Determination and factorization of the Cauchy–Green strain tensors from shears and stretches

Abstract

In a finite deformation at a particle of a continuous body, we consider any triad of infinitesimal material line elements in the undeformed state. It is shown how the right Cauchy–Green strain tensor C may be determined assuming that the stretches along the edges of the triads are known together with the three shears of the pairs of edges. This leads in a natural way to a factorization of the strain tensor C and a corresponding decomposition of the deformation gradient F . This decomposition generalizes the right “extended polar decomposition” introduced in a previous paper [Ph. Boulanger, M. Hayes, Unsheared triads and extended polar decompositions of the deformation gradient, Int. J. Nonlinear Mech. 36 (2001) 399–420]. Alternatively, we may consider the same triad after deformation. It is shown how the inverse left Cauchy–Green strain tensor B - 1 may be determined assuming that the “resiles” (inverses of the stretches) along the edges of the triads are known together with the three “reverse shears” (opposites of the shears) of the pairs of edges. This leads in a natural way to a factorization of the tensor B - 1 and a decomposition of the inverse deformation gradient F - 1 . The corresponding decomposition of F generalizes the left “extended polar decomposition” of [Ph. Boulanger, M. Hayes, Unsheared triads and extended polar decompositions of the deformation gradient, Int. J. Nonlinear Mech. 36 (2001) 399–420]. The left and right decompositions of F associated with the triads are then related. Particular attention is given to the case of plane strain. In this case the results are made more explicit.