Quasistatic crack growth in 2d-linearized elasticity

Abstract

In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation (GSBD). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the GSBD setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

“…However, our primary purpose comes from the study of free-discontinuity problems defined on the space GSBD p , see [38], which has obtained steadily increasing attention over the last years, cf., e.g., [30,31,32,33,34,35,46,47,48,50]. We have indeed already mentioned before how the analysis of partition problems has proved to be a relevant tool in the study of freediscontinuity problems on SBV .…”

“…To this aim, an additional tool is required when dealing with the case L = R d×d skew , namely a careful decomposition of sets (Lemma 7.4) for which an additional rest set R aux has to be introduced. Our construction is inspired by similar techniques used in [49,Theorem 3.2] and [50,Theorem 4.1]. While Lemma 7.1 can be proved directly in the case L = SO(d), so that a reader only interested in this case can now already skip to its proof, we need two auxiliary lemmas to deal with the case L = R d×d skew .…”

Existence of strong solutions to the Dirichlet problem for the Griffith energy

“…As a concluding remark, we observe that in dimension 2, the authors of [7] prove the existence of a strong quasistatic evolution, namely minimising the antiplane version of (G) with respect to its own (closed) jump set at any time t. The starting point is therein the existence result [29], that has been recently extended to planar elasticity by Friedrich and Solombrino in [34] in dimension 2. In the present context it is immediate to combine our density lower bound with the geometrical 2d argument in [7,Proposition 5.5] to get that the sequence of piecewise-constant in time evolutions u k in the Francfort-Marigo approach (obtaining by dividing the given time interval [0, T ] by k+1 nodes t i k = i T k and interpolating in time the solutions to the incremental minimum problems in the nodes) satisfies a density lower bound uniform in time and in k. However, the improvement of the evolution in [34] seems delicate. Indeed, the tool of σ p -convergence, developed in [22] and crucial in [7], is not directly applicable now, as we do not work in SBV p .…”

A compactness result in $$GSBV^p$$ and applications to $$\varGamma $$-convergence for free discontinuity problems

“…The recent paper [35] provides the first compactness and existence result in GSBD for the Griffith energy in dimension two without any a priori bounds or fidelity terms. A related result [32] has been obtained in the passage from nonlinear-to-linear energies in brittle fracture by means of Γ -convergence (see also [34] for a discrete-to-continuum analysis).…”

“…To prove the main compactness result, we follow the strategy devised in [32,35]: given a sequence of functions, we pass to suitable modifications whose energies coincide with the original ones up to an error of order θ. Subsequently, we let θ → 0 and apply carefully a diagonal sequence argument (see Section 3.4).…”