Calculus of Variations and Geometric Measure Theory

M. Giaquinta - P. M. Mariano - G. Modica

Stress constraints in simple bodies undergoing large strains: a variational approach

created by modica on 27 Nov 2012
modified on 01 Dec 2014


Accepted Paper

Inserted: 27 nov 2012
Last Updated: 1 dec 2014

Journal: Proc. Royal Soc. Edinburgh
Year: 2014


We consider a simple body that is hyperelastic in large strain regime until the $3-$covector defining the first Piola-Kirchhoff stress, once projected on the appropriate space of second-rank tensor, reaches a threshold indicating critical states. No information is given on the post-critical behavior. We determine the existence of equilibrium configurations according to the constraint. Such configurations can have concentration of strain in regions with vanishing volume. The related stress appears naturally as a measure over the deformation graph. Once restricted to the regular part of the deformation, such a measure determines the first Piola-Kirchhoff stress tensor and may also be concentrated over sets with vanishing volume projections on the reference and current placements. These projections can be interpreted as dislocations or dislocation walls. We analyze explicitly specific cases.