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MATHEMATICAL MODEL FOR PREDICTING ANISOTROPIC EFFECTS IN PLASTICITY By CARL GOTTLIEB LANGNER A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1973 DEDICATION To my wife, Ann, whose constant help and encouragement made this work possible. ACKNOWLEDGMENTS The author wishes to thank Professor L. E. Malvern for his help in preparing this dissertation. The author heartily commends Professor Malvern for his courage and faith that enabled him to support and defend his student, the author, even when at times he was not entirely convinced of the worth of his student's unorthodox ideas. The author acknowledges discussions with the following persons which helped to congdal the basic ideas of the radial element theory: M. A. Eisenberg, I. K. Ebcioglu, E. K. Walsh, R. E. ReedHill, P. R. Paslay, N. Cristescu, Ion Suliciu, C. S. Ting, and Richard Johnson. The author received technical help from Guy Demoret, J. D. Macmillan, and Bill Luckhurst. Thanks are also due the National Science Foundation, which helped support the author by Grant No. GK23452, and to the University of Florida, which provided the majority of computer time used in this study. iii TABLE OF CONTENTS Acknowledgments . . . List of Figures . . . Page 111 Abstract . .. . vii Chapters I Introduction. .. . ..... 1 II General Concepts of the Radial Element Theory 17 III A Class of Elemental Stress Vectors for Predicting Elastic Material Behavior 29 IV A Class of Elemental Stress Vectors for Predicting ElasticPlastic Material Behavior. 41 V StressStrain Behavior for Axial Loading and Reversed Axial Loading. . ... 52 VI StressStrain Behavior for the Class of Loadings Involving Fixed Principal Axes .. .68 VII StressStrain Behavior for Biaxial Loadings and for a General Loading History . .. 91 VIII Comparisons between the Theory and Experimental Results for Commercially Pure Aluminum. ... .108 IX Conclusions and Recommendations for Future Work. 135 Bibliography . . 145 Appendices A Computer Programs . . B Tabulated Results . . C Yield Surfaces in Principal Stress Space. . D Yield Surfaces in Principal Strain Space. . S152 . 162 . 170 . 184 Biographical Sketch. . .. ..... 198 LIST OF FIGURES Figure Page 1 Basic Concepts of the Theory. . 20 2 Spherical Coordinates .... .. .34 3 Distribution of Elemental Stress Vectors for Axial Loading from an Initially Isotropic State . . 54 4 StressStrain Behavior for Axial Loading, Small Strain .. ... .. .. 57 5 StressStrain Behavior for Axial Loading, Large Strain. ... '. .... . 58 6 Distribution of Elemental Stress Vectors for Reversed Axial Loading from an Initially Plastic State . ..... .. .. 61 7 Axial Loading and Reversed Axial Loadings for X =0 65 8 Axial Loading and Reversed Axial Loadings for X=1.5. . . .. 66 9 StressStrain Behavior for Shear Loadings 74 10 Numerical Errors for Axial Loading; Ordinary Trapezoidal Integration, X =0 . .. 76 11 Numerical Errors for Axial Loading; Ordinary Trapezoidal Integration, X =1 .... .... 77 12 Numerical Errors for Axial Loading; Modified Trapezoidal Integration, =0 . .. .78 13 Numerical Errors for Axial Loading; Modified Trapezoidal Integration, =1 . 79 14 Surfaces of Constant Offset Strain for an Initially Isotropic Material. .......... ... 84 15 Surfaces of Constant Offset Strain after an Axial Loading. . . .85 Figure Page 16 Surfaces of Constant Offset Strain after a Pure Shear Loading. . . 86 17 StressStrain Curves for Biaxial Strain Loadings 98 18 StressStrain Curves for Biaxial Stress Loadings 99 19 Loading Paths for Biaxial Strain with el2 Constant. . . .100 20 Loading Paths for Biaxial Strain with ell Constant. . .. .101 21 Loading Paths for Biaxial Stress with s12 Constant . .... 102 22 Loading Paths for Biaxial Stress with S11 Constant . .... 103 23 Apparatus for TensionCompression Experiments. 112 24 Specimen for TensionCompression Experiments 113 25 Comparison of Experiment and Theory for Tension CompressionTension Tests. . ... 115 26 Comparison of Experiment and Theory for CompressionTensionCompression Tests. 116 27 Comparison of Experiment and Theory for Axial Loading and Reversed Axial Loading .. 117 28 Comparison of Experiment and Theory for Axial Loading and Reversed Axial Loading .. 118 29 Testing Machine for Biaxial Stress Experiments 121 30 Strainometer and Specimen for Biaxial Stress Experiments. . . ... 122 31 Comparison of Experiment and Theory for Biaxial Tests, Axial Stress versus Axial Strain. .. .. 127 32 Comparison of Experiment and Theory for Biaxial Tests, Shear Stress versus Axial Stress. .. .. 128 33 Comparison of Experiment and Theory for Biaxial Tests, Shear Strain versus Axial Strain. .. .. 129 34 Ratio of Stresses Compared with Slope of Strain Path for Large Strains .. . .130 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MATHEMATICAL MODEL FOR PREDICTING ANISOTROPIC EFFECTS IN PLASTICITY By Carl Gottlieb Langner August, 1973 Chairman: Lawrence E. Malvern Major Department: Engineering Science, Mechanics, and Aerospace Engineering Existing theories of plasticity do not adequately describe the anisotropic effects of plasticity. Among the effects not easily incorporated into existing theories are the Bauschinger effect and the observed changes in the yield surface during complex loadings. A new theory of plasticity, the radial element theory, is introduced in this dissertation, which predicts stressstrain curves that are similar in shape to the actual stressstrain curves of many metals. The theory predicts a Bauschinger effect of the Masing type that compares closely with experimental data. The theory also agrees fairly well with experimental data for biaxial loadings of the torsion tension type. In addition, yield surfaces are predicted by vii this theory that in some ways are similar to those observed experimentally by others. The most unusual feature of the theory is the occurrence of a nonsymmetric stress tensor for loadings that involve rotations of the principal strain axes relative to the material. The general form of the radial element theory is not restricted to small deformations or to any one type of material behavior. Hence, it may find some use in other fields as well as in plasticity. viii CHAPTER I INTRODUCTION This work is a contribution to the theory of plasticity, or more specifically, to the theory of isothermal, time independent, stressstrain relations for polycrystalline metals. A new theory is introduced here for predicting elasticplastic stressstrain behavior. The new theory is designed primarily to predict the threedimensional anisotropic effects observed in metals for reversed loadings and for loading histories involving combined stresses. These are precisely the effects that are not predicted very well by existing theories of plasticity. We begin our discussion of plasticity with some brief definitions of the more frequently used terminology. For detailed discussions of the concepts, terminology, and no tations of plasticity theory the reader should consult one or more of the general expositions [15].1 Elasticity is the property of a solid material once deformed to return exactly to the shape it had before deformation when all loads 'Numbers in brackets designate reference articles and books listed in the Bibliography. are removed. Elasticity implies a onetoone functional relationship between stress and strain. Plasticity is the property of a material whereby permanent or residual strain develops and is maintained when a certain yield condition is satisfied or exceeded. Yield condition defines the boundary between elasticity and plasticity; it is the set of stress or strain states situated at the outer limits of elastic behavior. Stress space and strain space are mathematical vector spaces having, respectively, the nine components of the stress or strain tensors as coordinates. In stress or strain space the yield condition forms a hypersurface called the yield surface. Loading point is the point in both stress and strain space that describes the current state of stress and strain. Residual strain point is a hypothetical loading point, corresponding to an elastic unloading from the current loading point, defined by zero stress. A loading is any motion of the loading point that involves plasticity. Loading path is the locus of the loading point as it moves in both stress and strain space during a loading. Direction of loading is the tangent to the loading path in either stress or strain space. Radial loading is a loading such that the loading path is a straight line passing through the origin in either stress or strain space. Reversed loading is a loading, following after an initial loading, in which the loading point retraces the original loading path back to and perhaps beyond the origin in either stress or strain space. Workhardening is the phenomenon whereby the yield stress increases during any approximately radial loading. More generally, it is the tendency of the yield surface to expand in directions surrounding the direction of loading in stress space. Bauschinger effect is the phenomenon whereby the yield stress, for a hypothetical reversed loading, decreases or follows the loading point during any approximately radial loading. More generally, Bauschinger effect is the tendency of the yield surface to shrink or follow the loading point in directions roughly opposite the loading direction in stress space. The concepts of workhardening and Bauschinger effect also often include the functional stressstrain relationship that occurs during a radial loading and a reversed radial loading, respectively. Two simple hypotheses regarding work hardening are isotropic hardening, whereby the yield surface in stress space expands uniformly while maintaining its original shape and position, and kinematic hardening, whereby the yield surface in stress space translates according to a certain rule while maintaining its original shape and size. In this work, the material under discussion will always possess a domain of elasticity in which the stressstrain relationship is linear and isotropic relative to a current residual strain. The elastic domain always includes the current loading point but does not necessarily include the current residual strain point. Initially the residual strain is zero and the yield condition is isotropic, so that the stressstrain behavior for a given loading history is inde pendent of the initial direction of loading. Subsequent yield conditions generally are not isotropic, so that loading directions are important in determining the stressstrain relations after initial plastic yielding. In practice, the initial isotropic state of a polycrystalline metal can be restored by annealing or by melting and careful solidification. Throughout this work we assume that the yield condition is not influenced by mean stress and that the residual or plastic part of the strain occurs at constant volume. The first assumption was confirmed experimentally for several metals by Crossland [6]. From these assumptions it follows that mean stress and mean strain are elastically related. Rate effects (creep and relaxation) are ignored; either the material is inviscid or all loadings occur at the same constant rate. Thermal effects are ignored; either the material is insensitive to temperature changes or deformations occur at the same constant temperature. Since the present subject is stressstrain relations, we can simply disregard the equations of motion and the energy relations of a continuum. There are two main classes or categories of plasticity theories. The first class we shall call the phenomenological theories, since the basic ideas of this class of theories 5 are directly concerned with the relationship between stress and strain. The phenomenological theories were developed in a long series of papers [7 28] by many authors. Early examples of this class of theories are: (1) the theory of perfectplasticity which neglects elastic strains, developed by Saint Venant [8], Levy [9], and von Mises [10]; (2) the theory of perfectplasticity which includes elastic strains, developed by Prandtl [11] and Reuss [15]; and (3) the total deformation theory of Hencky [12]. These early perfect plasticity theories assumed an isotropic yield surface and no workhardening, and they required the plastic strain increment to be proportional to the deviator stress. Because they ignored workhardening, these theories do not represent very accurately the stressstrain behavior of metals. The Hencky theory related stress to the strain rather than to increments of strain and it included workhardening. This theory has been shown to be unsuitable for describing plastic behavior for any loadings other than radial and nearradial loadings [1]. Because of their simplicity, and in spite of their shortcomings, the abovementioned theories have been applied extensively to technological problems [1, 2]. The perfectplasticity theories were generalized by Ros and Eichinger [16], Melan [17], Prager [18], Hill [19], Drucker [20], and others to include workhardening. The resulting family of theories is known as plastic potential theory, because the plastic strain increment is expressed as a derivative of a potential function with respect to stress. The simplest and most common form of plastic potential theory combines the ideas of initial elasticity, a smooth initial yield surface (usually the Mises condition), isotropic workhardening, and a flow rule defined by the requirement that the direction of the plastic strain increment be the same as the normal to the yield surface at the loading point. Further generalizations were made by Koiter [21], Sanders [22], and Hodge [26], who developed the idea that more than one yield surface can be operating simultaneously; this idea permits the formation of corners from an initially smooth yield surface. The concept of kinematic hardening was proposed by Ishlinskii [23] and Prager [24, 25], and a modified form of kinematic hardening was proposed by Ziegler [27]. Mroz [28] proposed a theory which