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.

中文翻译：

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使用变形梯度的 QR 分解导出的兼容性条件的简单实用表示

每当使用变形梯度 $${\mathbf {F}}$$ 的 Gram-Schmidt (QR) 分解时，本文研究了有限变形场的存在性和唯一性的条件。首先，推导出兼容性条件，前提是规定了右柯西-格林张量 $${\mathbf {C}} = {\mathbf {F}}^T {\mathbf {F}}$$。众所周知，在这种情况下，黎曼曲率张量 $${\mathbb {R}}$$ 的消失确保了有限变形场的兼容性。我们通过这种兼容性条件推导出对拉普拉斯拉伸 $$\varvec{{\mathcal {U}}}$$ 施加的限制，这是由变形梯度的 QR 分解产生的。拉普拉斯拉伸的导出条件是明确的，因为正确的柯西-格林张量的 Cholesky 分解确保了唯一的拉普拉斯拉伸的存在。尽管黎曼曲率张量的消失从纯几何的角度提供了一个充分必要的相容性条件，但从某种意义上说，这种条件缺乏直接的物理解释，即人们无法识别这种条件对一个可以很容易地理解的量施加的限制。从实验测量。另一方面，我们的兼容性条件限制了拉普拉斯拉伸的组件对参考配置中某些空间变量的依赖性。与从 $${\mathbf {F}}$$ 的传统极坐标分解获得的对称右柯西-格林拉伸张量 $${\mathbf {U}}$$ 不同，拉普拉斯拉伸的分量可以通过实验测量。因此，这个新导出的兼容性条件为传统使用的兼容性条件的某种抽象概念提供了物理意义，即黎曼曲率张量的消失。代表剪切和伸长率的拉普拉斯拉伸的某些分量之间的耦合在推导这种条件时起着至关重要的作用。最后，讨论了这种兼容性条件的含义。