Theory of Elasticity and Plasticity [Reprint 2014 ed.] 9780674436923, 9780674432055

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T H E O R Y OF E L A S T I C I T Y AND PLASTICITY

H A R V A R D

M O N O G R A P H S

A P P L I E D

IN

S C I E N C E

N U M B E R

3

THEORY OF ELASTICITY AND PLASTICITY H. M. WESTERGAARD GORDON MCKAY PROFESSOR OF CIVIL ENGINEERING HARVARD UNIVERSITY

1 9 5 2

Cambridge,

Massachusetts

HARVARD UNIVERSITY PRESS New

York:

J O H N W I L E Y & SONS, INC.

COPYRIGHT

1952

BY

OF PRINTED

IN

LIBRARY

THE

PRESIDENT

HARVARD

THE OF

UNITED

STATES

CONGRESS

NUMBER

AND

FELLOWS

COLLEGE OF

CATALOG

AMERICA CARD

52-6348

Editorial Committee F. V. Hunt, Chairman Howard W. Emmons L. Don Leet P. Le Corbeille!· E. Bright Wilson

F O R E W O R D

Fundamental research in applied science is characterized by diversity. Problems arise that overlap the tidy categories of academic science and demand the use of source material and methods of investigation drawn from many fields of knowledge. A wide range of applicability of the results further distinguishes this kind of research from the specialized objectivity of development work in engineering technology. With characteristic unpredictability, research in applied science is often of greatest potential value to those it might not reach through conventional specialized channels of publication. The Harvard Monographs on Applied Science are designed to provide a medium for publishing the results of University research to a wider audience than would be reached by individual professional journals. The size and style of the monographs seem also to be well adapted for presentation of research results with due regard for a critical orientation of the work in its technical background. The term "Applied Science" can be interpreted broadly enough to include all phases of science that influence the lives of men. Without accepting any narrower limitation of objective, this series of monographs will be devoted primarily to reports of significant research in the applied physical sciences, with especial emphasis on topics that involve intellectual borrowing among the academic disciplines. T H E EDITORIAL COMMITTEE

PREFACE

This book is an outcome of lectures that I began giving at the University of Illinois in 1925 and continued to give at Harvard University during most of the years since 1936. Naturally the content has changed and developed as the years went by. I feel that the content has matured so as to justify publication in a book. The book deals mostly with the theory of elasticity and to a much lesser extent with the theory of plasticity. Much is common to the two theories, and I have found it desirable to include some consideration of plasticity. Most attendants of my lectures on the theory of elasticity and plasticity have been students of civil or mechanical engineering in the first year of graduate work, which actually is a very suitable time to study the subject. The book caters to that group, though I certainly hope that it will find readers in other groups as well. When a colleague asked me at what level the book was to be, the best answer that I could give was that the trail begins at the foot of the mountain. With this in mind I have not hesitated to insert rather full explanations of mathematical processes that are to be used and may be new to some readers. "Notes" on purely mathematical processes will be found every now and then. The advanced reader will know how to skip. The book contains a chapter of historical notes, Chapter Π. Ordinarily the proper place for an historical chapter might well be at the end of a book; but I am convinced that this time it is at the beginning. Historical notes can be a natural and most suitable means of presenting ideas, provided that the ideas can be explained in terms that are understood. Many important ideas in the theories of elasticity and of plasticity can be explained in simple terms to any one who already has some familiarity with mechanics of materials; most students of elasticity have that, or, if not, they are likely to have something equivalent. The historical notes in Chapter II extend into some subjects that are adjacent to the central subject of the book, for example, to the theories of elastic instability, but this is suitable in order to give the proper setting to the restricted subjects of elasticity and plasticity. I should like to have the historical chapter considered as an integral part of the presentation of the whole subject, rather than as an appendix.

vili

PREFACE

I have made it a policy to move as quickly as possible toward the treatment of problems in three dimensions. The contents of Chapters V and VI bear witness to that. Systematic treatments of two-dimensional problems are relegated to later chapters. When I began lecturing on the theory of elasticity in 1925, the twovolume book Drang und Zwang (R. Oldenbourg, Munich and Berlin, ed. 1, 1920) by A. Föppl and L. Föppl was of very great help. It was also of great help that I had previously had the privilege of attending a course of lectures given by Ludwig Prandtl in Göttingen in 1913 on the theory of elasticity; and later, after my arrival at the University of Illinois as a graduate student in 1914, a course given by George A. Goodenough. Professor Goodenough went as far as could reasonably be expected in a semester using the treatise The Mathematical Theory of Elasticity (Cambridge University Press, ed. 2, 1906) by A. E. H. Love. No one who has been seriously concerned with the theory of elasticity during the twentieth century can be without a feeling of indebtedness to Love. I am furthermore indebted to Stephen Timoshenko, whose Theory of Elasticity (McGraw-Hill Book Company, Inc., 1934) has been of great help in preparing this book even though I have treated most topics very differently; evidence of this gratitude will be found in many footnotes. I am indebted to the University of Illinois for the opportunity to give a course on the subject, and to Harvard University for the opportunity to continue this interest after I came here in 1936. I am particularly grateful to two of my colleagues at Harvard University, Richard von Mises and Gustav Kuerti, because of their willingness at any time to let me talk to them about problems that confronted me, especially of a mathematical nature, and to give me advice. Professor von Mises answered some particularly puzzling questions. Professor Kuerti offered to read my manuscript; I am indebted to him for many constructive comments. I am extremely grateful to Harry J. Macke, who served as Teaching Fellow in Civil Engineering at Harvard University from 1948 to 1950, assisting me in my courses, and continued in residence during 1950-1951. His meticulous scrutiny of the manuscript and many other services have been most helpful in the preparation of this book. HARALD MALCOLM WESTERGAARD

Division of Engineering Sciences Harvard University Cambridge, Massachusetts

C O N T E N T S

xiii

Introduction by Gordon M. Fair CHAPTER

I:

SCOPE

1. 2. 3. 4. 5.

Elasticity and Plasticity Homogeneity Isotropy Theory and Experiments Relation to "Mechanics of Materials," "Structural Mechanics," and "Soil Mechanics" 6. One-, Two-, and Three-Dimensional Problems of Elasticity

CHAPTER

II:

HISTORICAL

5 6

NOTES

Works on the History 8. Origins, 1636-1820 9 . Period of the Founding of the Three-Dimensional Theory of Elasticity, 1820-1830 10. Period of the Classics, 1830-1900 11. Books since 1900 12. Trends since 1900 13. Two Examples of Experimentation 14. The Strangely Late Discovery of the Shear Center in Beams 15. Stress Functions, Strain Functions, and Analogies 16. Use of Principles of Minimum Energy 17. Photoelasticity 18. Bending of Slabs or Plates 19. Instability, Buckling, and Stabilization by Load 20. Theory of Plasticity 21. Deformation Method in Structured Mechanics, and the Methods of Successive Approximation by Moment Distribution and Relaxation 22. Analysis of sui Arch Dam as an Example of a Satisfactory Solution of a Complex Problem by Imperfect Methods 7.

1 2 3 4

8 9 11 13 20 21 21 22 23 28 31 32 37 39 41 43

CONTENTS

ζ

CHAPTER

III:

STRESS

23. 24. 25. 26. 27.

Notation for Stress Uniform State of Stress Uniform Plane State of Stress The Dyadic Circle Extension to Certain Problems of a Three-Dimensional State of Stress 28. Note on Vectors 29. Uniform Three-Dimensional State of Stress 30. Stress Tensor 31. Principal Stresses and Lamé's Stress Ellipsoid 32. Shearing Stresses 33. Mohr's Circles 34. Mohr's Hypotheses on Limits of Stress 35. Octahedral Stress 36. Principal Stresses and the Octahedral Stress Determined from Six Components of Stress 37. Stress Deviations 38. Direction of the Octahedral Stress 39. Stress Quadric and Two Additional Surfaces That Represent the Stress Tensor 40. Differential Equations of Equilibrium of a Nonuniform State of Stress

46 48 49 51

CHAPTER

OF

IV:

ELASTICITY,

STRAIN, AND

THE

HOOKE'S SIMPLEST

LAW,

THE

POSSIBLE

BASIC LAWS

EQUATION

OF

55 55 57 58 59 61 61 64 66 67 69 72 73 75

PLASTICITY

41. Displacement, Strain, and Detrusion 42. Hooke's Law 43. Vector Strain, Analysis of Strains sis If They Were Stresses, and the Strain Tensor 44. Determination of Strain Tensor from Measured Strains 45. Note on Laplace's Operator 46. The Basic Differential Equation of Elasticity and the Principle of Superposition 47. Love's Notation and Lamé's Constants 48. The Special Case in Which Poisson's Ratio Is £ 49. Plane State of Stress in a Plate 50. Compatibility of Strains and Detrusions 51. Cylindrical Coordinates

78 79 83 85 87 87 88 89 90 91 91

CONTENTS 52. 53. 54. 55.

zi

Octahedral Strain Plastic Strain Deviations and Plastic Detrusions The Simplest Conceivable Laws of Plasticity Comments on Departures from the Simplest Laws of Plasticity

CHAPTER

V:

STRAIN

CYLINDERS

POTENTIAL

AND

SPHERES,

THERMAL

AND APPLICATIONS INERTIA

FORCES,

TO

95 95 95 97

HOLLOW

AND

STRESSES

56. Strain Potential 57. Lamé's Formulas for Stresses in Hollow Cylinders; Example of Stress Concentration 58. Lamé's Formulas for Stresses in Hollow Spheres 59. Disk with Circumferential Shear at Inner and Outer Concentric Circular Edges 60. Resultant Force on Surface of Revolution When Stresses Are Symmetric about Axis 61. Two Examples 62. Body Forces 63. Circular Disk Rotating about Its Axis of Revolution 64. Thermal Stresses 65. Examples of Thermal Stresses

CHAPTER

VI:

GALERKIN

APPLICATIONS

66. 67. 68. 69. 70. 71. 72. 73. 74.

VECTOR,

INCLUDING

TWINNED

EFFECTS

GRADIENT,

OF A S I N G L E

100 102 104 106 107 108 111 113 114 116

AND

FORCE

Galerkin Vector Relation to the Strain Potential Galerkin Vector When Poisson's Ratio Is £ The Galerkin Vector Supplies a General Solution of the Basic Equation of Elasticity Equivalent Galerkin Vectors Note on Harmonic and Biharmonic Functions Galerkin Vector kZ and Love's Strain Function for Solids of Revolution Example of a Vertical Load on the Horizontal Surface of an Extended Solid Kelvin's Problem of a Single Force Applied in the Interior of an Extended Solid

119 122 123 123 124 125 129 131 133

zìi

CONTENTS

75. Kelvin's Problem When Poisson's Ratio Is £ 135 76. Boussinesq's Problem of a Normal Force and Cerruti's of a Tangential Force on the Plane Surface of a Semi-infinite Solid When Poisson's Ratio Is 4 136 77. The Twinned Gradient and Its Application to Determine the Effects of a Change of Poisson's Ratio 137 78. Boussinesq's Problem When Poisson's Ratio Has Any Value 139 79. Cerruti's Problem When Poisson's Ratio Has Any Value 141 80. Mindlin's Problem, Part I: Vertical Force Applied at Some Distance below the Horizontal Surface of a Semi-infinite Solid 142 81. Mindlin's Problem, Part II: Horizontal Force Applied at Some Distance below the Horizontal Surface of a Semi-infinite Solid 145 82. Thick Disk Rotating about Its Axis of Revolution 148 83. Strain Potentiell φ(Λ, a) and Love's Strain Function in Polar Coordinates 151 84. Stress Concentration at a Small Spherical Hole 154 85. Deflections of the Surface of a Semi-infinite Solid under Various Normal Loads 157 86. Pressures Transmitted between Two Solids with Smoothly Curved Surfaces through a Small Area of Contact 163 Index of Authors

171

Index of Subjects

173

I N T R O D U C T I O N

Harald Malcolm Westergaard, the author of this treatise, long cherished the thought of writing a textbook on elasticity and plasticity. Into it he hoped to distill the experience of his many years of teaching. In it he wished to bring together the significant contributions that he himself had made to the exposition of his field of learning. Beginning in the spring of 1949, he devoted himself to the preparation of the manuscript. But he was not to be permitted to complete the task he had assigned to himself. (As one sentence in his preface indicates, he planned to include chapters on two-dimensional problems.) The symptoms of his terminal illness manifested themselves all too soon. Although he remained hopeful to the end, which came on June 22, 1950, he knew that his chance of finishing the book was not good. A man cast in the heroic mold, he therefore bent every effort to bring the first half of his manuscript into definitive shape. That he succeeded in doing so is evidenced by the publication of this book. GORDON M. FAIR Cambridge, Massachusetts December 1951

CHAPTER

I

Scope 1. Elasticity and Plasticity Both the theory of elasticity and the theory of plasticity deal with stresses and deformations in solid bodies. When loads—consisting of forces—are applied to a solid body or structure and are increased gradually from zero to final values, internal forces—stresses—and changes of shape—deformations, or strains in the general sense of that word— are created. If, when the loads are removed, the solid body or structure resumes its original shape, the property of elasticity has been displayed; but if deformations remain, plasticity has been displayed. Elasticity can be defined as the property of a solid body or structure by which the deformations are functions only of the temperatures and the stresses within some range of the temperatures and stresses. In a majority of cases of practical importance these functions can be stated as linear functions. Then ordinarily the deformations will also be linear functions of the forces that are applied as load; and if these forces are increased in the same ratio, the deformations will be found to be proportional to the load. The last statement expresses the law that was discovered in 1660 by Robert Hooke and announced by him in 1678 in the form: Ut tensio sic vis, "As the extension so the force," or, the force is proportional to the extension. By appropriate courtesy to this scholar of seventeenthcentury England, any applicable statement of the linearity of general relations between stresses and strains is now called Hooke's law. In the range of plasticity the relations between stresses and strains are more complex. The strains are functions not only of the momentary stresses and temperatures but depend also on items of the past history of stresses and temperatures, especially on past increases and decreases of the stresses. This book deals mostly with elasticity but occasionally with problems of plasticity. The reason for including consideration of plasticity is that

2

THEORY OF ELASTICITY AND PLASTICITY

much is common to the theory of elasticity and the theory of plasticity. Many problems of practical importance involve both elasticity and plasticity. This is likely when concern with strength and safety of a structure, or parts of a structure, or parts of machinery is a major reason for analysis of stresses and deformations. A notable exception in which elasticity is involved but not plasticity occurs when the stresses that are contemplated are repeated many times, as in moving machinery of various kinds; then failure may take place by fatigue, at some stress within the range of elasticity. Another exception occurs when the major concern is with stiffness within the range of elasticity, as in many problems of vibration and stability. There are also some problems involving primarily only plasticity, especially those concerned with creating new permanent forms, as in rolling or extruding. Sometimes a formal analysis is made only within the range of elasticity, but then there are added some qualitative considerations and estimates of the redistribution of stress that would take place if some of the stresses should enter the plastic range. Such a redistribution may involve a favorable "borrowing of strength" by the parts stressed most from those stressed less. Very often, however, the problem at hand calls for quantitative studies, first with all parts considered to be within the range of elasticity, and thereafter with some parts admitted to be in the range of plasticity. In their present stage the theories of elasticity and plasticity are concerned mostly with problems of equilibrium. Unless stated otherwise, the discussion will be of loads and stresses in equilibrium—within the realm of statics. 2. Homogeneity If a vertical cylinder is loaded on the top by a uniformly distributed vertical pressure and is supported similarly at the bottom, and the question is asked, what stresses occur on a horizontal section drawn through the cylinder, the natural first answer is: a uniformly distributed pressure. If the cylinder is of steel, the material has a grained structure consisting of crystals of several kinds, with many different orientations, and with irregular boundaries between the grains. This suggests a modification of the first answer: the stresses must be quite irregular, with discontinuities at the boundaries; the diagram of distribution of stress must be ragged within microscopic parts. If the material is concrete, made of stones or gravel, with spaces filled by sand, and cement paste in the remaining

SCOPE

3

voids, the diagram of stress must be visibly ragged within larger parts, and furthermore microscopically ragged within each stone, which itself has a crystalline structure. If one thinks of the atomic structure of each crystal, one must superpose ultramicroscopic irregularity in the complete picture. On the other hand, if the area of the cross section of the cylinder is divided into a hundred equal parts, one may count on nearly equal pressures on each of these hundred parts: the statistical averages on parts not too small are uniform, as if the material were homogeneous, that is, uniform and continuous in all parts. Thus the picture of the cylinder as homogeneous serves the purpose of a realistic analysis of the distribution of stress, provided that it is kept in mind that the uniform stresses are statistical averages that do not represent the actual stresses on microscopic parts. The picture of homogeneity of the material can be applied generally in the theories of elasticity and plasticity, and is so applied. This eliminates in the analysis of stresses and deformation the dilemma of discontinuities within microscopic parts; makes it possible to attach a definite meaning to the stress per unit of area on an area of dimensions dz by dy; and makes it possible to analyze the stress by means of calculus and differential equations. For special purposes, such as the exploration of the basic laws of stress and strain, one may return to a picture of the grained structure with random distribution of the axes of the crystals. Also, sometimes the analysis under the assumption of homogeneity and elasticity will indicate a stress concentration with an infinite stress at a point or along a line and with great stresses within a small surrounding space. A sharp reëntrant corner is such a point or line. The stress concentration thus obtained by analysis may indicate failure, or it may suggest a revision in terms of plasticity; but it may call for an interpretation involving the grained structure. Except for such special situations the picture of the material as homogeneous is adequate and dependable. 3. Isotropy Usually one may assume that the orientations of the crystals and grains constituting the actual material have a random distribution. This leads to the assumption that any part of the material that is large enough to contain a considerable number of grains will display the same properties of over-all stress and strain regardless of the directions in which the part has been cut or is loaded. The quality of equal properties

4

THEORY OF ELASTICITY AND PLASTICITY

in all directions is called isotropy, and a material possessing that quality is called isotropic. The equivalent imagined homogeneous material will be isotropic in all parts no matter how small their size. Nonisotropic materials, such as a single crystal or wood, are sometimes called aeolotropic. Unless stated otherwise specifically, the material will be assumed to be isotropic. 4. Theory and Experiments The subjects of elasticity and plasticity are physical and mathematical, experimental and analytical. The basic physical laws of elasticity are simple; they are the linear equations expressing Hooke's law for the various materials. The constants in these equations can be determined by laboratory experiments. A single crystal involves a number of such constants, but when the material can be considered to be isotropic, only two independent constants need be determined, and the required laboratory experiments are simple. With Hooke's law stated in terms of two constants, the analysis by theory of elasticity proceeds by mathematical means: by statics or dynamics and by geometry of continuity. The tools are equations, including differential equations. Under these circumstances the results possess the certainty of the mathematical processes; but even then a verification by experiments is desirable and welcome, both to the person who made the analysis and to those who did not but have some concern with the problem. In other problems of the theory of elasticity it is expedient or necessary, if a solution is to be produced at all, to introduce assumptions or approximations; then certainly accompanying experimental evidence is desired. The experiments may be made with models or on full scale. One may look on the model or the structure itself as a machine that solves its own differential equations; but proper instrumentation is required to bring out the results. A useful experimental method is that of photoelasticity, by which polarized light sent through transparent models produces fringe patterns that can be interpreted as patterns of stress. Experimental elasticity and the theory of elasticity work side by side to produce evidence. The basic laws of plasticity, which are more complex than those of elasticity, have not yet been established in all aspects. Much experimentation and some analysis are still needed to complete the business of establishing these basic laws, including the laws of rupture by separation and of failure by sliding along some surface. From whatever basic laws

SCOPE

S

may be assumed to apply, the theory of plasticity proceeds by mathematical means; but to obtain results at all a code is accepted of permitting much greater freedom of assumption and approximation than in the theory of elasticity. Correspondingly, there is a much greater dependence on accompanying experimental work. Whereas experimental stress analysis has a notable role in connection with the theory of elasticity, it has a vital role in connection with the theory of plasticity. 5. Relation to "Mechanics of Materials," "Structural Mechanics," and "Soil Mechanics" Like the theories of elasticity and plasticity, the subject commonly known as "mechanics of materials" or "resistance of materials" or "strength of materials" deals with stresses and deformations in solids. There is no sharp dividing line between mechanics of materials and the theories of elasticity and plasticity. The distinction is in part only a matter of tradition. A way of explaining the difference is to say that mechanics of materials permits a greater freedom of assumption and approximation than do the theories of elasticity and plasticity; thereby simplicity is gained, but in return there is a greater dependence on experimental verification. Sometimes the needed critical examination is obtained by the more complex but also more accurate processes of the theory of elasticity. For example, the assumption in mechanics of materials that within the limit of elasticity the normal stresses on the cross section of a straight beam are proportional to the distance from a neutral axis is shown in the theory of elasticity to be usually very adequate and nearly correct though not exact; small corrections are evaluated in the theory of elasticity, but the important conclusion is that generally they are small. In its first phase, mechanics of materials deals with such matters as bending of beams and columns and twisting of shafts. It is the natural and usual order to make oneself familiar with this first phase of mechanics of materials before engaging upon the theories of elasticity and plasticity, but this sequence is not necessary, because the theories of elasticity and plasticity begin with fundamentals. After experience with the theories of elasticity and plasticity one may return to the mechanics of materials, where, because of the freedom of the empirical approach, one may tackle problems that defy the more precise processes of the theories of elasticity and plasticity. In its advanced phases, mechanics of materials may emerge as a more advanced subject than the theories of elasticity and plasticity.

