Non-local elasto-viscoplastic models with dislocations in finite elasto-plasticity. Part I: Constitutive framework

Abstract

In the first part of the paper we elaborate on a theoretical framework for elasto-plastic materials, based on the decomposition of the second-order deformation, associated with the motion, into its elastic and plastic second-order components, and applied to solids with lattice defects at the micro-structure level. Only the dislocations have been considered as possible lattice defects, which are modelled through the non-zero torsion of the plastic connection as a tensorial measure of the dislocations and through the scalar dislocation density and its gradient. The non-local evolution equations for the scalar dislocation density as well as for the plastic distortion have been derived to be compatible with the principle of the imbalance of the free energy. A coupling term dependent on the gradient of the scalar dislocation density influences the evolution of the plastic distortion. The internal power included the power dissipated by micro-forces, which consist from micro-stress and micro-momenta satisfying their own balance equations and being related not only with the plastic distortion but also with the scalar dislocation density. The free energy is defined to be dependent on the elastic strain and on the second-order plastic deformation, as well as on the scalar dislocation density and its gradient. Different constitutive non-local dissipative models have been derived and compared with the already existing ones.

Disclinations and GND tensor effects on the multislip flow rule in crystal plasticity

“…For this definition, we refer to Cleja-Ţigoiu [27]; see also Appendix C. Remark 2. The fields introduced by (6) define the incompatibilities of the mathematical elements, which characterize the existing defects inside the materials.…”

“…Various extensions of classical models of crystalline plasticity concern: extensions of Schmid law in order to describe the evolution of plastic distortion [25, 26], the presence of tensorial variables, such as the GND tensor [23, 27] and non-local effects [28, 29].…”

“…The so-called bend-twist, a second-order tensor field that measures the incompatibility of small elastic rotation (see de Wit [6] and Fressengeas et al [7]), or different fields that generate the non-zero curvature tensor (see also the comments by de Wit [8] and Clayton et al [32]) and that lead to a non-zero Frank vector, can be identified with the disclination density. Different tensorial and scalar measures for dislocations and disclinations have been proposed by de Wit [6, 8], Kondo and Yuki [10], Arsenlis and Parks [33], Clayton et al [32, 34], Steinmann [19], Cleja-Ţigoiu [18, 27], Cleja-Ţigoiu et al [36], and many others.…”

“…The appropriate local forms of the reformulated second laws are to be strongly dependent on the postulated expressions for the energy density function and for the internal expended power. The free energy function is expressed by an elastic strain energy (as in classical plasticity), to which is added a defect energy , which depends on different measure of the defects; see Gurtin [24, 31], Teodosiu [19], Cleja-Ţigoiu [27] and Cleja-Ţigoiu and Paşcan [29].…”

“…Proposition 1. The virtual power dissipated by driving forces related to the plastic mechanism during the virtual plastic process associated with the multislip flow rule (equation (27)) is given by…”