Optimal Regularity and Structure of the Free Boundary for Minimizers in Cohesive Zone Models

Abstract

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are C 1,1/2 , and that near non-degenerate points the fracture set is C 1,α , for some α ∈ (0, 1).Here, [v] = v RT − v LT , where v RT and v LT are the right and left traces on {y = 0} of v | R n ×(0,A) and v | R n ×(−A,0) , respectively, and g ∈ C 2 [0, ∞) ∩ C 3 (0, ∞) is strictly increasing, bounded, concave, with g(0) = 0 and g ′ (0 + ) ∈ (0, +∞). The parameter g ′ (0 + ) has an important physical Date: December 22, 2017.

Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy

“…Evolutionary models (prescribing the crack path) have been studied in [DMZ07, BFM08, Cag08, CT11, LS14, Alm17, ACFS17, NS17, TZ17, NV18, CLO18], see also references therein. See [DMG08,CCF20] for further results on the topic.…”

Cohesive Fracture in 1D: Quasi-static Evolution and Derivation from Static Phase-Field Models

“…To be more precise, we need to consider the lower semicontinuous envelope of the energy and a minimizer will belong in general to BV(0, 1): indeed the presence of the lower-order term allows for the possibility of having more than one jump, while some regularity can still be proved, see Proposition 3.2. In the case γ = 0 it is well-known that minimizers in dimension one are in SBV with a single jump, see [15]; see also [16,25] for further regularity results for cohesive energies. Graph of g 0 (s) (bottom curve) and g(s, s ) for m s = 0.3, 0.5 and 0.7 (from bottom to top) using the function f b 1 (s) defined in (6.5) with = 1.5, 1 = 0.2.…”