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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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