/Ilr. J .. \icH:·Unf'IJr M ..-dumic-,. Vol. 8. pp. 261-~77, Pergamon Prn.~ 1913. Printed in Great Britain
A UNIFIED THEORY OF THERMOVISCOPLASTICITY OF 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 shown that a therrnodynamies for 'viscoplastic' materials can be accommodated within the framework of modem mechanics 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 of plasticity 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 theory itself involves generalizations of ideas proposed early in the development of modern continuum 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]"' thermodynamics of 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 the internal state variable (hidden variable) approach to the thermodynamics of non-linear viscoelastic materials have been explored extensively by Schapery and by Valanis in a number of papers (e.g. [11-14]). The notion of internal or hidden variables was, at first. not easily reconciled within the framework of modern continuum mechanics. and a number of alternate approaches to the 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 is shown that certain materials can be characterized by only two constitutive functionals, one describing the free energy, which is independent of temperature gradient. and the other describing the heat flux. A key feature of Coleman's theory is that the stress. entropy and the internal dissipation are determined as Frechet differentials of free energy functional. More recently. the thermodynamics of non-linear materials with internal state variables has 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 that their approach [18] to continuum thermodynamics is but one of the several approaches including 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 unify them. 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 the qualitative properties of Coleman's theory of materials with memory [15]. It is shown by Lubliner [22J that under specific but fairly broad conditions, the principle of fading memory of 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 of Coleman [15] and the stronger results of Coleman are valid under equivalent constitutive hypothesis. General theories of thermoviscoelasticity derived from, say. Coleman's thermodynamics of simple materials [15]. are generally regarded as adequate for describing non-linear behavior 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 response of 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 state variables (independent constitutive variables) must be introduced in the 'plasticity' phe- nomena such as yielding. strain hardening. etc., are to be encompassed by the constitutive cquations. Some macroscopic measure of the influence of dislocations immediately arises as a likely candidate for such additional measures. Coleman-Gurtin type thermodynamics for the study of elastoplastic materials has been 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 crystalline solids to interpret various internal state variables. Arguments have been made that dis- locations. their arrangements and their interactions in crystalline solids play the role of internal state variables. The present study follows a pattern similar to that of Kratochvil and Dillon [23] in that we use Kroner's arguments from dislocation theory to justify the inclusion of "'hidden state variables"' which manifests itself in the form of second-order tcnsors Alil. Howevcr. we carry the study a bit deeper by also investigating the relationship of 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 state variable. Recently Oden and Bhandari [28] prcsented a theory of thermoplastic materials with 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 idea of 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 the functional theory of Oden and Bhandari [28] can be obtaincd from the evolutionary theory prcscnted herein by introducing certain plausible assumptions concerning properties of the equations of evolution. In the present paper what might be called a Coleman-Gurtin type thermodynamics of non-linear materials with internal state variables is dcveloped and is applied to a combincd treatment 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 of materials with memory. The basic physical concepts are dcrived from the consideration of dislocation behaviour of crystalline solids, and emphasis is placed on what appears to be a logical identification of internal state variables. Finally. the relationships of this approach to 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 simple materials, it is necessary to choose a sct of state variables suitable for describing plastic phenomena. In the following, we briefly summarize elcments of the theory of dislocations which are useful in choosing appropriate state quanti tics for the class of materials studied here. 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 by most solids with crystalline structurc. The basic feature of a crystalline solid is tbe regularity and periodicity of crystal lattice structure. A dislocation is a line discontinuity in the atomic lattice: it represents a defect in the regularity or the ordered state of an otherwise perfect lattice. One important result of the microscopic theories of plasticity of crystallinc solids 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 their motion 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. the basic 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 direction of such slip movements in crystals: the magnitude of the vector indicates the amount of slip occurring and the direction indicates the direction of relative movcment undergonc by two originally contiguous points. Advances in solid state physics and metal physics (e.g. see [34-36]) have shown that any dislocation can be constructed from edge and screw segments and any motion resolved into components in the slip plane (glide) and normal to the slip plane (climb). The gliding of dislocations causes layers of crystals to slip over one another producing the now process during plastic deformation. The amount of slip in one plane is always equal to a multiple of 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 