combines the concepts of isotropic hardening, kinematic hardening, multiple yield surfaces, and a field of workhardening moduli; this theory can be made to predict workhardening and a piecewiselinear Bauschinger effect. The phenomenological theories possess two advantages over other plasticity theories. (1) Most of these theories are simple enough to use in the solution of problems involving engineering structures. (2) Numerous experimental studies [29 32] have demonstrated the validity of the plastic potential theories for loadings in which the plastic deformation continues without interruption. See Hill [1, pp. 2232] for references to early experimental work. These experimental studies generally avoided reversed loadings and unloading followed by reloading in a different direction, situations in which the phenomenological theories do not provide accurate predictions. There are two principal objections to the phenomenological plasticity theories. (1) The theories lack the unity of a truly predictive theory. By this is meant that as new ideas and new experimental data become available the phenomenological theories must be repeatedly modified and made increasingly complex with new assumptions in order to account for each new phenomenon. A more satisfactory theory would not require this layering of assumption upon assumption, but would predict all essential phenomena with the simplest and most basic form of the theory. (2) There does not appear to be any easy way to include in these theories a realistic Bauschinger effect or any of the related anisotropic changes in the yield surface as a function of the loading history that have been observed experimentally. The first objection is posed here by the author. The second objection, which concerns mainly the continuing lack of an adequate hardening hypothesis, has been stated before in a variety of places; see Zizicas [3(b)], Naghdi [4, p. 144], Green and Naghdi [5, p. 253], Iwan [33, p. 612], and Batdorf and Budiansky [34, p. 323]. The following passage from Iwan [33] is pertinent: One such theory, the incremental theory of plasticity, describes the threedimensional yielding behavior of a material in terms of a yield surface in stress space, along with a flow rule and a workhardening law. To date, this theory has been applied quite successfully to the rather large class of problems which have to do with the monotonic loading behavior of materials. However, when the theory is extended to the analysis of a cyclic and hysteretic behavior, certain difficulties arise. For the most part, the difficulties result from the requirement that the theory be capable of accounting for a Bauschinger effect. Furthermore, experimental results indicate that this Bauschinger effect should have a specific form; namely, that the stressstrain curves associated with onedimensional symmetrical closed hysteresis loops should be of the same form as those of stabilized initial loading curve (or cyclic stressstrain curve for cyclic hardening or softening systems) except for an enlargement by a factor of two. The idea that hysteresis loops should have this form was first suggested by G. Masing and is often referred to as Masing's hypothesis. The recent development of the concept of kinematic hardening by Prager has provided a means for introducing a Bauschinger effect into the incremental theory of plasticity but, even with this extension, stressstrain behavior satisfying Masing's hypothesis has only been obtained for the special case of a linear work hardening law. When the workhardening is not linear, it would appear that some additional assumptions need to be added to the kinematic hardening hypothesis but as yet these assumptions have not been set forth. The reader who wishes further information on cyclic loading is referred to two recent papers by Krempl [35] and Feltner and Landgraf [36]. Masing's hypothesis on the form of the stabilized hysteresis curve was originally proposed in the article [37]. Numerous recent papers [32, 38 53] have reported experimental investigations of the initial and subsequent yield surfaces in metals. The metals tested included copper, brass, mild steel, and various alloys of aluminum. All papers but one [47] reported biaxial tests performed on tubular specimens. In [38, 40, 50] the tests involved tension and internal pressure, and in [42, 48] the tests involved torsion and internal pressure. The remaining papers reported combined tension and torsion loadings. Most of the papers dealt with the question of whether or not a corner is formed in the yield surface at the loading point. Two methods have been used to study this question. The first method consisted of probing with the loading point in various directions in stress space and thereby detecting a portion of the yield surface according to some given definitionof yielding. This was usually done both initially and after a certain prestrain. The second method consisted of imposing on the material a zigzag loading path in stress space and then measuring and plotting the corresponding plastic strain increments. According to Drucker [20], if the yield surface is smooth then the direction of the plastic strain increment must be normal to the yield surface and independent of the stress increment. If abrupt changes are observed in the direction of the plastic strain vector corresponding to the abrupt changes in the stress vector, then a corner must exist at the loading point. The following general conclusions can be drawn from the experimental papers [32, 38 53]. The initial yield surface is closer to the Mises condition than to the Tresca condition. All subsequent yield surfaces are smooth and convex, and the direction of the plastic strain increment is always very close to the yield surface normal at the loading point. The plastic strain increments appear to depend only very slightly upon abrupt changes in the directions of the stress increments. Subsequent yield surfaces exhibit a region of high curvature surrounding the loading point, but a sharp corner is not formed. A flattened region generally forms on the side of the yield surface opposite the loading point for data plotted in the shear stressaxial stress plane. The experimentally observed characteristics of the yield surface depend intimately upon which definition of yield is used. A proportional limit definition of yield [32, 40, 41, 52, 53] gives yield surfaces which translate away from the origin in stress space (in the direction of loading) and which drastically change shape while showing little or no crosseffect (lateral expansion or contraction) during a radial loading. A backwardextrapolation definition of yield [42, 48, 49] leads to yield surfaces which expand nearly uniformly while translating very little; here work hardening and a large crosseffect occur simultaneously. The sensitivity of the yield surface to the yield definition led one author [47, 50] to report his results as families of curves of constant offset strain; this procedure avoids the subjectivity of the proportional limit definition and the arbitrariness of any other definition. The second class of plasticity theory may be called the physical theories. Examples of the physical theories are presented in the papers of Batdorf and Budiansky [34], Besseling [54], Hutchinson [55], Lin and Ito [56, 57], and Wells and Paslay [58]. Studies of the microstructure of metals has led to the following conclusions. (1) Crystallo graphic slip is the principal process of plastic deformation in facecenteredcubic metals at low and intermediate temperatures [57]. (2) The phenomenon of workhardening and the Bauschinger effect are due to lockedin stresses produced by large numbers of dislocations that accumulate and become stuck at grain boundaries and around alloying elements in the material [59, Chapter V]. In various ways each of the physical theories has incorporated the ideas of crystalline slip and lockedin stresses. The most notable achievements of the physical theories are prediction of a developing corner in the yield surface at the loading point during plastic deformation [34, 56, 57] and prediction of a Bauschinger effect of the Masing type [54 58]. None of the physical theories has been entirely successful. Either the theories are too specialized and complicated to reliably and economically predict stressstrain behavior for general loading paths [55, 56, 57], or they require speci fication of special functions such as the initial yield function and the stressstrain function for a monotonic pure tension or pure shear loading [34, 54, 58]. In the latter case the physical theories are conceptually not too much different from the phenomenological theories, most of which also require specification of various functions, presumably to be obtained from empirical data. The theory of Lin and Ito [56, 57] deserves a special discussion as it is at present the most highly developed of the physical theories. In this theory the material is idealized as a cubical array of sixtyfour differently oriented crystals of cubic shape. Each crystal is assumed to have one slip plane with three equally spaced slip directions. Heterogeneous slip and stress fields are calculated from single crystal slip properties subject to conditions of equilibrium and compatibility. The computed initial yield surface of this aggregate is very close to the Tresca condition, whereas the surface in stress space corresponding to a finite offset strain is close to a Mises condition. This observation offers one possible explanation why experimentally observed initial yield surfaces usually appear closer to the Mises than to the Tresca yield condition. The computed yield surface after an initial axial loading is found to have a corner at the loading point, but this corner is "blunt" in the sense that a corresponding surface of constant offset strain has a rounded nose instead of a sharp corner. The theoretical results presented in this dissertation corroborate the theoretical results of Lin and Ito [56, 57], as discussed above, including prediction of a Bauschinger effect of the Masing type. Coincidentally, both the present theory and the LinIto theory require the specification of four constants, two elasticities and two plasticity constants, in order to be completely determined. However, the basic ideas of the two theories are very different: the LinIto theory is based on the ideas of slip in a threedimensional crystalline aggregate, whereas the present theory does not make any appeal to the crystalline microstructure of metals. The present theory has two advantages over the LinIto theory, namely, greater accuracy and greater efficiency. Greater accuracy is achieved since the present theory can be thought of as representing an infinite rather than a finite polycrystalline aggregate. Greater efficiency is achieved since the present theory contains less structural detail, and this allows a greater quantity of results to be computed from a given amount of computer time. For completeness, we mention three other currently accepted theories of plasticity that do not fit very well into either of the two categories discussed above. The first of these is the continuum theory of dislocations [60, pp. 8892], which was developed largely by Kr6ner [61]. This theory is concerned with the exact mathematical description of dislocations in an elastic body, which is accomplished by replacing the usual Euclidean metric tensor with a Riemarnnian metric tensor. The second of these theories is the theory of anisotropic fluids proposed and developed by Ericksen [62]; here a fluid is assumed to consist of molecules shaped like dumbbells, This theory has been promoted as a kind of plasticity theory because in the solution for Poiseuille (pipe) flow [62(b)] the material at the center of the flow is found to move as a rigid plug. The third of these theories is the mathematical theory of plasticity introduced by Ilyushin [63]. Functional analysis of the type expounded by Riesz and Nagy [64] is applied to general loading paths in stress and strain space. Relationships are assumed to exist between the curvatures of the loading paths in the two vector spaces. Lensky [65] attempted to connect this theory with experimental data. The author's brief study of these three theories has failed to reveal any concrete predictions of matters vital to plasticity, such as predictions of stressstrain relations and/or changes in the yield surface during plastic deformation. In this chapter we have briefly outlined most of the existing theories of plasticity, and we have shown that none of these theories is entirely satisfactory for predicting all aspects of experimentally observed stressstrain behavior. This is true even for the restricted situation in which temperature and time effects can be ignored. We conclude that there is still need for further theoretical work in plasticity. It is in the spirit of Zizicas [3(b), p. 448] that we undertake the development of a new theory. Instead of waiting for the physical theories (which take into account the full details of crystallographic structure) to come all the way to meet the mathematical theories, it may prove faster for the mathematical theories to move and meet the physical theories some where halfway between the two. One should be prepared to examine, or even replace completely if necessary, the fundamental assumptions of inr'ceeneltal theory. Certainly this theory, in its present form, should not be used as a basis for criticizing or dis couraging research work at variance with its predictions. Such a pointofview may delay important developments. The formulation of a proper set of mathematical stressstrain relations should be guided primarily by the aim of improving as much as possible the description of the physical behavior of materials. If approximations have to be made for mathematical convenience, they can be much better justified after the compli cations arising from a generally acceptable set of stressstrain relations are clearly demonstrated. A