On finitely deforming rigid-plastic materials

International Journal of Plasticity. Vo[. 2. pp. 2~.'-27". 1986 07-t9-6..t.19~ 86 53.00 ~ .00 Printed in the U.S.A. ~ 1986 Pergamon Journals Ltd.

ON F I N I T E L Y D E F O R M I N G R I G I D - P L A S T I C M A T E R I A L S *

JAMES CASEY

University of Houston-Universi ty Park

Dedicated to the memory of Aris Phillips

Abstract-- In the context of a Lagrangian strain-space formulation of finitely deforming elastic- plastic materials, a theory of rigid-plastic materials is first derived as a limiting case. Harden- ing, softening, and perfectly plastic responses are discussed. Consequences of a physically plausible work assumption are examined, and special classes of materials are studied. An a pr ior i theory of finitely deforming rigid-plastic materials is then proposed.

I. INTRODUCTION

For ductile solids, one can readily conceive of deformations in which elastic strains are negligible in comparison with plastic strains. In practice, deformations of this type occur in many metal forming processes, for example, in extrusion, drawing, rolling, and forg- ing. It was the unanticipated behavior of solids during such processes that captured the imaginations of the founders of our subject, and that rapidly culminated in a set of governing equations for finitely deforming solids. The extensive experimental investi- gations of Tresca,'[" during the period 1864-1872, on the behavior of various solids during punching and extrusion, established that under sufficient pressure, solids flow in the manner of fluids.~ On 7 March 1870, Saint-Venant presented a (slightly incomplete) set of equations for the two-dimensional motions of what we would now identify as a special finitely deforming rigid-perfectly plastic incompressible material. On 20 June 1870, L&y extended these results to the three-dimensional case.

The salient features of the theory of SAI~T-VENAN'r [1870] and L/~vY [1870] are: (a) the use of an algebraic condition which stress must satisfy during plastic flow (Tresca's yield condition); and (b) the postulation of a proportionality between deviatoric stress and the symmetric part of the velocity gradient. A similar set of equations, but with a new and simpler yield condition was proposed by Vow MisEs [1913]. Detailed discussions of the St. Venant-L&y-Mises equations and their applications are contained in HILL [1950], PRAGER [1956], and GEIRINGER [1973].

A general theory of finitely deforming elastic-plastic materials was presented by GREEN & NAGHDI [1965, 1966].§ These authors introduce plastic strain as a primitive kinematical variable and adopt constitutive equations of the rate-type. Also, influenced by developments in the classical formulation of infinitesimal plasticity, Green and

*An oral summary of this work was presented at the 2nd Joint ASCE/ASME Mechanics Conference, held on 23-26 June 1985 at the University of New Mexico, Albuquerque.

t A survey of Tresca's work, together with an extensive bibliography, can be found in BELL [1973]. Two excellent review articles in English by Tresca himself are also available: TRESCA [1867, 1878].

*While Tresca was certainly aware of a period of work-hardening that separated elastic behavior from perfectly plastic behavior, it was with the latter, which he termed "the period of fluidity," that he was primar- ily concerned.

§The theory of GREEN & NAGHDI [1965, 1966] is formulated in a thermodynamical setting. The corresponding purely mechanical theory may be obtained by specializing the thermodynamical theory to the isothermal case. The remarks that follow apply specifically to the mechanical theory.

247

Naghdi utilize yield surfaces in stress space, together with loading criteria, a flow rule, and a hardening rule that involve the time rate of stress. But such a stress-space formulation has a shortcoming, which was recognized by NAOHDI & TR.~P [1975a]: it does not reduce directly to the theory of elastic-perfectly plastic materials. NAOHDI ,~ TRAPP [1975al proposed an alternative strain-space formulation of plasticity in which: (a) yield surfaces in strain space are employed; (b) the constitutive equations for plastic strain rate and the rate of the work-hardening parameter are linear in the rate of Lagrangian strain; and (c) loading criteria are stated in terms of the inner product of the rate of Lagrangian strain and the outward normal to the yield surface in strain space. Once the loading criteria of the strain-space formulation are adopted as primary, certain associated conditions in stress space can be derived. As shown by CASEY ~ NAGHDI [t981, 1983], it is then possible to define three distinct types of material response, namely hardening, softening, and perfectly plastic behavior. Geometrically, the yield surface in strain space is always moving outwards during loading, whereas the yield surface in stress space concurrently moves outwards during hardening behavior, inwards during softening behavior, and remains stationary during perfectly plastic behavior.

In the theory of elastic-plastic materials, a response function which relates the stress to the strain, plastic strain, and the work-hardening parameter is always available. This function can be used to transform the yield function in stress space into the yield function in strain space. Likewise, its inverse can be used to accomplish the inverse trans- formation. It is therefore clear that as far as yield surfaces are concerned either space may be used. However, the relation between rate o f stress and rate o j strain is not always invertible (CASEY & NAGHDI [1984b]), and it is essential to express the rate constitutive equations and the loading criteria in terms of the rate of strain rather than the rate of stress.-

In the present paper, the status of rigid plasticity is evaluated in the light of the fore- going developments in the theory of elastic-plastic materials. An essential feature of rigid plasticity is the loss of the constitutive equation for the stress tensor. This causes a severe disjunction between events in strain space and those in stress space. For example, one can no longer calculate the yield function in strain space from that in stress space. A number of basic questions immediately arise, such as: How should loading criteria be defined in the absence of a yield surface in strain space? What form does the flow rule take? What are the geometrical and algebraic conditions that characterize hardening, softening, and perfectly plastic behavior?

The approach taken in the present paper is to examine first the reduction that takes place in the theory of elastic-plastic materials in the l imit of rigid plasticity. Guided by the results obtained by the limiting process, we then proceed to postulate an a priori

theory of rigid-plastic materials. Section II of the paper contains a summary of the theory of elastic-plastic materi-

als, and is based on the papers of GREEN & NAGHDI [1965, 1966], NAGHDI 8.: TRAPP [1975a], and CASE¥ & NAOHDI [1981, 1983, 1984b]. Further background information can be found in the review papers (CAsEY & NAGHDI [1984a, 1984c]), the second of which contains a useful mathematical appendix. Constitutive restrictions obtained from a physically plausible work assumption which was proposed by NAGHDI ~ TRAPP [1975b] are also discussed in Section II. In Section III, constitutive relations and constitutive restrictions for rigid-plastic materials are obtained as a limiting form of those given in Sec- tion II. In Section IV, for concreteness, the procedure of Section III is illustrated with

+For a related discussion, see CasEV ,~ NAGHDI [1984C].

Finitely deforming rigid-plastic materials 2-*9

reference to a special class of materials. These include materials of the St. Venant-L~vy- Mises type. Finally, in Section V, an a priori constitutive theory o f rigid-plastic materials is postulated, its basic ingredients are: (a) a yield funct ion which, for fixed values of strain and work-hardening parameter, describes a yield surface in stress space; (b) loading criteria in strain space; and (c) character izat ion o f hardening, softening, and perfectly plastic behavior. Because the loading criteria involve the rate o f strain, the

formulat ion is a strain-space one . t

Ii. ELASTIC-PLASTIC MATERIALS

Consider a deformable elastic-plastic body moving in a three-dimensional Euclidean space. Denote the Lagrangian strain and symmetr ic P io la-Kirchhoff stress tensors by E and S, respectively, and let these tensors be regarded as points in six-dimensional Euclidean strain space and stress space. Let E p denote the plastic strain tensor, and let ,~ be a work-hardening scalar parameter .

II. 1 Various representations of response functions

The stress response may be written in the equivalent forms

S = S(E,EP,~) = S ( E - EP,EP,~) . (I)

For present purposes, the second form is especially useful. It is to be noted that for finite deformat ions , the tensor E - E p is merely a strain difference and should not be confused with "elastic strain." At fixed values o f E p and K, the inverses of the response functions in eqn (1) are

E = E(S,EP, K) (2a)

and

E - E p : E(S,EP, K) .

