/Ilr. J .. \icH:·Unf'IJr M ..-dumic-,. Vol. 8. pp. 261-~77, Pergamon Prn.~ 1913. Printed in Great Britain

A UNIFIED THEORY OF THERMOVISCOPLASTICITYOF CRYSTALLINE SOLIDS

D. R. BHANDARI and J. T. ODEN

Department of Engineering Mechanics. University of Alabama. Huntsville. Alabama. U.S.A.

AblitnIet-A unified Iheory of thermoviscoplasticity of crystalline solids is presented. In parlicular it is shownthat a therrnodynamies for 'viscoplastic' materials can be accommodated within the framework of modemmechanics of materials with memory. The basic physical concepts are derived from the consideration of disloca-tion behaviour of crystalline solids. Relationships of Ihe present approach 10 several of Ihe cxisling [heories ofplasticity arc examined.

I. INTRODUCTION

TIiL'; paper describes a rather general theory of thcrmoviscoplasticity of crystalline solids,along with the necessary thermodynamic considerations that must underline it. The theoryitself involves generalizations of ideas proposed early in the development of moderncontinuum thermodynamics, principally in the linear theories of irreversible thermo-dynamics of materials with memory of Biot [1. 2] and Ziegler [3. 4]. In their theory,dissipation was included in the governing functional as a quadratic form in the rates-of-change of certain internal state variables. Their development was based on Onsager's work.with its characteristic symmetries, and is often referred to as the "classica]"' thermodynamicsof irreversible processes. Similar concepts were used for special thermodynamic descrip-tions of certain elastic, viscoelastic and plastic materials by Drucker [5]. Dillon [6],Vakulenko [7). Kluitenberg [8,9]. and Kestin [10]. among others. Extensions of theinternal state variable (hidden variable) approach to the thermodynamics of non-linearviscoelastic materials have been explored extensively by Schapery and by Valanis in anumber of papers (e.g. [11-14]).

The notion of internal or hidden variables was, at first. not easily reconciled within theframework of modern continuum mechanics. and a number of alternate approaches tothe development of a thermodynamic theory of materials with memory were initiated.Perhaps the most prominent among these was the work on dissipative media by Coleman[15J and Coleman and M izel [16]. which extended the earlier isothermal theory of Cole-man and Noll [17]. ]n Coleman's thermodynamic theory of simple materials [15J. it isshown that certain materials can be characterized by only two constitutive functionals, onedescribing the free energy, which is independent of temperature gradient. and the otherdescribing the heat flux. A key feature of Coleman's theory is that the stress. entropy andthe internal dissipation are determined as Frechet differentials of free energy functional.

More recently. the thermodynamics of non-linear materials with internal state variableshas been studied by Coleman and Gurtin [18]. In their approach. the collection of con-stitutive equations describes what is generally referred to as a material of the evolution type.Here a separate constitutive equation is given for the rate of change of hidden variable

261

262 D. R. BHA.NDARI and J. T. ODFN

which is often referred to as an equation of evolution. Coleman and Gurtin point out thattheir approach [18] to continuum thermodynamics is but one of the several approachesincluding those based on constitutive equations of differential type (e.g. Colcman and Mizel[19]. Schapery [20]. Perzyna and Olszak [21] and others).

All these approaches are generally regarded as independent of one another. However.attempts have been made by Coleman and Gurtin [18] and by Lubliner [22] to unifythem. For example. if it is assumed that the solutions of the evolution equations are stable.then the stability postulate in the theory of Coleman and Gurtin [18J gives most of thequalitative properties of Coleman's theory of materials with memory [15]. It is shown byLubliner [22J that under specific but fairly broad conditions, the principle of fading memoryof Coleman [15] is obeyed by non-linear evolutionary materials. Furthermore, the genera-lized stress relations derived by Coleman and Gurtin [18J are implicit in the work ofColeman [15] and the stronger results of Coleman are valid under equivalent constitutivehypothesis.

General theories of thermoviscoelasticity derived from, say. Coleman's thermodynamicsof simple materials [15]. are generally regarded as adequate for describing non-linearbehavior in most polymers and even in certain metals if no dislocations take place (rather.if the dislocation density is small). Since such theories generally assume that the responseof the material is governed by some mcasure of the gradient of the motion (c.g. F. Cor i').the absolute tempcrature 0, and the temperature gradient g, it seems logical that statevariables (independent constitutive variables) must be introduced in the 'plasticity' phe-nomena such as yielding. strain hardening. etc., are to be encompassed by the constitutivecquations. Some macroscopic measure of the influence of dislocations immediately arisesas a likely candidate for such additional measures.

