Fully discrete approximation schemes for rate-independent crack evolution

Abstract

Over the past two decades, several distinct solution concepts for rate-independent evolutionary systems driven by non-convex energies have been suggested in an attempt to model properly jump discontinuities in time. Much attention has been paid in this context to the modelling of crack propagation. This paper studies two fully discrete (in time and space) approximation schemes for the rate-independent evolution of a single crack in a two-dimensional linear elastic material. The crack path is assumed to be known in advance, and the evolution of the crack tip along it relies on the Griffith theory. On the time-discrete level, the first scheme is based on local minimization, whereas the second scheme is a regularized version of the first one. The crucial feature of the schemes is their adaptive time-stepping nature, with finer time steps at those points where the evolution of the crack tip might develop a discontinuity. The set of discretization parameters includes the mesh size, crack increment, locality parameter and regularization parameter. In both cases, we explore the interplay between the discretization parameters and derive sufficient conditions on them ensuring the convergence of discrete interpolants to parametrized balanced viscosity solutions of the continuous model. To illustrate the performance of the approximation schemes, we support our theoretical analysis with numerical simulations. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle

“…Open questions in the contact mechanics concern non-smooth behaviour, e.g. owing to non-coercive [2,3] and non-convex functions [4], nonlinear constitutive equations [5][6][7], time-discontinuous evolution [8] and geometry singularities [9].…”

Non-smooth variational problems and applications

“…The cases of stochastic optimal control [ 7 ] and coefficient identification from measured data at the boundary [ 8 ] belong to the field of inverse problems, which are known to be ill-posed. Well-posedness analysis of the underlying problems provides further investigation with respect to issues of stability and long-time behaviour [ 9 ], rate-independent time evolution [ 10 ], asymptotic behaviour due to damage [ 2 ], variation of geometric and physical parameters, singular perturbations [ 11 ], multi-scale analysis and periodic homogenization [ 12 ]. The theoretical results are supported by construction of efficient computational techniques, finite and boundary element methods and numerical simulation[ 7 , 10 ].…”

“…The theme issue studies the unilateral contact appearing for inclusions subject to delamination with cracks [ 2 , 3 ], non-smooth slip [ 9 ], frictional contact [ 4 , 13 ] and degenerating Robin-type transmission conditions for the thin reactive heat-conducting inter-phases [ 11 ]. The rate-independent evolutionary systems driven by non-convex energies have been suggested [ 10 ], which are successful to model properly jump discontinuities in time during quasi-brittle crack propagation. The elaborated mathematical and mechanical description gives an impact to practise testing methodologies by rigid punch indentation [ 5 ], in fracture mechanics and seismology [ 13 ].…”