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hf. J . Solids Structures, 1974, 10, pp. 313-319. Pergamon Press. Printintedn Gt. Britain.

A SIMPLE THEORY OF PLASTICITYJ. LUBLINER

Department of Civil Engineering, University of California,Berkeley, California 94720(Received 23 March 1973)

Abstract-An internaI-variable model of rate-independent plastic behavior, based on loading-unloading irreversibility, is proposed. The model is compatible with thermodynamics andassumes no yield or loading function, stability p%ostulate or specific nature of the internalvariables. It is shown that current theories of plasticity are restricted forms of the proposedtheory.

1. INTRODUCTIONThe purpose of this note is to expose a theory of plasticity which is an outgrowth of somerecent work of mine[l, 21 on the thermodynamics of materials i&which irreversible processesare characterized by internal variables. The bulk of the literature on such materials[3-S]deals with internal variables that are governed by local rate equations of the form

4, =f&tate>, a=l,...,n, (1)where n is the number of internal variables, and the state is given locally, for example,by the temperature@, the Cauchy-Green deformation tensor C, and the n-vector q whosecomponents are the internal variables qo1 which may be scalars or components of tensorsthat are unchanged by rigid-body motion). Clearly the solutions of equation (1) are notinvariant under time reversal. This kind of irreversibility, typical of rate processes,characterizes viscoelastic and viscoplastic materials, the difference between these two typesof material behavior residing in the structure of the functions f, .

Rate-independent plastic flow, on the other hand, is characterized by a different kind ofirreversibility. It may be called a one-u*ay process: it occurs when the external process(mechanical work and heating) goes in some direction (called loading) but not when itis reversed (unloading). If the state of external process is defined locally by C and 8, then wemay assume the existence of a tensor A, a scalar a, and an n-vector r, all functions of thestate variables C, 8 and q, such that the loading rate # is given by

# = tr(AC) + a8 (2)and the internal variables are governed by

il=r(rb> (3)where (u} = uH@), H(u) being the Heaviside step function.

Alternatively we may take as external variables the temperature 0 and the Piola-Kirchhoffstress tensor P, and assume the existence of B, b and r, respectively tensor, scalar and n-vector functions of P, 8 and q such that313

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314 J. LUBLINER&J= tr(BP) + be (4)

and q is governed in form by equation (3) as before.The proposed model of rate-independent plasticity is embodied in either equations (2)

and (3) or (3) and (4). It does not require (but does not preclude) the existence of yield orloading surfaces, or of stability postulates, and it leaves the number and nature of theinternal variables unspecified. The relationship between these traditional concepts and thepresent model will be explored in section 3. Since the model is one of irreversible behavior,it must first be examined for compatibility with the second law of thermodynamics.

2. COMPATIBILITY WITH THE SECOND LAWThe second law will be expressed under the guise of the material local form of the Clausius-

Duhem inequality[9]. Since the functions characterizing plastic flow, as defined in thepreceding section, are independent of temperature gradient we may limit the inequality toits form for vanishing temperature gradient. The existence of entropy and Helmholtz freeenergy per unit mass, denoted respectively by q and (i/, as functions of C, 8 and q will beassumed on the grounds that at any point in state space there are neighboring points thatcan be reached by unloading, that is, a process in which only C and 0 vary; hence any stateis one of constrained equilibrium.

With p,, denoting mass density in the reference configuration and y irreversible entropyproduction per unit mass, we have

- p,($ + 18) + * tr(PC) = p. ey 2 0. (5)With $ a C function, this becomes, upon application of the chain rule and substitution ofequations (2) and (3)- p. [ 1 I[+ tr (6)The inequality is satisfied for all 4 if and only if?

p=2po(7)

provideda*a4=r, I 0 (8)

and01;111, (9)

t See Appendix for necessity proof.

