A Global Method for Relaxation in W 1,p and in SBV p

The Variational Approach to Fracture

“…In contrast to what was encountered when using Griffith's energy, the above functional does not admit a minimizer in SBV (R 2 ). A relaxation process occurs whereby, for high enough gradients, it is energetically cheaper to replace those by many infinitesimally small jumps; see (Bouchitté et al, 1995), (Braides et al, 1999), (Bouchitté et al, 2002). We will return to this point in Subsection 4.2 below.…”

Quasistatic evolution of a brittle thin film

“…What is missing is either a bound on the jump part of the derivative (this bound was appearing naturally in [9,6] because the authors were considering a surface energy density with linear growth with respect to the jump of the deformation), or an L ∞ -bound on the minimizing sequences. In the spirit of [17], we define for all (u; A) ∈ BV (Ω;…”

“…x 3 ), ξ = (ξ α |ξ 3 ) for some matrix ξ ∈ R 3×3 and z = (z α |z 3 ) for some vector z ∈ R 3 . Such studies have been performed in [5] where the authors imposed to the crack to live on the mid-surface of the plate and also in [6,9]. Unlike the latest references, we are working with materials satisfying Griffith's principle.…”

“…The originality of our work comes from the fact that we are directly dealing with Griffith surface energy, which was proscribed in [6,9] as explained above. We propose here an alternative proof of this result, deriving the Γ-limit by an argument very closed to that used in the proof of Theorem 2.1 in [17].…”

“…It permits us to show that for a deformation belonging to L ∞ (Ω; R 3 ), there is no loss of generality in requiring that any minimizing sequence is uniformly bounded in L ∞ (Ω; R 3 ) and thus, weak SBV p (Ω; R 3 )-compactness follows. The two-dimensional surface energy remains of Griffith type whereas the bulk energy follows the one obtained in [6,9]: the stored energy density is given by the function QW 0 : R 3×2 → R where…”