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.
Nonlinear Elasticity Material Tensor Parallel Derivative Frame Invariance Trajectory Manifold
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Cauchy, A.L.: Recherches sur l’equilibre et le mouvement intérieur des corps solides ou fluides élastique ou non élastique a condensation et la dilatation des corps solides. Bull. Soc. Philomath. Paris 9–13 (1823)Google Scholar
Cauchy A.L.: Sur les équations qui expriment les conditions d’equilibre ou les lois du mouvement intérieur d’un corps solides élastique ou non élastique. Ex. de Math. 3, 160–187 (1828)Google Scholar
Green G.: On the laws of reflection and refraction of light at the common surface of two non-crystallized media. Trans. Camb. Philos. Soc. 7, 1–24 (1839)Google Scholar
Green G.: On the propagation of light in crystallized media. Trans. Camb. Philos. Soc. 7, 121–140 (1841)Google Scholar
Simó J.C., Ortiz M.: A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput. Methods Appl. Mech. Eng. 49, 221–245 (1985)CrossRefMATHGoogle Scholar
Simó J.C.: On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60, 153–173 (1987)CrossRefMATHGoogle Scholar
Bruhns O.T., Meyers A., Xiao H.: On non corotational rates of Oldroyd’s type and relevant issues in rate constitutive formulations. Proc. R. Soc. 460, 909–928 (2004)MathSciNetCrossRefMATHGoogle Scholar
Bruhns O.T., Xiao H., Meyers A.: A weakened form of Ilyushin’s postulate and the structure of self-consistent Eulerian finite elastoplasticity. Int. J. Plast. 21, 199–219 (2005)CrossRefMATHGoogle Scholar
Rivlin, R.S.: Review of “The Foundations of Mechanics and Thermodynamics: Selected Papers. W. Noll, 324 p. Springer, 1974”. American Scientist, January–February (1976)Google Scholar
Kollmann F.G., Hackenberg H.P.: On the algebra of two point tensors on manifolds with applications in non-linear solid mechanics. Z. Angew. Math. Mech. 73, 307–314 (1993)MathSciNetCrossRefMATHGoogle Scholar
Giessen E., van der Kollmann F.G.: On mathematical aspects of dual variables in continuum mechanics. Part 1: mathematical principles. Z. Angew. Math. Mech. 76, 447–462 (1996)MathSciNetCrossRefMATHGoogle Scholar
Giessen E., van der Kollmann F.G.: On mathematical aspects of dual variables in continuum mechanics. Part 2: applications in nonlinear solid mechanics. Z. Angew. Math. Mech. 76, 497–504 (1996)MathSciNetCrossRefMATHGoogle Scholar
Truesdell, C.: A First Course in Rational Continuum Mechanics. Academic Press, New York (1977–1991)Google Scholar
Baig M., Khan A.S., Choi S-H., Jeong A.: Shear and multiaxial responses of oxygen free high conductivity (OFHC) copper over wide range of strain-rates and temperatures and constitutive modeling. Int. J. Plast. 40, 65–80 (2013)CrossRefGoogle Scholar