The Relaxation of Two-well Energies with Possibly Unequal Moduli

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

“…Then the passage from this length scale to the larger length scale is achieved by relaxing the elastic energy; this is the topic of §3d. The key point to note here is that by neglecting interfacial energy, we obtain a problem that is tractable, indeed that has been rigorously solved [53].…”

“…However, the most-explicit expression available forŴ is cumbersome to write down. Instead, here we present a less explicit expression and refer to [53] for details:Ŵ…”

A generalized Cahn–Hilliard equation incorporating geometrically linear elasticity

“…The (analytical) computation of W is non-trivial for dimensions larger than one. For two dimensions, an explicit (albeit involved) expression was derived in [18]. We reproduce it here; this requires first defining some intermediate quantities.…”

“…with α i , ε * i ∈ R. Partial results are available for three dimensions [18]. Explicit expressions are unknown (although arguably not necessary) for dimensions larger than three.…”

“…In the present article we shall address this problem by combining the Cahn-Hilliard model with recent results in [18] where the relaxation of two-well elastic energies with given volume fraction is studied. We restrict ourselves to the (one-and) two-dimensional case, as only partial relaxation results are available in three dimensions.…”