Remarks on the Determinant in Nonlinear Elasticity and Fracture Mechanics




Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
email: fonseca@andrew.cmu.edu



Jan Malý
Department KMA, Charles University,
Praha 8, CZ-18675, Czech Republic
email: maly@karlin.mff.cuni.cz



ABSTRACT:

The role of the determinant in ensuring local invertibility of Sobolev functions in $W^{1,N}(\Omega; \mathbb R^N)$ is studied. Weak continuity of minors of gradients of functions in $W^{1,p}(\Omega;\mathbb R^N)$ for p<N is fully characterized. Properties of the determinant are addressed within the framework of functions of bounded variation, and a change of variables formula is obtained. These results are relevant in the study of equilibria, cavitation, and fracture of nonlinear elastic materials.


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