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Continuity of a deformation as a function of its Cauchy-Green tensor. (English) Zbl 1030.74003

Summary: If Riemann-Christoffel tensor associated with a field \(C\) of class \(\mathcal C^2\) of positive-definite symmetric matrices of order 3 vanishes in a simply connected open subset \(\Omega\subset\mathbb R^3\), then this field is Cauchy-Green tensor field associated with a deformation \(\Theta\) of class \(\mathcal C^3\) of the set \(\Omega\), and \(\Theta\) is uniquely determined up to isometries of \(\mathbb R^3\). Let \(\dot\Theta\) denote the equivalence class formed by all such deformations, and let \(\mathcal F : C \to\dot\Theta\) denote the mapping defined in this fashion. We establish here that the mapping \(\mathcal F\) is continuous, for certain natural metrizable topologies.

MSC:

74A05 Kinematics of deformation
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