force and couple are transmitted into the wire. For the objective (bodily movement of the segment) to be achieved, the couple (torque) must be transferred to the anterior teeth; hence, the arch wire must be rectangular or have torquing loops incorporated into it. Figure 9-28 shows a free-body diagram of an anterior section of arch wire with the hook attached, indicating the force and couple delivered to

= Pb(e)

e

FIGURE 9-26. A buccal view of straight-pull-headgear force to the maxillary incisal segment.

355


C re

J-hook

Arch wire

FIGURE 9-27. A buccal view of high-pull-headgear force to the maxilla,y incisal segment.

the wire; also shown is the action-reaction transfer of the force system from the wire to the segment. Similar force systems may also be produced by a straight- or cervical-pull appliance, but differing from that in Figure 9-28 in the vertical component. The anchorage assembly is chosen according to the desired direction of the applied force; controlling the magnitude of the torque delivered are the magnitudes of the applied

T0

H0

F0 =

Ta = F0(e)

FIGURE 9-28. Force diagrams in the buccal view showing headgear-generated torque in an anterior section of a rectangular arch wire (left) and the transfer of that torque to the incisal pair (right).

356


and its moment arm with respect to the center of the wire crosssection. Also, if the hook extends occlusally from the wire, the sense of the torque is reversed from that of Figure 9-28. Note that the couple generated in the arch wire differs conceptually and, therefore, in the manner of interpretation with regard to displacement potential, from the couple obtained in moving the line of action of the extraoral force directly to the center of resistance of the segment. For example, in Figure 9-27 the extraoral force delivers a torque to the arch wire, but the mechanically equivalent force system at the center of resistance includes no couple. Furthermore, the displacement potential of an extraoral force is highly influenced by any labiolingual action or resistance from the arch wire; again using the example of Figure 9-27, with anteroposterior holding force generated by intraoral mechanics exerted against the anterior crowns (in addition to the headgear action), the anticipated displacement is lingual root-torque rather than bodily movement. To obtain extraoral force transmission to the buccal segments using J-hooks, because the J-hooks cannot reach these segments directly, either the force must be transmitted through the arch wire or sliding jigs must be used to deliver the force to the premolars (or to the retracted canines) and carried distally by crown contact. The former procedure has been discussed previously; the advantage of the latter is the facility to keep the extraoral force, in the absence of friction, from affecting the anterior teeth. The headgear assembly including the sliding jigs may be used to drive the buccal segments distally, with the arch wire as a guide, or to reinforce the anchorage provided by those segments in response to intraoral mechanics. Generally, a straight-pull appliance is used in this particular application; any vertical components of delivered force will be transferred to the arch wire at the points where the J-hooks contact the mesial extents of the sliding jigs. (To transmit action having a vertical component to the posterior segments from extraoral activation, a face bow must be used.) Although the examples cited in this section may seem to restrict the delivery of extraoral force through J-hooks to the maxillary arch, headgear of this type may also interact with the mandibular dentition. Figures 9-29 and 9-30 illustrate force transmission to produce incisal-segforce

ment uprighting and to reinforce posterior anchorage, respectively. Note that, although in Figure 9-29 the high-pull force cannot pass through the center of resistance, a cervical- or straight-pull headgear to a pair of hooks affixed gingivally to a rectangular arch wire might substantially reduce the potential for lingual crown-tipping and, with the arch wire providing a holding force, third-order rotation opposite in sense to that suggested by the figure can be achieved. Although posteriorly di-

rected active-force components exist with delivery to either arch, the contrasts in effects (in one arch versus the other) lie in part in the vertical components of force and the line-of-action locations with respect to centers of resistance. General considerations of headgear to the mandibular arch are discussed in the following section.

357


FIGURE 9-29. A buccal view of high-pull action to the mandibular incisal segment.

FIGURE 9-30. A buccal view of straight-pull action to reinforce anchorage in the mandibular arch.

358


Delivery of Extraoral Force to the Mandibular Arch Although the extraoral appliance most often delivers force to the maxil-

law arch wire and dentition—rightly given principal attention in this chapter—extraoral force also may be directed to the mandibular arch. Many of the applications are counterparts to extraoral actions to the maxillary dentition: retraction of canines, anterior retraction, and reinforcement of posterior anchorage, for example. In several of the preceding sections, some contrasts between maxillary- and mandibular-arch

delivery of extraoral force have been mentioned. In the view into the occlusal plane the differences in the force diagrams are not highly notable. The comparable lateral widths are smaller in the mandibular arch, which results in slightly larger fractions of activating force projected into

the coronal plane compared to their maxillary-arch counterparts, but this differential is unlikely to be clinically significant. Comparisons as viewed from a buccal perspective, however, suggest somewhat more substantial differences.

Generally, using the cervical-pull force to the maxillary arch as a reference, the angulation of the line of action of the headgear force is steeper to the mandibular arch than to comparable locations in the maxillary arch, even with the teeth in occlusion. With the mouth opened, not only the angulation, but also the magnitude of the headgear force may be increased, depending on the location and orientation of the activating force. Differences in line-of-action locations with respect to centers of resistance are also noteworthy. To approach the center of resistance of the maxilla, superior to the line of brackets, a high-pull headgear must deliver force to the arch wire anterior to the center of resistance (see Fig. 9-25). To align the active force with the center of resistance of the mandibular arch, gingival of the bracket line, the high-pull appliance must attach to the arch wire posterior to the center of resistance. When a vertical component of headgear force exists, for example with high-pull action, an intrusive displacement potential is created with force delivery to the maxillary arch, but an extrusive tendency is developed with activation of this headgear to the mandibular arch. The analysis approach to determining the displacement potential of a given appliance or choosing the particular headgear to produce a desired displacement is the same in format, whether the appliance engages the maxillary or mandibular arch. Of particular interest, however, is a comparison of internal forces in the dentofacial complex between the dentition and the head cap or neck pad. Activation of an extraoral appliance to the maxillary arch creates an internal force system within the bones of the skull. For example, the high-pull headgear generates internal compression, shown in simplistic form in the schematic in Figure 9-31. In effect, because both the active and responsive portions of the headgear force system are against the same structure (the head), the force transmitted into the maxilla travels posteriorly and internally

359


Fsuperior

F posterior

FIGURE 9-31. A buccal view depicting the internal torces generated in the skull by a high-pull headgear to the maxillanj arch.

through continuous tissue structure to the anchorage location. The consequences of the internal compression developed are not substantial due to the sizable volume of bone through which the force is distributed.

The extraoral force delivered to the mandibular arch is carried through the dentition and into the mandible, and then it must also be transmitted to the head cap or neck pad. One or a combination of three paths must be taken by this force: through the dentition (only possible when the teeth are occluded); through the masseter, temporal, and pterygoid muscles, having skull and mandible connections and resistance only to tensile loadings (similar to an elastic); and through the temporomandibular joints (TMJ). Figure 9-32 shows a free-body diagram in the

buccal view of the half mandible (midsagittal-plane symmetry assumed), with a straight-pull headgear delivering force. The position of the buccal-plane component of the headgear force Pb, with respect to the TMJ, tends to rotate the mandible open. In this configuration there is little or no transfer of the headgear force through the dentition; the force of occlusion would be vertically downward and, therefore, could not oppose In essence, as shown in the figure, the responsive force system with forms a pair of opposing couples in the buccal view: the

360


—.

(Ftm,)v

(Ftmj)a

FIGURE 9-32. A free-body diagram in the buccal view of the mandible with the mandibular arch subjected to straight-pull extraoral action.

anteroposterior component of the TMJ force with Ph and the vertical component of the TMJ force with the resultant of the tensions generated in the musculature. Hence, the resultant force against the TMJ is larger than this headgear pushes the condyle posteriorly and superiorly, compressing the articular disc against the postglenoid process. A similar free-body diagram in the buccal view of the mandible subjected to high-pull extraoral force is displayed in Figure 9-33. With this headgear arrangement, the contribution to the responsive force system of the resistance of the musculature is apparently small and a portion of the response is through the occlusion, wherever it exists. The senses of the TMJ-force components are the same as in Figure 9-32, but the total TMJ-force magnitude is less than that of Pb. The dashed rectangle at the condyle suggests the relative magnitudes of the vertical and anteriorly directed responsive components there; the force triangle (upper right) shows the relationship among the three balanced forces and suggests the effect upon the two responsive forces of a change in angulation of Note that an increase in the angle of with the occlusal plane creates an

361


Ph

FIGURE 9-33. A free-body diagram in the buccal view of the mandible with the mandibular arch subjected to high-pull-headgear force.

increase in the magnitude of and a decrease in the size of Ftmj. If the angulation of Pb is lessened such that its line of action passes below

the TMJ, a jaw-opening moment is present, disappears, the muscle forces appear to counteract the potential jaw opening, and the force diagram begins to resemble that of Figure 9-32. In summary, the use of headgear to the mandibular arch may be effective in producing wanted dental-unit segment displacements. Force-displacement analyses would proceed along lines identical to those for headgear to the maxillary arch. Of some concern in delivering extraoral force to the mandibular arch, however, may be the effects of jaw position on the activating forces and the responsive-force return paths, particularly through the TMJ.

Dual-Force Headgear Assuming clinical acceptability of extraoral action to either arch, consid-

eration may be given to the delivery of headgear force to both arches simultaneously. Potential applications include buccal anchorage reinforcement for concurrent interarch mechanics, simultaneous retraction of canines in both arches, and concurrent retraction of incisal-anterior segments. The activating forces, two to each side of the arch, result, generally, in an increase in the resultant magnitude of responsive force at anchorage compared to that generated by headgear to the single arch.

362


One head cap or a head cap to support the maxillary-arch activation and

a neck pad to support the force system to the mandibular arch might be employed. The larger, total responsive force suggests that the area over

which the anchorage force is distributed should be increased; this is particularly important when a dual-force headgear assembly is supported by just a head cap. If complementing displacements are to be produced in the two arches, the magnitudes of force to the individual dentitions must be coordinated to avoid interarch interferences. Practical limitations exist with regard to the directions and locations of the lines of actions of forces to the two arches. For example, arranging a high-pull activation to the mandibular arch and a cervical- or straightpull force system to the maxillary arch would require the lines of action of activating forces on each side to cross one another. The overall analysis of a dual-force headgear is accomplished, if one pair of forces is transmitted to each arch, by separate force-displacement

analyses of the actions to the individual arches. In this approach, the procedures discussed in preceding sections of this chapter are employed. Conceptually, the application of dual-force action to a single arch may be considered. To be determined initially is whether the actions of the two forces (per side) are (1) separable in their effects on different portions of the dentition or (2) complementary toward a more uniform distribution of force through a segment or the arch. If the former is the case, two individual analyses are undertaken; if the latter is

the situation, the resultant of the two active forces (on each side) is obtained and a single analysis is carried out within the procedures already discussed.

The Chin-Cap Assembly Although ordinarily categorized as an orthopedic appliance, discussion of the chin-cap assembly is warranted here because of its geometric and

mechanical similarities to orthodontic headgear and its possible inclusion in the reverse-pull (extraoral) appliance examined in the succeeding section. The components of the assembly include a molded or fitted cap that extends superior to the point of the chin without encroaching on the lower lip and also posterior to distribute force (pressure) frontally and beneath the chin. Because sizable magnitudes are generally employed, the force is generated most often by a pair of traction bands; springs or heavy elastics may also be used. The activating elements are connected to a head cap that transmits the responsive force to the cranium. Deferring, as mentioned, consideration of its potential role in reverse-pull extraoral mechanics, the chin-cap assembly has one of two functions: as an orthopedic device to attempt to retard or restrict the growth of the mandible in individuals exhibiting Class III tendencies, or as an orthodontic appliance to produce or reinforce posterior forces of occlusion toward intrusion of the buccal segments and closure of the anterior open bitef

363

Extraoral Appliances P0

F0

Chin cap

p0

FIGURE 9-34. A free-body diagram of the chin cap in an 0cc/usa! view.

Figure 9-34 shows an occiusal-plane view of the force system exerted on the chin cap. Little of note appears in this sketch except for the midsagittal-plane symmetry. The force F0 is the action-reaction counterpart of the resultant of the posteriorly directed component of force distribu-

tion of the chin cap against the chin; its magnitude depends on the amount of traction and the direction of the activating force with respect to the occiusal plane. The right-side buccal view of Figure 9-35 shows one-half of the chin

Pb

Chin cap I I

FIGURE 9-35. A buccal view of the chin cap subjected to high-pull-headgear force.

364


subjected to active and responsive sagittal-plane force components. High-pull action gives rise to both anteroposterior and superior force distributions between the cap and chin. The design of the assembly should be such that, in this view, the line of action of the active force passes through the center of resistance of the mandible; this establishes the need for a vertical force component. Because the activating element contacts the chin cap posterior to the point of the chin, the fit must be good and the material of the cap must extend substantially to the posterior beneath the chin so as to counteract any tendency for cap rotation (clockwise in this view) and dislodgement. Figure 9-36 shows verticalpull action intended to produce occlusion and, by action and reaction, cap

intrusion of the molars and, possibly, the premolars. The action is meant

to be perpendicular to the occlusal plane and the force is directed through the center of resistance of the set of dental units to be intruded. To this end the force of the chin cap upon the chin is to be relatively concentrated beneath the posterior portion of the mandible and the ma-

terial of the chin cap must, therefore, extend posterior of the line of action of the active force.

As with the extraoral force delivered to the mandibular arch, the activated chin-cap assembly creates internal, responsive force systems that travel between the locations of active-force delivery and the anchorage. In the appliance shown in Figure 9-36 the responsive force is intended to be transmitted from the mandible entirely through the buccal segments; no force is meant to be carried through the TMJ. With the orthopedic appliance activated by high-pull action to oppose growth,

Chin cap FIGURE 9-36. A buccal view of the chin cap subjected to vertical-pull action.

365 Extraoral Appliances

however, the location and direction of the line of action of the active

force is similar to that of Figure 9-33; hence, activation appliance produces notable responsive forces in the TMJ. As long as the lines of action of the active forces to the chin pass superior to the condyles,

tend to rotate the lower jaw open and, conse-

the appliance does quently, the inuscl responding to it.

not generally assume any role in

Reverse-Pull Appliances With conventional extraoral anchorage at cranial or cervical locations,

the principal action in the direction of force delivery with headgear is toward the posterior. The attraction of extraoral anchorage combined with the clinical need at times for anteriorly directed active force has led to the development of "reverse-pull" headgear. Because the anteropostenor action of the reverse-pull appliance is opposite in sense to that of other headgear, and since the transfer of responsive force to extraoral anchorage must be through pressure, the principal anchorage for the reverse-pull device must be at the front of the head. Specifically, three locations are available over which the responsive force may be distributed: the forehead, the region between the nose and the upper lip and encroaching into the cheek regions, and the front (protuberance) of the chin.

Several reverse-pull designs have been marketed, and all use some form of head cap. A view into the transverse plane highlights the essentials of the reverse-pull appliance. Figure 9-37 schematically illustrates P0

Bridge

a0

2F0 + 2P0

a0

P0

FIGURE 9-37. A free-body diagram in the occiusal view showing action and response of the reverse-pull headgear.


the typical occiusal-view components of the force system exerted on the

usually symmetric headgear. The two pairs of forces posteriorly directed are the forces F0 exerted by the activating elements and the forces P0 directed from the head cap against the anchorage element. The force Q0

from pressure contact between the anchorage element and the delivery region of the responsive force. Neglecting any minor lateral components of the force system, in this view an anteroposterior force balance and action-reaction indicates that the anchorage force is greater than the activating force by the magnitude of force transmitted from the head cap. The head cap exists only to provide overall stability to the device, but its presence places an additional burden of force on the anarises

chorage. Head band

\

Pb

/

\\

Distribution of responsive force from forehead

Plastic bar

Cheek pad

Fb

FIGURE 9-38. A free-body diagram in the buccal view of a "cantilevered," reverse-pull extraoral appliance.

367


Three basic designs of reverse-pull appliances are currently marketed. Not mentioned here in any particular order, one is a device having a plastic bar cantilevered from a stiff head cap and, perhaps, partially supported by cheek pads. Elastics are activated from the "free" end of the bar and attached intraorally. A free-body diagram in the buccal view of one-half of the (symmetric) appliance (halved at the midsagittal plane) is presented in Figure 9-38. To keep the force R against the cheek minimal, a sizable couple Cb must be developed as shown; actually, together Pb and Cb form the resultant of a nonuniform pressure distribution as illustrated, demanding substantial vertical width and overall stiffness and stability in the head band. Figure 9-39 provides a free-body diagram

F,,,

Forehead pad

F9, = F, h +

=

1-I

—

+ a fraction of F9 H

r L___ Head strap

Bridge

h

Chin cap

FIGURE 9-39. A free-body diagram in the buccal view of a "split-anchorage," reversepull, extraor& appliance.

368


in the buccal view of a design that divides the frontal anchorage between

the chin and the forehead (or encircles the face much like a baseball catcher's mask). Although perhaps providing the greatest area over which to distribute the responsive force, considering the three designs, allowance should be incorporated for the normal relative movement of the mandible. The wire bridge between the head band and chin cap should, therefore, be free to slide vertically with respect to the head band. The inclined strap (shown dashed) must be relatively flexible, but must carry sufficient force F9 to hold the appliance in place. A static analysis of the horizontal forces is given within the figure; the responsive force magnitudes and depend on both the elastic and strap fractions.

Figure 9-40 shows the force systems in buccal view exerted on a reverse-pull appliance that uses a chin cap for anchorage. The force system on the left exists before the elastics are in place; after the elastics have been activated (F,,) the force system is that shown on the right in the figure. Both sketches are provided to show the differences primarily in the magnitude and location of Q,,, resulting from placement of the elastics. The connection between the wire bridge and the chin cap must

be rigid since F,, tends to rotate the bridge about that location. This potential for rotation is then transferred into and through the chin cap to the chin; hence, the design of the chin cap, the location of the connection of the high-pull fraction band to the chin cap, and the band traction

must all be coordinated to prevent the rotation. (A cervical strap attached to the chin cap can help offset the rotational potential.) Here again, the head cap primarily provides stability and the resultant pressure delivered to the anchorage is the superposition of the contributions from the elastics and the head cap (through the fraction bands). In designs where the chin is used partially or totally as anchorage, the potential exists for interference with normal growth of the mandible; also, internal forces are created within the TMJ that would not otherwise be present (see Fig. 9-33). The force transmitted from the wire bridge to intraoral locations is obviously restricted to an extent in direction. Vertical components are virtually nonexistent. If the elastics are not to cross the occlusion, lateral

Bridge

Chin cap FIGURE 9-40. Free-body diagrams in the buccal view of a reverse-pull appliance incorporating a chin cap before (left) and after (right) activating the elastic(s) anchored to the bridge.

369


force

components may be present, largely depending on the points of

delivery of the force. Without crossing the occiusal surfaces and deliver-

ing force to the lingual crown surfaces, the buccal segments are not directly accessible; however, force could be carried to posterior locations by the arch wire or by sliding jigs. Beyond these apparent limitations, the reverse-pull appliance may be used in a variety of treatment protocols calling for anteriorly directed force. Force may be transmitted to either arch (although of some concern must be the transmission of force to the maxillary dentition from the chin cap in that jaw movements alter characteristics of that force). Incisors crown-tipped lingually or canines distally inclined may be uprighted. Torque may be induced in the anterior portion of a rectangular arch wire by delivering the force occlusally or gingivally (with respect to the arch wire) to affixed spurs or hooks (see

Figs. 9-27 and 9-28), and labiolingual root movement may be accomplished with the incorporation of a posteriorly directed holding force generated intraorally. Although requiring a sizable magnitude of force, the reverse-pull headgear may be used to advance the maxilla and, with the third of the discussed designs above, simultaneous, potential resistance to growth or protrusion of the mandible is inherently incorporated. Although usually two elastics, equidistant from the midsagittal plane, would be used to deliver anteriorly directed force, thereby maintaining the symmetry of the appliance, if not so large as to impair the stability of

the device one force could be generated in an elastic attached to the bridge and delivered toward alignment of a single, malposed tooth.

Synopsis This chapter investigated the force systems created in the activation of

the extraoral appliance. Discussion has centered on the displacement potential of each type of headgear examined. Although references have been made to intraoral holding forces and anchorage reinforcement, detailed evaluations of orthodontic-appliance assemblies incorporating both intraoral and extraoral components were purposely excluded. Examples of the various classifications of headgear received attention, but not all combinations of activating elements, delivery mechanisms, and anchorage units were discussed. Not specifically undertaken, for example, was an examination of the delivery of high-pull action to a face bow. The force-displacement analysis of this particular headgear, however,

should not be difficult for the reader who understands the procedures and results expected from the individual applications of cervical-pull face-bow and high-pulliJ-hook appliances. Although perhaps differing significantly in appearance and composition, several headgear designs may fit a specific clinical need. No attempt is made here to critically evaluate designs that appear to be mechanically similar; however, for a particular application, the reader should now be prepared to carry out such a comparison, and to impart

370


the facility to do so is a principal objective and thrust of this chapter.

With the possible exception of the rather recently developed reversepull devices, the basic headgear discussed herein are firmly entrenched in the accepted array of orthodontic appliances. It is necessary to comprehend their potential and shortcomings with regard to use alone, with an arch wire merely to distribute the delivered force, or together with active intraoral mechanics. In the chapter to follow, a variety of orthodontic procedures discussed in a force-and-structural analysis setting and the interaction between intraoral and extraoral appliances in simultaneous use are explained.

Reference Drenker, E.W.: Unilateral cervical traction with a Kloehn extraoral mechanism. Angle Orthod., 29:201—205, 1959.

Suggested Readings de

Alba, J.A., Chaconas, S.)., and Caputo, A.A.: Orthopedic effect of the extraoral chin cup appliance on the mandible. Am. J. Orthod., 69:29—41, 1976.

de Alba, J.A., Chaconas, S.J., and Emison, W.: Stress distribution under highpull extraoral chin cup traction. Angle Orthod., 52:69—78, 1982.

Armstrong, M.M.: Controlling the magnitude, direction, and duration of extraoral force. Am. J. Orthod., 59:217—242, 1971.

Badell, M.C.: An evaluation of extraoral combined high-pull traction and cervical traction to the maxilla. Am. J. Orthod., 69:431—466, 1976. Baldini, C.: Unilateral headgear: Lateral forces as unavoidable side effects. Am. J. Orthod., 77:333, 1980.

Baldini, G., Haack, D.C., and Weinstein, S.: Bilateral buccolingual forces produced by extraoral traction. Angle Orthod., 51:301—318, 1981. Baldridge, J.P.: Unilateral action with headcap. Angle Orthod., 31:63—68, 1961.

Barton, J.J.: High-pull headgear versus cervical fraction: A cephalometric comparison. Am. J. Orthod., 62:517—539, 1972. Block, A.J.: An analysis of midline and off-center extraoral force. Angle Orthod., 32:19—26, 1962.

Fischer, T.J.: The cervical facebow and mandibular rotation. Angle Orthod., 50:54—62, 1980.

Greenspan, R.A.: Reference charts for controlled extraoral force application to maxillary molars. Am. J. Orthod., 58:486—491, 1970.

371


Haack, D.C., and Weinstein, S.: The mechanics of centric and eccentric cervical

traction. Am. J. Orthod., 44:345—357, 1958.

Hershey, H.G., Houghton, C.W., and Burstone, C.J.: Unilateral facebows: A theoretical and laboratory analysis. Am. I. Orthod., 79:229—249, 1981. Jacobson, A.: A key to the understanding of extraoral forces. Am. J. Orthod., 75:361—386, 1979.

Jarabak, J.R., and Fizzell, J.A.: Technique and Treatment with Light-wire Edgewise Appliances. 2nd Ed. St. Louis, C.V. Mosby, 1972, Chapter 7.

Kloehn, S.J.: An appraisal of the results of treatment of Class II malocciusions with extraoral forces. In Vistas in Orthodontics. Edited by B.S. Kraus and R.A. Riedel. Philadelphia, Lea & Febiger, 1962, pp. 227—258.

Kuhn, R.J.: Control of anterior vertical dimension and proper selection of extraoral anchorage. Angle Orthod., 38:340—349, 1968. Merrifield, L.L., and Cross, J.J.: Directional forces. Am. J. Orthod., 57:435—465, 1970.

Oosthuizen, L., Dijkman, J.F.P., and Evans, W.G.: A mechanical appraisal of the Kloehn extraoral assembly. Angle Orthod., 43:221—232, 1973.

Osvaldik-Traph, M., and Droschl, H.: Upper headgear versus lower headgear, yokes, and Class II elastics. Angle Orthod., 49:57—61, 1979. Perez, C.A., de Alba, J.A., Caputo, A.A., and Chaconas, S.J.: Canine retraction with J hook headgear. Am. J. Orthod., 78:538—547, 1980.

Ringenberg, Q.J., and Butts, W.C.: A controlled cephalometric evaluation of single-arch cervical traction therapy. Am. J. Orthod. 57:179—185, 1970.

Tabash, J.W., Sandrik, J.L., Bowman, D., Lang, R.L., and Klapper, L.: Force measurement and design of a torquing high-pull headgear. Am. J. Orthod., 86:74—78, 1984.

Thurow, R.C.: Edgewise Orthodontics. 4th Ed. St. Louis, C.V. Mosby, 1982, Chapter 19. Thurow, R.C.: Atlas of Orthodontic Principles. 2nd Ed. St. Louis, C.V. Mosby, 1977, Chapter 16. Watson, W.G.: A computerized appraisal of the high-pull face-bow. Am. J. Orthod., 62:561—579, 1972.

Worms, W.W., Isaacson, R.J., and Speidel, T.M.: A concept and classification of centers of rotation and extraoral force systems. Angle Orthod., 43:384—401, 1973.