6

THEORY OF ELASTICITY AND PLASTICITY

Structural mechanics deals with stresses and deformation in structures, their strength, stiffness, and stability. A structure is usually conceived of as made up of several solid parts—a truss is an example—but often a single object is considered to be a structure. For example, the theory of arches, which might very well be treated in mechanics of materials, is usually assigned to structural mechanics; this is because of the direct applicability of methods developed in structural mechanics. Again, there are no sharp dividing lines with mechanics of materials nor with the subjects of elasticity and plasticity, and there is overlapping. Problems often come to the theory of elasticity from structural mechanics; and some developments in structural mechanics, such as Castigliano's method of minimum internal stress energy, are directly applicable to the theory of elasticity and have become a part of it. The subject of soil mechanics, which has come to be of great practical importance, deals to a considerable extent with the plastic state, but complication is added by the role of moisture under changing stress and by the geological aspects. The subject is experimental and theoretical. Often one phase of a problem of soil mechanics is a problem of elasticity. Some of the theory of soil mechanics is theory of plasticity. Some experiments and observations in soil mechanics may prove to be of value to the theory of plasticity of other substances. 6. One-, Two-, and Three-Dimensional Problems of Elasticity and Plasticity In some aspects the problem of the bending of beams is one-dimensional : significant quantities such as the deflection, the curvature, and the bending moment are functions of one independent variable, the distance measured along the beam; but a beam is a three-dimensional object, and the complete problem of stress and strain in beams is three-dimensional. Euler founded the one-dimensional theory of the bending of beams in the eighteenth century. The one-dimensional problems are now usually considered to belong to mechanics of materials. Similarly, in significant aspects the problem of the bending of plates or slabs is two-dimensional: deflections, curvatures, and twists are functions of two coordinates in the plane of the plate. Lagrange founded this branch of the theory of elasticity in 1811 by stating the partial differential equation that governs the deflections. Still, when all possible stresses in the plate or slab are included for consideration, the problem is three-dimensional. The problem of torsion of a straight bar with con-

SCOPE

7

stant, noncircular cross section, solved basically by Saint-Venant at the middle of the nineteenth century, begins as a three-dimensional problem but is reduced to a two-dimensional problem of determining the warping of the cross section and the accompanying shearing stresses. Two-dimensional problems of various kinds receive much attention in the theories of elasticity and plasticity, if for no other reason, because they are more accessible to complete analysis than the three-dimensional ones; the latter must often be left to the approximate or empirical methods of mechanics of materials if an answer is to be produced at all. The fundamentals, however, of both the theory of elasticity and the theory of plasticity must necessarily be developed in terms of three dimensions. This was achieved for the theory of elasticity in the 1820's in France by Navier, Cauchy, and Poisson, who are thus the founders of the theory of elasticity. A number of specific problems of three dimensions that cannot be reduced to two have been solved in the course of the years and are now a part of the theory of elasticity. Three-dimensional theory of plasticity is still in the process of being founded. The fundamentals are three-dimensional but the case studies are still mostly twodimensional.

CHAPTER

II

Historical Notes 7. Works on the History Extensive historical information about the development of the theory of elasticity and mechanics of materials is found in the following works: "Historique abrégé des Recherches sur la résistance et sur l'élasticité des corps solides," pp. xc-cccxi of the notes by Barré de Saint-Venant in his annotated third edition (Paris, 1864) of the section "De la résistance des corps solides" of Navier's Résumé des leçons données à l'École des Ponts et Chaussées sur l'application de la mécanique à l'établissement des constructions et des machines. The preceding pages xxxix-lxxxiii are on the life and works of Navier. Isaac Todhunter and Karl Pearson, A history of the theory of elasticity and of the strength of materials (Cambridge University Press, Cambridge; vol. 1,1886, 936 pp.; vol. 2,1893,1320 pp.). The existence of this monumental work is amazing evidence of previous vigorous growth of a specialized subject which has kept its vitality through a long period afterward. A. E. H. Love, The mathematical theory of elasticity (Cambridge University Press, Cambridge; ed. 1,1892,1893; ed. 4,1927, 643 pp.), especially the " Historical introduction," pp. 1-31. This treatise has proved its lasting value. Encyklopädie der mathematischen Wissenschaften (Teubner, Leipzig), vol. 4, subvol. 4 (1907-1914), "Elastizitäts- und Festigkeitslehre," pp. 1-770. This is particularly valuable as a work of reference. Hans Lorenz, Technische Elastizitätslehre (Oldenbourg, Munich and Berlin, 1913) ; the last chapter, pp. 644-683, is on the history. F. Auerbach and W. Hort, eds., Handbuch der physikalischen und technischen Mechanik (Barth, Leipzig), vol. 3 (1927), vol. 4, subvols. 1 and 2 (1931). As a work of reference this handbook supplements the Encyklopädie by virtue of its later dates of publication.

HISTORICAL NOTES

9

A brief account is given in a paper by H. M. Westergaard, "One hundred fifty years advance in structural analysis," Trans. Am. Soc. Civil Engrs. 94, 226-240 (1930). A summary of the history of the theory of plates is given in a paper by H. M. Westergaard and W. A. Slater, "Moments and stresses in slabs," Proc. Am. Concrete Inst. 17, 415-538 (1921), especially pp. 417-423. The notes that follow are intended only as a brief sketch. 8. Origins, 1636-1820 None less than Galileo Galilei (1564-1642), founder of modern science, originated mechanics of materials in an item of his work. In his last publication, Two new sciences, completed in 1636 and published in 1638, he included a discussion of the failure of a beam cantilevered from a wall by rupture in the section next to the wall. Galileo conceived of the material as rigid. His assumption of compression concentrated at the lower edge of the section of rupture and tension distributed uniformly over that section does not agree with Hooke's law but would be correct under some assumed law for a plastic stage. He arrived at correct ratios of strength for similar cross sections and derived the design of a cantilever of uniform strength. Robert Hooke (1635-1702) announced his law of proportionality of deformation and force in 1678. In France Edme Mariotte arrived at the same law, experimented with beams in 1680, and located the neutral axis between stretching and shortening at the middle of the beam. The assumption that a plane cross section of a beam remains plane during bending is sometimes named after James (or Jacques) Bernouilli (1654-1705); he introduced the problem of determining the curve into which a beam bends, the elastic curve. Leonhard Euler (1707-1783) used a principle of least action for the bending of beams, found elastic curves in a number of problems, and derived his famous formula for the axial load causing a slender column to buckle. He published these analyses in an appendix, "De curvis elasticis," to a mathematical work on curves possessing properties of maximum or minimum (1744), and in a paper, "Sur la force des Colonnes" (1757), published by the Berlin Academy in 1759. In these studies Hooke's law appears in the form that the bending couple is equal to the product of the curvature and a "moment of stiffness" (this quantity would now be stated as the product of the modulus of elasticity and the moment of inertia of the cross section about the neutral axis). Euler suggested that the moment of stiffness could be determined experimentally by supporting the beam

10

THEORY OF ELASTICITY AND PLASTICITY

or column as a cantilever and loading it at the end, in which case the deflection is expressed by a simple formula. On a later occasion (1779) Euler developed the theory of transverse vibrations of beams. The items contributed by Euler to the origins of the theory of elasticity are of lasting importance, even though they are but a small and incidental part of his whole work. Among the contributions to the subject in the eighteenth century, besides Euler's, those of Charles Augustin Coulomb (1736-1806) are the most important. Coulomb began his career as a military engineer, and his interest in the subject of stress, strength, and deformation was direct and practical as well as scientific. The volume for 1773 (published 1776) of Mémoires de Mathématique et de Physique, Présentés à l'Académie Royale des Sciences, par divers Savans, et lûs dans ses Assemblées, contains on pages 343-382 a paper by Coulomb entitled "Essai sur une application des règles de maximis et minimis à quelques problèmes de statique, relatifs à l'architecture." This paper presents for the first time a completely adequate analysis of the fiber stresses in a beam with rectangular cross section; Hooke's law is assumed for the fibers; the equilibrium of forces on the cross section and external forces is established; the neutral axis is placed correctly by correct reasoning; and the stresses are evaluated. Coulomb noted that at rupture under some circumstances the neutral axis may be at a different place, thus indicating thought of the plastic stage. Considering failure, he introduced the conception of sliding by shearing deformation. And he presented his theory of earth pressure on a retaining wall, according to which a wedge of earth slides when the friction and cohesion in a plane section become insufficient. In a later paper (1784) Coulomb originated the theory of torsion of wires and shafts. Coulomb's paper of 1773 is of extraordinary merit. Unfortunately, his theory of bending of beams did not exert all the influence that it deserved. In spite of occasional experimental work, attempts by others to present a theory of bending of beams remained confused, until Navier brought order to mechanics of materials in the 1820's. Coulomb and Navier are the founders of mechanics of materials. Thomas Young (1773-1829), naturalist, physician, and philologist, in his Course of lectures on natural philosophy and the mechanical arts (London, 1807), introduced the modulus of elasticity. This is an elastic property of the material itself and not merely a constant applicable to an elastic solid of particular dimensions, such as Euler's moment of stiffness of a beam. The "modulus" defined by Young is a certain column of

HISTORICAL NOTES

11

the material in question, but his conception has afterward been restated in a more convenient form with the result that the length of the column multiplied by the weight of the material per unit volume is what is now known as the modulus of elasticity or Young's modulus. In the same book Young introduced the idea of the shearing strain or detrusion as an elastic deformation; Coulomb had considered it as a deformation at failure. In 1811, incidentally to the judging of a paper submitted for a prize, Joseph Louis Lagrange (1736-1813) stated the partial differential equation of fourth order, named after him, that governs the bending, including vibration, of plates or slabs. The paper, which dealt with "elastic surfaces," was by Sophie Germain; she corrected it in accordance with Lagrange's criticism, and won the prize in 1815; her paper was published in 1821. The interest in vibrations of plates had been aroused by Chladni's spectacular experiments (1787) in which "nodal figures" were produced by sand on the plate. In 1816 David Brewster (1781-1868) published in the Philosophical Transactions of the Royal Society of London (1816), pp. 156-178, his discovery that deformations produce double refraction in glass. By transmitting polarized light through glass plates subjected to forces he obtained patterns of colored fringes. He stated that this method can be used to measure forces throughout the glass as well as external forces and suggested that it be used for the study of arches and other structures. He thus laid the foundation for the modern experimental method of analysis of stresses by photoelasticity. 9. Period of the Founding of the Three-Dimensional Theory of Elasticity, 1820-1830 As has been mentioned, Coulomb and Navier founded mechanics of materials—Coulomb through his paper of 1773, and Louis Marie Henri Navier (1785-1836) through his lectures published in 1826 as Résumé des leçons données à l'École des Ponts et Chaussées sur l'application de la mécanique à l'établissement des constructions et des machines. This book by Navier is the first great textbook on mechanics of engineering. It exerted widespread influence. Navier published a second, revised edition in 1833. Especially important is the treatment of the bending of beams with any cross section. Navier used Bernouilli's assumption, now often called Navier's hypothesis, that a plane cross section of the beam remains plane during bending. He referred extensively to experimental results

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available at the time, as is appropriate in mechanics of materials. His analysis included various problems of static indeterminacy. As indicated before, the founders of the three-dimensional theory of elasticity are Navier and the two eminent mathematicians Augustin Louis Cauchy (1789-1857), and Siméon Denis Poisson (1781-1840). The pertinent papers were presented to the Paris Academy by Navier in 1821, by Cauchy in 1822, and by Poisson in 1828. Cauchy's most important publications on the subject are dated 1827 and 1828. Poisson's paper was published by the Paris Academy in 1829 (Mémoires 8, 357-570, 623-^627). Interest in the constitution of matter expressed in terms of forces acting between molecules or material points played a major role in these papers, as did interest in vibrations and optics. The assumptions and arguments made then about the intermolecular forces have since been replaced or discarded. Cauchy, however, derived in 1822 the basic differential equations used today for displacements in an isotropic material. These equations contain two constants of elasticity which can be stated in terms of Young's modulus and the quantity now known as Poisson's ratio, which is the ratio of transverse contraction to longitudinal extension per unit length under simple unidirectional tension. Navier's corresponding equations contain only one constant of elasticity. Poisson concluded from analysis of intermolecular forces that the ratio now named after him should be This value is within the range of possibilities, but for steel, for example, the true value is close to 0.3. Cauchy also considered nonisotropic materials and showed the possibility of 21 constants of elasticity for a crystal; he reduced the assumed maximum number to 15 by premises that have not been found tenable. Poisson added to the discussion of the fundamentals and gave the solutions of a number of specific problems. One of his contributions is the analysis of longitudinal pressure waves and transverse shear waves; Ke derived the ratio of the velocity of the former to that of the latter as V3, which is correct when Poisson's ratio is J. In the list of founders of the theory of elasticity one may well include the name of the distinguished English mathematician George Green (1793-1841) who, in a paper on optics dated 1837 and published in 1839, brought new clarity to the question of the possible 21 constants of elasticity in a crystal, and two in an isotropic material, by expressing the elastic energy per unit volume as a quadratic function of the deformations. It should be added that Navier and especially Poisson made notable contributions also to the two-dimensional theory of bending of elastic

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plates, for which Lagrange had furnished the governing equation in 1811. Navier, in a paper submitted to the Paris Academy in 1820, solved Lagrange's equation for a rectangular plate with simply supported edges. Poisson included studies of the bending and vibrations of plates in his paper of 1828-29. Here he published the first adequate derivation of Lagrange's equation for the bending of plates (Lagrange had merely stated it). He also solved problems of the vibration of circular plates and obtained a satisfactory agreement with the nodes found experimentally by Savart—an early example of the relation of the theory of elasticity to experimentation.

10. Period of the Classics, 1830-1900 After having been founded in France, the theory of elasticity advanced there and in England in this period; advances were also made in Germany and in Italy. The progress in France is identified in particular with the names of Gabriel Lamé (1795-1870), Barré de Saint-Venant (1797-1886), and Joseph Boussinesq (1842-1929). About 1828 Lamé and Clapeyron submitted a paper on elasticity to the Paris Academy. It was published in 1831 in Crelles Journal and in 1833 by the Academy in Mémoires présentés par divers savans (vol. 4). In this paper the general differential equations are derived again in the manner of Navier by not entirely tenable considerations of intermolecular forces; the ellipsoid known as Lamé's stress ellipsoid, which defines all vector stresses on sections at a point as radius vectors from the center, is established, together with the ellipsoid or hyperboloid known as Lamé's stress-director surface, which defines the directions of stresses at a point and the sections on which they act as the directions of radius vectors and tangential planes respectively; the theory is applied to a series of simple cases and includes the derivation of the formulas known as Lamé's for stresses in hollow cylinders and spheres; finally, some solutions of more general type are given. In this paper the material is assumed to be isotropic, with only one constant of elasticity, corresponding to the special value | of Poisson's ratio. In his later work Lamé correctly established two constants of elasticity; in the form in which they appear together in the equations they are now known as Lamé's constants. Especially important in this later work are the book on elasticity that he published in 1852, Leçons sur la théorie mathématique de l'élasticité des corps solides, 335 pages, and a book on curvilinear coordinates published in 1859, which includes applications to elasticity. An example of the

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notable features of the book on elasticity is the use of potentials to obtain certain types of solution. In this book Lamé credits to Clapeyron the theorem of equality of external and internal work. The theorem is stated in an equation in which the internal work is expressed in terms of the stresses. Clapeyron is remembered also for his "Theorem of three moments," Comptes Rendus 45, 1076 (1857), for analysis of continuous beams. The works of Barré de Saint-Venant are eminent among the classics of the theory of elasticity. Todhunter and Pearson devote 326 pages of their History of the theory of elasticity and of the strength of materials to his works, and volume 1 of this great treatise is dedicated "To the memory of M. Barré de Saint-Venant, the foremost of modern elasticians." His most important single contribution is his general solution of the problem of torsion of straight bars with constant noncircular cross section, with exact solutions of many particular problems. With characteristic consideration for those who would use his results he solved these problems with a completeness that included many numerical coefficients and graphical representations. The paper in which he presented this analysis of torsion deals also with the flexure of beams and with the general theory of elasticity. It was presented at a meeting of the Paris Academy in 1853 and published by the Academy in Mémoires des savants étrangers 14, 233-560 (1855), under the title "Mémoire sur la torsion des prismes," with the explanatory addition to the title, "avec des considérations sur leur flexion, ainsi que sur l'équilibre intérieur des solides élastiques en général, et des formules pratiques pour le calcul de leur résistance à divers efforts s'exerçant simultanément." Navier had made the mistake of extending to the problem of torsion the assumption that the plane cross section remains plane, but Saint-Venant analyzed the warping of the cross section and established its importance. His analysis of the bending of beams in this paper and in a subsequent 101page paper published in 1856 (in the Journal de Mathématiques de Liouvülé) is also important; here the most significant result is that the corrections of Navier's analysis are generally minor, so that they may usually be ignored. Saint-Venant advocated the theory that the maximum elongation or shortening is the proper criterion of nearness of failure, on which computations of strength should be based; Lamé and Clapeyron had taken for granted that the greatest tensile or compressive stress determined the nearness of failure; Coulomb had assumed the greatest shearing strength,

HISTORICAL NOTES

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sometimes supplemented by internal friction, to govern. Of these three hypotheses that of Coulomb comes closest to the truth in defining conditions of plastic deformation in ductile materials, but a settlement of this question was not approached until the twentieth century. In papers published in 1870 and 1871 Saint-Venant adopted a view in accordance with Coulomb to which experiments in the 1860's by Tresca with many materials had given new support: he assumed that plastic flow occurs under a constant shearing stress, and he founded in these papers the mathematical theory of plasticity in two dimensions; these papers were published in Comptes Rendus 70 (1870) and Journal de Mathématiques 16 (1871). His disciple Maurice Lévy added a set of basic equations for three dimensions (1870, 1871). Like other productive scholars in the theory of elasticity in the nineteenth century, Saint-Venant concerned himself with intermolecular forces and elasticity of nonisotropic materials. His work on elastodynamics included studies of impact. Saint-Venant is one of the examples in the history of science of the scholar who combines power for original work with a feeling for the heritage of the past and capacity to attract disciples of outstanding quality. As a historian of his science he called attention to the merit of Coulomb's paper of 1773. He expressed his respect for Navier by publishing in 1864 a third, annotated edition of Navier's Leçons sur l'application de la mécanique, 1st part, "De la résistance des corps solides." The extensive notes and additions make this an important new work of 1164 pages, including a valuable new historical section of 221 pages. In 1862, the German mathematician Alfred Clebsch (1833-1872) had published a 424-page book, Theorie der Elasticität fester Körper, which includes consideration of Saint-Venant's works on torsion and bending. Saint-Venant and his disciple Flamant translated this book, and Saint-Venant added extensive notes that more than doubled the content. The annotated French edition of Clebsch's book was published in 1883, and this was Saint-Venant's last major work. Among the disciples of Saint-Venant the one who contributed most to the theory of elasticity is Boussinesq. In the field of elasticity he is best known for his analysis of stresses and deformations in a large solid loaded on an extended plane surface. He also published papers on the theory of earth pressures and on plasticity. His field of research included several other branches of mechanics and mathematical physics. In Great Britain William John Macquorn Rankine (1820-1872), author of several books on engineering and of many papers including a number

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dealing with elasticity, wrote the first great textbook in the English language on mechanics of engineering including mechanics of materials; it is his Mantidi of applied mechanics, of which the first edition was published in 1858, the twentieth in 1919. The first edition of this book presents for the first time the approximate determination, now current, of the shearing stresses in beams. The empirical formula for the strength of columns, named after Rankine though credited by him to Tredgold (1788-1829) as originator and to Gordon for having revived it, has served well, even though it is not the best formula for columns. The fair applicability of the formula is explained by the inevitable imperfections of the material, of the shape, and of the manner of loading, and by the role of the plastic stage in shorter columns. It was Rankine who introduced and established the clear distinction between strain and stress as technical terms: strain being always proportional to deformation, and stress to force. In 1862 George Biddell Airy (1801-1892), the Astronomer Royal at Greenwich Observatory, introduced a new idea the importance of which was not realized fully until much later; it is that of a "stress function." A genuine stress function, such as the one that Airy established, defines stresses in equilibrium by its values and derivatives regardless of requirements of deformations; such requirements can be imposed in a second step by specifying conditions that the function must satisfy. Airy's function defines stresses in a two-dimensional state by its second derivatives with respect to the coordinates χ and y. Airy's paper on the subject, which was published in 1863 by the British Association for the Advancement of Science, was criticized because of the neglect of the matter of compatibility of deformations. But the very fact that an Airy function defines stresses in equilibrium independently of deformations makes it applicable not only to the elastic but also to the plastic stage. In 1870 James Clerk Maxwell extended the idea to three dimensions by combining three Airy functions of the three coordinates x, y, and z. And in one of his papers of 1871 Saint-Venant actually used an Airy function for the plastic stage, and he stated the differential equation that governs the function under the assumed circumstances. He credited the device of the stress function, which he calls an "auxiliary unknown quantity," to Maurice Lévy, who had used it in the study of earth pressures in a paper awaiting publication. The Airy function is now used widely for the elastic stage and occasionally for the plastic stage. In the course of the years other stress functions, defining stresses directly regardless of elasticity or plasticity,

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have been established and serve various purposes. Functions of a different type which define deformations directly and stresses only indirectly through Hooke's law are frequently also called stress functions, but it is preferable to call them strain functions. This usage is adopted here, and the name "stress function" is reserved for those which like Airy's define stresses directly. James Clerk Maxwell (1831-1879), most famous for his electromagnetic theory, had published a paper on elasticity of no small merit as early as 1853, in the Transactions of the Royal Society of Edinburgh 20, 87-120 (1853); this paper includes an account of photoelastic experiments and a discussion of the interpretation of the fringes obtained with circularly polarized light. In 1864 Maxwell made three important and fundamental contributions to structural mechanics in two papers in the Philosophical Magazine [J¡\ 27,250,294 (1864) : the diagram of stresses in trusses and the theorem of reciprocal deflections, both named after him, and the first systematic method of analysis of statically indeterminate structures. It is not strange that new developments in the theory of elasticity should be found in the treatise Natural philosophy (first edition, 1867; second 1879 and 1883) by William Thomson—later Lord Kelvin (18241907)—and Peter Guthrie Tait (1831-1901); in Theory of sound (first edition, 1877-78) by Lord Rayleigh (1842-1919) ; and in papers by Kelvin and by Rayleigh. For example, in Natural philosophy Thomson and Tait settled a controversy of ideas of Poisson and Kirchhoff on the conditions at the edge of a plate or slab; established a hydrodynamic analogy for torsion; and presented important applications (by Kelvin) of the theory of elasticity to problems of geophysics, including the problem of tides in the solid earth and the rigidity of the earth. "Rayleigh's method" of approximate analysis of vibrations of solids by consideration of energies is used widely. Rayleigh discovered by theory the surface waves in elastic solids that are now called Rayleigh waves. He published this theory in the Proceedings of the London Mathematical Society 17 (1887), and he forecast the importance of these waves in seismology. In 1891 C. A. Carus Wilson, then at McGill University in Montreal, published a study by photoelasticity of stresses in beams under concentrated loads, in the Philosophical Magazine [5] 32,481-503 (1891). During the 75 years since 1816, when Brewster published his basic discovery, optical research had clarified the governing physical laws. The foundations had been laid for a development of photoelasticity as a method of