Thus, in finite plasticity, E - E p may depend explicitly on plastic strain E p. In t roduce four th-order tensors

and

(2b)

a~, ag g = a--E = a ( E - E p) (3a)

a~: a ~ 9E=-~ =-~ . (3b)

Relative to a basis formed f rom tensor products of three o r thonorma i vectors el, e2, e 3, eqn (3a)l has a representat ion

fit should be noted that, as in the elastic-plastic case, the conditions in stress space that attend loading are not equivalent to the loading criteria of the traditional stress-space formulation of plasticity.

250 3. C ,,::;E ~

0SKL (3C)

0Eti , .

s ummat ion being implied over repeated indices. The componen t s or" 13 sat isfy tiae symmetry cond i t ions £~,Lw.v = 2 L x . t t , . = £KL.,. . ,4. Similar results hold for 91~.

F r o m eqns (1)~, (2a), and (3a,b), it fo l lo~s that

13N-~; = ~,N;13

= O'EKL. t t \ ,£ ,W, \ ,pQex@ e L @ e p @ % (4a)

= ~ ,

where the fou r th -o rde r tensor 9 is def ined in terms of the Kronecker del ta by

5 = 1/2(6,~'M6L,v + 6x,v6L.tl)e~,-@ el_ @ e.~/@ e~ . (4b)

We also observe that

a5 as as as O E I~ = '1:3 + OFJ ~ 0,~ c),~ '

of o~:; of 0E OE p OE I' ~)~ 0~

(Sa)

and

al;; [of] as £ --E-5 + - - = 0 £ + - - = 0 , a 3E p ' - ~ 0~

0g o f = o ~z[ °g] o~: =o ~ ; a-E-~ + OE - - -7 ' L J - ~ + 0-7 '

(5b)

and

a f

aS

5 - U + - -

+ - - = 0 0E r ' O~ '

o f =o ~ os + - - = o . mE p ' 0~:

(5c1

The yield funct ion may be specified in terms of any of the three separa te sets of vari-

ables that a p p e a r in eqns (1) and (2a). Thus ,

g(E,EP,~c) = g ( E - EP, EP, K) =f(S,EP,~c) , (6)

Finitely deforming rigid-plastic materials 251

in which g is the yield function in strain space, f is the yield function in stress space, and $ is introduced here for later reference• The constitutive equations (1)1._,, (2a), and (2b) may be used to obtain the functions g, g and f from one another. At fixed values of E p and K, the equations

g = 0 , f = 0 (7)

describe yield surfaces in strain space and in stress space, respectively. The function g is so chosen that g < 0 describes the region enclosed by the yield surface in strain space. For physically realistic constitutive equations, f < 0 will then correspond to the region enclosed by the yield surface in stress space•

From eqns (3a, b) and the identities (6)1.2 follow the relations

0 ! = a: 3E LaS J as,wx ~ M,X:KL eK (~ eL ,

OS '

(8a)

as well as

OEP OEP ' ' L J \ O E p / -~ -- OS,wN ~ ex(~eL

= - ~ , a v : ! ~ '

Og Of aS O f _ Of OS,w~v OK OK OK as osM,~, oK

a~ ag OK OE '

(8b)

and

Og Og 0E O(E - E p ) '

Og _ a g Og Og_ 3g OE p 0E p O(E - E p) ' OK OK '

o g _ a f = a g of 0~ ag OK OK OK aS O~ OE '

(8c)

where a superscript T signifies the transpose of a tensor•

252 J. C,sE',

11.2 Rate-type constitutive equations

In the strain-space formulation of plasticity, the time-rates of change of E;' and ,; are of the form

and

f 0 if g < 0, (a) i~ v = 0 if g = 0 and ~ < 0, (b)

0 if g = 0 and a = 0 , (c)

rr~a# if g = 0 and a > 0, (d}

(9)

f 0 if g < 0 (a) 0 if g = 0 and oa < 0 (b)

~ = 0 ifg=Oand a = 0 (c)

7r~A if g = 0 and ,5 > 0. (d)

(lO)

In eqns (9) and (10), a superposed dot denotes material time-differentiation, and

= I a g . E , (ll) 8E

while ~- and k are scalar-valued functions of (E,EP, K) and p is a symmetric second- order tensor-valued function of (E,EV, K). Equivalently, ,-r, A, and p can also be written as functions of (E - EV, EP,~) or (S,EV, g).

The strain-space formulation of plasticity was introduced by NAGHI)t & TRAeP [1975a]. The constitutive equations (9) and (10) are of the same form as those of NAGHDI e, TRAPP [1975a] except that the function ), in eqn (10d) was previously written as an inner product of p and another second-order tensor. As noted by CASEY & NAGHDI [1984b; p. 236], there is an essential arbitrariness in specifying/~ through such a product. The form in eqn (10d) is therefore preferable, in the present development, we have factored out the scalar function re for later convenience. Thus, the functions rrk and 7:,o of the present paper correspond respectively to/~ and # of CASEY & NAGHD[ [1984b], and subsequent quantities correspond accordingly.

The conditions involving g and ~ in eqns (9) and (10) describe (a) an elastic state; (b) unloading from an elastic-plastic state; (c) neutral loading from an elastic-plastic state; and (d) loading from an elastic-plastic state. These are the loading criteria of the strain- space formulation of plasticity. For a geometrical interpretation of these conditions, see CASEY & NAGHDI [1981] and CASEY & NAGHDI [1984a].

It is stipulated that during loading g remains equal to zero, and hence a = 0 also. It then follows with the use of eqns (11), (10d), and (9d) that

Og Og ) l+ , - r A ~ + p . ~ - E - ~ = 0 . (12)

Assuming that the coefficient of re in eqn (12) is finite, we infer that 7r is nonzero, and hence without loss in generality, we take re > 0.


During loading, the rate of change of stress may be expressed as

Og -- ~ [ ~ : - ~;~1 + ~-o® ~-~ [~:1 ,

with

a$ a~; or = a ~ + ~ - ; [ o ] = ,

= ~ + , 8 [ p ] ,

(13)

(14)

in which use has been made of eqns (1)1.2 , (9d), (10d), (11), and (5a)L.2. Tensors similar to # and # may also be defined in connection with the response functions i~ and in eqns (2a) and (2b). Thus,

a t a~: = ~ ~ + ~ -~ [p] = 5:",

a t a t gr (15) ~ = a ~ + b-ga to] =

= 5 - 0 ,

where eqns (5a)3. 4 and (4b) have been employed. In view of the relations (5b,c) and the definitions (14) and (15), it is evident that

o = - . e [ 5 ] , 5 = - ~ [ o ]

o = - . e [ ~ ] , ~ = - ~ ' ~ [ ~ ] . (16)

II.3 Hardening, softening, and perfectly plastic behavior

In order to discuss the conditions in stress space that accompany the various loading conditions in strain space, let

and

Of f--- ~-~ "S (17)

~/' = 7rP , (18a)

254 J. C,~s~

with

a a7 • / lSb) p = _ U of

In an elastic state, during unloading, and during neutral loading, j~= a and the strain- space criteria in statements (9a,b,c) imply' f < 0, ( f = 0, f < 0), and ( f = 0, f - 0), respectiveIy.