it changes the shape of crystals without destroying its crystallinity. The shape change due to slip is generally referred to as plastic distortion. Another important aspect in dislocation theory is the multiplication and intersection of dislocations. The density and distribution of dislocation lines in a crystal usually increase during plastic deformation by dislocation multiplication processes such as Frank-Reed sources and multiple cross glides. The increase in dislocation density raises the internal energy of the crystal and thereby facilitatcs the plastic now. but at the same timc dcvelops resistance (drag) forccs to further dislocation motion. This interaction of dislocations in a crystal 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-crystalline solids. From the above discussion, it seems obvious to assume that for microscopic theories of such materials, crystal defects (dislocations), their exact arrangements and motion are the important parameters in describing plastic phenomena. Then the gross behaviour of these parameters must constitutc the macroscopic plasticity phenomena, the main intcrest of our study. To complctely specify these dctails on a macroscopic level would naturally 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 not necessary to use an infinite set of such measures to construct a reasonable continuum thcory of plasticity. Kroner points out that dislocation arrangements can be describcd in a macroscopic way by taking the mean values and the first. second. etc .. moments of the dislocation distribution. In fact, the order of moments may be determined by cxperimental investigations. and experimental cvidence seems to support the notion that only a small number of these is sufficient to adequately describe dislocation arrangements. We follow the 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 tcnsors of differcnt rank. since these consist of various orders of moments reprcsenting various levels of approximation. However, in this formulation. to be consistent with our other basic variables. we shall consider the Ali) to be second order tensors. For example, in the macroscopic theories, the plastic strain, denoted ". may be interpreted as a limit of the average of the local geometrical changes in a volumc element. These physical observations and results from dislocation theory guide us in the present work to postulate that the inelastic strain" and a Iinite set of dislocation arrangement tensors Ali) constitute the internal (structural) statc variables suitable for describing the plastic behaviour in crystalline solids.
3. KNEMA TICS
Most of the usual kinematical relations are assumed to hold. We consider a material body ~. the elements of which are material particles X. We wish to trace the motion of the body relative to a reference configuration Co in three-dimensional space E3-i.e .. at some reference timc r = 0, the particles X are in onc-to-one correspondence with spatial point (places) x. When convenient, we shall associate with each x a triple Xi of rectangular coordinates 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 which
instantaneously 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 configurations C, c E3. As is customary. we denote as the deformation gradient at time I with respect to matcrial 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 Venant strain 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 during its 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) (3.6) and
13.7) wherc F is given by 13.5) and i = V),1. Introducing (3.7) into (3.3), we see that
(3.8) which can be written in the form
(3.9) wherc
(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 though we rccognize that at this point" is a purely kinematical quantity and that y - " may embody strains which are permanent in the usual sense of the term. The tensor ~ = )' - " shall be referred to as the difference slrai/llensor. In most crystalline solids, plastic deformation (i.e.. yielding in thc sense of permanent deformation) is attributed to a flow process of crystalline lattice defects normally described in terms of devclopmcnt and propagation or dislocations. In such situations we shall interpret 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 homogeneous motion of dislocations. The latticc distortion is restorable, and on rcstoration thc latticc distortion 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 = ii (3.11 ) where
i is thc homogeneous latticc distortion i 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 is that 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 part of 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 rcview briefly here certain notations. now fairly standard. on thermodynamic processes. We shall assume that couple stresses and body couples are absent in the body !!4 and that there is no diffusion of mass in tM. A thcrmodynamic proccss of tM can then be described by a set of 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 unit mass, 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 all time t is called a thermodynamic process in fJI if and only if it is compatible with the laws of balance of linear momentum and conservation of energy (cf. [40]. pp 295). Under appropriate smoothness assumptions, the local forms of thcse Jaws are
Div (Fa) + ph = pii (4.1 ) and
tf(ayT) - p(cp + sO + sO) + Div q + ph = 0 (4.2) where p is the mass density in the reference configuration Co and the superimposed dots indicate 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 point X 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 entropy supply 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
pl" = ps. - [ph0 + Dlv. (q/O) ] . (4.3)
The Clausius-Duhem inequality asserts that the rate-of-production of entropy is non- negative: r ~ o. (4.4)
This implies that (4.3) can be written in the form
. . 1 0 p8s - ph - DIV q + B q . 9 ~ (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 (4.6) where we call the quantity (1* the internal dissipation. Clearly, our (1* is defincd by
(1* = rr((1)·T) - p(¢ + sO). (4.7) The inequality (4.6) is then called the general dissipation inequality.
