A Piecewise Korn Inequality in SBD and Applications to Embedding and Density Results

Abstract

We present a piecewise Korn inequality for generalized special functions of bounded deformation (GSBD 2 ) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that GSBD 2 functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in GSBD 2 in dimension two. Moreover, for all d ≥ 2 we show GSBD 2 (Ω) ⊂ (GBV (Ω; R)) d and the embedding SBD 2 (Ω) ∩ L ∞ (Ω; R d ) → SBV (Ω; R d ) into the space of special functions of bounded variation (SBV ). Finally, we present a Korn-Poincaré inequality for functions with small jump sets in arbitrary space dimension.2010 Mathematics Subject Classification. 74R10, 49J45, 70G75, 26D10.

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

“…The Griffith energy (G) is approximated by the functionalŝ Ω (v + η ε )f p (e(u)) + (1 − v) 2 4ε + ε q−1 |∇v| q dx , lim ε→0 η ε ε p−1 = 0 , (G ε ) for u ∈ W 1,p u 0 (Ω; R n ) := W 1,p (Ω; R n )∩{u : tr ∂Ω (u−u 0 ) = 0 on ∂ D Ω}, v ∈ W 1,q 1 (Ω; [0, 1]) := W 1,q (Ω; [0, 1]) ∩ {v : tr ∂Ω v = 1 on ∂ D Ω}, and +∞ otherwise: such approximation has been proven without any a priori assumption on u, for any p > 1, and in any dimension in [25], together with compactness for minimisers (see [26]), assuming that O δ,x 0 (∂ D Ω) ⊂ Ω for δ ∈ (0, δ), (1.2) for some δ > 0 and x 0 ∈ R n , where O δ,x 0 (x) := x 0 + (1 − δ)(x − x 0 ). This generalises [21,22,51], assuming a priori u ∈ L 2 and p = 2, [28], requiring u ∈ L p , p > 1, and [43], obtained in dimension 2 (see also e.g. [45,18,53] for the antiplane shear case and different approximations).…”

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

“…When L = R d×d skew , one can equivalently characterize P R L (Ω) as the subspace of GSBD functions (see [38]) whose symmetrized approximate gradient e(u) equals zero L d -almost everywhere. For a proof we refer to [36,Theorem A.1] and [48,Remark 2.2(i)]. In a similar fashion, in the case L = SO(d), P R L (Ω) coincides with the GSBV functions whose approximate gradient satisfies ∇u(x) ∈ SO(d) for L d -a.e.…”

“…However, our primary purpose comes from the study of free-discontinuity problems defined on the space GSBD p , see [38], which has obtained steadily increasing attention over the last years, cf., e.g., [30,31,32,33,34,35,46,47,48,50]. We have indeed already mentioned before how the analysis of partition problems has proved to be a relevant tool in the study of freediscontinuity problems on SBV .…”

“…For a proof we refer to [48,Theorem 2.3], although the argument can be retrieved in previous literature (see for instance [4,23]). The additional property that L ∞ -caps are preserved by the approximation, which was not stated explicitly there, is a direct consequence of the proof.…”

A compactness result in $$GSBV^p$$ and applications to $$\varGamma $$-convergence for free discontinuity problems

“…The compactness result relies on the idea that a control on a sequence of functions can always be ensured by subtracting suitable piecewise rigid motions. Using a piecewise Korn inequality [33,35], it can be shown that such a modification can be performed without essentially increasing the energy of the configurations.…”