Coleman-Gurtin type thermodynamics for the study of elastoplastic materials hasbeen used by Kratochvil and Dillon [23. 24]. Tseng [25]. and Hahn [26]. These investiga-tors have employed certain basic concepts from the theory of dislocations in crystallinesolids to interpret various internal state variables. Arguments have been made that dis-locations. their arrangements and their interactions in crystalline solids play the role ofinternal state variables. The present study follows a pattern similar to that of Kratochviland Dillon [23] in that we use Kroner's arguments from dislocation theory to justifythe inclusion of "'hidden state variables"' which manifests itself in the form of second-ordertcnsors Alil. Howevcr. we carry the study a bit deeper by also investigating the relationshipof this theory to others existing or recently proposed in the literaturc. Perzyna and Wonjo[27] also developed a thermodynamic theory of viscoplasticity by introducing a second-order tensor A called the inelastic strain tensor. which played the role of a hidden statevariable. Recently Oden and Bhandari [28] prcsented a theory of thermoplastic materialswith memory based on an extension of Coleman's thermodynamics of simple materials[15]. Among features of their work were that their theory does not make use of the ideaof yield surfaces and it can be reduced to either Coleman's theory [15] or the Green-Naghdi theory of plasticity [29] as special cases. In the present work it is shown that thefunctional theory of Oden and Bhandari [28] can be obtaincd from the evolutionarytheory prcscnted herein by introducing certain plausible assumptions concerning propertiesof the equations of evolution.

In the present paper what might be called a Coleman-Gurtin type thermodynamics ofnon-linear materials with internal state variables is dcveloped and is applied to a combincdtreatment of rheologic and plastic phenomena. A theory of viscoplasticity of crystalline

A uniJled Theory of Ihermol'iscoplasTicily o( crysTalline solid., 263

matcrials is constructed which is unified. in that a thcrmodynamics for "viscoplastic"materials is accommodatcd within thc framcwork of modern continuum mechanics ofmaterials with memory. The basic physical concepts are dcrived from the considerationof dislocation behaviour of crystalline solids, and emphasis is placed on what appears to bea logical identification of internal state variables. Finally. the relationships of this approachto sevcral of the existing theories of plasticity are examined.

2. 50\1E PHYSICAL ASPECTS FRO" DISLOCATION THEORY

Before stating specifically the constitutive equations for our thermoplastically simplematerials, it is necessary to choose a sct of state variables suitable for describing plasticphenomena. In the following, we briefly summarize elcments of the theory of dislocationswhich are useful in choosing appropriate state quanti tics for the class of materials studiedhere. For a detailed discussion of dislocation behaviour in crystalline solids. see. for example.the articles of Bilby [30]. Taylor [31]. Kondo [32]. Mura [33]. and books ofCottrel [34].Read [35]. Kroner [36]. and Gilman [37].

It is well established in crystal physics that "plasticity"' phenomena such as yielding,creep and work-hardening, etc .. are the fundamental mechanical properties exhibited bymost solids with crystalline structurc. The basic feature of a crystalline solid is tbe regularityand periodicity of crystal lattice structure. A dislocation is a line discontinuity in theatomic lattice: it represents a defect in the regularity or the ordered state of an otherwiseperfect lattice. One important result of the microscopic theories of plasticity of crystallincsolids is that among the crystal lattice defects such as impurity atoms, vacancies. grain-boundaries, etc .. dislocations play the most important role: it can be argued that theirmotion and generation in crystals account for nearly all the plasticity phenomena. Further-more. the plastic (or irrecoverable) deformation in crystalline solids is a now process. thebasic now mechanism being a slipping of crystals caused by the motion of dislocations.Burger's vector B [34] is frequcntly cmployed to charm;terize the magnitude and directionof such slip movements in crystals: the magnitude of the vector indicates the amount ofslip occurring and the direction indicates the direction of relative movcment undergoncby two originally contiguous points.

Advances in solid state physics and metal physics (e.g. see [34-36]) have shown thatany dislocation can be constructed from edge and screw segments and any motion resolvedinto components in the slip plane (glide) and normal to the slip plane (climb). The glidingof dislocations causes layers of crystals to slip over one another producing the now processduring plastic deformation. The amount of slip in one plane is always equal to a multipleof the Burgers vector. so that the crystal lattice pattern after slip always retains its regu-larity. Thus. one significant feature of plastic deformation in a crystalline solid is that itchanges the shape of crystals without destroying its crystallinity. The shape change dueto slip is generally referred to as plastic distortion.