more accurate set always serves as a reference for comparison with approximate relations. This dissertation introduces a new plasticity theory, hereafter referred to as the radial element theory. This theory is related to the physical theories mentioned above in the sense that special importance is attached to the lockedin stresses. In fact, a new concept is introduced  a set of elemental stress vectors depending on both the direction in space and the loading history at a material point which exploits the directional aspects of the locked in stresses in determining the overall stress at a point in a material body. The radial element theory is a purely mathematical theory; its only connection with empirical data is through specification of four constants: two elasticities (Young's modulus E and Poisson's ratio v, say), one yield strain ey, and a workhardening parameter X. Among the successes that have been achieved with the radial element theory are: realistic predictions of work hardening and a Bauschinger effect of the Masing type (Chapter V); relatively simple calculations to obtain anisotropic yield surfaces for complex loading paths (Chapter VI); a method of calculating stressstrain behavior for any arbitrary loading path (Chapter VII); and good agreement with experimental data on commercially pure aluminum (Chapter VIII). The experimental data are somewhat limited in quantity and accuracy, but they include rwo different loading situations, cyclic tensioncompression loading of a rod and torsiontension loading applied to a thinwall tube. In summary, the most pertinent properties which the radial element theory aims to predict are: (1) isotropic elasticity relative to the current residual strain; (2) isotropic initial yield condition; (3) subsequent yield conditions which are anisotropic, with the anisotropy determined by the history of deformations; (4) inviscid plastic flow, also determined by the deformation history. For tests conducted at a steady loading rate and at constant temperature, these properties describe fairly well the mechanical behavior of most polycrystalline metals. CHAPTER II GENERAL CONCEPTS OF THE RADIAL ELEMENT THEORY In the theory presented here, the yield surfaces, flow rules, and hardening laws are not specified explicitly, as is true of most plasticity theories, but rather they are determined by the theory once a specific definition of an elemental stress vector is adopted. The key to the theory is the relationship that is assumed to exist between the elemental stress vector and the deformation of a radial element, which is the material element that reflects the deformation history in a given direction. Different material stressstrain behavior results if different relationships are assumed between the elemental stress vectors and defor mations of the radial elements. Stress is defined in terms of certain integrals of the elemental stress vectors over an infinitesimal ellipsoid. Suppose we wish to study the stressstrain behavior at a point P in a material body. The mathematical theory for this study can be developed as follows: Consider a spherical surface embedded in the body in its initially undeformed condition and centered at the given point P. The radius R of the sphere may be finite if the body under goes only homogeneous deformations. However, if nonhomogeneous deformations are allowed, then the sphere must be considered as infinitesimally small R + 0. Throughout this dissertation we assume the material is a "simple material", that is, one for which the stress tensor at each material point is uniquely determined by the history of the deformation gradient. During an arbitrary sequence of deformations the sphere will deform continuously into a sequence of ellipsoids of various shapes and orientations. Mathematical description of the undeformed sphere and of any one of the sequence of deformed ellipsoids, relative to a fixed coordinate system, affords the means of defining various finite strain tensors. For any such strain tensor, the principal directions in the deformed configuration must coincide with the principal axes of the ellipsoid, and the principal strain magnitudes must be related by an invertible function to the principal diameters of the ellipsoid. The sequence of strain tensors computed in this manner from a given sequence of deformations comprises the strain history at the given material point P. For definiteness and because it is useful, we introduce here one example of a finite strain tensor: the logarithmic strain, also sometimes called the natural strain. Let n (a=1,2,3) be unit vectors directed along the three principal axes of the deformed ellipsoid; let d be the corresponding principal diameters; and let "log( )" signify the natural logarithm function. Then X = d/2R are the principal stretches and a = log (X ), a =1,2,3 3 (2 .1) _= n n log (X) are the logarithmic principal strains and the logarithmic strain tensor, respectively. The logarithmic strain has the following two advantages over other strain measures for use in connection with the theory of plasticity: (1) mean and deviator components represent exact measures of volume type and distortiontype deformations, respectively, and (2) experimental results for tension and compression tests give the same stressstrain curve (up to necking or failure) when logarithmic strain is used. The latter fact allows the possibility that an initially isotropic plasticity theory can agree with experimental data. For the many further details concerning finite strain measures the reader is referred to Truesdell and Toupin [66, 241324] or Malvern [67, pp. 154182]. The first fundamental concept of the present theory is the radial element. Consider a material element which initially is a differential cone or filament radiating outward from the center P to a point Q on the surface of the undeformed sphere. See Figure 1. This radial element, corresponding to a given fixed direction in the undeformed state, will elongate (or contract) and shear as the sphere deforms into a sequence of ellipsoids during a given defor mation history. elemental stress vector deformation deformed ellipsoid undeformed sphere resultant stress vector FIGURE 1. BASIC CONCEPTS OF THE THEORY The second fundamental concept of the theory is the elemental stress vector f, which we define to act at each point Q on the surface of the deformed ellipsoid. This elemental stress vector is assumed to correspond directly to the radial element along the line PQ. Let n be a unit vector in the direction of PQ. Then the set of all elemental stress vectors f(n) corresponding to the set of all radial elements forms a vector field over the ellipsoid. Again refer to Figure 1. The vector f is assumed to depend exclusively on the deformation history of the individual radial element upon which it acts. The overall stressstrain behavior of the material is governed by this relationship which we assume to exist between the elemental stress vector and the defor mations of the corresponding radial element. For example, elastic behavior is obtained if this relationship is assumed to be linear and onetoone, as proved in Chapter III. Elasticplastic behavior is obtained if the relationship is linear and if the magnitudes of the elemental stress vectors are limited in the manner specified in Chapter IV. Suppose we are dealing with a material that exhibits elasticplastic behavior, and suppose that the material has experienced plasticity in certain directions. That is, suppose we computed the elemental stress vectors elastically and we truncated those elemental stress vectors f that exceeded a certain yield criterion. Then the plasticity will be expressed as a reduction of the stress components associated with those directions from the values predicted by elasticity. Anisotropic effects as well as plasticity are introduced into the theory in this manner. These ideas will become clearer to the reader as he progresses through Chapters III and IV. At this point we must draw a vital distinction between the elemental stress vector used in the radial element theory and the ordinary stress vector introduced by Cauchy [66, p. 537]. Cauchy proved that the stress vector t acting on an elemental surface is related to the surface normal unit vector n by a homogeneous linear function t(n) =Tn, where T is the stress tensor. This result follows from the principle of linear momentum and is known as the Cauchy stress principle. In the radial element theory f(n) is generalized so that it no longer satisfies the Cauchy stress principle. This generalization of the stress vector is necessary in order to exploit the individual response of material elements oriented in various directions. If we wish to preserve intact the classical theory of continuum mechanics, it becomes necessary to consider the elemental stress vectors as an entirely separate concept from the Cauchy stress. The elemental stress vectors can be viewed as a structural entity of the material similar to microstresses in a dislocation theory or they can be viewed simply as a mathematical device to be used in generating stressstrain relations. Regardless of how they are viewed we shall continue to develop and use the elemental stress vectors as if they indeed possessed a rigorous mechan ical basis. Ultimately, however, we shall want to use the stress strain relations generated by the radial element theory in the solution of boundary value problems, such as determination of the bending or twisting of some engineering structure under a set of loads. For this latter purpose we must ignore the details of the radial element theory and accept only the overall stressstrain relations. Finally, we must combine these stressstrain relations with the established principles of continuum mechanics [67, Chapter 5], including the Cauchy stress principle, in order to obtain a meaningful, wellset mechanical problem. The following equations, which resemble force and moment equilibrium equations, are necessary conditions for the con struction of a unique stress tensor. F= f dA=0 M== rxfdA=0 (2.2) A A Here r is the radius vector from the center to the ellipsoid surface and A is the total surface area of the deformed ellipsoid. Equations (2.2) represent restrictions on the permissible class of elemental stress vectors; the specific constitutive assumptions for f reported in Chapters III and IV do satisfy these restrictions. The third fundamental concept of the radial element theory comprises the resultant stress vector t and the couple stress vector c, defined as follows: !t(N) =  f dA, c(N = rx f dA .(2.3) N AN N AN Here N is a unit vector normal to the crosssectional area aN, and AN is onehalf the surface area of the deformed ellipsoid; both areas aN and AN are determined by N as shown in Figure 1. Unfortunately, the functions t() and c(N) are nonlinear, in general, which eliminates them as direct candidates for the stress tensor and couple stress tensor, respectively. By comparing equations (2.2) and (2.3), we see that equations (2.2) are equivalent to (N = t (N) and c (N) = cN) (2.4) Equations (2.4) state that the resultant stress on one side of a given plane surface is equal and opposite to the resultant stress acting on the other side of the same surface. If these equations were not true then the resultant stress would not be unique, since we would not know which resultant stress vector, i.e., t(N) or t(N to assign to a direction N. The following procedure has been tried in conjunction with elasticplastic behavior for a variety of loadings and seems to yield a unique stress tensor. Suppose we are given a set of elemental stress vectors f(n) distributed over a known deformed ellipsoidal surface. The function f(n) can be quite general, with discontinuities occurring in the derivatives and in the function itself, provided the vectors are real and provided equations (2.2) or (2.4) are satisfied.1 Consider any orthogonal triad of unit vectors N (a =1,2,3), and compute by means of equation (2.3)1 the three resultant stress vectors t =t(N ) corresponding to these N In a Cartesian coordinate system with N as base vectors, the components of t form a matrix T= [tl ], (i,a=1,2,3). That is, the components of ti= (tl,t21,t31), t 2= (t12,t22,t32 ) t = (t13,t23,t 3) form the matrix tll t12 t13 T t21 t22 t23 31 32 t33 Note that the second subscript identifies the plane on which the vector acts and the first subscript identifies the component. Define a positive definite norm in terms of the offdiagonal elements of T by the expression 2 2+ 2 M = (t12+ t21) (t23+ t32+ (t31+ t13 (2.5) We can now search for and obtain an orthogonal triad of unit 2 vectors N' by the requirement that the norm M be minimum when ay IThe maximum generality of the function f(n) that satisfies equations (2.2) and that will permit construction of a stress tensor has not yet been established. 2The orthogonal triad of vectors IN obtained by minimizing the norm M probably is unique for any reasonably wellbehaved radial element theory. However, a proof of uniqueness is not available at this time. the resultant stress vectors t' =t(N') or the matrix T'=T(N') are computed relative to N'. In fact, for all specific distri a butions of f(n) considered in this dissertation, the norm M can be made zero by a proper choice of N'. Then T' becomes a as diagonal as possible, with t2 =t', ti3 t2' t = t3 and tl' t2, ti3 are the "principal" resultant stresses. We now define a resultant stress tensor T by the require ment that the components of T coincide with the elements of the matrix T' in the special coordinate system with N' as base ar vectors. That is, we define t! =t! where T'=[t ] is the la i'  tensor T referred to the coordinate system defined by N'. a Relative to any fixed Cartesian coordinate system the components of N' form a rotation matrix R=[N' and the resultant stress tensor T satisfies the transformation equations T= R*T'*R T'= RTTR .