Approximation results by difference schemes of fracture energies: The vectorial case
Let \(\Omega\) be a bounded open subset of \({\mathbb R}^3\) with Lipschitz boundary, \(N\in{\mathbb N}\), and let \(SBV(\Omega;{\mathbb R}^N)\) be the space of the special functions with bounded variations defined in \(\Omega\) and with values in \({\mathbb R}^N\). For every \(u\in SBV(\Omega;{\mathbb R}^N)\), \(J_u\) denotes the set of the jump points of \(u\), \(u^+\) and \(u^-\) the approximate upper and lower limits of \(u\), and \(\nu_u\) the normal unit vector normal to \(J_u\). Let \(\psi: {\mathbb R}^{N\times 3}\to[0,+\infty[\) be a quasiconvex function satisfying \(C_1| z| ^p\leq\psi(z)\leq C_2(1+| z| ^p)\) for every \(z\in{\mathbb R}^{N\times 3}\) and some \(C_1, C_2>0\) and \(p>1\). Let \(g: {\mathbb R}^N\times S^1\to[0,+\infty[\) satisfy standard symmetry, subadditivity, and lower semicontinuity assumptions. The paper provides a variational approximation, in the sense of De Giorgi's \(\Gamma\)-convergence, by finite-difference schemes of the brittle fracture energy \[ u\in SBV(\Omega;{\mathbb R}^N)\mapsto\int_\Omega\psi(\nabla u)dx+\int_{J_u}g(u^+-u^-,\nu_u)d{\mathcal H}^2, \] by means of functionals of the type \[ u\mapsto\int_{{\mathcal T}_\varepsilon\cap\Omega}\psi_\varepsilon(\nabla u)dx, \] where \(\psi_\varepsilon\) is a non-convex interaction potential taking into account a separation of scales, \({\mathcal T}_\varepsilon\) is a suitable regular triangulation of \({\mathbb R}^3\), and \(u\) is affine on each element of the assigned triangulation.
scientific article; zbMATH DE number 2066888