Do - Fa,(El,, a. condition of von Mises-considered here as the extreme case of an inequality, body.2

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1 720 ENGINEERING: T. Y. THOMAS PRoc. N. A. S. that he had previously employed co-ordinate systems in which the rotation vanished, as an aid in the construction of invariant relations. See R. S. Rivlin and J. L. Ericksen, "Stress-Deformation Relations for Isotropic Materials," J. Rat. Mech. and Anal., 4, , Op. cit., p Op. cit., p. 27. COMBINED ELASTIC AND PRANDTL-REUSS STRESS-STRAIN RELATIONS BY T. Y. THOMAS GRADUATE INSTITUTE FOR MATHEMATICS AND MECHANICS, INDIANA UNIVERSITY, BLOOMINGTON, INDIANA Communicated July 29, Introduction.-The concept of stress-stain relations valid for both the elastic and the plastic domain has certain theoretical advantages which appear to compensate for the fact that such relations are necessarily somewhat more complicated than the corresponding relations for either of these domains treated separately.' For example, there is the possibility of a more gradual transition from the elastic to the plastic state corresponding to the actual behavior of materials; also, the use of a single system of equations eliminates the troublesome matter of matching the elastic and the plastic solutions, which arises in the process of unloading when distinct systems of equations are assumed to govern the elastic and plastic behavior of a body.2 The discussion in this paper is based on the assumption of a set of stress-strain relations of the form Do - Fa,(El,, a..., t33; lla,.. *, 33), (1) in which the F's are the components of a tensor invariant of the stress tensor ur and the rate-of-strain tensor e; the quantities Do,,/ are the components of the absolute time derivative of the stress tensor.3 Briefly stated, we have attempted to construct the tensor invariant F so as to obtain the utmost simplicity consistent with the usual assumptions, such as isotropy of the material, homogeneity, and the quadraticyield condition of von Mises-considered here as the extreme case of an inequality, since relations (1) are intended to embrace both the elastic and the plastic domain as commonly conceived. While this set of stress-strain relations is of differential character it reduces approximately on integration, as shown in section 9, to the usual algebraic relations of classical elasticity theory for small displacements about the natural or unstressed state of the body. In the extreme case of yield, the stressstrain relations of this theory become identical with the condition of incompressibility and the Prandtl-Reuss relations (sec. 8) which have been taken as basic in the recent book by Prager and Hodge.4 2. Definitions.-Rectangular co-ordinates will be employed exclusively, so that there will be no distinction between the covariant and contravariant indices of tensorial quantities, and covariant differentiation, denoted as usual by a comma, will be

2 VOL. 41, 1955 ENGINEERING: T. Y. THOMAS 721 identical with partial differentiation relative to these co-ordinates. The use of such systems involves, of course, the implicit assumption that a rectangular system can be found for which the equations of the theory will be valid and will lead to results which approximate the results of experiment. An essential part of the solution of any problem is therefore the selection of the proper rectangular co-ordinate system. The velocity is defined by its components va = dxa/dt, i.e., the components of velocity are the total time derivatives of the co-ordinates xa of the material particles. We also denote by o-aa the components of the symmetric stress tensor. Other quantities which will enter the discussion can now conveniently be defined as follows: Stress deviation: o,*= -l/3aiba Rate-of-strain tensor: ea,@ = 1/2(Va.,, + Vpa), Rate-of-strain deviation: 4at = fag-1/3(f4a where it is understood that repeated indices, unless the contrary is stated, are to be summed over the values 1, 2, 3, and that bag denotes the usual Kronecker delta. 3. Dynamical Relations.-The following dynamical equations must be satisfied in a continuous medium which is not subjected to body forces, namely, dp -- + PVaICV = 0 (equation of continuity), (2) dt O' = dv (equations of motion), (3) where p denotes the density. Since the first of these relations is a consequence of the assumption of the conservation of mass, it follows, by putting p = 1, that we obtain va a = 0 or Eaa = 0 as the condition for incompressible flow. Equations (2) and (3) above are invariant under the group of orthogonal coordinate transformations which relate the preferred co-ordinate systems of our Euclidean metric space. These equations are also invariant under the group of uniform translations, in conformity with the principle that it is not possible from dynamical considerations to distinguish between two reference frames which are moving relative to each other with uniform velocity. 4. Simplification of the General Stress-Strain Relations.-As the first step in the process of simplifying relations (1), we assume that these relations have the form- Du = A ad 4Eik + Bad ike (4) a a ik where A and B are tensors which can be taken without loss of generality to be symmetric in their contravariant and also in their covariant indices. One could assume the coefficients Aik and Bik in these relations to be functions strictly of the coordinates xa and the time t, in which case the right-hand members of the relations will be linear and homogeneous in the quantities Eik and U'ik. But certain of the following requirements will be satisfied with greater facility by sacrificing this linearity and allowing these coefficients to be expressed as tensor invariants of e and o. We shall find, however, that this can be done in such a way that only simple quadratic expressions in the quantities e4e and v* are involved in the components of the tensors A and B.

