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

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.

Well-posedness of solutions to a phase-field model for the martensitic phase transformations

“…For the investigation of phase-filed model, phase-field method is widely used, and it has been regarded as one of the most significant computing methods [6][7][8][9]. The spatial and temporal evolution of the variables is governed by the Cahn-Hilliard equation and/or the Allen-Cahn equation [10][11]. The two well-known equations can describe the ordering of atoms during the phase separation, in which the order parameter is conserved or not conserved.…”

Global existence of weak solution to a phase‐field model on martensitic phase transformations

“…The Cahn-Hilliard/Allen-Cahn equations are the two well-known models for temporal evolution of microstructures, respectively, to the cases that the order parameter is conserved and not conserved. [7][8][9][10] These two kinds of order parameters are also presented in the articles, [9][10][11][12] and the property of solutions to the parabolic problems are investigated in the phase-field models. [13][14][15][16] In this article, we investigate a phase-field model of transformations between martensitic variants and multiple twinning within martensitic variants, which is developed for lattice rotations and large strains.…”

“…In these phase‐field models dealing with interface problems, the phase‐field approach is widely utilized. The Cahn–Hilliard/Allen–Cahn equations are the two well‐known models for temporal evolution of microstructures, respectively, to the cases that the order parameter is conserved and not conserved 7–10 . These two kinds of order parameters are also presented in the articles, 9–12 and the property of solutions to the parabolic problems are investigated in the phase‐field models 13–16 …”