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A simple theory of plasticity 315since the inequality (6) then reduces to

It is significant that, unlike the models studied by Coleman and his collaborators[6, 9-l 11,for the present model the Clausius-Duhem inequality is not sufficient to determine con-stitutive equations for stress and entropy. To eliminate the indeterminacy presented by theparameter E.we must additionally argue that the unloading process is quasi-reversible, that is,y = 0 when 4 < 0. Then, by (lOa), we have I = 0, and equations (7) may be replaced by theclassical relations

a*(11)

where E = +(C - I), while the irreversible entropy production is given by(12)

restricted to be non-negative by the inequality (8).Analogous considerations apply to the complementary model given by equations (3)

and (4), with P, 8 and q as state variables. The complementary free energy per unit mass, x,may be defined by the Legendre transformation

x = i tr(PE) - 11/. (13)We then have

Ecp,axap 'ax=&and

(14)

3. RELATIONSHIP TO OTHER THEORIES OF PLASTICITYAmong the salient ad hoc assumptions which are features of current theories of plasticity

are the following:(1) The existence of a yield criterion, given by a yield function, say F(P, 13,q), such thatno plastic flow occurs when F < 0, is assumed. In the context of the present model a yieldfunction may be incorporated into the structure of the r., i.e. r = 0 for F < 0.

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316 J.LUBLINER(2) The existence

rate is given byof loading surfaces, given by G(P, 8, q) = 0 such that the loading

(16)in other words, the existence of a function G, such that

(17)is assumed. G may or may not coincide with F. The former view is the classical one, asformulated, for example, by Green and Naghdi[l2], while the latter view was proposedoriginally by Melan[l3] and more recently by Eisenberg and Phillips[l4].

(3) The internal-variable vector is usually assumed to consist of the plastic strain tensor EPand the hardening parameter rc[12], although, in place of the former, such other symmetricsecond-rank tensors as the contracted tearing curvature tensor of Kondo[lS] or the dis-location loop density of Krbner[l6] may represent more closely the internal processes.EP arises naturally if the strain-stress equation (14a) has the property

a*E- = 0,ap aqa2E- = 0,deaq

since thenE = E(P, 0) + EP(q) (19)

(Equation (18) means that the thermoelastic moduli in terms of P and E are independent ofplastic deformation.) If n 2 6, then the components of EP may be used as six of the internalvariables (otherwise they will be subject to constraints).

Since by virtue of equation (14)

equation (19) impliesaE, aEkleah, =q

necessitating the existence of a function x(P, 0) such thatwE = p,, ap-(hence both E and EP are symmetric), and x is given by

x = xe + k tr(PEP) + ~~(6, a>.If furthermore the specific heat (at constant P and q), given by

(20)

(21)

(22)

(23)

(24)--()dZXa82

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A simple theory of plasticity 317is also assumed independent of q, then xp is at most linear in 8, and x is given by

1x = xe + T tr(PEP) + @VP(d) sP(q),PO

(25)where .sp is the additional internal energy (strain energy) and qp is the configurationalentropy due to plastic deformation[l7]. Since the Clausius-Duhem inequality (15), with xgiven by (25), reduces to

tr(PfiP) + po(eQp - dP)2 0 (26)it is clear that such an entropy (increasing with plastic deformation) must exist if theBauschinger effect (entailing the possibility that tr Ptip < 0) is to be compatible with thesecond law of thermodynamics.

With q given by {EP, K} , r is given by, say, {M, m}. Two definitions of K are common:the first as the plastic work, so thatB = tr(Pkp), (27)

implyingnz = tr(PM), (28)

and the second as given by

so that

Note thatdegree) in

li- = [tr(I?pI?p)]2, (29)

nz = (tr M)l/. (30)Green & Naghdi[l2] assume II linear (and not merely homogeneous of the firstBP, precluding the definition (29).

