A simple and practical representation of compatibility condition derived using a QR decomposition of the deformation gradient

Abstract

This paper examines a condition for the existence and uniqueness of a finite deformation field whenever a Gram–Schmidt (QR) factorization of the deformation gradient $${\mathbf {F}}$$ is used. First, a compatibility condition is derived, provided that a right Cauchy–Green tensor $${\mathbf {C}} = {\mathbf {F}}^T {\mathbf {F}}$$ is prescribed. It is well-known that under this condition a vanishing of the Riemann curvature tensor $${\mathbb {R}}$$ ensures compatibility of a finite deformation field. We derive a restriction imposed on Laplace stretch $$\varvec{{\mathcal {U}}}$$ , arising from a QR decomposition of the deformation gradient, through this compatibility condition. The derived condition on Laplace stretch is unambiguous, because a Cholesky factorization of the right Cauchy–Green tensor ensures the existence of a unique Laplace stretch. Although a vanishing of the Riemann curvature tensor provides a necessary and sufficient compatibility condition from a purely geometric point of view, this condition lacks a direct physical interpretation in a sense that one cannot identify the restrictions imposed by this condition on a quantity that can be readily measured from experiments. On the other hand, our compatibility condition restricts dependence of components of a Laplace stretch on certain spatial variables in a reference configuration. Unlike the symmetric right Cauchy–Green stretch tensor $${\mathbf {U}}$$ obtained from a traditional polar decomposition of $${\mathbf {F}}$$ , the components of Laplace stretch can be measured from experiments. Thus, this newly derived compatibility condition provides a physical meaning to the somewhat abstract idea of the traditionally used compatibility condition, viz., a vanishing of the Riemann curvature tensor. Couplings between certain components of the Laplace stretch representing shear and elongation play a crucial role in deriving this condition. Finally, implications of this compatibility condition are discussed.