Asymptotic Analysis of Ambrosio--Tortorelli Energies in Linearized Elasticity

Abstract

Abstract.We provide an approximation result in the sense of Γ-convergence for energies of the form Ω Q 1 (e(u)) dx, where Ω ⊂ R n is a bounded open set with Lipschitz boundary, Q 0 and Q 1 are coercive quadratic forms on M n×n sym , a, b are positive constants, and u runs in the space of fields SBD 2 (Ω); i.e., it's a special field with bounded deformation such that its symmetric gradient e(u) is square integrable, and its jump set Ju has finite (n − 1)-Hausdorff measure in R n . The approximation is performed by means of Ambrosio-Tortorellitype elliptic regularizations, the prototype example being where (u, v) ∈ H 1 (Ω, R n )×H 1 (Ω), ε ≤ v ≤ 1, and γ > 0.

A Derivation of Linearized Griffith Energies from Nonlinear Models

“…For the sake of a simplified mathematical description the investigation of fracture models in the realm of linearized elasticity is widely adopted (see for example [2,4,7,13,14,34]) and has led to a lot of realistic applications in engineering as well as to efficient numerical approximation schemes (we refer to [5,6,12,26,38,39,43] making no claim to be exhaustive). On the contrary, their nonlinear counterparts are usually significantly more difficult to treat since in the regime of finite elasticity the energy density of the elastic contributions is genuinely geometrically nonlinear due to frame indifference rendering the problem highly non-convex.…”

Korn-Poincare inequalities for functions with a small jump set

“…The three terms in (1.1) are mutually orthogonal and have a clearly distinct physical 1 interpretation: e(u) represents the regular strain, [u] the crack opening or the plastic slips, E c u some notion of diffuse damage. The modeling of fracture in linear elasticity [FM98,BFM08] focuses on the interplay between the regular and the jump part, and is normally restricted to the special functions of bounded deformation SBD, defined as those u ∈ BD for which E c u = 0, see also [Cha03,SFO08,FI14,Iur14]. In fracture models it is natural to relate e(u) L 2 to an elastic energy and the total area of the crack H n−1 (J u ) to the fracture energy, considering functionals of the type e(u) 2 L 2 + H n−1 (J u ) which constitute the vectorial counterpart to the Mumford-Shah functional from image segmentation [AFP00].…”

Phase field approximation of cohesive fracture models

“…The only approximations with quadratic volume energy densities available so far in literature have been obtained for energies which are linear [38] or affine in [u] [7,26,32], and have in common that the profiles of u and v in the optimal-transition problem giving g(|[u]|) can be decoupled.…”