(4) Druckers postulate of material stability[l8] takes, in terms of P and E, the formtr PBP r 0. (31)

In conjunction with the concepts of yield and loading surfaces, this postulate has beenused to prove the convexity of these surfaces and the normality of ep to the loading surface.However, the postulate itself is independent of the existence of any such surfaces and evenof the plastic strain EP, since a plastic strain rate fip may always be defined as

= M(4),where

M=$,.01

(32)

(33)

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318 J. LUBLINERTaking (31) as valid isothermally, we have

tr(M@tr(BP) > 0,which requires, in order to be valid for all P,

M=pBwhere ,u > 0. The normality rule follows if B is given by (17a).

(34)

(35)

REFERENCES1. J. Lubliner, On the thermodynamic foundations of non-linear solid mechanics. ht. J. Non-linear

M echani cs 7, 237-254 (1972).2. J. Lubliner, On the structure of the rate equations of materials with internal variables: Acta mech.17, 109-119 (1973).3. H. Ziegler, An attempt to generalize Onsagers principle, and its significance for rheological problems.

Z. angew. Math. Phys. 9,748-763 (1958).4. R. A. Schapery, Application of thermodynamics to thermomechanical, fracture and birefringentphenomena in viscoelastic media. J. appi. Phys. 35, 1451-1465 (1964).5. K. C. Valanis, Thermodynamics of large viscoelastic deformations. J. Math . Phys. 45, 197-212 (1966).6. B. D. Coleman and M. E. Gurtin, Thermodynamics with internal state variables. J. them. Phys. 47,597-613 (1967).7. J. Lubliner, On fading memory in materials of evolutionary type. Acta mech. 8, 75-81 (1969).8. J. Kestin and J. R. Rice, Paradoxes in the application of thermodynamics to strained solids. A Crit icalRevi ew of Thermodynamics. (Edited by E. B. Stuart et al.) Mono Book Corp. 275-298 (1970).9. B. D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction andviscosity. Arch. ration. Mech. Anal. 13, 167-178 (1963).10. B. D. Coleman, Thermodynamics of materials with memory. Arch. rat ion. Mech. Anal. 17, 146 (1964).11. B. D. Coleman and D. R. Owen, On the thermodynamics of materials with memory. Arch. ration.Mech. Anal. 36,245-269 (1970).12. A. E. Green and P. M. Naghdi, A general theory of an elastic-plastic continuum. Arch. rat ion. M ech.Anal . 18, 251-281 (1965).

13. E. Melan, Zur plastizitat des Raumlichen Kontinuums. Ing-Arch. 9, 116-126 (1938).14. M. A. Eisenberg and A. Phillips, A theory of plasticity with non-coincident yield and loading surfaces.Acta mech. 11, 247-260 (1971).15. K. Kondo, On the analytical and physical foundations of the theory of dislocations and yielding by thedifferential geometry of continua. ht. J. Engng. Sci. 2, 219-252.16. E. Kriiner, Dislocation: A new concept in the continuum theory of plasticity. J. Math. Phys. 42, 27-37(1963).

17. A. H. Cottrell, Di slocati ons and Plastic Flow in Crystals. Oxford University Press, 39-40 (1953).18. D. C. Drucker, A more fundamental approach to plastic stress-strain relations. Proc. 1st U.S. nat n.Congr. appl. Mech. 487-491 (1951).

APPENDIX-PROOF OF NECESSITY OF EQUATIONS (7), {A, a}, and {c, 6} be denoted by the 7-vectors u, v and w,

respectively, and ra at+b/aq, by -k. With this notation the Clausius-Duhem inequality maybe written as

u.w+k(v.w)>O, (Al)and equations (7) are equivalent to the assertion that

u = - Ikv , W Ithat is, u is parallel to v. To prove that (Al) implies (A2) we assume the contrary of thelatter, that is, we suppose that u also has a component normal to v:

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A simple theory of plasticity 319u= - ikv+z,

v-z=o. (A3)Now suppose w = cz, where c may be any real number. (Al) and (A3) together imply

cz * z > 0. (A4)For (A4) to hold for all real c, z must vanish, that is, (A2) is necessary.

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