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THEORY OF ELASTICITY AND PLASTICITY

determining stresses in problems of interest by precise measurements. The work of Carus Wilson marks one of the beginnings of this phase. In Italy Alberto Castigliano (1847-1884) developed in the 1870's the method named after him of minimum internal energy expressed in terms of stresses, the "stress energy" (the stress energy is distinguished here from the "strain energy," which is expressed in terms of deformations). The method is of primary importance in the analysis of statically indeterminate structures and has become an important tool in the theory of elasticity. Castigliano presented the method first in a thesis in Torino in 1873, thereafter in papers in 1875 and 1876, and finally in a notable book, Théorie de l'équilibre des systèmes élastiques et ses applications (Turin, 1879). Also in Italy, E. Betti published in 1872 a generalized form of Maxwell's theorem of reciprocal deflections; in the more general form it is often called Maxwell's and Betti's theorem. Whereas Kelvin had previously solved the problem of a concentrated force applied at a point of the interior of a large elastic solid (1848), and Boussinesq had solved the corresponding problem of a normal force at a point of an extended plane surface of a large solid (about 1878), V. Cerniti in Italy was the first to solve the corresponding problem of a tangential force at a point of the plane surface (1882). The three problems are now called Kelvin's, Boussinesq's, and Cerruti's problems, respectively. In 1892 Beltrami stated six general equations of compatibility that stresses in an isotropic elastic solid must satisfy in addition to the three equations of equilibrium. Structural mechanics was advanced greatly and systematically in Switzerland by C. Culmann (1821-1881) and later by W. Ritter; and in Germany by Otto Möhr (1835-1918), Heinrich Müller-Breslau (18511925), and F. Engesser. One of Mohr's contributions is known as "Mohr's circles"; it is a graphical representation of a uniform three-dimensional state of stress, which he used in 1900, and which many have used after him, to interpret the circumstances of yielding and rupture. There was also considerable activity in experimental work. Particularly important are A. Wöhler's experiments, conducted over the twelve-year period 1858-1870, on fatigue of metals; J. Bauschinger's explorations of loading into the plastic stage, unloading or reversal of loading, and reloading; tests of columns by Bauschinger and especially by L. von Tetmajer; and tests, continued into the twentieth century, of materials, structural parts, and plates and slabs by C. Bach. Lectures on the theory of elasticity given at different times from 1857 to 1874 by the physicist, mineralogist,

HISTORICAL NOTES

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and mathematician Franz Neumann (1798-1895) contributed especially to the theory of crystals and exerted influence; the lectures were published in a 374-page book in 1885. By painstaking experiments with specimens made of single crystals the physicist W. Voigt determined the sets of elastic constants of crystals of various materials (these investigations were published in 1887-1889 in Wiedmanns Annalender Physik und Chemie). His results prove that the premises from which Cauchy had derived the relations that would reduce the maximum number of constants from 21 to 15 are not valid, and he thereby settled a question that had been controversial for a long time. The physicist Heinrich Hertz (1857-1894), famous for his experiments with electromagnetic waves, made important contributions to the theory of elasticity, especially (in 1881 and 1882) his theory of pressures by contact between two solids, of impact, and of hardness. The buckling of a column under axial load is the basic case of elastic instability, but other forms attracted attention. In his Cours de mécanique appliquée (1859-1866), Bresse analyzed the buckling of a circular or nearly circular tube under external pressure by use of the theory of curved beams. In a paper published in the Proceedings of the Cambridge Philosophical Society in 1889 (pp. 287-292), G. H. Bryan dealt with the same problem by consideration of energy; and he added later a study of the buckling of plates in the Proceedings of the London Mathematical Society 22, 54-67 (1890-91). The simple experiment of twisting a short piece of thick string suggests that a column will buckle under a reduced axial load when it is also subjected to torsion. A. G. Greenhill solved this problem for shafts in a paper that he published in the Proceedings of the Institution of Mechanical Engineers (1883), pp. 182-209. In Germany, F. Engesser dealt with lateral instability of trusses in 1893; and in his early work Kipp-Erscheinungen (75 pp.), dated 1899, Ludwig Prandtl analyzed the tilting of beams by torsional instability. The necking— localized contraction—of a bar of ductile metal before it breaks in tension must have been observed since the earliest days of testing. This is an example of plastic instability. Closely related to problems of elastic instability are those of increased stability under load—as exemplified by a long beam in tension or a tube with internal pressure. An important example is the stiffness of a suspension bridge by virtue of its weight. Joseph Melan published a solution of this problem in 1888, in Eandbuch der Ingenieurwissenschaften (Leipzig, 1888), vol. 2, "Brückenbau," subvol. 4, pp. 1-144, especially pp. 38-42.

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Among the classics of the theory of elasticity one must certainly count two of the works that were mentioned at the beginning of these notes: the History of the theory of elasticity and of the strength of materials by Todhunter and Pearson, and the treatise by Α. Ε. H. Love. The latter should be considered as both a classic and a modern work. 11. Books since 1900 It is suitable to begin the notes on developments in the twentieth century by listing some of the books on the subject that belong to the period and have exerted influence. The coming into existence of these books is no small part of the history of the theories of elasticity and plasticity. The order of listing is chronological, except that books by the same author are listed together. Augustus Edward Hough Love (1863-1940, who began his career at Cambridge University and served as a professor at Oxford University from 1898), The mathematical theory of elasticity (Cambridge University Press; first ed., vol. 1, 1892, vol. 2, 1893; revised and expanded second ed., in one volume, 1906; third ed., 1920; fourth ed., 1927, 643 pp.). The additions and revisions in the third and fourth editions are relatively few. Α. Ε. H. Love, Some problems of geodynamics (Cambridge University Press, 1911,180 pp.). Encyklopädie der mathematischen Wissenschaften, vol. 4, "Mechanik," ed. by Felix Klein and Conrad Müller (Teubner, Leipzig, 1904-1935). Subvolume 4 of vol. 4 is dated 1907-1914; pages 1-770 are under the general heading "Elastizitäts- und Festigkeitslehre." August Föppl (1854-1924), Technische Mechanik (Teubner, Leipzig; vol. 5, 1907, 391 pp.). This volume has the subtitle "Die wichtigsten Lehren der höheren Elastizitätstheorie." August Föppl and (his son) Ludwig Föppl, Drang und Zwang, Eine höhere Festigkeitslehre für Ingenieure (Oldenbourg, Munich and Berlin; 2 vols.; first ed., 1920; second ed., vol. 1, 1924, 359 pp.; vol. 2, 1928, 382 pp.). Hans Lorenz, Technische Elastizitätslehre (Oldenbourg, Munich and Berlin, 1913; 692 pp.). A. Nádai (1883, in the United States since 1929), Die elastischen Platten (Springer, Berlin, 1925; 326 pp.). A. Nádai, Der bildsame Zustand der Werkstoße (Springer, Berlin, 1927; 171 pp.). A revised and enlarged edition in English has the title Plasticity, A mechanics of the plastic state of matter (McGraw-Hill, New York, 1931 ; 349 pp.).

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F. Auerbach and W. Hort, eds., Handbuch der physikalischen und technischen Mechanik (Barth, Leipzig; vol. 3, 1927; vol. 4, subvols. 1 and 2, 1931). E. G. Coker and L. N. G. Filon, Photo-elasticity (Cambridge University Press, Cambridge, 1931; 720 pp.). Stephen Timoshenko (1878T , in the United States since 1922), Theory of elasticity (McGraw-Hill, New York, 1934; 416 pp.). This book is partly based on an earlier book of his (1914) in Russian on the same subject. S. Timoshenko, Theory of elastic stability (McGraw-Hill, New York, 1936; 518 pp.). S. Timoshenko, Theory of plates and shells (McGraw-Hill, New York, 1940; 492 pp.). R. V. Southwell, An introduction to the theory of elasticity for engineers and physicists (Oxford University Press, London, 1936; 509 pp.). Max Mark Frocht, Photoelasticity (Wiley, New York; vol. 1, 1941, 411 pp.; vol. 2, 1948; 505 pp.). I. S. Sokolnikoff, Mathematical theory of elasticity (McGraw-Hill, New York, 1946; 373 pp.). Enrique Butty, Tratado de elasticidad teorico-tecnica {Elastotecniá), vol. 1, "Teoria general. Problemas elásticos planos y espaciales" (Centro Estudiantes de Ingeniería de Buenos Aires, 1946; 1004 pp.). In this work a scholar has drawn on many sources and has produced a distinguished book in Spanish. 12. Trends since 1900 In the twentieth century there has been an increasing interest in the applications of the theories of elasticity and plasticity. The most extensive clientele of these theories is now among civil and mechanical engineers. The books that have just been mentioned have contributed much to make this possible. Testing of materials and structures and experimental stress analysis have undergone great developments. The United States came into prominence, first in experimentation, later in the theories. The comments that follow will be limited to examples that will illustrate the trends, types of activity, and types of ideas. 13. Two Examples of Experimentation Two widely different examples of activity in experimentation will be mentioned. Since 1914 Herbert F. Moore at the University of Illinois and

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his collaborators have conducted extensive systematic experiments on fatigue of metals. This subject is important in itself, and it has a bearing on the question of the extent of applicability of the theory of elasticity. The results have been published mostly in the Bulletins of the University of Illinois Engineering Experiment Station. The second example is the experimentation, since 1904, by the physicist P. W. Bridgman at Harvard University to determine the behavior of substances under extremely high pressures. He has reached pressures corresponding to a depth underground of as much as a quarter of the radius of the earth. Some of his experiments are of fundamental importance for the understanding of the plastic state and of rupture. 14. The Strangely Late Discovery of the Shear Center in Beams One might surmise that all the simple basic laws of mechanics of materials that deal with the bending of beams had been discovered before the end of the nineteenth century. Yet one matter had been overlooked, and the principle involved was not discovered and clarified until the 1920's—which is strangely late. If a channel (a bar with U-shaped cross section) is to be used as a beam with the web vertical (so that the plane of symmetry is horizontal), and if the loads are to be in a vertical plane, the question may arise where this plane should be located if the beam is to bend without twisting. The answer that had been given intuitively many times before was that the plane of the loads should contain the centers of gravity of the cross sections. Actually the plane should be located so as to contain another center which is identified with the cross section and is called the shear center. This center lies at some distance back of the web, on the side opposite that of the center of gravity. The distance from the center of the web to the shear center is a little less than one half of the width of the flange. The strikingly simple explanation is that the plane of loading must contain the resultant of the shearing stresses on the cross section. The position of this resultant can be determined with good approximation by simple computations by rules of mechanics of materials. The precise determination is a problem for the theory of elasticity. The discovery of the shear center and the clarification of the matter are due to two Swiss engineers, A. Eggenschwyler1 and R. Maillart.2 They found the theory confirmed by tests made by 1 A. Eggenschwyler, Schweiz. Bauztg. 76,206, 266 (1920); Eisenbau, 207 (1921); Zentr. Bauverwallung 41, 501 (1921); Bauingenieur 3, 11 (1922). ! R. Maillart, Schweiz. Bauztg. 77, 195 (1921); 83, 109 (1924).

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Bach with steel channels. But one may also perform a convincing experiment with a channel made of cardboard. The theory of the shear center applies to beams of any cross section.

15. Stress Functions, Strain Functions, and Analogies In 1900 the Australian mathematician John Henry Micheli (18631940) published a series of four papers on elasticity which contain many new ideas. One of these papers3 deals with uniform torsion and bending of curved bars with circular center line. In the problem of uniform torsion the shearing stresses on the cross section, which are the important stresses, are required to have as their resultant a single force along the axis of intersection of the cross sections, which is the axis of revolution. Micheli devised a stress function that defines these shearing stresses by simple formulas, established the differential equation that governs the stress function, showed that the stress function must be constant at the edge of the cross section, stated a simple integral through which the stress function defines the resultant shearing force, and showed how the displacement in the direction of the axis may be determined. Having in mind the application to stout helical springs of small pitch, he investigated numerically some cases of a nearly circular cross section and concluded that the maximum stresses would be greater than in the corresponding problem of a straight bar. Furthermore, he gave a solution for a rectangular cross section in terms of an infinite series. For the problem of uniform bending of the curved bars he established a strain function and derived the governing differential equation. In 1903, in Germany, Ludwig Prandtl, whose later work in aerodynamics and hydrodynamics is of great importance, published a short paper on torsion of straight bars with constant cross section.4 He introduced a stress function that might be obtained from Michell's stress function simply by dividing the latter by the square of the radius of the center line of the curved bar and thereafter assuming this radius to be very great. Prandtl, however, invented his stress function independently, and it is proper that the stress function in this form has been named after him. Both Michell's and Prandtl's stress functions may be represented by the surface of a hill with a base of the same shape as the cross section. Prandtl's stress hill has particularly simple properties, which were noted • J. H. Micheli, "The uniform torsion and flexure of incomplete tores, with application to helical springs," Proc. London Math. Soc. 31, 130-146 (1899-1900). * L. Prandtl, "Zur Torsion von prismatischen Stäben," Physik. Ζ. 4,758 (1903).

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by him: the directions of the shearing stresses are defined by the contour lines of the hill; their values are equal to the slopes of the hill; and the twisting couple is twice the volume of the hill. Prandtl showed that when the material of the bar is isotropic and elastic, and when the vertical scale of the hill is chosen small, the shape of the hill may be produced by inflating a soap film—or membrane with uniform tension—from a hole shaped like the cross section by a uniform pressure from below. The explanation is that the stress function and the deflection of the film obey the same differential equation. Here is an example of a stress function and a corresponding analogy. (The hydrodynamic analogy for torsion introduced by Kelvin and Tait has been mentioned previously.) In general, Prandtl's stress function leads to simpler solutions than Saint-Venant's method. The stress hill and the film have helped the visualization; and films and membranes have been found useful both for demonstrations and for measurements. In the course of the years other useful analogies have been added, and new stress functions have been devised. Thus in 1923 A. Nádai 5 introduced the "sand-hill analogy" and the "roof-and-membrane analogy" for torsion of bars in the plastic stage. The stress function involved is still Prandtl's, which, as a genuine stress function, defines stresses in equilibrium and applies in the plastic as well as the elastic range. The constant slopes of the sand hill represent yielding under a constant shearing stress. The roof under consideration also has constant slope, so that it will have the same shape as the sand hill, except that the vertical scale is required to be small enough to make the roof rather flat. There must be no rafters, and the thickness of the roof is constant. What would be the floor of the attic is shaped like the cross section of the bar. The floor is replaced by a membrane with the same tension in all horizontal directions. The membrane is inflated by air pressure from below. When the pressure is still small, the membrane will have no contact with the roof and will be shaped according to Prandtl's analogy; this represents the elastic stage. Under higher pressure some part or parts of the membrane will assume contact with the roof, and as the pressure increases further, the areas of contact will grow in size; these areas represent the beginning and progress of plastic yielding under a constant shearing stress, while the remaining areas of no contact represent the parts of the bar that are still in the elastic range. 6 A. Nádai, "Der Beginn des Fliessvorganges in einem tordierten Stab," Z. atigew. Math. u. Mech. 3, 442 (1923).

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In his paper on uniform torsion and bending of curved bars J. H. Micheli included a brief section on the torsion of a circular shaft with variable diameter.6 The interesting stresses in such a shaft—or solid of revolution —are the shearing stresses on the meridian section. Micheli showed that his stress function for uniform torsion of curved bars also applies here, with the only difference that a constant term in the governing equation vanishes and that the stress function has a constant value—equal to the twisting moment divided by 2 π—at the outer boundary and is zero at the axis (or at the inner boundary if the shaft is hollow). In 1905 A. Föppl 7 presented a solution of the same problem in terms of strain functions, and he added considerations of an analogy with flow. In 1925 L. S. Jacobsen 8 showed that when the strain function is stated as the angle of twist at each point it can be produced experimentally as an electric potential in a conductor shaped like a hollow-ground razor blade, the area of which covers one half of the meridian section of the shaft; the sharp edge is along the axis of the shaft; the thick edge follows the meridian curve; the thickness is proportional to the cube of the distance from the sharp edge; and the current is in the direction of the length of the blade. Jacobsen used this electrical analogy to determine by measurements the stress concentrations at the transition from a smaller to a larger diameter of a shaft for various values of the radius of the transition fillet and the two diameters of the shaft. Like most of the other analogies that can be used in experiments, this one serves also the important purpose of visualization of solutions and of aid to intuition. Besides the stress function and soap-film or membrane analogy for the torsion of bars with constant cross section, there have also been devised a stress function and a membrane analogy for the bending of beams. The beam under consideration is loaded and supported at the ends only. The load may include a twisting couple and does include a transverse force. The stresses are assumed to be within the elastic range. Saint-Venant solved the basic problem in his paper on flexure of 1856. He showed that in such a beam, except near the ends where irregularities may occur, the normal stresses will be distributed in accordance with β

Reference 3, pp. 141-142. A. Foppl, "Ueber die Torsion von messer," Sitzber. math.-physik. Klasse (1905). 8 L. S. Jacobsen, "Torsional-stress section and variable diameter," Trans. 7

26).

runden Stäben mit veränderlichem DurchAkad. Wiss. München 35, 249-262, 504 concentrations in shafts of circular crossAm. Soc. Mech. Engrs. 47, 619-638 (1925-

26

THEORY OF ELASTICITY AND PLASTICITY

Navier's hypothesis; they will be proportional to the distance from some neutral axis, and their computation involves no difficulty. The remaining significant stresses are the shearing stresses on the cross section, and it is these that the stress function defines. The stress function may take various forms, but each form is easily modified into the others. The idea was developed, evidently independently, three times: by Vening Meinesz in the Netherlands, 1911; by S. Timoshenko, then in Russia, 1913; and by A. A. Griffith and G. I. Taylor in England, 1917-18.® The surface representing the stress function is produced in Timoshenko's variant of the method by deflecting a membrane from a hole in a plane plate; the hole is shaped like the cross section; and the load is a function of the distance from the plane of the bending moment. In the variant defined by the others the membrane is unloaded but the supporting plate is curved in one direction. Griffith and Taylor used the analogy successfully in extensive experiments. A prismatic bar may be called a beam or a column according to whether it is loaded primarily by transverse forces or by axial forces. Similarly, a flat object bounded by two parallel planes may be called either a slab or a slice. It is useful for the present purposes to adopt the following distinction: a slab is loaded primarily by forces causing bending, whereas a slice is loaded primarily by forces having resultants that are contained in the middle plane; the plane sides of the slice may be purely imagined sections through a solid. The behavior of the slab is expressed by its deflections; and the important stresses in the slice can be stated in terms of an Airy function. In the elastic range the deflections of the slab are governed by Lagrange's equation; and the Airy function is governed by an equation that happens to be of precisely the same form when the slab is loaded at the edges only. Therefore in the governing equation one may replace the deflection of the slab by an Airy function, and vice versa. The result is an analogy of slabs and slices that has proved to be very useful. Wieghardt10 used this analogy in an experimental investigation of stresses in a sharply curved beam with rectangular cross section. Such a beam is a slice in the sense that was indicated; he obtained the results by bending a plate. The analogy was also used in experimentation with rubber slabs, begun in 1930, in studies for the design of Hoover Dam, • See S. Timoshenko, Theory of elasticity (McGraw-Hill, New York, 1934), footnotes, pp. 288 and 301. 10 K. Wieghardt, "Ueber ein neues Verfahren, verwickelte Spannungsverteilungen in elastischen Körpern auf experimentellem Wege zu finden," Mitt. Forschungsarbeit. Gebiete Ingenieurwes. 49, 15-30 (1908).

HISTORICAL NOTES

27

which for a period was called Boulder Dam; this dam, with its unprecedented height of 726 ft, is an important structure.11 The more important usefulness of the analogy, however, does not lie in its applicability to experimentation but in the fact that any analytic solution for slices applies to slabs, and vice versa, slabs and slices being about equally important structural elements. The thinking of the bending of a plate or slab is an aid to the solution of a corresponding problem of a slice, and vice versa. Each of the stress functions that have been mentioned has served in a restricted yet fairly general type of problem, and the analogies have been especially useful as an aid to intuition. So much for stress functions and analogies, but there is more to be said about strain functions. In the second edition of his treatise The mathematical theory of elasticity (1906), Α. Ε. H. Love introduced12 a strain function for solids of revolution in the elastic stage under loads possessing axial symmetry so that all meridian sections remain plane and are stressed and strained alike. Love's strain function differs from Michell's, already mentioned, for uniform bending of curved bars. The displacements appear as combinations of second derivatives of the strain function, and the stresses as combinations of third derivatives. In spite of this complexity, Love's strain function has been found useful: first, in simplifying the derivations in problems already solved, such as Kelvin's of a single force in the interior of an infinite solid, and Boussinesq's of a normal force on the plane surface of a large solid; and second in the solution of new problems. In 1930 B. Galerkin13 in Russia extended the idea of Love's strain function by setting up three strain functions for the general problem of three-dimensional elasticity. If there is symmetry about an axis, two of these functions may be taken as zero, and the third will be Love's strain function. Each of the three functions is required to satisfy a differential equation that is of fourth order but has the merit of containing neither of the other two functions. In subsequent papers Galerkin14 demonstrated 11 United States Department of the Interior, Bureau of Reclamation, Boulder Canyon Project, Final Reports, Part V, Technical Investigations, Bulletin 2, "Slab Analogy Experiments" (1938), 184 pp. » P. 262; ed. 4 (1927), p. 275. 18 B. Galerkin, "Contribution à la solution générale du problème de la théorie de l'élasticité dans le cas de trois dimensions," Comptes Rendus 190, 1047 (1930); and a paper in Russian published in Compt. Rend. Acad. Sci. U.R.S.S. (1930), p. 353. 14 B. Galerkin, Comptes Rendus 193, 568 (1931); 194, 1440 (1932); 195, 858 (1932); and several papers in Russian.

28

THEORY

OF E L A S T I C I T Y

AND

PLASTICITY

the applicability of his method. It happens that Galerkin's three strain functions lend themselves to interpretation as three components of a vector: 16 if the directions of the three coordinates x, y, and ζ are changed but the vector as a function of the position of the point in space is left unchanged, the three functions will be changed so as to represent three new components of the vector, but the state of stress and strain defined by the functions remains the same. I t is appropriate, therefore, to speak of the "Galerkin vector," which is the vector function having Galerkin's three functions as components. Moreover, the idea of the Galerkin vector can very well be used in an «-dimensional space,16 if anyone wishes to go beyond the cases of η — 2 or 3, as one may for the purpose of obtaining new perspectives into three-dimensional space. In contrast, Maxwell's three stress functions (which are three Airy functions) do not lend themselves to interpretation as components of a vector. In a paper published in 1936 Raymond D. Mindlin 17 (1906), at Columbia University, utilized the Galerkin vector to solve the problem of stresses and displacements produced in a semi-infinite elastic solid (occupying all space on one side of a plane surface) by a single force applied in any direction at an interior point, at some finite distance from the surface. This achievement completes a series of related problems that have been solved; the series can now be stated as Kelvin's, Boussinesq's, Cerruti's, and Mindlin's problems, respectively. All of these problems are of practical interest, not least in soil mechanics.

