On the Allen-Cahn/Cahn-Hilliard system with a geometrically linear elastic energy

TL;DR

This paper extends the Allen-Cahn/Cahn-Hilliard system to include geometrically linear elastic energy, proving existence of solutions and providing explicit relaxed energy formulas for specific dimensions.

Contribution

It introduces a new elastic energy model within the Allen-Cahn/Cahn-Hilliard framework, including existence proofs and explicit formulas for relaxed energies in certain dimensions.

Findings

Existence of weak solutions for the new model.

Explicit formulas for relaxed energy functionals in D=1 and D=3.

Uniqueness of solutions in scalar-valued deformation cases.

Abstract

We present an extension of the Allen-Cahn/Cahn-Hilliard system which incorporates a geometrically linear ansatz for the elastic energy of the precipitates. The model contains both the elastic Allen-Cahn system and the elastic Cahn-Hilliard system as special cases and accounts for the microstructures on the microscopic scale. We prove the existence of weak solutions to the new model for a general class of energy functionals. We then give several examples of functionals that belong to this class. This includes the energy of geometrically linear elastic materials for D<3. Moreover we show this for D=3 in the setting of scalar-valued deformations, which corresponds to the case of anti-plane shear. All this is based on explicit formulas for relaxed energy functionals newly derived in this article for D=1 and D=3. In these cases we can also prove uniqueness of the weak solutions.