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Finite Elasticity and Weak Diffeomorphisms

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Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete / 3. Folge. A Series of Modern Surveys in Mathematics book series (MATHE3, volume 38)

Abstract

Let Ω ⊂ ℝ3 be a bounded domain which is taken to be the reference configuration or the rest state of a perfectly elastic body. Classically a deformation of the body, which in its rest position occupies the region Ω, is described by a smooth map u : \(\Omega\to \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Omega }\subset {\mathbb{R}^3}\) that is orientation preserving and globally invertible. According to our idealized situation of a perfectly elastic body no heat transfer occurs in the process of loading and unloading a perfectly elastic material, and such loading and unloading process is completely reversible. Therefore it is reasonable to assume that all mechanical properties of a perfectly elastic material are characterized by a stored energy function W(x, G),which depends on the infinitesimal deformations G = Du(x), and that in terms of it the total internal energy stored by the body which undergo the deformation u is given by
$$\varepsilon (u,\Omega ): = \int\limits_\Omega W (x,Du(x))dx$$
(1)
Materials whose mechanical properties are characterized by a stored energy functions are often called hyperelastic materials.

Keywords

Elastic Body Coercivity Condition Store Energy Density Area Formula Store Energy Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Dipartimento di Matematica ApplicataUniversità di FirenzeFirenzeItaly
  3. 3.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Čzech Republic

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