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 F\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {F}}$$\end{document} is used. First, a compatibility condition is derived, provided that a right Cauchy–Green tensor C=FTF\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {C}} = {\mathbf {F}}^T {\mathbf {F}}$$\end{document} is prescribed. It is well-known that under this condition a vanishing of the Riemann curvature tensor R\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}$$\end{document} ensures compatibility of a finite deformation field. We derive a restriction imposed on Laplace stretch U\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{{\mathcal {U}}}$$\end{document}, 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 U\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {U}}$$\end{document} obtained from a traditional polar decomposition of F\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {F}}$$\end{document}, 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.
