Material growth and remodeling formulation based on the finite couple stress theory

Abstract

The mathematical formulation for material growth and remodeling processes in finite deformation is developed based on the couple stress theory. The generalized continuum mechanics of couple stress theory is capable of capturing small-scale cellular effects and of modeling mass flux in these processes. The frame-indifferent balance equations of mass, linear and angular momentums, as well as internal energy together with the entropy inequality are first introduced in the presence of the mass flux based on the finite couple-stress theory. Then, within the framework of material uniformity the Eshelby and Mandel stress tensors as driving or configurational forces for local rearrangement of the first- and second-order material inhomogeneities are determined for the Cauchy stress tensor as well as the couple stress tensor. In the next step, the basic kinematic tensors are multiplicatively decomposed into elastic and anelastic parts, and by utilizing the derived entropy inequality, the hyper-elastic constitutive equations with respect to both reference and current configurations are obtained. Additionally, an admissible form for each of the two evolution laws of classical and higher-order material transplant tensors of material growth which satisfy the general formal restrictions are developed as a function of classical and hyper versions of the Mandel stress. Moreover, in a numerical study the effects of presented evolution laws on the growth of a cubic materially isotropic object under a specific oscillating external loading, corresponding to some diagonal classical stress and skew-symmetric couple stress tensors in the reference configuration, are investigated.