A derivation of Griffith functionals from discrete finite-difference models
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Calculus of Variations
A derivation of Griffith functionals from discrete finite-difference models Vito Crismale1 · Giovanni Scilla2 · Francesco Solombrino2 Received: 3 January 2020 / Accepted: 31 August 2020 © The Author(s) 2020
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 δ is smaller than the ellipticity parameter ε, we show the Γ -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. Mathematics Subject Classification 49M25 · 49J45 · 74R10 · 65M06
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . 3 Discrete models and approximation results . . . 4 Compactness . . . . . . . . . . . . . . . . . . . 5 Semicontinuity properties for the Griffith energy 6 The upper limit for the Griffith energy . . . . . . 7 The non-interpenetration constraint . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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Communicated by J.Ball.
B
Francesco Solombrino [email protected] Vito Crismale [email protected] Giovanni Scilla [email protected]
1
CMAP, École Polytechnique, CNRS, 91128 Palaiseau Cedex, France
2
Dipartimento di Matematica ed Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia Monte Sant’Angelo, 80126 Napoli, Italy 0123456789().: V,-vol
123
193
Page 2 of 46
V. Crismale et al.
1 Introduction In this paper we provide a variational approximation by discrete finite-difference energies of functionals of the form 2 λ |E u(x)| dx + μ |div u(x)|2 dx + Hd−1 (K ), (1.1) Ω\K
Ω\K
Rd ,
where Ω is a bounded subset of K ⊆ Ω is closed, u ∈ C 1 (Ω\K ; Rd ), E u 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. Functionals as in (1.1) are widely used in the variational modeling of fracture mechanics for linearly elastic materials, in the framework of Griffith’s theory of brittle fracture (see, e.g. [29]). Here Ω stands for the reference configuration and u represents the displacement field of the body. The total energy (
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