A piecewise korn inequality in SBD and applications to embedding and density results - Friedrich, Manuel (SIAM Journal on Mathematical Analysis, 2018)
We present a piecewise Korn inequality for generalized special functions of bounded deformation (GSBD2) 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 GSBD2 functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in GSBD2 in dimension two. Moreover, for all d = 2 we show GSBD2(ω) C (GBV (ω; R))d and the embedding SBD2(ω) L1(ω; Rd) → SBV (ω; Rd) into the space of special functions of bounded variation (SBV). Finally, we present a KornPoincaŕe inequality for functions with small jump sets in arbitrary space dimension.
CRIS-Team Scopus Importer:2021-10-21
Keywords: Brittle materials; Coarea formula; Functions of bounded deformation; Korn's inequality; Korn-Poincaŕe inequality; Variational fracture