A three-field based finite element analysis for a class of magnetoelastic materials

Abstract

A simple yet effective material model was proposed by Zhao et al. (2019) and demonstrated to be capable of modeling the shape transformations of various planar and three-dimensional material samples programmed with the so-called “hard-magnetic soft materials”. Based on the aforementioned material model, this paper aims to further accomplish the following two tasks. First, a detailed analysis is performed to investigate the impact of the multiplicative volumetric-distortional split of the deformation gradient tensor applied to the material magnetic energy. Through a trivial boundary value problem, the impact of the volumetric-distortional split is quantified for the strictly incompressible material and the nearly incompressible material, respectively. Second, a finite element procedure based on the three-field variational principle, or the mixed displacement-Jacobian-pressure formulation (Simo et al., 1985; Simo and Taylor, 1991), is developed for the magnetoelastic materials programmed with complex magnetic patterns. Even though the finite element formulation based on the three-field variational principle is a standard and widely adopted technique in the literature, the introduction of the multiplicative split of the deformation gradient into the magnetic energy makes the derivation of the specific finite element terms less trivial. In this work, the finite element formulation and the consistent linearization of the coupled system are derived in detail for the Newton–Raphson iterations. Through the theoretical analysis and numerical examples, the approach based on the distortional part of the deformation gradient is shown to possess computational advantages over its counterpart in the context of a relatively simple penalty method. Moreover, the convergence behaviors of the finite element simulations and the impact of the finite element spaces on the pressure oscillation are analyzed in detail.