A Unified Theory of Thermoviscoplasticity of Crystalline Solids

/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 dislocation 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 thermodynamics 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 constitutive 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' phenomena 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 dislocations. 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

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