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 four response 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 consider a material which is characterized by the following system of constitutive equations:
cp = (PlY, 8. g. aY)) (S.la)
(1 = n(y. O. g. lX(i)) (5.1b)
q = ti(y. (], g. IXlil) (5.1c)
lil S = Sly. O. g, IX ). (5.1d)
In addition, the internal state variables IXU) are assumed to be given by a set of functional relationships of the type
(5.le)
The influence of the histories of y, (] and possibly even 9 on the current responsc can often be introduced through equations of the type in (5.1e): equations (5.1e) are sometimes refcrrcd to as equatio/ls of So far, the constitutivc assumptions (5.1) arc csscntially of the typc studied by Colcman and Gurtin [18]. The rcmarkable feature of this approach is that these equations apply to almost all materials irrespective of their constitution. In fact, as discussed carlier, the constitutive properties of the material depend on !X(i) which characterize the internal state of the body. In crystalline solids the behavior of dislocations, their distribution and their interactions. play the role of internal state variables. In accordance with our previous discussion of section 2 and motivated by the physical results from dislocation theory of crystalline solids we now postulate that the internal state variables !XH) consist of a second order tensor 'I called the 'plastic' (or inelasticl strain and a set A(i) of dislocation arrangemcnt tensors, 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) (5.3a) A Ii) = A HI(y, 0, 'I, A H)). (5.3b) Constitutive assumptions (5.3) are the immediate consequences of the basic physical results of dislocation theory: that is the plastic flow in crystalline solids is a dissipative and ti!11e-dcpendent process determined by the dynamical motion of dislocations. It is assumed that the values of'l and AH) at time t are uniquely determined by the solution of(5.3) subject to the initial conditions (say) '1(0) = 0 and A(i)(O) = A~). We now require that for every admissible thermodynamic process in 14, the response functions appearing in (5.1aH5.1d) and (5.3) must be such that the postulatc (4.6) of positive entropy production is satisfied at each point X of ~ and for any time t. This is equivalent to the inequality • . A 1 (r(ayT) - p(cp + su) + 0 q • g ~ O. (5.4) 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 (5.5) 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 a result of(5.3)) but y, 0 and 9 are left arbitrary. Thus for the inequality (5.5) to hold independent of the signs of y, 8 and g, their coefficients must vanish. Consequently. we obtain q,(.) = 0 (5.6a) o9 0'= po,.ljl(.) (5.6b) S = -8eljl(.). (5.6c) The Clausius-Duhem inequality (5.5) reduces to 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) (5.8) we can write the intcrnal dissipation inequality (5.7) in the form u*(y, e. 't, AliI) ~ O. (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) (5.lla) (f = &(y. O. fl. A(i)) (5.1 Lb) s = S(},. 0, 't.Ali)) (5.llc) (ii) ip determines stress (f through the relation (f pD (p()'. 0, fl. A fiI) (5.12) = 7 (iii) cP determines entropy S through the relation S = - De(p(y. O. '1. AliI) (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 takes the form (5.14a) 270 D. R. BHANDARI and J. T. ODES (1 = o-(y, 0, fl, A IiI) = (le/p( ,) (5.14b) s = Sly. 0, fl. Ali)) = - 0e4J(.) (5.14c) q = q(y, 0, g,,,. Ali)) (5.14d) and ;, = q(y, e, fl. Ali)) (5.15a) .4(i) = Ali)(y, O. fl, A(i)). (5.15b) 6. RELATIONSHIP WITH OTHER EXISTING THEORIES Several thermodynamic theories of elastic-plastic materials can be obtaincd as special cases 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 those state quantitics from the set AliI which arc responsible for viscous effects in crystalline solids. 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 also assume that a single state variable A (say the dislocation loop density tensor) is sufficient to includc effects like Bauschinger effect. Then the constitutive equations (5.15a) are replaced by a quasi-linear transformation ofli into : u Aij (6.1 ) Relation (6.1) may be referred to as a dislocation production law. and thc fourth order tensor QjjU as the dislocation production tensor [38]. The implication of (6.1) is that it simply restricts the occurrence of dislocation densities to the process in which plastic deformations occur. Constitutive equations (5.14) with (6.1) are not sufficient to formulatc a determinate problem in the sense of classical theory of