Force and Structural Analyses of Representative Orthodontic Mechanics tul© The preceding chapters introduced the concepts and procedures neces-

sary to undertake force and structural analyses of orthodontic appliance elements. Although all preparations were discussed with the intended application in mind, that application has not been examined in its totality to this juncture. It is appropriate to indicate initially that the analyses that follow are exemplary; although specific analyses may be more interesting and meaningful to some than to others, the intended emphasis is on the analysis procedure and on the logical, orderly succession of steps within it. Discussions are carried out with pertinent, accompanying diagrams. Of the sample mechanics examined, the overall procession is from the relatively straightforward to the more complex, wherein the level of complexity is associated with that of the involved dentition, that of the appliance and the force system exerted upon it, and the sophisti-

cation of the modeling that must be undertaken to ensure an understandable and self-validating solution. At the outset, it is helpful to recollect several of the more prominent concepts previously introduced. The tooth crowns are subjected to mechanical force systems exerted by the appliance; the characteristics of these crown force systems determine the potential displacement patterns of the dental units (Chap. 5). The appliance, as activated, exists in a state of quasi-equilibrium which implies an almost exact balance of force and moments within the total force system (Chap. 2). The forceanalysis problem is inherently indeterminate and, therefore, active configurations relative to passive constraints and stiffnesses (Chap. 3) are keys to the completion of a solution. Differential force and anchorage are to be clearly understood, inasmuch as the engaged orthodontic structure is attached to a nonrigid "foundation," the dentition. Finally, the assem372

373

Force and Structural Analyses of Representative Orthodontic Mechanics

bly under study is, strictly speaking, a dynamic one, which means that

time is a principal independent parameter and deactivation is an ongoing, between-appointments process that must not be overlooked while carrying out instantaneous analyses. Each individual analysis follows a common procedural outline. An overview of the as-activated force system is undertaken in the context of the intended objectives of the appliance. As necessary and warranted, the actual appliance and the involved dentition or segment thereof are then modeled with care; the intent is the subsequent accomplishment of an analysis in the absence of undue involvement and in the presence of

the principal aspects of the problem. The analysis itself is pursued, using the four-step approach outlined in Chapter 6. Following the discussion of active, reactive, and responsive force systems exerted on and

existing within the appliance, exerted on the dentition, and existing within the periodontium, the significant structural characteristics of the appliance may be considered and, possibly, a rationale for improving the design established. Toward quantification of the components of the force systems, needed measurements at activation must be indicated before the computations of force magnitudes are begun. Although each example pertains to only a portion of the total therapy plan for the patient, the individual tooth and segment displacements considered must be examined within the framework of overall treatment objectives. The functional requisites are to achieve, or at least approach to the extent possible, ideal tooth positions and orientations as well as ideal arch-form geometry. Proper interdigitation and relationships of dentition to basal bone are sought. The positions and orientations of the maxilla and mandible, with treatment completed, are expected to reflect facial harmony and skeletal balance. Operational relationships— occlusion and TMJ function—must be proper at the conclusion of active treatment. Finally, correctional procedures are undertaken to promote

retention of the as-treated, realigned, dental-unit configurations and supporting-tissue positions. The implicit, biomechanical objectives of this final chapter are two in each example. First, given the force system generated at the activation site(s) by the appliance or a portion of it, an overall force analysis involving the appliance structure and the supporting dentition is undertaken toward examination of all potential dental-unit displacements. Both the displacements desired and intended and those unwanted—generally associated with the responsive force system—are scrutinized. Second, a structural evaluation of the appliance and its elements may be pursued with a view toward potential failures and, perhaps, ways in which the design might be positively modified. Although any one of the examples to be considered might easily be subjected to discussion to the point of devoting the equivalent of a complete chapter to its structural analysis, including numerous alternative designs, clearly some limitations must be envoked. In general, detailed numerical involvements are avoided. Most studies are through instantaneous, quasi-static analyses, although references may be made to rele-

374


vant processes and long-term expectations. Little detailed attention is

given to periodontal response and remodeling beyond that discussed in Chapter 5. All examples pertain to fixed appliance therapies and associated auxiliaries; removable or functional appliances, retainers, and orthopedic devices (such as palate-splitting mechanisms) are excluded (although the analysis procedures developed in this text may be applied to them). Numerous appliances and auxiliaries are in common clinical use to accomplish a specific displacement objective; examples are taken from several treatment philosophies and there is no intention to rate or rank the individual appliances or approaches to the mechanics or to favor or support one treatment technique over another. As previously indicated,

the examples are intended to illustrate the analysis procedures and, hopefully, to enable the individual practitioner to analyze existing or proposed appliances and mechanics toward judging their suitability to accomplish specific treatment objectives.

Individual Tooth Malalignments This section of the chapter focuses on sample mechanics used to correct

individual, malposed teeth. The objective is the delivery of a specific force system to the tooth crown by the appliance to produce the displacement necessary to bring the malaligned unit into proper intra-arch position, correcting crown location, long-axis angulation, and first-order orientation as warranted. Assumed is the existence of space in the arch to accommodate each desired displacement. Activation to the extent of producing inelastic behavior, anywhere within the appliance, is generally considered for purposes of analysis to be a structural failure. In addition to maintaining the integrity of the appliance in its interconnected members and the attachments to teeth, awareness must exist of the distribution of responsive, anchorage force and displacement control during the desired tooth movement. Leveling Displacements To begin, a premolar requires a facially or lingually directed, simpletipping displacement to achieve proper alignment. A round arch wire is

engaged into the facially placed brackets of the premolar and, minimally, the teeth immediately mesial and distal to the premolar. The dental units ordinarily adjacent to the premolar are assumed present. Given

a choice of brackets and reasonable first-order orientations of all involved teeth, a wire is assumed available which, when fully engaged and activated, will not have been inelastically bent as a consequence of that activation; in short, the faciolingual malalignment is not excessively severe.

375


Suppressing the arch-wire curvature in the vicinity of the premolar,

the structural analysis is that of a passively-straight wire beam, supported on either side of the malalignment site and elastically deflected in the activation process. Because its faciolingual stiffness at the premolar

is nonzero, the deflection to engage produces the wanted, simple-tipping force. The direction (and sense) of that force corresponds to that of the deflection, the location is that of the engagement of wire into premolar bracket, and the initial magnitude of the force depends on the amount of the malalignment and the localized bending stiffness (which is, in turn, dependent on a host of parameters discussed in Chapter 7). A reasonable, additional assumption is that of similar mesiodistal bracket widths, and neglected are any small differences between interbracket distances on either side of the premolar long axis. The top sketch of Figure 10-1 shows a lingually malposed premolar, the adjacent teeth, and the engaged arch wire. The center sketch presents the force system exerted on the three teeth by the activated wire. Symmetry associated with the assumption of equal interbracket distances yields mirror-image responsive forces (Q) and couples (C) against the crowns neighboring the premolar. The magnitudes Q are individ-

ually not more than one-half that of P (see Fig. 7-1); teeth mesial and distal to the anchorage units shown may also share the responsive-force burden, but those units adjacent to the activation site receive the largest portions of the response. The couples arise from the curvature induced in the wire by the activating deflection; they exist only upon elimination of any first-order clearance between wire and ligated brackets. The freebody diagram of the involved wire segment is shown in the lower sketch of Figure 10-1. The force system is the action-reaction counterpart of that

in the middle sketch except that a distribution of the responsive force system to more than two crowns is suggested. The first-order curvature effects decay more rapidly than those associated directly with the activating deflection; accordingly, only the two crowns adjacent to the premolar are shown subjected to responsive couples. The quasi-static analysis is straightforward, aided by the symmetry; the force balance

requires the total response on one side of the activation site to equal one-half of P and the moment balance demands that the responsive couples be equal in magnitude. Three of the four steps in the analysis procedure introduced in Chapter 6 are covered in the foregoing narrative. The potential side effects of the process of aligning the premolar are lingually directed simple tip-

ping and first-order angular displacements of the adjacent teeth. The former can apparently be handled by making the initial magnitude of P as small as practicable and by ensuring a broad distribution of the responsive force. Note, however, that the larger malpositions require high elastic range and bending flexibility. Continuous wires exhibiting low stiffness will concentrate the bending deformations at the activation site and, thus, cause the two teeth neighboring the premolar to carry nearly all of the responsive force. Moreover, the more flexible the wire, the

376


0

j- (P)

Response decay

P

FIGURE 10-1. 0cc/usa! views of a lingually ma/aligned premolar, adjacent teeth, and an engaged leveling wire (top), the force system exerted on the premo/ar and split anchorage by the arch wire (center), and the free-body diagram of the wire segment (bottom).

377

Force and Structural Analyses ot Representative Orthodontic Mechanics

greater the tendency for occurrences of first-order couples that can pro-

duce rotational displacements. A second example of this section is that of moving a "high" canine into proper occlusogingival alignment. To provide adequate space and reduce arch-length discrepancy, the adjacent first premolar has been extracted. A continuous arch wire is placed and, again, its curvature in the arch-form plane is ignored in the analysis. Because required movements of five or more millimeters are not uncommon, the extrusive force is generated by stretching a relatively flexible, "elastic" element between the canine bracket (affixed to the facial crown surface) and the arch wire. Activation of the elastic induces the extrusive force against the canine shown in views from the facial and mesial perspectives within Figure 10-2. The latter reflects the eccentricity of the force and suggests that a

lingual-crown-tipping displacement might accompany the extrusive movement (see Fig. 5-16). A free-body diagram of the elastic is shown in the upper right of the figure; from action-reaction and quasi-static analyses the magnitudes of the three forces encountered thus far, all approximately parallel to the long axis of the canine, are equal at any time. The

initial magnitude of Fe is best measured directly with an appropriate force gauge. A free-body diagram of the arch wire is shown in the lower portion of Figure 10-2. The symmetry of the previous example is absent here because of the extracted premolar, but the anchorage is again split and a force balance requires the sum of the two, resultant, responsiveforce magnitudes to equal that of the active force. From a moment balance with respect to the attachment location of the elastic, the resultant responses are inversely proportional to their individual distances from

the line of action of the elastic force (in the absence of second-order couples as part of the response). Note that the force delivered to the anterior anchorage is greater than—approximately twice the magnitude of—that delivered to the posterior anchorage (see Fig. 2-28). The prudent arch-wire choice in this example is that exhibiting the highest bending stiffness possible, given the bracket-slot size. The arch wire need not be deflected in the activa-

tion of the appliance. The greater the wire stiffness, the more nearly uniform is the distribution of force within the anchorage unit. The anterior segment in this example must sustain a responsive resultant approximately two-thirds the magnitude of the force applied to the canine bracket. The lateral-incisor root is smaller than that of the canine and, if the arch wire is flexible, an active force of sufficient size to produce the desired canine displacement would undoubtedly move (intrude, tip?)

the incisor as well. Use of a full size rectangular wire has the added benefit that, when the canine has been displaced occlusally to the extent that the elastic may be discarded and canine-bracket engagement is possible (without excessive force or inelastic material behavior), a torsional couple may be delivered to the canine crown to correct any third-order malpositioning caused by the eccentricity of the active, extrusive force. Moving a terminal molar into proper alignment presents a somewhat different problem in that, unlike the two previous examples, restricting

378


Fe

I Fe

V

Fe

segment

Fa

FIGURE 10-2. Mesiodistal (upper left) and facial (upper center) views of a high" (impacted) canine subjected to an elastic force, a free-body diagram of the stretched elastic (upper right), and a free-body diagram of the arch-wire segment that transmits the responsive elastic force to split, intraoral anchorage (bottom).

procedures to intra-arch mechanics demands that anchorage be entirely

mesial to the molar. (A palatal bar might be used depending on the nature of the malalignment, but this option is not examined here.) The desired displacement may be attempted by cantilevering an arch-wire segment (or an appropriate auxiliary) from the adjacent buccal-segment teeth, unitized to distribute the responsive force system. The cantilever

379


might be positioned on the facial or lingual aspect or the mechanics

divided between both sides of the segment; only minor differences in the three analyses exist, and facial-surface mechanics are assumed in this discussion. The top sketch in Figure 10-3 shows the malaligned terminal molar, with its root structure tipped mesially, in need of assistance to erupt completely. Also indicated schematically is the anchorage together with an arch wire and a cantilever. Although the continuous arch wire might extend distally to the malposed molar, in these example mechanics that wire terminates just distal to the bracket affixed to the crown of the tooth immediately mesial to the molar. In the left center of the figure the molar is shown subjected to crown loading (exerted by the cantilever) including an extrusive force and a second-order couple to correct the long-axis angulation. Not seen in this buccal view is a third-order couple, possibly present to maintain or correct the long-axis alignment in the mesiodistal perspective in light of the eccentricity of with respect to the cre of the molar. The bottom sketch of Figure 10-3 presents a free-body diagram of the cantilever in the buccal perspective. The action-reaction counterpart of the active force system exerted on the cantilever is carried through it

mesially and transferred to the anchorage unit. The force balance is straightforward; the moment balance leads to the second relationship given next to the diagram. The sketch in the right center of the figure shows the responsive force system delivered to the anchorage. The analysis thus far indicates that the force systems against the molar crown and anchorage are similar; the forces are equal in magnitude but opposite in sense and, therefore, in displacement tendency. The second-order couple exerted on the anchorage is, however, larger than the active couple. Because the tooth mesial to the terminal molar has a comparable (if a first or second molar) or smaller (if a premolar) root-surface area, the anchorage unit must include at least two teeth and, if available, three. The continuous arch wire should exhibit a high bending stiffness to distribute the responsive force more widely. As noted earlier, the cantilever could simply be an extension of the continuous wire if the malalignment is minor. If the occlusogingival malposition is substantial, however, necessitating a sizable activation, the bending flexibility and elastic range must be high. Concurrently, sizable localized second- and third-order stiffnesses may be warranted for displacement control. This combination of characteristics is perhaps best obtained by maximizing the mesiodistal length (C) of the cantilever while choosing a "working" (medium-size, rectangular) wire cross-section. Another consideration is delivery of the responsive force system at the mesial end of the cantilever near the middle of the anchorage unit to more uniformly distribute that response.

Rotational Corrections Changing views, a common individual-unit malalignment is the rotated tooth. To attempt the correction, the active force system must pro-

380


Anchorage

Ct

Fb

Cb -

Fb = Cb =

+

F,,

all FIGURE 10-3. Buccal views of a terminal molar to be aligned and intraoral anchorage (top), the crown force system exerted by a cantilever on the molar (center left) and on the anchorage (center right), and a free-body diagram of the cantilever (bottom).

381


a moment about the desired axis (the cro in the occiusal view) having the sense (clockwise or counterclockwise) to oppose the malposilion. If the cro is to be on the long axis, the resultant of the active force system should ideally be a couple in an occlusogingival plane. Such a duce

force system might nearly be produced by an arch wire engaging a bracket slot, but angulated with respect to the slot. If a distolingual rotation is desired, for example, the wire would push against the distal extent of the bracket slot and against the ligation at the mesial extent of the slot. Two less than efficient aspects of this approach are the dependence, in part, on the ligation to maintain the activation and the small moment arm equal to the mesiodistal width of the bracket slot. An increase in the size of the moment arm is the improvement demonstrated in the mechanics illustrated in Figure 10-4. If a small interproximal space exists, an elastic may be stretched between an eyelet or button (affixed to the band or directly bonded to the crown surface in the lingual position shown) and the arch wire. The bracket attached to the facial surface is assumed aligned occlusogingivally with the arch wire, but perhaps not initially engaging the wire. On the upper left in the figure is an occlusal view showing the tooth, bracket, lingual button, and the elastic. In the upper right view the active force system against the crown is just the

Fe

Fe

Here, Fe = F,,, the forces form a couple, C0.

FIGURE 10-4. Occlusa/ views of a rotated tooth and the use of an elastic and arch wire to attempt the correction.

382


force in the absence of bracket-wire contact. The displacement tendency includes clockwise rotation (from the moment of the force elastic

about the long axis) and facial crown tipping (from the force itself). The

lower left view shows the added responsive force from the arch wire following tipping and bracket-wire contact. As time proceeds and movegrows. If ment occurs, F, diminishes and arises as a secondary activation as indicated, its magnitude will never exceed that of Fe. The sketch on the lower right in the figure depicts the ideal situation (if pure, long-axis rotation is desired) with the two forces equal and forming the

couple C0. In comparison with the first-order couple formed in the bracket and mentioned earlier, these mechanics exhibit an activating element with substantial elastic range (the elastic) and a moment arm approximately equal to one-half the mesiodistal width of the tooth. The arch wire used should have sizable bending stiffness to prevent tipping toward the labial beyond that to engage the bracket and wire. When a rotational correction is desired with little or no change in the faciolingual, long-axis position, and the full couple cannot be instanta-

neously generated, a sizable moment with a small, active force is needed. This combination requires a large moment arm, and the auxiliary of Figure 10-5 has the wanted characteristic. As suggested by the top sketch in the figure, the "rotation lever" might be positioned on the lingual and one end rigidly affixed to the crown of the rotated tooth. The lever is activated with a ligature tie or an elastic to the facially placed arch wire, several teeth mesiodistal from the malposed unit. The activating deformation is partially in the lever; its passive configuration is shown dashed. A free-body diagram of the lever is given in the center of the figure; the action-reaction counterpart of the couple is the wanted

force system, but it cannot exist in the absence of the two forces of magnitude f as indicated by a quasi-static analysis. For f to be light despite a sizable activation, the lever must exhibit high bending flexibility and elastic range; not only should D be large, but the lever cross-section must be small. The lever is better tied to the arch wire with an elastic than with a segment of ligature wire because of the difference in their extensional stiffnesses. The force system delivered to the rotated tooth

crown is shown in the lower portion of the figure together with the expression for the net active moment with respect to the long axis. The responsive force does not exist in the absence of crown contact with the arch wire.

Another means of correcting a rotated tooth uses a "V-spring" ("wedge"). The tooth, arch wire, spring, and tie are depicted in the top sketch in Figure 10-6. The spring is activated by elastically reducing the passive angle between its "legs," wedging the auxiliary between the arch wire and tooth crown, and tying the spring to the bracket. To maintain the activation, the arch wire must have substantial bending stiffness and must be secured in the bracket affixed to the crown. A free-body diagram of the spring and the force system delivered to the tooth crown are also shown in the figure. The desired wedge force is exerted by one leg of the spring on the crown. This force, together with a compo-

383


— —

__—

—

0

—

— Passive lever

.p.1

= f(D)

Effective couple equals F(D — d)

FIGURE 10-5. 0cc/usa! views of a lingual-lever system to correct a rotation. Shown are a composite sketch of the mechanics (top), a free-body diagram of the lever (center), and the force system transmitted to the tooth crown (bottom).

of the tie force, forms the couple meant to correct the rotation. Also shown in sketches are forces from the ligature and friction. The size of the frictional force between arch wire and bracket will be the principal influence on the location of the axis of rotation relative nent (parallel to

384


Friction F,,,

Friction

FIGURE 10-6. 0cc/usa! views of a 'wedge" used to correct a rotation. Shown are a schematic of the mechanics (top), a free-body diagram of the wedge (lower left), and the force system delivered to the tooth crown (lower right).

to the long axis. If this force is sufficiently great, the rotation could be, in

the absence of crown contact (on the side opposite to that of the spring position), about an occlusogingival axis through the bracket. The effectiveness of the spring depends on the combined flexibilities of the tie

and spring in the presence of adequate elastic range. Increments of movement following activation will be small if the tie is a segment of ligature wire and the spring stiffness is high. The last rotational-correction scheme to be examined is similar to the first in this series in that the activation is again tangential to the crown. The activating force here, however, is directed mesiodistally. The upper portion of Figure 10-7 depicts the presence of crown contact from the adjacent tooth; the driving force at the facial surface is accompanied by the crown-contact response. These two forces form a couple and, in the absence of significant friction, the first-order rotation induced will be counterclockwise about the long axis of the malposed unit. In the lower portion of the figure there is no crown contact and a rolling displacement should be expected with the center of resistance moving in the direction of the force and the center of rotation located lingual to the long axis. In both instances here the arch-wire influence is suppressed and it is assumed that the tooth is not ligated to the wire. In all of these examples of rotational correction, if the couple is to be formed, the "second" force arises from a secondary activation. When "in-place," first-

order rotations are wanted, it is important to note the location and source of the secondary force together with gauging the appropriate stiffness of the element producing that force, so that the format of the expected, actual movement can knowledgeably be assessed.

385

Force and Structural Analyses of Representative Orthodontic Mechanics Factive

Potential displacement is long-axis rotation.

No adjacentcrown contact

Potential displacement is a generalized, rolling rotation.

FIGURE 10-7. Occlusal views of an attempted rotational correction, activated by a mesiodistal force against a bracket with (top) and without (bottom) crown contact between the rotated tooth and the adjacent dental unit.

Bilateral Action Another class of displacements appropriate to this section encompasses the intra-arch, bilateral movements of individual teeth in a symmetric manner with respect to the midsagittal plane. As an initial example, consider the closing of space between the maxillary central incisors. The appliance consists of an arch-wire segment, gabled between two brackets if warranted, and an elastic encircling the pair of incisor brackets. Appropriate sketches are given in Figure 10-8. If the action is truly bilateral, then two activation sites and no anchorage exist; action-response is effectively replaced by "action-action." The forces at either end of the elastic drive the central incisors toward each other and the ligated arch-wire segment is expected to produce first- and/or secondorder couples to control or rectify the angulations of the teeth during the

movements. The free-body diagrams in the figure are balanced by paired, mirror-image force-system components. If no tipping is to occur

during the space closure, the "guiding" wire segment must possess high bending stiffness. To prevent mesiolingual "rolling" the ligation must be snug and the ties stiff, but concurrently not induce sizable frictional resistance.

386

Bioengineering Analysis of Orthodontic Mechanics Elastic

-4

0' F0

FIGURE 10-8. The closing of space between maxillary central incisors: the teeth and the mechanics (left), tree-body diagrams of the elastic and the wire segment (center), and the delivered crown force systems (right), all in facial views.

A second bilateral-action example involves the terminal molars of either arch, and the mechanics were part of the general analysis of facebow therapy in Chapter 9. The innerbow could be fabricated prior to insertion to constrict or to expand the arch width at the molars and/or to

induce first-order couples (through placement of toe-in or toe-out bends) as depicted in occlusal perspective in Figure 10-9. If the inner bow is, in effect, a separate auxiliary and engages only the molars, as in the previous example no anchorage exists and only action at both molars. The buccal tube in its mesiodistal length provides a relatively large inherent moment arm for the couple and improved first-order control compared to any bracket, thus allowing the operator latitude in choice of bow-arm or wire size as it affects bending stiffness. Figure 10-10 shows posterior views of active, bilateral, buccolingual forces and third-order couples. The couples necessitate the use of rectangular wire (and tube slots) or torquing spurs. If an arch wire is continuous and engages all brackets anterior to the molars, with bilateral action present so also is bilateral anchorage. A proper analysis in this instance would involve either half arch; although ideally every wire-bracket connection mesial to the molars could be mechanically passive, from a practical standpoint contact (and, therefore, force transfer) will exist at most, if not all, such connections. Another means of generating mirror-image, bilateral, force systems to posterior teeth in the maxillary dentition is the transpalatal bar. Spanning the palatal vault, this appliance typically interconnects canines or molars through attachments on their lingual crown surfaces. Buccolingual forces, third-order couples, and, perhaps, also first-order couples may be activated on both sides of the maxillary arch. A free-body diagram of the transpalatal bar and, through action-reaction analysis, typical, delivered force systems to the crowns are given in a posterior view in Figure 10-11. The transpalatal bar is fabricated from a member having a substantial cross-section in order that it maintains its shape and stabil-

387


/ / L0

Co

Outer bow arm

L0

Co

FIGURE 10-9. lntra-arch bilateral action to the terminal molars from an 'inner bow': force systems against the arm ends (top) and their action-reaction counterparts exerted on the molar crowns (bottom) in 0cc/usa! views.

ity when activated. The bar is usually formed to give the tongue ample

room, but without exerting undue pressure on the hard palate. The stiffnesses at the crown-contact locations are likewise substantial; hence,

activating displacements must be relatively small. With its bulk the

388


Tp

FIGURE 10-10. Posterior views of bilateral action against the terminal molars: actionreaction force systems exerted on the appliance (top) and the molars (bottom).

transpalatal bar is more often used to maintain arch width and/or long-

axis orientations as an auxiliary to other mechanics than to directly generate tooth movements.

lnterarch Mechanics Individual teeth may experience displacements induced by interarch mechanics. Perhaps the simplest example is in a finishing step in therapy using the "up and down" elastic to effect proper interdigitation. The mechanics are similar to those illustrated earlier to extrude the "high" canine, but the anchorage here is provided by teeth in the opposing arch. Figure 10-12 shows a maxillary premolar subjected to an occlusally directed force from an elastic stretched and placed around the bracket.

The elastic also encircles brackets affixed to the facial surfaces of the opposing teeth of the mandibular arch. A free-body diagram of the elastic and an illustration of the responsive forces delivered to the involved mandibular teeth complete the figure. Extrusion of all three teeth may

389


I

FIGURE 10-11. A free-body diagram of a transpalatal bar, as viewed from the posterior (top), and the force systems delivered bilaterally to first maxillary molars (bottom).

depending on the magnitudes of the elastic forces and the individual bending stiffnesses of the engaged arch wires. If extrusion of only occur,

the maxillary premolar is desired, the maxillary arch wire, if present, should exhibit occiusogingival bending flexibility to permit the desired displacement. In addition, because of the eccentricity of the active force, a labial crown torque from the wire would be helpful. The mandibular arch wire should be stiff to provide restraint against occiusally directed displacement and tipping and to distribute the responsive force system

beyond the two mandibular teeth shown.

390


A t

Q4

H5

FIGURE 10-12. Mechanics to complete interdigitation of a maxil!asy premolar (left), a free-body diagram of the involved elastic (center), and the forces exerted on the crowns through elastic-bracket and adjacent crown contacts (right).

Interarch mechanics may also be used to correct a unilateral crossbite. As an example, consider a mandibular premolar that is improperly tipped lingually and must be uprighted. If other mandibular teeth are not involved, a button or eyelet might be affixed to the lingual crown surface of the premolar and an elastic stretched over the occlusal surface and attached to the maxillary arch wire. Appropriate sketches are given in Figure 10-13. Because the elastic force has a vertical component, in the absence of a controlling, mandibular arch wire, some extrusion should be expected to accompany the wanted labial crown tipping. The maxil-

lary arch wire should be stiff in bending to distribute the responsive force, equal in magnitude to that of the active force, to at least several of the maxillary teeth neighboring the elastic-wire connection. Two additional concerns are noteworthy in this example. First, the relative positions of the arches influence the size of the elastic force here (and in the preceding example); e.g., the mean force is expected to be greater in a mouth-breathing patient. Second, with the elastic running across an occlusal surface, any occlusion or mastication tends to wear or cut the elastic, reducing its life. With their low cost and relaxation tendencies, however, such elastics should be routinely replaced every day, if not more often.