With the aid of eqn (t2), the relations (8b), and eqns (14)l, (IS)i, and (18a,b), the function ~ can be written as

= ! - ~ r : l , (19a)

where

Of c?g

= - ° o g = o-g (19b)

It follows from eqns (18a) and (19a) that

F + A = - > 0 , ~ = = -= (20) ,-r [ ' + ,'1

It was demonstrated by CASEY • NAGHDI [1981, 1983] that '/5 can be used to define hardening, softening, and perfectly plastic behavior according as

(a) ~ > 0 , (b) ~ < 0 , (c) ~ = 0 , (2l)

respectively. • is called a measure of strain-hardening. By taking a material time- derivative of the first and last members of eqns (6)and making use of eqns (9d), (10d), (12), and (18a,b), it can be deduced that during loading

/ - = ~ . ( 2 2 )

During loading, the yield surface in strain space is always moving outwards locally (since g = 0 and ~ > 0). Concurrently, by virtue of the relations (6), (21a,b,c), and (22), the yield surface in stress space is moving outwards if the material is hardening, inwards if it is softening, and remains stationary if it is perfectly plastic.

Since rr > 0, and also since during loading

a? ]~ (23)

in view of eqns (18a) and (22), we note that the function P can be used instead of ¢b for the purpose of characterizing hardening, softening, and perfectly ptastic behavior; also, the same geometrical interpretations are associated with f / ( ~ ) greater than, less than or equal to zero, as with corresponding conditions on f/~. This observation will


be of crucial importance in our discussion of the strain-hardening behavior of rigid- plastic materials.

During loading in a region of hardening or softening, the constitutive equations (9d) and (10d) may, with the help of the inequalities (21a,b) and eqn (23), be expressed in stress-space form:

I~ p f ~A (24a) = ~ 0 , ~ = r "

During loading in a region of perfectly plastic behavior, the constitutive equations (9d) and (10d) reduce to

g A (24b) t P = 3 ° ' ~=-X '

by virtue of eqns (21c) and (19a). Returning now to eqn (13)~, we may write S in the form

= ~3[ !~1 , (25)

where the fourth-order tensor ~£ is defined by

JC = ~ + 7tO ® __Of (26) 0S '

and eqns (8a). and (4b) have been invoked. As shown by CASEY ~; NAGHDI [1984b], ,~ has a sixth-order determinant which satis-

fies the equation

det 3£ = ~ . (27)

In a region of hardening or softening, eqn (25) may be inverted to yield

where

i~ = 0l~0l; [SI , (28)

af (~--# o) , ~=~-~o®~ ,

J£0"~ = 0Z:t£ = ~ , (29)

and use has been made of the relations (22), (4a,b), (17), (21a,b), and (26).

II.4 Consequences o f a work postulate

For elastic-plastic materials, the work assumption of NAGHDI & TRAPP [1975b] may be stated as: The external work done on an elastic-plastic material in any sufficiently

256 d. CAs~Y

smooth homogeneous cycle of deformation is nonnegative. This postulate has several impor tant consequences, which will now be summarized.

(i) Existence of a potential. The work assumpt ion implies the existence o( a potential • ~ f rom which stress can be calculated (see CaSEV & NaGHD~ [1984b]). Thus,

= '~(E,E", K) = ~(E - EP,E ;', ~) (30a)

and

o~ 0f S - mE - O(E - E") (30b)

By virtue o f eqns (30b)l and (4a),

£ T = £ , 01Z T = 0 E . (31)

With the use of the inverse functions 1~ and E, we may define functions 0 and 0, such that

O=O(S,E",K) = ~ - S.E , (32a)

0, = 0 , ( S , E " , K ) - - ~ - S ( E - E " ) , (32b)

and

a~ aO, as as

E - E") . (32c)

The following relations hold between various partial derivatives of ~,, f , 0, 0, :

a~ o~ - - = - - + S , 0E" 0E"

a~ a~ a~. of - ( 3 3 )

O E " = O E " ' O E " O E " '

of a~ a0 a0. OK OK OK OK

(ii) Normality of O and i. The work postulate also leads to the constitutive restriction

, ag 6 - = - 4 / mE ' (3' _>0) (34)

in which 3'* is an undetermined scalar funct ion of the independent variables in eqn (1)j, or equivalently, o f the independent variables in eqns (1)2 or (2a). The normal i ty con-


dition (34) was first established by NAGHDI & TRAPI' [1975b], and an al ternat ive simpler p r o o f o f the result was given by CASEY [1984].

It follows f rom eqns (34), (Sa)l, (16)1, (4a)l, and (31), that

,3f (35) a - g "

By virtue of eqn (34) or (35), the relat ion (19b) reduces to

Og Of (36)

Similarly, f rom eqn (26) one obta ins

ag of 3¢ = ~ - ~-y*~-~ ® - - 0 s

(37)

(iii) A constitutive restriction related to convexity of yield surfaces. It was demonstrated by CASEY & TSENG [1984] that the work postula te leads to a const i tut ive restriction on the first part ial derivatives of the funct ion ~ (or the associated funct ions f , 0, 0,) . For a special class o f mater ia ls , to be discussed in subsect ion II .5, this res- trition ensures convexity o f the yield surface in stress space. We record the constitutive restriction o f CASEY & TSENG [1984] in the closely related forms

(38a)

and

['-ffK-K -- \ OK 0 + [ OEp -- \ aEp 0 -- ( S - S o ) " p < O • (38b)

In inequalities (38a,b), the subscript "0" stands for evaluation of a function at an elastic point.I" Thus ,

g (E0,E p, K) = f ( S 0 , E p, K) < 0 . (39)

All other funct ions in (38a,b) are evaluated at an elastic-plastic point (g = f = 0).

II. 5. A special class of elastic-plastic materials

A special class o f mater ia ls o f considerable practical interest arises when the potential ~b is o f the form~:

"}'Also, the symbol 0 corresponds to r of CASEV ~, TSENO [1984]. .*A thermodynamical treatment of the class of materials encompassed by eqn (40) can be found in G~EN

~, N^OHOt [1965: Section 6].

258 J, C ~

= " ' ( E - EP) + ¢"(E~',~) . (40)

From eqns (40) and (30a,b), we obtain

de' S - (414)

d(E - E v)

Hence, recalling eqn (1):, we see that

aS aS - 0 , - - = 0 , (41b)

aEP O<

so that the response function g in eqn (1): is independent o f its second and third argu- ments. On account o f the results (41b)~._,, it follows f rom eqns (14)2 and (16)a that

6 - = 0 , ~ = 0 . (41c)

In view of (41c): and (15),, the normal i ty condi t ion (35) reduces to

, Of p = ' ~ ~-g , ( v * > 0 ) . ( 4 2 )

Thus, for the class of materials for which the assumption (40) holds, p is directed along the normal to the yield surface in stress space. It then follows from eqn (9d) that during loading Ev also lies along the normal to the yield surface in stress space. Addit ion- ally, in the light of eqns (40) and (42), the inequality (38a) reduces to

aj (S- S0 ) '~ -> 0 (43)

which ensures convexity of the yield surface in stress space. *

L inear s tress response. A more restrictive special case occurs when ~b' in eqn (40) is given by

¢ ' = ! ( E - E p ) - £ [ E - E p] , (44)

in which 13 is a now c o n s t a n t tensor satisfying the symmetry condit ion (31)~. In view of eqns ( 4 4 ) , (30b), and (4a),

S = 1 3 [ E - E p] , E - E p=gR, [S] . (45)

Thus, the stress response is now linear in E - E p. From eqns (43), (45)1, and (8a)t, it can be deduced that

i'CASE',' & TSENO [1984: Section 71 discussed con'~exity for" tile case in which g" in eqn (--tO) is zero: this restriction is unnecessary.


