Length scale effects in finite strain micromorphic linear isotropic elasticity: finite element analysis of three-dimensional cubical microindentation

Abstract

A three-dimensional finite strain micromorphic materially linear isotropic elastic model is formulated in two ways for finite element implementation: (i) direct finite strain elasticity; and (ii) rate form with semi-implicit time integration. The model is based upon the finite strain isotropic micromorphic elasticity model proposed by Eringen and Suhubi in 1964. For (i), the direct formulation, the constitutive equations are calculated in the reference configuration, and the resulting stresses are mapped to the current configuration. For (ii), the rate formulation, the constitutive equations are integrated in time in the current configuration using the Truesdell objective stress rates. After formulating the coupled weak form, the balance of linear momentum and the balance of first moment of momentum are linearized to construct the consistent tangent for solution by the Newton–Raphson method. Three-dimensional numerical examples are analysed to compare the two formulations (i) and (ii) for standard finite strain isotropic elasticity, and the formulation (i) is used to demonstrate the elastic length scale effects that come through the higher-order couple stress in the micromorphic theory.