16. Use of Principles of Minimum Energy Two principles of minimum energy, each of which is the basis of a general method in the theory of elasticity, were known and in use before the end of the nineteenth century, but the applications of them have undergone new developments in the twentieth century. One is the principle of minimum potential energy by variation of shape: if all geometrically possible combinations of deformations are contemplated, the one that makes the potential energy minimum also makes the accompanying stresses statically consistent and thus represents the true solution. The internal potential energy is the strain energy, but to this must be added a Η. M. Westergaard, "General solution of the problem of elastostatics of an «-dimensional homogeneous isotropic solid in an »-dimensional space," Bull. Am. Math. Soc. (October 1935), pp. 695-699. le Reference 15. 17 R. D. Mindlin, "Force at a point in the interior of a semi-infinite solid," Physics 7, 195-202 (1936).

HISTORICAL NOTES

29

the potential energy of the external forces, which may be stated as the negative of the work of the external forces when they are considered to be constant during the change of shape. The other principle is Castigliano's of minimum stress energy by variation of the state of stress: if all statically possible combinations of stresses consistent with the external loads are contemplated, the one that makes the stress energy minimum also makes the accompanying deformations geometrically consistent— establishes geometric continuity—and thus represents the true solution. A paper published in 1908 by the Swiss mathematician Walter Ritz 18 gave new impetus to the application of these principles, especially to obtain approximate solutions. If it can be assumed that a nearly true solution can be expressed in terms of a limited number of chosen functions with a limited number of unknown parameters attached to them, then the pertinent principle of minimum will furnish the most plausible values of the parameters, and in general, with these values of the parameters, the best possible solution in terms of the functions chosen. Since Rayleigh used this idea and Ritz stated it in a more general form, the use of it to obtain approximate solutions is sometimes called the method of Rayleigh and Ritz. Each time a new and useful strain function or stress function is devised, new potentialities arise thereby for use of this method of minimum; for the strain function or the stress function might be stated approximately in terms of a finite number of well-chosen functions with unknown parameters attached; then the potential energy or the stress energy may be expressed in terms of the parameters, and these may be determined so as to produce the minimum. The books by Lorenz (1913), Α. Föppl and L. Föppl (1920, 1924,1928), and Timoshenko (1934, 1936, 1940) Usted in Sec. 11 contain examples of the application of the principles of minimum to problems of elasticity. Beyond the range of Hooke's law, if the stresses can be stated as definite functions of the strains, and vice versa, the principle of minimum by variation of shape can still be applied, provided only that the strain energy be interpreted as the internal work of deformation in terms of strains, regardless of whether it is recoverable energy or not. The statement of the principle of minimum by variation of the state of stress, on the other hand, will require an amendment: instead of the stress energy the quantity required to be a minimum is one called the "complementary energy" or "complementary work." Whereas the work of a force Ρ over 18 W. Ritz, "Ueber eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik," Crelles J. rein. u. an gew. Math. 135,1-61 (1908-09).

30

THEORY OF ELASTICITY AND PLASTICITY

the distance χ is the integral of Ρ dx, the corresponding complementary work is the integral of χ dP, and the internal complementary energy is expressed similarly as the sum of integrals of deformation times increment of stress. Within the range of Hooke's law the complementary energy can be stated as equal to the stress energy; thus the principle of minimum complementary energy includes Castigliano's principle of minimum stress energy as a special case. Engesser19 discovered the principle of minimum complementary work and published a paper ön it in 1889. Since the interest in structural mechanics continued for many years afterward to be centered on the elastic stage, it is not strange that Engesser's paper remained nearly unnoticed for a long time. Yet a principle so fundamental and simple as that of the complementary energy has certainly been rediscovered several times. For example, the present writer arrived at it in 1939, but fortunately found Engesser's paper before publishing his own on the subject.20 At about the same time A. W. Adkins at the Massachusetts Institute of Technology discovered the same principle, used the phrase "complementary work," which corresponds exactly to Engesser's "Ergänzungsarbeit," and presented a paper on the subject, with his own derivation of the principle, at a meeting of the Applied Mechanics Division of the American Society of Mechanical Engineers in 1942. Both the principle of minimum potential energy and the principle of minimum complementary energy can be of service in the solution of some problems of plasticity. However, for general and direct applicability in the theory of plasticity differently stated variational principles are needed. Gustavo Colonnetti,21 P. Hodge and William Prager,22 19 F. Engesser, "Ueber statisch unbestimmte Träger bei beliebigem Formänderungs-Gesetze und über den Satz von der kleinsten Ergänzungsarbeit," Z. Architekt u. Ing.-Ver. Hannover 35 (1889), columns 733-744, especially 738-744. 20 H. M. Westergaard, "On the method of complementary energy, and its application to structures stressed beyond the proportional limit, to buckling and vibrations, and to suspension bridges," Proc. Am. Soc. Civil Engrs. (February 1941), republished in Trans. Am. Soc. Civil Engrs. 107, 765-793 (1942). Engesser's paper is mentioned by M. Grüning in Encyklopädie der mathematischen Wissenschaften, vol. 4, subvol. 4 (1907-1914), pp. 453-464. 21 G. Colonnetti, "De l'équilibre des systèmes élastiques dans lesquels se produisent des déformations plastiques," J. Math. Pures Appliq. [9] 17, 233-255 (1938). 22 P. Hodge and W. Prager, "A variational principle for plastic materials with strain-hardening," J. Math. Phys. 27, pp. 1-10 (1948). Also a paper by W. Prager presented at the Sixth International Congress for Applied Mechanics held in Paris in 1946.

HISTORICAL NOTES 23

H. J. Greenberg, such principles.

31

and others have contributed to the formulation of

17. Photoelasticity The experimental techniques for determining stresses by means of polarized light and the accompanying theories by which to interpret the fringe patterns constitute the field of knowledge now called photoelasticity. Some of the theories involved are part of optics, others are part of the theory of elasticity. Because of the close relation of photoelasticity to the theory of elasticity, some brief comments on its development are in order here. Brewster's discovery, announced in 1816, of double refraction due to stress, Maxwell's early work, dated 1853, and Carus Wilson's use of photoelasticity to measure stresses, published in 1891, have already been mentioned in Sees. 8 and 10. These beginnings were important, but it is since 1900 that photoelasticity has gained most of its momentum. An outstanding twentieth-century representative of the French tradition in the theory of elasticity, structural mechanics, and structural engineering, Augustin Mesnager (1862-1933), gave the first impetus to this development through two papers that he published in 1901 on elasticity and photoelasticity (one through the International Association for Testing Materials, Congress at Budapest, the other in Annales des Ponts et Chaussées, 4th trimester of 1901, pp. 129-190). He returned to the subject in several later publications. In a paper of 191324 he developed a theory of interpretation of the fringe patterns; this theory contains certain theorems now named after him. In England, Ernest George Coker took up photoelasticity and made it his life work. He published many papers on the subject. The treatise Photo-elasticity that he published in 1931 with L. N. G. Filon25 is of great value as a reservoir of clear information about achievements up to that time in theory, technique, and case studies; many of the achievements are due to the two authors. ** H. J. Greenberg, "On the variational principles of plasticity," issued by Graduate Division of Applied Mathematics, Brown University, Providence, R. I., and under auspices of Office of Naval Research, March 1949, 93 pp. typewritten and 5 charts. M A. Mesnager, "Détermination complète, sur un modèle réduit, des tensions qui se produiront dans un ouvrage," Ann. Ponts et Chaussées, M im. et Doc. [5] 16, 133-186 (1913). 85 E. G. Coker and L. N. G. Filon, A treatise on photo-elasticity (Cambridge University Press, Cambridge, 1931).

32

THEORY OF ELASTICITY AND PLASTICITY

In the United States, Max M. Fracht 28 published in 1941 and 1948 a book in two volumes entitled Photoelasticity. His own contributions toward accuracy of technique and accumulation of case studies are substantial. The second volume contains an account of a phase that has been developed after 1930, namely, three-dimensional photoelasticity. Three-dimensional photoelasticity became possible by the invention of plastics in which strains produced by loads can be "frozen"; the threedimensional solid is cut into slices that produce fringe patterns when they are placed in the apparatus for two-dimensional elasticity. The interpretation of these patterns with due consideration of the laws of optics is not simple, Raymond D. Mindlin at Columbia University and D. C. Drucker27 have contributed to the solution of this general problem of interpretation. Photoelasticity has the advantage that the required apparatus is relatively inexpensive. Experiments can be repeated without too much difficulty. The result is that photoelastic laboratories have multiplied, and photoelasticity has served as a means not only of exploration but also of demonstration. The demonstrations are often both beautiful and convincing. 18. Bending of Slabs or Plates Flat elastic solids of constant thickness, extensive in the directions of the flat surfaces and subject to bending, are called plates or slabs, unless they are so thin and flexible that the word membrane is appropriate. Slabs are distinguished from slices: as mentioned in Sec. 15, the word slice is used when the resultant loads are in the middle plane and no bending is involved. When bending is involved, the following classification is useful for purposes of the theory of elasticity: (1) thick slab or thick plate; (2) medium-thick slab or medium-thick plate, or simply slab or plate; (3) thin plate with great deflections, or simply thin plate; and (4) membrane. To explain the distinctions one may say (1) that a slab or plate is called thick if the energy of the transverse shearing stresses is great enough to prevent linear or nearly linear distributions of the normal bending stresses through the thickness at each point. (2) A slab or plate is called medium-thick if the only stress energy that need be considered is that of bending and twisting by stresses that are propor56

M. M. Frocht, Photoelasticity (Wiley, New York; vol. 1, 1941, vol. 2, 1948). D. C. Drucker and R. D. Mindlin, "Stress analysis by three-dimensional photoelastic methods," J. Applied Phys. 11, 724-732 (1940). 27

HISTORICAL NOTES

33

tional to the distance from the middle plane. (3) A plate is called thin if the deflections produce appreciable stresses in the middle surface, thus making it necessary to take into account both the energy of bending and the energy of stretching. (4) In a membrane only the energy of stretching need be considered; the energy of bending is negligible. The analysis is simplest—essentially two-dimensional—for medium-thick slabs and for membranes; the medium-thick slabs satisfy Lagrange's equation. Thick slabs, on the other hand, present complexities of three-dimensionality and thin plates those of nonlinearity of relations of loads, displacements, and stresses. The distinction between the types is not sharp; it is dependent on the accuracy with which one desires to compute. Also, a slab may be considered to be medium-thick within most of its area, but may have to be analyzed as a thick slab in the region around a fairly concentrated load. Some authors, including Α. Ε. H. Love, use the phrase "thin plates" for those that are here classified as medium thick. Theories of thick plates were developed by Saint-Venant (in the annotated translation, mentioned in Sec. 10, of Clebsch's book), J. H. Micheli,28 J. Dougall,29 A. E. H. Love,30 A. Nádai,31 S. Woinowsky-Krieger,32 and others. By the precise theory of thick plates Dougall obtained substantial support for the assertion that the simpler theory—in which the energy of the transverse shearing stresses is ignored and Lagrange's equation governs—gives results of good accuracy for plates of medium thickness except at places where loads are concentrated. This corresponds to SaintVenant's finding that the ordinary theory of the bending of beams as derived in mechanics of materials gives good accuracy for beams of ordinary proportions. Nádai's analysis has been useful for studies of the special conditions in the region of a load that is concentrated over a small area when the plate can be treated otherwise as medium-thick. In 1907, in volume 5 of his Technische Mechanik (pp. 132-144) August Föppl pioneered in the analysis of thin plates with large deflections. 28

J. H. Micheli, "On the direct determination of stress in an elastic solid, with application to the theory of plates," Proc. London Math. Soc. 31, 100-124 (18991900). 2 · J. Dougall, "An analytical theory of the equilibrium of an isotropic elastic plate," Trans. Roy. Soc. Edinburgh 41, 129-228 (1904). 10 A. E. H. Love, The mathematical theory of elasticity, ed. 2 (1906), p. 444; ed. 4 (1927), p. 465. S1 A. Nádai, " Die Biegungsbeanspruchung von Platten durch Einzelkräfte," Schweiz. Bauztg. 76, 257-260 (1920). 32 S. Woinowsky-Krieger, "Der Spannungszustand in dicken elastischen Platten," Ingenieur-Arch. 4, 203-226, 305-331 (1933).

34

THEORY OF ELASTICITY AND PLASTICITY

S. Timoshenko contributed to the subject in 1915 in a paper in Russian, and chapter 9 of his Theory of plates and shells, published in 1940, is devoted to it. Nádai's book of 1925, Die elastischen Platten, contains original contributions to the subject (pp. 284-308). This problem is important in the structural mechanics of aircraft. The remaining comments will be restricted to the theory of mediumthick slabs or plates. For convenience of discussion, the slab will be assumed to be horizontal. The forms of support that are considered most frequently are the "simple support," which prevents deflection at a Une or point of support by vertical reactions only, and the "fixed edge," also often called "clamped edge" or "built-in edge," at which both deflection and rotation are prevented; but elastic supports of various types have also been considered. In the early investigations, such as Poisson's published in 1829, the interest was directed toward deflections and vibrations. Since 1900 there has been an increasing interest in stresses, bending moments, and twisting moments, and in strength and structural safety. This is natural in view of the applications of steel plates in structures including ships, of reinforced-concrete slabs in buildings and bridges, and of plain-concrete slabs in pavements. Investigations that have been carried through to the point of obtaining tables of numerical coefficients or simple formulas for design based on such coefficients are often the ones that have been appreciated most; but it is not forgotten that invention or adaptation of mathematical processes comes first. In order of increasing complexity the problems that have attracted most interest may be listed as follows: (1) Rectangular slab simply supported at three corners and loaded by a single force at the fourth. The problem may be interpreted as one of torsion. Kelvin and Tait in their Natural philosophy (ed. 1, 1867;, dealt with it and used it to elucidate their theorem of conditions at edges. (2) Elliptic slab with fixed edge and under uniform load. The deflection is expressed by a simple polynomial. (3) Circular plate with symmetry of loads and supports about the center. The theory dates back to Poisson in 1829. Kelvin and Tait gave examples in their Natural philosophy (1867). (4) Long (or "infinitely long") slab with two parallel simply supported edges, loaded at a point. This problem appeared to call for infinite series,

HISTORICAL NOTES

35

33

but in 1921 A. Nádai obtained expressions in finite form for the bending moments and twisting moments. (5) Rectangular slab with four simply supported edges and any load. Navier stated the solution of this problem in 1820 in terms of double Fourier series. This problem is a special case of the next, through which simpler computations have been obtained. (6) Rectangular slab with two simply supported parallel edges; either of the remaining edges can be simply supported or fixed or free; the load may have any distribution. Maurice Levy 34 stated in 1899 a solution in general form in terms of single infinite series containing products of trigonometric and hyperbolic functions. E. Estanave applied this method in a dissertation submitted in Paris and dated 1900. Lévy's form of solution has been used by many of the investigators who have produced the valuable numerical results. Outstanding examples are found in the works of H. Hencky 38 (1913), A. Nádai 36 (1915 and later), and B. G. Galerkin37 (1915, 1916, and later). The computations by Galerkin of numerous tables of coefficients for this and other classes of slabs are a notable service. In his paper of 1921 Nádai 33 added a new method of computation by single infinite series, based on his solution for the infinitely long plate under a single concentrated load. (7) Rectangular slab with four fixed edges. This problem is much more difficult than those in the preceding group. When series are used, there is generally a rapidly mounting complexity in the computation of each new term. Yet workable methods have been devised and results have accumulated, with satisfactory agreement between numerical values obtained by different investigators when they dealt with the same problem. The results are due especially to the following investigators: M A. Nádai, "Ueber die Spannungsverteilung in einer durch eine Einzelkraft belasteten rechteckigen Platte," Bauingenieur 2, 11-16 (1921). M M. Lévy, "Sur l'équilibre élastique d'une plaque rectangulaire," Comptes Rendus 129, 535-539 (1899). 54 H. Hencky, Der Spannungszustand in rechteckigen Platten (Munich and Berlin, 1913), 94 pp. ,β Α. Nádai, "Die Formänderungen und die Spannungen von rechteckigen elastischen Platten," Forschungsarbeit. Gebiete Ingenieurwes. 170-171 (1915), 87 pp.; also his book Die elastischen Platten (1925). ST B. G. Galerkin, papers in Russian on "rods and slabs," Vestnik Engeneroß (1915), No. 19, and on "flexure of rectangular slabs and thin walls," Bull. Poly' tech. Inst. Petrograd (1916 and 1918); and a book on "elastic plates" (Moscow, 1933).

36

THEORY OF ELASTICITY AND PLASTICITY

H. Hencky86 (1913), J. G. Boobnov38 (1914), H. Happel39 (1914), A. Nádai36 (1915 and 1925), B. G. Galerkin37 (1915), A. Mesnager « (1916), H. Leitz41 (1917), N. J. Nielsen42 (1920), I. A. Wojtaszak43 (1937), S. Timoshenko44 (1938, 1940), and Dana Young45 (1939, 1940, and a related problem of three clamped edges, 1943). (8) "Flat slab." A reinforced-concrete slab supported directly on column capitals without the assistance of girders is called a flat slab. Like many other problems in structural mechanics, this one begins naturally with some simple considerations of statics. It is to the credit of John R. Nichols46 that he established, in a paper dated 1914, the simple rules of statics that govern the sums of the numerical values of certain positive and negative bending moments; the issue was controversial at the time. A paper by the present writer, with Willis A. Slater,47 published in 1921, in which coefficients of moments were computed by the theory of elasticity and compared with results of many tests that had been made previously, was of some help in establishing rational rules of design. ss

J. G. Boobnov, book in Russian on "theory of structure of ships," St. Petersburg, vol. 2 (1914), p. 465; this reference is quoted from S. Timoshenko, Theory of plates and shells, preface and p. 222. " Hans Happel, "Ueber das Gleichgewicht rechteckiger Platten," Nachr. Kgl. Ges. JFm. Güttingen, Math.-physik. Klasse (1914), pp. 37-62 (problem of concentrated load at center). 40 A. Mesnager, "Moments et flèches des plaques rectangulaires minces, portant une charge uniformément répartie," Ann. Ponts et Chaussées, Partie Tech. (1916), subvol. 3, pp. 313-438. 41 H. Leitz, "Berechnung der eingespannten rechteckigen Platte," Ζ. Math. Physik 64, 262-272 (1917). 42 N. J. Nielsen, Bestemmelse af Spandinger i Plader ved Anvendelse af Differensligninger (Copenhagen, 1920), 232 pp. Difference equations applied to a variety of problems including problems of four fixed edges. 43 1. A. Wojtaszak, "The calculation of maximum deflection, moment, and shear for uniformly loaded rectangular plate with clamped edges," J. Applied Mechanics (Trans. Am. Soc. Mech. Engrs.) 4, A173-A176 (1937). 44 S. Timoshenko, "Bending of rectangular plates with clamped edges," Proc., Fifth Internat. Cong. Applied Mechanics (held at Cambridge, Mass., 1938) (1939), pp. 40-43; and Theory of plates and shells (1940), pp. 222-232. 45 Dana Young, "Clamped rectangular plates with a central concentrated load," J. Applied Mechanics (Trans. Am. Soc. Mech. Engrs.) 6, A114-A116 (1939); "Analysis of Clamped Rectangular Plates," Ibid. 7, A139-A142 (1940); and "Deflections and Moments for Rectangular Plates with Hydrostatic Loading," Ibid. 10, A229-231 (1943). 4 ® J. R. Nichols, "Statical limitations upon the steel requirement in reinforced concrete flat slab floors," Trans. Am. Soc. Civil Engrs. 77, 1670-1681 (1914). 47 Η. M. Westergaard and W. A. Slater, "Moments and stresses in slabs," Proc. Am. Concrete Inst. 17, 415-538 (1921).

HISTORICAL NOTES

37

(9) Slabs of other shapes. The method of difference equations was applied successfully to a variety of problems by N. J. Nielsen.42 Besides this method and the principle of minimum energy of strains and loads, the method proposed and used by Biezeno and Koch48 is suitable for simple as well as complex conditions of the edges. By their method the total area is divided into parts, and4 the load on each part, expressed as an integral, is made to assume the right value by proper combination of functions expressing the deflections. 19. Instability, Buckling, and Stabilization by Load The basic problem of instability by lack of stiffness, or buckling, is the problem of the column. By the end of the nineteenth century the formulas of Rankine and others, which serve in conjunction with Euler's and are represented by a straight line or a parabola in the diagram of load and slenderness, had been compared with results of tests, for example, by A. Ostenfeld49 in Denmark. Furthermore, Engesser,60 Considère,61 and Jasinsky62 had attempted another realistic approach to the problem of columns, namely, by theory in which the stress-strain relations in the plastic range were considered. Research on the strength of columns has been continued in the twentieth century with much activity. In the long history of research on columns three events stand out: first, Euler's original analysis; second, the introduction of Rankine's formula, imperfect though it is; and third, the work, published in 1910, of Theodor von Kármán,63 who was then in Göttingen. Here one finds a combination of consideration of the work of the past, new painstaking significant experimentation, and new theory that accounted well for the observed facts. Some basic ideas of the theory had appeared in the discussions by Engesser, Considère, and Jasinsky, but the development of 48 C. B. Biezeno and J. J. Koch, "Over een nieuwe methode ter berekening van vlakke platen, met toepassing op eenige vor de techniek belangrijke belastingsgevallen," Ingenieur (Netherlands) (13 January 1923), 12 pp. 4e A. Ostenfeld, "Exzentrische und zentrische Knickfestigkeit, mit besonderer Berücksichtigung der für schmiedbares Eisen vorliegenden Versuchsergebnisse," Z. Ver. deul. Ing. (1898), pp. 1462-1470; (1902), pp. 1858-1861. 50 F. Engesser, Ζ. Architek.- u. Ing.-Ver. Hannover 35 (1889), column 455; Schweiz. Bauztg. 26, 24 (1895); and Z. Ver. deuil. Ing. 42, 927 (1898). el A. Considère, Congrès international des procédés de construction (Paris, 1891), p. 371. 6S F. Jasinsky, Schweiz. Bauztg. 26, 172 (1895). 63 T. von Kármán, "Untersuchungen über Knickfestigkeit," Mitteil. Forschungsarbeit. Gebiete Ingenieurwes. 81 (1910), 44 pp.