Og (E - E0)" 7--~_ >- 0 , (46)

oIL

so that the yield surface in strain space is now also convex. When ~,' is of the form (44), it follows with the aid of eqns (32b), (40), (45), and (4a)

that

0. = - ~ s . ~ [ s ] + f " ( E p , ~ ) . (47)

III. THE RIGID-PLASTIC LIMIT

In this section, we obtain constitutive results for a rigid-plastic material by taking the limit of a sequence of elastic-plastic materials of the type discussed in Section II. The basic concept involved in this limit is that as the kinematical condition E--, E p is enforced, the constitutive functions can change in such a way that various stress-space conditions are unaffected. In particular, we will suppose that as E - E p---, 0 the value of the stress tensor is no longer determined by a stress response function. However, as a result of external loads, a definite stress still exists at each point of the rigid-plastic material,'[" and the limit of the elastic-plastic stress response function, though indeterminate in form as a function of strain variables, must approach this value. Thus, in the rigid-plastic limit, the value of stress must be obtained independently of the stress constitutive equations (1). In general, functions of strain-space variables will undergo a change in functional form in the rigid-plastic limit. Functions of stress-space variables may or may not do so: we will assume that the functions p, ~, and f and the partial derivatives o f f have definite (finite-valued) limiting forms, which may possibly be the same as in the elastic-plastic case.

III.1 The general case

As E - E p---, 0, the function E in eqn (2b) has a limiting value:~

lim E(S,E p, x) = 0 . (48)

with the stress retaining its current value S. Assuming sufficient smoothness of E, we deduce from eqns (2b), (48), and (3b) that

lim 0E 0E l i m ~ = 0 , - ~ = 0 , l i m ~ = 0 . (49)

From eqns (49)~ and (4a), it follows that

.g ~ ~, (50)

as E - E p--, 0 (i.e., the norm of ,13 becomes unbounded), while eqns (49)2.3 and (5a)3.4 imply that

lRecall that the existence of the stress tensor is a consequence of the balance of linear momentum (see TRUESDEtL 8, TOUP*.'," [1960: Section 203]), and is therefore independent of constitutive notions.

++Appropriate norms are understood to be available both for second-order and fourth-order tensors.

2 6 0 J . CASE '~

A

l i m ~ = 9 lim °r~ = 0 . (51) ' OK

We assume that the response funct ions in eqns (1) become inde te rmina te forms as E - E* ' - , 0 but that they have a l imit ing value equat to the current value o f stress at each poin t o f the ma te r i a l (which of course mus t be de t e rmined independent ly) . We therefore wri te

S, g---, i nde te rmina te forms .

[ i m g = l i m g = S . (52)

Thus , as in any m e d i u m with in ternal cons t ra in t s , there is an essential i nde te rminacy in the stress response func t ion (see TRt~ESDEL~. & NOEL [1965: Sect ion 30] and GREEY

et al. [1970]); In the present case, the c o n s t r a i n t - r i g i d i t y is ex t remely severe and none of the six independen t componen t s o f stress is de te rmined by the const i tut ive equat ions (1)l o r ( 1 ) 2 .

It is c lear f rom eqns (49)~.2.3 and (5c)3,4 that the der ivat ives o f g with respect to E p and • may tend to any finite values, or may poss ib ly be infini te . There fo re , we write

os 8EV - - --* a rb i t r a ry finite, or inf ini te ,

os - - --+ a rb i t r a ry finite, or inf ini te . OK

(53)

Then , in view of eqns (5a)t,2 and (50),

- - - ' + 0 0

OE p

0g - - --, a rb i t r a ry finite, or inf ini te .

(54)

To construct the yield funct ions g and g in eqns (6) f rom the yield funct ion f requires a knowledge o f the stress response funct ions in eqns (1). But, as we have just seen, the la t ter become inde te rmina te in the r ig id-p las t ic l imit . We there fore suppose that

g , ~ i n d e t e r m i n a t e forms ,

l i m g = l i m B = l i m f . (55)

As in the case o f the l imits (52)2 and (53), the l imits (55) are de t e rmina b l e only f rom st ress-space cond i t ions . The yield sur face in s t ra in space , descr ibed by eqn (7) 1, must co l lapse to a poin t : as E - E p --+ 0 any change in s t ra in wilt induce a change in plas t ic s train, so there can be no elastic region in s train space. As regards stress, any value that satisfies l i m f < 0 is poss ible p r io r to yielding. Since it is no longer a p p r o p r i a t e to refer to the region lira f < 0 in stress space as "e las t ic ," we call it the prep&st& region in stress space.


The derivative Of/OS has been supposed to have a definite limit as E --. E p. We will assume this limit to be non-vanishing at l i m f = 0. Hence, with the use o f eqns (8a)2 and (49)t, we deduce that at l img = 0,

Og - - --* oo . ( 5 6 ) OE

Similarly, since, by assumption, the derivatives Of/OE p and 0f /0~ also have finite limits, it can be argued with the aid o f the relations (Sb)t, (8c)1.3.4.5, (51)t, (53)1, (54)1,

and (56) that

O$ Og ....-*00 - - . - - -~00

O(E - E p) ' OE p

O g --, arbi t rary finite, or infinite , (57) OE p

O_._gg = __Og --, arbi trary finite, or infinite . OK OK

It follows from the relations (11) and (56) that, apart f rom one exceptional case which is treated in Appendix A, in the rigid-plastic limit

--, oo (58)

during loading. We have assumed that the constitutive functions 0 and A in eqns (9d) and (10d) have definite limits. Since we are only concerned with physical situations in which k and !~ p remain finite, it then results f rom the relation (58) and either o f eqns (10d) or (9d) that

lim rr = 0 ,

lim 7r~ = finite scalar-valued function = 7, say . (59)

In the rigid-plastic limit, we can utilize the primitive strain-space loading criteria:

E = 0 : non- loading , (a)

I~ :~ 0 : loading . (b) (60)

Non- loading can occur in either o f two ways, namely

i~ = 0 and lira f < 0 (a)

and (61)

i~ = 0 and l i m f = 0 . (b)

262 J. CASES

The first is a preplas t ic state and c o r r e s p o n d s to the elastic state g = f < 0. The second refers to yield at a s t a t iona ry value of s t ra in; it co r r e sponds to a tr ivial ly special

case of neut ra l loading . In the r ig id-plas t ic l imit , neutra l load ing with g ~ 0 is impos- sible, and there is also no ana logue to un load ing from an elas t ic-plas t ic state.

The cons t i tu t ive re la t ions in eqns (10a,b ,c) are replaced by

E = 0 = l i m k = 0 , (62)

while the re la t ions in eqns (9d) and (10d) become

E ~ o = E = ~ l imp , {a)

!~ :¢: 0 = lira ~ -- 7 l i m a , (b)

where use has been made of eqn (59)2. For loading to be possible at all, both l imo and

7 must be non-van ish ing , so wi thout loss in genera l i ty we take ~ > 0. In the e las t ic-plas t ic case, load ing can occur only when f = g = 0. The co r r e spond-

ing s ta tement in the r ig id-plas t ic case is

I£ :¢: 0 = l i m j ' = 0 . (64)

We observe that by vir tue of the f low rule in eqn (gd), toge ther with eqns (11) and

(59)2,

(Og.o) =l (65) lim rr ~-~

The l imit ing form of eqn (12) is

lim rc A ~ + 0 " ~ = - t . (66)

As regards the tensors 8", 0, i , and ~-, it is clear f rom the re la t ions (15)j.2,~, (49)2.3,

(14)~.2, (53), and (54) that

l i m g = 0 , l i m i = l i m o ,

0--* a rb i t r a ry finite, or inf ini te , (67)

~" - - + O O .

It is evident f rom eqns (18a,b) , (19a,b) , and (59)t that as E + E p,

l i m e / ' = 0 , l i m ' u A = 1 , ][___,~ . (68)

and (63)


Thus, the funct ion ,/' cannot be used in the rigid-plastic limit to character ize strain- hardening behavior. However , by virtue of eqns (23) and (59),, the quanti ty f / (Trg) has a limit, namely

lim f - l i m f _ l i m p (69) ,'r~ 7

which can be utilized to characterize s t ra in-hardening behavior as follows:

(a) l i m p > 0 : hardening ,

(b) l i m p < 0 : softening , (70)

(c) l i m p = 0 : perfect ly plastic .

