A piecewise Korn inequality in SBD and applications to embedding and\n density results

2016/04/28 by Manuel Friedrich, Friedrich, Manuel · 1 citation
Engineering · #Elasticity and Material Modeling #Topology Optimization in Engineering #Composite Structure Analysis and Optimization

paper · pdf · doi:10.48550/arxiv.1604.08416

Abstract

We present a piecewise Korn inequality for generalized special functions of\nbounded deformation (GSBD²) in a planar setting generalizing the classical\nresult in elasticity theory to the setting of functions with jump\ndiscontinuities. We show that for every configuration there is a partition of\nthe domain such that on each component of the cracked body the distance of the\nfunction from an infinitesimal rigid motion can be controlled solely in terms\nof the linear elastic strain. In particular, the result implies that GSBD²\nfunctions have bounded variation after subtraction of a piecewise infinitesimal\nrigid motion. As an application we prove a density result in GSBD².\nMoreover, for all d \≥ 2 we show GSBD²(\Ω) \⊂ (GBV(\Ω; Bbb\nR))^d and the embedding SBD²(\Ω) \∩ L^\∞(\Ω; Bbb R^d)\n hookrightarrow SBV(\Ω; Bbb R^d) into the space of special functions of\nbounded variation (SBV). Finally, we present a Korn-Poincar 'e inequality for\nfunctions with small jump sets in arbitrary space dimension.\n

Cited by

• Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane

Related

• A Korn-type inequality in SBD for functions with small jump sets
• A Korn-Poincar 'e-type inequality for special functions of bounded\n deformation
• A quantitative geometric rigidity result in SBD
• A sharp Korn's inequality for piecewise H¹ space and its application
• On the optimal constants in Korn's and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions