Calculus of Variations and Geometric Measure Theory

A. Braides - A. Piatnitski

Overall properties of a discrete membrane with randomly distributed defects

created on 20 May 2004
modified by braidesa on 14 Jun 2008


Published Paper

Inserted: 20 may 2004
Last Updated: 14 jun 2008

Journal: Arch. Ration. Anal. Mech.
Volume: 189
Pages: 301-323
Year: 2008


We consider a 2D nearest-neighbour square lattice system with energy densities being either `weak springs' as in Chambolle's treatment of Blake and Zisserman's weak membrane, or `strong springs' (simple quadratic interactions) with probability $p$ and $1- p$, respectively. We prove that for $p<p_c$ (the percolation threshold) the effect od the weak springs is negligible and the overall energy is expressed by the Dirichlet integral, while for $p>p_c$ the effective energy is a Griffith brittle fracture type energy. We use the notation and techniques of $\Gamma$-convergence and $SBV$ function spaces to express this result.

Keywords: Homogenization, discrete systems, percolation theory, Gamma-limits