Modeling dislocations and disclinations with finite micropolar elastoplasticity

Abstract

Aspects of a constitutive model for characterizing crystalline metals containing a distribution of dislocation and disclination defects are presented. Kinematics, balance laws, and general kinetic relations are developed – from the perspective of multiscale volume averaging – upon examination of a deforming crystalline element containing a distribution of displacement discontinuities in the form of translational and rotational lattice defects, i.e., dislocations and disclinations. The macroscopic kinematic description is characterized by a three-term multiplicative decomposition of the deformation gradient. The micro-level description follows from an additive decomposition of an affine connection into contributions from populations of dislocations and disclinations to the distortion of the lattice directors. Standard balance equations apply at the macroscopic scale, while momentum balances reminiscent of those encountered in micropolar elasticity (i.e., couple stress theory) are imposed at the micro-level on first and second order moment stresses associated with geometrically necessary defects. Thermodynamic restrictions are presented, and general features of kinetic relations are postulated for time rates of inelastic deformations and internal variables. Micropolar rotations are incorporated to capture physics that geometrically necessary dislocations stemming from first order gradients of elastic or plastic parts of the total deformation gradient may alone be unable to reflect, including evolution of defect substructure at multiple length scales and incompatible lattice misorientation gradients arising in ductile single crystals subjected to nominally homogeneous deformation.