Abstract
We analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step delta is smaller than the ellipticity parameter varepsilon , we show the varGamma -convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no L^p fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.
Highlights
In this paper we provide a variational approximation by discrete finite-difference energies of functionals of the form λ |Eu(x)|2 dx + μ|div u(x)|2 dx + Hd−1(K ), Ω\K (1.1)where Ω is a bounded subset of Rd, K ⊆ Ω is closed, u ∈ C1(Ω\K ; Rd ), Eu denotes the symmetric part of the gradient of u, div u is the divergence of u and Hd−1 is the (d − 1)dimensional Hausdorff measure
In the appropriate functional setting, u is a generalized special function of bounded deformation, for which the symmetrized gradient Eu and the divergence div u are defined almost everywhere in an approximate sense, and the set K is replaced by the (d − 1)-rectifiable set Ju, the jump set of u
Coming back to the problem of providing discrete approximations of the Griffith functional, we mention the finite-elements approximation in [30] and focus on the discreteto-continuum analysis performed by Alicando et al [1]
Summary
Page 3 of 46 193 proposed by Braides and Yip [14], this analysis has been recently developed by Bach, Braides and Zeppieri in [6] for (1.2). While in the scalar-valued case controlling the total variation along d independent slices of uε is enough to provide BV -compactness, no analogue procedure is at the moment known in G S B D (whose definition [27, Definition 4.1] in principle requires a uniform control of the symmetrized slices on a dense set of directions in the unit sphere, cf [27, Remark 4.15]) Such issue prevents us to get a uniform bound in G S B D from a control on the slices corresponding to the directions of the lattice vectors, that could be obtained from the discrete functional as in [6,17]. We are able to prove that a continuous Ambrosio–Tortorelli functional, defined on the standard piecewise affine interpolations uε of the uε and on suitable piecewise constant interpolations vmin,ε of the vε (different than the standard ones), bounds from below the discrete energies (1.4) To this aim, taking the additional (NNNN) interactions is crucial in dimension d = 3.
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