(2.6) The deviator stress tensor S =[s.] is defined by the equations S= + P1, p = (tll+ t22+t33)/3 (2.7) where 1=[ ij] is the unit tensor and 3p is the trace of the tensor T or the matrix T'. The ordinary stress tensor Z=[cr. . S1J can be written as E = S + i = T + (+ p) 1 (2.8) where c is the mean stress to be determined by a separate relationship between mean stress and mean strain. Equation (2.7) will prove to be most useful in Chapters V, VI, and VII of this dissertation. The above procedure forms the basis of a general computer program for predicting the stress from a given arbitrary strain history. This computer program is described in Chapter VII and is one of the programs listed in Appendix A. The above pro cedure is complicated, and it may seem to the reader to be an arbitrary and unnecessarily elaborate exercise of algebraic manipulations. The author has considered this matter in detail and has concluded that the procedure presented is the simplest and most direct method of constructing a stress tensor from a general distribution f(n) of elemental stress vectors. The most controversial feature of the above procedure is the possibility of predicting a nonsymmetric stress tensor. I could have defined the deviator stress as the symmetric part of the deviator of the tensor T, and this would have given the ordinary symmetric stress tensor. However, my present purpose is to raise the possibility that the stress tensor may actually be nonsymmetric for certain loadings, and so I retain the definitions (2.7) and (2.8). In Chapter VII it is demonstrated that this procedure does predict a nonsymmetric stress for plastic deformations that involve rotations of the principal strain axes. It may be possible to resolve the couple stress vectors c(H) into a couple stress tensor by a procedure similar to that proposed above for the resultant stresses. However, as our primary interest in this dissertation is with the resultant stresses, hereafter we shall ignore the couple stresses. Another definition of stress which possibly could be more useful for describing material behavior during a finite strain involves first transferring (via the deformation function) the elemental stress vectors f from the deformed ellipsoid to corresponding points on the undeformed sphere. Then the resultant stress vectors t(N ) are defined as integrals of the vectors f over the hemispheres of radius R that are symmetric about a given set of orthogonal unit vectors N The stress tensor, in this case analogous to the second PiolaKirchoff stress tensor, is found from the vectors t(N ) as before. The two definitions of resultant stress are not the same, but they can be made equivalent by appropriate choices of the relationship between the elemental stress vectors and the deformations of the radial elements. In the case of stress calculated relative to the undeformed sphere, we must correlate this stress with one of the strain tensors associated with the undeformed state. CHAPTER III A CLASS OF ELEMENTAL STRESS VECTORS FOR PREDICTING ELASTIC '.'FERIAL BEHAVIOR The general theory presented in Chapter II was valid for finite strains, and in principle it is possible to con tinue this analysis without invoking small strain approximations. However, by making such approximations the theory is greatly simplified. Questions of which stress and strain measure to use are avoided, and computations of yield surfaces are brought within the scope of a modest computational program. Therefore, in the remainder of this work we shall restrict our attention to the case of infinitesimal strains. For this case, the difference between the deformed ellipsoid and the undeformed sphere is negligible. The elemental vectors f(n) and all integration to obtain the resultant stresses t() may be referred to the surface of the undeformed sphere. For simplicity we shall always consider this sphere to have unit radius R=1. Let a.. be the components of stress as defined by equation (2.8) and let e.. be the components of the infini 1J tesimal strain tensor [ 67, pp. 120135]. The deviator components s..,e.. of stress and strain are defined by the following decomposition: s.. =c.. o6. ej .= e.. 6ij (i,j =1,2,3) 1 1 0 = 1(l+ 22+ 033) E = ('ll+ 22+ 33) Here a,e are the mean stress and mean strain and 6.. is the 13 Kronecker delta. These definitions imply Sll+ s22+ 33 = 0, ell+ e22+e33 =0 . Throughout the remainder of this work we shall be concerned mainly with relationships involving the deviator components s.. and eij, instead of the "actual" or "true" stresses and strains, since, as usual, we assume plastic behavior to be confined to these variables. The deviator strain tensor, with components e. is real and symmetric by definition, and is assumed known. The deviator stress tensor, with components s i, is real but not necessarily symmetric, and is the entity that we seek to compute. We shall have use of the elasticity constants E, v Young's modulus, Poisson's ratio = E/(1 2v) volume or bulk modulus } (3.1) S= E/(l + v) shear modulus(3 and the following special constants depending on a parameter X: 4 2X 4_X 4 2 U X X=AL(43 ), 7=A( 3), C=x+?= =. (3.2) Our assumption (page 4) that mean stress and mean strain are elastically related can be written simply as o=se. 'The shear modulus is customarily written as 2p or 2G; throughout this dissertation I write it simply as l to save repeatedly writing the "2." We require this elastic volumecompressibility relationship to hold at all times. A specific constitutive assumption for the dependence of the elemental stress vector f on the deviator strain tensor E is now introduced, which predicts linear elastic behavior when the stress is computed. f =xE.n+ t(nEn) n (3.3) Here E=[e..] is the deviator strain tensor, n is a general 13 unit vector, and x,7 are the constants defined by equations (3.2). The motivation for considering f in this form is that equation (3.3) can be conveniently modified to predict elasticplastic behavior, which is the main purpose of this dissertation. The modified form of equation (3.3) will be presented in Chapter IV, with further developments presented in Chapters V, VI, VII. We can prove that f in the form of equation (3.3) depends directly on the deformation of a radial element in the direction of n, which is one of the requirements of the theory expressed in Chapter II. Since E is the deviator strain tensor, the inner product En represents the deviator strain vector associated with the unit vector n. A further contraction of En with n gives the magnitude nEn of the normal component of the deviator strain vector, and the vector (nEn)n is the normal component. Hence, equation (3.3) defines f to be linearly related to the deviator strain vector and the normal component of the deviator strain vector associated with the radial element in the direction of the unit vector n. Further observations can be made by deriving expressions for the normal and tangential (shear) components of f. The equations f = (f n)n= an f =ff =x(E.n1an) (3.4) t nI where a= nE.n show that the normal component of f is proportional to the normal component of the deviator strain vector in the direc tion of n, and that the tangential component of f is proportional to the maximum shear component of strain associ ated with the direction n. From equations (3.2) and (3.4) it seems reasonable to limit the parameter to the range 0< X< 2, since for this range stretch in a given direction is accompanied by a tensile normal component f and shear n' in a given pair of directions is accompanied by a tangential component ft corresponding to the same directions, with f t oriented so as to support the shear. To simplify the calculations we now refer the vectors and tensors to a Cartesian coordinate system that coincides with the principal strain axes. Thus e 0 0 E= 0 e2 0 n = (cos 81, cos 82, cos 83) 0 e3 where e.(i= 1,2,3) are the principal deviator strains and 6.(i =1,2,3) are the angles between the vector n and the coordinate axes. Substituting these results into equations (3.3) and (3.4) yields f=[ (xel+ 7a)nI, (xe2 + ?)n2, (xe3 +r a)n3] f =Can, ft =x(el )nl,(e2)n2 (e3)n3] (3.5) 2 2 2 = nEn= e ln + e2n2 +e n2 1 1 2 2 3 3 Additional expressions for f and f are given by n  3 3 f = ne. cos 28 f = E 9.e. sin29. n 2 1 t 2 1 1 1 where 1 = (sin 81, cot 9 cos 82,cot l cos 83 1 1 2 s(3.6) 2 = (cot 92 cos 81, sin 82, cot 62 cos 93) A3 = (cot 83 cos 81, cot 83 Cos 82, sin 83) The 8. (i=1,2,3) are unit vectors tangent to the unit 1 sphere at the point Q determined by the unit vector n. They each lie in a plane containing one principal strain axis and the vector n, and they are directed away from the point Q. Equivalence of the two expressions for f and f can be n t shown by substitution. We now prove that the overall force and torque vanish as required by equations (2.2). Equation (3.5)1 for the elemental stress vectors is used, which is equivalent to equation (3.3) expressed in principal strain coordinates. Introducing spherical coordinates n =cos 8, n2 =sin cos 0, n3 =sin sin as shown in Figure 2, and imposing the integration limits 00 <0 21, 0 O8 s, we obtain FIGURE 2. SPHERICAL COORDINATES FI = F d0 d8 sin (fl) = el d d sin cgos (x + cos28) + 7f dO d sin38 cos 9 (e2 cos20 + e3sin2 = el[2 [x (sin28) + ( cos4 8) 1 + ?[(e2+ e3)0+ (e2 e3) sin 20 ]2 [ sin [83= 0 MI = d0 dG sin B (f3cos 8e2 f2cos 83) = x(e3 e2) dOJ d9sin38 sin O0cos0 = (e3 e2[ [sin2 2 cos 8 (2 +sin28) =0 . (e3e2)[sn 0 [ 3 0 .Proof that the vector equations (2.2), namely F=M=0, are valid can be completed by repeating the integration for the other two axes. A simpler proof is obtained by symmetry, since we can simply permute the indices without affecting the above results. We now compute the resultant stress vectors with respect to the coordinates along the principal strain axes, using the definition (2.3). We start with the vector t1 = (tllt21,t31) corresponding to the elaxis, and again we use the relations nl = os n2 = sin cos 0, n3 =sin sin 0. This time the integration limits are 0 0 < 27r, 0G'9 /2. tll = j dO j dEsin E) (fl) = eL[0x2 [x sin2 ) + ( cos8) i)T/2 + ~[(e2+ e3)0+ (e2e3) sin20]21T [sin o/2 = el(x+) + (e2+e3) = (x+ )el = e t2 1 Idl dG sin (f2) 21 IT = dj d9 sin 9 cos 0 (x + sin 9 cos 0) + ~ d dB sin28 os (elcos 8 + e3sin 9 sin2f) = 0 t31 d de sinG (f3) e3 d 2 2 2 e= d dO sin B sin 0 (X +7 sin2 sin 0) + 7 dj d9 sin28 sin 0 (elcos 2 + e2sin2 os 2) = 0 The remaining expressions for the resultant stresses can be obtained simply by a permutation of indices. S= (t12't22t32) = (0, pe2,0) 3 = (t13t23't33) = (0, 0,Le3) Following the procedure of Chapter II, we can write the above results in matrix form. tll t12 t13 e1 0 0 T = t21 t22 t 23 = 0 e2 0 (3.7) t31 t32 t33 0 0 e3 The norm M defined by equation (2.5) vanishes for the matrix T when computed relative to the coordinate system used here. Hence, by definition, the axes of this coordinate system are the principal stress axes and by equation (2.7) the quantities s.=t.. (no summation) are the principal deviator stresses. These quantities determine a deviator stress tensor S=[sij] which, because of equation (3.7), is related to the deviator strain tensor E=[e..] by the classical elasticity equation 21 S = pE (3.8) Note that the principal axes of stress and strain coincide, a result that is true in general if and only if the material behavior is isotropic elasticity. Obviously, the definitions (3.2)1,2 of the constants x,7 were chosen so that the present results, equations (3.7) and (3.8), would emerge. Equation (3.7) is valid only for resultant stresses computed relative to the principal strain coordinates. It is important in the present theory to know how the resultant stress vectors transform during a change of coordinates. Consider an arbitrary Cartesian coordinate system, related to the principal strain coordinates through an arbitrary orthogonal transformation. Equation (3.3) can be written in component form as 3 f.= x e..n.+7Yn., (i=1,2,3) j=1 (3.9) 3 3 a F = ejkn nk j=1 k=l Introducing spherical coordinates n1 =cos 8, n2 = sin 8 cos , n3 = sin 9 sin0, we can rewrite equations (3.9) as fl = (xell+7C)c +x (cos 6+x(e+ el3sin) sin 8 f2 = (xe22+ ra)sin 9 cos 0+x(e21pcos 8+ e23sin 8 sinf) (3.10) a = e cos +e22sin 2 cos 0+e sin 2sin 0 + e23sin2 sin20+e31sin28 sin+ el2sin 28 cos . After a lengthy evaluation of integrals having the limits 0Q gs2i, 08 stress vector t = (tllt21t31) in the new coordinate system as follows: t = id9 d8 sin (fl) = (x+ ))ell= ell t21 = d70J dsin8 (f2) = (x+ )e21= e21 t31 = l d sin e (f3) = (x + :)e31= e31 where S = 2L/(3 X). By repeating the integration with respect to the other two coordinate axes, or simply by permuting the indices, we find L2 = (t12,t22,t32) = 3 = (t13't23,t33) = The matrix form T = shows that the strain tensorial. (e12 pe22, e32) (e13, 3e23 ,4e33) tll t12 131 pell 1 e12 e 13 t21 t22 t23 = e21 Ae22 23 (3.11) t31 t32 t33 e31 ge32 pe33 the resultant stress T is linearly related to E, but that in general the relationship is not The exception occurs when X=l, giving simply T=LE. Since strain transforms as a tensor, it follows that in general the resultant stress T does not. The relationship between the stress and strain tensors and the resultant stress matrix T =[tij as computed from equations (2.3) and (3.3) will now be made definite. Let S=[s..] and E=[e ij be the deviator stress and strain tensors, related here by equation (3.8). If primed and unprimed quantities refer to two different Cartesian coor dinate systems, then we have the following transformation equations: 3 3 e..= F F a. a = s. = t/i 13 k=l i=1 ak 3R kR ij 3 3 e'.. = a kia e =s / = t!./ (3.12) i3 k=l =1 ki j ki ij / 1iij ij = [p if i= j or =2p/(3 X) if i /j] Here A=[a..] is an arbitrary rotation matrix, the elements 13 of which obey the orthogonality conditions [67, p. 27]. 3 3 . Sa a = a =6 = if i k=l ik jk k=l ki kj ij =0 if ij j Although the resultant stress matrix T does not transform as a tensor, we can always construct the deviator stress tensor S by first finding the principal axes of T, which are identical to the principal stress axes. Then the elements of S can be determined from the elements of T by equation (2.7). The procedure of Chapter II can be rephrased as an eigenvalue problem. Thus, the principal stress axes N can be found by solving the equation (if a solution exists) C (t..