Acta Mechanica

, Volume 225, Issue 11, pp 3199–3235 | Cite as

The geometry of nonlinear elasticity

Article

Abstract

Elasticity is the prototype of constitutive models in Continuum Mechanics. In the nonlinear range, the elastic model claims for a geometrically consistent physico-mathematical formulation providing also the logical premise for linearized approximations. A theoretic framework is envisaged here with the aim of contributing a conceptually clear, physically consistent, and computationally convenient formulation. A reasoning about the physics of the model, from a geometric point of view, leads to conceive constitutive relations as instantaneous incremental responses to a finite set of tensorial state variables and to their time rates along the space-time motion. Integrability of the tangent elastic compliance, existence of an elastic stress potential, and conservativeness of the elastic response, under the conservation of mass, are given a brand new treatment. Finite elastic strains have no physical interpretation in the new rate theory, and referential local placements are appealed to, just as loci for operations of linear calculus. Frame invariance is assessed with a consistent geometric treatment, and the clear distinction between the new notion and the property of isotropy is pointed out, thus overcoming the improper statement of material frame indifference. Extension of the theory to elasto-visco-plastic constitutive models is briefly addressed. Basic computational steps are described to illustrate feasibility and convenience of calculations according to the new theory of elasticity.

Keywords

Nonlinear Elasticity Material Tensor Parallel Derivative Frame Invariance Trajectory Manifold 
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.

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Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  • Giovanni Romano
    • 1
  • Raffaele Barretta
    • 1
  • Marina Diaco
    • 1
  1. 1.Department of Structures for Engineering and ArchitectureUniversity of Naples Federico IINaplesItaly

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