In these last two examples of this section a pertinent question is "Why not use intra-arch mechanics instead?" With the "up and down" elastics the intent may well be to produce some extrusion in both arches.

391

Force and Structural Analyses of Representative Orthodontic Mechanics Maxillary arch wire

Responstve force delivered to the maxillary arch

Angle less than 900 yields potential for extrusion

A portion of the elastic

FIGURE 10-13. Mechanics to correct a mandibular premolar in cross-bite: mesiodistal views of force systems exerted on the elastic, the premolar crown, and the maxillary arch wire. -

In the cross-bite example, a larger activation is possible resulting in more

displacement potential between appointments and appliance adjustments. Moreover, the anchorage is more unifiable in the maxillary arch. In summary, there are advantages and disadvantages to consider with either approach.

392


lntra-arch Vertical Positioning This section undertakes analyses of mechanics in one arch designed to

produce displacements that are primarily occlusogingival in direction and are intended to move two or more teeth concurrently. Examples include correcting the anterior open or deep bite and modifying the curve of Spee. In all instances, geometric and mechanical symmetry with respect to the midsagittal plane is assumed. Views of the teeth, appliance, and force systems are from the buccal perspective. Analyses may, then, involve just one-half of the arch and appliance in each example. One-half of the U-shaped arch model introduced in Chapter 6 is used and within the L-shaped half arch the dentition is considered as two or three segments, depending on the mechanics. Examined first is the "utility" arch, formed in conventional wire to engage brackets affixed to the facial crown surfaces of the four or six anterior teeth, bypassing the teeth of the buccal segment mesial to the

terminal molars, and engaging buccal tubes attached to the molar crowns. As appropriate or warranted, each buccal segment may be unified by means of a separate wire segment and ligation. Shown schematically in the top sketch of Figure 10-14 is a mandibular utility arch. Two segments of the half dental arch are involved: the half anterior segment

including the central and lateral incisor and, perhaps, the canine, all represented by the incisor drawn, and the buccal segment including available premolar(s), molar(s), and, possibly, the canine. In this initial

example the intent is the intrusion of the anterior teeth; the passive configuration of the arch, with the ends of the wire engaging the buccal tubes, is shown dashed. A free-body diagram of the as-modeled, half

utility arch is given in the center of the figure. Engagement into the anterior brackets creates intrusive force as the arch attempts to restore its passive configuration. If the utility arch is formed in round wire and no torquing spurs are included or attached, and no anteroposterior activation is present, the half anterior segment exerts only the vertical of mag-

nitude Fa. The resultant force system from the buccal tube, the only other site of contact with the half arch, is obtained through a quasi-static analysis; required for force and moment balances are a vertical force and a second-order couple in this view. The action-reaction law and the balanced free-body diagram yield the overall force system exerted by half of

the appliance on half of the mandibular dentition as indicated in the lower sketches of Figure 10-14; also given are the quasi-static relationships involving the parts of the force system and the anteroposterior distance from anterior brackets to buccal tube. (Recall a previous, similar analysis given in Figure 2-29.) The utility arch in Figure 10-14 is activated by elastic bending, displacing the anterior section of the arch occlusally to enable bracket engagement. The amount of force induced (Fa) depends on a localized,

393


Posterior segment Half anterior segment

op

t

Cp = Fa(e)

FIGuRE 10-14. A utility arch wire in the mandibular arch: buccal views of the passive (dashed) and activated (solid) mechanics (top); a free-body diagram of the half arch wire (center); and the force system delivered to the terminal molar (lower left) and half anterior segment (lower right).

394


vertical stiffness of the arch wire and the amount of vertical displace-

ment necessary to engage the anterior segment. The stiffness is dependent on the wire from which the arch is formed and the distance 1?. The passive position of the anterior section of the arch is independent of the stiffness and is controllable by the clinician in, for example, the size of the tip-back bend placed mesial to the buccal tube. The displacement(s) following activation depend on the magnitudes of F,,, E, and the involved root-surface areas. If intrusion of the incisors is desired, the canines are not engaged and, if F,, is sufficiently light, the terminal molars alone may provide enough anchorage; a conservative operator would, however, reinforce the anchorage with an additional molar or premolar on each side. In the lower right sketch of Figure 10-14, note the position of the line of action of the intrusive force with respect to the center of resistance; apparently, the intrusion is likely to be accompanied by labial crown

tipping. One means of counteracting this additional (assumed unwanted) displacement component is illustrated in the maxillary, utility-

arch analysis of Figure 10-15. As suggested in the top sketch in the dashed, passive configuration, the intent here (as in the previous example) is to open the anterior bite—in this instance through intrusion of

the maxillary anterior teeth. The analysis procedure through the sketches of Figure 10-15 follows that of the foregoing example. Note in the center sketch that the anteroposterior forces must be equal in magnitude; they are essentially independent of other components of the force system and their size is determined by a longitudinal stiffness, a function largely of the characteristics of "tie-back" element and the "omega loop" bent into the arch, and the amount of anteroposterior activating deformation. In the representations of the anterior and posterior segments and the exerted force systems, the moments about the centers of resistance of the forces P,, and oppose those of F,, and C,,, respectively. Note also, because the roots of the maxillary central incisors are substantially larger than their counterparts in the mandibular arch, two-tooth posterior segments are necessary, minimally, to provide anchorage for

intrusion of four incisors and are insufficient if the involved anterior segment also includes the canines. Returning to the mandibular arch, the treatment of an anterior open bite may call for extrusion of the mandibular incisors and a suitable utility arch may be engaged. The vertical forces and second-order couple

of Figure 10-16 are reverse in sense from their counterparts in Figure 10-14, but added in the new figure is a third-order anterior couple, generated, perhaps, through use of rectangular wire and appropriate activation. The extrusive force tends to produce lingual crown tipping because of its position in relation to the cre. If the labiolingual long-axis orientation is to be maintained, or if labial crown tipping is to accompany the extrusion, the torsional couple (or a labially directed force component) must be present. Directing attention to the posterior response, usually particularly important when creating anterior extrusion (in either arch) is the use of more than sufficient anchorage or previous anchorage prep-

395


F,,,

Fa

Pa

cp L

= F,,(L)

=

Pa

= Fa

FIGURE 10-15. A tied-back, maxillary utility arch: a buccal view with the passive anterior position dashed (top); a free-body diagram of the half arch wire (center); and the force systems exerted on the posterior (lower left) and half anterior (lower right) segments.

aration; the sense of the second-order couple is toward mesial crown

tipping and the reduction of arch length. Noting, however, from the free-body diagram in the center of Figure 10-16 the moment-balance equation for the half arch, although the solution is indeterminate, the relationship suggests that the presence of Ca serves to reduce the magni-

396


Cp

I

Co

•1

F8

Fa = C8 =

—

Ca

FIGURE 10-16. A mandibular utility arch incorporating third-order, anterior-segment control: passive (dashed) and activated (solid) configurations (top); free-body diagram of the arch (center), and the force systems delivered to the posterior (lower left) and half anterior (lower right) segments.

397


tude of the posterior couple. In selecting a rectangular wire from which

to prepare the utility arch, if the torquing couple is to be secondarily activated in holding action, seemingly a thin, wide, arch wire is appropriate toward maximizing the ratio of third-order stiffness to vertical stiffness in the anterior segment. A small occlusogingival dimension enhances the bending flexibility while a substantial faciolingual dimension adds to the potential for torsional stiffness. The examples of utility-arch use to this point suggest that a variety of anterior-segment displacements are possible. The relative ease of analysis, in the presence of the assumed symmetry, should now be apparent. The validity of the modeling is not at question; the involved segments are well defined and, practically speaking, unitized if individually leveled prior to engagement of the utility arch. In a final example utilizing the appliance, the roles and locations of action and anchorage are reversed. Noting the relative magnitudes of root-surface areas, anterior segment versus terminal molars, under spe-

cific conditions it is reasonable to use the utility arch to extrude depressed molars with the six anterior teeth providing support. Figure 10-17 is in the format of the preceding several figures. The free-body diagram of the half arch shows the necessary, third-order component of the anterior force system (provided through use of rectangular wire, for example). As is also suggested in Figure 10-14, the portion of the anterior vertical resultant against the incisors reflects a tendency toward flar-

ing these teeth; the torsional component provides a counter-tipping moment, thereby giving added stability potential to the anchorage. Even though the combined root area of three anterior teeth exceeds that of the terminal molar, however, the differential is small enough that care must be taken in choosing the wire size and amount of activating deflection to minimize the tendency for reciprocal actions. Because of the differences in root areas of the two sets of central incisors, the potential anchorage is "stronger" in the maxillary than in the mandibular arch. The appliance

is more attractive in this application if some intrusion of the anterior segment can be tolerated or is desired together with the molar displacement. The second category of tooth movements noted in the opening paragraph of this section is restricted to occlusogingival displacements, but the appliance contacts all dental units in one arch. The specific mechanics intend to produce, enhance, reduce, or eliminate a curve of Spee, the pattern of collective occlusal-surf ace heights as viewed from the buccal perspective. For this analysis, in the presence of midsagittal-plane symmetry, the half arch is divided into three segments: an anterior half-segment including central and lateral incisors and, possibly, the canine, a middle segment including one or both premolars and, if not part of the anterior half-segment, the canine, and a posterior segment including the

erupted molars and, occasionally, the second premolar. If a tooth has been extracted and site closure has not yet occurred, the modeling will generally place the extraction space between two segments. After prepa-

398


7 Fa

F8 =

Ca = F8(€')

FIGURE 10-17. A utility arch to extrude mandibular terminal molars: buccal views of the activation and anchorage sites and the arch (top); a free-body diagram of the half arch wire (center); and the force systems exerted on the terminal molar (lower left) and half anterior segment (lower right).

399


ration of the continuous arch, its passive shape should be superimposed

over the line of brackets in the buccal view. In an attempt to gauge just how the activated arch will be configured occiusogingivally after working for a time, the relative positions (wire to brackets) should be such as to distribute the localized activations approximately evenly over the half arch; the engaged wire will attempt to assume a minimum-energy (of deformation) orientation (Chap. 4). Figure 10-18 exemplifies such a superimposition; the objective is to reduce or eliminate a distinct curve of Spee. In a first example, the intent is to reduce the curvature in the mandib-

ular arch through placement of a flat arch wire. The relevant sketches are those of Figures 10-18 and 10-19. Assumed is the arch wire fabricated in round wire of a size relative to the bracket slots such that no second-

order couples are developed upon activation. The free-body diagram within Figure 10-19 shows a resultant force for each segment; the quasistatic analysis indicates that Fmjd is in magnitude the sum of and Fa. Typically (with second molars erupted), the posterior-segment, root-surface area exceeds that of the anterior segment (and, perhaps, also that of the middle segment). Aware, in addition, that extrusion is, generally, more easily accomplished than intrusion (with moderate-size forces), the likely displacement of substance will occur in the middle segment (if not blocked by occlusal contacts). Light forces generated by a wire of small diameter will enhance the possibility of anterior-segment intrusion. An examination of similar mechanics in the maxillary arch is of interest. The sketches in Figure 10-20 are comparable to those in Figure 10-19; the magnitude relationship among the three forces is the same, but directions (senses) of potential displacements are reversed. In the maxillary arch, in the absence of posterior occlusion, extrusion of the molars is possible (particularly if the second molars do not engage the wire). To deter a similar displacement in the anterior segment, unitizing the six

anterior teeth through appropriate ligation is warranted. If the group-

/

Buccal tube

anterior

posterior

Passive arch wire

\

/

Brackets

FIGURE 10-18. Buccal view of the superimposition of a mandibular leveling arch wire and the line of brackets prior to engagement. The purpose of the superimposition is to anticipate the occlusogingival displacements to be generated by the arch wire over a working time period.

400


Fa

Fa

FIGURE 10-19. A free-body diagram in the buccal view of the activated arch wire of Figure 10-18 (top), and the occiusogingival forces delivered to the posterior, middle, and half anterior segments of the mandibular dentition.

ings of dental units are such as to define the middle segment as the two adjacent premolars, from the relative size of Fmid some intrusion of those teeth is inevitable also. Within the initial assumptions made, the displacements achieved in mechanics intended to modify the curve of Spee depend on the relative sizes of the root-surface areas of the segments, occiusal contacts with opposing teeth, the magnitudes of the forces generated in activation, and, to some extent, the type of displacement (intrusion versus extrusion). The sizes of forces produced are governed by the characteristics of the arch wire that determine bending stiffnesses and the difference in

401


Pt

Pa

Fa Fmid

Incisal pair

Terminal molar Pt

FIGURE 10-20. Reducing the curve of Spee in the maxillary arch with a round arch wire, as activated anteroposteriorly passive (but stopped against posteriorly directed, relative displacement through the buccal tubes): free-body diagram of the half arch wire (top) and occiusogingival forces delivered to the segments of the half arch (center) immediately after activation; and antero posterior forces generated secondarily against the terminal molar (lower left) and half anterior segment (lower right) with on going leveling. Action-reaction counterparts appear dashed in the free-body diagram (top).

severity of curves exemplified in Figure 10-18. The clinician may influence all pertinent parameters and must consider each in designing the appliance and segmenting the dentition.

402


In addition to the possible appearance of couples in the force system,

due to absence of second-order clearances, another secondary effect must be mentioned before concluding this section. In the process of reducing the severity of a curve of Spee, the deactivation of the appliance results in an increase in its effective anteroposterior length. Ligations in the anterior segment and, if present, stops or friction at the terminal molars will induce anteroposterior force in the manner illustrated (dashed) in the free-body diagram in Figure 10-20. As the deactivation continues the forces grow; a horizontal force balance yields equal

magnitudes of force delivered to the anterior and combined posterior segments. These forces (for one-half of the arch) are shown in the lower sketches of the figure. Because the anteroposterior stiffness of the wire is substantial, the horizontal force components can become sizable with only a small curvature reduction. A consequence of this development is the flaring of the incisors possibly accompanied by distal crown tipping

of the terminal molars. To counteract this generally undesirable side effect, the clinician must either ensure relative freedom of the arch wire

to slide through the molar tubes as the curve is flattened, or initially activate the appliance horizontally in opposition to the expected effect of curvature reduction by tying back posterior ends of the arch wire with flexible auxiliaries. With the latter approach, initial horizontal forces are developed upon activation that are opposite (in sense) to those shown in Figure 10-20; as the appliance deactivates, these forces decay along with the vertical components. Careful control of the magnitude of this horizontal action is necessary if interproximal spaces exist at this time.

Intra-arch Retraction Mechanics Anteroposterior therapy procedures to close spaces, correct procum-

bency, reduce overjet, and eliminate extraction sites often may be gener-

ally categorized as "retraction mechanics." Such therapy procedures may involve interarch action and response (subsequently examined in this chapter), extraoral appliance delivery (discussed in Chapter 9) or— the present focus of attention—treatment within the single arch. Either a continuous- or segmented-arch approach might be undertaken. The usual common characteristics of the various procedures include posteriorly directed driving forces as the primary actions, posterior-segment anchorage, and, often, bilateral action in the presence of midsagittalplane (geometric and mechanical) symmetry. Two example displacements are analyzed in this section: the retraction of a canine into a space

created by the extraction of a first premolar, and the retraction of an incisal segment. In conventional, continuous-arch, edgewise therapy the retraction of a canine is attempted with the tooth subjected to a distal-driving force induced by a stretched elastic or helical spring and the dental unit experiencing movement with respect to a stationary, guiding arch wire. With

403


the driving-force element attached at midcrown level, occiusal to the cre of the canine, the potential tipping displacement can be transformed by

the arch wire, generating a second-order, counter-tipping couple, to a movement characterized by little, if any, change in long-axis angulation. If the line of action of the driving force is buccal or lingual of the long axis (i.e., the distally directed force system is not split faciolingually and two driving elements used), the bracket-wire connection must also provide first-order control (unless long-axis rotation is a desired part of the overall displacement). Shown in Figure 10-21 are sketches associated with the analysis of a

process on a guiding arch wire. Providing the distaldriving action is a compressed, linear, helical-coiled spring, stabilized through its winding about the arch wire and maintained in the activated state by a segment of ligature wire tied between the mesial end of the spring and the buccal tube affixed to the terminal molar. Note first in the figure a composite illustration of one-half of the arch wire, the involved dental units, the spring, and the piece of ligature wire mentioned. A free-body diagram of the compressed spring is also shown. The magnitude F, of the spring force is the product of the modulus of the spring and the deformation from the passive state (Chap. 4). The ligature-wire loop, as indicated in its free-body diagram, like the spring, is a two-force

member; mesiodistal force balances in the two diagrams, with the action-reaction law, indicate that all four force magnitudes are equal. Action-reaction between the ligature loop and the buccal tube yields the

force shown, directed mesially, against the molar crown. Through crown contact, that responsive force is distributed with a portion of it transferred to the adjacent second premolar. (If the second molar was erupted and aligned, through a ligature tie to it the responsive force could be further distributed.) The second-order couple should be present, resulting from a tip-back bend placed in the arch wire, to reinforce the anchorage by amending a tipping potential toward a bodilymovement format. A sketch of the canine shows the contact force from the distal end of the spring against the mesial extent of the canine bracket; action-reaction indicates that the distal-driving force also equals F, in size. The countertipping couple is exerted by the arch wire. Although if total bodily movement is desired, must be activated simultaneously with the spring (and in the proper couple-force ratio), and consideration must be given to the potential impedance to movement of a placed second-order bend along the "path" to be taken by the canine bracket. The alternative is to allow C, to be a secondary activation, growing in magnitude from zero following the disappearance of second-order clearance between canine bracket and arch wire. Some tipping will then occur, the amount depending on the spring characterisEics, the initial bracket-wire clear-

ance, and the bending stiffness of the arch wire; the couple will approach an upper bound equal to the concurrent moment of the driving force about the cre of the canine, but the inherent deterrent of a permanent bend in the wire is not present.

404


-4

/ (_

Spring

)

N

Pr F,

F5

F,

FIGURE 10-21. Buccal-view sketches associated with the force analysis of a canine-retraction procedure on a guiding arch wire. The distal-driving force is provided by a compressed, coiled spring.

405


A free-body diagram of one-half of the arch wire is also given in

Figure 10-21. Through the action-reaction law once again, the couples exerted by the canine and the posterior segment are obtained. Because

neither the spring nor the ligature loop have direct contact with the wire, those forces do not appear in the diagram. The placement of the lip-back bend, however, may produce in the arch wire in the molar region, upon activation, an occlusogingival force between wire and buc-

cal tube. That force on the wire and its action-reaction counterpart against the molar tube are shown dashed in the two sketches. An occlusogingival force balance in the free-body diagram of the half arch wire requires the existence of another vertical force, also shown dashed. If these vertical forces exist as secondary effects, a moment balance in the wire free-body diagram results in the two couples, although opposite in sense, being of different magnitudes. Note also that, depending on the character of the tip-back bend, the vertical forces may be as drawn or all have senses opposite to those shown. These mechanics exemplify the differential-force concept. Whatever the level of activation of the spring, the mesiodistal forces against the canine and anchorage crowns are equal in magnitude and bodily-movement potentials are established, or nearly so, at both activation and support sites. If only intraoral anchorage is used (although it could well be reinforced with headgear), the differential must be in the root-surface areas. Accordingly, inclusion of as many teeth as are available for support is prudent. If desired, through appropriate placements of stops— the arch wire against the posterior attachments—distribution of the responsive force can be directed into the arch wire, transmitted anteriorly

through it, and the incisal segment can be added to the anchorage. Before leaving this example, two additional noteworthy facets are illustrated in the occlusal-view sketches of Figure 10-22. Indicated on the

left is the division of the distal-driving force from the spring into that part actually reaching the crown of the canine and the frictional component that enters the arch wire. Although present in the actual, clinical situation, this friction—arising from the canine-bracket and ligation contact with the arch wire—is ignored in Figure 10-21 and the accompanying discussion. To quantify the division of the spring force is virtually impossible, but a reasonable assumption if employing light to moderate force magnitude is to expect that approximately one-half of the spring action actually reaches the canine. The presence of the friction tends to displace the arch wire toward the posterior with the canine. The incisal segment is, then, vulnerable to displacement if the wire is not properly stopped against the posterior, anchorage teeth. (Refer to the Frank and Nikolai (1980) paper, not only for the discussion of bracket-wire friction,

but also for the comments regarding the selection of the guiding arch wire.) The sketches on the right in Figure 10-22 exhibit the substance of the first-order effects of the buccal location of the driving-force element and the arch wire. Recalling the immediately preceding comments about friction, perhaps a counter-rotation tie mesial to the canine is preferable to

406


actually delivered to canine Portion of

Carried posteriorly by friction

1= I

Arch wire FIGURE 1 0-22. 0cc/usa/-view sketches associated with the force analysis of a canine-retraction procedure on a guiding arch wire: frictional-force analysis (left) and first-order effects (right).

dependence on the ligation to generate the moment to oppose the po-

tential for long-axis rotation. Tracing the force analysis through the arch wire and into the supporting dentition, the responsive first-order couple is best generated in the posterior segment—with the aid of a long buccal tube—and kept away from the lateral incisor.

A somewhat different approach to intraoral canine retraction replaces the distal-driving element and the guiding arch wire by a single

auxiliary activated between the canine and the buccal segment. Although two, or perhaps four, canines may be moved concurrently into adjacent extraction spaces, these mechanics in any one quadrant are virtually independent of ongoing therapy elsewhere in the oral cavity. Two or three teeth of a posterior segment are ligated together and the desired canine displacement is initiated within the hypothesis of differential-force treatment. An example auxiliary, fabricated from a straight segment of ortho-

dontic wire, is shown from a buccal perspective in the top sketch in Figure 10-23. If bodily movement is wanted, as is typical, the design of the auxiliary is intended to yield mesiodistal flexibility (to enable generation of a relatively light driving force in an activation of several millime-

ters), a corresponding, second-order stiffness level (in an attempt to maintain the as-activated, couple-force ratio nearly constant over time as

movement takes place), and, assuming buccal placement of the auxiliary, to deter distolingual rotation of the canine, a high first-order stiff-

407


Ccc

10-23. Buccal views of segmented-arch, canine-retraction mechanics: the fabricated retraction spring (top); a free-body diagram of the activated spring (center); and the crown force systems exerted on the canine and the posterior-segment anchorage (bottom). FIGURE

A free-body diagram of the "retraction spring" is shown in the center of the figure. A horizontal force balance requires and to be equal in magnitude. As noted in the previous example, to generate a ness.

second-order couple without some accompanying occlusogingival action is difficult; such secondary forces, shown dashed in Figure 10-23, must be mutually opposite in direction against the auxiliary (from a ver-

tical force balance), but may be as sketched or with both senses re-

408


The moment balance, in the presence of the vertical forces, yields unequal C magnitudes. If the opposing molars are in occlusion, perhaps the better alternative in vertical-component senses is that shown in the figure; accordingly, if some distal crown tipping occurs in the retraction process, a deterrent to extrusion is present. The action-reaction law permits the proper illustrating of the crown force systems exerted by the auxiliary on the involved dentition in the buccal-view sketches of the lower portion of Figure 10-23. Unlike the previous example, here the distal-driving force and counter-tipping couple against the canine crown are individually activated. Maintaining the proper couple-force ratio concurrent with a translational displacement is, strictly speaking, impossible; the magnitude will decrease as the canine crown moves toward the posterior while diminishes only if distal root movement occurs. Accordingly, mesiodistal flexibility of the auxiliary must be high, as noted previously, to minimize the rate of decay with movement, and the initial couple-force ratio should be less than that required for bodily movement (one-half of the total canine length; see Chap. 5), observing that the ratio will grow as the displacement progresses. The sketch of the terminal molar in Figure 10-23 indicates that the moments, with respect to the cre, of the mesially directed and secondorder responsive components are opposite to one another as desired; versed.

the couple works against the mesial-crown-tipping potential of the force. These components, together with the vertical responsive force, represent the resultants against the posterior-segment anchorage. If arch-length preservation is critical, as much root-surface area as is avail-

able should be used in distributing the responsive force system. Note that, in the absence of the continuous arch wire, the incisors cannot be made part of the anchorage. Note also that, although the friction of the continuous-arch retraction mechanics is absent in this approach, perhaps more attention needs to be focused on displacement control. For the auxiliary, a rectangular arch wire is suggested; the faciolingual dimension should be sizable to enhance occiusogingival-plane bending stiffness and first-order control while the occlusogingival dimension should be small to minimize the necessary complexity of the buccal-view geometry of the retraction spring. Intra-arch therapy designed to retract the anterior or incisal segment exhibits features similar to those of canine-retraction larly in buccal-view evaluations, but both obvious and subtle differences exist as well. For the arch as a whole, an analysis begins by consideration of three segments: an anterior segment and two posterior segments. The canines belong to the posterior segments if just the tour incisors are

to be retracted; if six teeth are to be moved, the canines are part of the anterior segment. Each of the three segments is unitized; of crowded incisors, if necessary, and all leveling have been previously accomplished. Although the responsive force system is bilateral and the retraction of the anterior teeth is intended to be undertaken in unison, midsagittal-plane symmetry presents the opportunity for a one-half-

409


analysis. Involved in the discussion from this point, then, are two or three anterior teeth and one associated posterior segment. As with canine retraction, both continuous and segmented-arch-wire procedures are used, and both are examined subsequently. Of substantial concern, particularly in the simultaneous retraction of incisors and canines, is the stability of the posterior-segment anchorage. Anterior retraction encompasses all displacements of the four or six arch

teeth characterized by some lingual movement of the crowns and/or root

apices. Depending on the initial positions and angulations of anterior teeth (as viewed from a buccal perspective), one of several possible orthodontic displacements may be required. Using the U-shaped archwire model—halved at the midline—the displacements are categorically transverse and first-order control need not provoke the degree of concern inherent in canine-retraction mechanics, particularly when only the incisors are to undergo displacement.

