Three-dimensional finite element analysis of finite deformation micromorphic linear isotropic elasticity

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

“…2): 27 nodes for u h and 8 vertex nodes for F h (Q27P8), where h is the discretization parameter. For the rate form (ii) of the model, (19) and (20) are written in Galerkin form [23], and the time-integrated stress equations (14) to (16) are used. The mixed formulation is employed in recognition that F h is a microdisplacement tensor (no gradient is calculated on a micro-displacement u 0h , but F h is analogous to ∂u h =∂X when com-paring F h and x h ), where the gradient of a quadratic interpolation is linear, such that $u h and F h would be of approximate same order interpolation.…”

“…The weak form and finite element formulation used to implement the direct elasticity model (i) is presented in [20]. The finite element implementation of the rate form (ii) with semi-implicit time integration is summarized in this section, since it is new.…”

“…2): 27 nodes for u h and 8 vertex nodes for F h (Q27P8), where h is the discretization parameter. For the rate form (ii) of the model, (19) and (20) are written in Galerkin form [23], and the time-integrated stress equations (14) to (16) (21) where j = ½j h z are the natural coordinates, N u a (j) the tri-quadratic interpolation functions, N F b (j) the tri-linear interpolation functions, d e k(a) the displacement vector at node a of element e, and f e kK (b) the micro-displacement tensor at node b in element e. In the finite element implementation, the terms not linearized in (19) and (20) will yield the residual vectors R d and R f respectively. The terms involving du and dF in (19) result in the associated consistent tangent matrices K dd and K df , respectively.…”

“…Aside from Eringen and co-workers [3], limited research efforts have formulated, let alone conducted, a finite element implementation for micromorphic elasticity or elasto-plasticity models at finite strain [5, 6, 13-18, 19, 20]. To the best of the authors' knowledge, based on the formulations in [5,6] for a multiplicative decomposition of the deformation gradient F and micro-deformation tensor x into elastic and plastic parts, none have attempted a full Total Lagrangian 3D finite element implementation [20]. This paper illustrates this implementation for two formulations (i) and (ii), and an application to cubical microindentation, for the finite strain micromorphic linear isotropic elastic model, which is the precursor to future 3D finite element implementations of the finite strain elasto-plasticity models presented in [5,6].…”

“…An outline of the remainder of the paper is as follows: section 2 summarizes the kinematics of a micromorphic continuum, section 3 summarizes the constitutive equations for each formulation (i) and (ii). The weak form, summary of linearization, and finite element equations for solution by a Newton-Raphson algorithm are given in section 4 for formulation (ii) (FE implementation for formulation (i) is presented in [20]), a one-dimensional (1D) uniaxial strain formulation is presented in section 5, and numerical results and conclusions are presented in sections 6 and 7, respectively. Figure 1 illustrates the mapping of the macroelement and micro-element in the reference configuration B 0 to the current configuration B through the deformation gradient F and microdeformation tensor x.…”