The Mathematical Foundation of Plasticity Theory

ADVANCES IN APPLIED MECHANICS, VOLUME 34

The Mathematical Foundation of Plasticity Theory WE1 H. YANG The University of Michigan Ann Arbor; Michigan

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11. Minkowski Norms and Holder Inequality . . . . . . . . . . . . . . . . . . . . . .

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............................

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IV. Constructing the Dual Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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V. Application to Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 11

111. Generalized Holder Inequality

VI. A Duality Theorem for Plane Stress Problems . . . . . . . . . . . . . . . . . . . 313

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 315

Abstract Supporting the mathematical theory of plasticity, two pillars constituting the modeling of yield behavior are the convexity and normality conditions on a yield function. The models so established have been time tested to hold for a wide class of ductile materials in many practical applications of the theory. Their logical basis was the belief that a material does not release energy during plastic deformation (nonnegative dissipation). Even though the theory is sound, the lack of a mathematical proof on normality still causes some researchers to call it a postulate. In this paper, we shall dismiss this notion of assumption by introducing a generalized Holder inequality. The normality relation becomes a requirement for the inequality to be sharp (equality inclusive). This inequality also reveals the primal-dual relation of stress and strain rate. Even before this connection of the mathematical and physical implications, different duality relations were used in many known minimax theorems in plasticity. We shall claim that convexity, normality, and duality, all properties of the generalized Holder inequality, complete the foundation for the constitutive modeling of the mathematical theory of plas303 ISBN 0-12-002034-3

ADVANCES IN APPLIED MECHANICS. VOL. 34 Copyright Q 1998 by Academic Press. All rights of reproduction in any form reserved. 0065-2165/98$25.00

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ticity. By using these models, the minimax formulations of plasticity problems can be derived in a systematic and unified approach.

I. Introduction Convexity and normality are the two pillars of a constitutive model that is the basis of the mathematical theory of plasticity (Hill, 1950). The yielding behavior of metals has been modeled satisfactorily by convex yield functions more than a century ago (Tresca, 1864,1868). Such models remain valid, even in light of new materials and experimental results. Although it was conjectured earlier (Prandtl, 1924; Reuss, 1930), a logical basis of normality was not in place until 1959 (Drucker, 1959). Based on the existing foundation, we shall add another pillar to complete this useful constitutive model. First, we shall present a yield function f as a norm (or a seminorm) on the 3 x 3 symmetric stress matrix B. An initial yield criterion may be modeled by

f (c)= llall - 0 0 5 0 ,

or

l b l l 5 009

(1)

where 00 is a material constant. Both the form of the norm and the material constant are allowed to vary with plasticity deformation history to model subsequent yield behavior. This representation not only gives a sense of bounded stress, but will also lead to a foundamental inequality that encompasses the Cauchy-Schwarz and Holder inequalities as special case. These inequalities pertain to the inner product of vectors, matrices, and functions. We shall restrict our discussion to finite dimensional spaces that involve only vectors and matrices. The upper bound of an inner product is expressed in terms of dual norms of vectors and matrices. Although the Cauchy-Schwarz inequality applies only to Euclidean norms, the Holder inequality extends the applications to a family of Minkowski norms. Norms that model yield behavior are generally non-Euclidean. The inner product in plasticity involves the stress and strain rate matrices. The generalized Holder inequality for such an inner product may take the form

where the colon denotes the inner product operator between two matrices and the subscripts of the norms, ( p ) and ( d ) , suggest the primal-dual relation. A primal norm is determined to fit experimental data. Its dual norm is determined mathematically to establish the theorem of generalized Holder inequality. We shall first

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introduce the background leading to the theorem, then present its application to plasticity. The well-known normality relation in plasticity between the strain rate and stress becomes the natural sharpness condition of this generalized Holder inequality. This new inequality is fundamental to all minimax (duality) theorems in plasticity. A special case for plane stress problems will be presented as an example. The familiar Cauchy-Schwarz inequality for the Euclidean vector space is commonly stated as

where x, y are vectors in Rn; T transposes a vector, and clidean norm. The equality holds if y=ax,

aER,

11 . 112 denotes the Eu(4)

