Measure-valued solutions for models of ferroelectric material behavior

TL;DR

This paper introduces measure-valued solutions for ferroelectric material models, proving their existence under various conditions, including rate-dependent and rate-independent cases, with and without coercivity assumptions.

Contribution

It defines measure-valued solutions for ferroelectric models and establishes their existence in both rate-dependent and rate-independent scenarios, even without coercivity.

Findings

Existence of measure-valued solutions in rate-dependent case with coercivity.

Existence of solutions in rate-independent case with regularized energy functional.

Framework applicable to nonlinear ferroelectric behavior modeling.

Abstract

In this work we study the solvability of the initial boundary value problems, which model a quasi-static nonlinear behavior of ferroelectric materials. Similar to the metal plasticity the energy functional of a ferroelectric material can be additively decomposed into reversible and remanent parts. The remanent part associated with the remanent state of the material is assumed to be a convex non-quadratic function $f$ of internal variables. In this work we introduce the notion of the measure-valued solutions for the ferroelectric models and show their existence in the rate-dependent case assuming the coercivity of the function $f$. Regularizing the energy functional by a quadratic positive definite term, which can be viewed as hardening, we show the existence of measure-valued solutions for the rate-independent and rate-dependent problems avoiding the coercivity assumption on $f$.