Book review: Marcelo Epstein, The Geometrical Language of Continuum Mechanics
Reuven Segev
Journal of Geometric Mechanics 2011, 3(1): 139-143 doi: 10.3934/jgm.2011.3.139
Intended mainly for continuum mechanicists, Epstein's book introduces modern geometry and some of its applications to theoretical continuum mechanics. Thus, examples for the mathematical objects introduced are chosen from the realm of mechanics. In particular, differentiable manifolds, tangent and cotangent bundles, Riemannian manifolds, Lie derivatives, Lie groups, Lie algebras, differential forms and integration theory are presented in the main part of the book. Once the reader's familiarity with continuum mechanics is used for the introduction of basic geometry, geometry is used in order to generalize notions of continuum mechanics. Integration of differential forms is used to formulate flux theory on manifolds devoid of a Riemannian structure. More specialized topics, namely, Whitney's geometric integration theory and Sikorski's differential spaces are used to relax smoothness assumptions for bodies and fields defined on them. Finally, an overview is given of the work that Epstein and co-workers carried out in recent years where the theory of inhomogeneity of constitutive relations is developed using the geometry of principal fiber bundles, G-structures and connections.
keywords: principal fiber bundles Continuum mechanics integration connections. inhomogeneity differentiable manifolds
The co-divergence of vector valued currents
Reuven Segev Lior Falach
Discrete & Continuous Dynamical Systems - B 2012, 17(2): 687-698 doi: 10.3934/dcdsb.2012.17.687
In the context of stress theory of the mechanics of continuous media, a generalization of the boundary operator for de Rham currents---the co-divergence operator---is introduced. While the boundary operator of de Rham's theory applies to real valued currents, the co-divergence operator acts on vector valued currents, i.e., functionals dual to differential forms valued in a vector bundle. From the point of view of continuum mechanics, the framework presented here allows for the formulation of the principal notions of continuum mechanics on a manifold that does not have a Riemannian metric or a connection while at the same time allowing irregular bodies and velocity fields.
keywords: balance equations de Rham currents Continuum mechanics boundary operator vector valued currents differential operators.

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