An approximation result for special functions with bounded deformation

被引:118
作者
Chambolle, A [1 ]
机构
[1] Univ Paris 09, CEREMADE, CNRS, UMR 7534, F-75775 Paris 16, France
来源
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES | 2004年 / 83卷 / 07期
关键词
functions with bounded deformation; free discontinuity problems; brittle fracture;
D O I
10.1016/j.matpur.2004.02.004
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
A "special displacement with bounded deformation" is a function u: Omega subset of R-N --> R-N whose symmetrized gradient is a bounded measure which coincides, outside a (N - 1)-dimensional rectifiable "jump set" J(u), with a summable function e(u). We show that in dimension N = 2, when u and e(u) are square integrable, and the total length H-1 (J(u)) is finite, then such a displacement is approximated with a sequence (u(n))(ngreater than or equal to1) of piecewise continuous displacements whose jump sets J(un) are (relatively) closed, with u(n) and e(u(n)) converging strongly in L-2, respectively to u and e(u), and the lengths H-1(J(un)) converging to H-1 (J(u)). As an application, we approximate with a sequence of elliptic functionals a functional which appears in the theory of brittle fracture in linearized elasticity. (C) 2004 Elsevier SAS. All rights reserved.
引用
收藏
页码:929 / 954
页数:26
相关论文
共 25 条
[1]  
Alberti G, 2000, CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, P95
[2]   APPROXIMATION OF FUNCTIONALS DEPENDING ON JUMPS BY ELLIPTIC FUNCTIONALS VIA GAMMA-CONVERGENCE [J].
AMBROSIO, L ;
TORTORELLI, VM .
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 1990, 43 (08) :999-1036
[3]  
AMBROSIO L, 1989, B UNIONE MAT ITAL, V3B, P857
[4]  
AMBROSIO L, 1992, B UNIONE MAT ITAL, V6B, P105
[5]   Fine properties of functions with bounded deformation [J].
Ambrosio, L ;
Coscia, A ;
DalMaso, G .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 1997, 139 (03) :201-238
[6]  
Ambrosio L., 2000, OX MATH M, pxviii, DOI 10.1017/S0024609301309281
[7]  
AMBROSIO L, 1995, GAKUTO INT SER MATH, V9, P1
[8]  
Bellettini G, 1998, MATH Z, V228, P337, DOI 10.1007/PL00004617
[9]   Computing the equilibrium configuration of epitaxially strained crystalline films [J].
Bonnetier, E ;
Chambolle, A .
SIAM JOURNAL ON APPLIED MATHEMATICS, 2002, 62 (04) :1093-1121
[10]   Numerical experiments in revisited brittle fracture [J].
Bourdin, B ;
Francfort, GA ;
Marigo, JJ .
JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, 2000, 48 (04) :797-826