Finite difference approximation of energies in Fracture Mechanics

We provide a variational approximation by finite-difference energies of functionals of the type defined for u E SBD(Q), which are related to variational models in ’ fracture mechanics for linearly-elastic materials. We perform this approximation in dimension 2 via both discrete and continuous functionals. In the discrete scheme we treat also boundary value problems and we give an extension of the approximation result to dimension 3. Mathematics Subject Classification (2000): 49J45 (primary), 49M25, 74R10 (secondary). 1. Introduction In this paper we provide a variational approximation by discrete energies of functionals of the type defined for every closed hypersurface K c Q with normal v and u E B K; Ilgn), where Q C is a bounded domain of Here -emu = + . 2 denotes the symmetric part of the gradient of u, [u ] is the jump of u through K along v and is the (n I)-dimensional Hausdorff measure. These functionals are related to variational models in fracture mechanics for linearly elastic materials in the framework of Griffith’s theory of brittle fracture Pervenuto alla Redazione il 10 novembre 1999 e in forma definitiva il 13 aprile 2000. 672 (see [33]). In this context u represents the displacement field of the body, with SZ as a reference configuration. The volume term in (1.1) represents the bulk energy of the body in the "solid region", where linear elasticity is supposed to hold, JL, À being the Lame constants of the material. The surface term is the energy necessary to produce the fracture, proportional to the crack surface K in the isotropic case and, in general, depending on the normal v to K and on the jump [u]. The weak formulation of the problem leads to functionals of the type defined on the space SBD(Q) of integrable functions u whose symmetrized distributional derivative Eu is a bounded Radon measure with density Fu with respect to the Lebesgue measure and with singular part concentrated on an (n I)-dimensional set Ju, on which it is possible to define a normal vu in a weak sense and one-sided traces. The description of continuum models in Fracture Mechanics as variational limits of discrete systems has been the object of recent research (see [17], [19], [20], [15] and [36]). In particular, in [19] an asymptotic analysis has been performed for discrete energies of the form where Rs is the portion of the lattice sZ’ of step size > 0 contained in S2 and u : Rs R’ may be interpreted as the displacement of a particle parameterized by x E In this model the energy of the system is obtained by superposition of energies which take into account pairwise interactions, according to the classical theory of crystalline structures. Upon identifying u in (1.3) with the function in L 1 constant on each cell of the lattice the asymptotic behaviour of functionals can be studied in the framework of r-convergence of energies defined on L 1 (see [25], [23]). A complete theory has been developed when u is scalar-valued; in this case the proper space where the limit energies are defined is the space of SB V functions (see for instance [24]). An important model case is when w) In this case we may rewrite He as where R~ is a suitable portion of R, and D! u (x) denotes the difference quotient ~ (u (x + 8~) M(~r)). Functionals of this type have been studied also in [22] in the framework of computer vision. In [22] and, in a general framework, in [19] it has been proved that, if f (t) = min{t, 1 } and p is a positive function with 673 suitable summability and symmetry properties, then R, approximates functionals of the type defined for u E SB V (S2), which are formally very similar to that in (1.2). A similar result holds by replacing min{t, 1 { by any increasing function f with /(0) = 0, > 0 and f (oo) = b -E-oo. Following this approach, in order to approximate (1.2), one may think to "symmetrize" the effect of the difference quotient by considering the family of functionals ..----.......---... 11 ....... By letting 8 tend to 0, we obtain as limit a proper subclass of functionals (1.2). Indeed, the two coefficients It and k of the limit functionals are related by a fixed ratio. This limitation corresponds to the well-known fact that pairwise interactions produce only particular choices of the Lame constants. To overcome this difficulty we are forced to take into account in the model non-central interactions. The idea underlying this paper is to introduce a suitable discretization of the divergence, call it div~u, that takes into account also interactions in directions orthogonal to ~, and to consider functionals of the form with 8 a strictly positive parameter (for more precise definitions see Sections 3 and 7). In Theorem 3.1 we prove that with suitable choices of f, p and 9 we can approximate functionals of type (1.2) in dimension 2 and 3 with arbitrary and (D satisfying some symmetry properties due to the geometry of the lattice. Actually, the general form of the limit functional