During loading in a region of hardening or softening, equat ions (24a)~.2 have rigid- plastic limits

l i m f l i m f IS = l i ~ l imo , l imk - l i m p l ima . (71)

In the rigid-plastic limit, the relat ions (24b)t.2, which hold for perfect ly plastic behavior, revert to the forms in eqns (63a,b) in view of eqns (59)2 and (68)2.

Since g must be finite during loading, it follows f rom eqns (13)i, (63a), and (65) that

O + £ [ 0 ] -* finite (72)

and, hence, in view of the second set o f equat ions in (14), the relation (67) 3 now becomes

--, a rb i t ra ry finite . (73)

We note that the condi t ion (72) is equivalent to

a £ ~ [0] -" finite . (74)

However , a£ itself does not tend to zero (see Appendix B). We also observe that by virtue o f eqns (26), (19a,b), and (68)1,

. . . r [ Of ] t m ~ [~-~ = 0 . (75)

I I I .2 Implications of the work postulate in the rigid-plastic limit

In this subsection, we examine the forms which the consequences of the work assumption of NAGHDI & TRAPP [1975b] take in the rigid-plastic limit.

Just as we did previously for the functions in the relations (52) and (55), we suppose that as E --, E p the funct ions g) and ¢ in eqns (30a) become indeterminate fo rms with limits which are determined by stress-space quantit ies. Thus,

264 J. CAS>'~

7 , v~---" indeterminate forms ,

l i m ~ = l i m ( ~ = l i m t ) . = l i m 0 + S . E (76a)

where use has been made of eqns (30a) and (32a,b). Likewise, in accordance with eqns (30b) and (52),

0 E - - --, indeterminate forms 0E

lim °EO ----:-~ = lira 0 (E - E ~')

- S .

(76b)

Additionally, the partial derivatives o f ]~ and ~ in eqns (33) may be supposed to become indeterminate forms with limits which can be obtained f rom the corresponding derivatives o f 0 and 0, in stress space:

- - - . indeterminate forms , 0E p ' 0E ~; ' O--~ ' 0~

ag oO o0, l i m ~ - - l i m ~ = l i m ~ - S = l i m - - - S (76c)

OE a

lim 0 ~ = lira Otf O0 00. 0-7 ~ = lim --- = lira - -

OK OK

The derivatives in eqns (32c) have limits

- l i m ~ : E = E r' , l im 0--S- = 0 . (76d)

Proceeding to the normal i ty condit ions discussed under item (ii) o f subsection II .4, we deduce f rom the relations the relations (35), (69)2, and the fact that l imo ¢: 0, that?

~-~ (lira3`* > 0) ,

and hence in view of the relation (63a),

(77a)

i~ = lira 3'7 ~-S (77b)

during loading. Thus, in the rigid plastic limit, the work assumpt ion requires lira p to lie along the limiting normal to the yield surface in stress space, and E is constrained to be in this direction also. It should be emphasized that the results (77a,b) have been

tFor a special class of rigid-plastic materials (included in subsection III.3 below) the results (77a,b) were obtained by NACHDt ~, TRAPP [1975C1.


obtained without restricting attention to the special set of constitutive equations discussed in subsection II.5, whereas the corresponding normality conditions for elastic- plastic materials have been deduced only under the restrictions (40),.:.

From eqns (36) and (68)z it is clear that

{ .Og Of) lim ~r~, ~-~.~-~ = 1 . (78)

Also, because of the relations (74), (77a), (31),, and (8a)l,

JC ~-~ --, finite . (79)

In the rigid-plastic limit, the constitutive restriction (38b) of CASEY & TSENO [ 1 9 8 4 ]

becomes

lim IA I00, ( 0 0 , ~ ( 0 0 , ( 0 0 , (80)

It is evident from the limit relations m eqns (76c) that the restriction (38a) also has the limiting form given by inequality (80).

III.3 Rigid-plastic limit of the class of materials discussed in subsection 11.5

For the class of elastic-plastic materials defined by eqn (40), it follows from eqns (47) and (49),, together with the limit relations (76a)_,, that

l i m ~ = l i m ~ = l i m 0 . = l i m ~ , " ( E P , ~ ) , l i m g / ( E - E p ) = 0 , (81)

lim0 = lim ¢,"(EP, K) - S . E .

In view of eqns (41b),.2 and (14)2,

a~ ag l i m f f - ~ = 0 , lim~-x = 0 , l i m # = 0 . (82)

Thus, for the class of materials being considered, the functions in relations (53)m and (41c), actually have vanishing limits. Once eqns (82)E.2 hold, the limits (57)3,4 are finite and are given by

O~o Of Og O$ Of l i m ~ = l i m I o E p , l i m ~ = l i m ~ = l i m - - o ~ ' (83)

where eqns (8C)4,5,3 have been invoked. The limiting value of the normality condition (42) agrees with that given in eqn (77a)

which, as was mentioned before, holds for a broader class of materials than that defined by eqn (40).

In the rigid-plastic limit, the inequality (43) becomes

266 I C t>.E~

(S - S0)-lira ~?j" - - > 0 , ( 8 4 ) OS

which, o f course, is a special case of the result (80). Consider ing the sub-class o f materials that satisfy eqn (44), we observe that in vie~

of the result (50), the linear stress response in eqn (45)~ clearly becomes an indeterminate fo rm in the rigid-plastic limit. Also, we note that the limit of ~ ' in eqn (44) is zero: this agrees with eqn (81):.

IV. ILLUSTRATION: SPECIAL CONSTITUTIVE EQUATIONS

For concreteness, we now examine the procedure discussed in Section tli with reference to a special set o f const i tut ive equat ions which are appropr ia t e for the elastic- plastic behavior of metals. These equations are a special case of those recently employed by CASEY & SULLIVAN [1985]. They belong to the class of materials discussed in subsection II .5.

Let the tensors E, E p, S be decomposed into deviatoric parts 7, y; ' , r and spherical parts O I, gv I, ~ I, respectively, where I is the second-order identity tensor. Thus, for

example

S = r + s l , . ~=~ , t rS , (85)

in which " t r " s tands for the trace opera tor .

IV. 1 Elastic-plastic case

We suppose that ~, is specified by eqns (40) and (44), so that the stress response is linear, as indicated in eqn (45)~. Assuming that the material is isotropic in its reference conf igura t ion , we may express the tensor 13 in the form

k ) 1 3 = 2 # ~ 1 + - ~ # ! @ ! (86)

in which the constants # ( > 0 ) and k (>0 ) are, respectively, the shear modulus and bulk modulus of elasticity. With the aid of eqns (85)~,2 and (86), the stress response in

eqn (45)1 becomes

r = 2 u ( y - y ' ) , ~ = 3 L ' ( 6 - J " ) . (87)

We specify the yield funct ion in stress space by

f = r . r + 3~0 g2 + c~"yv' ' t ' v - K , (88)

in which ~'0( >- 0) and o~ are constants . Thus, f includes explicit dependence both on mean normal stress g and on deviatoric plastic strain.? The yield function in strain space may be ob ta ined at once f rom eqns (6), (87), and (88):

tFor discussions and applications of equations of this type, see C.~,sEY ~ SULLD,'A.',' [19851, CASE~" & J.affqED-

MOTLAGH [1984] and also C.asE¥ • NAGHDI [1981 and 1984a1.


g = 4/x2(3( -- 3 ' P ) " (3( -- 3(P) + 274/ok2( # - #p)2 + ot3(p .3(p _ K (89)

The funct ion ~ is obviously also specified by the r ight-hand side of eqn (89). The various partial derivatives of f , g, and g are given by

Of 2(r + ~o~I) Of = 20t3(p aS OE p

Og O"--E = 2{4#2(3( -- vp) + 9~'b°k2(#- #P)i} = 2{2/~r + 3,~okgl} ,

Og Og - + 2c~3( p , (90)

OE p OE

Og ag Og O ( E - E p) OE ' OE p = 2°t3(P '

o f Og og oK OK oK

The quanti t ies f a n d ~, defined in eqns (17) and (11), now take the forms

f = 2 ( r . "/" + 3",,bog~) , (91)

g = f + 2{2~r.~, p + 9ffokg~ "p} .

