On the existence, uniqueness and completeness of displacements and stress functions in linear elasticity
type: Item
instance of: scholarly article
title: On the existence, uniqueness and completeness of displacements and stress functions in linear elasticity
zbMATH Open document ID: 0677.73006
DOI: 10.1007/BF00041692
author: K. Hackl
author: Udo Zastrow
published in: Journal of Elasticity
publication date: 1988-01-01T00:00:00Z
review text: This article is devoted to the question of representability of strain tensors, resp. stress tensors by means of a displacement, resp. a stress function. More precisely, given tensor fields E and S that satisfy the compatibility conditions (1) \(\nabla \times E\times \nabla =0\) and (2) \(\nabla \cdot S=0\) in an open subset \(\Omega\) of \({\mathbb{R}}^ 3\), the question is whether or not these tensor fields can be obtained from a displacement u and a stress function tensor A by the classical formulas (3) \(E=(\nabla u+\nabla u^ T)/2\) and (4) \(S=\nabla \times A\times \nabla\). The answer is known to be positive when \(\Omega\) is simply connected for strains, and when \(\Omega\) is simply bordered for stresses [\textit{M. E. Gurtin}, The linear theory of elasticity. In: Handbuch der Physik, Vol. VIa/2 (1972; Zbl 0277.73001) and \textit{G. Rieder}, Abh. Braunschw. Wiss. Ges. 12, 4-65 (1960; Zbl 0126.406)]. The authors treat the general case of a multi-connected, multi-bordered open set \(\Omega\). Let \({\mathcal E}(\Omega)\) denote the quotient space of the space of tensor fields E satisfying (1) by the space of tensor fields satisfying (3). They show that \({\mathcal E}(\Omega)\) is isomorphic to \({\mathcal H}^ 1(\Omega)^ 6\), where \({\mathcal H}^ 1(\Omega)\) is the first de Rham cohomology space of \(\Omega\). Thus, the dimension of \({\mathcal E}(\Omega)\) is \(6b_ 1\), where \(b_ 1\) is the first Betti number of \(\Omega\). In other words, a tensor field satisfying (1) can be uniquely decomposed as the sum of \(6b_ 1\) particular tensor fields satisfying (1) but not satisfying (3), i.e., non-integrable, plus the strain of a displacement u. The isomorphism between \({\mathcal E}(\Omega)\) and \(H^ 1(\Omega)^ 6\) is explicitly constructed. In a very similar fashion, the quotient space of the space of tensor fields \({\mathcal S}\) satisfying (2) by the space of tensor fields satisfying (4) is shown to be isomorphic to \({\mathcal H}^ 2(\Omega)^ 6\), where \({\…
Mathematics Subject Classification ID: 74B05
Mathematics Subject Classification ID: 74G30
Mathematics Subject Classification ID: 74H25
Mathematics Subject Classification ID: 74B10
zbMATH DE Number: 4110217
zbMATH Keywords: multiply-connected bodies
zbMATH Keywords: representability of strain tensors
zbMATH Keywords: quotient space
zbMATH Keywords: isomorphic
zbMATH Keywords: first de Rham cohomology space
zbMATH Keywords: second de Rham cohomology space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4110217