2021/02/19 by Annika Bach, Bach, Annika, Roberta Marziani +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2102.09872
We study the limit behaviour of singularly-perturbed elliptic functionals of the form \mathcal Fk(u,v)=∫A v2 fk(x,∇ u)dx+(1)/(εk)∫A gk(x,v,εk∇ v)dx , where u is a vector-valued Sobolev function, v ∈ [0,1] a phase-field variable, and εk>0 a singular-perturbation parameter, i.e., εk → 0, as k→ +∞. Under mild assumptions on the integrands fk and gk, we show that if fk grows superlinearly in the gradient-variable, then the functionals \mathcal Fk Γ-converge (up to subsequences) to a brittle energy-functional, i.e., to a free-discontinuity functional whose surface integrand does not depend on the jump-amplitude of u. This result is achieved by providing explicit asymptotic formulas for the bulk and surface integrands which show, in particular, that volume and surface term in \mathcal Fk decouple in the limit. The abstract Γ-convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.