F rom eqns (42) and (90)t, we obtain

p = 2 y * ( r + ~ b o g l ) , ( 3 ' * > 0 ) . (92)

We suppose that 3. in eqn (10d) is specified by

3. = 203r + ~bgl) .p (93)

= 43'*(/3r. r + 3~o~bg 2) ,

where eqn (92) has been utilized. With the aid of eqns (93), (90)7,2,1,3 , and (92), the functions P and _d in eqns (18b)

and (36) can be wri t ten as

P = 3 ' * f ' , A=3'*A , (94)

with

F = 4(/3r. r + 3ffo~bg 2 - otr-3(P)

A = a__gg. Of = 4 ( 2 # r . r + 9~bo2k.q :) > 0 . OE as

(95)

268 J. Cas~¥

From eqns (94)i._, and (20)t.: it follows that

1 F F + . l - > 0 , 4 , - (96)

7":r Y + .I

Therefore, recalling the relations (21a,b,c), we observe that hardening, softening, and perfectly plastic behavior can now be characterized by

(a) F > 0 , ( b ) [ ' < 0 , (c) F = 0 , (97)

respectively. During loading in a region of hardening or softening behavior. I~; and k can now

be obtained from the stress space relations

t~ p 2 f = ~ ( r + G j I ) ,

(98)

= ~-S(~r.r + 3'~oOg 2) ,

where f and F are given by eqns (91)l and (95)1, and where use has been made of eqns (24a)t.2, (94)1, (92), and (93). Similarly, for loading in a region of perfectly plastic behavior, eqns (24b)1._,, (94)2, (92), and (93) lead to the expressions

g~ = 2~ (~ + G~I) zl

= - - 4 ~ ( f l r . r + 3 ~ o O g 2) . (99)

The four th-order tensor 3~ in eqn (37) takes the form

3C = ~I - 4rry*(2/xr + 3g/okgl) @ (r + .~ogl) , ( t00)

where use has been made of eqns (90)3.~ and where rrT" can be calculated f rom eqns

(96)1 and (95)1,2.

IV.2 Rigid-plastic limit

In order for both - / - yv and # - ~v to approach zero in such a way that the values of r and g remain fixed, it is obvious f rom eqns (87)t._, that

/x--+ce , k - - + ~ (101)

The relations (101)~.2 are a special case o f the result (50), and may also be deduced f rom eqns (49)1, (4a), and (86). With regard to eqn (88), we suppose that the constitutive coefficients ¢~o and c, are unchanged during the limiting process,l" in which case

tlf , for example, c~ were to change, then fwould clearly undergo a change in form at the rigid-plastic limit. We have allowed for such changes in form in the development in Section II1.


l i m f = f = r . r + 3~og 2 + c ~ . ~ - K . (102)

It is evident f rom eqns (87), (89), and (101) that the yield functions g and g become indeterminate forms as ~ , - ~'P and g - ~ P approach zero. In accordance with eqns

(55)2, however, the limits o f g and ~ are given by eqn (102). F rom eqns (90)t_7 and (101)1.2, it follows that

Of = 2( r + ~0gl) lira ~-g

o f og lim f f -~ = 2 ~ = lim 0E p .... ,

Og Og Og , - - . ~ ¢ : ~ 0 , . . . . . * t O O

O'--E - O ( E - E p) ' OE p

(103)

Of Og 04 - 1 lim ~ = lira ~ = lira 0--~ = "

It should be noted that the results (103)2.5 are in accordance with eqns (83)j.2. In the rigid-plastic limit, f , ~, and 7r~ become

l i m f = 2 ( r . . ? + 3 ~ b o g g ) ' ~--,oo ,

3' = lim(Tr~) = 2Aim{ r(2#r. ' i ,p + 9~bokg~P)} ,

(104)

where use has been made of the relations (91)1.2, (101)1,2, and (59)1.2. The results (104)2

and (58) obviously agree. We assume that the constitutive coefficients 3 and q~ in eqns (93) are unaffected by

the limiting process. It is then obvious f rom eqns (92) and (93) that

l ima = 2( r + d/ogI)lim 3,* :g 0

l ima = 4(Br.r + 3d/o~bg2)lim 3 ,* (105)

From the relations (94)t,2, (95)j,~, (101)1,2, and (96)2, we deduce that

l i m F = 4(f lr .r + 3¢~o~b-g 2 - c~r.~,) ,

l i m p = lim 3,* lim/" ,

A ~ o o , A ~ o o ,

(106)

lim ,/' = 0 .

The results (106)4.5 are special cases o f the relations (68)3.1. Invoking eqn (68)2, we deduce f rom eqns (94)2 and (95)2 that

lim(Tr v - A ) = 4 lim{Trv*(2#r.~" + 9~kdkg-')} = l . (107)

270 J. CTA>~:~

It is clear f rom the relations (97a,b,c) and (t06)t that in the rigid-plastic limit, hardening, ~oftening, and perfectly plastic behavior may be characterized by

(a) l i m F > 0 , (b) l i m F < 0 , (c) l i m F = 0 , (108)

respectively. These results are consistent with the relations (70a,b,c) taken together with eqn (106)2 and the fact that l imT* > 0.

In the rigid-plastic limit, the quantities I~ and lim k, evaluated during loading, are given by

I~ = 27( r + ~,ogl)lim7" ,

lim k = 47(f3r. r + 3~50Og-')lim 7" ,

(109)

where use has been made of the relations (63a,b) and eqns (105)1._,. In view of eqns (109)~.:, we may express the factor y l imT* as

0 < ~/ limit* = 2 ( r . r + 3~og-"

S-E = 2 ( r . r + 3gog:

(t10)

F rom eqns (I09)~ and the first o f eqns (110) it is clear that only the direction of E is determined in the consti tutive theory. It is also of interest to note that in view of the results (110), the stress power S . IS is positive during loading.

In a region of hardening or softening behavior , instead o f eqns (109),.e, we may employ the stress-space forms

E = r"/" + 3~bogg (r + ~ogl) , 3 r ' r + 3~o0g 2 - oz'r-'y

(r-:r + 3'~og~)(3r'r + 3~00g 2) lim k = 2

3 r . r + 3"~o 0g-" - oer.-},

(111)

where eqns (71),.z, (104h, (105)1.2, and (106)1,2 have been invoked. In a region of perfectly plastic behavior, only the forms (109)~.2 can be utilized.

Special case: ce = 4~ = 0. By virtue o f the relations (108,a,b,c) and (106)1, hardening, softening, and perfectly plastic behavior are now determined by

(a) 3 > 0 , (b) 3 < 0 , (c) 3 = 0 , (112)

respectively. During loading,

r . r + 3 ~ o g 2 = K ,

lim ~? = ;: = 4 d r - r 7 limT* = l i m f ,

(1~3)


in which appeal has been made to the relations (64), (102), (63b), and (105)2. If ~bo > 0, the yield surface described by eqn (I 13)t is an ellipse in the plane of g and +_ i[71[. It is clear from eqns (i 13),. and the conditions (112a,b,c) that: (a) during loading in a region of hardening behavior, the ellipse expands; (b) during loading in a region of softening behavior, it shrinks; and (c) during loading in a region of perfectly plastic behavior, the ellipse remains stationary and the stress trajectory is tangent to it. (For a related discussion, see CASE'C & LIN [1983].

During loading in a region of hardening or softening, l~ is given by eqn (11 I)1 with c~ = 0 = 0, while eqn (111)2 reduces to eqns (113)2.

