On the existence, uniqueness and completeness of displacements and stress functions in linear elasticity
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 \({\mathcal H}^ 2(\Omega)\) is the second de Rham cohomology space of \(\Omega\). It is thus of dimension \(6b_ 2\), where \(b_ 2\) is the second Betti number of \(\Omega\). Finally, a few examples of multi-connected bodies are considered.
scientific article; zbMATH DE number 4110217