A generalized Cahn–Hilliard equation incorporating geometrically linear elasticity

Abstract

We consider a generalisation of the Cahn-Hilliard equation that incorporates an elastic energy density which, being quasiconvex, incorporates microstructure formation on smaller length scales. We prove global existence of weak solutions in certain microstructural regimes in (one and) two dimensions and present sufficient conditions for uniqueness. Preliminary numerical computations to illustrate some characteristic properties of the solutions are presented and compared to earlier Cahn-Hilliard models with elasticity.

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

“…Theorem 1 (Existence of weak solutions). Let the mobility tensor M be positive definite and continuous for all a, b satisfying (10), W fulfill (A), ψ be given by (7) and the initial data (a 0 , b 0 ) satisfy (10) and (3.16). Then there exists a weak solution (a, b, u) to (1)…”

Cahn–Hilliard equations incorporating elasticity: analysis and comparison to experiments

“…This question arises because the interfacial energy in our model, being on the larger length scale, is associated not with the physical interfaces in the morphology but with variations in the homogenized volume fraction. Preliminary computational investigations in [6] appear to show that, yes, they do. It is intriguing to ask whether an analysis can rigorously show this.…”

“…The two-dimensional Cahn-Larché system was studied in [6]. The main result was an existence theorem.…”

“…Based on lemma 3.6, the proof of existence of weak solutions with the stated regularity properties follows the key steps of the proof of the two-dimensional version of theorem 3.2 in [6].…”

“…In an effort to advance the mathematical understanding of coarsening, in [6] we put forth a model that aims to account for both interfacial and elastic energies but nevertheless holds forth hope of being analytically tractable. It achieves this by incorporating these energies on different scales.…”