Generalized gradient flow structure of internal energy driven phase field systems

TL;DR

This paper develops a unified variational framework for thermomechanical phase field systems, revealing a gradient flow structure driven by internal energy, and proves existence and uniqueness of solutions.

Contribution

It introduces a generalized abstract formulation that captures the thermomechanical coupling via a gradient flow driven by internal energy.

Findings

Established a gradient flow structure for the PDE system.

Proved global existence of weak solutions.

Demonstrated uniqueness under smoothness assumptions.

Abstract

In this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals.