Abstract
Let $\Omega$ be a bounded Lipschitz domain in $\BBbR^n$. The Cauchy-Green, or metric, tensor field associated with a deformation of the set $\Omega$, i.e., a smooth-enough orientation-preserving mapping $\bTh\colon\Omega\to\BBbR^n$, is the $n\times n$ symmetric matrix field defined by $\bnabla\bTheta^T(x)\bnabla\bTheta(x)$ at each point $x\in\Omega$. We show that, under appropriate assumptions, the deformations depend continuously on their Cauchy-Green tensors, the topologies being those of the spaces $\bH^1(\Omega)$ for the deformations and $\bL^1(\Omega)$ for the Cauchy-Green tensors. When $n=3$ and $\Omega$ is viewed as a reference configuration of an elastic body, this result has potential applications to nonlinear three-dimensional elasticity, since the stored energy function of a hyperelastic material depends on the deformation gradient field $\bnabla\bTheta$ through the Cauchy-Green tensor.
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Ciarlet, P., Mardare, C. Continuity of a Deformation in H1 as a Function of Its Cauchy-Green Tensor in L1. J Nonlinear Sci 14, 415–427 (2004). https://doi.org/10.1007/s00332-004-0624-y
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DOI: https://doi.org/10.1007/s00332-004-0624-y