elastoplasticity. This is due to the fact that when in (6.1) ;, = 0, which also means A = 0, the response may be reversible, and the theory reduces to one for thermoelastic materials. Thcrcfore, in order to formulate a determinate problem (that is, to determine a "plastic" stress-strain relation). one needs an additional postulate of yielding. This postulate can be considered as a consequence of physical assumptions of defining a limit point on the strain path before irreversible displacements take place (i.e. when;' :F 0). A convenient (but not essential) way of expressing this threshold character of the response function q is to make thc assumption of the existence of yield surface. Since the postulate of yielding and its consequences are well known, we shall not elaborate on these here. It has been shown by Owen [41,42] that a good theory of plasticity can be constructed without introducing eXplicitly the notion of yield functions. In his theory, Owen introduced the concept of an "elastic-range" which seems to be equivalent to using the yield function to express the threshold character of inelastic deformations described by Kratochvil and Dillon [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 theory is that the governing plastic conslitulive cquations arc homogeneous in time of the first degree in the "state variables": whercas thc plastic evolutionary equations (15.5a. b) in the present paper include time dependent plastic behavior and hence belongs to viscoplasticity theory. 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 present formulation if the plastic evolutionary equations (15.5) are made homogeneous in time of the first degree. Once this is done. the collection of constitutive equations togethcr. with the usual assumption of the existence of a yield surface. leads to results similar to those of Green 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 constitutive variables 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 for t e(or pr) andi are assumed to be functions of (/, F ', () 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 thermoplastic materials with memory. Following the arguments of Lubliner [22]. we observe that under suitable 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 ~=r Am(t) = )7t(i) {['(r)} (6.2) r= .- 00 where for the sake of conciseness we have used the notion r = (y. O. "" and the dependence of 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 satisfy Am( - '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 notation adopted in [28]. we obtain the functional forms a) '" (/ = ~[~(S):f(I)] = pey [r~: r] (6.3b) 5=0 5=0 ~ ~ S = fI [r~(s): ru)] = -a,,$ [r~: r] (6.3c) 5=0 s=1l 272 D. R. BHANDARI and J. T. ODE:- 00 q = f2 [r~(s): nt}. g(t)] (6.3d) .<=0 00 ;, = K[r~(s): no]. (6.3e) .<=0 We note that the stress a and entropy S are derivable from the free energy functional 00 cD [:] which is independent of 9 as implied by (6.3b) and (6.3c). .<=0 The general dissipation inequality in this case is: 00 ,7) 1 Ctl - p{tr[(o~ [:]);'1"] + bf cD [-Ir~]}+ 0 f2 [:]. 9 ~ 0 (6.4) .=0 5=0 .<=0 The internal dissipation and heat conduction inequalities are now given by :T.I rS) a* = -O'-l{tr[(o~ 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 based on Onsager's relation. a move which, to some. has proved to be controversial (see, Truesdell and Toupin [45]). furthermore. Valanis's work appears to be restricted to infinitesimal deformations and isothermal processes. We note that under appropriate additional assumptions Valanis's work can be obtained as a special case of the general formulation given by (6.3). To prove that this is so, consider the special case in which the free energy functional (6.3a) has the following form , I 1 I oy.. oy f)(P = cp + - f f A')k..(t - t' t - t") ~ (r')..-M (I") dr' dt" o 2 'or' at" o 0 I I ?y 'J.~ al + B It - t', t - t") ~ (t') ---!l (t") dt' dl" II at' at" o 0 A ll/l(fied Iheor.v of thermol'iscoplasticity of crystalline solid, 273 I I + ~f f Cijk/(r - t' r - r") ~ (t') atlu (t") dt' dr" 2 . at' 01" o 0 I I " a",. 00 + D'J(t - t', t - t")~(t')-- (t")dt' dt" f f ot' at" o 0 I I .. 0'/, . 