Existence of strong solutions to the Dirichlet problem for the Griffith energy

Abstract

In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo in 1998 [30]. In a recent paper [16] we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti, Focardi and Iurlano [17] and Chambolle, Conti, Iurlano [14].

Korn and Poincaré-Korn inequalities for functions with a small jump set

“…The foundations of the function spaces S B D and G S B D were laid down in the papers [3,5], and [22]. Several research avenues have stemmed from them: the derivation of regularity properties for functions in (G)S B D p , and in particular of minimisers of the Griffith's energy, in the spirit of the celebrated result [23] by De Giorgi, Carriero and Leaci for the Mumford-Shah energy (see [4,10,12,13,15,17]); of Korn and Poincaré-Korn inequalities with various degrees of generality ( [9,[26][27][28]); of approximation and density results ( [7,11,[18][19][20]29]); of integral representation for functionals in (G)S B D p [16].…”

“…2), where one can prove compactness and existence of minimisers under physical assumptions (see, e.g., [12,13,15]).…”

An FFT‐based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

“…For prescribed boundary displacements , depending on (pseudo‐)time, the Francfort‐Marigo model seeks a displacement field , such that, for any t ∈[ t 0 , t 1 ], u (·, t ) minimizes the energy where S u denotes the set of jump points of u (away from the boundary ∂Ω) and ∇ s stands for the symmetrized gradient, subjected to prescribed boundary displacements u (·, t )= u 0 (·, t ) on ∂Ω and the constraint that cracks may only grow, and an initial crack set . To render the model Equation well‐defined for each instant of time, the displacement field u needs to be chosen in a suitable function space of discontinuous functions, see Chambolle et al…”

“…and an initial crack set S u(⋅,t 0 ) . To render the model Equation (1) well-defined for each instant of time, the displacement field u needs to be chosen in a suitable function space of discontinuous functions, see Chambolle et al 51,52 To treat the Francfort-Marigo model numerically, it is customary to discretize the quasi-static evolution 53 in time by a backward Euler scheme, that is, to subdivide the time interval [t 0 , t 1 ] into K+1 increasing time instants t 0 = t 0 , t 1 , t 2 , … , t K−1 , t K = t 1 and to minimize, for each k = 1, 2, … , K, the functional (1) subjected to the boundary condition u k = u 0 (⋅, t k ) on Ω and the constraint S u k ⊇ S u k−1 , where u k and u k−1 denote minimizers of the functional (1) corresponding to the time steps t k and t k−1 , respectively.…”

Non-local approximation of the Griffith functional

“…This space is denoted by GSBD p (Ω), where the exponent p refers to the integrability of Eu. After that the existence of weak minimizers has been achieved, one can actually show that the jump set thereof is closed (up to a H d−1 -negligible set), and prove well-posedness of the minimization problem for (1.1) (see [17,19,22]).…”