+t..26..t) N = (j,a=1,2,3) (3.13) i=l 13 ji 1] a t ia and the eigenvalues t are related to the principal deviator stresses s by the equations s = t +t+t3)'3, ( = 1,2,3) .(3.14) In solving equation (3.13) it must be recognized that, in general, as for example when plasticity is involved, T is nonsymmetric and T(N ) is a nonlinear function. The fact that T(N ) can be nonlinear requires, in general, an iteration procedure for calculating N involving solution of a sequence of linear eigenvalue problems of the form (3.13). This topic will be resumed in Chapter VII. There may exist definitions of the elemental stress vector f other than the one proposed by equation (3.3) which would satisfy the equilibrium requirements F=M=0, and also predict linear elasticity. For example, one could imagine the elemental stress vectors for a cubic lattice, or some other structured lattice, as being zero in all directions except the principal lattice directions. The vec tors f in these lattice directions, for a given nonzero deformation, would be defined so as to make the resultant stresses finite. Such theories, which may or may not be isotropic, would involve complicated expressions for the elemental vectors and the resultant stresses. In contrast the present theory, based on equation (3.3), involves only simple trigonometric functions. Moreover, studies have shown [60, p. 5] that various microstructural theories of a given class generally lead to the same macroscopic results. Accordingly, the present investigation is restricted to equation (3.3) and to one generalization of equation (3.3) that involves plasticity. CHAPTER IV A CLASS OF ELEMENTAL STRESS VECTORS FOR PREDICTING ELASTICPLASTIC MATERIAL BEHAVIOR Plasticity is now introduced into the theory by limiting in a special way the magnitudes of the elemental stress vectors on the unit sphere. Recall the definition of f adopted in Chapter III which predicts linear elasticity, namely f = x E.n+ 7(n.En)n (3.3) = A(34 ), 7 = A(3X_) (3.2)1 2 Here E=[e..] is the deviator strain tensor, n is a general unit vector, and x,7 are elasticity constants. For plasticity, the elemental stress vectors are not uniquely related to the strain, as in equation (3.3), but are determined by the (time independent) strain history. The deviator strain history E(m) at a material point can be defined by a sequence of strain increments dE(m); thus E(0) = 0 E(m)= E ml)+dE(m) m=1,2,...,M (4.1) This discrete representation of the strain history facilitates computations on a digital computer, as we shall see later. The limit of a continuously varying strain history can be approached by letting each increment dE(m) become very small and at the same time letting the endpoint integer M become very large. The most reasonable assumption regarding plasticity is to limit the magnitude of the tangential (shear) component f of the elemental stress vector f. This is in keeping t with the physical assumption that plasticity is a phenomenon involving shear stress and unaffected by uniform pressure [68]. Accordingly, equation (3.3) can be generalized to include plasticity as follows: f(0) =0 f(m)=f 1i if gt gl, m=1,2,3,... Y/gt ifgt > Y. 2) g=f +x dE n +n (n"dE(n) gt = [g (gn)2] where Y is a characteristic yield stress. Equations (4.2) represent a procedure of repeatedly calculating a vector g and then the magnitude gt of its tangential component and then the elemental stress vector f(m) corresponding to a given deviator strain increment dE(m). The initial condition f(0)= 0 implies that the material initially is in an isotropic annealed condition. Equations (4.2) yield a solution for f(m) in terms of the deviator strain E(m) only in the case of a radial loading. Then dE(m) and E(m) are proportional: dE(m) dE; E ( = (c1+. .+ e) dE. Substituting this result into equations (4.2) we find for radial loading f(m)= g if gt Y S/gtif > (4.3) g=xE(m).n+7nn(nE(m).n, g =[g.g (gn)2] Since g is computed elastically in terms of the (total) strain E(m) and then truncated to yield f(m equations (4.3) represent what is called a totaldeformation type theory. Such a theory is valid for radial loadings and approximately valid for nearradial loadings, but it certainly would not predict reasonable results for loading paths that include stress reversals [ 1, p. 47]. The above definition of f(m) could perhaps have been stated with greater mathematical elegance in terms of time derivatives and/or the strain invariants. However, the form presented here is most easily adapted to computer pro gramming. Hence, I have adopted equations (4.2), and in the case of radial loadings equations (4.3), as the basic definitions of f to be used throughout the remainder of this work. Equations (4.2) represent a procedure of computing and then truncating those elemental stress vectors that exceed a certain yield condition. The proposed yield condition tests the magnitude ft of the tangential component of the elemental stress vector against a characteristic yield stress Y, which is assumed constant for all elemental vectors and all time. The present theory is essentially an incremental type theory in which each elemental vector obeys the same elasticperfectlyplastic law. The proposed yield condition predicts an initial yield condition of the Tresca type when expressed in terms of stress or strain, as we shall demon strate below. To find the initial yield condition, we assume residual stresses and strains are zero initially, so that the material behaves elastically as discussed in Chapter III. Then we consider those states of stress or strain for which the maximum value of ft on the unit sphere is equal to Y, the characteristic yield stress. Symbolically sphere ) f(ek)] =Y (4.4) (sphere) t 13 where e.. represents a point on the initial yield surface in strain space. In terms of coordinates along the principal strain axes, we can write the tangential component of f as t = x[(ela)nl, (e2a)n2, (e3a)n3 2 2 2 f = (ft't)= xD e ne c2 (4.5) i=li 1 2 2 2 a = elnl + e2n2 +e3n3 Let us first investigate the yield condition for the special cases of pure shear and axial strain, whereby e2 =el, e3 =0, and e2 =e3 =el/2, respectively. In the case of pure shear 2 2 2 2 2 2 2 5 a = el(n n2), ft = xel[nf + n2 (n n )2 In the case of axial strain 2 3 2 & a = el(3n 1), ft xelnl(l n) . For both cases, the maximum value of ft occurs when n = (51, F1, 0), as proved below. Let ek, e denote the deviator yield strains for pureshear and axialstrain loadings, respectively, and let sk = ek, S = ey denote the corresponding deviator yield stresses. That is, at yield e =e1 =e2, e3 =0 for a pureshear loading, and e =e = 2e2 =2e3 for an axialstrain loading. Invoking equation (4.4) and the above expressions for ft, we obtain the following equivalence relations among the various measures of yielding: 3 Y 3 Y 3 X Y ek= y ey x' sk y x ) (4.6) We now investigate the yield condition for arbitrary strain conditions. Introducing spherical coordinates nl =cos8, n2 =sin cos0, n3 =sin8 sing, we can rewrite equations (4.4) and (4.5)2 as. ma [f(8, )/] = ek' where ft/x = (e cos 9+ e2 sin 9 cos 0+ e3 sin sin a ) (4.7) 2 2 2 2 2 3 [(el 2e2cos 0 2e3sin 0)elcos 8 e2e3sin 2 sin 20 2 S2 2 2 2 2 2 2 2 sn . +(lsin 8 cos )e2 cos 0+ (1sin 9sin 0)e sin2 0 Let us now seek the maximum values of the function ft(9,), where the range of variables is 0 < 8 0 <s 21. Note that ft =0 when =0 and 8 =7 because of the multiplicand "sin 8." But ft in general is nonzero. Hence max(ft) must occur in the restricted range 0< 8< i. Necessary conditions for max(ft) to occur at a given point (8,0) are af /39= 3ft/ 30=0. Thus 8 2 2 2 0 = (ft/x) = (A B cos C sin 0)sin 28 a 2 2 0 = (ft/x) = (B C)sin 6 sin 20 where A = (2ael)el, B = (2ae2)e2, C = (2ae3)e3 Suppose 0< 8 <, so that sin 80. Then (A B os2 Csin20)cos 8 = 0, (B C) sin20 = 0 . The first equation is satisfied if 8 =T/2, so that cos 8= 0. In this case, the second equation becomes 2(B C) sin20 = (e2 e3)2 sin 4 = 0 2 2 2 since BC = (2ae2 e2) (2ae3 e) = (e2 e3) cos 20 In general, e2 e3, which implies sin 4 = 0. Thus 0 must have at least one of the values 0=n7/4 where n=0,1,2,...,8. Plugging these values into the expression (4.7)3 we find ft/x = abs (e2 e3) sin 0 cos 0 0 if 0=0, i/2, i7, 31/2, 27r Sabs (e2 e3) if 0=T/4, 3T/4, 51/4, 77T/4 The initial yield condition in this case is ek= m (f/x) = abs (e2e3) provided the maximum value of ft occurs at the points 8=7/2, 0 =ni/4 with n=1,3,5,7. Continuing in this manner, or simply permuting indices, we find the following equation which we recognize as the Tresca yield condition. ek = max (Ie2 e3 e3 el el e2) (4.8) This equation states that yield first occurs when the maximum shear strain (or stress) reaches a critical value. The directions corresponding to each segment of equation (4.8) can be specified by unit vectors as follows: ek = 4le e : n = \ (1, +1, 0) ek = e2e31 : n = 4 (0, +1, +1) (4.9) ek = e3 ell : n = (+1, 0, +1) These unit vectors n indicate the points on the unit sphere at which yielding first occurs. If the principal strains have distinct values, then only one line of the expressions (4.9) can be valid at any one time. Suppose the first line is valid, which was true for the case of pure shear analyzed previously, and suppose the deformation is continued beyond initial yield. Then there will develop four regions on the unit sphere surrounding the four points n=1= (+1, +1, 0), inside of which the elemental vectors are restricted by the plasticity equation ft =Y. Outside of the four regions ft< Y, and the elasticity equations of Chapter III continue to govern the elemental stress vectors. In general, the boundary between the elastic and plastic regions on the unit sphere can be very complicated. There is only one situation known to the author in which a closed form expression for this elasticplastic boundary exists. If the principal strain axes are fixed in the material and the loading is radial (i.e., the strain magnitudes el,e2,e3 increase proportionally), then the boundary between the elastic and plastic regions is given by the following equations: 23 3 2 2 ek (ft/)2 = ( e cos ) ( I e cos .)2 (4.10) i=1 i=1 1 or 2 2 2 2 2 2 (el2e2cos 02e3sin )el cos 8e2e3sin 8 sin 20 2 = 2 2 2 2 2 2 2 2 sin 6 k +(1sin 8 cos 0)e2 cos f+(lsin 9 sin 0) e3 sin 0$ The complexity of the foregoing equations for the elastic plastic boundary on the unit sphere even for the restrictive situation of fixed strain axes and radial loading obviously limits the number and type of closedform solutions that may exist for the stresses when plasticity is involved. Two closedform solutions, obtained for pure axial loading and reversed axial loading, form the subject matter of Chapter V. Other solutions, which do not necessarily involve fixed axes or radial loadings, were obtained numerically via the digital computer and are presented in Chapters VI and VII. We now prove that if the principal strain axes are forever fixed in the material, then the principal stress axes are also fixed and the two sets of axes coincide. The proof follows from the symmetry of the function f(n) relative to the principal strain axes. Equations (4.3), which are valid for radial loadings, can be expressed in principal strain coordinates as follows: 2 2 2 2 2 2 f=gF(gt/Y), gt=x(e n1 +e2 2+e3 n3 ) 2 2 2 g=x(elnl,e n2,e3) + 770m, a= elnl + en + e3n3 (4.11) where F(x) = [1 if x 1; 1/x if x >1] . Let us calculate the elemental stress vectors corresponding to the four unit vectors n= (a,+b,+c), where a,b,c are 2 2 2 positive constants subject to the restrictions a +b +c =1. Thus n(l)= (a, b, c) : f(1)= [x(ela e2b, e3c)+7(a,b, c)]F(gt/Yl n(2)= (a,b,) : f(2)= [(ela,e2b,e3c)+ 7Q(a,b,c)]F(gt/Y) n(3)= (a,b,c) : f (3)=x(ela,e2b,e 3)+7a(a,b,c)]F(gt/Y) n )= (a,b,c) : f(4)= [x(ela,e2b,e3c)+ 7 (a,b,c)]F(gt/Y) 2 2 2 2 2 2 2 ! where g /Y=(e a +e2 b +e c a /e Consider now the sum of these four elemental vectors 4 Z fi) = 4a (xel+ a) F(gt/Y) (1,0,0) (4.13) i=l(i) This sum has a nonzero component in the eldirection only. From equation (2.3) we see that the resultant stress vector t is computed by integrating the differential vector 1 dt1 =fdA/i over the hemisphere defined by n1 0, where n= (nl,n2,n3) is an otherwise arbitrary unit vector. An equivalent vector t is obtained by integrating the differ ential vector dt = (fl) +f + f + f )dA/IT (4.14) 1 _(1) (2)+ (3) (4) where f are defined above; here each component f dA/7I (i) (i) is integrated over one quadrant of the given hemisphere. By comparing equations (4.13) and (4.14) we conclude that tl has a nonzero component in the eldirection only. Similarly, we find that t points in the e2direction and t3 points in the e3direction. We conclude that for a radial loading the resultant stress matrix T=[ti ] is diagonal (t. = Oif ia) when computed relative to fixed principal strain axes. Recall that the principal stress axes are identical to the principal axes of the matrix T. The same conclusion can be reached similarly for the case of fixed principal strain axes and arbitrarily varying strain increments del,de2,de3. Although I omit details in order to save space, I offer the following outline of this proof: First assume a given distribution f(0)(n) of elemental stress vectors which is symmetric in the sense of equations (4.12). Then introduce an arbitrary strain increment and compute the new elemental stress vectors f(1)(Q9. Proof follows by observing that the distribution f(l)(Q) is also symmetric in the sense of equations (4.12). The general case in which the principal strain axes rotate relative to the material is too complicated to allow such simple con clusions to be derived. The following is a discussion of the essential features of the radial element theory as regards computing elastic plastic stressstrain behavior from equation (4.2). This discussion should