38

THEORY OF ELASTICITY AND PLASTICITY

the theory by von Kármán was new and far-reaching. The theory took into account the behavior of the material in the plastic stage under reducing stress on the convex side of the deflecting column, and gave due consideration to the effect of eccentricity of the load; eccentricity is difficult to avoid even in the most carefully conducted experiment. As mentioned before (at the end of Sec. 10) other problems of buckling and elastic instability had received attention before the end of the nineteenth century. In 1913 Stephen Timoshenko,64 who had previously published almost exclusively in Russian, made himself known in the Western world by an extensive paper in French. Here he solved a variety of problems of practical importance involving elastic stability or instability or buckling. His work in this field has culminated in his treatise Theory of elastic stability (1936). This field of research has been especially important in the development of structural design of aircraft. The subject of stabilization by load is closely related to that of buckling.56 One need only think of a column under axial and transverse load. An increase of the axial pressure will cause an increase of the deflection, but a reversal, into a tension, will cause a reduction, a stabilization by virtue of the load; the same formulas continue to apply. This relation of buckling to stabilization becomes particularly clear when one considers not only the smallest load producing buckling but also the additional modes of buckling, possible though unstable, under a series of greater loads. The whole sequence of modes of buckling plays a role in the explanation of stabilization by reversal of the load causing buckling. The most spectacular example of stabilization by load is the stiffening of a suspension bridge by its own weight. Joseph Melan 58 had published his analysis of this effect in 1888 (as mentioned at the end of Sec. 10). Leon S. Moiseiff introduced Melan's ideas in the United States on the occasion of the design of the Manhattan Bridge (1909) ; and he and F. E. Tumeaure 67 at the University of Wisconsin extended the theory. Since 1925 there 54 S. Timoshenko, "Sur la stabilité des systèmes élastiques," Ann. Ponts et Chaussées, Mém. et Doc. (1913), subvol. 3, pp. 496-566; subvol. 4, pp. 73-132; subvol. 5, pp. 372-412. 65 H. M. Westergaard, "Buckling of elastic structures," Trans. Am. Soc. Civil Engrs. 86, 576-654 (1922), and "On the method of complementary energy," Ibid. 107, 765-793 (1942). 66 A translation by D. B. Steinman of the edition of 1906 of Melan's work has the title Theory of arches and suspension bridges (Myron C. Clark Publishing Co., Chicago, 1913; 303 pp.) 57 J. B. Johnson, C. W. Bryan, and F. E. Tumeaure, Modern framed structures (Wiley, New York, ed. 9, 1911), vol. 2, pp. 276-318.

HISTORICAL NOTES

39

has been a growing literature in the United States on the theory of suspension bridges, especially a number of papers in the Transactions of the American Society of Cimi Engineers; this interest is understandable when one considers the succession of great suspension bridges that have been built. 20. Theory of Plasticity The founding of the theory of plasticity by Saint-Venant and Lévy in 1870 and 1871 has been mentioned (in Sec. 10). The various investigations to determine the combinations of stresses in two or three directions that define the limits of elasticity, the yield point if it exists, and the strength have a direct relation to the theory of plasticity. These investigations have led to various tentative and sometimes conflicting conclusions. Thus Otto Möhr68 (1900) found support for his hypothesis that the limits are governed by limiting combinations of normal stress and shearing stress on any one section. James J. Guest69 (1900) concluded from his own tests of hollow cylinders of steel or brass in which he produced essentially two-dimensional states of stress that the greatest shearing stress governs. Theodor von Kármán60 (1912) and Robert Böker61 (1915), who tested cylinders of marble or sandstone under triaxial compression, found minor departures from Mohr's hypothesis. A. J. Becker62 (1916), who like Guest used hollow cylinders with internal pressure and axial tension, concluded that yielding is governed either by a critical strain, computed as if the material were still elastic, or by a critical shearing stress, the strain or shearing stress governing according to whichever limit is reached first; this is in conflict with Mohr's hypothesis. Richart, Brandtzaeg, and Brown63 (1928), who ,8 O. Mohr, Abhandlungen aus dem Gebiete der technischen Mechanik (Ernst, Berlin; ed. 2, 1914), pp. 192-235. 69 J. J. Guest, "On the strength of ductile materials under combined stress," Phil. Mag. [5] 60, 69-132 (1900). 60 T. von Kármán, "Festigkeitsversuche unter allseitigem Druck," Mitt. Forschungsarbeit. Gebiete Ingenieurwes. 118, 37-68 (1912). el R. Böker, "Die Mechanik der bleibenden Formänderung in kristallinisch aufgebauten Körpern," Miti. Forschungsarbeit. Gebiete Ingenieurwes. 175-176 (1915), 53 pp. a A. J. Becker, "The strength and stiffness of steel under biaxial loading," Bull. Univ. III. Eng. Exp. Sta., No. 85 (1916), 65 pp. M F. E. Richart, A. Brandtzaeg, and R. L. Brown, "A study of the failure of concrete under combined compressive stresses," Bull. Univ. III. Eng. Exp. Sta., No. 185 (1928), 102 pp.

40

THEORY OF ELASTICITY AND PLASTICITY

tested concrete under triaxial compression, observed minor departures from Mohr's hypothesis. Böker61 supplemented von Kármán's and his own experiments by a theory in which he considered the material to have a grained structure with random orientation of critical planes of the individual grains. Brandtzaeg64 modified and extended this theory and succeeded in accounting for some of the features of the observed relations of triaxial stress and strain in concrete beyond the elastic range. The analyses by Böker and by Brandtzaeg are early examples of the theory of the basic laws of plasticity. The performance of the crystalline structure has continued to require at least occasional attention in the development of the theory of plasticity. The work of Theodor von Kármán on columns (1910, mentioned in the preceding article), besides serving its main objective, is a significant example of a solution of a particular problem of plasticity. Other examples of solutions of particular problems or types of problems of plasticity are: Nádai's analyses of torsion by the sand-hill and the roof-and-membrane analogies (discussed in Sec. 15), and analyses by Prandtl 66 and by Hencky 66 of two-dimensional problems under the assumption of yielding under a constant shearing stress, including the problem of a bearing pressure on a semi-infinite solid or on the blunted top of a wedge. The general theory of plasticity was advanced in 1913 by Richard von Mises67 when he proposed a complete set of basic equations of plasticity. One of these equations expresses a function of stresses that he assumed to be particularly important in defining the conditions of flow; this function is the sum of the squares of the differences between the three pairs of principal stresses and may also be stated conveniently in terms of three normal stresses and three shearing stresses in the directions of x, y, and z; von Mises assumed this function to have a constant value during the yielding of the most important materials. The remaining basic M A. Brandtzaeg, "Failure of a material composed of nonisotropic elements, An analytical study with special application to concrete," Kgl. Norske Videnskab. Selskabs Skrifter, No. 2 (1927), 68 pp. 65 L. Prandtl, "Ueber die Eindringungsfestigkeit (Härte) plastischer Baustoffe und die Festigkeit von Schneiden," Z. angew. Math. u. Mech. 1, 15-20 (1921); "Anwendungsbeispiele zu einem Henckyschen Satz über das plastische Gleichgewicht," Ibid. 3, 401-406 (1923). M H. Hencky, "Ueber einige statisch bestimmte Fälle des Gleichgewichts in plastischen Körpern," Z. angew. Math. w. Mech. 3, 241-251 (1923). 67 R. von Mises, "Mechanik der festen Körper im plastisch-deformablen Zustand," Nachr. Kgl. Ges. Wiss. Güttingen, Math.-physik. Klasse (1913), pp. 582592.

HISTORICAL NOTES

41

equations express flow in terms of the deviations of the principal stresses from their average. Nádai's book on plasticity appeared first in 1927 and thereafter in 1931 in the enlarged edition in English. William Prager (1903) came to Brown University in 1941. With support by the Department of the Navy he and his collaborators have succeeded in creating there a center of enlightened productive scholarship in the field of plasticity. One of the services performed by this group has been the translation of some publications in Russian that are of interest. 21, Deformation Method in Structural Mechanics, and the Methods of Successive Approximation by Moment Distribution and Relaxation The analysis of statically indeterminate structures—in which geometric continuity must be considered as well as equilibrium—had been well systematized during the nineteenth century by Maxwell, Möhr, Müller-Breslau, and Castigliano. In the resulting general methods stresses, or simple functions of stresses such as bending moments, are the primary variables; equations between them remove false geometric discontinuities. One may equally well use certain deformations as primary variables and have equations between them remove false reactions. There were several solutions of particular types of problems by this scheme; but in the 1920's A. Ostenfeld68 in Denmark systematized the procedures into a general method which he called the deformation method. This method is analogous in many respects to the "stress methods" of Maxwell and Möhr, just as the principles of minimum potential energy and of minimum complementary energy are analogous in some respects. Hardy Cross69 (1885), who was at the University of Illinois from 1921 to 1937 and thereafter has been at Yale University, devised the moment-distribution method for the analysis of structural frames, and he began teaching it in 1924. He published the method in a 10-page paper in the Proceedings of the American Society of Civil Engineers in 1930. The 68 A. Ostenfeld, Die Deformationsmethode (Springer, Berlin, 1926), 118 pp.; also previous papers, 1920 to 1923. e> H. Cross, "Analysis of continuous frames by distributing fixed-end moments," Proc. Am. Soc. Civil Engrs. (May 1930); republished in Trans. Am. Soc. Civil Engrs. 96,1-10 (1932); discussions, pp. 11-156; also, H. Cross and N. D. Morgan, Continuous frames of reinforced, concrete (Wiley, New York, 1932; 343 pp.).

42

THEORY OF ELASTICITY AND PLASTICITY

subsequent discussions in the Proceedings by others cover 128 pages and the author's closure 18 pages. This method revolutionized the practice of analysis of structural frames of tall buildings. The moment-distribution method is related to the deformation method because false reactions are introduced, but instead of removing them by setting up a single set of many linear equations for rotations and deflections of the joints and solving that set by cumbersome algebra, the false reactions are removed by successive steps without computing the amounts of motion. The structure itself is under constant consideration. Let a frame of a tall building be considered in which, for reasons of symmetry of the load, the deflections of the joints of columns and girders may be ignored. Then the rotations of the joints are the governing deformations, and couples restraining these rotations are the false reactions. To, begin with, all joints are kept unrotated, "locked," by false couples that are computed readily. A joint is chosen, and it is unlocked by removing the false couple that held it. Thereby the bending moments at the joint and some bending moments at neighboring joints are changed; these changes and the resulting new values of the false couples supporting the neighboring joints are computed. The first joint is locked in the new position into which it rotated, and this completes the first step of redistribution of the moments. The second step is like the first except that another joint is unlocked and relocked after rotation. In the third step one may return to the first joint and unlock that again; but more likely a third joint is chosen. The process goes on until each joint has been unlocked several times. Convergence of the process is observed through the gradual reduction of the false reactions. With judicious handling of the method the convergence is ordinarily rapid. Several steps of the process may often be combined into one. When the false reactions include forces, the process remains essentially the same. It has sometimes been said that the moment-distribution method is merely a solution of equations by iteration; but it is more than that, because it is not the set of equations but the structure itself that is kept in mind as one proceeds. Hardy Cross advised against statements that the idea of the moment-distribution method can be applied to problems of other types, unless one has tried how it works. He himself has proved that the idea is applicable to analysis of flow in a network of conduits.70 70

H. Cross, "Analysis of Flow in Networks of Conduits or Conductors," Bull. Univ. IU. Eng. Exp. Sta. No. 286 (1936), 29 pp.

HISTORICAL

NOTES

43

71

Richard V. Southwell conceived of the "relaxation method" and developed its theory and practice. It is a method of numerical calculation by successive approximation and is applicable to a variety of problems in physical science, including mechanics. The moment-distribution method, which preceded it and of which Southwell was not aware at first, is a method within its scope, and so is the method of Hardy Cross for analyzing the flow in networks of conduits. The applicability of the relaxation method to problems of elasticity and plasticity has been proved and gives promise. 22. Analysis of an Arch Dam as an Example of a Satisfactory Solution of a Complex Problem by Imperfect Methods A high arch dam is a large mass of concrete that plugs a canyon and creates a lake upstream; it is arched upstream in order to make compressive stresses predominate. Its boundaries with the walls of the canyon are irregular, and the stress and strain created by its weight and by the pressure of the water extend a long way into the sides of the mountains. The problem of stress and strain in the dam is a complex problem of elasticity. Hoover Dam, on the Colorado River between Arizona and Nevada, is an arch dam, 726 ft high. Its predecessor in record height is Owyhee Dam in Oregon, about 400 ft high. John L. Savage, chief designing engineer of Hoover Dam, recognized the importance of careful analysis of the stresses that were to be expected. No formal solution by the theory of elasticity was available. The so-called "trial-load" method had been applied in a primitive form in the design of some previous dams. Under the direction of Savage this method was improved during the design of the Owyhee Dam and was developed still further in 1929 and 1930 in the process of determining the shape and dimensions that Hoover Dam was to have.72 Ivan E. Houk, Russell S. Lieurance and his team of assist71 R. V. Southwell, "Stress-calculation in frameworks by the method of systematic relaxation of constraints," Proc. Roy. Soc. (London) A 151, 56-95 (1935); 153, 41-76 (1935-36); and two books, Relaxation methods in engineering science, A treatise on approximate computation (Oxford University Press, London, 1940; 252 pp.), and Relaxation methods in theoretical physics, A continuation of the treatise relaxation methods in engineering science (Oxford University Press, London, 1946; 248 pp. and charts). n United States Department of the Interior, Boulder Canyon Project, Final Reports, Part V, Technical Investigations, Bulletin 1, "Trial load method of analyzing arch dams" (1938, 266 pp.); and Bulletin 3, "Model tests of Boulder Dam" (1939, 400 pp.). See also H. M. Westergaard, "Principles of analysis of

44

THEORY OF ELASTICITY AND PLASTICITY

ants, and Robert E. Glover deserve credit in particular for this advance in analysis and for the successful use of it. By the trial-load method, as it was developed, the dam is considered to be divided into numerous horizontal arches and numerous vertical cantilevers, so that each element of volume of the dam is occupied at the same time by an arch and a cantilever. The cantilevers increase in width upstream. Since the arches are thick, nonlinear distributions of the normal stresses and contributions of shearing stresses to the deflections of arches and cantilevers had to be taken into account; and the movements of the abutments had to be considered. For the purpose of explanation let only the horizontal components of the water pressures be taken into account. In the first adjustment radial deflections are considered; these are in the directions of normals of the arches and are contained in middle planes of the cantilevers. A guess is made of a part of the load, a trial load, which is supported by the cantilevers; the remainder must be supported by the arches. This is a guess of a function of two variables that define the place in the dam. If the guess is right, the arches and the cantilevers will have the same "radial" deflection at each place. The first guess cannot be expected to be right. From the discrepancies of the computed deflections an estimate is formed of revisions of the distribution of the load, for a second trial. This estimate is aided by computations in advance of the deflections due to simple patterns of load, or unit loads. Lieurance and his assistants acquired great skill in obtaining near equality of the radial deflections of arches and cantilevers after only a few revisions or trials of the distribution of loads. Such "radial adjustments" served to approach the final design of Hoover Dam; but further analysis was needed to determine the stresses in the dam thus designed in a preliminary way. The radial adjustment by which the deflections of the arches and cantilevers were brought into harmony created other geometric discrepancies, especially of two kinds: elements of the arches have rotated about vertical axes, and points of their center Unes have been displaced in the direction of the tangents to the center lines; the cantilevers have received no such motions, except through movements of their bases. These discrepancies required further adjustments of two kinds: adjustment for twist and tangential adjustment. Either of these could be undertaken next. arch dams by trial loads," Proc. Third Internat. Cong. Applied Mech. (held in Stockholm 1930) 2,366-372 (1931), and "Arch dam analysis by trial loads simplified," Eng. News-Record 106, 141-143 (1931).

HISTORICAL NOTES

45

In the adjustment for twists trial loads of a new type are to be guessed. They consist, first, of couples in horizontal planes applied everywhere to the arches, and equal and opposite couples twisting the cantilevers; these distributed couples are a function of two variables defining the place. Second, the trial load includes a set of twisting couples and forces acting on the cantilevers at the crest of the dam; the twisting couples are an estimated function of one variable; the forces are computed so that they will counteract the couples as loads on the crest, but they will act independently as loads on the individual cantilevers. Third, shearing stresses, mainly vertical components, are introduced on the sides of the cantilevers; these are to account for the twisting moments on the vertical sections and are computed from the loads already introduced. The loads are adjusted until the discrepancies of twists have been removed. The tangential adjustment is performed in a similar way by estimating appropriate "shear loads." The influence of Poisson's ratio may be taken into account by revisions in the method of radial adjustment. Each adjustment eliminates discrepancies of one kind but disturbs the results of any previously completed adjustment. However, the observation that the new discrepancies were generally much smaller than the original ones indicated that a sequence of successive adjustments—for example, in the order of radial, twist, tangential, radial, and so on— would be a rapidly convergent process. This was found to be true. Each adjustment is an elaborate operation, but the required number of adjustments is small. Also, after experience it was found that two adjustments could be combined into one. The method that has been described would apply with any accuracy that might be desired if the arch dam were a shell not too thick; but Hoover Dam has a maximum thickness of 660 ft at the base, and the method is therefore imperfect, even though the influence of the thickness was taken into account in various ways. Nevertheless, tests of models confirmed the computed deflections well. The results are satisfactory, and the example illustrates how one may manage when the history of the theory of elasticity does not yet include a formal solution of the problem at hand; the future may.

CHAPTER

III

Stress 23. Notation for Stress Figure 1 shows a solid body divided into two parts by an imagined plane section of which cL4 is an element of area. The force exerted by part I I on part I within the area dA is written s cL4, where s is the force per unit area. The bold face of printing designates s as a vector, having direction as well as magnitude; in writing or in a typewritten paper the same purpose is achieved most simply by underscoring the letter; s printed in italic or

FIG. 1. Vector stress s, normal stress σ, and shearing stress τ. written without underscoring denotes the magnitude of the vector s. One may imagine that the area dA was finite to begin with but has become infinitesimal; then under the assumption of homogeneity of the material (explained in Sec. 2) one may assume that, as the area shrank to the infinitesimal

(53)

and its directions make angles of 45° with the directions of the largest and the smallest principal stresses. (2) In a given state of stress at a point all possible combinations of a and r are represented by the area bounded by the three semicircles. Point G in Fig. 12 could not lie outside this area. Problem 8. Derive Eq. (53) from Eq. (44).

34. Mohr's Hypotheses on Limits of Stress In the paper that he published in 1900, in which he inquired about limits of elasticity and of strength and referred to many results of experiments, Otto Möhr 6 advanced the hypothesis that these limits are defined by limiting combinations of σ and τ on any section. This means that, if the state of stress is not to exceed the limit, no corresponding point in the στ-plane is permitted to lie outside a limiting curve. Then Mohr's circles, which he used in this study, may touch the limiting curve but may not have any segments outside the curve. If Mohr's hypothesis is correct, only the largest and the smallest principal stresses, which define the largest of the circles, have influence in relation to the limits, whereas the intermediate principal stress has no influence. If the hypothesis is nearly right, the intermediate principal stress has only a minor influence. Although a hypothesis is involved, it is of a general form and includes some of the specific hypotheses of limits of stress as special forms. Thus the idea, taken for granted by Lamé and Rankine, that the limits are defined by a greatest possible compressive stress — σι and a greatest 6 O. Mohr, "Welche Umstände bedingen die Elastizitätsgrenze und den Bruch eines Materiales?"; Ζ. Ver. deut. Ing. (1900), pp. 1524-1530, 1572-1577; also, with additions, in his Abhandlungen aus dem Gebiete der technischen Mechanik, ed. 2, pp. 192-235. See also the historical notes in the first part of Sec. 20.

STRESS

65

possible tensile stress σ3 would be expressed by saying that the limiting curve is the semicircle having the values of σι and σ3 at the ends of its horizontal diameter. Coulomb's idea, dating back to his paper of 1773, that some materials fail when the greatest shearing stress reaches some ultimate value is expressed by a straight line parallel to the axis of σ as limiting curve. On the other hand, the idea held, at least for a while, by Saint-Venant that the tendency to failure is measured by the greatest elongation or shortening per unit length would not lead to a single limiting curve or line in Mohr's diagram; the intermediate principal stress would have a considerable influence. When a hypothesis of limits of stress is in conflict with Mohr's, it is usually best to consider a limiting surface with the three principal stresses as coordinates. Mohr's ideas on this subject, including the use of his circles to represent experimental results, have been found very useful for the interpretation of tests, especially of brittle materials such as concrete, and of soil under triaxial compression. Sometimes it is found that the limiting curve may be drawn within some range as a straight Une, so that one may write τ ~ Α + Β(-σ),

(54)

in which — σ is a positive compressive stress. The corresponding limits of the principal stresses will be defined by another linear equation of the form - f f i = C + D(-e*), (55) in which C is the strength in simple compression if the range includes σ3 = 0. For example, on the basis of a series of tests by them of concrete under triaxial compression, Richart, Brandtzaeg, and Brown 6 arrived at the approximate value D = 4.1 for concrete when — σ3 has a range from 0 to 4,090 lb in. -2 In an extreme case of a specimen of a poor grade of concrete for which C was only 1,050 lb in. -2 (about half of the ordinary low standard of 2,000 lb in."2), the values - σ 3 = - σ 2 = 6,560 lb in."2 left the specimen still unbroken with — σι = 24,600 lb in. -2 and still strong enough to give — σι = 1,000 lb in. -2 when retested to failure in simple compression. In some of these tests σ2 was equal to σ3, in others equal to σι. This difference in the arrangement appeared to have a minor influence on the limiting curve, thus indicating a minor deviation from Mohr's 6 F. E. Richart, A. Brandtzaeg, and R. L. Brown, "A study of the failure of concrete under combined compressive stresses," Bull. Univ. III. Eng. Exp. Sta., No. 185 (1928, 102 pp.), especially pp. 60, 78, 82.

66

THEORY OF ELASTICITY AND PLASTICITY

hypothesis. Previous tests of marble by von Kármán 7 and by Böker 8 had indicated minor deviations in the opposite direction. Möhr included in the statement of his theory the assumption that failure occurs along planes on which σ and τ reach the dangerous combinations. It is preferable to regard this assumption as a separate hypothesis which may well be questioned even though the hypothesis of the existence of the limiting curve is accepted as approximately true for some important materials. Problem 9. Express C and D in Eq. (55) in terms of A and Β in Eq. (54).

35. Octahedral Stress The octahedral stress τ0 is the resultant shearing stress on a plane, called octahedral, that makes the same angle with the three principal directions. Eight such planes can form an octahedron. The square of each direction cosine of any of the normals is Then, with σι, σ2, and denoting the principal stresses, Eq. (44) gives W = (σι - σ2)2 + (σ2 - σ3)2 + (σ3 - σι)2.