Another important aspect in dislocation theory is the multiplication and intersectionof dislocations. The density and distribution of dislocation lines in a crystal usually increaseduring plastic deformation by dislocation multiplication processes such as Frank-Reedsources and multiple cross glides. The increase in dislocation density raises the internalenergy of the crystal and thereby facilitatcs the plastic now. but at the same timc dcvelopsresistance (drag) forccs to further dislocation motion. This interaction of dislocations in acrystal may account for work-hardening phenomena. It appears that the complex nature

264 D. R. BHASDARJ and 1. 1'. OnEN

of plastic phenomena in crystallinc solids is basically different from that of non-crystallinesolids.

From the above discussion, it seems obvious to assume that for microscopic theoriesof such materials, crystal defects (dislocations), their exact arrangements and motion arethe important parameters in describing plastic phenomena. Then the gross behaviourof these parameters must constitutc the macroscopic plasticity phenomena, the mainintcrest of our study. To complctely specify these dctails on a macroscopic level wouldnaturally require an infinite numbcr of these statc quantities. However. Kroner [38, 39]has shown that due to the randomness of dislocation distributions, it is generally notnecessary to use an infinite set of such measures to construct a reasonable continuumthcory of plasticity. Kroner points out that dislocation arrangements can be describcd in amacroscopic way by taking the mean values and the first. second. etc .. moments of thedislocation distribution. In fact, the order of moments may be determined by cxperimentalinvestigations. and experimental cvidence seems to support the notion that only a smallnumber of these is sufficient to adequately describe dislocation arrangements. We followthe suggestion of Kroner. and introducc a /inite number of quantities Alii (i = I. 2, .... N)for describing the dislocation arrangements. In gcncral, the quantitics Ali) will bc tcnsorsof differcnt rank. since these consist of various orders of moments reprcsenting variouslevels of approximation. However, in this formulation. to be consistent with our otherbasic variables. we shall consider the Ali) to be second order tensors. For example, in themacroscopic theories, the plastic strain, denoted ". may be interpreted as a limit of theaverage of the local geometrical changes in a volumc element.

These physical observations and results from dislocation theory guide us in the presentwork to postulate that the inelastic strain" and a Iinite set of dislocation arrangementtensors Ali) constitute the internal (structural) statc variables suitable for describing theplastic behaviour in crystalline solids.

3. KNEMA TICS

Most of the usual kinematical relations are assumed to hold. We consider a materialbody ~. the elements of which are material particles X. We wish to trace the motion ofthe body relative to a reference configuration Co in three-dimensional space E3-i.e .. atsome reference timc r = 0, the particles X are in onc-to-one correspondence with spatialpoint (places) x. When convenient, we shall associate with each x a triple Xi of rectangularcoordinates which give the location of x relative to a fixed spatial frame of referencc in Co:X = (Xl. X2• X3) denote labels (material coordinates) of a particle X at x in Co whichinstantaneously coincide with xj at r = O.

The motion of 8d relative to Co is given by the relation

x = I(x,n (3.1)

where I describes a mapping which carries the particle X onto its place x in E3 at time I.Effectively, I(X. I) is a one-parameter family of mappings of Co onto current configurationsC, c E3. As is customary. we denote as the deformation gradient at time I with respect tomatcrial particles X thc tensor

F(x. t) = VI(X, I) (3.2)

A IIl/i(/ed /Ileory of /hermoviscoplas/ici/y o( cry.~/(Jlline solids 265

and wc assume thaI dct F > 0 for cvery X and I. WC also introducc the Grecn--Saint Venantstrain tcnsor)' and the Cauchy-Green deformation tensor C by thc relalions

y = 1fC - l) and C = FTF (3.3a, b)

where I is the unil tensor and FT denotes the transpose of F.Consider a configuration Ci' 0 :::;t :::; I. intermediate between Co and C,. and define as

the place of X at time l

y = X(x'i) (3.4)

Formally, y = X(ic- I(X). i) where k(X) = X defines the place of particle X in C" thus C;(or, for that matter. Co and C.) need not be a configuration actually occupied hy.!Jf duringits motion. The deformation gradient at C; is then

F = VX(x'i). (3.5)

For fixcd t. wc assume that (3.4) is invertible so that we can write X = X-Ilyll,:/. Then

x = X(X(y). i) = i(y, t)

and

wherc F is given by 13.5) and i = V),1.Introducing (3.7) into (3.3), we see that

which can be written in the form

wherc

(3.6)

13.7)

(3.8)

(3.9)

(3.IOa, b)

and in which the dependence on X and t is understood.We shall refer to y as the total strain tensor. In the absence of a more appropriate term.

we follow the classical terminology and refer to " as the inleastic slrai/l tensor even thoughwe rccognize that at this point" is a purely kinematical quantity and that y - " may embodystrains which are permanent in the usual sense of the term. The tensor ~ = )' - " shall bereferred to as the difference slrai/llensor.