help the reader to understand the develop ments of the theory presented in the next three chapters. Consider a loading and unloading of an initially isotropic material that involves plasticity. Before loading the ele mental stress vectors f(n) associated with a given material particle are all zero; here n represents the set of unit vectors pointing in all possible directions. During loading the vectors f(n) vary with the strain according to equations (4.2), and the stress is computed in terms of these f(n) by the procedure of Chapter II. When unloading occurs the stress drops to zero; however, the strain and the set of elemental stress vectors fo(n) after unloading generally are not zero. This situation is analogous to a beam which, after plastic bending, has a residual curvature and lockedin stresses even when the bending moment is zero. (This beam analogy applies to many parts of the present theory.) The importance of the residual stress vectors fO(n) is that, for any fixed material particle and any moment of time, this set of vectors contains all the information about past deformation history sufficient to predict the future mechanical behavior. Suppose we are given a finite increment of strain A E along a known strain path. Then knowledge of f (n) and A E enables us to compute the elemental stress vectors f(n), and hence the stress, corresponding to this new increment of strain. Furthermore, from the set of vectors f (n), we are able to compute the entire yield surface at each step of a strain history, as will be demon strated in Chapter VI. CHAPTER V STRESSSTRAIN BEHAVIOR FOR AXIAL LOADING AND REVERSED AXIAL LOADING Consider a monotonic loading with fixed principal strain axes in which the principal deviator strains are maintained in the constant ratio e2 = e3 =e1/2. This case is called axial strain or axial loading, and includes the common tension and compression tests of a cylindrical specimen. Referring the deviator strain tensor to the prin cipal strain axes, we can define axial loading by the following expressions: 0e 0 e(0) =0 and e(t) 20 E = 0 e/2 (5.1) 0 0 e/2 or e(t) <0 for t>0 . For this case we can use equations (4.3) to compute the elemental stress vectors f. We begin by computing g= (gl 2' 33) 2 2 2 2 a = n.E.n = e(n n2/2n /2)= e(3n 1) 2 gl = (7c + xe)n3 = e n (3Bn1+ a) 2 (5.2) g2 = (770i xe/2)n2 = e n2 (3T7nl ) g3 = (T7a xe/2)n3 = e n3(3Tnl ) Here a = 2x 7 and p =x + 77. Normal and tangential components of g are given by n = Can = e(3nl 1) (nl,n2,n) 3 2 2 2 S= X(E.n"n ) = xe[ (1 nl)n, nln2, nln3] (5.3) ( 3 2 g= (= x e nl(ln n) . With these results the expression for f becomes = j if gt = e sin28 (5.4) S Y/gt if gt/Y > 1 Y y where 8 is the spherical coordinate defined by n =cos 8. We recall from equation (4.6) that e =4Y/3x is the yield strain for axial loading. For convenience we define three new variables 1, 2', *, as follows: e l 2 = + cos = (5.5) 1 for 0 8< or 2 Then f=g r (5.6) cos */sin29 for 1 O8< Q 2 or T 2 <<01 : l This situation is illustrated in Figure 3. We see that plasticity develops in two expanding parallel bands on the unit sphere, centered on the circles 8 = /4 and 8=31~/4, for the case of axial loading. Resultant stresses are computed as follows: tll 2I2 d. '/2 dB sin8(fl) o o S2 S d sin (gl) + cos i S2 d sec (g1) 4 2 e (5.7) cos4 8 sin28). ev .28sin28 = e[3( ) +a( 2 )]S + 26L +ae]8 = p(e +e y esin ) = s y( tan + sec ) mH n ft 4 0o (degrees) o0 u 30 60 90 120 150 80 .) , bO a Elastic loading e /e = 1.0 4y id n a r E8 (degrees) C 30 \60 .90 120 150 /80 Plastic loading el/e = 1.2 FIGURE 3. DISTRIBUTION OF ELEMENTAL STRESS VECTORS FOR AXIAL LOADING FROM AN INITIALLY ISOTROPIC STATE t22 /2 d T d sin 8 (f2) IT/2 o SI dS dB sine (g2) SJ d dB sec 8(g2) (5.8) o 1 o 2 e 2 2e 49sin49 2in4in26 26 + ey r3,7(sin ,8 log (Cos 8)] =[3??(32 S4 s[@ 3 ()+Clog(cos 6)JS2 = [ ) e + (t) sin (+ ( ) (log cos l1 log cos t2)] The integration limits in equations (5.7) and (5.8) are given in set theory notation as S= (', S1= ((0,81) (82t)), S2 (81' 2) Here we have used identities such as 2 2 sin =cos2 2 = (l sin ) sin 21= sin 22 =cos 2 2 sin2 2 =cos i = (1+ sin) cos 2l= cos 22 = sin . By symmetry considerations we find t33 =t22 t23 = t32= t31= t3 = t12 =t21 = 0 The principal stress axes coincide with the principal strain axes, and so we can use equation (2.7) or equation (3.14) to solve for the components of the deviator stress tensor. Letting p=(tll+ t22+ t33)/3 and L= log cos t log cos t2, we obtain s 6 1 4L p = tan + 2*see ( ) sin n ( 1 3IT 3 3 s= 2s= 2s = 2 Sy[ ( tan ) + (1 ) sec 21X1 4L X + T( 3 ) sin + (3 ) S(5.9) 1X/3 where s = e =Y( 1 is the yield stress for axial y y 1X/21 loading. Equation (5.9) and the following expression for the deviator strains are the desired results for predicting stressstrain behavior in the case of axial loading. el =22e2 =2e3 =e see i, 0 < 7/2 (5.10) Figures 4 and 5 show stressstrain curves calculated and plotted for various values of the parameter X. We see that the slope of the stressstrain curves and the stress relative to yield both increase with increasing values of X. Hence, we are justified in calling X a workhardening parameter. The data of Figure 4 lend support to the suggestion of Chapter III that X be limited to the range 0 X< 0 we see that the stress reaches a maximum and then falls with increasing strain. The material is unstable, which is an unrealistic and undesirable material property. Not obvious from Figure 4 but true nonetheless is the fact that the slope dsl/del is greater than the elastic modulus t for X>2 and for certain values of the strain. When this occurs, the residual strain becomes negative for a positive loading; here again the material is unstable [69]. The reader is referred to Tables Bl and B2 of Appendix B, where stress and strain are tabulated for five values of the workhardening parameter X. These tabular values, based on the closedform solution of equations (5.9) and (5.10), will be compared with the approximate numerical results of Chapter VI, in order to establish the accuracy of the latter. The 57 " l I I'* 7 i I : I I I i i. ; : : I .... : ' : ,i ! t > t [ l I ) :  ,: i ,' ,> '^' : l!' I ,i .I ; i I :i, h i' i i j i *1 l : i : *:: : i l .1 i i !, ; ..:i .. .,, ,: ,! .. ; ii .TI i . Deviator Sfriain, e^e ,,+:'. 14 8 10 12 AXIAL LOADING. SMALL STRAIN 2 4 6 8 10 12 FIGURE .' STRSSSTRAN BEHAIOR FO AXIL LADIG, MAL STAI :.ll 1:;. ii.. it * I . I It I: 1ii : htj : I  I i ,C  ' .* . I . .1_... __ , f  . . I' Z':1717 ' :  .. ' I I .i. " ...[_ t .T _ .". i 'i ~4 i    I  4I II :!....i ... ,si .; :: 1 i ... .; ... 1 .. ] .. I  lij i... .i .. ... .[ .. .. j ' Lr. II ii * I I .1] *1. I 71'  ___ ....Fmm 1 4 tP .. I: Satortess.. ... f14 T i  Ji T a or 0 :rirrI:::::rflII:: :fI1ss s ::;1_1I , .. .. .... L.IZ .: I .. I. + i .2 ..... _.NL _ \ .. .. ! i ..I. . .. i:L ....' 1._1 i .i : it_ T It L I '+ ....... .. I FIGURE 5. STRESSSTRAIN BEHAVIOR FOR AXIAL LOADING, LARGE STRAIN i reader is also referred to Program Ai of Appendix A, which is the computer program used to generate the data of Figures 4 and 5 and Tables B1 and B2. The programming language of Appendix A is BASIC. The prediction of various stressstrain curves raises the possibility of fitting the present theoretical results with experimental uniaxial test data. Perhaps there are certain metals and certain values of the theoretical parameter X for which the theory and experiments can be matched reason ably well. The reader is referred to Chapter VIII, where we have attempted to fit the theory with experimental results on commercially pure polycrystalline aluminum. The present theoretical results are expressed in terms of the deviator stresses and strains. In order to make comparisons with experimental data these quantities must be converted to true stresses and strains. The following results, easily obtained and applicable for either uniaxial tension or uniaxial com pression, are presented without derivation: C1 = 3 s /2, 02 = 03 = 0 (5.11) el = e +sl/20, e2 = 3 = el/2 + sl/2 We now investigate the case of reversed axial loading. As before, f(n) has rotational symmetry about the elaxis, and so depends only on the spherical coordinate angle 8. Assume a given material body is loaded axially into the plastic range so that the initial stressstrain behavior is predicted by equations (5.9) and (5.10) derived above. Figure 6a shows the distribution of the elemental stress vectors f as a function of the angle 8 for one such loading. We observe that plasticity has occurred because the slope of f(8) is discontinuous. Now superimpose an axial loading of the opposite sense (i.e., with reversed signs) onto the given loading, as shown in Figures 6b through 6d. Figure 6b is approximately the distribution of elementary stress vectors which gives zero resultant stress. Figures 6c and 6d show distributions of the elemental stress vectors after yield has occurred for the reversed loading. Note that Figure 6d is exactly a mirror image of Figure 6a. If the material were loaded cyclically between equal tensile and compressive limits, then the elemental stress vectors would alternate indefinitely between two mirror image distributions. In view of the behavior of the function f(8) shown in Figure 6 and described above, only two additional parameters are needed to describe the distributions of f during a reversed loading. We denote the two parameters a1 and *2' During the initial loading l1 is variable and equal to * used previously, and 2 = 0. During the reversed loading i1 is constant and equal to the maximum value of that occurred in the initial loading, and 12 is variable in the range 0 2 2 .1' Define a vector field h= (hl,h2,h3) by hi = (37) cos 28+ a) e cos 8 h2 = (3 cos28) e sin9 cos 0 (5.12) h3 = (3B cos28 ) e sin sing 61 .... ._ 8 (fdegreesl .. . 30 60 9120: :150 :180 r 0 ... ... ... ... .i ....: \ .: : ..  S(a) Initial plastic state e /e 2.3 O . (b) Elastic unloading el/e 0.5 : . i. I . 0 0 : : .. .' ' .. _^,^ ....__i_ i_: _ S(b) Elastic unloading e e .ldi S: .. .. e e I 1.2 0 r4 S .. .. I .. .. / ..... .. .i  3 0 c: .60: 120 : : 150. :. 180. S" . c: .I::(dc) Reversed plastic state H   Ci H e /e. .. ... FIGURE 6. DISTRIBUTION OF ELEMENTAL STRESS VECTORS FOR REVERSED AXIAL LOADING FROM AN INITIALLY PLASTIC STATE where 8,0 are the spherical coordinates of Figure 2 and 0B !iT, 080s 21, unless specified otherwise. As before, a=2x 7 and C =x +7. We can express the elemental stress vector f in terms of h as follows. 1. Initial elastic loading f=hcos 1, i T lI 0 (5.13a) 2. Elasticplastic loading 09s f=h sec J for eS11 (5.13b) Lcsc 2 for 8 eS2 3. Elastic unloading < 12 0 < (< 41 constant f=h Jsee1 for eSll 2heos2 (5.13c) csc28 for eS2 4. Reversed elasticplastic loading 0 2 < 1l< I, 1 constant f=h fsee 1 for eSi 2h see 2 for eS3 (5.13d) tcsc28 for 8 eS2 lcsc26 for 8 e S 5. Reversed elasticplastic loading 0 < r1 < 2< 1 , 1 constant f=h see *2 for 8 E:S3 (5.13e) esec 2 for 9 eSQ Here Sl'S2,SS3,S are sets of real numbers defined by I 1 L 2 !L 2 1 4 2 t2 4 2+ 1 4 2' 1 2 =4 2 S = ((Ol'),(,12 ),'(fT_ )), 2 = ((Jt 23 ,(T_2, l)) S3 = ((0,1), (C2, 2), ('Cl, )), S4 = ((Cl,2),I (C27T ) Resultant stresses are computed from the equations t.= (tlit 3 = f dA, (i =1,2,3) (5.15) li' 3i) f As before, since the principal strain axes are fixed in the material, we have the relations s tii (tll t22 t33)/3, ( =1,2,3, no sum) (5.16) Plugging appropriate expressions for f from equations (5.13) into equations (5.15) and (5.16), and performing the integration, we find the following closedform solutions. Here Ll=log os l logcos 2 and L2 logoss ~ logcos 2. Cases 1 and 2, having to do with initial loadings,were derived previously and are not repeated here. 3. Elastic unloading 2I 2 <01< < 1 constant el = 2e2 = 2e3 = ey(ee 1 2 cos c2) 2 2 )2 s = 2s2 = 2s3 = s y(1 tan a 1)+(l )see 1 (5.17) 2 1 4L1 X + ) sin 1+ ( )2 cos ] S3 1 3I3 2 cos 4. Reversed elasticplastic loading 0 2 1 Jr< I, *1 constant s1= 2s2 = 2s3 =s [ (* tan 1) 2 tan 2) (5.18) 21. 4 *2 2 1X +(1 ) sec *1 (2 ) sec 12+ () sin 1 4_1X 4 X (t) sin 2+ ( ) (Ll 2L2 5. Reversed elasticplastic loading 0 < 1 42< 1, 1 constant el= 2e2 = 2e3 = e sec 12 2 22 3 s =2s2 = 2s3 = (2 tan )+(l )see 2 (5.19) 2 1X 2 1 + W(j) sin 2+] 31 3a 3X Equations (5.17), (5.18), and (5.19) satisfy "Masing's hypothesis" [37], whereby the stressstrain curve of reversed loading is exactly double in size, although rotated 180 degrees and translated, compared with the stressstrain curve of the initial loading. Symbolically 1,2. Initial loading: s=s(e), O0e5 e m S s'=s 2s(e), sm =S(e) 3,4. Reversed loading: 2( sm=s(e) (5.20) e'=e 2e, 0 5. Reversed loading: s' =s(e), e' =e, eme For Case 5, where *2 41, the stressstrain curve of reversed loading is identical in size, but rotated 180degrees about the origin, compared with the continuation of the initial stressstrain curve. This phenomenon, too, is considered a part of Masing's hypothesis. The present result is to be expected because Masing's hypothesis is a general property of mechanical systems composed of elasticperfectlyplastic elements [33]. Figures 7 and 8 show computed results of axial and reversed axial loadings for X=0 and X=1.5, respectively. The large effect of the parameter X on the stressstrain behavior can be seen by comparing these graphs. Two related phenomena appear on both graphs, namely the presence of a Bauschinger effect and constancy of the elastic range. The Masing hypothesis, discussed above and embodied in equations (5.20), is readily discernible from Figure 7 or 8. A special s: '. i ' *l Hi ili !ti ifii ft iiUt !ii i i : I' II Itt; ii+iiii !iii: ii :i II I , ,.1,5 is .... i I 1.0 : i . 0.5 : i : ,  I: i1  I'.~. 4~~ ~ ~ + 4 , ' lii _ s i hl i; !r 9+ '.3*  Elastic limit for Reversed loadings 4!. . _.,  I  I s/s I_ I) ::; ; ~1'1 *I *. I... 4jj 4 ,!.... 