Compatibility conditions from a Gram–Schmidt decomposition of deformation gradient in two dimensions

Abstract

Planar deformations are analyzed in the context of a Gram-Schmidt decomposition of deformation gradient into a rotation and upper triangular stretch, where the latter consists of up to three possibly nonzero terms. Necessary—and sufficient under suitable restrictions—conditions for integrability of deformation gradient and stretch (geometrically, vanishing torsion and curvature of a certain affine connection) reduce to a system of three partial differential equations (PDEs). This system appears more convenient than corresponding compatibility conditions from polar decompositions, especially for motions involving simple shear. Admissible families of triangular stretch corresponding to simple shear, pure shear, dilation, and uniaxial strain are analyzed. Also considered are simultaneous simple shear and axial stretch, for which a non-trivial solution with non-vanishing rotation gradient is constructed.