In most crystalline solids, plastic deformation (i.e.. yielding in thc sense of permanentdeformation) is attributed to a flow process of crystalline lattice defects normally describedin terms of devclopmcnt and propagation or dislocations. In such situations we shallinterpret the homogeneous deformation 1(t) of (3.1) of the body PA as consisting of homo-gcneous lattice distortion and homogeneous shape distortion produced by homogeneousmotion of dislocations. The latticc distortion is restorable, and on rcstoration thc latticcdistortion disappears completcly (except locally at dislocation Jines) and the body fJB

266 D. R. BHANDARI and J. T. ODIN

occupics a differcnt configuration C;. Then in view of (3.7) wc may write

I = iiwhere

(3.11 )

i is thc homogeneous latticc distortioni is the homogeneous plastic distortion due to homogeneous motion of dislocation.That is. the total deformation I is the composition of two deformations i and i.where i isthat part of the deformation associated with plastic dislocations rather than lattice distor-tions and. therefore. not rccovcrable according to our hypothesis. We associate with this partof the deformation a strain tcnsor

(3.12)

where t = VI.

~. THER\lODY""\lIC PROCESSES

For the purpose of establishing notation and some rcsults for future reference. we rcviewbriefly here certain notations. now fairly standard. on thermodynamic processes. We shallassume that couple stresses and body couples are absent in the body !!4 and that there isno diffusion of mass in tM. A thcrmodynamic proccss of tM can then be described by a setof nine functions {I. a, h. cp, q. h, s. e. (Xlii} of the particle X and time t. The function I(X. t)defines the motion of fJd, a(X t) is the second Piola-Kirchhoff stress tensor (cf. [40]. pp.124]. h(X t) is the body force vector per unit mass, cp(X t) dcnotes the free energy per unitmass, q(X t} the heat flux vector, h(X t) the heat supply per unit mass per unit time, S(X. t)the entropy per unit mass, O(X, t) the absolute temperature, and a(i)(X, I) (i = 1,2 .... , n)are internal state variables. This set of nine functions defined for all X in fJI and for alltime t is called a thermodynamic process in fJI if and only if it is compatible with the lawsof balance of linear momentum and conservation of energy (cf. [40]. pp 295). Underappropriate smoothness assumptions, the local forms of thcse Jaws are

and

Div (Fa) + ph = pii

tf(ayT) - p(cp + sO + sO) + Div q + ph = 0

(4.1 )

(4.2)

where p is the mass density in the reference configuration Co and the superimposed dotsindicate time rates.

To specify a thermodynamic process it suffices to prescribe the seven functions {X, a. qJ.

q. s. e. 2(i)}. the remaining two functions band h are then dctermined from (4.1) and (4.2)A thermodynamic process in fM, compatible with the constitutive equations at each pointX of fJI and all time t is called an admissible process (cf. [40]. pp. 365).

TilE CLAUSIUS-DUllEr •• INEQUALITY

If qlO is regarded to be an "entropy flux" due to the heat flow and hiD to be the entropysupply due to the radiation (say), then the specific rate r of production of entropy is given by

A unified theory of {hermol'iscopla~{ici{y of cry.Hnlline solid., 267

(4.3). [ph . ]pl" = ps - 0 + Dlv (q/O) .

The Clausius-Duhem inequality asserts that the rate-of-production of entropy is non-negative:

r ~ o.This implies that (4.3) can be written in the form

. . 1 0p8s - ph - DIV q + B q . 9 ~

(4.4)

(4.5)

where 9 = grad 0. Now for each thermodynamic process, the energy-balance equation(4.2) enables us to write

1(1* + (jq.g ~ 0

where we call the quantity (1* the internal dissipation. Clearly, our (1* is defincd by

(1* = rr((1)·T) - p(¢ + sO).

The inequality (4.6) is then called the general dissipation inequality.

(4.6)

(4.7)

5. CONSTITUTIVE EQUATIONS

In the dcvclopment of constitutive equations for a non-linear theory of viscoplasticity,we shall assume that a simple crystalline material at point X is characterizcd by fourresponse functions {p, fr, q and .~.which determine the value of cp, (1. q and S when the Green-Saint Venant strain y, the absolute temperature O. the temperature gradient 9 and structural(or internal) state variables lX(i) are known at point X and time t. Specifically, we considera material which is characterized by the following system of constitutive equations:

cp = (PlY, 8. g. aY))

(1 = n(y. O. g. lX(i))

q = ti(y. (], g. IXlil)

S = Sly. O. g, IXlil).

(S.la)

(5.1b)

(5.1c)

(5.1d)

In addition, the internal state variables IXU) are assumed to be given by a set of functionalrelationships of the type

(5.le)

The influence of the histories of y, (] and possibly even 9 on the current responsc can oftenbe introduced through equations of the type in (5.1e): equations (5.1e) are sometimesrefcrrcd to as equatio/ls of <TOllitio/l, since they describc the evolution of the internal stateof the material over time. We shall further assume that the constitutive equations (5.1)satisfy the principle of material frame-indifference as postulated by Truesdell and Toupin[40].