Next, consider the case of perfectly plastic behavior (3 = 0). From eqns (113)2 it now follows that

K = constant = 2K'-, (say) , (114)

where K is positive, corresponding to an assumed positive initial value for K. The time rate of strain is given by eqn (109)~, while in view of eqns (I 14) and (113), the second of eqns (110) now becomes

s.E 3, lim3`* = 4K-" (115)

IV.3 St. Venant-Ldvy-Mises materials

A well-known special case of the constitutive equations considered in Subsection IVB may be obtained by setting

4 ' 0 = 3 = u = ~ = 0 . (116)

In view of eqns (116), (112c), and (114), only perfectly plastic behavior can be experi- enced by these materials and the yield function is of the Von Mises type:

l i m f = r • z - 2K 2 (117)

From eqns (109), (116), and a decomposition of the type (85), it follows that

5,=23`zl im3`* , g = 0 , (118)

where we have also appealed to the fact that g = 0 in the reference configuration of the body. By virtue of the relations (l18)t, (64), and (117), the quantity 3' lira3`* can be expressed as

0 < 3" limT* - 4--~-5. , = 2 , ~ K " (119)

From eqns (I 17) and (118)t, it is clear that plastic flow can be sustained by any stress whose deviatoric component has a magnitude 42K. The spherical component of stress is undetermined by the constitutive theory, but may be calculated f rom the balance of linear momentum together with appropriate boundary conditions.

272 J C.,,sE ~

It should also be noted that in the present case the stress power reduces to r.~, and is clearly positive during loading.

The class of materials considered in the present subsection is associated with the work of SA~xT-VEXAXT [18701, LfiVY [1970], and Voy NhSES [1913].-

V. AN A P R I O R I THEORY OF RIGID-PLASTIC MATERIALS

In Section III, we derived a theory of rigid-plastic materials as a l irmting case of the theory of elastic-plastic materials that was summarized in Section II, and specific exam- ples of this procedure were considered in Section IV. Motivated by these developments, we now postulate an a priori constitutive theory of rigid-plastic materials.

As before, E and S are the Lagrangian strain tensor and symmetric Piola-Kirchhofl" stress tensors. We assume that S and E are not connected by a constitutive equation. We admit the existence of a scalar-valued work-hardening parameter ,~ and a yield function f (S ,E, , : ) such that for fixed values of E and ~, the equation, +

f (S ,E ,~) = 0 (t20)

describes an orientable yield surface which separates stress space into two ciistinct parts. The region f < 0 is called the preplastic region. Points on the yield surface are called plastic. We assume that initially f_< 0.

Loading criteria are defined by the strain-space conditions in eqns (60a,b), which are repeated here for completeness:

!£ = 0: non-~oading , (a)

I~ e: 0: loading . ib)

(121)

We make the following assumptions:

1 ~ = 0 = ~ = 0 , ( a )

l ~ 0 = f = 0 , (b)

t ~ : ~ 0 = l ~ = - t a , ( ' r > 0 ) , tc)

(122)

l ~ : ~ 0 = k = y k , ( 7 > 0 ) , (d )

where the response functions p and A depend on the variables S, E, and K, and are, respectively, symmetric tensor-valued, and scalar-valued. Clearly,

O #: 0 . (123)

The scalar factor "/' is not prescribed by a constitutive equation.

tSee the discussion in Section I regarding the work of these authors. It should be mentioned that the original

equations are in Eulerian rather than Lagrangian form. +In the present section, which is logically distinct from Sections l-IV of the paper, we nov, use the sym ~

bols that were introduced before in connection with elastic-plastic materials to denote corresponding quantities for rigid-plastic materials. Strictly speaking, these symbols now stand for different functions than before.


We further assume that, during loading, the current stress continues to remain on the yield surface.t Hence, during loading

f = 0 . (124)

Thus, positive values of the func t ionfcan never be attained. In the light of this, it follows from the statements (121a) and (122b) that non-loading can occur only in two ways (anticipated in the relations (61a,b)):

1~ = 0 and f < 0 , (a)

l ~ = 0 a n d f = 0 . (b) (125)

From eqn (124) and the statements (122c,d) it follows that during loading

f- = P , (126)

where the notation of eqn (17) has been used and F is now defined by

P = - . (127)

The function ff in eqn (127) is the analogue for rigid-plastic materials of the function defined by eqn (18b) for elastic-plastic materials.

Hardening, softening, and perfectly plastic behavior of rigid-plastic materials are defined by

(a) P > 0 , (b) P < 0 , ( c ) P = 0 , (128)

respectively. During loading, the current strain has a non-zero velocity in strain space. Concurrently, in a region of hardening behavior, the yield surface in stress space expands locally; in a region of softening behavior, it shrinks locally; and, in a region of perfectly plastic behavior, it is stationary locally.

During loading in a region of hardening or softening behavior, the constitutive relations in statements (122c,d) can be written in stress-space form

f f (P~: 0) (129) E = T o , ,

During loading in a region of perfectly plastic behavior, the strain-space relations (122c,d) must be retained. It should be emphasized that in all three cases, the loading criteria are of strain-space form, as listed in the definitions (121a,b).

In order to check that one has the correct number of equations and unknowns, suppose that the motion X of the rigid-plastic body and the mass density field in the ref-

tThis is the "consistency condition" for rigid-plastic materials i.e., that loading from a plastic state leads to another plastic state.

274 t. C ', ~E'~

erence configuration are specified. Conservation of mass and the balance o~ linear momentum then provide four equations for four unknowns, consisting of mass density in the present configuration and the three components of body force, The Lagran- gian strain tensor and its rate can be calculated from X- The functions A and # are specified by constitutive equations. During loading, the constitutive theory provides eight equations for the eight unknowns consisting of 7, ,c, and the six components of S (i.e., eqns (120) and (122c,d)). (During non-loading, the constitutive theory provides no equation for S, but merely limits S to lie inside the yield surface, and also only motions for which E = 0 are admissible.)

Relative to the theory of elastic-plastic materials summarized in Section [I, we see that the theory of rigid-plastic materials lacks the six unknown components of the plastic strain tensor and the six component constitutive equations for stress. Also, the theory of elastic-plastic materials has one additional unknown, ~-, together with an additional equation (see eqn (12)).

Constitutive relations for rigid-plastic materials were previously proposed by HiLt [19621. However, these differ ira a number of important ways from those given in the present section: (i) HILL [19621 assumes that the strain rate lies along the normal to a "local yield surface element" in stress space; (ii) Hill does not enforce strain-space loading criteria?; (iii) Hill 's function h, which corresponds to P in eqn (127) above, is not similarly defined (Hill's theory has neither a yield function nor a hardening parameter).

REFERENCES

1867 TRESCa, H.E. , "On the "Flow of Solids", v~ith the Practical Application in Forgings, &c.," PFO- ceedings of the institution of blechanical Engineers (London), 114.

1870 SAINT-VEx,',.~T, A.J.C.B. DE-.', "Sur l'l~[ablissement des t~q.uations des Mou~ements lnterieurs Op&t}s dan les Corps Solides Ductiles au dela des Limites oO i'Elasticitd Pourrait les Ramener 5. leur Pre- mier Etat ," Comptes Rendus des Seances de [ 'Academie des Sciences, 70,473. See afso Journal tie .Mathdmatiques Pures et Appliquees, 16 (187I) 308 and 373.

1870 Ldvy, M., "M6moire sur les Equations G~Sndrales des Mouvements lntdrieurs des Corps Solides Duc- tiles au delS. des ILimites oO l't~lasticitd Pourrait [es Ramener a. leur Premier l~tat," Comptes Rendus des S~Sances de l 'Academie des Sciences, 70, 1323. See also Journal de ?',iathdmatiques Pures et Appliqu~es, 16 (1871), 369.