Shown schematically in the upper portion of Figure 10-24 are the incisors and posterior teeth engaging one-half of a continuous arch wire. The intent of the mechanics is to close space between the lateral incisors and the canines and, in doing so, to complete the displacement with the desired labiolingual, long-axis angulations of the maxillary incisors, The free-body diagram of the half arch wire and the force systems delivered to the half-incisal and posterior segments are shown in the center and lower portions of the figure, respectively. The anteroposterior, active

and responsive forces are ordinarily generated in one of two ways. A hook is affixed to or an "omega loop" may be bent into the arch wire mesial to the molar buccal tube (assuming facial-surface mechanics) or an appropriate opening or closing loop is incorporated in the arch distal to the lateral incisor. The arch is "tied back" either with flexible elastic elements or, in the presence of the loops (providing the mesiodistal flexibility), with segments of ligature wire. With the former method, illustrated in the figure, the posterior ends of the arch wire are left "free" to slide within the attachments to the buccal segments. The third-order couple Ca may be initially activated, secondarily activated for control only, or not present at all, depending on the character of the retraction displacement desired. Correspondingly, a responsive couple against the posterior segment may or may not exist. Vertical force components may also be present, as noted previously in the canine-mechanics discussion, but for simplicity they are not included here. Figure 10-25 presents appropriate illustrations for mechanics using a retraction spring to activate an anterior-segment force system. (An analogy is drawn here with Figure 10-23 and the discussion accompanying it.) Overall, in this approach, arch-wire segments individually engage

the incisal and posterior dental segments. To create incisal-segment torque (a moment about the bracket line), the retraction spring is activated against a hook affixed occlusally or gingivally of the wire-segment level, as discussed in Chapter 9 (see Fig. 9-28). The posterior end of the spring is attached to the buccal-tube assembly affixed to the terminal molar.

410


Ph

Cd

segment

Cd

FIGURE 10-24. Buccal views of continuous-arch mechanics to retract an incisal segment: representations of the buccal and half -incisal segments and the appliance (top); free-body diagram of the half arch wire (center); and active and responsive force systems delivered to the dental segments (bottom).

411


Fa

Fb

Retraction spring

Fb

Buccar segment

FIGURE 10-25. Buccal views of segmented-arch mechanics to retract an incisal segment: the buccal and half-incisal segments and the appliance (top); free-body diagram of the retraction spring (center); and representations of force delivery to the haif-incisal and buccal segments (bottom).

Because of the greater root-surface area, compared to that of the canine, a larger driving force is appropriate against the haif-incisal segment. Accordingly, the inherent potential for the responsive force system to displace the anchorage mesially is increased. Moreover, the

412


counter-tipping couple cannot be generated in the posterior segment with the mechanics illustrated in Figure 10-25. If all six anterior teeth are

to be retracted together, first premolars have been extracted, and third molars are unavailable for inclusion in the anchorage segments, unless mesial migration of the posterior teeth is a desired adjunct to the retraction process, an appropriate extraoral appliance (e.g., a face-bow headgear) is almost a necessity to transfer away at least a portion of the responsive force system. Alternatively, at minimum, posterior anchorage must be established and precede the retraction procedure. Posterioranchorage preparation is discussed in the section that follows.

Prepared, Posterior-Segment Anchorage In one or more phases of typical treatment mechanics the posterior segments ordinarily are recipients of responsive force. Moreover, these seg-

ments are expected to sustain such force systems without experiencing significant displacements. During canine or anterior retraction, for example, mesial movements of the buccal segments are often unwanted and may be intolerable. As mentioned in the preceding section,, headgear may be used to reinforce posterior anchorage, but the orthodontist cannot be assured of complete patient cooperation during the betweenappointments period. Accordingly, prior to the activation of mechanics against the posterior teeth, the clinician may "prepare" the buccal segments to withstand the responsive forces to come by making that anchorage as strong and resistive as possible. This section does not fully examine the process of posterior-anchorage preparation; rather, it focuses exclusively on the posterior segment. The procedures are nearly always bilateral and the preparation may be undertaken in either arch. Several biomechanics concepts form the basis for this process: (1) the strongest resistance is that which, upon action against it, responds with force carried to its supporting structure that approaches a uniform distribution; (2) the resorption of newly apposed bone is more difficult than the removal of mature bone; and (3) the most easily performed orthodontic displacement is that of extrusion. The objective of posterior-segment anchorage preparation is the distal crown tipping, without loss of arch length, of at least the terminal molar and, preferably, all of the teeth in the segment that contribute to the anchorage. The force to be resisted is likely delivered to the molar (and then distributed throughout the segment); it often originates with an interarch "elastic," and must not be directed with respect to the long axis of the molar such that the potential for extrusion is present. Anticipating that some loss of anchorage may occur in almost all instances of force applied against it, the amount or degree of anchorage preparation must be dependent on the magnitude of the responsive forCe system, the length of time the responsive force is expected to be present, and the format of anchorage displacement expected, should it be realized.

413


buccal-segment anchorage is most typically prepared for subsequent activation against it of inclined, interarch (Class III or Class Perhaps

II) elastics. Such an'elastic may be stretched between one hook affixed to an engaged arch wire (the hook located between lateral-incisor and canine brackets) and another hook, part of the buccal-tube assembly, affixed to the terminal molar on the same side but in the opposing arch.

With the mandible closed, typical distances between ends of the stretched elastic, parallel and perpendicular to the occlusal plane, are 22 to 30 mm and 8 to 9 mm, respectively. The angle of the elastic with the occlusal plane is, then, in the range of 15 to 20° with the teeth occluded. As shown in Figure 10-26, if the angle of interest is 18° and the force in the stretched elastic is 4 oz, the horizontal and vertical components are 108 g and 35 g respectively. Moreover, the 35 g is a sizable extrusive

force against the molar if its long axis is vertical. In addition, as the mandible opens, the elastic is further elongated and the angle between the stretched elastic and the long axis is diminished, producing a compound increase in extrusion potential. In partial preparation for interarch elastic force, the terminal molar, together with the adjacent teeth in the buccal segment, are displaced in distal crown tipping to establish angulation, relative to the anticipated stretched-elastic position and direction, similar to that of a tent peg. The angle between the occlusal extent of the long axis (of the "peg") and the elastic should be made greater than 90° as indicated in Figure 10-27 to eliminate the potential for extrusion. Subsequently, when the elastic is actually activated, creating the potential to tip the molar crown mesially and roll the tooth mesiolingually, a ligated, stabilizing arch wire is engaged to produce both counter-tipping (second-order) and counter-ro-

35g

4oz 108 g

4 oz

4 oz

FIGURE 10-26. Horizontal and vertical components of force delivered by a typical, 4 oz, Class II elastic force.

414


\' 90° pIus

\

FIGURE 10-27. A buccal view of a mandibular, terminal molar with crown tipped distally

so that the Class II elastic force does not include an extrusive component.

tating (first-order) couples and reinforce the anchorage. As noted previ-

ously, this provides as strong a resistance to movement as possible, considering the available root-surface area. To prepare the anchorage without losing arch length in the process, the proper displacement is crown movement as defined in Chapter 5. The desired angulation is to be achieved without mesial migration of the root apex and with minimum displacement of the crown. Accordingly, the cm of the tooth should be positioned at the level of the root apex. From a buccal perspective, a distal driving action is required together with a second-order couple to "hold" the apex. In the clinical setting, because achievement and maintenance of the necessary couple-force ratio is virtually impossible, perhaps an appropriate compromise incorporates an initial simple-tipping displacement followed immediately by bodily movement. The distal driving force may be created in several

415


ways, but it is a "pushing" action delivered by the arch wire to one of

the teeth in the buccal segment through a stop. This force is distributed within the segment through crown contact or interligation of the brackets and tube. Within the buccal portion of the arch wire, second-order bends are fabricated, to be passive upon wire engagement, thus permitting some initial tipping without interference as shown on the left in Figure 10-28. In the center sketch of the figure the tipping has eliminated the secondorder clearance, but some mesial displacement of apices has occurred. With the distal driving force still active, further potential tipping is inhibited by the secondary activation of second-order couples (sketch on the right in Figure 10-28) and, ideally, some bodily movement follows. Continued distal crown displacement is accompanied by distal movement of apices as well, the latter toward restoration of the initial apical positions (a procedure inherently difficult because of the need to resorb new bone at the apices). Note the intent to displace the posterior teeth not as a unit, but as individual "dominos," particularly during the tipping portion of the movement. An alternative approach to posterior anchorage preparation is again

in a sequence of two displacements, but in reverse order to that just described. Active second-order bends initially produce pure couple loading (in the idealized absence of friction); the crowns move distally and the apices mesially. Subsequently, the distal driving force is added toward alteration of the displacement to bodily movement. This procedure has its shortcomings compared to that described formerly with (1) the centers of rotation in pure, second-order torquing further from the

\ Arch wire passive Second-order clearance gone

Stabilizing, second-order bend

FIGURE 10-28. Distal crown movement in steps to prepare posterior anchorage: initial, second-order clearance between wire and bracket as the displacement is begun by a distal driving force (left); the second-order clearance eliminated by tipping (center); and subsequent activation by the wire of a counter-tipping couple (right).

416


than in simple tipping, (2) friction likely inhibiting the crown movement along the wire, (3) the addition of the distal driving force after the initial torquing displacement as a separate, not a secondary, activation, and (4) the control of the couple-force ratio toward produchon of bodily movement more difficult when the transverse force follows the couple rather than vice versa. Clinicians may describe the amount of anchorage preparation in a qualitative manner as "mild," "moderate," or "severe." Involved are the time interval over which these mechanics are continued and the initial force level together with the sizes of the second-order bends to apices

achieve a particular set of long-axis angulations with respect to vertical. The degree of preparation is jointly dependent on the expected responsive force system against the anchorage in magnitude, displacement-format tendency and time interval of existence of that response, the size of the anchorage (in terms of root-surface area) relative to the action, and the amount of "slipping" of anchorage that can be tolerated. At the risk of being repetitive, an additional, important consideration is that new, uncalcified bone is more difficult to resorb than mature bone; hence, the elimination ("burning") of anchorage generally requires more time than the establishment of it—an inherent asset not to be overlooked.

Following preparation of anchorage it is put to use. Typically this anchorage in the maxillary arch must resist Class III elastics and, in the mandibular arch, Class II elastics. To prevent the responsive forces from the elastics from tipping the prepared posterior teeth, stabilizing, second-order bends are placed in the wire segments or buccal portions of

the continuous arch wire engaging these teeth. Moreover, to further protect the terminal molars in particular (to which the elastics are attached) against possible extrusion, arch wires are fabricated such that, upon engagement, intrusive forces are induced when possible in addition to the second-order force system from the tip-back bends. As time proceeds following anchorage preparation, the newly laid-down bone in the mesial, crestal regions adjacent to the posterior teeth matures and becomes more susceptible to resorption. Concurrently, the time during

which the anchorage will continue to be "in use" is growing shorter. As the wanted displacements proceed toward their desired extents, the active force can be increased in magnitude. The accompanying responsive force will likewise grow, heightening the potential for the "slipping" of anchorage. Because the anchorage is not to remain as prepared following treatment—rather, the buccal teeth will need to be uprighted—the beginning of the elimination of the long-axis inclinations in

the closing stages of Class III or Class II mechanics, for example, is appropriate. Care must be taken, however, to ensure that the anchorage "burning" procedure approaches the exact reversal in format of the anchorage-preparation process. If a portion of the preparation remains following discontinuation of the mechanics against the anchorage, a com-

bination of the "memory" of the periodontal-ligament fibers and ongoing physiologic processes will help to restore the upright orientations of the posterior teeth. "Up and down" elastics can be used subsequently, as necessary, to achieve the desired intercuspation.

417


Class III Mechanics For potentially good occlusion and dentofacial harmony, the maxillary and mandibular arches should be properly positioned, anteroposteriorly, with respect to each other. Without attempting a strict definition or detailed description, a Class III tendency or malposition generally suggests that the location of the mandibular dentition is improperly anterior with respect to the maxilla. The mechanical objective of Class III mechanics is often to exert posteriorly directed force against the mandibular arch. Alternatively, this phase of therapy may intend to impart action to advance the maxillary dentition. Class III mechanics, however, typically focus on the mandibular arch and may be designed, for example, to produce posterior displacement of the incisors, the six anterior teeth, or the entire arch to establish posterior anchorage (as described in the previous section), or to exert holding force against (or attempt to reverse) undesired, excessive, forward growth of the mandible. Class III mechanics generally require attachment of appliances to dental units in both arches, interarch force transfer is ordinarily present, and headgear may be used. The action and response are bilateral; geometric and mechanical symmetry relative to the midsagittal plane is typi-

cal and is so assumed in the analyses to follow. The U-shaped arch model is appropriate, and usually necessary is the division of each half arch from a buccal perspective into three segments: an incisal or anterior (half) segment, a middle segment, and a posterior segment. The teeth

typically included in each of the three segments have been noted in a previous section of this chapter. The interarch elastics, which are the principal activating elements of intraoral Class III mechanics, are each stretched between a gingival hook affixed to the buccal tube integral to the crown of a maxillary terminal molar and a hook attached to the mandibular arch wire just distal to the lateral-incisor position. A free-body diagram of the right-side elastic is shown in buccal view in Figure 10-29. The magnitude of tensile force in the activated elastic is symbolized by Fe and the forces exerted on the ends of the auxiliary have been decomposed into horizontal (occlusal-

plane) and vertical components. The second molar is assumed to be included in the maxillary buccal segment; the angulation of the elastic is Fe

vi"

= 3.7 (V,,,)

(with the elastic angulated 15° trom the occiusa] plane)

H,,,

Fe

FIGURE 10-29. A free-body diagram in the buccal view of a right-side? Class Ill elastic.

418


approximately 15° with the occlusal plane. With this orientation the hori-

zontal components are about 3.7 times as large as the vertical components and, for practical purposes, are equal to the elastic force itself. Ordinarily engaging the maxillary dentition is a continuous, stabiliz-

ing arch wire for the purpose of distributing the force of the elastic against the terminal molar. Although no direct connection between the elastic and wire exists, the elastic force can be transferred to the arch wire by the buccal-tube assembly. If the maxillary arch, in particular the maxillary buccal segment, is to provide anchorage in these mechanics, stabilizing second-order bends are placed in the posterior portion of the

arch wire including tip-back bends, perhaps active, at the terminalmolar positions. The second-order couples are intended to prevent mesial crown tipping of the posterior teeth and, often, the tip-back bend is sufficiently severe to also create an intrusive force against the terminal molar. In addition, an extraoral appliance to the maxillary arch, perhaps including a face bow with direct attachments to the terminal molars, may be used. Again, if anchorage reinforcement is the intention, the headgear principally transfers away the horizontal component of force originating with the elastic. Although dependent on the objective of the mechanics to some ex-

tent, typically a full-size arch wire is placed in the mandibular arch. From the hooks affixed to the wire, through which the elastic force is carried to the mandibular dentition, the horizontal component may be directed anteriorly, posteriorly, or both, depending on clearances and stops. If the objective is to establish posterior mandibular anchorage, the horizontal component is transmitted directly to the posterior segment—

perhaps first to the terminal molar, then distributed from that site throughout the segment. In this instance the vertical component is unwanted, so neutralization is attempted through concurrent activation of the arch wire itself (to be detailed subsequently) together with distribuHon of the component mesially and distally from the site of the hook to involve as much root-surface area as possible. The force and structural analyses of Class III mechanics is a formidable undertaking, but feasible after study of the previous chapters and, in particular, with understanding of the examples cited thus far that consider component parts of the mechanics and their analyses. The model has been proposed, several reasonable assumptions have been made, and the appliance elements have been discussed. The specific application to be examined in some detail is that of anchorage preparation in the mandibular arch. Referring again to Figure 10-29, of the four components present only one, the horizontal component delivered to the mandibular arch, is wanted. A significant portion of the mechanics is de-

voted to the suppression of the potential effects of the other three components. The initial magnitude of the elastic force, as chosen, must take into account the number of posterior teeth to undergo distal crown tipping, some expected relaxation of the elastic, the angulation of the elastic, the presence of friction, and the possible additional "loss" of force to the anterior teeth.

419


The maxillary force system is illustrated in the buccal views in Figure 10-30. To be noted first is the tendency of the elastic force to extrude the terminal molar and to tip the crown mesially. The maxillary arch wire and, perhaps, molar occlusion are available to resist extrusion, in addition to the periodontal support. In developing the force analysis of the

maxillary half arch, three separate sketches are presented to highlight the three-part activation of the wire. (None is a complete free-body diagram.) From top to bottom, the first shows the secondary action which is a direct outcome of activating the elastic; with a stop mesial to and contacting the buccal tube, the elastic force tends to displace the posterior

extent of the wire and the molar downward and forward. The second sketch illustrates the action-reaction counterpart to the primary activation of the tip-back bend, producing resistance to both mesial crown tipping and extrusion. The third sketch depicts the primary activation of a straight-pull headgear. The composite free-body diagram of the maxillary half arch wire is shown in the upper portion of Figure 10-31. Force-system components, exerted by the dentition, have been added to those activating the half arch to balance the free-body diagram. The headgear force is intended to

offset the horizontal component of the elastic force transferred to the wire through the buccal attachment to the molar and the stop, although some of the elastic action may find its way to the incisors (H1), partially

because of the high mesiodistal stiffness of the wire. Although Vmid could alone provide the balance of vertical forces, the combination of the From the Class IN elastic

CP

From the tip-back bend

From the straight-pull headgear

4

U

FtGURE 10-30. Right-side buccal views of the maxillary-arch force systems associated with Class Ill mechanics intended to prepare posterior anchorage in the mandibular arch.

420

Bloengineering Analysis of Orthodontic Mechanics vi

cp I-li

Fhg Vrnid

Midarch segment

Posterior segment

Incisal half-segment

v,

vw Vmid

(?)

FIGURE 10-31. A right-side, buccal-view, free-body diagram of the maxillaiy half arch wire in Class Ill mechanics (top). The crown force systems delivered to the posterior, midarch, and haif-incisal segments (bottom).

force and couple associated with the tip-back bend necessitates the pres-

ence of a vertical-force component in the anterior region to obtain a moment balance and complete the quasi-static analysis. The action-reaction law now enables an examination of the force system induced against the maxillary dentition (lower portion of Fig. 10-31).

Looking once more at the terminal molar, which also represents the posterior segment, the horizontal component of the elastic force is carried first into the arch wire and then is transferred to extraoral anchorage by means of the headgear. The second-order couple opposes the mesial

tipping potential of the elastic force, particularly important when the headgear is absent. The tendency of the elastic to extrude the molar must be countered by the arch wire and/or occlusion. The combination of the tip-back bend and the bending stiffness of the wire must be capa-

421


of providing the principal resistance to extrusion in the absence of posterior occlusion. The elastics tend to promote occlusion; however, note that posterior extrusion is potentially manifested in anterior biteopening that may or may not be a desirable adjunct to the mechanics. Depending on the relative magnitudes of the elastic and headgear forces, the horizontal force delivered anteriorly, if any, can have either sense. The sketches in Figure 10-31 assume that the headgear force dominates, which yields the senses of the horizontal incisal components shown. The vertical elastic-force component combined with the tip-back action induces the vertical components exerted on the middle and halfincisal segments. In the sketch representing the half-incisal segment the moments of the two force components about the cre oppose one another; in the absence of the headgear force and with the wire stopped as previously described, there is a tendency to flare the incisors. An interincisal force from the mandibular arch would add to the flaring potential if exerted on the lingual surfaces. The middle segment of the half arch wire may function as a fulcrum; the elastic action tends to rotate the arch wire and the maxilla counterclockwise as viewed from the right side. ble

The anteroposterior position of the middle-segment force depends largely on the characteristics of the posterior-segment force system; the senses of the couple and vertical forces of the posterior segment suggest that the resultant response in the middle segment is positioned more posterior than anterior in that segment. The appliances under discussion are facially affixed; hence, in addition to the tendencies seen in the buccal views of Figure 10-31, the elastic action has the potential to also rotate and lingually tip the molars. The arch wire must provide the resistance to these displacement tendencies, too, and toe-in bends may be placed to generate countering, first-order

couples. Moreover, the arch wire should be rectangular, not only to enhance bending stiffness potential relative to cross-sectional shape, but also to enable creation of stabilizing, third-order couples to counter any labiolingual, crown-tipping potential. Before proceeding to an analysis of the mandibular-arch mechanics, a review of the action and response associated with tip-back bends may be worthwhile; this force system may exist in both arches. A free-body diagram of one half of a mandibular arch wire was illustrated in Figure 6-27. The active tip-back bend was accompanied by an intrusive force; this is the combination of force and couple shown in Figures 10-30 and 10-31. Recall that, although a responsive force from the middle segment of the dentition would provide a vertical force balance, these two forces would form a couple with the same sense (counterclockwise as viewed) and, therefore, a moment balance is impossible. Thus, a third vertical force must be present, anterior to the middle-segment force, likely small because of its distance from the terminal molar, but creating a substantial, clockwise moment with respect to the buccal tube. Note further that the activation of a prepared second-order bend, depending on the mesiodistal position of the bend with respect to the bracket or tube and the overall occlusogingival passive position of the wire relative to the ortho-

422


dontic attachment, may produce a vertical force of either sense accompa-

flying the couple. Shown in Figure 6-26 was the second-order couple in combination with an extrusive force. This force system can be balanced by just the addition of one responsive force as long as the couple-force ratio is not extreme. The location of the responsive force is anterior of the buccal-tube force system by an amount equal to the couple-force

ratio. The couple formed by the two vertical forces is clockwise and opposite in sense to that of the tip-back couple. In both sketches the specific location of the middle-segment response depends on the relationship between the magnitudes of the force and couple exerted by the buccal tube. The analysis of the right half of the mandibular working arch wire is illustrated in Figure 10-32. Initially, on the left, separate principal actions against the wire—from the elastic and second-order effects—are shown.

These are then combined and the free-body diagram of the half arch wire is presented. The desired, horizontal component of the elastic force is meant to be wholly transferred, by the arch wire, distally to the posterior segment to become the crown-movement driving force. A portion of that horizontal component can reach the incisors, however, if not ini-

tially then after some tipping of the posterior teeth has occurred. The location of the connection of the elastic to the arch wire, together with the elastic angulation and the small root-surface areas of the teeth, creates a strong tendency to extrude the incisors. Accordingly, a reverse

F1

(?)

v;

v,

From the elastic

From the second-order bends

Posterior segment

I-"

Incisal half-segment

• VIII

I

H,' I

VP

I

vi

if

FIGURE 10-32. Right-side buccal views of the mandibular-arch force systems associated with Class Ill mechanics designed to prepare posterior anchorage in the mandibular arch.

423


curve of Spee in the wire is necessary to induce intrusive force against

the incisors and counter the vertical-component action of the elastic force. The potential to extrude the middle segment is created simultaneously; occlusion is needed to effectively counter this tendency. The secmay or may not be active initially, and it may ond-order couple,

appear with the opposite sense after some tipping has occurred, as noted in the previous section. With appropriate changes in some magnitudes and, perhaps, some slight alteration in the position of the middlesegment force, both vertical-force-component and moment balances can be achieved with or without the second-order effects present. Once again, action-reaction enables transfer of attention from the appliance to the dentition. The distal driving force is transmitted with the aid of a stop from the arch wire to the buccal tube and molar, then distributed to other teeth within the posterior segment. The source of the intrusive force is the curvature placed in the wire; the tip-back couple, perhaps initially active, is also shown and further represents second-order contact within the segment. Occiusal force from the maxillary arch may also be present and is indicated. Already noted is the tendency for the vertical force, from the passive curvature of the wire, to extrude

the middle segment. Responsive components against the haif-incisal segment include the opposing vertical forces, one arising with the elastic action and the other from the wire curvature, and the horizontal component that tends to produce lingual crown tipping. (Unlike the horizontal force against the maxillary incisors, this component can have only one sense.) If the horizontal component is unwanted, labiolingual clearance must exist initially between wire and incisor brackets; as that clearance disappears with the posterior displacements, the arch wire must be adjusted or replaced to recreate the clearance. An alternative approach is to deliver the elastic force not by the arch wire, but, instead, to a sliding jig (see Chap. 9) that by-passes the middle segment and directly transfers the horizontal component to the terminal molar. As mentioned previously, these mechanics are generally characterized by the presence of Class III elastics, and in the example just detailed only one of the four components (see Fig. 10-29) provides the desired action. It follows, then, that various additional applications of Class III

mechanics can be categorized with reference, at least partially, to the forces transmitted by the activated elastics. Considering another example in which the action is in the mandibular arch, the vertical component

of the elastic force might be used to help reduce anteroposterior arch curvature through extrusion of the middle segments. (A light, mandibular arch wire would be engaged to concentrate the elastic action.) The incisal-anterior segment might be retracted with the horizontal component providing the driving action, but the accompanying potential of the activated elastics to extrude those teeth must be recognized and appropriately dealt with, depending on the need or desire to close the bite through this displacement. Reversing the action-response roles of the "ends" of the elastics, movements in the maxillary arch are also possible with anchorage pro-

424


——

——

Unified dentition

FIGURE 10-33. An orthopedic force system, in a right-side buccal view, associated with maxillary-arch mechanics intended to reduce an anterior deep bite by rotating the maxilla as a unit.

vided in large measure by the mandibular dentition. With the mandibular posterior teeth providing some stability against undue extrusion of their interarch counterparts, previously prepared maxillary, posterior anchorage may be eliminated. In fact, advancement of the posterior segments anteriorly in the maxillary arch may be undertaken. The limit to whole arch forward movement is in the sufficiency of the mandibular-

arch anchorage and, although headgear may be used to augment the anchorage potential, extraoral-appliance application to the mandibular arch demands extraordinary considerations, in comparison to maxillaryarch attachment, which have been discussed in Chapter 9. Another important application of Class III mechanics in deep-bite cases is the coupling of the elastic force in the posterior to that from an anterior-positioned, high-pull headgear to rotate the entire maxilla. A heavy, continuous arch wire must be used to provide vertical unification of the maxillary dentition through as large a bending stiffness as possible. The active force system is shown in Figure 10-33. The two forces appearing in the buccal view should be parallel and equal in magnitude if the rotation is to be about the cra (center of rotation of the maxilla), partitioning the displacement effectively between the anterior and the posterior segments. Note, however, that the existence of posterior occlusion can hinder the downward movement of the posterior portion of the maxilla and, in effect, move the cra posteriorly.

Class II Mechanics The intraoral appliance assembly used in typical, edgewise Class II mechanics appears, excepting the headgear, to be the mirror image in the buccal view of the Class III assembly discussed in the previous section.