268 D. R. BHANDARI and J. T. ODEN

So far, the constitutivc assumptions (5.1) arc csscntially of the typc studied by Colcmanand Gurtin [18]. The rcmarkable feature of this approach is that these equations apply toalmost all materials irrespective of their constitution. In fact, as discussed carlier, theconstitutive properties of the material depend on !X(i) which characterize the internalstate of the body. In crystalline solids the behavior of dislocations, their distribution and theirinteractions. play the role of internal state variables. In accordance with our previousdiscussion of section 2 and motivated by the physical results from dislocation theory ofcrystalline solids we now postulate that the internal state variables !XH) consist of a secondorder tensor 'I called the 'plastic' (or inelasticl strain and a set A(i) of dislocation arrangemcnttensors, so that

(5.2)

Then in view of (5.le) and (5.2) our plastic evoilltionary eqllations are written in the form

;, = ~(y,O. 'I.Alii)

A Ii) = A HI(y, 0, 'I, A H)).

(5.3a)

(5.3b)

Constitutive assumptions (5.3) are the immediate consequences of the basic physicalresults of dislocation theory: that is the plastic flow in crystalline solids is a dissipative andti!11e-dcpendent process determined by the dynamical motion of dislocations. It is assumedthat the values of'l and AH) at time t are uniquely determined by the solution of(5.3) subjectto the initial conditions (say) '1(0) = 0 and A(i)(O) = A~).

We now require that for every admissible thermodynamic process in 14, the responsefunctions appearing in (5.1aH5.1d) and (5.3) must be such that the postulatc (4.6) of positiveentropy production is satisfied at each point X of ~ and for any time t. This is equivalentto the inequality

• . A 1(r(ayT) - p(cp + su) + 0 q • g ~ O.

As a consequence of(5.la) and (5.2) we can rewritc (5.4) as follows:

(r[(a -pil,.q,)YT] - p(S + 0eq,)8 - POgq, . 9

- p{tr[(a,q,);,T] + tr[(o)illjlIATCil]} +~q.g ~ 0

(5.4)

(5.5)

where o.,q,. oeq,. iJ,q" etc .. denote the partial differentiation of q, with respect [0 y. O. and 'I,respectively. We now follow the arguments similar to those of. say. Coleman and Gurtin[18]. i.e. we observe that by fixing y, 0, g. q and A(l) at time ( we also fix;' and A(i) (as aresult of(5.3)) but y, 0 and 9 are left arbitrary. Thus for the inequality (5.5) to hold independentof the signs of y, 8 and g, their coefficients must vanish. Consequently. we obtain

o q,(.) = 09

0'= po,.ljl(.)

S = -8eljl(.).

The Clausius-Duhem inequality (5.5) reduces to

(5.6a)

(5.6b)

(5.6c)

A unified theory of rhamolJiscoplasriciry (!( cryHallint' solidI 269

which is called the qeneral dissipation inequality.The general dissipation inequality (5.6d) implies that when 9 = 0, the illtemul dissipation

inequality

(5.7)

holds. Now if we define the internal dissipation a* by

a* = U(y. e. 't.A (il)

we can write the intcrnal dissipation inequality (5.7) in the form

u*(y, e. 't, AliI) ~ O.

(5.8)

(5.9)

Further, for any fixed value of a* we can arbitrarily vary the last term of the incquality(5.6d) by varying g. Hence, it follows that

q.g ~ 0 (5.10)

which is heal conduction inequality.Summarizing. from the results of(5.6), (5.7) and (5.10) we have the following consequences:

(i) The response functions (p. iT and S are independent of the temperature gradient g:namely

cp = (p(y. O. fl. A(il)

(f = &(y. O. fl. A(i))

s = S(},. 0, 't.Ali))

(ii) ip determines stress (f through the relation

(f = pD7(p()'. 0, fl. A fiI)

(iii) cP determines entropy S through the relation

S = - De(p(y. O. '1. AliI)

(5.lla)

(5.1 Lb)

(5.llc)

(5.12)

(5.L3)

(iv) (p. r" A(il and q obey the general dissipation inequality (5.6d). The internal dissipa-tion and heat conduction inequalities hold and are given by (5.9) and (5.10).

Thus. the complete set of constitutive equations for thermoviscoplastic materials takesthe form

(5.14a)

270

and

D. R. BHANDARI and J. T. ODES

(1 = o-(y, 0, fl, A IiI) = (le/p( ,)

s = Sly. 0, fl. Ali)) = - 0e4J(.)

q = q(y, 0, g,,,. Ali))

;, = q(y, e, fl. Ali))

.4(i) = Ali)(y, O. fl, A(i)).