4. , ! I : j I j :^I . i : ::. II FIGURE 7. AXIAL LOADING AND REVERSED AXIAL LOADINGS FOR X= 0 1 ' iiiiii ce ':'' ' t i i i i I I I ' ' ;i::I:::: :I: i __. ..: *iM K i 15 iii:ii:: : ~: ; i  : ' ':::: riii: ii ~.:1 iiii i:i ' li' L:l:: i / i: i : Ij:ffl : i : i I : j i 1 '~'' i : iilii t : : i r : : i i: r r. :' iii:iiiii i : : ']:: i: j: N i: I, i .1 I i fl :i~i ;. L : 1 : : : : : !? : e/ey I y i '20 '' i"~ ; : : : tlLiii "' i r ii i! : : : i , t  r r:  i I. i II I i; It ii "; iiili.i ii ,:i ~'i i i  i " Si ..i I i I : i:,:. . SElatic limri fbr revirsed load ings' . . S I t l! S1:i::, 4 1 41 e*55 .tt i' t: .. .... . :Qj i;:i I Iil ::i:l^: i ;r:; *[__ I ... ,., 5. i . L  3.0 2.5 2.0t 1.5  1.0 0^ 0 K  ..i I 1.5 2 ,C 2.E 1~ .1 ,< ;'.. ^!!;'! !; i : :l i:l : I . ... .. '' '.'. .. ~ i i :' ; ; : : '. ': : *:i I: . .1 I *L....:1[ li :1<.1> > i 10 k ; 15 y.  .. .  L1. ~'' ' j: Iliil : ~~ : i,: 1:;:: ',.;fir' 1_111~ : : : 1::: .::I: ijljlj i : .....:.... i. : : r :..... T i ...... ,,' i :: i I : II 1 ii !!!ii i~ ~ :: :i 7 7 *I e/e Y I *:i Ii y 25 i iI li I.. * I : : *iii i ; : ;::L ... i ri: :i : ii tti1 H I 0 ', w 7 K:! ![:1! ... ... t t FIGURE 8. AXIAL LOADING AND REVERSED AXIAL LOADINGS FOR X=1.5 Si I r  j::i: I : :::.: : : ': : f : :: : t I iiliiii : : ; r : : / I I' ' r T "  f  &  1 I L ' . l I i 5 i i : ....,. .. 1 I : i i  ::i : : : : : I: ):.:/:: : li:i/. :::: i;::: A phenomenon appears in Figure 8 but not in Figure 7, namely the hysteresis loop of unloading and reloading that involves plasticity. The reloading curves may be obtained either from a straightforward extension of the closedform results of Chapter V, or by a direct application of the numerical methods of Chapter VI. Note that the hysteresis loops of unloadingreloading are present only for certain values of X and certain values of strain, as defined by the condition s/s > 2. One further result relating to uniaxial stressstrain behavior can be derived from equations (5.9) and (5.10). Introducing identities such as sin = cos(26), sec tan =tan6 log cos 2 = log sin6 = log 6 6 6 /180... 6 sec =csc(26) = + 62/3 + 76 /15+... where 6 = 1 = T/4 /2, we can easily show that lim 1 lim 1 7 4 3 2X lim X log81 52 el c sy 680 Sy 33T3 6 80 3 X) (.) We conclude from equation (5.21) that a limiting stress corresponding to large values of strain exists if and only if X=0, in which case lim s 7T 4 el~oa = 3+3 = 1.4716107.... y CHAPTER VI STRESSSTRAIN BEHAVIOR FOR THE CLASS OF LOADINGS INVOLVING FIXED PRINCIPAL AXES Consider a class of loadings in which the principal strain axes are fixed relative to the material, but where the principal strain magnitudes can vary arbitrarily. i(0) =0, ie= ei(t) for t> 0 (6.1) ei = i (l 2 + e3)/3, (i =1,2,3) This class of loadings is more general than the axial load ings studied in Chapter V; axial loadings are included here as a special case. The following equations govern the elemental stress vectors and the resultant stresses: f(0) =0, =g 1 if gt m=1,2,3,... } Y/gt if gt Y (6.2 f(m) (,m)(m) [ 2 g f(ml)+ dE (mn + n (ndE () gt gg (g.n 1 27T 1T/2 1T/2 f t1= O d o d8 fsin8, t= d0 def sinB L o o 0/2 o T 7 (6.3) t3= d d f sin . 0o Here f(ml) and f(m) are the elemental stress vectors before and after application of a given strain increment dE(m). Again we refer vectors and tensors to the principal strain coordinates and we use spherical coordinates 8,0 where convenient. In Chapter IV we proved that if the principal strain axes are fixed in the material throughout a deformation, then the principal stress axes are fixed and coincident with the strain axes. Since these conditions apply in the present case, we can compute the principal deviator stresses s. from the equations s=tii (tll+t22+t33)/3, (i=1,2,3, no sum) S/2 (6.4) t.= ( t )= d, d f sin9 1 (tli't2i't3i)= J Equation (6.4)2 follows from equations (6.3) by symmetry considerations similar to equations (4.12). As noted in Chapter IV, the boundary between elastic and plastic regions on the unit sphere can be very complicated for a general loading. In fact, due to the complicated nature of this elasticplastic boundary, there probably does not exist a closedform solution for the class of problems discussed here. Therefore, we shall be forced to derive a numerical procedure, or algorithm, to be used in conjunction with a digital computer, for solving equations (6.1) through (6.4). Consider a finite grid of points covering one octant of a unit sphere and based on the spherical coordinates 8,0. Thus 8.=iP, j =jP, (i,j =0,,...,N) (6.5) where N is a gridspacing integer and P =7/2N. The integral of a function f(8,0) over a spherical octant can be approximated by the trapezoidal rule formula as follows: /2 /2 N N1 SdZf d8fsin8 = Z a.[ 10 iN + Z f.j]+ e o o i=0 2 j=1 where fi = f(8e,0) i = (P/2) (cos 8i. cos 8.i) (6.6) o = (P/2) (1 cos 81), N = (P/2) cos N1 If the function f(8,0) is continuous and possesses continuous first and second derivatives, then the truncation error e for the trapezoidal integration formula is bounded. Smax (7T/2) 2 2 abs E< 12N2 (ee + f sin 8)sin 8 (6.7) Here f and fs are partial derivatives and 0 <8,0 Oi/2. Equations (6.6) can be derived by summing the product of the area and the average functional value for each patch of the finite grid. The inequality (6.7) is derived by expanding the function f(8,0) in a twodimensional Taylor's series [70, p. 187] and substituting the result into equation (6.6). As an alternative integration formula consider the following modified trapezoidal rule: T/2 7/T2 N ,+ N1 dOj d f sin89= a .[ fiO iN + E f] + e' o o i=0 2 j=l 1 where f!.=f(8,01)), 8 =01 =iP for i=l,2,...,N1 (6.8) 98 =0 =P/6, 98=0 = 7/2 P/6 Here a.(i=0,...,N) are defined in equations (6.6). It can be shown that the truncation error e' for equation (6.8) is of order N3, instead of N2 as in the case of equation (6.6). That is, the integration formula (6.8) converges faster than (6.6) as the finite grid is refined. Proof _3 that e' is order N3 follows by combining Taylor's series and the EulerMacLaurin summation formula [70, p. 154], both suitably generalized to two dimensions. Now consider numerical integration of the elemental stress vectors f over a spherical octant, as required by equations (6.4). In particular, let us apply equations (6.6) to the three components fk(k=l,2,3) of f relative to the principal strain coordinates. Initially during a loading, the material is elastic and the functions fk(8,J possess continuous derivatives of all orders. Here the in equality (6.7) provides a bound on the truncation error. After plasticity occurs the functions fk (,) still are continuous, but the derivatives 8m 80 fk of all orders are discontinuous along the boundary between elastic and plastic regions on the unit sphere. Hence, in general, the inequality (6.7) does not hold true after plasticity has occurred. Equations (6.6) and (6.8) are the basic numerical integration formulae to be used in the remainder of this work, even though the functions to be integrated generally will have discontinuous derivatives. When plasticity occurs we simply are forced to establish the numerical accuracy by comparisons with the exact solutions of Chapter V, instead of by inequalities such as (6.7). As we shall see, the numerical error can be made arbitrarily small by using a sufficiently large gridspacing integer N. Let us comment briefly on the choice of integration formula. I could have chosen a more complicated, higher order polynomial formula of the Gauss type, say instead of the trapezoidal rule. The discontinuous derivatives of f(8,0) again would have rendered invalid the usual error estimates. Experience with equations (6.6) and (6.8) has shown that, except for integration in the elastic range, there is very little to be gained by using an equation with truncation error of order N3 instead of N2. Therefore, I doubt whether any significant improvement of accuracy can be achieved from any higherorder integration formula when the first derivative of the integrand is discontinuous. The only direct way of improving accuracy is to refine the grid. The decisive reason for choosing the trapezoidal rule formula is that, because of its relative simplicity, for a given computational effort it allows one to compute the integrand at a larger number of grid points. Assume that we are given a sequence of deviator strain increments de k (m=1,2,3,...) representing a strain history. The following equations govern the elemental stress vectors at points (8ij.) of the finite grid: f(0) = f(m) g 1 if gtY m=1,2,3,... ijk ijk k Y/g if gt>Y J S ml ,) xdek(m) nk+ n(de(m)n+ de (m2 + 3 n3) (6.9 k' ijk Imk nkdek ( 2 2 +de3(m) (2 2 2 2 n 2 gt=g+ g2 +3 (glnl+ g2n2+ g3n3 nl = os i, n2 = sin 8icos 0, n3 =sin i sin j where i,j =0,1,2,...,N and k=1,2,3 (no summation). The deviator strains and stresses are computed from the equations e(O)= 0, e de(m)+ e(m1), s (t+ t2+ t3)/3 k k k k k k 123 N NI t = a!( i0k iNk + D f ), (k=1,2,3) ki=0 2 j=l (6.10) where a =Nl(cos Bi cos 9i+) i= 1,2,...,N1 1 1 S=N(1 cos 1), = =N cos 9N Equation (6.10)2 can represent either the ordinary trape zoidal rule or the modified trapezoidal rule, depending on whether the coordinates i.,0. in equations (6.9) are defined by equations (6.5) or (6.8)3,4,5. Equations (6.9) and (6.10) have been incorporated into two computer programs, Programs A2 and A3 of Appendix A, representing the ordinary trapezoidal rule and the modified trapezoidal rule, respectively. These programs have been used to predict a variety of stressstrain behavior within the class of loadings defined by fixed prin cipal axes. Figure 9 shows stressstrain curves for sheartype loadings. The curves were computed for the case of pure shear, defined by el =e22 =e(t) and e j=0 if ij ll or 22, which is included in the class of loadings discussed here. The results apply equally to the case of simple shear, defined by e12 = e21=e(t), eij =0 if ij/12 or 21. Simple shear is equivalent to pure shear plus a rotation, but it cannot be computed directly from Programs A2 or A3, which I+, L t: : : : . 11 ii L1 I, i l. .... ... .. _I I . Fif. ... .. i' l SI "n uti t1 i I ' :: : +. ' II II II II  I I J... i , 1 t 1. l V L { S1 "" ;;+ i+: :+:. ;t :::+ ... : ;:+ + :;; :;:+ :++ i+ < I * I: . i  42 4 I ii i: 1i .i1 i r : i ^i . .' 7'T"i:i ! ' I  i Ii::;:}:: i i U, II II LU cLu ' ; ' i, :" ': i; : 11 *I , 0) 4,; I. LU i:. LU LU rI . II N ,r II ,I 1 02 LU rl L) a LU *H C/) i ii44i I77 , ) ..i + :+ .. Ct " I ... . t ? ' ; ? ir f :t : i. . , ...  I I' ; I i i :+; i : ,! 4 + ...... ...J'  .. ... .. .;:: !   *  .. I I 2 O'Z S'I 0" S' N mJ cn P: C3 n F4 input only the principal strains. Note that the stress strain curves in shear depend on the workhardening parameter X in a manner similar to the case of axial loading, but that the degree of workhardening is not as great. That is, the stress compared to yield and the slope of the stressstrain curve for a given moderate value of strain are less for shear loadings than for axial loadings. The question of numerical errors now arises. How accurate are the numerical results, such as presented in Figure 9, that are computed using ProgramsA2 or A3? To answer this question we generated a large quantity of numerical results for the case of axial loading, using Programs A2 and A3. These numerical results, some of which are contained in Tables B3 and B4 of Appendix B, then were compared with highly accurate results computed from the closedform solutions of Chapter V. Figures 1013 illustrate graphically the numerical errors for axial loading. The graphs show relative errors in the computed stress plotted against strain, for two values of X and for the two integration formulae discussed earlier. We observe that the numerical error is constant in the elastic range 0 : el/eki 4/3. However, after plasticity occurs the variation of numerical error with strain is very complicated, with many peaks and dips appearing. The peak numerical errors tend to increase proportionally with the strain; note that the numerical errors generally are less than 102 upper curves: lower curves: co  en Ga) ml  0 n V 4J H 0 I '( 0 CM\ I 0 m Il O ' FIGURE 10 NUMERICAL ERRORS FOR AXIAL LOADING; , .... ORDINARY TRAPEZOIDAL INTEGRATION, 1= 0    i *j *  ** _r, strain, el/ek :  :i*     I  A 100 i 7 f r O4 *H O EL  . I  , .. !  I I  I ' N= 20 N= 40 :: i "" l 77 SI 1.1 I I. I I I. I I I upper curves: N=20 2 lower curves: N= 40 U) .. f 1 * r M U0P\\ C)k ri 0 . .1 k r4 FIGURE 11 . .. NUMERICAL ERRORS FOR AXIAL LOADING; ao  ... ":. ORDINARY TRAPEZOIDAL INTEGRATION, = 1 1 0 10 H WXi EO) , H 1' 'i" : U l'*' C !Of' I U E 1 i;  II * UEIALERR O XA ODIG *** o I ;' RIAYTAEODLINERTO ,X=1,. .;.' ._ . j s r i ,eoe :^ :  *H t 0I L 1 lI f i i i 10 10 10 78 I I I I I I. 1 . 1 I I I ; upper curves: N=20 lower curves: N= 40 I I' II :I i I , I ij * i"' 'fI ! a.I I iI A i ) (UI ( rq U S4 >t Ii. r T7r.,1rt * A I. J NUMERICAL ERRORS FOR AXIAL LOADING; MODIFIED TRAPEZOIDAL INTEGRATION, X= 0   4  ..:, I:: v : Zstrain, el/ek  ~ L ~ h i ._ I ,. ,. I L ~ r I  S"  t 4  I I I 10 100 ;.. .  & ~ ~~~ '   '' I  i c I~_ I __~ I . upper curves: N=20 S lower curves: N= 40i PmW 41J C) H w e ECU I C  (0 i 4e1 IO  ir U)~ f> a I ii' i . . FIGURE 13 NUMERICAL ERRORS FOR AXIAL LOADING; MODIFIED TRAPEZOIDAL INTEGRATION, X= 1 strain, el/ek 1 t k I I I i I III I t I t t... 1 I I 1 ( I 1 I 1 II. (plotting accuracy) when the strain el/e is less than 10 and when the gridspacing integer is 20 or larger. The dips are explained by the fact that for axial loading the elasticplastic boundary coincides periodically with the finite grid. When this occurs the numerical integration involves only functions with continuous derivatives and hence the accuracy is greatly improved. For loadings other than axial loadings the dips in the numerical errors versus strain curves would not be as pronounced. The effect of the gridspacing integer N can be found by comparing the two sets of curves on each graph: the upper curve was computed with N=20; the lower curve represents N=40. When the integer N is doubled, we see that the numerical error is reduced by a factor of about 3.6, which 1 85 implies that the error is approximately of order N This rule applies independent of X and independent of the integration method. For X=0, the effect of the modified trapezoidal rule as compared with the ordinary trapezoidal rule is to reduce the numerical errors by factors of 9 and 3 in the elastic and plastic ranges, respectively. For X=l, the modified trapezoidal rule reduces the numerical error by a factor of 12 in the elastic range, and the numerical errors are approximately unchanged in the plastic range. From the latter observation we conclude that, in general, there is no clear advantage in using the modified trapezoidal rule, or other higherorder integration formulae, instead of the ordinary trapezoidal rule. The reader is referred to Table B5 of Appendix B, where values of stress and strain are tabulated for the pure shear type of loading. Here again we see that the numerically computed stress depends on the parameter X and on the grid spacing integer N. In this case, however, we cannot establish the numerical accuracy by comparisons with exact solutions, since there are no exact solutions for pure shear loadings. We recall that classical estimates of the truncation errors are invalid due to discontinuous derivatives in the functions fk (,0). Furthermore, classical methods such as Richardson's extrapolation technique [71, p. 186] are invalid due to the complicated variations in the numerical errors with N. Henceforth, in lieu of any rational methods, we assume that the peak numerical errors versus strain are the same as shown in Figures 1013 for any loading that is approximately radial. The strain measure for an arbitrary loading can be taken as the accumulated plastic strain, defined by t 3 3 e = dt [ (eij ij/p) (eij s/) (6.11) o i=1 j=1 In consequence of this assumption we require at least N=20 for any computed data which are to be plotted on a graph and at least N = 40 for any computed data which are to be included in a table. The cost penalty of using large values of N will be discussed later. We shall have need of the deviatoric plane (or vplane) representations of stress and strain, defined as follows: Consider a threedimensional vector space with the principal stresses al, 02, 3 assigned to a set of Cartesian axes. Any state of stress can be represented by a point a= (0 ,02,03) in this principal stress space together with a specification of the principal stress axes. Consider a plane in principal stress space, called the deviatoric stress plane, defined as the locus of points that satisfy the equation o1+ 02+ 03 = 0. The vector s= (sl,s2,s3) composed of the principal deviator stresses always is contained in the deviatoric stress plane. Any yield condition that is independent of mean stress can be represented at most by one closed curve in the deviatoric stress plane for each orientation of the principal stress axes. Principal strain space and the deviatoric strain plane are defined simply by substituting"strain, e, e" for "stress, a, s" throughout the preceding paragraph. The following quantities are associated with the deviatoric planes: \22 2 1 s = 3/ (s + s + ) deviatoric stress intensity 2 2 23 e = V3/8 (e1 + e +e ) deviatoric strain intensity 3 23 e = 3/8 [ e (ei si/t) residual strain intensity i=l This statement is true provided the stress tensor is symmetric. If stress is nonsymmetric, as occurs for certain loadings discussed in Chapter VII, then the stress couple must be specified also. The principal stress axes can be specified by a rotation tensor or by Euler angles. The quantities s, e, e are proportional to the magnitudes of the vectors s= (sls2,s3), e= (ele2,e3) eo=es respectively. The factor V3\/8 is included so as to eventually simplify plotting of the data. With these definitions s= (4/3,2/3,2/3)#.s=1 and s= (1,1,0)= s = v'/4, etc. The residual strain eo, as defined here, is a useful concept only when the principal axes of stress and strain coincide. One objection which might be raised against the theory presented in this dissertation is that the predicted initial yield condition is a Tresca condition, whereas the observed initial yield condition for most real materials appears to be better described by a Mises yield condition. See Naghdi, et al. [41], Mair and Pugh [49], and Phillips, et al. [52,53]. In the deviatoric plane representation the Tresca condition is an equilateral hexagon and the Mises condition is a circle; both are symmetric about the origin. We can remove the above mentioned objection by considering radial loadings in various directions and by defining yield as experimentalists must do in terms of an offset strain or proof strain. Let us temporarily define yield as the stress intensity s corres ponding to some fixed residual strain intensity e along any radial loading path. Figures 14, 15, 16 show curves of constant offset strain in the deviatoric plane for several values of e 0 Figure 14 shows curves of constant offset strain for an initially isotropic material, defined by f(n) =0 at graph number: offset strain, e /e : o y 0 1 2 3 4 5 6 0 .02 .05 .10 .20 .40 1.0 direction of loading: A(2,1,1), B(7,2,5), C(1,1, 0) workhardening parameter: 0 X=0, A X=1.0, 0 X=1.5 FIGURE 14. SURFACES OF CONSTANT OFFSET STRAIN FOR AN INITIALLY ISOTROPIC MATERIAL 85 K 7 i I r4 r` Ln o ) cD c 71 t : ? 7 ! .. .. .:. .. . ... .. ; ... 1.... _. : i 1 7. 1: 7. /..  ... ...  i . LL 77. 47 :7N . i ~// ../ // ~.... : .. .. .... .. L ..... .... ... ... ....... /LlT_.. L  I i  1: ci . i/i, I HC\n:D LiJA .1 U.. I i I _______ LI .L _ . .. ..  I .: i t7 I _l FIGURE 15. SURFACES OF CONSTANT OFFSET STRAIN AFTER AN AXIAL LOADING / i t.. / " =L . :2 ___ i.. ._ ;_5~_.I ; t 7:I7 _..i _!_F .. ; L _L_ ..... ZL _'_L.JL' ". t ...... ...'L . ... 'q "W  = w .  : I ~ I; : i ! i :. :i l " .... : C ...;.=:,i i_: ....... i ... r... .~... : : r , t.: ' ; = : ; '  ":. It..  I. j ; I ; ; ; i  FIGURE 15. SURFACES OF CONSTANT OFFSET STRAIN AFTER AN AX(IAL LOADING FIGURE 16. SURFACES OF CONSTANT OFFSET STRAIN AFTER A PURE SHEAR LOADING time t=0. The curves are shown as 120degree segments rather than as complete closed curves in order to include data for three values of the workhardening parameter X on one graph. If completed,the curves would be symmetric about the origin in the same sense as the Tresca hexagon. We see that the "initial yield surface" can assume a variety of shapes depending on the offset strain used to define yield. It is apparent that any definition based on an offset strain in the range .02< e /ey< 1.0 will predict an initial yield surface that is closer to a Mises circle than to a Tresca hexagon. This is approximately the range of offset strain encountered in most experimental determinations of yield. Figures 15 and 16 show curves of constant offset strain after the material has experienced plasticity during an initial axial loading and an initial pure shear loading, respectively. Here the data are computed for X=0 only. We see that the "subsequent yield surfaces" are not symmetric about the origin in the deviatoric stress plane, which implies the yield conditions are anisotropic. We observe in each case that the yield surfaces have expanded in the directions of loading and contracted or flattened in the directions opposite the loading. This phenomenon may be called the generalized Bauschinger effect. We also observe that the theoretical yield surface, defined by zero offset strain, exhibits a sharp corner at the point of loading, but that each yield surface defined by a finite offset strain, typical of an experimental determination, exhibits a "blunt corner" at the point of loading. The latter observation agrees with conclusions by the majority of experimentalists who have investigated the question of whether there are corners in the yield surface. See Ivey [32], Bertsch and Findley [45], Mair [51], and Phillips, et al. [52,53]. An interesting and informative presentation of elastic plastic stressstrain behavior involves the plotting of yield surfaces in the deviatoric planes for a sequence of stress or strain increments. Such a presentation can suggest rulesofthumb that might be applicable in a general loading situation. Program A4 of Appendix A is a modification of Program A2 that computes a sequence of yield surfaces from a given sequence of strain increments with fixed principal stress and strain axes. The program first computes a given state of stress and strain, and then probes in various direc tions to find the yield surface defined as the exact boundary of the elastic domain. Then the procedure is repeated for the next increment of stress or strain. Program A4 was used to generate yield surfaces for several loading paths as described below. Appendices C and D present yield surfaces computed for S=0 and for thirteen different loading paths. Three of the loading paths are purely radial and ten involve abrupt changes of direction. The yield surfaces are plotted in the deviatoric plane in both principal stress and principal strain space, which enables one to construct the complete stressstrain histories. The plots were machinemade directly from the output of the computer program, which at least eliminates human errors and the tedium of handplotting. In some ways the yield surfaces in principal strain space, contained in Appendix D, are the most interesting. Here the yield surfaces gradually develop and pull away from the former yield surfaces when a new loading direction is taken. By studying Appendices C and D, we see that most of the yield surfaces are composed of both straight and curved line segments. A sharp corner with curved sides tends to form around the point of loading, whereas the back sides of the yield surfaces generally retain the character of the initial yield surface (the Tresca hexagon). The generalized Bauschinger effect mentioned above is evident throughout Appendices C and D. For axial loadings the yield surfaces tend to shrink laterally, but for pure shear loadings the lateral shrinkage, or cross effect, is small. For all radial loadings the elastic range in the direction of loading is constant. Phillips and coworkers investigated experimentally the effects of temperature and plastic deformation on the shape of the yield surface for annealed, commercially pure aluminum. The changes in the yield surfaces observed experimentally by Phillips, et al. [52,53] and Ivey [32] during plastic loadings are similar in some respects to the theoretical results contained in Appendices C and D. Among the experimental observations are: (1) increased curvature at the point of loading and flattening on the backside of the yield surface, (2) translation of the yield surface away from the origin in the direction of loading in principal stress space, (3) no cross effect for either shear or axial loadings, (4) reduction of the elastic range in the direction of loading for both shear and axial loadings. While these experimental observations do not entirely agree with the theoretical results, the differences are not serious. The author believes that further computations could be made that would bring the present theory into much better agreement with the experimental results of Phillips, et al. Item (1) above is essentially in agreement with the present theoretical results. Item (2) could be predicted theoretically by choosing a value of X in the range 1 lX 2. The differences between item (3) and the theoretical results could perhaps be traced to the fact that the experimental results are plotted in the shearstress versus axialstress plane whereas the theoretical results are plotted in the deviatoric planes. Item (4), dealing with reduction of the elastic range, is more serious. Prediction of this effect would require a major modification of the theory, which is not justified at this time since the simpler present version of the theory has not yet been fully explored. CHAPTER VII STRESSSTRAIN BEHAVIOR FOR BIAXIAL LOADINGS AND FOR A GENERAL LOADING HISTORY Suppose we are given an arbitrary sequence of deviator strain increments comprising a deviator strain history at a material point. E() =0, E(m)=E(m)+ dE(m), m=1,2,3,... dE (m) de(m)], (i,j=1,2,3) (7.1) The following equations govern the elemental stress vectors and the resultant stresses. The notation is the same as in previous chapters. f )(n)=, f(m= 1 if gt m=1,2,3,... Y/gt if gt > Y (7.2) g=f+ xd(mn+nl(n. dE n) = [g(g (g gn)2 ] ,ff 7T/2 7fT/2 7r/2 l 2 dOl dB fsin t= d/2O dO f sin 9 SffJto 0 lo 7T /2 lo IT f (7.3) t3= f dj de fsin 8 3 o o We assume all vectors and tensors are referred to two coordinate systems: a given fixed Cartesian coordinate system and an By a fixed coordinate system we actually mean a set of coordinate axes fixed relative to the material in the undeformed state. For use in the deformed state we require the coordinate axes to be rotated by the same rotation that carries the principal strain axes from the undeformed state to the deformed state. This distinction was not necessary in Chapters V and VI as all vectors were referred to the principal strain coor dinates and these axes were fixed in the material. 91 arbitrary Cartesian coordinate system specified by the base vectors N (c =1,2,3). The spherical coordinates 9,O in a equations (7.3) are defined relative to the arbitrary coordinate system only. By symmetry considerations we can rewrite equations (7.3) as follows: l=I1 + 1 + 13 + 4 t2 = I1 12 13 + 14 t =I I I I t3= 1 +2 3 4 L3 =I1 + 12 13 14 1 /2 S> (7.4) where I.= dI dB fsine, (i=1,2,3,4) 1 S = (0,i/2), S3 = (T,37/2) S2= (,1/2,7), Sq= (3u/2,2) The reason for considering the decomposition (7.4) is that it allows us to reduce the integration domain from seven octants to four octants on the unit sphere. Computer storage as required for the numerical integration is reduced accord ingly. For computational purposes we introduce a finite grid of points over one hemisphere of the unit sphere, and we introduce a set of unit vectors defined by these grid points. If P= n/2N, then 6. =iP, j =j P, (i=0,...,N; j =0,...,4N) 1 J (7.5) n..= (cos 9., sin .cos .,9 sin .sin .) ij i' i We are now in a position to describe a numerical pro cedure for computing the stress history corresponding to a 