or the two vectors are colinear. The inequality (3) is often used in the upper bounding process of a mathematical analysis. A “sharp” upper bound that includes the equality case is vitally important in the field of functional analysis and its applications. Obviously, a function that is bounded above has a finite supremum. The supremum of a function is contained in the range of a sharp upper-bound function. Therefore, a search for the least upper bound will recover the maximum of the original function. This indirect method of finding the maximum of a function will fail if its upper-bound function is not sharp. The Cauchy-Schwarz inequality does not hold in a general non-Euclidean space. Holder introduced an extension so that a modified inequality is valid in a vector space in which a family of Minkowski norms (Birkhoff and MacLane, 1953) is defined. The colinearity condition (4) for sharpness must also be modified. We shall use a simple example to illustrate these points. Consider first the norm i n R 2 , Ilxllm = max(lx11, Ix21}.TwospecificvectorsxT = (1, 1) andy’ = (1, 1) produce the inner product xTy = 2. This inner product cannot be less than or equal to llxllmllyllm = 1. Now let y be measured by another norm, llylll = Iy1( 1 ~ 2 1 It . is well known that lx‘yl I IlxmIIIIyII1 for all x, y E R2,and that equality holds under a certain condition that, in general, is not the colinearity of x and y. This can be illustrated in geometric terms. Let

+

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Clearly, all such vectors lie on two “unit circles,” one in the shape of a square for x and the other in the shape of a diamond for y, as shown in Figure 1. The well-known inequality T

Ix YI I llXllo0llYlll = 1

(6)

holds for any x and y defined in (5). Here, the vectors x and y are said to be a pair of dual variables. Consider a specific vector, x=(l

a)?

o
(7)

The vector y in the direction of this x has the form T

so that it has unit length as defined in (5). The inner product of these two colinear vectors can be computed to yield xTy = (1 +a2)/(1+ a ) < 1.

(9)

Hence the sharpness condition in (4) is not valid in general. The intuition that an inner product of two vectors attains a maximum (or minimum) when they are colinear needs qualification. The sharpness condition for the dual vectors defined in ( 5 ) does exist and arises from certain specific pairing other than the colinearity of the two vectors. The pairing depends on the norm measures defined. We shall explain such pairing in a slightly more general setting than the 1-norm and 00norm considered so far.

FIG .

I . Unit circles for vectors in R2with different norm measures.

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11. Minkowski Norms and Holder Inequality

x

The family of Minkowski norms (Birkhoff and MacLane, 1953) for a vector R"is defined by

E

(called a p-norm), which includes the familiar cases p = 1,2, and 00. A pair of norms, II . [Ip and 11 . [Iq, in this family are said to have a dual relation (Birkhoff and MacLane, 1953) if (11) The Holder inequality holds for any pair of such dual norms so that lXTYl

I IIXllplIYlly.

It is well known that the equality case holds if either yj = a)xi)p-'sign(xi) or = alyilqp'sign(yj), a E R,1 I p , q 5 00, i = 1, 2 , . .., n. This sharpness result can be obtained by

Xi

Y = aVllxllp

or x = aVlIYIIq,

(13)

where V is the gradient operator. When operated on a differentiable function of n variables, V produces a gradient vector in Rn.A gradient vector is normal to the level sets (contours) of the norm function. Thus the expression in (13) will be called the normality relation. The vectors satisfying (13) belong to a dual pair. To avoid the issue of differentiability of certain norms, we postpone the discussion on the cases ( p , q = 1,00) in which the norms are Co functions. We shall show the equality case of (12) under one of the normality relations of (13) by first choosing a p-norm (1 < p < 00) for x and then obtaining the components of y by differentiation,

where k > 0 is chosen as the proportional factor, and the range of the sums over j and the index i are implied to span from 1 to n. Using (14), we can form the q-norm of y from its components and, with some simplification, obtain

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which gives the proportional factor k an explicit expression. Again, using (14) and (1 9 , we can produce the inner product for the pair of dual vectors,

The equality is thus obtained and the normality relation is the desired sharpness condition. The absolute bound on xTy is not needed because k is chosen to be positive. Although the crucial relation (1 1) is only alluded to here as a reference (Birkhoff and MacLane, 1953), it will be made evident during the construction of dual norms in a later section. Inequalities (3) and (6) are special cases of (12). The normality relation between dual vectors is not restricted to the family of Minkowski norms. General non-Euclidean norms have a dual structure for pairing vectors that participate in an inequality more encompassing than that of Holder.

