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Continuity of a deformation in $$H^{1}$$ as a function of its Cauchy-Green tensor in $$L^{1}$$. (English) Zbl 1084.53063
Summary: Let $$\Omega$$ be a bounded Lipschitz domain in $$\mathbb R^n$$. The Cauchy-Green, or metric, tensor field associated with a deformation of the set $$\Omega$$, i.e., a smooth-enough orientation-preserving mapping $$\Theta:\Omega\to\mathbb R^n$$, is the $$n\times n$$ symmetric matrix field defined by $$\nabla\Theta^T(x)\nabla\Theta(x)$$ 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 $$\mathbf H^1(\Omega)$$ for the deformations and $$\mathbf L^1(\Omega)$$ 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 $$\nabla\Theta$$ through the Cauchy-Green tensor.

##### MSC:
 53C80 Applications of global differential geometry to the sciences 74B20 Nonlinear elasticity
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