(5.14b)

(5.14c)

(5.14d)

(5.15a)

(5.15b)

6. RELATIONSHIP WITH OTHER EXISTING THEORIES

Several thermodynamic theories of elastic-plastic materials can be obtaincd as specialcases of (5.14) and (5.15) by imposing further restrictions on the constitutive equations(5.15). For example. onc important class of materials results from excluding all thosestate quantitics from the set AliI which arc responsible for viscous effects in crystallinesolids. In other words, wc neglect those quantitics which describe grain boundary sliding,internal slipping of grains. twinning etc. In agreement with Kroiier's conclusions, we alsoassume that a single state variable A (say the dislocation loop density tensor) is sufficientto includc effects like Bauschinger effect. Then the constitutive equations (5.15a) arereplaced by a quasi-linear transformation ofliu into Aij:

(6.1 )

Relation (6.1) may be referred to as a dislocation production law. and thc fourth ordertensor QjjU as the dislocation production tensor [38]. The implication of (6.1) is that itsimply restricts the occurrence of dislocation densities to the process in which plasticdeformations occur.

Constitutive equations (5.14) with (6.1) are not sufficient to formulatc a determinateproblem in the sense of classical theory of elastoplasticity. This is due to the fact that whenin (6.1) ;, = 0, which also means A = 0, the response may be reversible, and the theoryreduces to one for thermoelastic materials. Thcrcfore, in order to formulate a determinateproblem (that is, to determine a "plastic" stress-strain relation). one needs an additionalpostulate of yielding. This postulate can be considered as a consequence of physicalassumptions of defining a limit point on the strain path before irreversible displacementstake place (i.e. when;' :F 0). A convenient (but not essential) way of expressing this thresholdcharacter of the response function q is to make thc assumption of the existence of yieldsurface. Since the postulate of yielding and its consequences are well known, we shall notelaborate on these here.

It has been shown by Owen [41,42] that a good theory of plasticity can be constructedwithout introducing eXplicitly the notion of yield functions. In his theory, Owen introducedthe concept of an "elastic-range" which seems to be equivalent to using the yield functionto express the threshold character of inelastic deformations described by Kratochvil andDillon [23].

Green and N aghdi's theory oj plast icit y [29].The continuum theory of plasticity developed by Green and Naghdi characterizes the

A unified theory of thermori.fcop/asticity (~f crystalline solids 271

rate-independent plastic behavior of crystalline solids. The charactcristic of their theoryis that the governing plastic conslitulive cquations arc homogeneous in time of the firstdegree in the "state variables": whercas thc plastic evolutionary equations (15.5a. b) in thepresent paper include time dependent plastic behavior and hence belongs to viscoplasticitytheory. Although. from a theoretical point of view. these two models arc different. a rate-independent theory similar to Green and Naghdi can be constructed from the presentformulation if the plastic evolutionary equations (15.5) are made homogeneous in time ofthe first degree. Once this is done. the collection of constitutive equations togethcr. withthe usual assumption of the existence of a yield surface. leads to results similar to those ofGreen and Naghdi [29].

nellY'S theury (!f" elastic-plastic crystallille solids [25].Tseng. in his constitutive theory of clastic-plastic crystallinc solids [25] assumed that a

scalar quantity ct, called the internal structural density, is to be added to the constitutivevariables of thermo-elasticity. In Tseng's work, the constitutive equations for thermo-elasticity are functions of FI'». 0, g and :x; whereas the plastic evolutionary equations fore(or pr) andi are assumed to be functions of (/, Ft', () and ct. It appears the forms (15.5a. b)arc more convenient for making a thermodynamic analysis.

Theory of thermoplastic materials with mel/lOr)' [28].We shall now show that the results of Section 5 in thc present theory are nearly identical

to those presented by Oden and Bhandari [28] in their earlier work on thermoplasticmaterials with memory. Following the arguments of Lubliner [22]. we observe that undersuitable assumptions the present value of A(iI(e) from the evolutionary equation (5.15b)can be determined in terms of the 'past histories' ofy. 0 and". Then the solution of(5.15b)can be written in the form of a functional

~=rAm(t) = )7t(i) {['(r)}

r= .- 00

(6.2)

where for the sake of conciseness we have used the notion r = (y. O. "" and the dependenceof A(i) and r on X is understood. Thcn r'(r) denotes the restrictions of nr) to r < t. Func-tions ne) (t > - (0) permitting uniquc, continuous solutions Alil(l) of (5.15b) which satisfyAm( - 'l.:') = 0 arc considered admissible. In fact. for Alil(tl to be continuous. A'Ii) nt),Ali» and hence r(t) need not be continuous in t (see [22J).