3 722 ENGINEERING: 7'. Y. THOMAS PROC. N. A. S. The above assumption (sec. 1) that the right-hand members of relations (1) are the components of a tensor invariant of the stress and rate-of-strain tensors implies that the material is isotropic. Now, as generally understood, the requirement of isotropy means that at any point x and at any time t the properties of the material are independent of direction; this is expressed mathematically by the condition that the stress-strain relations retain their strict form under proper orthogonal coordinate transformations. It follows that A and B must be isotropic tensors. As is well known, the components of the symmetric tensors A and B must therefore have the form Aas = X6 ik + J( ink + ia k) ik = Sjikba# +.(5ink + S ibak) where X, IA,, and r are scalars. Substituting these expressions in equation (4), we now have D =- (Xejijba + 2jtea#) + (Q0iibao +2± ao). (5) We shall wish in particular to consider the single equation obtained from equation (5) by summing on the indices a and A. We shall likewise need to deal with the set of equations which result when this latter equation is combined with equation (5) in the manner indicated in section 2 for the formation of a deviation tensor. We thus obtain the following relations: =a_ (3X + 2.u)Esi + (30 + 2v)o-i (6) Dao 24* + 2Do* (7) a which are readily seen to be equivalent to the system (5). 5. The Fundamental Inequality.-It will be assumed that there exists a (positive) material constant K, regarded here as one of the quantities characterizing the behavior of the material, such that for all flows we have clafc4. K. (8) A point x of the material at which the equality holds in relation (8) will be called a yield point at the time t in question. A region of the material over which the equality holds will be said to be in a state of yield. The strict equality in condition (8) above is the well-known yield condtion6 of von Mises. For a region in a state of yield it follows from equation (7) and the definition of the absolute time derivative' that *doa Do-* a# dt 0 a# = 2AEaoa + 2oraa#a = 0. Hence we can write * * fana K a6.fr aj<a8=k (9)

4 VOL. 41, 19)55 ENGINEERING: 7'. Y. THOMAS 723 Now t/,u is a scalar, and it can therefore be considered quite generally to be a function of the co-ordinates Xa and the time t. However, from the standpoint of the structure of the theory it is advantageous to represent this scalar by writing v = ala*cf + ba afo$ + c, (10) where the quantities a, b, c, are scalar functions of position and time. When the yield condition is satisfied, the right-hand members of equations (9) and (10) must be equal. Assuming this equality to be of the nature of an identity, it follows that a = -1/K and bk + c = 0. Hence equation (10) takes the form K - - K)' (11) in which y now appears as a scalar function of the co-ordinates Xa and the time t. When yield occurs, this relation reduces to relation (9) above. We can now use equation (11) to eliminate the quantity r from the right-hand members of equation (7). Hence * g = 22_ [KKi)] - : A } (12) a# L K +K/ a# 6. Condition of Incompressibility.-It is generally considered that one can assume with sufficient accuracy that volume changes do not occur during plastic flow, i.e., when the material is in a state of yield (sec. 5). This condition of incompressibility can be realized automatically by choosing the coefficients in equation (6) to be suitable scalar invariants.7 Let us write equation (6) in the form where A D = e + Bo-i, (13) 1 _32_ + A= 3X + 2, 3X+ 2A' We now assume that A 0 and B 0 as or* -fr, K at any point x independently of the time t. It follows immediately from this assumption and from equation (13) that the condition of incompressibility Eii = 0 is satisfied for plastic flow.8 Corresponding to the procedure in section 5, let us represent the scalars A and B, as can be done without loss of generality, by expressions of the form A = a +* b4*4*; B = c + do-*4* in which a, b, c, and d are functions of the co-ordinates xa and the time t. Then, for plastic flow, we have a = -bk and c = -dk by the above assumption. Hence A and B become A = -bk + bou- ; B = -dk + do4.