425


The appliances include arch wires, generally both continuous, individu-

ally engaging the maxillary and mandibular dentitions; the desirable structural characteristics and detailing of the wires depend on the clinical objectives. In Class II mechanics, force is transferred between arches

by bilateral elastics that are individually stretched between hooks at mandibular-posterior and maxillary anterior sites. These mechanics, like those of the preceding section, receive their name from the type of malocclusion they often help to correct: the maxillary dentition is forward of the mandibular teeth, with respect to the ideal dentofacial-complex configuration, and the visible manifestation is known as "overjet." In Class II mechanics the stretched elastics may provide one or more active components of the desired force system. If the intent is to retract

the incisors or the six anterior teeth or, perhaps, the entire maxillary dentition, the elastics exert at least part of the posteriorly directed driving force. Torque against the anterior teeth, to hold or alter the labiolingual, long-axis angulations, may be a part of the active force system. An extraoral appliance may be used concurrently to provide additional driving action and, possibly, third-order action as well. In particular, to offset the often unwanted extrusive potential of the vertical component of the elastic force against the maxillary arch, a high-pull headgear may augment the intraoral mechanics. On the other hand, if the malocclu-

sion includes an anterior open bite and that correction has not been undertaken earlier, the vertical component of the Class II elastic force is welcome if the initial position of the maxillary-anterior teeth relative to

the soft tissue is such that extrusion can be tolerated as a corrective measure. The action of the appliance may be intended to be at the mandibular

posterior attachment sites, alternatively, with anchorage provided by the maxillary arch. The clinical objective may be to upright the mandibu-

lar molars or the "burning" of prepared anchorage that is no longer needed. Protraction of the mandibular dentition to aid or augment growth of a potentially deficient mandible may be attempted at the appropriate time. Another possible objective might be that of posterior bite closure through extrusion and, perhaps, accompanying mesial crown tipping of the mandibular posterior teeth. As noted in the discussion of Class III mechanics, the interarch elastic is a two-force member with each force considered the resultant of components parallel and perpendicular to the occlusal plane; maybe just one, or perhaps all four of the components of the forces transmitted by the inclined elastic may be desired as active initiators of displacement in a specific situation. The detailed analysis undertaken here, however, is that of the Class II mechanics used to produce controlled, posteriorly directed, bodily movement, displacement potential delivered to the maxillary incisal segment. In anticipation of the forthcoming Class II mechanics, posterior-anchorage preparation may have been completed in the mandibular arch; the degree of preparation is dependent on the available root-surface area, the expected magnitude and duration of the elastic forces, and the tolerable amount of anchorage "slipping" concurrent with the Class II

426


mechanics. A full-size stabilizing wire is ordinarily engaged in the man-

dibular arch for two principal reasons: (1) to unify on each side the posterior teeth toward distribution of the long-axis-parallel component of the elastic force tending to extrude the terminal molar (if and when the occlusal direction is less than perpendicular to the stretched elastic), and (2) to permit creation of sizable second-order couples with little wire deformation and, therefore, inhibit mesial crown tipping of the posterior teeth in favor of establishing bodily-movement potential. Second-order bends may be placed in the posterior sections of the wire to create immediately active, counter-tipping couples along with intrusive action from the arch wire, particularly at the terminal-molar sites. Should the elastics begin to produce mesial crown tipping of the terminal molars, crown contact will distribute that effect anteriorly and sequentially involve the teeth as far forward as the canines in the resistance. If some flaring of upright, mandibular incisors is a desired displacement adjunct, a portion of the anteriorly directed effect of the elastic may be diverted into the arch wire by a judiciously located stop mesial to the buccal tube and carried within the wire to the incisal segment. The activated, Class II elastic pulls on a gingivally directed hook affixed to the maxillary arch wire (or against a loop bent into that wire) located in the interbracket space distal to the lateral-incisor position. (If the six-tooth anterior segment or the entire maxillary arch is to be retracted, the elastic attachment may be distal to the canine site.) This wire is rectangular, to permit activation of third-order couples, but is a working wire having sufficient flexibilily to enable moderate, activating deflections and twists without creating force systems having excessive initial magnitudes. Two maxillary-arch-wire designs are rather common: one with gingivally oriented, simple, closing loops and the other without loops. If the wire contains no loops, then elastics (perhaps together with headgear) generate the wanted, posteriorly directed force components that

are carried by the wire to the incisors. To permit retracting displacements, the arch wire must be permitted to sllde through the posterior brackets and buccal tubes. To enhance clearances in an effort to minimize frictional resistance, the posterior sections of the maxillary arch wire might be reduced in an acid bath prior to engagement. Furthermore, for the same reason, permanent bends are not placed in the posterior sections of the wire (which means that no stabilizing couples are possible without eliminating the clearance and inducing resistance to the retraction effort). The placement of loops in the maxillary arch, alternatively, changes the picture substantially if those loops are to be activated, either initially or secondarily (and serve other than just an elastic attachment). Bilateral stops are affixed to the wire mesial to the buccal tubes. Depending on the positions and stability of the maxillary posterior segments, and perhaps other, more subtle considerations, the arch wire may be tied back to open and activate the loops somewhat and generate the initial posteri-

orly directed force against the incisors. Subsequent activation of the

427


II elastics augments the wanted force initially. As retraction occurs, the ioop force deactivates more rapidly because of its greater mesiodistal stiffness compared to the extensional stiffness of the elastic and, Class

in time, the loop force may change sense and oppose the horizontal component of the elastic force. The initial mesial-movement potential against the maxillary posterior teeth is changed then to a distalization tendency. On the other hand, if the loops are passively open and the stops are initially up against the buccal tubes, activating the elastics tends to close the loops, secondarily activating them. A division of the horizontal component of the elastic force occurs; a portion of it is carried

to the incisors and the remainder to the posterior segment, and both responses to the elastic force are posteriorly directed from the start. The

loop attempts to return to its passive configuration and, as time proceeds, the ratio of force magnitudes delivered to the segments, incisal compared to posterior, diminishes. Accordingly, this last arrangement would be most appropriate for whole-arch retraction. Figure 10-34 schematically compares the changes in forces to the seg-

ments over time, resulting from immediate, opening activation versus

Passive, partially closed

T

Passive,

open

' Fe

FIGURE 10-34. Active and delivered mesiodistal forces in the maxillary arch arising fmm the combined use of Class II elastics and simple, vertical loops. Right-side, buccalview, free-body diagrams of the half arch wires, and changes in the force magnitudes with time (top to bottom), comparing mechanics with the loop initially opened elastically (left) and initially passive (right).

428

Bioengineering Analysis of Qrfhodontic Mechanics

secondary closing activation. Not to be overlooked in this discussion is

the effect of the loops on the localized flexibilities (other than the mesio-

distal influence) of the anterior portion of the arch wire relative to the posterior portions. Depending on ioop geometry, a substantially increased tendency for the vertical components of the elastic forces to extrude the incisors may or may not be created. The third-order flexibility of the anterior arch-wire section, however, is invariably enhanced by the presence of any loops (see Chap. 8). Considering further the vertical-component influence on the maxillaw incisors, three counter-extrusive procedures, when warranted, are commonly used, either individually or together. First, if the tied-back, looped arch is used, the legs of each loop may be gabled with respect to one another. The result is the activation of an intrusive-force distribution against the incisors with engagement of the wire and prior to placement of the interarch elastics. Second, active second-order bends in the postenor segment together with intrusive action against the terminal molars concurrently induces, in response, intrusive force against the incisal segment. An accompanying feature of this force system, specifically in the created second-order couples, is the strengthening of the posterior-segment anchorage in the maxillary arch. The couples tend to counter the tipping potential of the mesiodistal force attempting to restore the passive state of the opened loop. Third, an auxiliary-appliance deterrent to incisor extrusion is present in the inclusion in the mechanics of a highpull headgear. This extraoral appliance, delivering force directed upward and rearward, not only helps with the posteriorly directed action, but also opposes the vertically downward component of the Class II elastic action. As indicated in Chapter 9, the J-hooks delivering the extraoral force should transfer their action to the arch wire at locations between the lateral and central incisors. The line of action of the headgear force may be directed centrally through the arch wire or occlusal or gingival to it, depending on the hook configuration. The potential then exists to create, with the extraoral force, third-order, incisal action in either direction or not at all. The force and structural analysis of Class II mechanics, used to bodily retract the maxillary-incisal segment, is initiated under the assumption of complete symmetry with respect to the midsagittal plane. The maxillary and mandibular dentitions on one side of the midline are each partitioned into three segments with the first maxillary premolars assumed in this example to have been extracted to accommodate the anticipated reduction in overjet. The incisal half-segment consists of a central and adjacent lateral incisor in each arch. Crowded incisors in the mandibular arch led in part to the decision to extract first premolars there also. Retraction of four canines was previously completed and each mid-

dle segment, then, includes a canine and a second premolar. With the second molars erupted and appropriately positioned in all quadrants, the posterior segments in the modeling become the molar pairs; buccal tubes are affixed to the second (terminal) molar crowns. Before beginning this example analysis, it is worthwhile to recollect the intra-arch, anterior-retraction mechanics discussed previously.

429

Force and Structural Analyses ot Representative Orthodontic Mechanics

The Class II elastic, like the elastic that identifies Class III mechanics, is recognized as exerting four significant components of force. All four appear in a buccal view; two buccolingual components are very small and, therefore, neglected. A free-body diagram of the right-side, Class II elastic would look much like that in Figure 10-29 except for the angulahon and, perhaps, slight differences in the relative magnitudes of the components parallel and perpendicular to the occlusal plane. In addition, if anchorage "slipping" in the mandibular arch and extrusion of the maxillary incisors are unwanted, only one action-reaction component of the elastic forces developed—the posteriorly directed component delivered to the maxillary arch wire—is desired. Accordingly, the mechanics must incorporate provisions to suppress the displacement potentials of the other three components. The Class II elastic is stretched on one end against a hook that is part of the buccal-tube attachment to the mandibular second-molar crown; tipping and extrusive displacement potentials are thereby created. Figure 10-35 includes right-side, buccal views of the free-body diagram of

the Class II elastic and the force system from the elastic against the terminal-molar crown. The sketch of the second molar also contains force-system components arising from contact with the arch wire, the adjacent first molar, and the occluding maxillary molar. The mandibular arch wire is continuous and full size (stabilizing). In this figure the force system exerted by the posterior segment on the wire is first shown divided into that associated with the elastic and the components arising from second-order bends. The two portions are then combined in the adjacent sketch and the entire force system exerted on the half arch wire is completed through force and moment balances. The horizontal component from the activated elastic is distributed throughout the posterior and middle segments through crown contact (if not from the instant of activation, then following sequential tipping

v. v; H;

From the Class II &astic

v'

H;

From the tip-back bend VPp

FGURE 10-35. Free-body diagram ot a right-side Class II elastic and right-side buccal views of the mandibular-arch force systems associated with Class II mechanics.

430


displacements until the contacts are established) and, via the arch wire

(through friction and, possibly, a posterior stop) to the incisors as well. The vertical components against the posterior teeth, together with the second-order couples, generate the vertical responses against the middle and incisal segments. Of some concern, because of the size of the force relative to the root-surface area of the as-modeled segment, is the extrusion potential against the middle segment. Accordingly, careful

adjustment of the arch wire in the buccal region may be prudent to "transfer" the first molar from the posterior to the middle segment. In addition, some displacement control should be expected from occlusion of the middle-segment teeth. Action-reaction involving the elastic, the posterior segment (the second molar), and the half arch wire must be used carefully in completing the force analysis. Force and moment balances and action-reaction permit the description of the force system delivered to the incisal seg-

ment. The second-order effects between the wire and second molar result in a small, intrusive force divided between the central and lateral incisors. If the posterior- and middle-segment anchorage is insufficient to contain the mesial-driving potential of the elastic force, some labial crown tipping of the incisors is possible. Note in the sketch of the in-

cisal segment that the moments of the two force components about the segment crc have the same sense; thus, a compound tendency for incisor flaring is possible. One partial remedy is not to place the posterior stops in the arch wire at all, thereby allowing transmission of the horizontal response forward beyond the canines only by means of friction.

Four separate bilateral actions may exist in these mechanics against the maxillary working arch wire. First, the elastics exert force on the wire

through hooks or loops distal to the lateral-incisor locations. Second, torquing couples are delivered to the incisors. Third, a headgear force system may be transmitted to the incisal segment. Fourth, if the arch wire contains active opening or closing loops, the posterior segments (minimally the second molars) exert force on the wire. In the specific example mechanics under analysis here, assumed is the potential existence of bracket-wire clearances distal from the canines, and no stops affixed to the wire to transfer horizontal force between arch wire and buccal-segment crowns. This eliminates the fourth source of force in the above list from consideration in this example. On the left in Figure 10-36 are shown, separately, the actions on half

of the maxillary arch wire of the Class II elastic and J-hook from the high-pull headgear, and the response to torsional activation in the incisal segment, all from the right-side buccal perspective. Note that the J-hook force is shown with its line of action piercing the incisal section of

the arch wire; actually, the line of action is the dashed line. The headgear force has been "moved" and the compensating couple placed is largely, if not totally, that shown in the lower left sketch. In the lower right in the figure, the three contributions have been joined, together with balancing components, to complete a free-body diagram of the

431


From the Class II elastic

ci F11

H, V, = V11 —

From the high-pull headgear

V,,9

H

Cb

From the lingual root torque

H11

vu

FIGURE 10-36. Right-side buccal views of the maxillary-arch force systems associated with Class II mechanics intended to bodily retract the maxillaly incise! segment.

right half of the continuous arch wire. The horizontal components (parallel to the occlusal plane) from the elastic and headgear add and gener-

ate the distal driving force against the incisors. The driving force is smaller than that sum by only the frictional resistance to the sliding of the wire through the buccal attachments. Assuming the third-order stabilizing couple is produced entirely by the eccentrically (gingivally) attached J-hooks, and no activating twist is placed in the wire prior to its

engagement, the net vertical force against the incisal segment has a sense dependent on the individual magnitudes and angulations of the headgear and elastic forces. Responsive vertical forces against the buccal segments are negligible due to the proximity of the headgear and elastic attachments to the incisal segments. By action-reaction, the crown force

systems against the segments are obtained. No forces of any consequence, neglecting friction, exist against the molars or middle segments, so they have not been pictured. The sketch representing the incisal pair (Fig. 10-36, upper right) shows the resultant lingual driving force, the counter-tipping, third-order couple, and a resultant of the vertical components of the elastic and headgear forces which often ideally should be zero.

The example mechanics under discussion are designed to bodily move the 4 maxillary incisors lingually to reduce an overjet. Assuming a continuous force system, reasonable light-force figures are 155 to 165 g and 1700 to 1800 g-mm for the driving force and counter-tipping torque, respectively, for the segment. (The couple-force ratio must be approximately one-half of the average tooth length; see Chap. 5.) Achieving the

432


driving force of nearly 3 oz per side may seem straightforward; however, the expected relaxation of elastics over time and the division of the

force between the Class II elastics, likely worn continuously, and the headgear, probably engaged intermittently, must be considered. Example calculations with sketches are presented in Figure 10-37, suggesting that, as a basis, the product of the vertical force component and time worn be equated for the elastic and headgear action on each side. Now note once again that, in this example, the torque is delivered through a pair of eccentrically attached J-hooks. The third-order couple induced is the product of the total headgear force and the perpendicular distance from the line of action to the center of the wire (e in Fig. 10-36). With the headgear force calculated in Figure 10-37, the torque is deter-

mined. Two difficulties are now apparent: (1) the dependence of the torque on the driving force; and (2) the inability to achieve sufficient torque magnitude (without intolerably long hooks attached interincisally). The only answer, then, is the establishment of at least a portion

of the total torque requirement through third-order activation of the rectangular arch wire, which brings with it the secondary considerations discussed in Chapter 8. The overall problem is complex and perplexing, and to date it has been approached clinically by retracting the incisors in the presence of some torque, expecting that the long-axis angulations

Projected root-surface area = 2(0.5 + 04) = 1.8 cm2 Force = 3.5(Area) (60)12.5

= 155 g against 4 incisors Couple-torce ratio = 0.5 (average total tooth length) = 105 cm For each side of arch:

/

Distal driving force = 80 g = 2.8 or It elastics worn continuously and headgear for 12 hr each day. Vh2 = 2(V11) (vertical balance)

It the Class Il elastic makes a 20° angle with the ocolusal plane, as does the headgear force, H11 = 40 g, = 15 g, F11 = 43 g, To accompany the 155 g against Hh2 = 80 g, Vhg = 30 g, Fhg = 85 g the incisal segment, needed Total headgear force (both sides) is a couple = 155(1 0.5)

=1650g-mm

=170g

To generate this couple with the headgear force, e must be 10 mm. FIGURE 10-37. The force system delivered to the maxillary incisal segment from the combined action of elastics, headgear, and arch wire, designed to bodily move the incisors toward the posterior without vertical displacement.

433


will have to be corrected after the desired crown positions have been achieved. With alterations, the mechanics under discussion may be used to produce posteriorly directed, en masse movement of the maxillary dentition. First, assuming that any mesiodistal spaces have been previously closed, the entire dentition should be unitized through appropriate in-

terbracket ligations. Stops should probably be placed mesial to the brackets or tubes affixed to the first molars so that the arch wire, in addition to crown contact, is used to help distribute the driving force. Second, the driving force, if partially delivered by Class II elastics, should probably be transferred to the arch wire just distal to the canines. This is somewhat anterior to the center of resistance of the maxilla and is a compromise, noting that the activating forces should be as nearly hori-

zontal as possible. On the other hand, consideration must be given to the position of the line of action of the resultant of the headgear and elastic forces on each side; the length of the moment arm of the resultant, with respect to the cra, must be minimized to suppress the potential for rotation of the maxIlla if bodily movement is wanted. Third, the magnitude of the resultant driving force must be increased to perhaps in excess of five times the figure determined for the incisal segment (depending on the root-surface area present), to be divided between the two sides of the arch. Again, if Class II elastics are used, responsive forces carried to the mandibular arch are enlarged, and more severe

anchorage preparation than that for incisal-segment retraction

is

needed. A worthwhile, last example of this section is the use Class II mechanics in protraction of the mandibular dentition, perhaps in an attempt to aid growth at the proper time when an otherwise retrognathic mandible is expected. In these mechanics the active and responsive ends of the elastics are reversed. The action is in the mandibular arch; the anteriorly directed elastic component delivered to the mandibular terminal molar

is the desired one of the four components generated. A series of diagrams in Figure 10-38 illustrate the essentials of the force system. The driving-force component must be distributed anteriorly to the incisal segment by the arch wire; crown contact is relied upon in the buccal segments. A unitized, maxillary dentition is the first line of anchorage, but reinforcement with extraoral assistance is warranted. To prevent unwanted anterior bite closure a high-pull headgear is attached to the maxillary arch wire at the same anteroposterior positions as the elastic attachments. Because the resultant force against the maxillary arch tends

to move it posteriorly, this force must be distributed as uniformly as possible throughout the entire maxillary dentition; moreover, the elastic and headgear forces must be relatively light to minimize the potential for reciprocal displacements in the two arches. To provide some differential in the root-surface areas, second molars should be left out of the mandibular strap-up (and moved mesially at a later time). A suitable reverse-

pull headgear, one not involving a chin cap, could help reinforce the maxillary-arch anchorage.

-

434

Bioengineenng Analysis of Orthodontic Mechanics

--—

Vt F11

Mesial to the terminal molar

Hm,d

H1

FIGURE 1 0-38. Right-side buccal views of the force systems intended to produce protraction mechanics in the mandibular arch.

435


Synopsis Beginning with displacements of individual teeth and proceeding to

controlled movements of dental segments, starting with an individual arch wire and advancing to complex appliances incorporating wires in both arches, interarch elements, and extraoral auxiliaries, the intent of this chapter was to gain familiarity with orthodontic-mechanics analysis from a structural standpoint. Example analyses were purposely chosen from existing treatment procedures so that, typically, the reader would begin with some clinical understanding of the mechanics examined. The appliances were disassembled and the active and responsive force systems, exerted on both the appliance members and the dentition, were completed and scrutinized. The analyses were more qualitative than quantitative in studying these examples, and a principal objective was the identification of the force-system components created throughout the appliance resulting from its activation. All forces and couples delivered to the dentition have the potential to move teeth and a good appliance design enables selective, controlled displacements where they are desired while impeding or suppressing unwanted tooth movements elsewhere. Not discussed were a substantial number of procedures and devices, and brief comments about several are appropriate before concluding this chapter. Positioners and retainers may be expected to perform minor tooth movements. They use the palate, the mucosa, and the gingiva as anchorage sites, providing broad distributions of responsive forces and strict displacement control. The emphasis in the set of example mechanics examined was on labially affixed appliances. Lingual mechanics in contrast are characterized by sizable and distinct wire curvatures in the arch-form plane, potentially reduced interbracket distances (although this depends on mesiodistal bracket widths, and narrower brackets are often used to accommodate lingual tooth anatomy), and pretorqued brackets because of the third-order inclinations of lingual surfaces of the anterior teeth in particular. No fundamental differences, however, are inherently present in force- and structural-analysis procedures, lingual versus labial appliances. Functional appliances, on the other hand, are unique because they are removable, labiolingual in their contacts with the dentition, and activated in part against the oral and facial musculature. These appliances can be complex in their designs, and the developments of this text are only indirectly applicable to their analyses. Palate-splitting appliances are symmetric, employ reciprocal-force (bilateral) action, and are truly orthopedic devices. They attempt to transmit heavy, distributed force into the halves of the hard palate to open the median suture (to catalyze bone apposition there) without tipping the posterior teeth buccally. The overall force analysis is straightforward; the difficulties in the design pertain to the path and distribution of the active force and the stiffness of pertinent elements of the device with respect to the step-wise ad justments of the appliance following incremental displacements.

436


Although the mechanics just mentioned and others did not receive detailed attention in this chapter, the patterns of analysis in the examples presented—the modeling, the force-displacement correlations, the free-body diagrams, the force and moment balances, the differentialforce concept, the four-step analysis procedure, the emphasis on the role of stiffness, the consideration of modes of failure, and the other "tools" from the previous nine chapters—should enable the conscientious reader and student or practitioner of orthodontics, regardless of the amount of clinical experience, to begin to undertake force and structural analyses of most mechanics of interest. The subsequent step is the consideration of appliance-design modifications, as warranted, toward improving the quality of orthodontic care. This text was prepared primarily toward closure of an apparent gap in the orthodontic educational literature—discussion of the appliance as a structural mechanism from a bioengineering viewpoint. Comments on the textual material of a constructive nature, submitted in writing to the author or publisher, are encouraged.

Reference Frank, C.A., and Nikolai, R.J.: A comparative study of frictional resistances be-

tween orthodontic bracket and arch wire. Am. J. Orthod., 78:593—609, 1980.

Glossary of Terms U APPENDIX

activation—The process of deforming an appliance member from its passive state (e.g., the stretching of an elastic) and completing its engagement to produce an intraoral force system transmitted by the appliance to the dentition.

activation site—The intraoral location of the activating process, often where the orthodontic force system is to be transmitted to the dentition.

.

aging—A long-term chemical process in which molecular (secondary)

bonds are broken concurrent with oxidation, resulting in the appearance of cracks and embrittlement that proceed with time from the material surface inward. alveolar bone—The hard tissue that locally provides support for the in-

dividual teeth.

alveolus—The composite support, including alveolar bone and periodontal ligament, that envelops the root of the tooth. amorphous—Descriptive of an irregular material microstructure, usually with the molecule as the basic structural unit.

angle of twist—The measure of torsional (third-order) deformation of a shaft or an orthodontic wire. anneal—A heat-treatment process employing a relatively high temperature that results in recrystallization of microstructure and produces marked changes in mechanical properties, including substantially reduced resilience, of metallic materials.

anode—The electrode of a galvanic cell having the larger negative potential and from which electrons flow, leaving positive ions, and dissolving (corroding) in the long-term oxidation process. 437

438


apical fibers—The fibers of the periodontal ligament extending into the cementum at the root apex. apposition—The cellular process resulting in the addition of new alveolar bone at the interface of the periodontal ligament and the tooth root.

atom—The basic unit of microstructure exhibiting the nature of the chemical element.

available energy—Energy retained in a body owing to existing deformational or thermal constraints that, upon release of the constraints, would be transferred from the body and could be controlled to produce a desired effect.

axisymmetric—descriptive of symmetry in two dimensions relative to a line (axis) perpendicular to the plane. basal bone—The hard tissue of the maxilla or mandible that blends into the alveoli and supports the dentition as a whole.

beam—A structural member characterized by a large longitudinal dimension (usually oriented horizontally) and small cross-sectional dimensions, and loaded (activated) perpendicular to the longitudinal dimension. beam (bending) axis—A reference line interconnecting the geometric centers of all of the right cross-sections of a beam. beam deflection—A characteristic, linear deformation of a beam measured from and perpendicular to the beam axis at a specific cross-section, ordinarily with reference to the passive configuration of the structural member. beam fiber—A characteristic element of a solid modeled beam, parallel to the beam axis, having the same length as the beam and a very small cross-section compared to that of the entire member. beam rotation—A characteristic, angular beam deformation: the change in orientation of a specific cross-section of the beam during activation as measured in the plane of bending.

bending couple—A couple in the plane of bending, either internal or external to a beam, associated with its deformation relative to the passive shape.

bending strain—The normal strain of a beam fiber at a particular location on a specific right cross-section of the structural member.

439

Glossary of Terms

bending stress—The normal stress induced in a beam fiber at a particu-

lar location along its length.

bioengineering—Engineering principles and practice applied to living systems. bodily movement—Translational displacement of a tooth or dental segment, ordinarily understood to be perpendicular to the long axis. braided—Descriptive of a multistrand orthodontic wire suggesting nonparallel, interwoven strands.

buckling—A structural failure of a long, slender member characterized by sudden, transverse deformation.

cathode—The electrode of a galvanic cell having the algebraically greater potential and the receiver of electron flow from the anode. cementoblasts—The cells that specifically contribute to sustaining and rebuilding cementum. cementoepithelial fibers—The fibers of the periodontal ligament that extend into the cementum and over the alveolar crest into the free and attached gingiva.

center of resistance—A singular point in a body through which the line of action of a resultant force must pass to produce translation of that body. center of rotation—The point about which a body rotates in a nontranslational plane (two-dimensional) displacement.

cervical-pull—Descriptive of an extraoral appliance having the responsive force delivered to and distributed over the back of the neck.

chin cap—An element of an extraoral appliance that transfers force to the chin in dentofacial orthodontic or orthopedic therapy.

circumferential fibers—Periodontal-ligament fibers located in the attached gingiva that encircle the tooth without extending beyond the ligament itself. Class II elastic—An appliance element engaged intraorally and activated between anterior maxillary and posterior mandibular locations on

one side of the dentition in Class II mechanics therapy. Class II mechanics—A phase of orthodontic treatment using appliances

ordinarily designed to retract maxillary anterior teeth and/or protract mandibular posterior teeth.

