A Thermomechanical Eulerian Formulation of a Size-Dependent Elastic-Inelastic Cosserat Continuum

Abstract

A thermodynamically consistent theory for finite deformation size-dependent elastic-inelastic response of a Cosserat material with a deformable director triad \({\mathbf{d}}_{i}\) and a single absolute temperature \(\theta \) has been developed by the direct approach. A unique feature of the proposed theory is the Eulerian formulation of constitutive equations, which do not depend on arbitrariness of reference or intermediate configurations or definitions of total and plastic deformation measures. Inelasticity is modeled by an inelastic rate tensor in evolution equations for microstructural vectors. These microstructural vectors model elastic deformations and orientation changes of material anisotropy. General hyperelastic anisotropic constitutive equations are proposed with simple forms in terms of derivatives of the Helmholtz free energy, which depends on elastic deformation variables that include elastic deformations of the directors relative to the microstructural vectors. An important feature of the model is that the gradients of the elastic director deformations in the balances of director momentum control size dependence and are active for all loadings. Analytical solutions of the small deformation equations for simple shear are obtained for elastic response and strain-controlled cyclic loading of an elastic-viscoplastic material.