A dimension-reduction model for brittle fractures on thin shells with mesh adaptivity

Abstract

In this paper, we derive a new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small thickness, to express the three-dimensional energy in terms of the variational model of brittle fracture in linear elasticity, and to study the [Formula: see text]-limit of the functional as the thickness tends to zero. The numerical discretization is tackled by first approximating the fracture through a phase field, following an Ambrosio–Tortorelli like approach, and then resorting to an alternating minimization procedure, where the irreversibility of the crack propagation is rigorously imposed via an inequality constraint. The minimization is enriched with an anisotropic mesh adaptation driven by an a posteriori error estimator, which allows us to sharply track the whole crack path by optimizing the shape, the size, and the orientation of the mesh elements. Finally, the overall algorithm is successfully assessed on two Riemannian settings and proves not to bias the crack propagation.

Brittle membranes in finite elasticity

“…We believe our result may pave the way to further reduced theories with different geometries, e.g. shells [2,17], or constraints, such as incompressibility [21,51] and weaker growth assumptions than (H 5 ).…”

“…In the framework of variational fracture mechanics [14], few dimension reduction results have been obtained in linearized [2,3,10,31,36,44] and nonlinear elasticity [9,49]. In the mentioned papers an explicit non-interpenetration constraint in the form of positive sign of the Jacobian of the deformation was not considered in the membranes setting.…”

Brittle fracture in linearly elastic plates

“…In the context of fracture mechanics, the study of the -limit of free discontinuity functionals of the form (1.1) has been considered, for instance, in [2, 7, 8, 10, 22, 32]. In particular, [7, 10] are concerned with the nonlinearly elastic case, in which the stored elastic energy density obeys a -growth condition of the form which is incompatible with linear elasticity.…”

Brittle fracture in linearly elastic plates

“…In the context of fracture mechanics, the study of the Γ-limit of free discontinuity functionals of the form (1.1) has been considered, for instance, in [2,6,7,9,19,27]. In particular, [6,9] are concerned with the nonlinearly elastic case, in which the stored elastic energy density obeys a p-growth condition of the form W (F ) ≥ C(|F | p − 1) which is incompatible with linear elasticity.…”

“…In particular, [6,9] are concerned with the nonlinearly elastic case, in which the stored elastic energy density obeys a p-growth condition of the form W (F ) ≥ C(|F | p − 1) which is incompatible with linear elasticity. The papers [2,27] consider the antiplanar case, where the energy is in the form (1.1) but the displacement u is supposed to be orthogonal to the middle surface ω, so that the dimension reduction problem becomes scalar and is described in terms of GSBV -functions (see, e.g., [4,Section 4.5]). In [19] the authors considered the convergence of quasistatic evolutions in the vectorial case, under the assumption that the crack path is known a priori, is transversal to the middle surface ω, and cuts the whole of Ω ρ .…”