440


Class HI elastic—An appliance element engaged intraorally and acti-

vated between anterior mandibular and posterior maxillary localions on one side of the dentition in Class III mechanics therapy. Class III mechanics—A phase of orthodontic treatment using appliances

ordinarily designed to retract mandibular anterior teeth and/or protract maxillary posterior teeth.

closing loop—An auxiliary fabricated of orthodontic wire that, upon mesiodistal-pulling activation, exhibits a reduction in the plane area enclosed within its geometry.

cold working—Inelastic (plastic) deformation of a metallic member at normal (room) temperature to alter the member geometry and/or change mechanical properties related to the amount and distribution of energy stored in the member. collinear forces—Concentrated forces that share the same line of action (but not necessarily the same point of application).

component (of a vector)—That portion of the vector associated with a reference line or axis angulated with respect to the line of action of the given vector. compressive deformation—The shortening of a characteristic dimension of a body associated with pushing force.

compressive strain—Internal compressive deformation per unit length.

compressive stress—Internal pushing force per unit area, the intensity normth (perpendicular) to the associated area. concentrated force—An external contact force with the area of contact very small compared to the total surface area of the body subjected to the force.

concurrent forces—Several concentrated forces exerted at a common point.

constant—A quantity that does not change within the context of its involvement in a problem or discussion. continuous orthodontic force—Action of an appliance against the dentilion that decreases little in magnitude during the between-appointments period. corrosion—The dissolving of a metallic material in a chemical, oxidation reaction.

441

Glossary of Terms

couple—A pair of concentrated forces having equal magnitudes, the same direction but opposite senses, and noncollinear (parallel) lines of action.

couple-force ratio—The ratio of magnitudes of the crown couple to the crown force, having net units of length (e.g., mm), in the two-dimensional analysis of transverse tooth movement. covalent bond—The bond between two or more atoms achieved by a sharing of electrons.

creep—Continuing deformation over time of a body subjected to a constant load.

crestal fibers—Periodontal-ligament fibers that emerge from the cementum and extend into the gingiva directly adjacent to the alveolar crest.

critical cross-section—The right cross-section of a beam, shaft, or wire where the internal bending or torsional couple has its maximum magnitude. crown movement—The transverse tooth displacement for which the cen-

ter of rotation coincides with the root apex (or the level of root apices for a multirooted tooth).

crystalline—Descriptive of the organized, latticed microstructure of a granular material. decomposition of a (concentrated) force—Replacing the given force by a mechanically equivalent set of concurrent force components, each of which has a line of action coinciding with an axis of a reference

frame originating at the point of application of the given force. deformation—Any change in the geometry (size and/or shape) of a body produced by the application of force.

dental segment—Two or more adjacent teeth in the same arch. differential-force—Pertaining to the distributions of two related forces,

equal in magnitude (and ordinarily in the same direction, but of opposite sense), over areas of differing sizes, thus yielding unequal average force intensities.

dimensions—The fundamental descriptions of measurement (mass, length, time, force, and temperature). direct resorption—The removal of alveolar bone at the bone-ligament interface catalyzed by the application of compressive force.

442

Bioengineering Analysis ot Orthodontic Mechanics

direction angle—The angle between a line or a vector and a reference axis.

displacement—Any movement of a particle or a body as a whole. distributed force—A force that involves a substantial portion of the surface area or volume of the body acted upon. dual-force appliance—An extraoral appliance characterized by the bilateral delivery of force to two separate intraoral locations on each side of the midsagittal plane.

eccentric activation—An orthodontic-appliance activation that creates a resultant force against the facial or lingual surface of a crown with a line of action not piercing the long axis of the tooth.

elastic—A flexible appliance element, ordinarily activated in two-point extension, that exhibits substantial elastic range (noun); descriptive of material behavior such that, upon unloading from a deformed state, recovery is totally to the configuration prior to loading (adjective).

elastic core—The inner portion of the cross-section of a beam, shaft, or wire that remains elastic (while the outer portion is strained beyond the elastic limit) when the member is subjected to substantial loading (activation) in bending or torsion.

elastic deformation—A deformation not sufficiently severe to take the most strained element of a body beyond the elastic limit of the material.

elastic limit—The limit of load, stress, deformation, or strain beyond which the loaded (activated) body will exhibit permanent deformation (a new passive shape) upon complete unloading (deactivation). elastic range—The deformation or strain coordinate of the elastic limit. elastic strength—The load or stress coordinate of the elastic limit. electrode potential—The relative ease or difficulty with which a metal or alloy will give up its valence (outer-shell) electrons in, for example,

a galvanic cell. electron—A negatively charged particle of an atom that is in continual

motion around the nucleus. endurance limit—In a dynamic, cyclic loading pattern, the greatest amplitude of load, stress, deformation, or strain that does not cause a

443 Glossary of Terms

failure within the expected fatigue life of the structural member; the

horizontal asymptote of a fatigue-life plot. energy—The capacity of a body to produce an effect. engineering strain—Strain computed with reference to the undeformed

(passive) geometry of the strained body.

engineering stress—Stress calculated using undeformed (passive) geometry of the stressed body. excessive force—Force delivered by an orthodontic appliance of such magnitude that it effectively ankyloses a tooth and absolutely no desired movement is detectable following a between-appointments period.

extraorat appliance—An orthodontic or orthopedic device activated against the dentofacial complex that induces its responsive force system outside of the oral cavity. extrusion—A translational form of tooth displacement with movement occiusally directed and parallel to the long axis.

face bow—An appliance member that transmits force, delivered extraorally to it on each side of the face, to the maxillary first or second molar crowns. failure—The development of excessive stress or deformation or a breakage (fracture, rupture) that interferes with or prevents the functioning of a structure or machine. fatigue—The deterioration or gradual loss of desirable structural characteristics, possibly culminating in failure, of a member subjected to fluctuating (cyclic) loading.

fiber—An actual or modeled (e.g., periodontal-ligament or beam) element, characterized by a substantial length compared to the dimensions of its cross-section, that transmits longitudinal force, often tensile force. fiber strain—Unit longitudinal deformation of a fiber. fiber stress—Longitudinal fiber force divided by the cross-sectional area of the fiber.

fibroblasts—The cells that specifically contribute to the rebuilding and remodeling (lengthening) of the periodontal-ligament fiber.

444


first-order—Pertaining to a rotational tooth movement or displacement

potential visible in an occlusal view.

force—A mechanical action of one body on another that tends to displace and/or deform the body receiving it. fracture—A failure of a structural or machine member through interruption or loss of a force transmission path. free end—One end of a beam, shaft, or wire not constrained in any way from displacement resulting from an applied force system. I

tree-body diagram—The figurative removal from a structure or machine

of a member or a portion of a member, depicting in an isolated sketch the geometry and force system exerted, preparatory to a force and/or structural analysis of that part of the assembly. friction—A resistance to the relative displacement of contacting bodies in

a direction tangent to the plane of contact, owing principally to surface roughnesses and contact pressure. frontal resorption—See direct resorption.

function—A mathematical quantity that assumes a value upon assignment of magnitudes to one or more variable in the problem or discussion; the interaction of two or more bodies toward a collectively produced effect. functional force—Force arising from relative displacements of interconnected bodies producing contact(s) between surfaces during routine tasks (e.g., chewing/masticatory force).

generalized rotation—A whole body, nontranslational displacement characterized by a center of rotation not coincident with the center of resistance of the body. gingival fibers—The extra- or supra-alveolar group of longer, periodontal-ligament fibers that are embedded in the gingival tissue and extend into the cementum.

ground substance—The matrix of the composite-material, periodontal ligament that provides the principal resistance to compressive force (pressure) exerted on the ligament.

hardening heat treatment—A heat-treatment process designed to increase the surface hardness and, perhaps, to suitably affect other properties of a material.


-

hardness—The resistance of a material surface to indentation.

head cap—The member of an extraoral appliance that transfers responsive force to the cranium.

headgear—A synonym for "extraoral orthodontic appliance(s)." heat—A form of energy transfer between two bodies or systems owing to a difference in temperatures, one from the other, in the absence of a perfect insulator between them.

heat sink—A receptor of energy transfer in the form of heat from a thermodynamic system.

heat source—A provider of energy transfer in the form of heat to a thermodynamic system. heat treatment—A process characterized by the transfer of energy in the form of heat to a metallic material, and subsequently from it, to alter its mechanical and/or thermal properties. heavy force—Orthodontic force induced by appliance members exhibiting relatively high characteristic stiffness(es). high-pull—Descriptive of the activating bilateral force, directed posteriorly and superiorly, of an extraoral orthodontic or orthopedic appliance. hinged face bow—A face bow having the outer and inner bows interconnected by a hinge that permits rotation of one relative to the other

about an axis perpendicular to the plane of the bows. Hookean material—A crystalline material exhibiting a linear relationship

between induced normal or shear stress and corresponding strain when subjected to relatively small levels of activation. horizontal fibers—The principal, periodontal-ligament fibers extending from cementum to alveolar bone that are radially directed relative to the passive orientation of the long axis of the tooth. hot working—Inelastically deforming a metallic member in an elevated-

temperature environment toward desired changes in geometry and/or mechanical characteristics of the material. incisal pair—Adjacent central and lateral incisors. inelastic deformation—Deformation of a member sufficiently substantial to exceed the elastic limit of the material and, upon total unloading

446


(deactivation), resulting in a "permanent set" (a change in geome-

try/shape from that of the passive state prior to the deformation). inner bow—The intraoral portion of the engaged face bow. intermittent force—An active, orthodontic force that decays to zero magnitude, or nearly so, prior to the end of a between-appointments

period. interrupted force—An orthodontic or orthopedic force that is inactive for

intervals of time during the between-appointments period, often exhibiting a cyclic, long-term, magnitude-time pattern (e.g., force exerted by an extraoral appliance worn only at night). intrusion—A translational form of tooth movement directed apically and parallel to the long axis. ionic bond—The interatomic bond involving unlike elements and cre-

ated by the transfer of valence (outer-shell) electrons from one atom to another.

J-hook—The element of an extraoral appliance in the shape of the letter "J" that transfers force between extraoral and intraoral locations. labial root (lingual crown) torque—The third-order couple of a transverse

force system applied to the crown(s) of a tooth or dental segment having the potential for rotational displacement resulting in labial movement of the root(s) and/or lingual movement of the crown(s). leading root surface—That portion of the tooth root under compression (pressure) during a transverse tooth movement.

leveling—That phase of orthodontic therapy in which occiusogingival and/or faciolingual malalignments are eliminated toward ideal arch form (in the occlusal plane) and the desired shape of the curve of Spee.

leveling wire—Orthodontic wire, ordinarily circular (in cross-section) and exhibiting low flexural stiffness, used in the leveling phase of treatment. light force—Orthodontic force produced by an appliance element having relatively low characteristic stiffness(es).

line of action—A line having the same direction as the associated concentrated-force vector and passing through the point of application of the force.


lingual root (labial crown) torque—The third-order couple of a transverse

force system applied to the crown(s) of a tooth or dental segment having the potential for rotational displacement resulting in lingual movement of the root(s) and/or labial movement of the crown(s). long-axis rotation—A nontranslational tooth movement characterized by the coincidence of the long axis and the axis of rotation.

longitudinal axis—The line composed of the centers of all of the right cross-sections of a beam, shaft, or wire. machine—A mechanism that produces a desired output when receiving a necessary input.

maximum force—An orthodontic force having the greatest magnitude within a range of force values that will produce tooth movement at a clinically acceptable rate. mechanical work—A form of energy transfer characterized by force acting through a distance.

mechanics—The branch of physics that focuses on force systems and their effects on stationary and moving bodies; in orthodontics, descriptive of the procedures and/or appliance(s) used in a specific phase of therapy. mer—The characteristic molecule of a polymer. metallic bond—The interatomic bond characterized by the release of valence (outer-shell) electrons, yielding an organized array of positive ions and an electron cloud ("gas") with electrostatic forces maintaining the shape of the material.

midsagittal plane—The imaginary, vertical plane that separates the left and right sides of the dentofacial complex. modulus of elasticity—The ratio of change in normal stress to accompa-

nying change in normal strain between states below the proportional limit within a Hookean material; the slope of the normalstress versus normal-strain diagram below the proportional limit. modulus of resilience—The amount of work done per unit volume on a material in the energy-transfer process that takes the material from a passive state to its elastic-limit configuration; the area under the stress-strain diagram up to the elastic limit. modulus of rigidity—The ratio of change in shear stress to accompanying

change in shear strain between states below the proportional limit

448


within a 1-lookean material; the slope of the shear-stress versus shear-strain diagram below the proportional limit. modulus of toughness—The amount of work done per unit volume on a material in the energy-transfer process that takes the material from a passive state to fracture; the total area under the stress-strain diagram. moment (of a concentrated force)—The measure of the rotational potential of a force with respect to a specific line (axis) or point; in magni-

tude the product of the size of the force and the distance (measured perpendicular to the line of action) from the force to the moment axis or moment center. moment of inertia (second moment of area)—A property of a body de-

pending on mass, mass distribution, size, and shape that is characteristic of the resistance of the body to motion; also a property of the cross-sectional area of a beam or wire, referenced to a line (axis) in the plane of the cross-section, that depends on the shape and size of the cross-section and the specific reference line. neck pad—The element of the cervical-pull, extraoral appliance that

transfers the distributed responsive force to the back of the neck. neutral axis—The intersection of the neutral surface with the right crosssection of a beam or wire activated in flexure.

neutral surface—The collection of beam fibers that are unstrained in plane bending (flexural) activation. neutron—A basic, uncharged particle within the nucleus of an atom.

normal deformation—The deformation of a body accompanying the application to it of pulling or pushing force; tensile or compressive deformation perpendicular (normal) to a reference area of a body.

normal strain—Normal deformation per unit length of an element of a loaded (activated) body. normal stress—The intensity of internal force (force per unit area) per-

pendicular to a reference area within a loaded (activated) body. oblique fibers—The principal fibers of the periodontal ligament that are embedded in the cementum and the alveolar bone and are inclined toward the root apices, providing hammock-like response and protection for the root apex (apices) of a tooth subjected to intrusively directed (e.g., masticatory) force.

449

Glossary of Terms

opening loop—An auxiliary fabricated of orthodontic wire that, upon

mesiodistal-pulling activation, exhibits an enlargement of the plane area enclosed within its geometry.

origin—The intersection of the coordinate axes of a reference frame. orthodontic attachment—The element affixed to the tooth crown that transmits force from the arch wire or auxiliary to the dentition (e.g., a bracket).

orthodontic displacement—Tooth movement achieved through biomechanical remodeling of the periodontal ligament and/or the alveolar bone.

orthodontic force—Force generated by the orthodontic appliance that contributes to the correction of a malocclusion. orthopedic force—Force generated to assist or retard bone growth, to displace teeth other than through remodeling of alveoli, or to move bones of the dentofacial complex to catalyze sutural remodeling. osteoblasts—The cells that specifically contribute to the apposition of bone in regions of the ligament-bone interface under tension.

osteoclasts—The cells that specifically contribute to the resorption of bone in regions of moderate compression (pressure). outer bow—The extraoral portion of the engaged face bow. oxidation—A chemical reaction with oxygen contained in the reactants

and yielding a metallic or nonmetallic oxide or hydroxide as a product.

parallelogram law—The fundamental, graphical procedure used to replace two concurrent, concentrated forces by a single, mechanically equivalent point force; the basic rule of vector addition.

passivation—The process by which a material surface is protected against corrosion.

passive—The configuration of a body unconstrained externally and not subjected to mechanical or thermal loading (activation). -

ligament—The soft tissue surrounding the root of a tooth and overlaying the alveolar bone.

periodontal

periodontal-ligament fibers—The material components of the composite ligament that give it resistance to failure when subjected to moderate tensile (stretching) action.

450


physiologic force—Force inherent in a biologic system (e.g., the pressure of the blood against the vessel walls; the force working in the tootheruption process).

plane of bending—The plane defined by the passive and activated configurations of the beam axis in two-dimensional bending (flexure). polar moment of inertia (polar second moment of area)—A property of a body depending on mass, mass distribution, size, and shape that is characteristic of the resistance of the body to motion; also a property of the cross-sectional area of a shaft or wire, referenced to a line (axis) perpendicular to the plane of the cross-section, that depends on the shape and size of the cross-section and the specific reference line. polygonal law—An extension of the triangle law, a corollary of the paral-

lelogram law, through which a system of more than two concurrent, concentrated forces may be replaced by a single point force having the same mechanical effect as that of the system. polymer—An amorphous material consisting of long chains and networks of repeating molecules. preangulated bracket—An orthodontic bracket having its slot inclined to the ordinary mesiodistal direction, permitting the generation of second-order action from a passively straight wire.

pretorqued bracket—An orthodontic bracket having its slot rotated with respect to the ordinary faciolingual direction, permitting the generation of third-order action from a passively straight (untwisted) rectangular wire of suitable cross-sectional dimensions. primary bond—A covalent, ionic, or metallic bond between two atoms. primary response—A response in the same plane as or a parallel plane to

that of the appliance activation.

principal fibers—The shorter, periodontal-ligament fibers generally directed in a radial manner with regard to the long axis of the tooth and embedded in the cementum and the alveolar bone. process—The change of state experienced by a thermodynamic system as energy is transferred to and/or from it.

projected root area—The plane area enclosed in the outline of a tooth root as viewed from a specific direction. projection (of a vector)—The magnitude of the vector component associated with a specified line or reference axis; the product of the vector

451

Glossaiy of Terms

magnitude and the cosine of the angle between the vector and the

specified line or axis. property—A characteristic of a material or body that partially describes it.

proportional limit—The upper bound of the linear portion of a load-deformation or stress-strain plot for a Hookean material.

proton—A basic, positively charged particle within the nucleus of an atom.

pure bending—Flexural action produced by couple loading such that no shear stresses exist on right cross-sections of the beam or wire. pure rotation—A displacement of a body, produced by couple action, characterized by the center of rotation coinciding with the center of resistance.

pure shear—The existence of shear stresses in the absence of normal stresses associated with a specified area within the activated body. quasi-static—Descriptive of the configuration of the appliance and the dentition in an instantaneous or short-term analysis between appointments: velocities, accelerations, and inertias are sufficiently small to be negligible such that the principles of statics (mechanical equilibrium) may be used in the analysis. rear resorption—The decomposition of bone within the alveolus, beginning away from the bone-ligament interface where the compressive

stress (pressure) induced by the activated appliance is reduced (through distribution) from the more substantial intensities delivered by the tooth root. reference frame—A set of two or three intersecting axes, ordinarily mu-

tually perpendicular, used as a basis for locations of points and directions of lines, graphically expressing relationships among variables, or undertaking vector analyses, in a plane or in space.

relaxation—The decrease over time of force internal to a body maintained under conditions of constant strain. resilience—The amount of energy transferred by mechanical work to take a body from an initial, passive state to its elastic limit; the area under the load-deformation plot up to the elastic limit. resultant—A force system—in its simplest form, a single concentrated force, a couple, or a force and a couple—that is mechanically equivalent to the given, more complex system of forces.

452


reverse-pull—Descriptive of an extraoral appliance that, when activated,

exerts anteriorly directed force on the dentition. right cross-section—A cross-section of a beam, shaft, or wire that is per-

pendicular (at right angles) to the longitudinal reference axis (and the external surface[s]) of the member at the location of the crosssection.

rigid body—A model of analytical mechanics that ignores potential deformations resulting from the application of force systems, focusing attention on possible or actual whole-body movement. root movement—The transverse tooth movement characterized by little displacement of the crown and the center of rotation located on the long axis and at or near bracket level.

rotation—Any orientation-changing, whole-body displacement; a displacement characterized by the change in angulation of some line(s)

in the body with respect to a specified reference frame. scalar—A mathematical quantity possessing only magnitude. scale—The means of conversion between the magnitude of a specific variable and its graphical representation as a coordinate or length. second moments of area—See moment of inertia and polar moment of inertia.

secondary bonds—Forces that hold molecules together in a solid, and also known as van der Waals bonds; these bonds are weaker than interatomic (primary) bonds.

secondary response—A response in a plane angulated to that of the appliance activation, occurring in part because of the curvature of the dental arch. second-order clearance—The angle through which an engaged arch wire may be tipped within the bracket slot, relative to the "slot-parallel" configuration, before making contact with the occlusogingival slot surfaces.

section modulus—The ratio of a second moment of area to the depth of a

beam or wire; often symbolized by "Z" and the geometric parameter in the flexural-strength equation. shaft—A long, slender, machine member, ordinarily straight and of constant cross-section (most often circular), that carries torque.


shear deformation—A change in shape as a result of loading (activation);

often characteristically, a rectangular element assumes the form of a parallelogram. shear strain—A unit shear deformation; the change in angle (in radians) between two intersecting, passively perpendicular lines.

shear stress—The intensity (force per unit area) of the tangential component of internal force.

simple tipping—The tooth movement produced by a labiolingually directed, concentrated crown force. sliding jig—An auxiliary guided by an arch wire that transfers a mesiodistal component of force, delivered by a J-hook, directly to a posterior site unreachable by the J-hook itself.

spring-back—-The recovery exhibited by a beam, shaft, wire, or wire loop upon its unloading (deactivation) from a state at or beyond its elastic limit. stabilizing wire—A stiff, rectangular arch wire that "fills" the bracket slot

and is used ordinarily to distribute a responsive force system throughout a dental segment. state—Defined by the collective values of the properties (characteristics) of a thermodynamic system.

statics—That portion of mechanics (of physics) that concerns force systems exerted on bodies in mechanical-equilibrium states. stiffness—The ratio of change in load to accompanying change in deformation of a member fabricated of a Hookean material and activated within its elastic limit; the slope of the load-versus-deformation plot beneath the elastic limit.

stop—An auxiliary affixed to the arch wire that prevents relative mesiodistal movement upon contact with a bracket; also a means of transferring mesiodistal force between wire and bracket. straight-pull—Descriptive of a type of extraoral appliance having the activating forces parallel to the occiusal plane.

strain—Unit deformation. strain energy—Energy of a system characterized by constraints maintaining the system in a deformed configuration compared to a reference (passive) state.

454


hardening—A process in which a body is subjected to inelastic (permanent) deformation resulting in the material incurring me-

strain

chanical as well as geometric property changes (e.g., an increase in surface hardness). strand—A longitudinal element of a multistrand wire.

stress—The intensity of internal force; internal force per unit of associated area. stress-relief—A heat-treatment process intended to reduce the magnitudes of residual ("locked in") stresses induced by inelastic deformation.

structural analysis—An engineering procedure in which a structure is examined analytically, graphically, or through modeling to determine how large a load it will carry without failure or if it will carry the known, desired load without failing. structure—An assembly of members designed to withstand a specific loading and to transfer that loading into a foundation (e.g., a frame-

work, bridge, or building). surroundings—The collection of bodies or objects that might interact, through energy transfer, with a defined thermodynamic system. tensile deformation—Deformation directly associated with a pulling (extensional) force. tensile strain—Tensile deformation per unit length at an internal location of the loaded (activated) body.

tensile stress—Intensity of internal force (force per unit area) pulling

perpendicular to an

area

within the loaded (activated) body.

thermal energy—Energy possessed by a body because its temperature is higher than that of its surroundings.

thermodynamic system—The body or bodies, separated from interacting surroundings by a defined boundary, that experience changes in energy levels during a process. thermodynamics—That portion of physics that concerns interactions between physical systems and their surroundings, specifically processes involving energy exchanges through heat and/or mechanical work. third-order clearance—The angle through which an engaged rectangular arch wire may be rotated about its longitudinal axis, relative to the


"aligned-surfaces" configuration, before the edges of the wire make

diagonal contacts with the occlusogingival slot surfaces. third-order couple—A couple, located in a mesiodistal plane, transmitted

by a rectangular arch wire or torquing spur to an orthodontic bracket.

threshold force—The minimum magnitude of force needed to produce a desired orthodontic displacement.

torque—An internal force system, carried longitudinally through a shaft or wire, and its resultant at any location is a couple in the plane of the right cross-section.

torquing auxiliary—An element of an orthodontic appliance, separate from the arch wire, used to transmit third-order action to the dentihon. torquing displacement—The rotational movement of a tooth resulting from the existence of an active second- and/or third-order couple in the crown force system.

torquing spur—A loop formed in round wire, either integral with or affixed to the arch wire, that when activated imparts third-order action to the tooth crown. torsional couple—An active or responsive external couple exerted on a

shaft or wire and directly associated with the turning and/or twisting of the member about its longitudinal axis.

toughness—The maximum amount of energy, referenced to a passive state, transferred to a body in the form of mechanical work, that the body can absorb prior to structural failure (ordinarily by fracture or rupture); the total area under the load-deformation plot. traction band—A rectangular band of woven elastic materials, activated

by longitudinal stretching, and an element of some extraoral appliances. trailing root surface—That portion of the root surface subjected to perio-

dontal-ligament tension in a transverse tooth movement. translation—Any whole-body movement in which no line of the body changes orientation (angulation) with respect to a specified, stationary, reference frame.

transmissibility—The principle that permits the moving of a concentrated force to any convenient location along its line of action without changing the mechanical effect of the force.