111. Generalized Holder Inequality All norms share a common property as convex, nonnegative, and homogeneous functions of degree one. One may arbitrarily define a norm as long as it satisfies the conditions

for all a E R and x, y E R". The condition (i') is omitted for the definition of a seminorm. Usually, the definition of a primal norm (or seminorm) arises naturally from a specific application. A dual norm must be matched to the primal norm so that a theorem for the generalized Holder inequality (Clarke, 1983) may be stated below.

THEOREM.For any two vectors x,y E R", where x is measured by a properly dejined primal norm (or seminom), (1 . there exists a dual norm (or seminorm) 11 . 11 ( d ) such that the inequality

holds. The case of equality is attained when the normality relation

is satisjied.

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The theorem, of course, covers the original Holder inequality (12) as a special case. A proof of the theorem and the method of construction for a dual norm are given in Yang (1991) for differentiable as well as Co norm functions. We shall review the dual norm construction and provide an example in plasticity.

IV. Constructing the Dual Norm The specific form of Ilyll(d) can be determined from a given l l ~ l l ( ~If) .the primal norm is given as a set of experimental data defining the “unit sphere,” as may be the case in an engineering problem, then a local gradient can be computed approximately at each data point. These gradient vectors will trace the “unit sphere” of the dual norm. This numerical method, although practical, may not be a satisfactory demonstration of dual norm construction. We shall give a close-form example. Let the primal norm be a C’ function such that its first derivatives can be computed everywhere. Dividing (19) by llyll(d), then the equation represents a map from x to y. If I ( ~ l l (is~ )strictly convex, the inverse map exists and eq. (19) can be solved for the components of x in terms of those of y. By taking the (p)-norm of x from its components and setting it to unity, an expression for llyll(d) is obtained in terms of the components of y. If Ilxll(p)contains a linear portion, the points on the linear portion map to a single point on llyll(d). In such cases, the map is no longer one to one, but construction of dual norm can still proceed. In a previous section we have verified the Holder inequality (12) without questioning the origin of the dual relation (1 1) of the Minkowski family of norms. For a given primal norm IIxIIp (1 < p < GO), we shall now derive the form of its dual norm llyll(d) without resorting to (11). Rewriting (14) by letting llxllp = 1 and k = llyll(d), we have

from which we may solve for

It is an easy matter to form

(JXII(~)

from the components xi and to obtain

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From the above equation,

is obtained to give the explicit form of Ilyll((/)in terms of yi and the definition of the primal norm. Now letting p / ( p - 1) = q , we confirm the dual relation (1 1) and conclude that q=- P P-1 is the dual norm for the primal norm lIxllp, 1 < p < 00.Construction of the dual norm for the other non-Euclidean C’ primal norms is analogous. The cases p = 1, 00 correspond to Co norms that are not differentiable in the usual sense at certain places of the independent variable. They can be regarded as the limiting cases, and the inequality (6) is also confirmed from the above analysis with q = 00, 1 obtained for the corresponding dual norms by taking limits from the p-norms in C’ . Although there is no mathematical difficulty in demonstrating the generalized Holder inequality and its normality relation for Co norms, it is desirable to offer an interpretation for the theorem at those points where the norm function is not differentiable. We shall call those points the vertices of a norm. Consider again the example in R2 shown in Figure 1. The gradient of JIxlloo has only four distinctive values, (1,0), (0, l), (- 1 , O ) and (0, - l),each evaluated along a respective edge, D A , A B , B C , and C D , on the unit circle of the primal norm, as shown in Figure 2(a). These four values are shown as four points ( A ’ ,B’, C’, and D’in Figure 2(b) on the “unit circle” of the dual norm, llyll(d) = 1, yet to be constructed. Because the dual norm must be a convex and continuous function, these four points must be connected. Let us connect the four points by straight lines to form a diamond, A’B’C’D’. Then no points of the “unit circle” should fall inside the diamond, or

F I G . 2. Duality and normality of 1-norm and m-norm.

