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Applications of algebraic topology in elasticity. (English) Zbl 1444.74009

Segev, Reuven (ed.) et al., Geometric continuum mechanics. Cham: Birkhäuser. (ISBN 978-3-030-42682-8/hbk; 978-3-030-42685-9/pbk; 978-3-030-42683-5/ebook). Advances in Mechanics and Mathematics 43. Advances in Continuum Mechanics, 143-183 (2020).
Summary: In this chapter we discuss some applications of algebraic topology in elasticity. This includes the necessary and sufficient compatibility equations of nonlinear elasticity for non-simply-connected bodies when the ambient space is Euclidean. Algebraic topology is the natural tool to understand the topological obstructions to compatibility for both the deformation gradient \(\mathbf{F}\) and the right Cauchy-Green strain \(\mathbf{C}\). We investigate the relevance of homology, cohomology, and homotopy groups in elasticity. We also use the relative homology groups in order to derive the compatibility equations in the presence of boundary conditions. The differential complex of nonlinear elasticity written in terms of the deformation gradient and the first Piola-Kirchhoff stress is also discussed.
For the entire collection see [Zbl 1446.74007].

MSC:

74B20 Nonlinear elasticity
55N99 Homology and cohomology theories in algebraic topology
55Q99 Homotopy groups
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