(56)

τ02

I t will be shown in the next section that may also be expressed conveniently in terms of the six components σχ, σν,. . . , jzy of any stress tensor. In 1913 R. von Mises 9 proposed the hypothesis that yielding of some of the most important materials occurs at a constant value of the quantity in the right-hand member of Eq. (56), so that this quantity will enter accordingly in the basic equations of plasticity. The idea that this quantity may serve as a measure of tendency to failure has also been credited to M. T. Huber,10 who published a paper on this idea in Polish in 1904. Huber assumed the criterion to apply only when the sum of the principal stresses is negative. 7 T. von Kármán, "Festigkeitsversuche unter allseitigem Druck," Mitt. Forschungsarbeit. Gebiete Ingeniewrwes. 118, 37-68 (1912). 8 R. Böker, "Die Mechanik der bleibenden Formänderung in kristallinisch aufgebauten Körpern," Mitt. Forschungsarbeit. Gebiete Ingenieurwes. 176-176 (1915), 53 pp. • R. von Mises, "Mechanik der festen Körper im plastisch-deformablen Zustand," Nachr. Kgl. Ges. TTí'm. Göttingen, Math.-physik. Klasse (1913), pp. 582-592. See also See. 20. 10 M. T. Huber, Czasopismo technizne (Lwov, 1904). This reference is quoted from S. Timoshenko, Theory of elasticity (McGraw-Hill, New York, 1934), p. 137. See also A. Föppl and L. Fôppl, Drang und Zwang (Oldenbourg, Munich and Berlin; vol. 1, ed. 2, 1924), p. 50.

STRESS

67

The quantity in the right-hand member of Eq. (56) has attained importance in the theory of plasticity. Since it happens to be proportional to the square of the octahedral stress, it has been found convenient to talk about it and about the hypothesis of von Mises in terms of the octahedral stress. The square of the octahedral stress is proportional to the sum of the areas of Mohr's three semicircles. The assumption of its importance is therefore in conflict with Mohr's hypothesis as applied to yielding but not necessarily as applied to rupture. In accordance with Eq. (28) the normal stress σο on the octahedral plane is the average of the three principal stresses. One might call this stress the octahedral normal stress, but it is preferable to refer to it merely as the average normal stress and reserve the phrase "octahedral stress" for the shearing stress. Problem 10. Express to as a function of the two quantities .

e™

and

Equations (270) are derived by writing the expressions for ex, and fxy in terms of ξ and η and then eliminating £ and η by treating the operators d/dx and d/dy as if they were constant coefficients in algebraic equations. The elimination is effected by applying the three operators shown in Eqs. (270) to the three expressions and adding. Equations (271) are obtained similarly. 61. Cylindrical Coordinates It is often desirable to use cylindrical coordinates r, θ, ζ instead of the rectangular coordinates x, y, z. The relations between the two sets of coordinates are defined by the equations R = ix + jy + kz = rr + kz, (272) and î = i cos θ + j sin Θ. (273)

92

THEORY

OF

ELASTICITY

AND

PLASTICITY

An additional unit vector is needed. It is denoted by 9 and pronounced "unit Θ." It is perpendicular to r and k and points in the direction of increasing θ when laid off from point R. Accordingly it can be stated as 9 = — i sin θ + j cos θ.

(274)

It is observed that d20

dì

a d =

, and

. î =

S d2f -dô="d^·

(275)

The components of displacements are defined by the equations Ρ

=

+ IV + kf =

îpr

+ 9 pe + kf.

(276)

Figure 16 suggests the derivation of an expression for the strain ee in the direction of 9. The motion and stretching of the element 9r d0 can be

FIG. 16. Strain E«. achieved in four steps, three of which are indicated in the figure. In the first step, to position 1, the element slides and stretches along the original circle with radius r, receiving the strain (l/r)(dpe/d0). In the next step, to position 2, the element moves to the circle with radius r + pr, receiving the strain pT/r. In the third step, to position 3, the element merely turns through a quasi-infinitesimal angle, thereby contributing only a quasiinfinitesimal added strain which can be ignored. The fourth step adds displacements in the direction of z, but the corresponding additional strain can be ignored. These considerations lead to the second of the following three equations; the first and the third are obvious: dpr =

These three equations give

«

=

Pr , 1 dpe 7 + rW

, and

9f * = äT

roT7\ (277)

93

STRAIN

The detrusion y^ is obtained by noting in Fig. 17 the angles in the directions of +0 that a pair of elements originally Sr dô and rdr will make with the pertinent tangent and radius after the displacement. This gives 7r9

r ΘΘ ' dr

r

r ΘΘ ' ' dr\r

The other two detrusions are, in agreement with Eqs. (206), - - i + i l

-

—

l + t -

W»

\Wr όβ

Fig. 17. Detrusion The operator V in V4> is (in agreement with the expression in terms of rectangular coordinates),

Equation (278) is reproduced by writing V 0 ρ for div ρ and referring to Eqs. (275). The operator div is defined through Eqs. (278) by noting that ρ may be replaced by any vector function. Laplace's operator V2 is now expressed by the following computation in accordance with Eqs. (281) and (278) : VΦ This gives

.

UIV νφ

' γJ Qf

a2 , 1 d . 1 θ2 . a2

f ßff \r dB] ^ ÒZ ÔZ

K /000.

Equations (278), (281), and (283) give new meaning to the basic equation of elasticity, Eq. (252). The expressions for the strains and detrusions open the way to corresponding expressions of Hooke's law.

'

94

THEORY OF ELASTICITY AND PLASTICITY

The stress dyadic, which defines the stress tensor, is stated as follows, in accordance with Eqs. (30) and (31) : Τ = Bri

+ SgS

+

(284)

Szk,

or Τ = Orti + Trefe + T„kf + τβτΐ9 + agÔQ + T0zk9 + T„ik + r j k + ajsk.

(285)

Equation (285) serves to denote all the components. One may very well derive the differential equations of equilibrium, Eqs. (110) to (112), from Eq. (109), div Τ + Κ = 0, provided that the divergence of a dyad Ff is defined in such a way that it becomes independent of the type of coordinates. This is achieved by accepting the following equation, which defines a vector: div (Ff) = (div F)f + (F o y) f.

(286)

That this equation serves the purpose is seen by stating the two vectors as F = ìFx + jFy + k f 2 and f = i f x + j f y + k/, and accepting obvious meanings of operators as expressed in the equations div (Fi/,) = i div (F/*)

(287)

and Fo v _

F

. £

+

F

,±

+

F.iL.

(288)

By Eqs. (285), (281), and (275) it is readily found that (Br ° V)r = 9 —, r

(se o v)9 = - Î - , r

and

(s2 o v)k = 0.

(289)

Then Eqs. (284), (286), and (289) give div Τ + Κ = f^div s, - ^

+ $ (div s e + y ) + k div s* + Κ = 0. (290)

By referring to Eq. (278), which defines the divergence in terms of cylindrical coordinates, it will be seen that Eq. (290) reproduces directly the three differential equations of equilibrium stated at the end of Chapter III, Eqs. (110) to (112). Problem 4- Show each step in the derivation of Eq. (286).

STRAIN

95

52. Octahedral Strain The vector strain [see Eq. (235)] associated with the octahedral normal that makes equal angles with the principal axes of strain will be denoted by e0. The octahedral plane of strain is perpendicular to the octahedral normal. The component of e0 contained in the octahedral plane of strain will be denoted by ^7o· This y0 is a detrusion or shearing strain, and it is called the octahedral strain. By the correspondence of stress and strain in which half-detrusions take the place of shearing stresses one finds, in analogy with Eq. (69), Ito 2 = (e* - O 2 + (e„ - ez)2 + (e, - e*)2 + f (y^ + y„? + yJ). (291) In analogy with the octahedral stress, 70 can also be expressed in terms of strain deviations. These are defined, in analogy with Eqs. (71), as eJ = €* - eo, . . . (*, y, 2),

(292)

in which eo is the average strain, equal to |eTOi. Thus one finds by referring to Eqs. (82) and (84) To2 =

Hiejy

+ (e/)2 + (e/)2] + f ( W +

ym* + yJ ) .

(293)

53. Plastic Strain Deviations and Plastic Detrusions These are defined as the actual strain deviations or detrusions minus the strain deviations or detrusions that would be computed if Hooke's law still applied. They are denoted by use of double primes, and are consequently defined by the equations tx" = ex'-

·~σχ'

and

yxy" = yxy - | r w

. . . ( * , y, z),

(294)

in which σχ is the stress deviation σχ — σο. Correspondingly the plastic octahedral strain yd' is defined in analogy with Eqs. (291) and (293) by the equations f(7o") 2 = W - e/')2 + («„" - e/')2 + W - O 2 + f [ ( W ) 2 + (yy/Ύ + (7~")21 and

(295)

(yo"y=m*y+(Ο2+(02]+itcwo2+^Ύ+(7«Ό2].

(296)

54. The Simplest Conceivable Laws of Plasticity Such laws have been discussed on several occasions by William Präger.® The "simplest laws" apply to an idealized material. • W. Prager, "The stress-strain laws of the mathematical theory of plasticity— a survey of recent progress," J. Applied Mechanics 16, 226-233 (1948); "Recent

96

THEORY

OF E L A S T I C I T Y

AND

PLASTICITY

A possible statement of such laws is the following: (a) When the plastic octahedral strain y0" increases beyond the greatest value that it has had previously, the octahedral stress TO is a definite function of 70". A definite curve of octahedral stress and plastic octahedral strain then exists, such as that shown in Fig. 18. To

_____

(- Elastic limit

A

FIG. 18. Octahedral stress and plastic octahedral strain. (b) When the octahedral stress and the plastic octahedral strain have reached any point A on this curve, the material can be considered to be a new material with A marking the elastic limit. If the octahedral stress drops below its value at A, the "new material" is elastic with the original constants of elasticity; it reënters the plastic stage when the octahedral stress at A is restored. The "new material" has the octahedral-stressstrain curve BAC, Β replacing 0. (c) Hooke's law is valid for the average strain eo and the dilatation 3eo in all stages. (d) When the plastic octahedral strain increases beyond its value at A, the increments of the plastic-strain deviations and the increments of the plastic detrusions defined by Eqs. (294) follow the law de/' _ de/' _ de/' _ àyxy" _ dyj' _ áyj' " P$-l{i~$)· and

+

(324)

-M^-'(rW)]·

«

At the inner surface r = a the values are στ = -p,

σβ =· p - 2q, and

p, = ^ (p -

qj,

(326)

and when r is large, the values are σ,= σ β = - ?

and

p ^ - Í L ^ ·

(327)

Equations (327) show that if values of r are kept large at the outer surface compared to the inner radius a, the outer surface may be replaced by any prismatic surface parallel to the axis of ζ and subjected to a constant pressure q. For example, the cross section of the outer surface may be a rectangle with edges in the directions of χ and y and with pressures — σχ = — σν = q at the edges. With the inside pressure p = 0, Eqs. (326) give — σβ = 2q at the surface of the hole, twice what would exist without the hole. Accordingly the hole is said to produce a stress-concentration factor of 2 under uniform external pressure. The corresponding concentration factor computed by σβ in Eqs. (322), with p = 0, for any value of the outer radius b would be the ratio of — σβ at r = a to the value of — σβ computed by the simplest means of statics; that value is the average of — σβ over the diametral section, which is qb/(b — a). Then Eq. (322) shows the concentration factor to be 2b/(b + a). This factor approaches 1 when b approaches a, as would be expected, and approaches 2, as was found, when b becomes large.

104

THEORY OF ELASTICITY AND PLASTICITY

The problem is now changed by specifying f = 0 instead of σ* = 0; that is, a plane state of strain is specified instead of a plane state of stress. In accordance with the observations made at the end of Sec. 49, after Eq. (269), the formulas that have been derived can be applied if μ is replaced by μ/(ί — μ). This changes the factor (1 — μ)/(1 + μ) in Eqs. (319), (323), and (325) to 1 — 2μ; the stresses στ and σβ are not changed; but a new constant stress σζ = μ(στ + σβ) is produced in the direction of z. By Eq. (323) the radial displacement at r = a now becomes =

1 + (1

2G[1 - (aW)] j[

~

i\P~

2(1

«

Let it be assumed that the value in Eq. (328) is kept equal to zero. This can be achieved byfillingthe hole of the hollow cylinder with a rigid cylinder. Then (329) *=l + (f-~2^)g> which is positive and greater than q if q is positive, as would be expected. Furthermore, either by substitution from Eq. (329) in Eq. (322), or by noting that eg = = 0 at r = a, one finds that (330) ~ σβ·Γ-° = 1 + (1 -β2μ)(α?/Ρ)' This also is a positive pressure when q is positive; σΙ,,_α is found to have the same value. Nevertheless, under practically these circumstances, except that the plastic stage would have been entered, Bridgman, in one of his early investigations of the effects of high pressures, obtained a failure by splitting along a radial section extending from the rigid surface at r = a, as if the failure had been by tension. Considerations of plasticity would not change the conclusion that σβ was negative. It can be inferred that it is a difference of principal stresses rather than actual tension that tends to produce rupture by splitting.

58. Lame's Formulas for Stresses in Hollow Spheres If ρ = Äp(J2),

(331)

the principal strains will be in the directions of R and of any tangent Τ to the sphere R = const.; and these strains will be €ß = ^

and e r - J ·

(332)

STRAIN POTENTIAL AND APPLICATIONS

105

Then

dÎV/,==

(â + I) p ·

«

Let 2Gp = V4>(R), which is è άφ/àR. Then Eq. (333) gives

It follows that ν*Φ(Φ = 0> so that φ is harmonic, if r φ = ö + const.

(335)

The corresponding principal stresses, in the directions of the radius R and tangent T, are dV _ 2C I d φ_ C , . = and (336) RdR~~ R* Let a hydrostatic state of stress vr = στ = D be superposed. This leads to the following formulas, which are comparable to Eqs. (320) and (319): 9Γ Γ σΒ = ψ3 + ϋ, στ ¿ +A (337) and

By proceeding as in the problem of the hollow cylinder, one arrives at Lamé's formulas for stresses in a hollow sphere subjected to the pressure ρ on the inner surface R = a and the pressure q on the outer surface R = b: _

,(b*m -(b*M

σΒ

P

and σ τ

~

Ρ

- 1 - 1~

(bs/2R*) + 1 (¿«/α») - 1 -

ι (am 1 - (a 3 /i 3 )

q

3

,1 + (a3/2R») 1 - (a*m ·

(339)

(

.

(M0)

.

Furthermore, the displacement is found from Eq. (340) by comparing Eq. (338) with the second of Eqs. (337); the value is p

R Γ β » / 2 # ) + ( 1 - 2 Μ ) / ( 1 + μ) ~ 2G |_ (b3/a?) — 1

9

(1-2μ)/(1 + μ)+(αν2^)Ί 1 - (α3/δ3) J' (341)

106

THEORY OF ELASTICITY AND PLASTICITY

Like the corresponding Eqs. (321) to (323) for hollow cylinders, Eqs. (339) to (341) assume simpler forms, comparable to Eqs. (324) and (325), when b is large compared to a. These forms continue to apply when the outer surface is replaced by any surface, provided that the values of R at the surface are kept large compared with a, and provided that the uniform external pressure q is maintained. Equation (340) shows that this q produces a stress-concentration factor of § at the surface of the hole. Problem 3. Show that with q = 0 and with a close to b Eq. (340) defines the average stress that may be computed by considerations of equilibrium of one half of the thin spherical shell. Problem 4. Let a spherical hole with radius R = a befilledwith arigidsphere with radius a. The pressure on the outer surface R = b is constant and equal to q. Determine the stresses at R = a. 59. Disk with Circumferential Shear at Inner and Outer Concentric Circular Edges The function φ = ce

(342)

serves to solve this problem. The function satisfies Laplace's equation [Eq. (312)], is therefore harmonic, and may be used as a strain potential. In fact, if one begins an exploration by writing down some harmonic functions of simple form with the intention of seeing afterward to what practical problems they might apply, φ in Eq. (342) would naturally be included among the first, and thus one would arrive at the problem. According to Eqs. (311) and (313) to (315), with φ as chosen, the only components that do not vanish are (343) The resultant of the stresses τ* on a cylindrical surface r = const, is a couple +2rC per unit distance in the direction of 2. The solution applies obviously to the problem of the disk; if the thickness is constant, these couples, transmitted at the outer and inner edges by circumferential shearing forces, are the only loads. In the list of harmonic functions of simple form one might well include Czd and Cz log r, but these do not appear to have any direct and simple practical applications.

STRAIN POTENTIAL AND APPLICATIONS

107

60. Resultant Force on Surface of Revolution When Stresses Are Symmetric about Axis If the strain potential ψ is a function of r and ζ only, φ possesses axial symmetry about ζ and will lend itself to the study of solids of revolution about the axis of z. The two problems of Lamé (in Sees. 57 and 58) are examples. Let the axis of ζ be vertical and positive downward, and let r point toward the right. Let a curve be drawn from point A to point Β in the rz-plane; this curve must not intersect itself, nor contain any point at which φ or r (δφ/dr) is infinite, nor contain any point having a negative value of r. Stresses on a curved section containing this curve will be considered to act on the part of the solid lying to the left of the curve from A to B. The curved section on which the stresses act is the surface of revolution obtained by revolving the curve AB about the axis of z. If the resultant of such stresses is not zero, it will obviously be a force along the axis of z. It will be shown that its value, positive upward, in the direction of —z, is (344) or 2a- times the increment of r (θφ/dr) from point A to point B. To prove this, consider a "curve" AB consisting of two elements: first î dr, and thereafter k dz extending from the end of the first element. The force to be determined acts on the part above the first element and outside the second. Its value, positive upward, is dQ = 2ττ(—σζ dr + τ„ dz).

(345)

With ν2Φ = 0, —

3» ^Te['r)·

[φ\

Tr9=

(370)

63. Circular Disk Rotating about Its Axis of Revolution The general problem in which the thickness is variable has received deserved attention. A complete procedure appears to have been developed first in 1894 by K. Ljungberg for purposes of steam turbines.2 A. Stodola 3 has contributed solutions, and his treatise Steam and gas turbines contains extensive accounts of the several analyses. The present discussion, however, will be limited to disks of constant thickness. The disk rotates about the axis of ζ with constant angular velocity ω. Let w denote the weight of the material per unit volume. The body force then is the centrifugal force per unit volume, namely, W(A?

Κ = r — r,

(371)

g

in which g is the acceleration due to gravity. The corresponding bodyforce potential is * =

(372)

Equation (368) will be satisfied by a solution of the form φ = Cr*, which gives ν2Φ = 16Cr2. Hence one may write (373)

Thereafter Eqs. (370) give _(3

+ MWr2

ffr=

and

g t

,_(l + 3

8g

(374)

og

Assume that the disk has a free edge at r = a. Then a uniform state of stress στ = + d f

+

âs2

dr2 + r ΛΛ 3 Pz

'• —

3 Pz« ω ?

2 ')]

(

SPrh 3Prz 2

=

5 0 3

>

,

.

(504)

Let a denote the angle from the axis of ζ to the position vector R. Let subscripts a designate components perpendicular to R in the rz-plane and

136

THEORY OF ELASTICITY AND PLASTICITY

in the direction of increasing a. Then, by referring to Eqs. (3) and (4) in Chapter III, one obtains the following further statement of stresses: 3 Pz fffi = Θ = — σβ = σα = ra, = 0. (505) It follows that er«, σ», and σα are principal stresses. Equations (505) indicate a particularly simple mechanism of transmission of force from the origin into the various parts of the solid. Imagine the solid divided into numerous cones or pyramids extending from a common vertex at the origin. Each of these pyramids will transmit its own radial force without reactions from the adjacent cones or pyramids. The displacements are defined most simply by the last expression in Eqs. (503). These simple relations apply only when Poisson's ratio is 76. Boussinesq's Problem of a Normal Force and Cerruti's of a Tangential Force on the Plane Surface of a Semi-infinite Solid When Poisson's Ratio Is \ It was mentioned in Sec. 10 that Boussinesq solved the problem of the normal force about 1878 and Cerniti the problem of the tangential force in 1882; these solutions are for any value of Poisson's ratio. It is clear from the results in the preceding section that with μ = | the answers to Kelvin's problem are answers to the other two. With the force kP acting at the origin of the coordinates on one half of the infinite solid that was considered in Kelvin's problem, Eqs. (503) to (505) are the solution of Boussinesq's problem when the semi-infinite solid occupies the space 2 > 0 and are the solution of Cerruti's problem when the semiinfinite solid occupies the space χ > 0, provided that μ = J. Boussinesq's problem with μ = $ requires no restatement of Eqs. (503) to (505). However, in order to prepare for the subsequent solution of Cerruti's problem with any value of μ it is desirable to restate Eqs. (503) to (505) in terms of rectangular coordinates, with the semi-infinite solid occupying the space ζ > 0 instead of χ > 0, and with the force at the origin equal to i Ρ instead of k P. The restatement is obtained by simple means and is Ρ

Cx

~

(. .

_ 3 Px? 2tRí'

x\

Λ

σ

"~

ZPxyz Tv

'~

2*R>'

r. _ 3 Pxf 2πΚ.6' _

T

"~

3Px _ ~

3Px?z 2 TR"'

3 Pxz* 2wR>' _

Txv

~

3Px?y 2TR6'

. {

. )

GALERKIN VECTOR, TWINNED GRADIENT

137

The stress OR is still a principal stress, and the other two principal stresses are zero. It will be observed that, when μ = a Galerkin vector of the form BR will define displacements and stresses in a cone or pyramid loaded by a force at the vertex, or in a wedge loaded by a force at a point of the edge; the magnitude and direction of Β can be determined by appropriate integrations. 77. The Twinned Gradient and Its Application to Determine the Effects of a Change of Poisson's Ratio 6 The procedure about to be described is suitable for certain problems, including Boussinesq's and Cerruti's, that have a simple solution when Poisson's ratio has a particular value m. The procedure serves to determine the effects of a change of Poisson's ratio from m to the appropriate value μ. Let primes designate values of displacements, stresses, and body forces obtained when Poisson's ratio is m. Then the bulk stress will be stated as W - S f í g W

(507)

Now let Poisson's ratio be changed to μ; let G remain unchanged; and for the time being let the displacements remain unchanged. The shearing stresses will not be changed thereby, but the normal stresses σ' receive the increment

Further changes must be provided. Let ρ", σ", τ", and Κ" denote the total additions to ρ', σ', τ', and Κ' that are needed to produce the final values ρ, σ, τ, and Κ. Then one may write, for example,

in which σ is defined by Eq. (508). It is reasonable to call the operator i

F + j f dx dy

k

| - = V-2k £ dz dz

v(510)

• Η. M. Westergaard, "Effects of a change of Poisson's ratio analyzed by twinned gradients," J. Applied Mechanics (September 1940), pp. A-113-A-116.

