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A variational approach to the modeling of compressible magnetoelastic materials

Barbora Benešová, Šárka Nečasová, Jan Scherz, Anja Schlömerkemper

Abstract

We analyze a model of the evolution of a (solid) magnetoelastic material. More specifically, the model we consider describes the evolution of a compressible magnetoelastic material with a non-convex energy and coupled to a gradient flow equation for the magnetization in the quasi-static setting. The viscous dissipation considered in this model induces an extended material derivative in the magnetic force balance. We prove existence of weak solutions based on De Giorgi's minimizing movements scheme, which allows us to deal with the non-convex energy as well as the non-convex state space for the deformation. In the application of this method we rely on the fact that the magnetic force balance in the model can be expressed in terms of the same energy and dissipation potentials as the equation of motion, allowing us to model the functional for the discrete minimization problem based on these potentials.

A variational approach to the modeling of compressible magnetoelastic materials

Abstract

We analyze a model of the evolution of a (solid) magnetoelastic material. More specifically, the model we consider describes the evolution of a compressible magnetoelastic material with a non-convex energy and coupled to a gradient flow equation for the magnetization in the quasi-static setting. The viscous dissipation considered in this model induces an extended material derivative in the magnetic force balance. We prove existence of weak solutions based on De Giorgi's minimizing movements scheme, which allows us to deal with the non-convex energy as well as the non-convex state space for the deformation. In the application of this method we rely on the fact that the magnetic force balance in the model can be expressed in terms of the same energy and dissipation potentials as the equation of motion, allowing us to model the functional for the discrete minimization problem based on these potentials.

Paper Structure

This paper contains 8 sections, 12 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and notation
  3. Model
  4. Notation
  5. Solution approach
  6. Weak solutions and main result
  7. Proof of the main result
  8. Appendix

Key Result

Theorem 4.1

Let the assumptions of Definition weaksolutionsmagnetoelastic be satisfied. Assume further that the energy densities $\tilde{\Psi}$ and $W$ satisfy the conditions 3063--3064. Then there exists a time $T' > 0$ such that the system 2949--2947 admits a weak solution $(\eta, \tilde{M})$ on $[0,T')$ in t

Figures (1)

  • Figure 1: A deformation $\eta(t,\cdot)$ mapping a (magneto)elastic material from its reference configuration $\Omega_0$ to its current configuration $\eta (t,\Omega_0)$.

Theorems & Definitions (26)

  • Remark 2.1: Transport of the magnetization
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5: Extension to inertial problems
  • Remark 2.6: Landau-Lifschitz-Gilbert equation
  • Remark 2.7: Ferromagnetic and martensitc materials
  • Remark 2.8
  • Remark 3.1
  • Remark 3.2
  • ...and 16 more