Incompatible Sets of Gradients and Metastability

Abstract

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an L 1 neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.

Effective Behavior of Nematic Elastomer Membranes

“…This set corresponds to the point D in Figure 2a. Therefore, by Proposition 5.6, we haveF ∈K (1) . Therefore, arguing as before, W rc 2D (F ) ≤ W mem (F ).…”

“…This set corresponds to the point B in Figure 2a. By Proposition 5.6, we haveF ∈K (1) . Therefore, there exists λ ∈ [0, 1] andG 1 ,G 2 ∈K with rank(G 1 −G 2 ) ≤ 1 such that…”

“…This corresponds to the point P in Figure 2a These sets correspond to the points Q and Z in Figure 2-(a) respectively. From Proposition 5.6, F ∈K (1) . Further, again by Proposition 5.6,K ⊂ K (1) .…”

“…Note thatG ± ∈ K, θ ∈ [0, 1] sinceδ ≤ d, rank (G + −G − ) = 1 andG = θG + +(1−θ)G − . Therefore, M w ⊂ K (1) . The proof of M s ⊂ K (1) is similar (also see [14, Theorem 3.1]).…”

“…Therefore, M w ⊂ K (1) . The proof of M s ⊂ K (1) is similar (also see [14, Theorem 3.1]). Again, using the polar decomposition theorem, we can takeG ∈ M s as a diagonal matrix.…”