Variational principles in dissipative electro‐magneto‐mechanics: A framework for the macro‐modeling of functional materials

Abstract

SUMMARYThis paper presents a general framework for the macroscopic, continuum-based formulation and numerical implementation of dissipative functional materials with electro-magneto-mechanical couplings based on incremental variational principles. We focus on quasi-static problems, where mechanical inertia effects and time-dependent electro-magnetic couplings are a priori neglected and a time-dependence enters the formulation only through a possible rate-dependent dissipative material response. The underlying variational structure of non-reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so-called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum-based modeling in macroscopic electro-magnetomechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle-point principles. Next, the implications on the formulation of boundary-value problems are addressed, which appear in energy-based formulations as minimization principles and in enthalpy-based formulations in the form of saddle-point principles. Furthermore, the material stability of dissipative electro-magneto-mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro-magneto-mechanics, an energy-based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy-based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials.

Constitutive modeling of nonlinear reversible and irreversible ferromagnetic behaviors and application to multiferroic composites

“…Phenomenological constitutive models, adapting multiple empirical parameters to experimental findings, have been developed with and without internal variables, in order to describe irreversible (Carman and Mitrovic, 1995;Kiefer and Lagoudas, 2004;Linnemann et al, 2009;Miehe et al, 2011aMiehe et al, , 2011bXu et al, 2013) or reversible ferromagnetic and magnetostrictive behavior. Concerning models for the nonlinear reversible behavior, being one part of this article, the simplest extension of linearity is given by the standard square model (Carman and Mitrovic, 1995;Wan et al, 2003;Wan and Zhong, 2004).…”

Measuring the magnetoelectric effect across scales

“…Thus, the indirect path through the left bottom corner in Figure 1 must be taken. In simple cases like 2-2-connectivity, this can be straight forwardly dealt with in modeling [121] (see Appendix), for the 0-3-case it becomes a highly difficult task [51,55,[132][133][134]. Irrespective of these modeling limitations, experiment can always define an effective coupling coefficient of the sample/device.…”

“…In this case, α is typically calculated directly from some finite element structure [55]. It has also been effective to embed the magnetostrictive and piezoelectric materials in an insulating polymer matrix for achieving converging models [51,132].…”

A Geometrically Exact Incremental Variational Formulation for Phase Field Models in Micromagnetics

“…Let Ω ⊂ R d denote a vacuum free space box with dimension d ∈ [2,3] and B ⊂ Ω the domain occupied by the material solid, as depicted in Figure 1 . Ω is considered to be large enough such that the magnetic field induced by the magnetization of the body B is decayed at its surface ∂Ω ⊂ R d−1 .…”

“…The time evolution of the magnetization is governed by the Landau-Lifschitz-Gilbert equation. We propose a new incremental variational principle similar to [2] and inspired by [3], that is consistent with the inherent geometric nature of the problem. …”