A dynamical theory of structures solids. I Basic developments

Abstract

Based on the well-accepted notion of a Bravais lattice of a crystal at the atomic scale and with particular reference to inelastic behaviour of materials, this paper is concerned with the construction of a macroscopic dynamical theory of solids which incorporates the effect of the presence of the atoms and their arrangements. The theory incorporates a wide variety of microstructural processes occurring at various physical scales and has a range approaching the atomic scale. These processes include the effect of the motion of individual dislocations, which are modeled here as continuous distributions at the macroscopic scale. The formulation of the basic theory, apart from the kinematical and kinetical variables employed in classical continuum mechanics, utilizes a triad of independent vector-valued variables — called directors — (or an equivalent tensor-valued variable) which represent the lattice vectors and are determined by additional momentum -like balance laws associated with the rate of change of lattice deformation in the spirit of a Cosserat (or directed) continuum. A suitable composition of the triad of directors and the ordinary deformation gradient is identified as a measure of perm anent or plastic deformation, the referential gradient of which plays a significant role in the kinematics of lattice defects. In particular, a uniquely defined skew-symmetric part of the gradient of plastic deformation is identified as a measure of the density of dislocations in the crystal. The additional momentum -like balance laws associated with the rate of lattice deformation include the effect of forces necessary to maintain the motion of dislocations, as well as the inertia effects on the microscopic and submicroscopic scales arising from the director fields. The basic theoretical developments also provide important clarifications pertaining to the structure of the constitutive response functions for both viscoplasticity and (the more usual) rate independent plasticity.