1878 TRESCa, H.E. , "On Further Applications of the Flow of Solids," Proceedings of the Institution of Mechanical Engineers (London), 301.

19t3 Vos MIsts, R., "?vlechanik der Festen K6rper im Plastisch-Deformablen Zustand," Nachrichten yon der Koniglichen Gesellschaft der Wissenschaften zu G6ttingen, Mathematisch-Physikalische Klasse, 582.

1950 HILL, R., The Mathematical Theory of Plasticity, Oxford University Press, Oxford. 1956 PRAGER, W., "Finite Plastic Deformat ion ," in ElatcH, F.R. (ed.), Rheology: Theory and AppIica~

tions, Academic Press, New York, 63. 1960 TRt,'ESDELL, C. and Tot:ee,', R.A., "The Classical Field Theories ," in FLiLGGE, S. ted.), Handbuch

der Physik, Vol. I I I / l , Springer-Verlag. 226. 1962 HILL, R., "Consti tut ive Laws and Waves in Rigid/Plastic Solids," J. Mech. Phys Solid~, 10, 89. 1965 GREEN', A.E. and NAGHDh P.M. , "A General Theory" of an Elastic-Plastic Con t inuum," Arch.

Ration. Mech. Analysis, 18, 251. 1965 TP.t.IESDELL, C. and NOEL, W., "The Non-Linear Field Theories of Mechanics," in Ft C(TGE. S. (ED.}.

HANDBUCH DER PHYS1K, VOL. I I , ~ , SPRINGER-VERLAG, I. 1966 GREE.',, A.E. and N.',GHDI, P.M., "A Thermodynamic Development of Elastic-Plastic Cont inua ,"

in PAP.~t:S, H. and SEDOV, L.I. (eds.), Proceedings of the IUTAM Symposium on Irreversible Aspects of Cont inuum Mechanics and Transfer of Physical Characteristics in Moving Fluids, Springer-Verlag, 117.

-In treating elastic-plastic materials, Hill does not enforce strain-space loading criteria either: see the discussion on page 291 of CasE'," s~ NAOHDI [1981].


1970

1973

1973

1975a

1975b

1975c

1981

1983

1983

1984 1984

1984a

1984b

1984c

1984

1985

GREEN, A.E., NAGHDI, P.M., and TRAPP, J.A., "Thermodynamics of a Continuum with Internal Constraints," Int. J. Engng. Sci., 8, 891. BELL, J.F., "The Experimental Foundations of Solid Mechanics," in FtOGGZ, S. (ed.), Handbuch der Physik, Vol. Via / l , Springer-Verlag, 1. GEI~,','GER, H., "Ideal Plasticity," in FtOooE, S. (ed.), Handbuch der Physik, Vol. Via/3, Springer- Verlag, 403. NAGr~DX, P.M. and TZAPe, J.A., "The Significance of Formulating Plasticity Theory with Refer- ence to Loading Surfaces in Strain Space," Int. J. Engng. Sci., 13, 785. NAGHDt, P.M. and TRAPP, J.A., "Restrictions on Constitutive Equations of Finitely Deformed Elastic-Plastic Materials," Q. Jl. Mech. Appl. Math., 28, 25. NAOHt)I, P.M. and TRAPP, J.A., "On the Nature of Normality of Plastic Strain Rate and Convex- ity of Yield Surfaces in Plasticity," J. Appl. Mech,, 42, 61. CASZY, J. and NAGHDI, P.M., "On the Characterization of Strain-Hardening in Plasticity," J. Appl. Mech., 48, 285. CASEV, J. and LIN, H.H., "Strain-Hardening Topography of Elastic-Plastic Materials," J. Appl. Mech., 50, 795. CASE't, J. and NAGHDh P.M., "A Remark on the Definition of Hardening, Softening, and Perfectly Plastic Behavior," Acta Mechanica, 48, 91. CASE'r, J., "A Simple Proof of a Result in Finite Plasticity," Q. Appl. Math., 42, 61. CASEY, J. and JAHEDMOTtAO~, H., "The Strength-Differential Effect in Plasticity," Int. J. Solids Struct., 20, 377. CASEV, J. and NAGHDh P.M., "Strain-Hardening Response of Elastic-Plastic Materials," in D~SAh C.S. and GAJ.tAGrmR, R.H. (eds.), Mechanics of Engineering Materials, Chap. 4, John Wiley & Sons, Ltd., London. CASEV, J. and NAGHDh P.M., "Further Constitutive Results in Finite Plasticity," Q. Jl. Mech. Appl. Math., 37, 231. CASEV, J. and NAGHDI, P.M., "Constitutive Results for Finitely Deforming Elastic-Plastic Materi- als," in WItLAM, K.J. (ed.), Constitutive Equations: Macro and Computational Aspects, p. 53, The American Society of Mechanical Engineers, New York. CASEY, J. and TSENG, M., "A Constitutive Restriction Related to Convexity of Yield Surfaces in Plasticity," J. Appl. Math. Phys. (ZAMP), 35, 478. CASEY, J. and SUttlVAN, T.D., "Pressure Dependency, Strength-Differential Effect, and Plastic Vol- ume Expansion in Metals," Int. J. Plasticity, 1, 39.

APPENDIX A

There is one highly exceptional case in which the constitutive functions may be such that during loading ~ remains finite (but indeterminate) in the rigid-plastic limit. This would occur if the limiting direction of Og/OE were perpendicular to the limiting direction of p. Thus, with co stariding for the angle between Og/aE and I~, we see that

3g =LI£11 cos . (Al)

In the rigid-plastic limit, 1~ becomes parallel to p during loading, whereas the magnitude of Og/OE approaches infinity. Consequently, if co were to tend to rr/2, ~ would remain finite. For the exceptional case,

Og a--E "p'-+ arbitrary positive finite ,

Og (A2) o---g p

Og - , 0 ,

2"6 J. C~s~'~

and instead of eqns (58) and (59)~, we would have

~ arbitrary positive finite .

~-~ positive finite . (A3)

Equations (59),_ through (67) would still hold. We note that ,'r would become indeterminate in view of eqn (65) and the relation (A2)~, and that the derivatives ag/& and Og/OE" would still be restricted by the conditions (57)a. : and (66). From the relations (A3):, (67), and (19b), it follows that

fl ---, arbitrary positive finite . (A4)

The function ,ib would then be finite, indeterminate, and less than unity, by virtue of the results (19a), (A3)e, and (A4). The relations (69) through (74) would still hold, but the tensor aCr [0f/as] in eqn (75) would no longer have a definite limit.

Once the work postulate is invoked, the conditions (A2)~.: of the exceptional case reduce to

c?f v_~_~, v j ~ arbitrary positive finite , aE aS

ag of (A5)

OE as

IJ Og - - - - , 0 ,

where use has been made of the results (77a). Thus, the limiting directions of Og/OE and 0f/as must be perpendicular to one another.

In the constitutive equations of Section IV, the exceptional case cannot arise, because the inner product of Og/OE and Of/0s tends to infinity, as indicated by the results (95)2

and (106)3.

APPENDIX B

In this appendix, we show that the tensor a~, defined in eqn (26), cannot approach zero in the rigid-plastic limit. For, suppose

timaC = 0 . (B1)

Then, by eqn (26),

lim frO@ ~ = - ~ (B2)


and hence

lim ~'# =

af 0S

a f : (B3)

By virtue of eqns (B2) and (B3),

0 f ® a f 0S aS

lira - 9 . (B4) 0f 2 ~g

Contracting both sides of eqn (B4) to obtain unity on the left-hand side, and utilizing eqn (4b), we obtain a contradiction. Therefore, eqn (B1) cannot hold.

Department of Mechanical Engineering University of Houston-University Park Houston, TX 77004, USA

(Received 27 September 1985 )

On finitely deforming rigid-plastic materials

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