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the dual norm function would be nonconvex. There exist, of course, convex functions passing through these four points but outside the diamond. We shall show that they cannot be the dual norm function either. Because duality is symmetric, the dual norm of the dual norm is the primal norm, and the symmetric normality relations are given in (1 1). The gradient of the dual norm on the “unit circle” llyll(d) = 1 must fall on the unit circle (the square A B C D ) of the primal norm. The diamond, satisfying this condition, is the only correct dual unit circle. The gradient of a Co norm function at the vertices is not unique and is interpreted as a point-to-set map. This generalized gradient (Clarke, 1983) takes the form of triangular fans, as shown in Figure 2, in which the lengths of all gradient vectors are proportionally drawn.

V. Application to Plasticity The concept of a norm is not restricted to vectors. There are matrix norms (Householder, 1954), function norms, and operator norms (Royden, 1988). In mechanics, variables may appear as matrices or matrix functions. A matrix norm has the same properties as that of the vector norms. The norms defined on stress matrices in the theory of plasticity are mathematical models of yield behavior based on experimental results and certain physical principles (Yang, 1980a, b). In plasticity, the natural primal variable is the stress denoted by u E R 3 x 3a, real symmetric 3 x 3 matrix representing the state of force per unit area at a point in the material. It may vary from point to point in a domain. It is called the stress field (a matrix function) in a body. Because no real material is infinitely strong, the strength of a material can be modeled by a bound, IIc II 5 co,presented in (1). The specific form of the norm is called the yieldfunction. The best known yield function is the von Mises yield function (Hill, 1950). To shorten the discussion, but still keeping all of the essentials relevant to the present exposition, the von Mises yield function is stated in a subspace of plane stress where an element is a 2 x 2 symmetric stress matrix

which represents a state of stress in a sheet of material located in the x, y plane of a Cartesian coordinate. The von Mises yield function can be stated in two equivalent forms,

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where a1 and a2 are the eigenvalues of the plane stress matrix in (25), and 11 . IIv denotes the von Mises norm. Another yield function, named after Tresca (1864), is physically more precise but is used less because of its nonsmoothness. When applied to the plane stress matrix, it takes the form

where the Tresca norm IIa IIT can also be expressed in terms of the stress components. One may not find these norms in a mathematics book, because they are derived from physical considerations. Nevertheless, they are valid norms or seminorms according to the conditions in (17). The dual variable in the context of plasticity is the strain rate, E , also a 2 x 2 symmetric matrix for the plane stress case. The inner product (a,r ) of the two matrices appears frequently in the field of mechanics. From a physical principle of nonnegative dissipation, Drucker (1959) had reached the conclusion that the plastic strain rate matrix is a constant multiple of the gradient of the yield function arranged in the form of a matrix, that is,

where the gradient is taken with respect to each of the stress components in (25) and arranged in a matrix form. This normality relation in plasticity is the sharpness condition of the generalized Holder inequality. What was not well understood in plasticity is that a plastic strain rate matrix and the stress matrix must also satisfy the inequality (2). The dual norm on strain rate was never introduced. Now the generalized Holder inequality gave the theory of plasticity a complete foundation. For the von Mises and Tresca primal norms, the corresponding dual norms on the plastic strain rate E in terms of its eigenvalues r1 and r2 are

116111 = max

{lei I , I E ~ I ,

+

I E ~ E~I},

respectively. They are derived from the corresponding primal norms in (26) and (27) by the constructive process described earlier. Because II . IIT is a Co norm, its gradient should be understood in a generalized sense for the construction of its dual I1 . I l l . For a graphic presentation of the results, the eigenvalues of the plane stress matrix ( a l , a 2 ) and that of the plastic strain rate matrix ( ~ 1 ~, 2 are ) regarded as

313

Mathematical Foundation of Plasticity Theory 11€11* = 1

IlOll" = 1

IIEIIL

=1

11611T

=1

FIG.3 . Geometric representation of primal and dual norms.

vectors in R2.The unit circles for the forms are shown in Figure 3, where the primal norms are shown in solid lines and the dual norms are in broken lines.