Substituting (6.2) into (5.14) and (5.15a), and making use of the more familiar notationadopted in [28]. we obtain the functional forms

<rj

cp = cP[r(c). Jlili){ F(r)}] = <I> [r~(s): r(cn1'=._ 00 .1=0

where r~(s)= ru - s)[o < S < 00] and r(t} = P(O). Similarly

a) '"

(/ = ~[~(S):f(I)] = pey <I> [r~: r]5=0 5=0

~ ~S = fI [r~(s): ru)] = -a,,$ [r~: r]

5=0 s=1l

(6.3a)

(6.3b)

(6.3c)

272 D. R. BHANDARI and J. T. ODE:-

00

q = f2 [r~(s): nt}. g(t)].<=0

00

;, = K[r~(s): no]..<=0

(6.3d)

(6.3e)

We note that the stress a and entropy S are derivable from the free energy functional00cD [:] which is independent of 9 as implied by (6.3b) and (6.3c).

.<=0The general dissipation inequality in this case is:

00 ,7) 1 Ctl

- p{tr[(o~ <I> [:]);'1"] + bf cD [-Ir~]}+ 0 f2 [:]. 9 ~ 0.=0 5=0 .<=0

(6.4)

<I)

where b,. <D [ - : I] denotes a frcchet differential linear in arguments to the right of.=0

vertical stroke:

The internal dissipation and heat conduction inequalities are now given by

:T.I rS)

a* = -O'-l{tr[(o~ <D [:]);'1'] + bf <D [-It~] ~ 0,=0 .=0

and

<Xlfl [~.r,9]·9 ~ O.s=o

(6.6)

(6.7)

The constitutive equations (63) together with the rcsults (6.4)--(6.7) are essentially of the typegiven in [28].

Valanis's 'emiochronic' theory of viscoplasticity [43]Valanis, in his recent work [43. 44]. has also used the concepts of hidden variables in

devcloping a functional "cndochronic" theory of viscoplasticity. His developmcnt is basedon Onsager's relation. a move which, to some. has proved to be controversial (see, Truesdelland Toupin [45]). furthermore. Valanis's work appears to be restricted to infinitesimaldeformations and isothermal processes. We note that under appropriate additionalassumptions Valanis's work can be obtained as a special case of the general formulationgiven by (6.3). To prove that this is so, consider the special case in which the free energyfunctional (6.3a) has the following form

, I

1 f f ..I oy.. oyf)(P = cp + - A')k (t - t' t - t") ~ (r')..-M (I") dr' dt"o 2 'or' at"

o 0

I I

II .~ ?y al+ B'J It - t', t - t") ~ (t') ---!l (t") dt' dl"at' at"o 0

A ll/l(fied Iheor.v of thermol'iscoplasticity of crystalline solid,

I I

+ ~f f Cijk/(r - t' r - r") ~ (t') atlu (t") dt' dr"2 . at' 01"

o 0

I I

f f" a",. 00+ D'J(t - t', t - t")~(t')-- (t")dt' dt"ot' at"

o 0

I I

f f .. 0'/, . 00+ E'J(t - t', t - t") ~ (t'l- ((")dt' dt"01" at"

o 0

, I

1 f f ae ae+ 2 F(r - t', t - t") at' (t')ar<n dt' dt".

o 0

273

(6.8)

Here Aijk/(.). BijUO etc. are material kernels. With the aid of (6.3b) and (6.3c). we obtainfrom (6.8) the stress and entropy:

, I

(J'ij = fAiikl(t - t') aYkl (r') d( + fBijk'(t - t') a,/ u (t') dt'at' at'

o 0

I

f" ae+ D'J(r - t')- (t') dt'at'

oI t

- pS = f Dij(t - r') ~ (t') dl' + r £ii(t - n~ (t') dt'at' • at'

o 0

I

f ao+ F(I - t') a1 (t') dt'.

o

(6.9)

(6.10)

For the sake of illustration. assume that the constitutive (evolution) equation for tiii isgiven in terms of the histories of Y and (] only: e.g.

