The effect of a surface energy term on the distribution of phases in an elastic medium with a two-well elastic potential

✍ Scribed by Michael Bildhauer; Martin Fuchs; Victor Osmolovskii

• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 247 KB
• Volume: 25
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.282

✦ Synopsis

Abstract

We consider the problem of minimizing

among functions u:ℝ^d^⊃Ω→ℝ^d^, u~∣∂Ω~=0, and measurable subsets E of Ω. Here f~h~^+^, f^−^ denote quadratic potentials defined on Ω¯×{symmetric d×d matrices}, h is the minimum energy of f~h~^+^ and ε(u) is the symmetric gradient of the displacement field u. An equilibrium state û, Ê of J(u,E) is called one‐phase if E=∅︁ or E=Ω, two‐phase otherwise. For two‐phase states, σ∣__∂__E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters __h__ϵℝ, σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J(u,E) in the limit σ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.