VI. A Duality Theorem for Plane Stress Problems Limit analysis for plane strain problems seems to dominate plasticity literature, only because a method of characteristics known as the slipline method (Hill, 1950) was developed. Under the new light of duality theorems, all such problems can be formulated and solved by computational optimization techniques. We depict the plane stress problems to demonstrate how the generalized Holder inequality can lead to a duality theorem. The plane strain problems can be treated in the same manner. It is well known that a stress distribution that satisfies the equilibrium equation, the yield criterion in a domain D , and the static boundary conditions on a part of the boundary aD, belongs to the set L of lower bound solutions (Yang, 1987). We may write L = S r l C, where S denotes the statically admissible set and C denotes the constitutively admissible set of the stresses. Considering the load factor, h E R,h 3 0. We seek the largest possible value of h. This limit analysis problem can be stated in the language of optimization as maximize subject to

h(a) V . 6 =0

a.n=ht llallv I 60,

in on

D aD,

(30)

where V. is the two-dimensional divergence operator applied to the plane stress matrix; n is the outward normal vector on the boundary of D ; t is a constant traction vector (preferably normalized) prescribed on a D,. The statement in (30)

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Wei H.Yang

forms the primal problem for which a unique A,,, = h* exists. The load factor h now has the form of a functional. The problem (30) has a dual whose minimum is equal to h*. To formulate this dual problem, we begin with the weak equilibrium equation,

SD

u.(V.o)dA=O

VUEK,

where u is an arbitrary function in the kinemtically admissible space K C R 2 ( D ) , which consists of all admissible velocity functions. Integrating (31) by parts and using the kinematic boundary condition on the complement to aD, (i.e., a D = aD, U aDk) and the static boundary condition in (30), one obtains an equivalent variational statement of equilibrium,

Lo:cdA=h

1,

t.udS

where c = (Vu+VuT) is the 2 x 2 strain rate matrix. Because u and, implicitly, E appear homogeneously in (32), we may scale u by a constant without altering the value of h. It we choose the integrals in (32) to be positive, the boundary integral can assume the value + I (normalization by scaling). Hence the functional h is equal to the domain integral on the left-hand side of (32). The primal problem (30) can be restated and the value of h bounded as follows:

. where the sequence of inequalities yields the sharp upper functional ~ ( u )We shall seek u E K to minimize this upper-bound functional. Cesari and his coauthors (Cesari et al., 1988; Cesari and Yang, 1991) have identified the correct function space where a minimizing u exists and yield the duality theorem: minp(u) = h* = maxh(o). ueK

U€L

(34)

Numerical solutions of plane stress problems using (34) were presented by Huh and Yang (1991). Similar duality theorems solved the problems of plane strain and plates (Liu and Yang, 1989; Yang, 1987).

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References Birkhoff, G and MacLane, S. (1953). A survey of modern algebra. Macmillan, New York. Cesari, L., Brandi, P., and Salvadori, A. (1988). Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions. Ann. Mat. Pura Appl. 152(4), 95-1 21. Cesari, L., and Yang, W. H. (1991). Serrin integrals and second order problems in plasticity. In: New solutions in differential equations and dynamic systems. Royal Society of Edinburgh, p. 1 17A. Clarke, F. H. (1983). Optimization and nonsmoorh analysis. Wiley, New York. Drucker, D. C. (1959). A definition of stable inelastic material in the mechanics of continua. J. Appl. Mech. 81, 101-106. Hill, R. (1950). The Mathematical theor): ofplasricify. Clarendon Press, Oxford. Householder, A. S. (1954). On norms of vectors and matrices. ORNL Rep. 1759. Huh, H., and Yang, W. H. (1991). A general algorithm for limit solutions of plane stress problems. Internat. J. Solid Structures 28,6. Liu, K. H., and Yang, W. H. (1989). Upper and lower bound solutions of an extrusion problem. Numer. Methods Indus. Form. Proc. 89. Prandtl, L. (1924). Proceedings of the 1st International Congress of Applied Mechanics, Delft, The Netherlands. Reuss, A. (1930). Z. Angew. Math. Mech. 10,266. Royden, H. L. (1988). Real analysis, 3rd edn. Macmillan, New York. Tresca, H. (1864). C. R. Acad. Sci. Paris 59. Tresca, H. (1868). C. R. Acad. Sci. Paris 64. Yang, W. H. (1980a). A generalized von Mises criterion for yield and fracture. J. Appl. Mech. 47, 297-300. Yang, W. H. (1980b). A useful theorem for constructing convex yield functions. J. Appl. Mech. 47, 301-303. Yang, W. H. (1987). A duality theorem for plastic plates. Acta Mech. 69. Yang, W. H. (1991). On generalized Holder inequality. Nonlinear Anal. 16(5).