5 724 ENGINEERING: T. Y. THOMAS PROC. N. A. S. When we substitute these expressions in equation (13), this equation can be given the form9 Dv~t = (a*'60- + ki, (14) Dl- 1 -a(a4/k1) where h and k are scalar functions of the xa and t. Equation (14) reduces to the condition of incompressibility eii = 0 for plastic flow. 7.- Homogeneous Material.-The material will be said to be homogeneous if the stress-strain relations are left strictly unchanged by co-ordinate translations; this condition is sometimes expressed by saying that the properties of the material will appear the same to observers at different points. Assuming that this invariance requirement applies independently of the time 1, it follows that the scalar functions,u, 7y, h, and k in the stress-strain relations (12) and (14) reduce to material constants for homogeneous material. The determination of the original scalars X, i, and r in terms of the above material constants IA, ay, h, and k, the yield constant K, and the invariants E*4* and r*,a*, can be obtained from equation (11) and the following two relations: 312+,U= *h 34+2v=k. 1-(ata60aO/K)' -~= k 3X + 2g = 1 8. Stress-Strain Relations for Plastic Flow.-In the case of plastic flow the stressstrain relations (12) and (14) become Da * 2* * * Ei= 0; =. - kei1akcza, 15 where u and K are material constants. There are only four independent relations in the second set of equations (15), (a) owing to the fact that these equations are identically satisfied when we sum on the indices a and A and (b) because the combination a*da4*#/ vanishes identically in view of the yield condition. Hence equations (2), (3), and (15), when combined with the yield condition, constitute a set of = 10 equations for the determination of the 10 dependent variables p, va and ae. 9. Small Stresses.-Let us now consider the situation for ordinary elastic displacements of isotropic and homogeneous material where the stresses are small in comparison with the stresses which produce yield. It can evidently be assumed here that terms in equations (12) and (14) containing the squares of components 4a are negligibly small in comparison with the value of the yield contant K. Assuming also that the terms which contain -y and k in equations (12) and (14) can likewise be neglected and that the absolute time derivative of the stress tensor can be approximated by the total time derivative for the usual elastic problems involving small stress, it follows that the stress-strain relations reduce to ay = 2,ue d = (3X + 2jA)ejj, aq dt dt where, is a material constant and the quantity X can be taken to have the constant value (h - 2,u)/3. These relations can readily be shown to be equivalent to the relations

6 VOL. 41, 1955 ENGINEERING: T. Y. THOMAS 725 = Xcii6aja + 2ttcEa,. (16) dt We assume that the co-ordinate system is chosen so that, relative to it, the body is at rest in its unstrained state. Then dua/dt = va, where the va are the velocity components and the ua are the components of particle displacements from the unstrained position. Hence But, aue + Ua<,,Vj = Va. (17) at a9t k Ouad _ (_) (va - Va,kVk),t, (18) on account of equation (17). Expanding the right-hand member of equation (18), it now follows readily that or dua,ft d = Va, - VakVk,jS, dt decip (Va kvk,# + V3,6kVka) (19) dt Eap where the eat are the components of the ordinary elastic-strain tensor. It is easily seen that the terms quadratic in the derivatives va,f, in the right-hand members of equation (19) can be neglected in comparison with the terms fall since the velocities and their derivatives can be assumed to be arbitrarily small for sufficiently small values of the stresses. Hence from equation (19) we have Using equation (20), equation (16) now becomes deant (approximately). (20) dacfft dei 2 dea,, d X,6 1 = Xibo+2Ac,) dt, dt a ~ dt dtxi~a.~a) Hence, by integration of these equations, we have a, = Xieizb,, + 2Isea, + Cal, (21) where the Cap are constants associated with the trajectory of a moving particle. But, if we suppose that initially the body is in its undisturbed state, with the stresses and displacements equal to zero, then Cap = 0 for each trajectory. Hence, under the above assumptions, considered to be realized for sufficiently small stresses, we see that the differential stress-strain relations (12) and (14) reduce to the well-known algebraic stress-strain relations of classical elasticity theory. ' Prepared for the Applied Mathematics Branch, Naval Research Laboratory, Washington, D.C. This paper was written while visiting the University of California, Numerical Analysis Research, Los Angeles, California.

