Abstract
A generalization of the Cauchy theory of forces and stresses to the geometry of differentiable manifolds is presented using the language of differential forms. Body forces and surface forces are defined in terms of the power densities they produce when acting on generalized velocity fields. The normal to the boundary is replaced by the tangent space equipped with the outer orientation induced by outward pointing vectors. Assuming that the dimension of the material manifold is m, stresses are modelled as m − 1 covector valued forms. Cauchy's formula is replaced by the restriction of the stress form to the tangent space of the boundary while the outer orientation of the tangent space is taken into account. The special cases of volume manifolds and Riemannian manifolds are discussed.
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Segev, R., Rodnay, G. Cauchy's Theorem on Manifolds. Journal of Elasticity 56, 129–144 (1999). https://doi.org/10.1023/A:1007651917362
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DOI: https://doi.org/10.1023/A:1007651917362