00 + E'J(t - t', t - t") ~ (t'l- ((")dt' dt" f f 01" at" o 0 , I 1 ae ae + 2 f f F(r - t', t - t") at' (t') ar Here Aijk/(.). BijUO etc. are material kernels. With the aid of (6.3b) and (6.3c). we obtain from (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 " ae + D'J(r - t')- (t') dt' (6.9) f at' o I t - pS = Dij(t - r') ~ (t') dl' + £ii(t - n~ (t') dt' f at' •r at' o 0 I ao + f F(I - t') a1 (t') dt'. (6.10) o For the sake of illustration. assume that the constitutive (evolution) equation for tiii is given in terms of the histories of Y and (] only: e.g. I I 'kl ayk/ -'j of) ,.., = -.'J (t - t') ~ dt' + f £' (t - t') - (t') dt' I'J f e at' ~ ot' . (6.11 ) o 0 Moreover. to obtain an explicit form of the constitutive equation for stress n. we further consider only isotropic materials for which ijkl A 0 t .I: 1 '.1: .1:') A = uil'kl + A (c\k(Jil + ut/5jk O Bijkl = B {) .. b + B I(S 'kc). + {)'le> 'k) I) kl . I Jl J J' 274 [). R. BIIA:-;DARI and J. 1'. ODfr-: 1 Cijkl = CObl'kl + C (c5jkc5}1 + c5i/()}k) E'i = E°c5.. I) Vij = D°c5.. etc. (6.12) I} Then with the aid of (6.11) and (6.12), we rewrite (6.9) in the form t t l oy" . kk a..} = f 2JI(1 - I') ~ (I') dl' + ().f K(t - n-°Y (t') dt' at' I} or' ° ° t + (j... I' I.(I - I ')-o£l ( I') dt' (6.13) I) 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) (6.14) 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 an intrinsic time scale, the lattcr being the property of the material at hand) by introducing a time scale z which is independent of t, the external time measured by clock. but which is intrinsically related to the dcformation and temperature. To illustrate this. we introduce in the manner of Pipkin and Rivlin [46] a non-negativc monotone increasing timc invariant parameter , z( r) = J [trW);) + (())2]t dr' (6.15) ° where superposed dot indicates dilTerentiation with respect to time r'. i.e. . d)', dO, Y = dr' (r I: B = dr' (1." ) (6.16) and z represents the arc length of the path in the ten-dimensional space of strain and temperaturc. Then introducing the time scale z into (6.13), we obtain . .- aii = 2 JI(Z - 7') ~ (z') dz' + c5 K("" - -'I OYkk (z') dz' f - oz' Ij f ~ - oz' °. ° + c5 } f ,t(z - z') ~O (z') dz'. (6.17) i oz' o A limped theory of thermoriscop/asticily of crystalline solids 275 We observe that hy introducing the "reduced"' time the form of the constitutivc equations does not alter and (6.17) is esscntially the one givcn by Valanis [43]. Prandt/-RellSS relations As 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 inlinitcsimal strains. reduces to a;j = 2 f µ(z - z') ~ (z') dz' o - d" au = 3 K(z - z') ~;~ (z') dz' (6.18) fo where a;j and l;j are the deviatoric stress and strain tensors and t au = 0'0 and j'u are the mean stress and the dilatation. By selecting (6.19) we see that - a;j = 2µo f (!t(:. :') di'iiz') (6.20) o which. when differentiated yields • (X • 1 . d)'.. = -dza ..+ -da ... (6.21) I) 2µ0 I) 2/10 I) Then using the relation (6.22a) we can write 1 = 2µ0 a;j and , (x. dll.· = dIll')' = -dza ... (6.22b) ') 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 REFERENCES [1] M. A. BlOT. Theory [381 E. KRljSER. DisloC;llion: A new concept in the continuulll theory of plasticity. J. lIIolh. PhI'.<. "2. 27-37 (1962). (391 E. KRO~ER. !low the internal state ofa plastically deformed body is to be described in a continuum theory. Proceedin~.< oIllre fimrllr IllIemotional Congress on Rheology. 1963 (Ed. 1'. H. LEE). Int('rscien('e. New York (1965). [401 C. A. TRul'sDRL and W. Noll.. The Non·lilwar Field Theories of Mechanics. Holltlhuck der I'hy. (Recei"ed 12 Seplelllber 1972) Resume-On presenle une lheorie unifiee de la thermoviscoplaslicite des solides cristallins. En partieulier nous pouvons montrer que la thermodynamique des materiaux "viscoplastiques" peut etre adaptee dans Ie cadre de la mecanique des milieux continus moderne des materiaux a memo ire. Lcs concepts physiques de base sont dCduits de la consideration du comportement des dislocations dans les solides cristallins. On regarde Ics rclalions entre 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 den Rahmen moderner Kontinuumsmechanik der Storre mil Gcdiiclllnis cingepasst werden kann. Die physikalischc Grundkonzepte werden durch die Betrachtung des Verhaltens von Versetzungen in kristallinen Fcslkorpern hergeleitet. Zusammenhange unserer Darstellung mit einigen bestehenden Plastilitiitstheorien werden untersucht. AlllltlTllltllll-lla.lOmClla e)\lIltall Teopllll TCp~IODHalmJlJraCTIl'lHOCTH l,pIICTaJlJIII'U'CHIIX Tl'.'t. B '1aCTIIOCTII. Moamo 1I0tmaaTL, 'ITO TI~lnlOil.lllta'IIlHa AJ/II "llfl3/WJI.'IaCTll'leCHI/X" M:lTeplla:/01l \lOmeT OhlTL 1I0CTpoella B pa\l1;ax conpe\lel/lloii \ICXaIlIlHI/ ell:JOlllllb/X cpeA il"n 'laTepl/a:101l e lIac.'lC;lCTBeH- I/OCThlO. OClloDHhle 1~lIall'leCHlle 1I0/lllTIIJI D1dnO;lRTCIl lIa paCC\JOTI)l'III1J1 nonCi\I'/I111l ;J.IIC:JOlia\lllft U t,pIICTa:I:III'ICC/iIlX TCJlax. 113Y'lCllhI naatOIOaaUIIC/OIOCTII IIpe;J..'laraI'MOrO 1I0ilXIlil