7 726 ENGINEERING: T. V. THOMAS PROC. N. A. S. 2 We have not seen this question adequately discussed anywhere in the literature, and, as far as we are aware, it has never been completely resolved. 3 For a discussion of the absolute time derivative of the stress tensor and related matters see T. Y. Thomas, "On the Structure of the Stress-Strain Relations," these PROCEEDINGS, 41, , See also C. Truesdell, "The Simplest Rate Theory of Pure Elasticity," Communs. Pure and Applied Math., 8, , 1955, and W. Noll, "On the Continuity of the Solid and Fluid States," J. Rat. Mech. Anal., 4, 3-81, 1955, where references to previous work are to be found. 4 W. Prager and P. G. Hodge, The Theory of Perfectly Plastic Solids (New York: John Wiley & Sons, 1951), p. 29. The second set of eqs. (15) constitutes the invariant form of the Prandtl- Reuss stress-strain relations, and these relations are strictly equivalent to the noninvariant form of the Prandtl-Reuss relations given by Prager and Hodge in view of the interpretation of these latter relations proposed by Thomas, op. cit., sec The right-hand members of eq. (4) do not represent the most general symmetric tensor invariant F. In fact, Rivlin and Ericksen have shown that the most general such invariant is given by a linear homogeneous combination of the Kronecker delta and a set of polynomial tensor invariants of e and a which may be said to constitute a complete set; for the three-dimensional case under consideration this complete set consists of seven invariants exclusive of the Kronecker delta. See R. S. Rivlin and J. L. Ericksen, "Stress-Deformation Relations for Isotropic Materials," J. Rat. Mech. Anal., 4, 421, This provides the opportunity for further generalization. Here we are concerned primarily with the construction of relations of the simplest possible formal character consistent with the requirements of this theory. 6 A derivation of the yield condition from physical or thermodynamic considerations would of course be very much worthwhile. In this connection it may be mentioned that a discussion of yield conditions based on a variational procedure, which may possibly have some bearing on this physical problem, has recently been carried out by us. See "Determination of the Plastic Yield Condition as a Variational Problem," these PROCEEDINGS, 40, , From a purely mathematical standpoint it appears that the yield condition is associated with a breakdown of the general independence of the stress-strain relations. See "Interdependence of the Yield Condition and the Stress-Strain Relations for Plastic Flow," these PROCEEDINGS, 40, , Instead of the assumptions made in this section, we might take the viewpoint that the coefficients 3X + 2p and r in eq. (6) are at most functions of the xa and t, i.e., that these coefficients are not actual scalar invariants of e and a. By the homogeneity condition of sec. 7 it then follows that 3X + 2A and 3t + 2r are material constants. This modification of our assumptions would allow for compressibility effects during plastic flow. 8 It is assumed implicitly here that the quanity aii and its derivative Dasi/Da do not become infinite. Dr. G. R. Irwin has suggested that the quantity 1 -a*r*/k in the denominator of the righthand member of eq. (14) be replaced by a power of this quantity in order to take account of the differences in speed at which different materials approach the yield state. A corresponding generalization of eq. (14) would also result by allowing the scalars A and B in sec. 6 to contain polynomial invariants of the third and higher degrees in the components vet. We have already mentioned the possibility of generalization of relations (12) in n. 5. In view of these possibilities, it would appear that our concept of the combined stress-strain relations for the elastic and plastic states could be made sufficiently general to account satisfactorily for the observed behavior of most materials. However, we feel that, before such generalizations are made, one should explore the consequences of the simpler stress-strain relations in the text and that extensions of these relations should be based, if possible, on variational principles or thermodynamic considerations.