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Mechanics of Generalized Continua: One Hundred Years After by Gérard A. Maugin (auth.), Gérard A. Maugin, Andrei V.

By Gérard A. Maugin (auth.), Gérard A. Maugin, Andrei V. Metrikine (eds.)

In their 1909 ebook Théorie des corps déformables, Eugène and François Cosserat made a historical contribution to fabrics technological know-how via constructing the elemental ideas of the mechanics of generalized continua. The chapters accumulated during this quantity exhibit the various parts of continuum mechanics that grew out of the foundational paintings of the Cosserat brothers.

The incorporated contributions supply a close survey of the newest theoretical advancements within the box of generalized continuum mechanics. the various issues coated contain: the houses of Cosserat media, micromorphic our bodies, micropolar solids and fluids, weakly- and strongly-nonlocal theories, gradient theories of elasticity and plasticity, disorder idea, everywhere-defective fabrics, our bodies with fractal constitution, in addition to different similar issues.

Key features:

- specializes in fresh advancements in continuum mechanics and fabrics layout, together with numerical examples and newly built models.

- offers numerical examples to illustrate the potency of assorted answer techniques.

- offers a distinct evaluate of generalized continuum mechanics, illustrating the real functions of this conception in numerous different clinical disciplines.

- encompasses a foreword written by way of popular physicist and fabrics scientist A.C. Eringen.

Mechanics of Generalized Continua can function an invaluable reference for graduate scholars and researchers in mechanical engineering, fabrics technological know-how, utilized physics and utilized mathematics.

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Example text

From this it follows that one symmetric stress tensor is not enough to represent the response of the continuum on the external loadings. The base of such a new continuum model generalizing the Cauchy’s model was known since Euler because he introduced two H. A. Eremeyev South Scientific Center of RASci & South Federal University, Milchakova St. A. V. 1007/978-1-4419-5695-8_3, c Springer Science+Business Media, LLC 2010 27 28 H. A. Eremeyev independent laws of motion: the balance of momentum and the balance of moment of momentum (see [18] among others).

2 Various Jets Given two smooth manifolds, M and N , of dimensions m and n, respectively, we say that two maps f, g : M −→ N have the same k-jet at a point X ∈ M if: (i) f (X) = g(X); (ii) in a coordinate chart in M containing X and a coordinate chart in N containing the image f (X), all the partial derivatives of f and g up to and including the order k are respectively equal. Although the above definition is formulated in terms of charts, it is not difficult to show by direct computation that the property of having the same derivatives up to and including order k is, in fact, independent of the coordinate systems used in either manifold.

Although the above definition is formulated in terms of charts, it is not difficult to show by direct computation that the property of having the same derivatives up to and including order k is, in fact, independent of the coordinate systems used in either manifold. Notice that, in order for this to work, it is imperative to equate all the lower-order derivatives. If, for example, we were to equate just the second derivatives, without regard to the first, the equality of the second derivatives would not be preserved under arbitrary coordinate transformations.

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