Deformations and stresses with and without microslip

Kinematical definitions of deformations with and without microslip are presented. Transformation properties for such deformations are shown to follow directly from their definitions, and Burgers vector is related to the deformation without microslip. A limit procedure provides a concept of stress without microslip and leads to a natural concept of elastic response. Various decompositions of local deformation into elastic and plastic parts proposed in the literature are shown to be compatible with this kinematical setting. INTRODUCTION One of the principal methods for incorporating the inelastic behavior of metals into continuum models of materials is to employ classical notions of deformation and stress in describing the kinematics and dynamics of an inelastic body, but to introduce additional measures of deformation, usually called elastic and plastic deformation, into the constitutive equations for the material under consideration. Although generally successful, this approach has drawbacks that merit consideration: 1) The variety of possible choices of elastic and plastic deformation and the difficulty in comparing models based on different choices has led to a lingering controversy in the literature; 2) the underlying classical kinematics does not directly incorporate the physical processes of slip at the microscopic level. My goal in this paper is to present a different method for incorporating the inelastic behavior of metals into a continuum model that is not subject to these drawbacks. This utilizes the research still in progress of Del Piero and Owen (forthcoming) on the kinematics of fractured continua in which classes of deformations broad enough to include explicitly slip at the microscopic level are defined and developed. In the second and third sections I describe a collection of deformations called invertible structured deformations that includes classical deformations, and I show how the results of Del Piero and Owen (forthcoming) lead to natural notions of deformation without microslip and deformation due to microslip. All of the considerations in the second and third sections are purely kinematical, so that these new notions of deformation are not variables that appear for the first time in constitutive equations. Among the kinematical properties of invertible structured deformations summarized in the second section is the Approximation Theorem, which shows that each invertible structured deformation is a limit of "piecewise classical deformations", or "simple deformations". It is this property of invertible structured deformations that permits one to interpret their effect on a body in terms of complicated slip mechanisms occurring in the approximating simple deformations. The other results in the second section motivate and justify not only multiplicative, but also additive decompositions of local deformation into parts without microslip and parts due to microslip. All of the terms and factors in these decompositions have definite transformation properties under changes in observer and reference configuration that are direct consequences of the kinematical definitions. In the fourth section, new results are given that lead to a notion of stress without microslip for an invertible structured deformation. This stress takes into account differences between a given element of surface area in the deformed configuration and a corresponding element of surface generated by an approximation to the given invertible structured deformation. A passage to the limit yields a simple formula for the stress without microslip in terms of the Cauchy stress and the left microslip tensor introduced in the third section. The discussion in the fifth section shows how the availability of both deformation without microslip and stress without microslip provides a natural concept of elastic response for a body undergoing invertible structured deformations: the stress without microslip is a function of deformation without microslip. If the body undergoes a classical deformation and, in particular, undergoes no microslip, then the elastic response reduces to the classical relation: the Cauchy stress is a function of the deformation gradient. The relation between stress without microslip and deformation without microslip is equivalent to an equation that gives the Cauchy stress as a function of deformation without microslip and of deformation due to microslip in which the dependence on deformation due to microslip enters through a multiplicative factor (the transpose of the left microslip tensor). The final section of the paper is devoted to showing how various decompositions of deformation into elastic and plastic parts that have been proposed in the literature (Lee and Liu, 1967), (Clifton, 1972), (Nemat-Nasser, 1979), (Green and Naghdi, 1965) fit naturally into the present framework of invertible structed deformations. If for each proposed decomposition the elastic deformation in that decomposition is identified as deformation without microslip, then the plastic deformation in that decomposition is easily related to one of the measures of deformation due to microslip introduced in the third section, and the transformation properties of both elastic and plastic deformation immediately follow. Thus, in the setting of invertible structured deformations, all the decompositions considered in the last section are consistent with one another: none has any special status, and each one identifies a particular measure of deformation due to microslip in the present theory. The present approach has some connections with theories of continuous distributions of dislocations and theories of defective crystals. The emergence in a natural way of Burgers vector, as shown in relation (10), shows that important physical ideas incorporated into theories of continuous distribution of dislocations (Kroner, 1958) have counterparts in the kinematics of fractured continua (Del Piero and Owen, forthcoming). Within the class of invertible structured deformations described here one can identify a smaller collection of deformations, the neutral deformations discussed by (Davini and Parry, 1991) (Fonseca and Parry, forthcoming): these correspond in the present theory to the invertible structured deformations whose right microslip tensor is the gradient of a vector field. INVERTIBLE STRUCTURED DEFORMATIONS Let a region ji in space represent a reference configuration for a body. An invertible structured deformation from <A is specified by giving a deformation g and a tensor field G. The fields g and G are required to satisfy