456


transseptal fibers—Periodontal-ligament fibers, located occlusogingi-

vally near the cementoenamel junction, that circumscribe roots and extend mesiodistally, interconnecting adjacent teeth.

transverse displacement—An orthodontic tooth movement characterized by displacements of points of the tooth at right angles (perpendicular) to the long-axis orientation. transverse rotation—A nontranslational tooth movement characterized by the axis of rotation positioned perpendicular to the long axis of the tooth. triangle law—A corollary to the parallelogram law; the two concurrent point forces and their resultant coincide with the three sides of a triangle. true strain—Strain computed with reference to the deformed state of the body.

true stress—Internal force per unit of associated deformed area within a body. twist—See angle of twist.

twisting deformation—The rotation about the longitudinal axis of the member of one cross-section of a shaft or wire with respect to another cross-section. ultimate strength—The maximum load imparted to a structural or machine member (or maximum stress induced in a material) preceding failure by fracture or rupture. undermining resorption—See rear resorption. unit bending stiffness—The product of the modulus of elasticity and the second moment of the right cross-sectional area with respect to the neutral axis for a beam or wire. units—The completion of a dimensional measurement or property value (e.g., 4 oz, 7 mm, 15 lb per square in.).

utility arch—An arch-wire appliance ordinarily engaging the four or six anterior teeth and the terminal molars, skipping the premolars (and often the canines) to create flexible, bilateral levers between anterior and posterior teeth.

variable—A mathematical quantity capable of assuming more than one value in a problem or during a discussion.


vector—A mathematical quantity exhibiting magnitude, direction, and sense.

work—See mechanical work. working wire—An orthodontic arch wire, often rectangular, used to exert crown force systems capable of controlling or imparting root torque

and the associated displacements in all three planes of space. yield point—The intersection of the stress-strain (or load-deformation) curve and a reference line with slope equal to that of the initial part of the cited curve and meeting the strain (deformation) axis at a specified offset location.

yield strength—The value of the load (or stress) at the yield point.

yielding—A substantial increase in strain with little or no increase in stress, occurring just above the elastic limit in some metallic materials.

List of Symbols APPENDIX

a

anterior; apical; distance from the center of resistance to

A

area

crown center; mesiodistal loop deformation

internal stressed area b

dimension of a rectangular beam, shaft, or wire cross-section; moment arm; a transverse inner-bow dimension

c

dimension of a rectangular shaft or a wire cross-section;

cra cre

one-half of the depth of a beam or wire; a transverse face-bow dimension carbon; couple center of a tooth crown center of resistance of a dental arch center of resistance of a tooth

cro

center of rotation

d

deformation; diameter; dimension from the center of re-

D

deg,°

sistance to crown center; displacement; moment arm lever arm degrees

e

distance; eccentricity; elongation; extensional deformation;

E

moment arm; strain modulus of elasticity (Young's modulus)

C

CC

elastic limit EL En

force exerted by an elastic (appliance element) total energy

f

facial; frictional component of force; lever force a concentrated, contact force

F f-e

iron faciolingual

9 G

gingival; grams center of gravity; modulus of rigidity

Fe

458

459 List of Symbols

h

H

HG HT

dimension of a rectangular beam or a wire cross-section; intensity of a distributed force; a vertical (height) dimension heat; horizontal component of force; hydrogen; a vertical (height) dimension headgear force

heat treatment inferior; inner-bow dimension

I

unit vectors associated with the x, y, z reference frame second moment of area (moment of "inertia")

J

polar second moment of area (polar moment of "inertia")

k

curvature

e

lingual; length

L

L0

lateral force; length; losses (of energy) passive length passive length

i, j, k

to

m

mesial

M

moment of a force or force system

m-d mm

mesiodistal millimeters external first- or second-order couple exerted on a beam or

M0

wire N

normal component of force

o

occiusal; outer-bow dimension

o o-g

origin of a reference frame; oxygen occiusogingival

OZ

ounces

p P pe

intensity of a distributed force; posterior; pressure a concentrated force proportional limit

Q

a

r

radius; radius of curvature

r R re

position vector radius of a circular shaft or wire responsive force; resultant force radius of the elastic core of a circular shaft or wire

S

arc length; curvilinear displacement; stress; superior

A

concentrated force

460


S

stiffness

SE

strain energy

t

time

T TE

temperature; torsional couple (torque) thermal energy

U

a

v

occiusogingival loop dimension; speed

V

V

velocity internal shear force resultant; vertical component of force

w W Wt

width work weight

x, y; x, y, z

mutually perpendicular reference axes; rectangular coordinates

z

transverse deflection of a beam or wire

a

empirically determined coefficient of torsion theory for a rectangular shaft direction angles

a, /3, y

coordinate on a beam or wire cross-section, locating points perpendicular to the neutral axis

/3

empirically determined coefficient of torsion theory for a rectangular shaft

6

transverse deflection of a beam or wire; small amount of; small change in change in

a-

normal stress

shear stress U

Ox, Oy,

angle between two lines; rotational deformation of a beam or wire cross-section direction angles angle between two lines; angle of twist of a shaft or wire in torsion normal strain

461

List of Symbols

shear strain

Subscripts

a

anterior; apical

at

anterior left

ar

anterior right

b

buccal

C

cc comp

canine; center of resistance; compressive; crown chin cap

crit

compressive critical

d

distal

e

elastic elastic; elastic limit

at

ft

facial; final forehead faciolingual

h

hinge; horizontal

hg, HG

headgear

I

fh

ir

incisal; initial; intrusive incisal left incisal right

t

lateral; left; lever; ligature; lingual

m

mesial mesiodistal

I

it

md mid

-

middle segment (between anterior and posterior segments)

n

normal

o OC 09

initial; occlusal

p

associated with a force P; periodontal; posterior

occlusion

occlusogingival

462


pr

permanent posterior left posterior right

q

associated with a force Q

r

resultant; right

S

sagittal; shear; spring; strap; stress

t

tensile; terminal molar tensile

perm p€

tens tmj

ligature tie temporomandibular joint

V

vertical

W

wire

o

initial; passive

0, 1

initial, final

tie

1, 2 1, 2, 3 1, 2

initial, final; left, right first-, second-, third-order 8 central incisor, lateral incisor,

1r

left central incisor right central incisor

2€ 2r

left lateral incisor right lateral incisor

45

premolar segment

II

III

Class IT elastic Class III elastic

a

normal

r

shear

1€

.

.

.

,

third molar

Superscripts F,

(primes)

forces or force components; comparable lengths; elements of a force system; responsive couple; responsive force; successive force values; successive positions of a point or a particle

comparable

Index

Page

numbers in italics indicate illustrations; page numbers followed by t indicate tables.

Abscissa, 4 Acceleration, 27 Action, 26 ADA Specification No. 32, 99—100, 100, 101, 258

Aging of elastomers, 109 Allotropism, crystalline, 76 Alveolar bone, anatomy of, 170

in shearing deformation, 79 of triangle, 10—11 of twist, 81, 82, 277, 283, 284—286

Annealing heat treatment, 139—140 Anode, 107 Apical fibers, 173, 174 Appearance of materials, 111 Appliances, extraoral, 322—371

apposition of, 175—176

canine retraction, with headgear,

deformation of, with tooth movement,

344—348, 346, 347, 348 cervical-pull, face-bow, 325—344 buccal view analysis of, 335—343, 336, 337, 338, 339, 340, 341, 342 coronal plane view of, 343—344,

147

density of, 184 orthodontic vs orthopedic forces on, 170

periodontal ligament and, 169, 175 remodeling of, 175—176

with continuous orthodontic force, 179, 180

with intermittent orthodontic force, 178—179

resorption of, 175—176 stress distribution in, 170—171, 171 Alveolar process, 169—170 Alveolus, 169

American Iron and Steel Institute, 2 American Society for Testing Materials, 72

Amorphous matter, 77 Anchorage, 187—188 Anchorage preparation, 412—416

Angle, conversions between unit systems, 22t cosine of, 11 direction, 14—15

measurement of, 10, 10

343

occlusal plane analysis of, 326—335, 327, 328, 329, 330, 332, 333, 334 reactive force system in, 326, 326 chin-cap assembly, 362—365, 363, 364 dassification of, 322—323

with delivery of force to anterior segment or entire arch, 348—357 in mandible, 356—357, 357 in maxilla, 348—356, 349, 350, 352, 353, 354, 355

with delivery of force to mandibular arch, 358—361, 359, 360, 361 reverse-pull, 365—369, 365, 366, 367, 368

interconnections and contact force systems, 64t intraoral, activation of, bending in, 257

control of, 229 deformation with, 79 463

464 Index

Appliances (Continued) elastics in, 215—216, 215, 216, 217— 219, 218 elements of, 211—223 energy in, 125—137

force exerted in, 71 minimum energy configuration with, 212 sites of, 211 soft tissue deformations with, 211

spacers in, 212—213, 213 springs in, 213—215, 214

strain energy in, 211 with tip-back bends, 224—228, 225, 226, 227 wire in, arch, 219—223, 220, 221, 222

wire in, ligature, 216—217, 217 anchorage location for, 187 attachment of, to dentition, 197—205 arch wires in, 201—205, 203, 204 brackets in, 197—198, 198, 199, 201, 202

buccal tubes in, 199 buttons, eyelets, and staples in, 199, 199

for distolingual rotation, 200, 200 for extrusion, 201, 201 force systems in, 200—202, 200 Begg torquing, 308, 308, 308t continuous arch-wire model of, 205— 208, 206, 207 control of, 228—231

deactivation of elements of, 211—223

strain energy in, 212 force through connections between parts of, 59 force through connections between tooth and, 59 function of, 196 for mandibular growth control, 191 materials for. See Materials, appliance in mechanical equilibrium, 57, 58 for palate splitting, 190 passive, 79 relaxation of, 212 structural analysis of, 194—232 as structure vs machine, 196—197 with tip-back bends, 224—228, 225, 226, 227 torque delivery by, 272—321 activating, isolation of, 288—289 in anterior segment, 299—310 with circular wire, 273—288 dentition response to, 289—294, 290, 291, 292, 293, 294

force systems and, 295—299, 298, 299

inelastic behavior in third-order mechanics and, 310—314, 311, 312

with rectangular wire, 299—305, 300, 301t, 303, 304t shaft theory and, 273—288 with spurs, 305—310, 306, 307, 308, 308t, 309 with wire ioops and springs, 314— 318, 314, 315, 317, 318

Arch(es), anterior segment of, displacement of, 397 extraoral force delivery to, 348—357 retraction of, 409—412, 410, 411 torquing of, 272, 299—310 torsional activation of, 290, 290

bilateral action on, for tooth alignment, 385

extraoral force on entire, 348—357

opposing, in tooth malahgnment correction, 388—391, 390, 391

posterior segment of, anchorage in, 412—416, 413, 414, 415

retraction mechanics within, 402—412, 404, 406, 407, 410, 411 utility, 392 mandibular, 392, 393, 394—397, 396, 398

maxillary, 394, 395 vertical positioning within, 392—402, 393, 395, 396, 398, 399, 400, 401 Atomic number, 73 Atomic weight, 73 Atoms, 73—74

bonding characteristics of, 74, 75 chemical composition of, 75 in crystalline structure, 76, 77 in density, 75 displacement of, 78 equilibrium position of, in solids, 78 forces between, 74, 75 in stress corrosion, 108 Available energy, 124—125 Availability of materials, 111 Axis, beam, 233, 235

long, in tooth displacement, 148, 149, 149, 150

moment, 38 reference, 3—8, 3

Basal bone, 169 Beam, arch-wire, activation of, 234, 235, 247

bending force systems on, 233 bending stress in, 243

465 Index

cantilevered, 236, 237, 255—257, 255

characteristics of, 233 cross-sectional rotation of, 233 deformation of, 236—237, 237 elastic range and strength in analysis of, 253 "fibers" of, 238—239, 238 force systems within, 241—248 framework for analysis of, 235—236 inelastic behavior of, in bending, 261— 264, 263

in-plane bending of, 236 longitudinal shortening of, 236, 237 neutral surface of, 239 section modulus for, 246 split-anchorage, 234—235, 234, 249

subject to in-plane couple loading, 253—255, 254

subject to transverse point load, 250, 251 stiffnesses of, 248—252, 249, 251

transverse deflection of, 233 Beam axis, 233, 235 Beam deformations, strains and stresses, 235—241, 237, 238, 240, 241 Beam stiffnesses, 248—256, 251 Begg therapy, 307—308 Bending, 80—81, 81

fatigue failure from, 103 load-deformation diagram for, 121 plane of, 236 of wire, activation process in, 130—135, 132, 133, 134

beam deformations, strains and stresses within, 235—241, 237, 238, 240, 241 beam stiffness in, 248—256, 251 cantilevered beam with, 253, 255— 257, 255

couple loading of split-anchorage beam with, 253—254, 254 deformation from, 82—85, 83

elastic beam theory application to, 257—261, 259t, 260t, 261t elastic range with, 252—253

elastic strength in, 247 force systems within beam with, 241— 248, 242, 244, 245, 246, 247 force-deformation diagram for, 142— 143, 142, 143 inelastic behavior in, 261—264, 263 loops and, 264—269, 264, 265, 266, 268, 269t material behavior with, 233—271

modulus of elasticity and yield strength for, 99—100, 102t springback for, 100, 102t

Bernoulli, Jean, 113 Bioengineering, orthodontic, mathematical topics in, 1—23 constants, variables, and functions, 1—3

dimensions and units, 20—21, 22t displacements of particles and solid bodies, 8—10, 9

frames of reference, 3—8, 5, 6, 7

measurements, computations, and numerical accuracy, 21—23 trigonometry, 10—16, 10, 11, 12t, 13, 14, 15 vector algebra, 16—20, 17, 18, 19 Bodily movement, tooth, 149, 152—155, 153, 154 Body forces, 25 Bonding in bracket-crown attachment, 198

Brackets, accessibility in placement of, 200

attachment of, to tooth, 197 attachment of, to bands, 198 bonding of, to tooth, 198 contact modes between wire and, 55, 56

in continuous arch wire appliance model, 205 couple between wire and, 220—222, 222 edgewise, 201—202, 202 equilibrating force on, 65, 65 force transmission through, 199—202, 200

friction between wire and, 55, 56 shear stress in bond to, 90, 91 stiffness of, 197 strength of, 197 universal design for, 202 wire ligation to, 201 Brinnell hardness scale, 100 Buccal plane, 8 Buccal tubes, 199 in bending of arch wire, 233 in cervical-pull appliance, 325 buccal view analysis, 336 occlusal plane analysis, 328, 329 in continuous-arch-wire appliance model, 205 with tip-back bends, 225 in utility arch, 392 -

Buckling, 98

Buttons, 199, 199

.,

Canine, occlusogirigival alignment of, 377—378

refraction of, with headgear, 344—348, 346, 347, 348

466 Index

Canine (Continued) process of, 402—408, 406, 407, 408 Cathode, 107

second-order, 235, 250—253 third-order, 300—302, 300, 301t Coefficients of friction, 54

Cement in bracket-crown attachment, 198 Cementoblasts, 176 Cementoepithelial fibers, 173 Cementum resorption, 176 Center of resistance, 46, 148 Center of rotation, 150, 762, 163t Ceramic material stress-strain

Cold working of metals, 138 Component, of vector, 17 Composition cell, 107 Compression tests, 97—98 Compressive deformation, 79, 80 Compressive strain, 94, 97 Compressive stress, 90, 97 alveolar bone response to, 171, 171 with beam deformation, 241 with bending, 81 in bodily-movement tooth displacement, 155 in determining physiologically proper orthodontic force, 183 in shaft theory, 279 in tipping, 157 Concentration cell, 108 Conduction, 122 Conservation-of-energy law, 122—124 Constants in structural analysis, 2 Contact forces, 25, 26 Convection, 122

relationship, 95—96, 96

Cervical-pull appliance, activation of, 325 buccal view analysis of, 337, 338 occlusal plane analysis of, 326—327, 327

asymmetric action with, 343—344 buccal view analysis of, 335—343 arch wire in, 340—341 center of resistance in, 338, 339

displacement tendencies in, 338 with face bow activation, 336—338, 336, 337, 338

in overjet and anterior open bite reduction, 341—343, 341, 342

components of, 325 in buccal view analysis, 335—343 free-body diagram of, 336, 336 coronal plane view of, 343—344 face bow of, 325 activation of, in occiusal plane analysis, 326—327, 327

in asymmetric action against molars, 330—335, 330, 332, 333, 334

in coronal-plane-view analysis, 343— 344, 343

in extrusion potential elimination, 339—340, 340

in overjet and anterior open bite reduction, 341—343, 341, 342 headgear for, 341—342 occlusal plane analysis of, 326—335, 327, 328, 329, 330, 332, 333, 334 for asymmetric action, 330—331, 330

buccal tubes in, 328, 329 with face-bow activation, 326—327, 327

force systems in, 327—328, 328 Change of state, 113—114 Chemical properties of materials, 72, 103— 110

Chin-cap assembly, 362—365 active force for, 323

components of, 362 with reverse-pull appliance, 368, 368 Circumferential fibers, 173 Class II mechanics, 424—434 Class III mechanics, 417—424

Clearance, bracket-wire, 202

Coordinates, 4—8

mutually perpendicular axes for, 5, 7— 8, 7 rectangular framework for, 6—7, 6 Corrosion, 104—109, 707 Cosecant, 11—13

Cosine, 11, 14 Cost of materials, 111 Cotangent, 13 Couple(s), 40—46, 40

between arch wire and bracket, 220— 222, 221

arch-wire transmission of, 202—204, 203, 204

in bending of arch wire, 234 in bending of beam, 245—246, 245, 246, 248

in bodily movement tooth displacement, 153, 753 computation of inherent moment of, 40, 41, 41

in equilibrium problems, 63 in free-body diagram, 58 in long-axis rotation, 166—167, 267

in loop activation, 266 for maxilla tipping, 190—191

mechanical work of, 117 mechanically equivalent, 43—44, 43

moving line of action of force through addition of, 45—46, 45

mutually perpendicular vectors of force and, 44, 44 in occlusal-plane analysis of cervical-

467 Index

pull appliance, 327—328

in physiologically proper orthodontic force, 183

in plane with vector representation, 42— 43, 42

in premolar alignment, 375 reduction of force and, 44—45, 45 in root movement, 158 in rotation, 47, 99—100, 202

in shaft theory, 275, 275 in split-anchorage beam loading, 253— 255, 254

in tipping, 155 torque vs, 272 transmission of, through brackets, 200, 201, 201

Couple-force ratio, 153, 163t Covalent bonds, 74 Creep, 101, 202 Crestal fibers, 173, 174 Critical cross-section, 196, 243, 247, 284 Cross-bite, 390, 391 Crown movement, 150 couple-force ratios and center-ofrotation location for, 161, 1631 forces in, 159—161, 160

periodontal ligament in, 151 stress values for, 161, 163t Crown-bracket attachment, 198—199, 199, 202, 202 Crystals, 76—77, 77, 78, 97

Curve of Spee, 206 modification of, 392, 397—402

in torquing, 309

Deformation, load vs, 78—87 for beam, 233 bending, 80—81, 81 compressive, 79, 80 definition of, 79 elastic, 81 of elastomers, 85—87, 86, 87 to fracture, 85 inelastic, 81 for linear spring, 214, 214 normal, 79, 80 in shaft theory, 275, 275 shearing, 79, 80 in strain hardening, 137—138 tensile or extensional, 79, 80 twisting, 81, 82 types of, 79—81 of wires, 82—85, 84

Density, 75 Dentition, appliance torque effect on, 289— 294, 290, 291, 292, 293, 294

attachment of appliance to, 197—205 arch wires in, 201—205, 203, 204 brackets in, 197—198, 198, 199, 201, 202

buccal tubes in, 199 buttons, eyelets, and staples in, 199, 199

for distolingual rotation, 200, 200 for extrusion, 201, 201 force systems in, 200—202, 200

in mechanical equilibrium, 57, 58 response of, to force, 146—169 with extrusion, 163—164, 264 with intrusion, 164—166, 265, 166 with long-axis rotation, 166—169, 167, 168, 169 single tooth, 148—151, 149, 150 with tooth displacement, 154—163, 154, 155, 156, 157, 158, 159, 160, 162

transverse crown force systems in, 151—154, 253

Dentofacial orthopedics, 189—191, 190, 192

Differential force, 187—188

Dimensions in orthodontic bioengineering, 20—21 Direction angles, 14—15

Direction cosines, 15, 16 Distributed force, 26 Dry corrosion, 108—109

Ductility, of ceramic materials, 96, 96 measures of, 96—97

of polymers, 97 of wire, 85 Dynamic friction, 53 Dynamics, 25

Elastic(s), in activation of appliance, 215— 216, 215, 216, 217—219, 218

in bending of arch wire, 233 in canine alignment, 377, 378 in canine retraction, with headgear, 345

process of, 402, 403 in cervical-pull appliance, 325 in asymmetric action against terminal molars, 331 in buccal-view analysis, 336 in chin-cap assembly, 362 in Class II mechanics, 417—418, 423 in Class Ill mechanics, 425 energy in activation of, 127—128, 127, 128

equilibrating force of, 65—66

in equilibrium, 63

468 Index Elastic(s) (Continued)

extension deformation with activation of, 80

in extraoral force delivery to maxilla, 351

force concentration with, 26 free-body diagrams of, 62—63, 62 in interarch mechanics, 388—389, 390 orthodontic force transmission through, 204

relaxation of, 128 in reverse-pull appliance, 367 in rotational movement, 381—382, 381 work in stretching, 117 Elastic bending, apptication of, to archwire activation, 257—261 range in, 252—253 stiffness and, 248—252, 249, 251

strength in, 247 theory of, 248 Elastic core, 262, 263, 311 Elastic deformation, 81 Elastic limit, of appliance elements, 230 of elastomers, 86 in load-deformation relationship, 84—85, 83

of polymers, 97 of shaft, 310 stress and strain within, 95 of wires, 84 Elastic range, 252—253

Elastic strength, 85, 95, 247 Elasticity, modulus of, 95 Elastomer, aging of, 109 internal force distribution of, 88, 89 load vs deformation of, 85—87, 86, 87 stress in, 88 Electrical conductivity, 75 Electrochemical cell, 105, 105, 108 Electrochemical corrosion, 108 Electrochemical oxidation, 104—107

Electrode potential, 105, 106t Electrons, 73—74

Elongation, percent, 96 Endurance limit, 103 Energy, 119 in activation and deactivation processes, 125—137

in activation of appliance, 212 analyses of, in orthodontics, 113—145 available, 124—125

in compressing open-coil, helical

displacement in relation to, 114—115, 115

in elastic module activation, 127—128, 128

heat transfer and, 121—122 internal vs external, 119 kinetic, 118, 119 measurements of, 20 mechanical work and, 115—117, 116 Newton's laws and, 117—118 potential, 119 in preparation of arch wire, 141—144, 142, 143

in relaxation process, 128—130, 129, 130, 131

strain, 119, 122 in activation of appliance, 211 in deactivation of appliance, 212 with elastic bending, 248 in preparation of arch wire, 141—142 in wire activation and deactivation, 131

in strain hardening and heat treatment of metals and alloys, 137—141, 139

thermal, 121—122

in wire activation with bending, 130— 135, 232, 133, 134 Equilibrium, definition of, 56, 57 distance for, between atoms, 74 energy in, 124—125 in orthodontic mechanics, 56—69

Extensometer, 94 Extra-alveolar fibers, 173 Extraoral appliances. See Appliances, extraoral Extrusion, via cervical-pull appliance, 339 in curve of Spee reduction, 399 force system for, 163—164, 264 as orthodontic translation, 149 proper orthodontic force for, 186 utility arch and, 394, 398 Eyelets, 199

Face bow, bilateral displacement action via, 386 for cervical-pull appliance, 325 activation of, in occlusal-plane analysis, 326—327, 327

in asymmetric action against molars, 330—335, 330, 332, 333, 334

spring, 125—126, 126 conservation of, 123—124

in coronal-plane-view analysis, 343—

conversions in unit systems, 22t in deformation of linear spting, 126,

in extrusion-potential elimination,

227

344, 343 339—340, 340

in overjet and anterior open bite

469 Index reduction, 341—343, 341, 342 deformations with activation of, 324

force transfer with, 322 Fatigue, material, 102—103, 203 Fibroblasts, 173 Flexure test, 100, 100 Force(s), active, 26, 27 analysis of, 24—70

areas or volumes of bodies associated with, 26 on beam activated, 234, 235 in bodily-movement tooth displacement, 152—155, 153, 154

body, 25 in canine retraction, 403, 404, 406 on chin-cap assembly, 363—365, 363, 364

concentrated, 26, 27, 28. See Point force contact, 25, 26 continuous, 179, 180 in continuous arch-wire appliance model, 206, 206, 207—208

conversions between unit systems, 22t in crown movements, 159—161, 160 definition of, 25 and deformation, 27, 81 as a dimension, 20—21 direction of, 27

and displacement, 27 distributed, 26 resultants

of, 50—53, 52

external, 26 in mechanical equilibrium, 57 in mechanical work, 117 of extraoral appliances, 322—323, 324 in extrusion, 163—164, 164 in free-body diagram, 58 frictional, 53—55, 54, 55, 56 interatomic, 74, 75 intensity of, 51 intermittent, 178, 180 internal, 26, 27 in mechanical equilibrium, 57 in mechanical work, 116—117 interrupted, 179, 191 in intrusion, 164—166, 265 line of action of, 27 in long-axis rotation, 166—169, 167, 168, 169

magnitude of, 27

on lever, 66—67, 66 orthodontic application of, 63—69 of two-dimensional parallel force system, 67—68, 67 vector characteristics of, 59—60, 59 in mechanical work, 115—117, 216 Newtonian laws concerning, 117—118 orthodontic, 146 differential, 187—188 excessive,

181

heavy, 178 light, 178 maximum, 188 optimum, 188 physiologically proper, 182—189, 183t, 289

threshold, 188 tooth displacement vs initial level of, 189, 189

orthopedic, 146, 189 between particles, 73 physiologic, 146 point, 26, 27, 28. See Point force reactive, 26, 27 in representative orthodontic mechanics, 372—436

responsive, 26 resultant, 30 graphic representation of, 30, 31 parallelogram law for, 30—31, 30 polygonal law for, 33, 34 on rigid bodies, 46—50, 48, 49 triangle law for, 31—32, 32 on reverse-pull appliance, 366 on rigid body, 28 in rotational movements of tooth, 157, 157, 382

sense of, 27 time vs, 28—29 with tip-back ends, 224—225 in tipping of tooth, 155—157, 156 transmission of, bracket-crown attachment in, 198, 199

vector addition and decomposition of, 29—35, 30, 32, 32, 34, 35 weight, 118, 119 work and, 123 Fracture, of bracket-to-crown attachment,

for maxilla tipping, 190—191

in mechanical equilibrium, 56, 57 of brackets, 65—66, 65

of cantilevered structural member, 68—69, 68 connections

between members and

supports, 60—62, 61t

188

force-time pattern for, 178—182, 180,

197—198

failure and, 103 percent elongation at, 96 percent reduction in area with, 97 slipping, 98 strain at, 95, 96 stress at, 95 fatigue

