Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity

Abstract

This paper concerns finite elasto-plasticity of crystalline materials with micro-structural defects. We revisit the basic concepts: plastic distortion and decomposition of the plastic connection. The body is endowed with a structure of differential manifold. The plastic distortion is an incompatible diffeomorphism. The metric induced by the plastic distortion on the intermediate configuration (considered to be a differential manifold) is a key point in the theory, in defining the defects related to point defects, or extra-matter. The so-called plastic connection is metric, with plastic metric tensor expressed in terms of the plastic distortion and its adjoint. We prove an appropriate decomposition of the plastic connection, without any supposition concerning the non-metricity of plastic connection. All types of the lattice defects, dislocations, disclinations, and point defects are described in terms of the densities related to the elements that characterize the decomposition theorem for plastic connection. As a novelty, the measure of the interplay of the possible lattice defects is introduced via the Cartan torsion tensor. To justify the given definitions, the proposed measures of defects are compared to their counterparts corresponding to a classical framework of continuum mechanics. Thus, their physical meanings can be emphasized at once.

Applications of vector bundle to finite elasto-plasticity

“…We mention now that only the geometrical background is discussed here. When an axiomatic characterization of crystalline materials is done, the definition of the plastic and elastic distortions has to be introduced simultaneously with constitutive and evolution equations (see, for instance, Cleja-Ţigoiu [10, 24]).…”

“…Further on, we propose to reappraise the theorem concerning the decomposition of the plastic connection, proved in Cleja-Ţigoiu [10]. Based on the definitions concerning the connection on vector bundles, all types of the lattice defects, dislocations, disclinations and point defects ought to be described in terms of the densities related to the elements, which characterize the decomposition theorem for plastic connection, following Cleja-Ţigoiu [23, 35].…”

“…The definition of the plastic distortion is the very key point of the theory. Many authors (see Le and Stumpf [8], Clayton et al [9], Cleja-Ţigoiu [10]) assume the existence of intermediate (time-dependent) configurations, denoted now by B int , and it is taken as differential manifolds. Thus, the plastic distortion is an anholonomic diffeomorphism acting on the tangent space of the body in its reference configuration with the values in the tangent space over the intermediate configuration.…”

“…In developing constitutive models in elasto-plasticity, the basic hypothesis concerns the existence of intermediate (time-dependent) configurations, which are not viewed solely as geometric structure, being related to the multiplicative decomposition of the deformation gradient. In Cleja-Ţigoiu [10], we assume:…”

“…Just this incompatibility or the non-integrability of the elastic and plastic distortions is associated with the presence of defects, such as dislocations. For instance, Noll’s measure of dislocation, ( F p ) − 1 curl F p , (where curl F p ≠ 0 is a measure of the non-integrability) and its effects on the multiple flow rule in crystal plasticity can be found within a non-local framework, for instance, in Cleja-Ţigoiu [10].…”