138

THEORY OF E L A S T I C I T Y AND P L A S T I C I T Y

a twinned gradient, because the part k(d/dz) in V is replaced by its "twin" - k {d/dz). Under some circumstances a twinned gradient will serve to determine the changes ρ", σ",.. . that must be added to the original values p', σ ' , . . . . The following form will be tried:

Í* " ^ + k έ) Φ = (" v + 2k I ) *•

=

This form gives r." -

- 0,

(512)

and 2G div ρ " = - ΨΦ + 2 0 ·

(513)

By substituting from Eqs. (508) and (513) in Eq. (509) one finds ' " = (1+V-2M)

Θ

' + Γ^Τμ

(S *

*

Let it be specified next, by a voluntary choice, that

φ

^ " - S - y

^

These stresses require that the additional body force be

Κ

"=(ίέ+^)ν2φ·

(519)

The solution is established if a function Φ can be found that satisfies Eqs. (516) and (519) simultaneously.

GALERKIN VECTOR, TWINNED GRADIENT

139

In a majority of significant problems both the original and the final body forces are zero; that is, Κ' = Κ" = Κ = 0. (520) Then the governing Eqs. (519) and (516) will be satisfied if V ^ = 0 and f î2 = r r l i f ® ' · θζ 1+ m

(521)

These two equations can be satisfied simultaneously because Θ' is harmonic. Furthermore, one finds by Eqs. (518)

and accordingly Θ = Θ' + Θ" = τ φ ® Θ'. 1+ m

(523)

Example. A simple example which affords an opportunity to test the procedure is furnished by the problem in Sec. 73, Eqs. (479) to (495), in which the pressure C cos ax cos by acts on the surface ζ = 0 of the semiinfinite solid ζ > 0. The derivations and the results are simplest when Poisson's ratio is zero. With m = 0 Eqs. (494) and (521) give — = 2 μ α = 2μϋ cos ax cos by er", oz

(524)

and accordingly φ

=

c

and ν2"*» = 0.

(525)

By substitution from Eqs. (525) in Eqs. (511), (518), and (523) all the terms containing μ in Eqs. (493) to (495) are reproduced readily.

78. Boussinesq's Problem When Poisson's Ratio Has Any Value The load k Ρ acts at the origin of coordinates on the semi-infinite solid occupying the space ζ > 0. The axis of ζ may be visualized as vertical and positive downward. As explained in Sec. 76, when Poisson's ratio is the problem has the simple solution stated in Eqs. (503) and (504). The solution for any value of μ is obtained most conveniently by the method of the twinned gradient. With m = f, Eqs. (504) give θ

' - ϋ ·

(526)

140

THEORY

OF E L A S T I C I T Y

AND

PLASTICITY

Equations (521) become

These equations are solved by i

i i ^

=

l o g ( s

+

>) .

(52S)

That this Φ is harmonic was shown in Sec. 61, Eqs. (349) to (352). In accordance with Eqs. (511) and (518), the displacements and stresses to be added to those applying when Poisson's ratio is § become8 „ =_

=_

Pt

(1 - 2μ)Ρ

2G dr

4ttG

r £(£ +

«)'

1 ¿)Φ _ (1 - 2μ) Ρ ί 2G dz 4τ¡GR ' „ _ 1 3Φ (1 - 2μ) Ρ 1 °"Γ rdr 2ττ £ ( £ + 2)' σβ "

_

θ

2

φ

_

1

θ

Φ

6

2

φ

dr2 r dr dz2 _ (¡jiMPf L _ 2ir L R(R + «)

+

. l

m

By adding the values in Eqs. (529) to those in Eqs. (503) and (504) and by Eq. (523) one obtains the following final values, which are the solution of Boussinesq's problem for any value of Poisson's ratio: Ρ ~rz (1 Pr = 4ΠGR R2 R+ z J Ρ f = 2(1- μ R2J' 4IRGR ( H -ß)P ζ Θ 7Γ

Λ"

Ρ Γ 3r*z , (1 - 2μ)#Ί σ = ' 2τ&1 R3 + R+ ζ J (1 - 2μ)Ρ \"ζ R 1 2v&~\_R~R + z\' 1 3 Ρζ3 ZPrz αζ=— η 6 Trz = & 2ttÄ " 2vR The remaining components vanish. • See reference 5.

(530)

(531)

GALERKIN VECTOR, TWINNED GRADIENT

141

Boussinesq's problem may also very well be solved by a combination of a Galerkin vector and a strain potential. The suitable forms are F = kZ = kBR and φ = C log (R + z). (532) The Galerkin vector kBR was used in Sec. 74 in solving Kelvin's problem, and the strain potential φ was investigated in Sec. 61, Eqs. (349) to (357). According to Eq. (499) and the last of Eqs. (353), the combination of F and φ causes the shearing stresses τ„ to vanish at the surface ζ = 0 if (1 - 2μ)Β + C = 0. (533) Furthermore, according to Eqs. (500) and (355), the combination will account for the total load if 4ir(l - μ) Β + 2irC = P. (534) Equations (533) and (534) give B =f and ΔΤΐ

ΖίΓ

(535)

Substitution of these values in Eqs. (498) and (499) and in Eqs. (351) and (353) and superposition of the results reproduce Eqs. (530) and (531). This solution is almost but not quite as convenient as that by the twinned gradient.

79. Cerruti's Problem When Poisson's Ratio Has Any Value The load i Ρ acts at the origin of coordinates on the semi-infinite solid ζ > 0. The axis of ζ may be visualized as vertical and positive downward. A simple solution is obtained by the method of the twinned gradient. Eqs. 506 apply when Poisson's ratio is Then 7 m=

\and

@ , =

- S ·

Equations (521) are satisfied when θ2Φ __ (1 - 2μ)Ρ χ dz* ~ 2τ R3' θΦ (1 - 2μ)Ρ χ (1-2μ)Ρ χ dz~ 2ττ R(R + z)' 2ττ R+ Z

0; these values are stated in Eqs. (530) and (531); (b) A Galerkin vector kΖ in which Ζ is Love's strain function; the function is

(c) The strain potential , _ Ρ m - 2μ)ε ~ 87Γ(1 — ίΟ L R

2chi] i?3J

+

(546)

Mindlin's Galerkin vector will be found to be correct because the combination of elements (a), (b), and (c) makes σ2 = rn = 0 at the surface ζ = c. In the derivation of displacements and stresses defined by Ζ in Eq. (545), it is useful to note that the items resulting from the terms having the factors Ri and R are available through the solution of Kelvin's problem with 2 Ρ replaced by Ρ and — Ρ respectively, and that the items resulting from the term containing dR/dz may be obtained as 2c times the derivatives with respect to ζ of the corresponding items resulting from the term containing R. Thus the more complicated parts of the solution are already mostly furnished. The contributions from the strain potential are established easily. By these means one obtains the following formulas, which agree with those given by Mindlin: _ pr

Γ

Pr Γζι_ _ 4(1 - μ)(1 - 2μ) (3 - 4μ)2ι 16τγ(1 - ß)G U i 3 R(R + z) "r R3 . 6cz(z — c)l + Ä« J' Ρ I~z¿ 3 - 4μ 16τγ(1 - μ)ΰ U i 3 + Ri + (3 - 4μ)ζ* - 2CZ + 2â + Ri

5 - 12μ + 8μ* R 6cz\z - c) 1 Rb J'

GALERKIN VECTOR, TWINNED GRADIENT _

Θ

.

,

8χ(1 -

2(1 + μ ) *

μ) L

, - 2 ( 1 + μ)(3 -

Ri3

4μ)ζ + 4 ( 1 +

145 μ)ΰ

R3

12(1 + m)CZ21

^

σ

=

'

J'

Ρ _ 2 ( 1 + μ)ζΐ , 4 ( 1 - μ ) ( 1 - 2μ) 5 8τγ(1 - μ) L-Ri Ri3 R(R + z) - 2 ( 5 - 7μ)ζ + 1 2 ( 1 - ß)e

+ +

Ri

3

3(3 -

4μ)ζ -

6(7 Ä8

2ju)cz2 + 24c 2 z

+

30cz 3 (z Ri

Ρ 4 ( 1 - μ ) ( 1 - 2μ) £ — Γ Γ: ( 1 - 32 μ Μ 8τγ(1 — - μ) L R i + ?) μ) L ( I - 2 μ ) ( 3 - 4μ)ζ - 6 ( 1 - 2μ)ΰ . 6 ( 1 + + R 3

c)'

ff« =

_

0

Ρ

r

'* — 8ττ(1 O í 1- μ).ΛL

+ -•3 ( 3 T

=

"

-

6c22~| ]

2μ)εζΐ R*

Μ ! —( ι - 2μ)ζι (1 - 2μ)(ζ - 2c) 6 5 "T RiD Ri3D 3 R3 8 2 3 4μ)ζ + 1 2 ( 2 - j f l r f - 18c z 30CZí3 (Z -- '

Rl

X i =

* = - 8 ι γ ( 1 - μ) Υ'

Ρ -8»(1-

Zt =

μ)Κ

8π(1 - μ) Ύ'

^

The function Χ 2 is harmonic and may be replaced, according to Eq. (419), by the strain potential dX2 _ (1 - 2μ)Ρχ 2ic{R + z)

(

.

(550)

The quantities Fi and fa are the same as F and φ in Eqs. (540) and (541), which define the solution of Cerruti's problem with the force iP acting at the origin of coordinates on a semi-infinite solid occupying the space ζ > 0. It should be added that Mindlin's statement of the Galerkin vector defining the solution of Cerruti's problem without the aid of a strain potential is Fi + iXi with Fi and X 2 as in Eqs. (549). The function Xb is harmonic and may be replaced by a strain potential. It is concluded that Mindlin's Galerkin vector defined by Eqs. (548) and (549) can be replaced by the combination of the following four elements: (a) Values obtained in Cerruti's problem with the force iΡ acting at the origin of coordinates on the semi-infinite solid occupying the space ζ > 0; these values are stated in Eqs. (538) and (539); (b) A Galerkin vector iX in which z

=

8 ^ 0

c. This singularity is that of Kelvin's problem for a force iP at R = 2kc. Thus the solution accounts correctly for the applied force. The further requirement of the solution is that σζ, rzx, and r2„ vanish at the surface ζ = c. The formulas that Mindlin obtained meet this requirement. The following displacements and stresses are obtained by combining elements (b), (c), and (d); these displacements and stresses produce the final values when they are superposed on the values in Eqs. (538) and (539) which represent the solution of Cerruti's problem: 4μ Í = 16ττ(1 - ß)G L*i3 _L 3 -Ri 4μ _ 3 — R - c) _ 6cx2(z - c) + -χ? + R2c(z 3 R5 Ρ _ 6cxy(z — c)"| V = 16x(l - μ)0 xy Rf R3 R6 J Ρ ~xzi _ xz + 2(3 — 4μ)a; _ 6cxz(z — c)"[ f = 16ir(l - μ)0 R¡3 R3 R& J' Px 0 contain the key to the problem. By the second of Eqs. (530), the deflection at the surface 2 =· 0 due to a normal load Ρ is at the distance r from the point of application _ (1 - μ)Ρ _ (1 - μ2)Ρ =

^Ër

,-Μ

(584)

Then, if a normal load p = p(x, y) per unit area is distributed in some way over an area A, the deflection of the surface ζ = 0 at the origin of coordinates will be

For convenience, the surface ζ = 0 will be considered to be horizontal and ζ positive downward. The following notation will be used: Ρ = Kf(r, Θ) and 5 = j f / ( r , Θ) dr,

(586)

in which Κ is a constant and η and π are the algebraically smallest and largest values of r, possibly negative, for any given value of Θ. The quantity 5 may be interpreted as the area of a radial vertical section in a solid

158

THEORY OF ELASTICITY AND PLASTICITY

erected from the surface ζ = 0 and with elevations / of the top surface. Then Eq. (585) may be rewritten in the forms (587) in which 0i and 02 must be chosen so that the whole solid with ordinates/ is included. If the origin of coordinates lies within the base of the solid with ordinates / , and if the sections S include negative values of r, the range from θι to 02 is x. In the first application, let the solid with ordinates / be a hemisphere with radius a. Let Κ be chosen so that the total load will be P; then Κ = 3Ρ/2πα 3 . At the center of the hemisphere 5 = S0 — and at either end of a diameter in the direction of χ one finds S = So cos2 Θ. Then at the center and at the edge of the loaded area Eq. (587) gives, by integration from 0 to ir,

(588) respectively. It is a simple matter to show by use of Eq. (587) that between the center and the edge of the loaded area the deflection is a quadratic function of the distance from the center. This conclusion will also be obtained by another method which will be explained next and has the advantage that it yields other information as well. The slope of the originally horizontal surface ζ = 0 may be interpreted as a vector Vfo which is horizontal because fo is a function of χ and y only. By Eq. (584) this vector slope due to a single force Ρ is directed toward the point of application of Ρ and is inversely proportional to the square of the distance from that point. Therefore the vector slope may be interpreted as a gravitational force per unit attracted mass, created by a gravitating mass proportional to Ρ and placed at the point of application of P. Furthermore, the values of fo may be interpreted as the values in the ry-plane of the corresponding gravitational potential that defines the force by its gradient. These conclusions continue to apply when instead of one force Ρ there are many forces, or when there is a distributed load such as ρ = Kf in Eqs. (586). It is well known that a gravitating thin uniform spherical shell produces no resultant gravitational force in its hollow interior, and that consequently the corresponding potential is constant in this space. The correct-

GALERKIN VECTOR, TWINNED GRADIENT

159

ness of this is verified readily by considering a double cone with vertex at any point in the hollow space, with infinitesimal vertex angles, and with bases on the shell. Since the two bases make equal angles with the axis of the double cone, the two base areas are proportional to the squares of their distances from the vertex; therefore the gravitational forces from the two bases cancel each other. The conclusion that there is no resultant gravitational force at any point in the hollow interior continues to apply when the thin shell is replaced by a thick uniform spherical shell of constant thickness; and it continues to apply when the thin or thick shell is distorted without change of density by changing the dimensions in the directions of χ, y, and ζ by constant ratios that stretch or squeeze the spherical shell into an ellipsoidal shell with the inner and outer ellipsoidal surfaces concentric and similar. From this it is concluded that at any point inside a gravitating uniform solid ellipsoid the gravitational force depends only on the mass contained inside an ellipsoid drawn through the point, concentric with and similar to the outer surface. Furthermore, inside the gravitating solid ellipsoid, at points on a radial line drawn from the center, the direction of the gravitational force must be constant, and its magnitude must be proportional to the ratio of the volume of the ellipsoid drawn through the particular point to the square of the major axis of that ellipsoid; that is, the magnitude of the force must be proportional to the distance from the center. Then the corresponding potential at such a line, inside the ellipsoid, must be a quadratic function of the distance from the center. In applications to deflections of the surface of a semi-infinite solid due to distributed normal loads on the surface it is preferable to consider the gravitating solid to be symmetric about the surface ζ = 0. A scale factor must be used that makes the dimensions in the directions of ζ exceedingly small so that the gravitating mass will be practically at the surface 2 = 0 . The gravitating ellipsoids are flattened out. In the application that led to Eqs. (588), in which the solid with ordinates / is a hemisphere with radius a, the gravitating ellipsoid is a flattened-out spheroid with equatorial radius a. Let the origin of coordinates be at the center of the loaded area. Then there will be symmetry about the axis of z; the gravitational forces in the plane ζ = 0 will be toward the center, and their magnitudes will be independent of the direction. Consequently, Eqs. (588) may be supplemented by the following expressions that apply within the whole loaded area:

160

THEORY OF ELASTICITY AND PLASTICITY

The last quantity is the curvature of the deflected surface in any direction; it is constant within the loaded area. In a second application let/, which defines the load p as Kf, be proportional to the vertical intercept between the inner and outer surface of a thin hemispherical shell with constant thickness and radius a; and let Κ be chosen so that the total load will be P. By computing as if the thickness of the shell were 1, the deflection f 0 at the center of the loaded area is obtained through Eqs. (587) as follows: J/

=

/ 2a Κ = 2 Va - r~>

S — τα.

To establish the corresponding gravitational potential, the hemispherical shell is first converted into a complete spherical shell. The latter is converted into a flattened-out ellipsoidal shell by multiplying all vertical dimensions by an exceedingly small ratio. The ellipsoidal shell remains hollow and the inner and outer surfaces remain similar and concentric. It follows that fo in Eqs. (590) is not only the deflection at the center of the loaded area but the deflection over the whole loaded area. The load Kf defined by Eqs. (590) produces the constant deflection f 0 within the loaded area. This is one of the results obtained by Boussinesq18 in his book published in 1885, in which he presented both his own comprehensive theories and his elucidation of other theories on the effects of surface loads contained within small areas. The distribution of pressure and the deflection defined in Eqs. (590) can be imagined to be produced by pressing the end of a rigid cylinder against the flat top of the semi-infinite solid. It has already been shown that inside a gravitating uniform ellipsoid the gravitational potential at points on any radial line drawn from the center is a quadratic function of the distance from the center. The following further fact about this potential is obtained in the theory of potentials. The gravitational potential within the whole volume of the gravitating uniform ellipsoid is a quadratic function of the rectangular coordinates.16 u J. Boussinesq, "Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques, principalement au calcul des déformations et des pressions que produisent, dans ces solides, des efforts quelconques exercés sur une petite partie de leur surface ou de leur intérieur; Mémoire suivi de notes étendues sur divers points de physique mathématique et d'analyse," Mim. Soc. Sci. Agrie. Arts Lüle [4] 13 (Lille, 1885), 722 pp., especially p. 212. " See, for example, O. D. Kellogg, Foundations of potential theory (Springer, Berlin, 1929; 384 pp.), especially p. 194.

GALERKIN VECTOR, TWINNED GRADIENT

161

It is concluded that if the solid with ordinates / defining the normal load? Kf is a semiellipsoid with center at the origin of coordinates and with principal radii a and b in the directions of χ and y, the deflections within the loaded area are of the form Jo- B-iCx>-$D?.

(591)

Heinrich Hertz 17 originated this analysis in the paper that he published in 1881 on the contact of elastic solids; there he established relations between the constants a, b, B, C, and D, and he presented a table of numerical coefficients; he extended the analysis in a paper in 1882.18 F. Heerwagen19 and L. B. Tuckerman20 have prepared fuller tables for numerical computations, and S. Timoshenko21 has illustrated their use. This matter will be discussed again in the next section. In a paper published in 1925 Vogt22 determined deflections, average deflections, average rotations, and average tangential displacements within a loaded rectangular area on the surface of a semi-infinite solid when the load is uniformly distributed over the rectangle, or has a linear antisymmetric distribution so that the resultant is a couple, or is a uniformly distributed tangential load; and he expressed the final results in tables and in simple approximate formulas. This work has been found particularly useful in design' studies of arch dams, including Hoover Dam.23 In a significant example the deflection is computed at the corner 0,0 of a rectangle with sides χ = 0, χ = a, y = 0, and y = b when the load is uniformly distributed over the rectangle and is ρ per unit of area and normal to the surface. By Eqs. (587) this deflection is 17 H. Hertz, "Ueber die Berührung fester elastischer Körper," J. reine u. an gew. Math. 92, 156-171 (1881); republished in his Gesammelte Werke (Leipzig, 1895), vol. 1, pp. 155-173. 18 H. Hertz, "Ueber die Berührung fester elastischer Körper und über die Härte," Verhandl. Ver. Beförderung Gewerbefleisses (Berlin, November 1882); republished in his Gesammelte Werke (Leipzig, 1895), vol. 1, pp. 174-196. " F. Heerwagen, "Kugellager. Erfahrungen aus dem Betriebe und Beiträge zur Theorie," Z. Ver. deut. Ing. 45,1701-1705 (1901). M See H. L. Whittemore and S. N. Petrenko, "Friction and carrying capacity of ball and roller bearings," Nat. Bur. Standards (U. S.), Technol. Papers, No. 201 (1921), 34 pp., especially p. 19. In the formulas on page 18 of that paper the letter 5 is used in two meanings: i 2 denotes the square of Poisson's ratio, but otherwise S denotes a stated function of the radii of curvature. 11 S. Timoshenko, Theory of elasticity (1934), p. 347. a F. Vogt, "Ueber die Berechnung der Fundamentdeformation," Avhandl. Norske Videnskaps-Akad. Oslo, l . Mat.-Naturv. Klasse (1925), No. 2, 35 pp. » See Sec. 22.

162

T H E O R Y OF E L A S T I C I T Y 2

(ι - μ )ρr r^-V'ade πΕ LJO eos θ + J

fo

=

0

AND

PLASTICITY

p ^ i d í M ] eos - Ö) J

[ e log » + ( * + »>» + * log « + ( « ; + * > » ] •

(592)

The center of the rectangle is the common corner of four rectangles of half the dimensions. It follows that the deflection there is twice the deflection in Eqs. (592). The deflection at any other point inside or outside the loaded rectangle is obtained similarly by combining the influences from four rectangles of other proportions but having a common corner at the point. Problems 15 to 21 deal with deflections of the surface of a semi-infinite solid loaded on the surface by normal forces. Problem 16. (a) Derive the first of Eqs. (589) by direct use of Eqs. (587). (b) Thereafter derive the deflection at a point outside the loaded circle in terms of the angle between the tangents drawn from the point to the circle, (c) What is the average deflection within the loaded circle? Problem 16. Determine the deflections outside the loaded circle in the problem in which the deflection inside the loaded circle is expressed in Eqs. (590). What can be said about the slope of the surface directly outside the loaded area? Problem 17. A total load Ρ is distributed uniformly over the area of a circle with radius a. Determine the deflections at the center, at the circumference, and at a point half way between.24 Problem 18. A total load Ρ is distributed within a circle r = a according to the law ρ = K f , in which / = (1 — r 2 /a 2 ) 4 ' 6 . Determine the constant K, the deflection at the center, and thereafter the deflection at any point within the loaded area. Problem 19. Determine the deflection at the origin of coördinates when the load p = Kx is applied (a) within a triangle bounded by y = 0, χ = a, and ay = bx; (b) within a triangle bounded by χ = 0, y = b, and ay = bx; and (c) within t i e rectangle composed of the two triangles (this problem is basic for Vogt's analysis of deformations of foundations). Problem 20. A total normal load Ρ is distributed in some way over an area A. The area is now expanded by multiplying all its dimensions by the same ratio, and A receives a new value, but the manner of distribution of the total load Ρ is not changed,/remaining the same at comparable points. Show that deflections at comparable points are proportional to P/A^. M Use, for example, B. O. Peirce, A short table of integrals (ed. 3,1929), formulas p. 66 and table p. 123 for elliptic integrals.

GALERKIN VECTOR, TWINNED

GRADIENT

163

Problem 21. Show that in principle—though not in practice—an extended elastic solid with horizontal top and some weights of small dimensions may be used as a planetarium reproducing the motions of the mass centers, including perturbations by interaction of planets, in a special imagined solar system in which all orbits are coplanar. The weights are proportional to the masses of the sun, planets, and satellites, are placed on the surface of the elastic solid in appropriate initial positions, and are given appropriate initial velocities without spin. It is specified that there must be no friction between the weights and the elastic solid.

