By Marcelo Epstein
ISBN-10: 0521198550
ISBN-13: 9780521198554
This publication provides the basic strategies of contemporary differential geometry in the framework of continuum mechanics. it truly is divided into 3 elements of approximately equivalent size. The ebook opens with a motivational bankruptcy to provoke upon the reader that differential geometry is certainly the ordinary language of continuum mechanics or, higher nonetheless, that the latter is a chief instance of the applying and materialization of the previous. within the moment half, the basic notions of differential geometry are provided with rigor utilizing a writing type that's as casual as attainable. Differentiable manifolds, tangent bundles, external derivatives, Lie derivatives, and Lie teams are illustrated by way of their mechanical interpretations. The 3rd half comprises the speculation of fiber bundles, G-structures, and groupoids, that are acceptable to our bodies with inner constitution and to the outline of fabric inhomogeneity. The summary notions of differential geometry are therefore illuminated through functional and intuitively significant engineering purposes
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Extra info for The Geometrical Language of Continuum Mechanics
Sample text
A good example of what can go wrong is provided by a linear spring: although the displacement caused by forces abides by the principle of superposition, the stored elastic energy does not. Indeed, the latter is a quadratic function of the forces. 6 An interesting observation pertaining to the method of finite elements in Solid Mechanics can be made: to express the stresses in components on the same basis as the displacements leads to stress fields with artificial discontinuities. Vector and Affine Spaces Similarly, the multiplication of a scalar α by a linear function f is, by definition, the linear function αf given by: (αf )(v) := α f (v), ∀ v ∈ V.
From the physical point of view the cotangent space is a veritable bonanza: it contains all those physical entities which produce linearly a real number δW, called virtual work, when acting on virtual displacements. And what can these entities be but generalized forces acting on the configuration κ? There can be no other meaning of a generalized force. This being so, we have, by merely following the natural geometric constructions arising from the very definition of the configuration space Q, stumbled upon the most general possible notion of generalized force in Continuum Mechanics.
For the case U = V, contrast this last result with the expression for the action of the covector on the vector. [Remark: note, in passing, that not every linear operator can be expressed as the tensor product of two vectors. ] We have spoken about the conceptual novelty of the tensor product. No less important is its notational convenience. In the case of a vector, say v ∈ V, it is obviously advantageous to be able to express the master concept of a vector in terms of the subsidiary notion of its components in a particular basis by simply writing: v = v i fi .
The Geometrical Language of Continuum Mechanics by Marcelo Epstein
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