is the following with W explicitly given; in particular we may choose W (~u (x ) ) and c = 2. We underline that the energy density of the limit surface term is always anisotropic due to the symmetries of the lattices The dependence on [u], vu arises in a natural way from the discretizations chosen and the vectorial framework of the problem. To drop the anisotropy of the limit surface energy we consider as well a continuous version of the approximating functionals (1.5) given by 674 where in this case p is a symmetric convolution kernel which corresponds to a polycrystalline approach. By varying f, p and 8, as stated in Theorem 3.8, we obtain as limit functionals of the form for any choice of positive constants ~c, À and y. This continuous model generalizes the one proposed by E. De Giorgi and studied by M. Gobbino in [31], to approximate the Mumford-Shah functional defined for U E SBV(Q). The main technical issue of the paper is that, in the proof of both the discrete and the continuous approximation, we cannot reduce to the 1-dimensional case by an integral-geometric approach as in [ 19], [22], [31], due to the presence of the divergence term. For a deeper insight of the techniques used we refer to Sections 4 and 5; we just underline that the proofs of the two approximations (discrete and continuous) are strictly related. Analogously to [19], in Section 7 we treat boundary value problems in the discrete scheme for the 2-dimensional case and a convergence result for such problems is derived (see Proposition 6.3 and Theorem 6.4). ACKNOWLEDGMENTS. Our attention on this problem was drawn by Andrea Braides, after some remarks by Lev Truskinovsky. We also thank Luigi Ambrosio and Gianni Dal Maso for some useful remarks. This work is part of CNR Research Project "Equazioni Differenziali e Calcolo delle Variazioni". Roberto Alicandro and Maria Stella Gelli gratefully acknowledge the hospitality of Scuola Normale Superiore, Pisa, and Matteo Focardi that of SISSA, Trieste. 2. Notation and preliminaries We denote by (., .) the scalar product in I. I will be the usual euclidean norm. For x, y E JRn, [x, y] denotes the segment between x and y. If a, b E R we write a A b and a V b for the minimum and maximum between a and b, respectively. If ~ = (~1, ~z) E II~2, we denote the vector in R2 orthogonal . to ~ defined (-~z, ~ 1). ’ If Q is a bounded open subset of and are the families of open and Borel subsets of Q, respectively. If it is a Borel measure and B 675 is a Borel set, then the measure It L B is defined as n B). We denote by .en the Lebesgue measure in R’ and by Hk the k-dimensional Hausdorff measure. If B c R’ is a Borel set, we will also use the notation IBI ] for L’(B). The notation a.e. stands for almost everywhere with respect to the Lebesgue measure, unless otherwise specified. We use standard notation for Lebesgue spaces. We recall also the notion of convergence in measure on the space L 1 (S2; R"). We say that a sequence un converges to u in measure if for every 17 > 0 we have limn e ~ : un (x) u(x)] I > = 0. The space L1(Q; when endowed with this convergence, is metrizable, an example of metric being for 2.1. BV and BD functions Let Q be a bounded open set of JRn. If u E L I (Q; we denote by Su the complement of the Lebesgue set of u, i.e. x §t Su if and only if for some .z E If z exists then it is unique and we denote it by E(x). The set Su is Lebesgue-negligible and u is a Borel function equal to u .en a.e. in SZ. Moreover, we say that x E Q is a jump point of u, and we denote by Ju the set of all such points for u, if there exist a, b E R" and v E sn-1 I such that a ~ b and where j A’ The triplet (a, b, v), uniquely determined by (2.1 ) up to a permutation of (a, b) and a change of sign of v, will be denoted by ( u + (x ) , u (x ) , vu (x ) ) . Notice that Ju is a Borel subset of Su. We say that u is approximately differentiable at a Lebesgue point x if there exists L E such that If u is approximately differentiable at a Lebesgue point x, then L, uniquely determined by (2.2), will be denoted by Vu(x) and will be called the approximate gradient of u at x. 676 Eventually, given a Borel set J C R’ , we say that J is if where ?-~n-1 (N) = 0 and each Ki is a compact subset of a C 1 (n 1) dimensional manifold. Thus, for a 1tn-1-rectifiable set J it is possible to define 1tn-1 a.e. a unitary normal vector field v. 2.1.1. BV functions ’ We recall some definitions and basic results on functions with bounded variation. For a detailed study of the properties of these functions we refer to [9] (see also [26], [30]). DEFINITION 2. l. Let u E L 1 (S2; II~N); we say that u is a function with bounded variation in Q, and we write u E B V (Q; if the distributional derivative Du of u is a N x n matrix-valued measure on Q with finite total variation. If u E BV(Q; R N), then u is approximately differentiable fn a.e. in Q and Ju turns out to be 1tn-1-rectifiable. Let us consider the Lebesgue decomposition of Du with respect to ,Cn, i.e., Du