470

Index Free-body diagram, 58

of arch wire in torsion, 222, 222 of beam activated, 234, 235 of bracket subjected to concurrent forces, 65—66, 65

of cantilevered beam, 255, 255 of cantilevered structural member loaded at free end, 68—69, 68 of chin-cap assembly, 363, 363 definition, 58 for elastic band, 215, 215 of elastic traction band and neck pad, 325, 326

for face bow of cervical-pull appliance, in activation, 326—327, 327

with asymmetric action against terminal molars, 330—331, 330, 332

in buccal view analysis, 336, 336 of half-box loop, 315, 315 of lever subjected to two-dimensional force system, 66—67, 66

of segmented beam, 242, 242 of spacer, 212, 213 of split-anchorage beam, 251 of spring for torquing, 317—319, 317, 318

of static elastic stretched between two points, 62—63, 62

of structural member supported at ends, 67, 67 of torquing associated with arch-wire activation, 291—292, 292 of transpalatal bar, 292—293, 292 of utility arch, 393 of wire loop, 267, 268

Friction, between bracket and wire, 55, 56 block-on-plane analysis of, 53—54, 54 coefficients of, 54 in deactivation process, 135 definition, of, 53

direction, magnitude, and sense of force of, 54—55

dynamic, 53, 55 in equilibrium, mechanical, 63 in extraoral force delivery to maxilla, 351

in mechanical work, 117 sliding, 53 static, 53, 55 Functional relationships, 3 Galvanic cell, 107, 107 Galvanic shock, 107 Gingival fibers, 173

Gram, in measuring mass, 20 Gravitational law, 25

Gravitational potential energy, 119 Gravity, 118, 119, 119

Hardening heat treatment, 140 Hardness, 100 numbers, 100 Head cap, 322 in canine retraction with headgear, 345 of chin-cap assembly, 362 in dual-force headgear, 362 in extraoral force delivery to maxilla, 351

with face-bow appliance, 325, 331 for reverse-pull appliance, 365 Headgear, active force vs time with usage of, 324. See Appliances, extraoral arch-wire interaction with, 324 canine retraction with, 344—348, 346, 347, 348 cervical-pull, 323, 341—342

chin-cap assembly and force of, 363— 364, 363

classification of, 323 design of, 322 dual-force, 323, 361—362 for extraoral force delivery to maxilla, 351, 352 high-pull, 323 intraoral appliance interaction of, 324 resultant force of, 30, 30 reverse-pull, 323, 365, 365 straight-pull, 323 Heat transfer, 121—122

Heat treatment of metals, definition of, 138

energy analyses of, 138—139, 139 types of processes of, 139—140 Hooke's Law, 95, 96 Horizontal fibers, 173, 174

Hot working of metals, 138 Hydrocarbon structures, 76, 76 Hypotenuse, 11, 13 Inelastic deformation, 81 Inertia, 56, 57 Inner bow, 325 Interatomic bonds, 74 Intermolecular bonds, 74 Interradicular fibers, 173 Intra-alveolar fibers, 173, 174 lntraoral appliances. See Appliances, intraoral Intrusion, cervical-pull appliance for, 341— 343, 342

in curve of Spee reduction, 400 definition, 149, 164

471

index force system for, 164—166, 765

as orthodontic translation, 149 proper orthodontic force for, 186 utility arch and, 394 Ion, 74 Ionic bonds, 74 Isomers, 75 1-hook(s), in canine retraction with headgear, 345—347, 346

in extraoral appliances, 322 in extraoral force delivery to maxilla, 351

interconnections and contact force systems of, 64t orthodontic force transmission through, 205

Joule, as measure of energy, 21

teardrop, 265, 265 in torquing, 314—316, 314, 315

Machine, 194—195, 196—197

Magnetic properties of materials, 72 Mandible, mandibular arch, in Class II mechanics, 433—434 in Class III mechanics, 421—424, 422 extraoral force delivered to, 356—361 from buccal perspective, 356, 357, 360—361, 360, 361 free-body diagram of, 360, 360

growth control of, 191, 362 utility arch and, 392, 393, 394—397, 396, 398

Mass, 20 Mastication, 172 Materials, appliance, appearance of, 111 availability of, 111 behavior of, 72—111

Kelvin, degree, 21 Kilogram, 20, 21 Kinematics, 25 Kinetic energy, 118 law of, 118, 123 particle, 118, 122 Kinetics, 25 Lagrange, J.L., 113 Lamina dura, 170 Law of conservation of energy, 122—124 Law of cosines, 13—14, 31

Law of kinetic energy, 118 Law of sines, 13—14, 31

Laws of thermodynamics, 123 Length, characteristic in bending, 250, 252

characteristic in torsion, 284 conversions between unit systems, 20, 22t

dimension of, 20 Lever, 66—67, 66 Long-axis rotation, 150, 166—169, 167, 768, 169 Loops, wire, 264—269 activated, 264, 264, 265, 266—267, 266 box, 265, 265

elastic ranges for, 267, 269 force vs deformation of, with activation, 265—266, 265 half-box, 314—316, 314, 315 helices of, 267, 268 L, 265, 265

passive, 264, 264 stiffnesses of, 265—266, 269 T, 265, 265

chemical and thermal influences on, 103—110, 105, 106t, 107 costs of, 111

handling and working of, 111 internal structure of, 73—78, 75,

76,

77

load-deformation behavior of. See Deformation, load vs mechanical stress in, 87—91, 89, 90, 91, 92

properties of. See Properties, material selection of, 110—111

standardized testing of. See Tests of appliance materials storage of, 111 types of, 71 Mathematical topics, 1—23 constants, variables, and functions, 1—3 dimensions and units, 20—21, 22t frames of reference, 3—8, 5, 6, 7

measurements, computations, and numerical accuracy, 21—23 trigonometry, 10—16, 70, 11, 12t, 13, 14, 15

vector algebra, 16—20, 17, 78, 19 Maxilla, maxillary arch, in Class II mechanics, 426—433 in Class III mechanics, 419—421, 420, 424

extraoral force delivered to, 348—356 from buccal perspective, 351—355, 352, 353, 354, 355 entire arch, 350, 350 free-body diagram for, 349—350, 349, 350 incisal segment, 349, 349, 353—354, 354

472

index

Maxilla (Continued) from occiusal perspective, 350—351 tipping of, 190—191, 190

utility arch and, 394, 395 Mechanical properties of materials, 72 elastic limit, 95, 97 modulus of elasticity, 95, 97, 99, 102t proportional limit, 95, 98 springback, 100, 102t standardized testing to determine, 91— 103

from static tension test, 92—94, 93 yield strength, 96, 99, 102t Mechanical system, 113—114 Mechanics, 24—25 equilibrium in, 56—69

principle of transmissibility in, 28, 29 Mer, 76 Metallic bond, 74 Meter, 20 Metric units, 20 Modulus of elasticity, 95 of crystalline solids, 97 for wire materials, 99, 102t Modulus of resilience, 96, 121 Modulus of rigidity, 98, 278 Modulus of toughness, 96, 121 Modulus, section, 246 Molecules, 74 bonding characteristics of, 74, 75 displacement of, 78 equilibrium position of, in solids, 78 isomers of, 75 Moment arm, 36, 38, 39 Moment axis, 38 Moment center, 36, 38, 39 Moment of force, 36—40, 37, 39 in free-body diagram, 58 within wire beam, 245—247, 245 within wire shaft, 280—282, 282 Monomer, 76, 76 Neck pad, 322 in canine retraction with headgear, 345

of cervical-pull appliance, 325 in dual-force headgear, 362 Neutral axis, 240, 245 Neutral surface, 238, 239, 240 Neutron, 73 Newton, 20, 21 Newtonian mechanics, 24—27, 56, 113, 117—118

Nucleus of atom, 73 Oblique fibers, 173, 174 Occlusal plane, 8

Omega loop, 394, 409 Open-bite correction, 392 Ordinate, 4 Orthodontic forces, 146 differential, 187—188 excessive, 188 force-time pattern for, 178—182, 180, 181

heavy, 178 light, 178 maximum, 188 optimum, 188 physiologically proper, 182—189, 183t, 189

threshold, 188 tooth displacement vs initial level of, 189, 189

Orthopedic forces, 146 Orthopedics, dentofacial, 189—191, 190, 191

Osteoblasts, 175 Osteoctasts, 175 Osteoid, 175 Outer bow, 325 Over-bite reduction, 341—343, 341, 342 Overjet reduction, 341—343, 341, 342

Palate splitting, 189—190 Parallelogram law of vector addition, 19— 20, 19

in decomposition of point forces, 30, 30 triangle law from, 31—32, 32 Particles, atomic structure of, 73 displacements of, 8—10, 9, 36 deformations vs, 79 in process laws, 114—115, 215

resistance to, 53 forces between, 73 kinematic characteristics of, with work, 118, 118 Pascal, 21 Passivation, 108

Periodontal ligament, alveolar bone and, 169

blood supply interruption in, 177 components of, 151, 170 in crown loading response, 151 in crown movements, 160—161, 160

deformation of, with tooth movement, 147, 172—173

in extrusion response, 164 fibers of, 173 force-time pattern of pressure on, 178 function of, 170 with intermittent orthodontic force, 179 in intrusion response, 165 loading of, by tooth root, 51

473 Index

in long-axis rotation, 167—169, 168 physiologically proper orthodontic forces and, 182—189, 183t, 189 pressure transmission through, 170

in root movement, 159, 159 thickness of, 170, 184 in tipping, 155—156 Periodontium, components of, 169—170 crown loading response of, 169—172 orthodontic force-time pattern effect on, 178—182, 180, 181 response of, to force, 169—177 alveolar bone, 170—171, 171, 175—176 periodontal ligament, 171—175, 174 Physical properties of materials, 72 Piezoelectric effect, 176 Plane of bending, 236

Plane of reference, 8 Point force(s), 26, 27 line of action of, 27 in mechanical work, 115—116, 116 moment of, 36—40, 37, 39

point of application of, 59, 59 in transverse bending deformation, 83 plot of vs deflection, 83, 83, 84 proportional and elastic limit for, 84— 85

work-energy analysis of, 120 yield strength, 96, 99, 102t Proportional limit, on load vs deformation plot, 84 of polymers, 97 on stress-strain diagram, 95 Proton, 73 Protraction of mandibular arch, 433, 434 Pythagorean theorem, 13, 18

Quantum mechanics, 24 Quasi-static state, vs mechanical equilibrium, 58—59

of orthodontic system, 27 Radiation, 122 Reaction, 26 Reference frame, 3—8 axes of, 3—4, 3

coordinates within, 4, 5 curves within, 4—5

mutually perpendicular coordinate directions within, 7—8, 7 planes associated with, 8 rectangular coordinates of, 5—6, 6 Reference frame fixed within tooth, 148— 149, 149

vector addition in decomposition of, 29— Relaxation, 101 35, 30, 31, 32, 34, 35 Resilience, from load-deformation Poisson effect, 80, 98 diagram, 85, 121 Polar second moment of area, 282 modulus of, 96, 121, 120 Polygonal law, 33, 34 Resultant of force system, 29—33, 30, 31, Polymerization, 76, 76 32, 46—50, 48, 49 Polymers, 76 Retraction mechanics, 402—412 Polymorphism, crystalline, 76 Reverse-pull appliances, 365—369 Potential energy, gravitational, 119, 123 in buccal view, 366, 367 Power, 274 cantilevered, 366, 367 Pressure, 20 with chin cap, 368, 368 Principal fibers, 173 in occlusal view, 365—366, 365 Process, 113 split-anchorage, 367, 367 Projected root area, 183, 183 Rigid-body mechanics, 28 Projection, of vector, 17 center of resistance in, 46 Properties, material, 71—72 resultants of force systems in, 46—50, of arch wires in bending, 258, 259t 48, 49 bonding characteristics of atoms and rotation in, 47 molecules in, 75 tooth in, 147 creep, 101, 102 translation in, 47 density, 75 Rockwell hardness scale, 100 elastic-limit strain, 95 Root, force system on occlusal-view "slice," 155, 155 elastic strength, 95 movement of, couple-force ratio and electrical conductivities, 75 center-of-rotation location for, hardness, 100 161, 163t modulus of elasticity, 95, 97, 99, 102t forces in, 157—159, 158 springback, 100, 102t maximum periodontal stress values stiffness, 75 strength. See Strength, material for, 161, 163t periodontal fiber strain with, 159, structure insensitive, 78 thermal conductivities, 75 159, 173

474 Index

Root (Continued)

in tooth displacement, 150 in periodontal ligament deformation, 147

periodontal ligament loading of, 51 projected areas of, 183t shape of, 184 spurs in torquing of, 306 surface area of, 183, 183t in tipping, 156 Rotational tooth movements, 149, 150 arch wire in, 381—382, 381 brackets in, 200—201, 200 from cervical-pull appliance, 340—341 in correction of malaligned tooth, 379— 385, 380, 381, 383, 384, 385

crown contact in, 384, 385 crown force system in, 157, 157/

in periodontal ligament, 163 in shaft theory, 278, 279—280, 280

torque and, 274 with twisting, 81 Shearing deformation, 79, 80, 98 SI units system, 21 Sine, 11 Sliding friction, 53 Slope, 6 Slug, 21 Solid body displacements, 8—10, 9 Solid material, bonding characteristics for, 75

cohesive forces in, 73 corrosion of, 104 crystals, 76—77, 77

imperfections in, 77 internal structure of, 73—78

elastics in, 381—382, 381

Spacer, 212—213

force system for, 166—169, 167, 168, 169

Spring, activation of, 125—126, 126, 127,

,generalized, 47 lever for, 382, 383 long-axis, 150 periodontal fiber strain with, 173 physiologically proper orthodontic force for, 186

,pure, 47 spring ("wedge") for, 382—384, 384 ,transverse, 150, 151

213

in canine retraction, 345, 402, 407, 407 for cervical-pull appliance, in asymmetric action against terminal molars, 331 in buccal view analysis, 336 in chin-cap assembly, 362 in extraoral force delivery to maxilla, 351

orthodontic force transmission through, 204

Scalars, 16 Scale, 3 Second, 20, 21

Second moment of area, 246 Section modulus, 246 Shaft theory, 273—299 application of, to arch wire, 288—299

active and responsive force systems, 295—299

appliance and dentition response to torsional activation, 289—294 isolation of activating torque, 288—289 with circular shaft, 274—284, 275, 276, 278, 279, 280, 281, 282, 283 deformation in, 275, 275

passive and activated configurations of shaft element in, 278, 278 with rectangular shaft, 285—288, 286, 287, 287t

rotational displacement in, 274, 277 shear stress in, 278, 279—280, 280 shear strain in, 277—278 torsional behavior in, 274—275 twist angle in, 283, 283 Shear strain, 98—99, 99 Shear stress, 90, 90, 98—99, 99 in bending of wire beam, 243—244, 244

relaxation of, 128 in rotational movement, 382, 384 torque via, 272, 316—318, 317, 318 with wire loops, 264 work done within, 119, 119 Springback, 100, 102t Spurs, torquing, 305—310 attachment of, 306 Begg, 308, 308 facial and sagittal views of, 306 force systems of, 306—307, 307 Stainless steel corrosion, 108 Staple, 199 Static friction, 53 Statics, 25 Stiffness(es), of Begg-torquing appliances, 308t bending, 84, 248—256, 251

of elastomers, 87 of incisal-segment torquing arches, 304t loop, 265—266, 269t

material, 75 torsional, 284—286, 295, 301t, 302 Storage of materials, 111 Strain, 94 Strain energy, 119, 122 in activated appliance, 211

475 Index in arch-wire activation and deactivation, 130—135

in deactivating appliance, 212 with elastic bending, 248 in preparation of arch wire, 141—142 Strain hardening, definition of, 137—138 energy analysis of, 138—139, 139

Strength, material, arch wires, 258, 259t in compression vs tension, 98 elastic, 85, 95 fracture, 95 of interatomic bonds, 75 ultimate, 95, 98 yield, 96 Stress in, alveolar bone, distribution of, 170—171, 171

remodeling changes in, 175—176

bonding forces and, 87, 88 in bracket-band vs bracket-base assemblies, 198 compressive. See Compressive stress conversions in unit systems, 22t definition of, 88 in elastomers, 88 from external loading, 87—88 normal, 90, 90 in shaft theory, 274—275 shear. See Shear stress

and strain, 94, 120, 120 tensile. See Tensile stress from twisting of wire, 311—314

vector components of, 90, 90 Stress cell, 108 Stress corrosion, 108 Stress-relief heat treatment, 139 Structural properties of materials, 72

obtained through standardized testing, 91—103

Structure, 194, 196—197

Supra-alveolar fibers, 173 Surroundings, 113, 117, 121, 124

as variable in structural analysis, 2 Temporomandibular joint, 359, 364, 365 Tensile deformation, 79, 80 Tensile strain, 94 with beam deformation, 241 in elastic traction band in cervical-pull appliance, 325 Tensile stress, 90 in alveolar-bone remodeling, 176 alveolar bone response to, 171, 171 in bodily-movement tooth displacement, 155 in crown movements, 160 in tipping, 157 in torqued shaft, 279 Tensile stress-strain diagrams, 94—97, 95, 96

Tests of appliance materials, 92—103 in ADA Specification No. 32, 99—100, 100, 101

compression, 97—98

creep, 101, 102 fatigue, 102—103, 103

flexure, 99, 100

-

hardness, 100

modulus-of-elasticity determination, 95, 102t

relaxation, 101 shear stress-strain, 98—99, 99 sprirtgback, 100, 102t tensile stress-strain, 91—98, 93, 95, 96 work-energy analysis in, 120 yield strength, 96, 102t Thermal conductivity, 75 Thermal energy, 121—122, 123, 124 Thermal properties of materials, 72, 103— 110

Thermodynamics, laws of, 123 Time, conversions between i.mit systems, 22t

in creep, 101 as a dimension, 20 distance vs, 3, 3 force vs. 28—29

Tangent, 11 Temperature, conversions in unit systems, 22t corrosion and oxidation affected by, 109

in creep, 101 as a dimension, 20 in equilibrium distancing between atoms, 74 in heat transfer, 122 in heat treatment of metals, 138 oral, 2 in strain hardening, 138 in tests of materials, 97

in headgear usage, 324 tooth location vs, 2 as variable in structural analysis, 2 Tipping, maxilla, 190—191, 190

with cervical-pull appliance, 328 with extrusion and intrusion, 166, 166 Tipping, tooth, 150 in canine retraction with headgear, 347 via cervical-pull appliance, 339, 339 couple-force ratio and center-of-rotation location for, 161, 163t crown force system in, 155—157, 156 physiologically proper orthodontic force for, 184—185

476 Index

Tipping (Continued) of premolar, 374—377, 376

reverse-pull appliance and, 369 stress values for, 161, 163t Tooth displacement, via bilateral action, 385—388, 386, 387, 388, 389

bodily movement, 149—150

couple-force ratio and center of rotation location for, 161, 163t crown force system in, 152—155, 153, 154

leading and trailing root surfaces in, 154

proper orthodontic force for, 185 stress values for, 161, 163t controlling force-time pattern in, 178— 182, 180, 181

crown movement, 150 in dentofacial orthopedics, 189—191, 190, 191 energy in, 135—137, 136

equilibrium of forces in, 57 extrusion. See Extrusion force system in, 147 intrusion. See Intrusion

rotation of shaft cross-sections vs, 283 shear stress and, 274 transmission of, along shaft, 276—277 Torquing, tooth, 150 appliances for, 272—321

proper orthodontic force for, 186 reverse-pull appliance and, 369 root movement and, 157 Torsional theory, 273 Toughness, in load-deformation behavior, 85

modulus of, 96, 121 Translation, 47 Transmissibility principle, 28, 29 Transpalatal bar, 291—292, 291, 292, 386— 388, 389

Transseptal fibers, 173 Triangle, definition of, 10—11 hypotenuse of, 11 oblique, 13—14, 13

Pythagorean theorem for, 13 right, 11, 11 Triangle law, 31—32, 32 Trigonometry, 10—16 Twisting, 81, 82, 310—311

leveling, 374—379, 376, 378

magnitude and duration of force in, 177—191

measurement of, 9, 9 mechanical response to force in, 148— 151, 149, 150

periodontal ligament response to, 172 physiologically proper orthodontic forces in, 182—189, 183t, 189

in quasi-static state, 27 root movement in, 150 rotation. See Rotational tooth movements tipping. See Tipping, tooth torque in, 272 as translation, 149 utility arch and, 394 Tooth eruption, 146 Tooth malalignments, 374—392 bilateral action in correction of, 385— 388, 386, 387, 388, 389 canine, 377, 378

interarch mechanics in correction of, 388—391, 390, 391

intra-arch mechanics for, 374—388 leveling displacements for, 374—379, 376, 378 molar, 378, 379, 380 premolar, 374—377, 376 rotational corrections of, 379—385, 380, 381, 383, 384, 385 Torque, definition of, 272

Unit bending stiffness, 248 Unit cell in crystalline representation, 76, 77

Unit vectors, 17 Units in orthodontic bioengineering, 20— 21

Utility arch, 392—397 mandibular, 392, 393, 394—397, 396, 398 maxillary, 394, 395

van der Waals bonds, 74, 76 Variables in structural analysis, 2—7 functional relationships between, 3

independent vs dependent, 2, 6 plotting graphs of, 4, 5 reference framework for, 3—7, 6 Vector algebra, 16—20 Vector(s), addition of, 19—20, 19

in decomposition of concentrated forces, 29—35, 30, 31, 32, 34, 35

resultant of, 29 definition of, 16, 17 direction of, 16 in equilibrium problems, 59—60, 59 magnitude of, 16 moment of, 36 perpendicular to plane of couple, 41— 42, 42

for point force, 27

477 Index

sense of, 16 stress, 90, 90 in three-dimensional framework, 18—19, 18

unit, 17

Wet corrosion, 108 Wire, orthodontic, activation of, 130—135, 132, 133, 134 in arch appliance, 219—223, 220, 221, 222 in ligature appliance, 216—217, 217 alloys of, 258 as beam, 233, 257—261

stiffness, range, strength of, 259, 260, 261

bending of. See Bending of wire in canine retraction with headgear, 345—347, 346

friction between bracket and, 55, 56, 135, 136, 405, 406

geometry of, 258, 259 headgear interaction with, 324 interconnections and contact force systems of, 64t ligation of, to brackets, 201—202, 202 with loops, 264—269 selection of, 258 as shaft, 288—289

active and responsive force systems, 295—299

appliance and dentition response to activational torsion, 289—294 isolation of activating torque, 288— 289

shear stress in, 90—91, 92

standardized testing of, 99, 100 stiffness of, 220, 258, 259t, 260t, 261t strain hardening and heat treatment in manufacture of, 140—141

in cervical-pull appliance, 340—341 characteristics of, 233

strength of, 258, 259t, 260t, 261t with tip-back bends, 224—228, 225,

contact modes between bracket and,

226, 227 twisting of, 310—314

55, 56

in continuous-arch-appliance model, 205—208, 206, 207

deformation of, with bending, 82—85, 83

description of, I ductility of, 85 elastic beam theory applied to

Work, as energy transfer, 123 forms of, 118 gravity and, 118, 119, 119 mechanical, 115—117, 116

power and, 274 in preparation of arch wire, 141—144, 142, 143

activation of, 257—261

elastic range of, 258, 259t, 260t, 261t force transmission through, 201, 202— 204, 203

Yield strength, 96, 99, 102t Young's modulus, 95

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0-8

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22-8

Bioengineering Analysis of Orthodontic Mechanics

Systemized Orthodontic Treatment Mechanics

Orthodontic Treatment of Impacted Teeth

Applications of Orthodontic Mini-Implants

Methods in Bioengineering: Systems Analysis of Biological Networks (The Artech House Methods in Bioengineering)

Removable Orthodontic Appliances

Mosby's Orthodontic Review

Functional analysis in mechanics

Functional Analysis in Mechanics

Functional analysis in mechanics

Functional Analysis in Mechanics

Functional analysis in mechanics

Functional Analysis in Mechanics

Functional Analysis in Mechanics

Functional Analysis in Mechanics

Carbohydrate Bioengineering

Mechanics and Analysis of Composite Materials

Mechanics and Analysis of Composite Materials

Introduction to Bioengineering

Mechanics and Analysis of Composite Materials

Geometric Aspects of Analysis and Mechanics

The Tip-Edge Orthodontic System

Design Analysis in Rock Mechanics

Design Analysis in Rock Mechanics

Design analysis in rock mechanics

Tensor Analysis and Continuum Mechanics

Limit Analysis in Soil Mechanics

Orthodontic Miniscrew Implants: Clinical Applications

Carbohydrate Bioengineering Interdisciplinary Apporaches

Bioengineering for Pollution Prevention

Methods in Bioengineering: Nanoscale Bioengineering and Nanomedicine (Artech House Methods in Bioengineering Series)

Bioengineering Analysis of Orthodontic Mechanics

Systemized Orthodontic Treatment Mechanics

Orthodontic Treatment of Impacted Teeth

Applications of Orthodontic Mini-Implants

Methods in Bioengineering: Systems Analysis of Biological Networks (The Artech House Methods in Bioengineering)

Removable Orthodontic Appliances

Mosby's Orthodontic Review

Functional analysis in mechanics

Functional Analysis in Mechanics

Functional analysis in mechanics

Functional Analysis in Mechanics

Functional analysis in mechanics

Functional Analysis in Mechanics

Functional Analysis in Mechanics

Functional Analysis in Mechanics

Carbohydrate Bioengineering

Mechanics and Analysis of Composite Materials

Mechanics and Analysis of Composite Materials

Introduction to Bioengineering

Mechanics and Analysis of Composite Materials

Geometric Aspects of Analysis and Mechanics

The Tip-Edge Orthodontic System

Design Analysis in Rock Mechanics

Design Analysis in Rock Mechanics

Design analysis in rock mechanics

Tensor Analysis and Continuum Mechanics

Limit Analysis in Soil Mechanics

Orthodontic Miniscrew Implants: Clinical Applications

Carbohydrate Bioengineering Interdisciplinary Apporaches

Bioengineering for Pollution Prevention

Methods in Bioengineering: Nanoscale Bioengineering and Nanomedicine (Artech House Methods in Bioengineering Series)

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