86. Pressures Transmitted between Two Solids with Smoothly Curved Surfaces through a Small Area of Contact The fundamental solution is due to Heinrich Hertz 25 in 1881. Let two equal spheres of the same elastic material and the same radius Ro be brought together so that without being under stress they barely touch each other at the origin of coordinates, with the horizontal plane z = 0 as a common tangential plane and the axis of + ζ as a common axis. Thereafter let the spheres be pushed together by a pressure Ρ having its resultant along the axis of ±z. An effect will be that the contact is extended from the original point to the area of g. small circle r = a. Because of the symmetry the spherical surfaces are flattened into a plane within the circle of contact. Since the radius Ra of the spheres is very large compared with the radius a of the circle of contact, the stresses and deformations in each sphere in the immediate vicinity of the area of contact will be essentially the same as in a semi-infinite solid under the same pressures within the circle of contact. The deformations of the lower sphere are such that at any point within the area of contact the curvature of the surface in any direction has changed from l/R0 to 0, a change of — l/Ro. Therefore the pressures on the semi-infinite solid must have such a distribution within the loaded circle that they produce the constant curvature — l/Ro in any direction within that area. The second of Eqs. (589) shows that such a change is produced by a pressure ρ = Kf when/is represented by the ordinates of a hemisphere with radius a. I t follows that i Ro

=

3(1 - M2)i\ 4Ea>

f593) m

Outside the circle of contact the deflections of the two surfaces will be such as to leave a space between the surfaces. Equation (593) permits the computation of a and thereafter of /»max. 25

See references 17 and 18.

164

THEORY OF ELASTICITY AND PLASTICITY

The two solids of the same material need not be equal spheres to produce this distribution of pressures of contact. Clearly it is sufficient that the surfaces be smooth in the vicinity of the initial point of contact, and that in this vicinity the initial distances between the two surfaces in the direction perpendicular to the common tangential plane 2 = 0 be the same as for the two equal spheres; at any point that distance is r2/R0. This distance can be obtained with two spheres with radii Ri and R¡, or instead with two cylinders with the same radius c crossing each other at right angles. It is sufficient that

¿=Κέ+έ) or

respectively. The spheres or cylinders need not be complete spheres or cylinders. If 2?2 = 00, the corresponding surface is plane. If R2 is numerically greater than i?i, the spherical surface with radius Ri may be concave; but then the value of R2 in the formulas will be negative. By substituting from the first of Eqs. (594) in Eq. (593) one finds Γ 3(1 — μ2) Ρ |_2£(i?i _ 1 +

1 [ZP&iRr 1 + J?2~1)ii"l

3P or

Pmxx

Ι» J '

2

~ 2ττα ~ π|_ 2

2(1 - μ2)2 1

/ W = Ο^Ο^Ε ^!" +

J

(595)

when μ = 0.3. It is a simple matter to extend this analysis to the problem in which the spheres with radii Ri and J?2 have different values of E and μ; namely, Ει, μι, and Eì, μ2 respectively. Then Eq. (593) is replaced by 3 P / 1 - Mx2

1 , 1

1 - μΛ

which defines a. Thereafter one finds _ 1

Pmax.

ePjRr 1 + flr1)2 Λ - Ml2 , 1 - μΛ' \ Εχ ^ Ε2 ) _

(597)

In the general problem of contact of two elastic solids with smooth surfaces, when the plane ζ = 0 is the initial common tangential plane and the origin of coordinates is the initial point of contact, the initial distance between the two surfaces in the vicinity of the point of contact will not be

G A L E R K I N VECTOR, TWINNED G R A D I E N T

165

proportional to r2 but must be assumed to be some other quadratic function of χ and y. By a proper choice of the directions of χ and y this quadratic function can be brought into the form Cx2 + Oy1. Then, if the two solids are of the same material, when pressure is applied, each solid will receive deflections of the form of Eq. (591) within the area of contact; and in accordance with the comments that accompanied Eq. (591) the area of contact will be an ellipse with principal radii a and b in the directions of χ and y. Hertz's theory 26 leads to the formulas that follow which are related to those applying when C = D. It is expedient to use the following notation: C0Sa

D — C = D+ c

, and

1 1 /I - μι2 , 1 - μΛ = + - Ε Γ ) ' Έ> 2[-ΈΓ

,_ otA (598)

in which Ει, μι, E%, and μι are the values of E and μ for the two solids; E' may be regarded as a modified average modulus of elasticity; fo,i is the deflection of the center of the area of contact toward the interior of the solid having the constants Ei and μι, the deflection being measured relative to a region of the interior in which the deformations are negligible in comparison with those at the area of contact; f0,2 is the deflection similarly defined for the other solid; accordingly both fo.i and fo,2 are positive, and they are measured in opposite directions and relative to two different regions; fo.i + fo,2 is the total approach of these regions toward each other; and finally Χ , Υ, Ζ are three functions of a in Eqs. (598), all pure numbers. Then one may write a =

fo.1 =

X\_2E'{C

(1

+ D)Ì'

Ô =

Y[?

[ I P*E'(Ç + D)].

£'(C + Z?)T'

(599)

The functions X , Y, and Ζ of α can be expressed in terms of elliptic integrals. Hertz gave a short table of numerical values of X and Y. Tuckerman's extension of Hertz's table 26 includes more values of X and Y and values of Z. Table 1 is Hertz's table as extended by Tuckerman. I t permits numerical computations by means of Eqs. (598) and (599). 26

See references 17 to 21.

166 TABLE

α 30° 35° 40° 45° 50° 55° 60° 65°

T H E O R Y OF E L A S T I C I T Y AND P L A S T I C I T Y 1. Hertz's table as extended by Tuckerman: values of X, Y, and Ζ in Eqs. (599) for different values of α in Eqs. (598), X 2.731 2.397 2.136 1.926 1.754 1.611 1.486 1.378

Y 0.493 0.530 0.567 0.604 0.641 0.678 0.717 0.759

Ζ 1.453 1.550 1.637 1.709 1.772 1.828 1.875 1.912

a 70° 75° 80° 85° 90° 95° 100°

Χ 1.284 1.202 1.128 1.061

Y 0.802 0.846 0.893 0.944

1.000

1.000

0.944 0.893

1.061 1.128

Ζ 1.944 1.967 1.985 1.996 2.000 1.996 1.985

For two spheres of the same material and with the same radius Ra one finds C = D = Ra-\ a = 90°, X = Y = 1, and Ζ = 2. Substitution of these values in Eqs. (599) leads readily to agreement with Eqs. (595) and (588). This serves toward confirming the forms of Eqs. (599). In his paper of 1881 Hertz used the total approach f 0,i + ¿"0,2 to investigate impact, especially of two spheres. This study included a determination of the duration of the impact. The two-dimensional problem of contact of two cylinders with parallel axes, or of a cylinder and a solid with plane top, may be regarded as a special form of Hertz's problem in which D = 0 and b = 00 so that the contact will be within a strip of width 2a; but to solve this problem the direct approach through two-dimensional theory is preferable. After the area of contact and the maximum pressure per unit area have been determined one may undertake to explore the state of stress in the vicinity of the area of contact. A significant result of such explorations 27 is that, when the area of contact is a circle, the greatest shearing stress occurs in the interior of the solid at a depth of about one-half the radius of contact, and its value is about 0.31pmaz when μ = 0.3. In his paper of 1882 on contact and hardness, Hertz realized the importance of his theory of elastic contact to the problem of hardness. Hardness of a material involves more than a limit of the elastic stage because, if any permanent indentation is created by contact under pressure, some stresses will have entered the plastic stage. Hertz's theory of contact, however, has been an example of how an analysis of the elastic stage may serve as a basis for discussion of the subsequent plastic stage and may be of help in the appraisal of experiments in which the plastic stage has begun. " See S. Timoshenko, Theory of elasticity (1934), pp. 344 and 350.

GALERKIN

VECTOR, T W I N N E D

GRADIENT

167

Problem 22. A spherical ball with radius R0 rests in a cylindrical groove with radius 1.52?0 and axis in the direction of y. The material is the same in the ball and the base. Determine the values in Eqs. (599) in terms of Ρ, Ε, μ, and Ra, and compare the results with those that would be found if the groove were replaced by a plane.

Indexes

AUTHOR Adkins, A. W., 30 Airy, G. B., 16 Auerbach, F., and W. Hort, 8, 21 Bach, L., 18 Bauschinger, J., 18 Becker, A. J., 39 Beltrami, E., 18 Bernoulli, J., 9 Betti, E., 18 Biezeno, C. B., 37 Böker, R., 39, 40, 66 Boobnov, J. G., 36 Boussinesq, J., 13,15,18,136,160 Brandtzseg, Α., 39, 40, 65 Bresse, 19 Brewster, D., 11, 31 Bridgman, P. W., 22, 97, 104 Brown, R. L., 39, 65 Bryan, C. W., 38 Bryan, G. H., 19 Butty, E., 21 Carus Wilson, C. Α., 17, 31 Casagrande, Α., 99 Castigliano, Α., 18, 30, 41 Cauchy, A. L., 7, 12, 60, 73, 74 Cerniti, V., 18,136 Chladni, 11 Clapeyron, Β. P. E., 13, 14 Clebsch, Α., 15 Coker, E. G., and L. N. G. Filon, 21,31 Colonnetti, G., 30 Considère, Α., 37 Coulomb, C. Α., 10 Cross, H., 41, 42 Culman, C., 18 Dougall, J., 33 Drucker, D. C., 32 Eggenschwyler, Α., 22 Engesser, F., 18, 19, 30, 37 Estanave, E., 35 Euler, L., 6, 9, 37

INDEX Filon, L. N. G. See Coker and Filon Flamant, 15 Föppl, Α., 20, 29, 33, 66 Föppl, L., 20, 29, 66 Frocht, M. M., 21, 32 Frye, R. N., 99 Galerkin, B. G., 27,28,35,36,120,130 Galilei, G., 9 Germain, S., 11 Glover, R. E., 44 Goodier, J. N., 154 Gough, H. J., 154 Green, G., 12 Greenberg, H. J., 31 Greenhill, A. G., 19 Griffith, Α. Α., 26 Guest, J. J., 39 Happel, H., 36 Heerwagen, F., 161, 165 Hencky, H., 35, 36, 40 Hertz, H., 19,161,163,165 Hodge, P., 30 Hooke, R., 1, 9 Hort, W. See Auerbach and Hort Houk, I. E., 43 Huber, M. T., 66 Jasinsky, F., 37 Johnson, J. B., 38 Kármán, T. von, 37, 39, 40, 66 Kellogg, O. D., 160 Kelvin, Lord, 17,18, 34,133 Klein, F., 20 Koch, 37 Lagrange, J. L., 6 , 1 1 , 1 3 Lamé, G., 13, 60, 64, 74, 89, 100, 102, 104 Land, R., 51 Leitz, H., 36 Lévy, M., 15, 16, 35, 39 Lieurance, R. S., 43, 44

172

AUTHOR

Ljungberg, K., 113 Loewenstein, L. C., 113 Lorenz, H., 8, 20, 29 Love, Α. Ε. Η., 8, 20,27, 33, 74, 88,89, 130 Maillart, R., 22 Mariotte, F. E., 9 Maxwell, J. C., 16, 17, 28, 31, 41 Meinesz, V., 26 Melan, J., 19, 38 Mesnager, Α., 31, 36 Micheli, J. H., 23, 33 Mindlìn, R. D 28, 32,142, 145 Mises, R. von, 40, 66, 67, 84 Mohr, O., 18, 39, 41, 51, 61, 64, 65, 66,67 Moiseiff, L. S., 38 Moore, H. F., 21 Müller, C., 20 Müller-Breslau, H., 18, 41 Nádai, Α., 20, 24, 33, 34, 35, 36, 40, 41 Navier, L. M. H., 7, 8,11, 35 Neumann, F., 19 Nichols, J. R., 36 Nielsen, N. J., 36, 37

INDEX Richart, F. E., 39, 65 Ritter, W., 18 Ritz, W., 29 Saint-Venant, B. de, 7, 8, 13, 14, 15, 25, 33, 39, 117,150 Savage, J. L., 43 Savart, 13 Slater, W. Α., 9, 36 Sokolnikoff, I. S., 21 Southwell, R. V., 21, 43, 154 Steinman, D. B., 38 Stodola, Α., 113 Tait, P. G., 17, 34 Taylor, G. I., 26 Tetmajer, L. von, 18 Thomson, W. See Kelvin Timoshenko, S., 21, 26, 29, 34, 36, 38, 66, 148, 154, 161, 165, 166 Todhunter, I., and K. Pearson, 8, 14, 20, 73, 74 Tresca, H., 15 Tuckerman, L. Β., 161, 165 Turneaure, F. E., 38

Ostenfeld, Α., 37, 41,113

Vogt, F., 141,161 Voigt, W., 19

Papcovitch, P. F., 120 Pearson, K. See Todhunter and Pearson Peirce, B. O., 127, 162 Petrenko, S. N., 161, 165 Poisson, S. D., 7, 12, 34, 80 Prager, W., 30, 41, 69, 95 Prandtl, L., 19, 23, 24, 40, 58

Westergaard, H. M., 9, 28, 30, 36, 38, 43, 61, 137, 140, 141, 148 Whittemore, H. L., 161, 165 Wieghardt, K., 26 Wühler, Α., 18 Woinowsky-Krieger, 33 Wojtaszak, I. Α., 36

Rankine, W. J. M., 15, 37, 64 Rayleigh, Lord, 17, 29

Young, D., 36 Young, T., 10, 80

SUBJECT Aelotropic materials, 4 Analogies, 23 Arch dam, analysis of, 43 Basic differential equation of elasticity, 87,90 solutions of, 100, 119 Beams, 26 bending of, 9, 11 Biharmonic functions, 122, 125 obtained from harmonic functions, 126 Biharmonic vector functions, 122, 128 Body forces, 75 in rotating disk, 113 in two-dimensional state of stress, 112

use of strain potentials to determine the effects of, 111 Boulder Dam. See Hoover Dam Boussinesq's problem of normal force on plane surface of semi-infinite solid, when Poisson's ratio has any value, 139 when Poisson's ratio is 136 solution by combination of Galerkin vector and strain potential, 141 Buckling, 37 Bulk stress, 68, 80 Centrifugal force in rotating disk, 113 Cerruti's problem of tangential force on plane surface of semi-infinite solid, when Poisson's ratio has any value, 141 when Poisson's ratio is 136 solution by combination of Galerkin vector and strain potential, 142 Column, 26 Compatibility of strains and detrusions, 91 Complementary energy, 29 Contact pressures between solids. See Pressures Creep, 98

INDEX Cylinders, crossing at right angles^ pressure between, 164 hollow, Lamé's formulas for stresses and displacements of, 102 stress concentration factors computed from, 103 plane state of strain in, 104 rupture of, 104 thermal stresses in, 117 Cylindrical coordinates, 91 Deflections of the surface of a semiinfinite solid under various loads, 157 interpretation of by gravitational field, 158 Deformation, 1, 78 ellipsoid of, 84 Deformation method in structural mechanics, 41 Detrusion, 79, 82, 84 compatibility of with strains, 91 in cylindrical coordinates, 92 plastic, 95 Dilatation, 79 Direction cosines, 57 Disk, with circumferential shear at inner and outer concentric circular edges, 106 rotating, thick, 148 thin, 113 thermal stresses in, 117 Displacement, 78 in cylindrical coordinates, 92 Dyad, 56 divergence of, 76, 94 Dyadic, 56 stress, 58 strain, 84 Dyadic circle, 51 Elasticity, 1 one-, two-, and three-dimensional problems of, 6 history of, 8

174

SUBJECT

Elasticity, limits of. See Limits of stress basic differential equation of. See Basic Electrical analogy for torsion, 25 Ellipsoid, of deformation, 84 gravitating, application to deflections on surface of semi-infinite solid, 159, 160 rotating about axis of revolution, 150 of stress, 13, 59 Energy, complementary, 29 potential, 28 principles of minimum of, 28 strain, 28 stress, 29 Equilibrium, differential equations of, 75 in cylindrical coordinates, 76 Extended solid, example of vertical load on the horizontal surface of, 131 with horizontal surface, used as a planetarium, 163 single force applied in the interior of, 133. See also Kelvin's problem, Semi-infinite solid Fatigue, 97 Galerkin vector, 28, 119 application of to problems in two dimensions, 122 displacements and stresses obtained from, 121 equivalent, 124 when Poisson's ratio is i , 123 proved to be general solution of basic equation of elasticity, 123 relation to strain potential, 122 for solids of revolution, 129 Gradient of scalar function, 56 in cylindrical coordinates, 93 Hardness, 166 Harmonic functions, 101, 125 methods of obtaining, 126 History of elasticity, books since 1900, 20 founding of three-dimensional theory, 11 origins, 9 period of the classics, 13 works of, 8

INDEX Hole, spherical, stress concentration at, 154 Homogeneity, 2 Hooke's law, 1, 9, 79 Hoover Dam, studies for design of, 26 analysis of, 43 Hydrostatic state of stress, 60, 68 Impact, 166 Instability, 37 Isotropy, 3 Kelvin's problem of single force applied in interior of extended solid, 133 when Poisson's ratio is i , 135 Lamé's constants, 13, 88 Laplace's equation, 101 Laplace's operator, 87 in cylindrical coödinates, 93 in polar coordinates, 152 Limits of stress, hypotheses on, 39, 64, 66 tests of, 65 Love's notation, 88 Love's strain function for solids of revolution, 129 in polar coördinates, 151 Membrane, 32 Membrane analogy, for bending of beams, 25 for torsion, 24 Mindlin's problem, force applied at some distance below horizontal surface of semi-infinite solid, part I : vertical force, 142 part II: horizontal force, 145 Modulus of elasticity, 10, 80 in shear, 82 Modulus of rigidity, 82 Mohr's circles, 61 Moment distribution, 41 Octahedral plane, 66 Octahedral strain, 95 plastic, 95 Octahedral stress, 66, 69 determined from six components of stress, 67 direction of, 72

SUBJECT Photoelasticity, 11, 17, 31 three-dimensional, 32 Plane state of strain, 91 Plane state of stress, 49 in a plate, 90 relation to plane state of strain, 91 Plastic detrusions, 95 Plastic flow of concrete and rock, 98 Plastic octahedral strain, 95 Plastic strain deviations, 95 Plasticity, 1, 39 departures from simplest laws of, 97 flow and deformation theories of, 97 one-, two-, and three-dimensional problems of, 6 simplest conceivable laws of, 95 Plates or slabs, 11, 13 bending of, 32 conditions of edge support of, 34 plane stress in, 90 problems of, 34 thermal stresses in, 116 Poisson's ratio, 12, 80 effects of a change of, 137. See also Twinned gradient limits of, for isotropic materials, 80 simple solutions for particular values of, 137 special case when value is 89, 123 Polar coordinates, 151 Potential energy, 28 Pressures transmitted between two solids with smoothly curved surfaces through small area of contact, 163 deflections caused by, 165 greatest shearing stress caused by, 166 Principal directions, 52, 59 Principal stress, 52, 59 determined from six components of stress, 67 Relaxation, 43, 98 Resultant force on surface of revolution when stresses are symmetric about axis, 107 Resultant stress, 46 Roof-and-membrane analogy, 24 Rotating ellipsoid, 150 Rupture, 96 of hollow cylinders, 104

INDEX

175

Saint-Venant's principle, 117, 150 Sand hill analogy, 24 Semi-infinite solid, deflections of surface of, under various normal loads, 157 horizontal force applied at some distance below the horizontal surface of, 145 normal force on plane surface of, 136. See also Boussinesq's problem tangential force on the plane surface of, 136. See also Cerruti's problem vertical force applied at some distance below the horizontal surface of, 142 See also Extended solid Shear center in beams, late discovery of, 22 Shearing strain. See Detrusion Shearing stress or shear, 47, 61 octahedral. See Octahedral stress Slab, 26. See also Plates Slabs and slices, analogy of, 26 Slice, 26 Spheres, hollow, Lamé's formulas for stresses and displacements of, 104 stress concentration factor computed from, 106 thermal stresses in, 118 Spherical hole, stress concentration at, 154 Spherical surfaces in contact, 163 Stabilization by load, 38 Strain, 1, 78 analysis of, as if stress, 84 compatibility of with detrusions, 91 in cylindrical coordinates, 92 large, 97 octahedral. See Octahedral strain true, 97 Strain deviations, 95 plastic, 95 Strain dyadic, 84 Strain energy, 28 Strain functions, 23 Strain gages, 85 Strain potential, 100 axially symmetric, 107 equations for, 100 * in cylindrical coordinates, 101 in polar coordinates, 151

176

SUBJECT

Strain potential, examples of, 108, 111 relation to Galerkin vector, 122 used to study body forces, 111 used to study thermal stresses, 115 Strain tensor, 84 determined from measured strains, 85 in a concrete dam, 86 Stress, 1, 46 in any direction, 49 bulk, 68, 80 components of, 47 compressive, 47 equilibrium of. See Equilibrium hydrostatic state of, 60, 68 limits of. See Limits normal or tensile, 4 7 j notation for, 46 octahedral. See Octahedral stress plane state of. See Plane principal. See Principal shearing, 47, 61 uniform state of, 48, 56, 81 Stress concentration, at cylindrical holes, 103 a t spherical holes, 106, 154, 156 Stress deviations, 69 principal, 69, 71 Stress dyadic, 58 in cylindrical coordinates, 94 Stress ellipsoid, Lamé's, 13, 59 Stress energy, 29 Stress functions, 16, 23 Stress quadric and two additional surfaces that represent stress tensor, 73 Stress tensor, 58, 68 invariants of, 68 Stress-director surface, Lamé's, 13, 74 Spherical holes, stress concentration at, 106, 154, 156

INDEX Successive approximation, methods of, 41 Superposition, principle of, 88 Suspension bridge, stiffening of, 38 Tensor, stress. See Stress tensor strain. See Strain tensor Thermal expansion, coefficient of, 114 Thermal stresses, 114 in disk, 117 in long cylinder, 117 in plate, 116 in sphere, 118 two-dimensional problems of, 115 Torsion, 14 Trial-load method for analysis of stresses in dams, 43 True strain, 97 Twinned gradient, 137 application of, to determine the effects of a change of Poisson's ratio, 137 example of application of, 139 solution of Boussinesq's problem by use of, 139 solution of Cerruti's problem by use of, 141 solution of problem of thick rotating disk by use of, 148 Uniform state of stress, 48, 56, 81 Vector strain, 83 Vector stress, 46 on any section, 53 Vectors, 55 divergence of, 57 operations with, 55, 56 position, 56 unit, 56 Young's modulus, 10, 80