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Published in:
2015 | OriginalPaper | Chapter
11. Static Elasticity in a Riemannian Manifold
Author : Cristinel Mardare
Published in: Differential Geometry and Continuum Mechanics
Publisher: Springer International Publishing
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Abstract
We discuss the equations of elastostatics in a Riemannian manifold, which generalize those of classical elastostatics in the three-dimensional Euclidean space. Assuming that the deformation of an elastic body arising in response to given loads should minimize over a specific set of admissible deformations the total energy of the elastic body, we derive the equations of elastostatics in a Riemannian manifold first as variational equations, then as a boundary value problem. We then show that this boundary value problem possesses a solution if the loads are sufficiently small in a specific sense. The proof is constructive and provides an estimation for the size of the loads.