I I

f -. 'kl ayk/ f -'j of),..,= e'J (t - t') ~ dt' + £' (t - t') - (t') dt'I'J at' ~ ot' .o 0

(6.11 )

Moreover. to obtain an explicit form of the constitutive equation for stress n. we furtherconsider only isotropic materials for which

ijkl A 0 t .I: 1 '.1: .1:')A = uil'kl + A (c\k(Jil + ut/5jk

Bijkl = BO {) .. bkl + B I(S 'kc). + {)'le> 'k)I) . I Jl J J'

274 [). R. BIIA:-;DARI and J. 1'. ODfr-:

Cijkl = CObl'kl + C1(c5jkc5}1 + c5i/()}k)

E'i = E°c5..I)

Vij = D°c5.. etc.I}

(6.12)

Then with the aid of (6.11) and (6.12), we rewrite (6.9) in the form

t t.. f oy" . f °Ykkal} = 2JI(1 - I') ~ (I') dl' + (). K(t - n- (t') dt'at' I} or'

° °t

. I' ( ')o£l ( ') d '+ (j.. I. I - I - I tI) at'

where the material kernels JI( .), and K(.) and i.(.) are now given by

2JI(t) = 2,4 1(1) + 4B'(r)*C1(t)

K(t) = AO(I) + 2B1(1)*('0(1)

i.(1) = 0°(1) + 2B1(I)*Eo(1)

(6.13)

(6.14)

and the symbol * in (6.14) denotes the convolution operator.Finally, we obtain Valanis's "cndochronic" theory of visco plasticity (i.e. a theory in which

stress, among other properties, is a functional of strain history. defined with respect to anintrinsic time scale, the lattcr being the property of the material at hand) by introducing atime scale z which is independent of t, the external time measured by clock. but which isintrinsically related to the dcformation and temperature. To illustrate this. we introduce inthe manner of Pipkin and Rivlin [46] a non-negativc monotone increasing timc invariantparameter

,z( r) = J [trW);) + (())2]t dr'

°where superposed dot indicates dilTerentiation with respect to time r'. i.e.

. d)', dO,Y = dr' (r I: B = dr' (1." )

(6.15)

(6.16)

and z represents the arc length of the path in the ten-dimensional space of strain andtemperaturc. Then introducing the time scale z into (6.13), we obtain

. -. .aii = 2 f JI(Z - 7') ~ (z') dz' + c5 f K("" - -'I OYkk (z') dz'- oz' Ij ~ - oz'

° °.+ c5i} f ,t(z - z') ~O (z') dz'.

oz'o

(6.17)

A limped theory of thermoriscop/asticily of crystalline solids 275

(6.18)

We observe that hy introducing the "reduced"' time the form of the constitutivc equationsdoes not alter and (6.17) is esscntially the one givcn by Valanis [43].

Prandt/-RellSS relationsAs a final example, we show that the Prandtl-Reuss relations of classical plasticity can

be obtained from (6.17) as a special case. For isothermal proccsscs and inlinitcsimalstrains. reduces to

a;j = 2 f µ(z - z') ~ (z') dz'

o-

f d"au = 3 K(z - z') ~;~ (z') dz'

o

where a;j and l;j are the deviatoric stress and strain tensors and t au = 0'0 and j'u are themean stress and the dilatation. By selecting

we see that

-a;j = 2µo f (!t(:. :') di'iiz')

o

which. when differentiated yields

• (X • 1 .d)'.. = -dza ..+ -da ...

I) 2µ0 I) 2/10 I)

Then using the relation

we can write

1= 2µ0 a;j

and

(6.19)

(6.20)

(6.21)

(6.22a)

(6.22b), (x.

dll.· = dIll')' = -dza ...') 2µ0 I)

We recognize these results as the familiar Prandt-Reuss equations of classical plasticity.

Acknoll'ledgeml'nt- The support of this work by the U.S. Air Force omcc of Scientific Rcscarch under Contrac[F44620·69·C·O 124 is gralefully acknowledged.

276 D. R. BHANDARIand J. T. ODm

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(Recei"ed 12 Seplelllber 1972)

Resume-On presenle une lheorie unifiee de la thermoviscoplaslicite des solides cristallins. En partieulier nouspouvons montrer que la thermodynamique des materiaux "viscoplastiques" peut etre adaptee dans Ie cadre dela mecanique des milieux continus moderne des materiaux a memo ire. Lcs concepts physiques de base sontdCduits de la consideration du comportement des dislocations dans les solides cristallins. On regarde Ics rclalionsentre notre approche actuelle et plusieurs lheories exislantes de la plastidte.

Zusammcnfassung-Einc einheitJiche Theorie der Thermoviskoplastizitat kristalliner Fc,tkorpcr wird dar·gcstcllt 1m besondcren war cs uns miiglich zu zeigcn, dass cine Thermodynamik "viskoplaslischer" slorrc in denRahmen moderner Kontinuumsmechanik der Storre mil Gcdiiclllnis cingepasst werden kann. Die physikalischcGrundkonzepte werden durch die Betrachtung des Verhaltens von Versetzungen in kristallinen Fcslkorpernhergeleitet. Zusammenhange unserer Darstellung mit einigen bestehenden Plastilitiitstheorien werden untersucht.

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