A complete theory of saturated nonlinear elasticity is formulated in terms of a relaxed energy function, whose existence and uniqueness are proved. Within this reversible context, it is shown that a normality rule for the post-saturated strains emerges naturally. Special consideration is given to saturation conditions expressed in terms of the logarithmic stress. Introducing the concept of material evolution, the passage to irreversible theories, such as plasticity and growth, is effected by means of a criterion of instantaneous healing imposed on any given underlying saturation scheme.
[2] Courant, R. and Hilbert, D.: Methods of Mathematical Physics, Vol II, Wiley, New York, 1962.
3.
[3] Epstein, M.: On the wrinkling of anisotropic elastic membranes. J. Elasticity, 55, 99-109 (1999).
4.
[4] Epstein, M.: On the anelastic evolution of second-grade materials. Extracta Math., 14, 157-161 (1999).
5.
[5] Epstein, M.: Towards a complete second-order evolution law. Math. Mech. Solids, 4, 251-266 (1999).
6.
[6] Epstein, M. and Forcinito, M.A.: Anisotropic membrane wrinkling: theory and analysis. Int. J. Solids Struct., 38, 5253-5272 (2001).
7.
[7] Epstein, M. and de León, M.: Geometrical theory of uniform Cosserat media. J. Geom. Phys., 26, 127-170 (1998).
8.
[8] Epstein, M. and de León, M.: Homogeneity without uniformity: towards a mathematical theory of functionally graded materials. Int. J. Solids Struct., 37, 7577-7591 (2000).
9.
[9] Epstein, M. and Maugin, G.A.: Sur le tenseur de moment matériel d’Eshelby en élasticité non linéaire. C. R. Acad. Sci., Paris, II310, 675-768 (1990).
10.
[10] Epstein, M. and Maugin, G.A.: The energy-momentum tensor and material uniformity in finite elasticity. Acta Mech., 83, 127-133 (1990).
11.
[11] Epstein, M. and Maugin, M.A.: The electroelastic energy-momentum tensor. Proc. R. Soc. London, A433, 299-312 (1991).
12.
[12] Epstein, M. and Maugin, G.A.: On the geometrical material structure of anelasticity. Acta Mech., 115, 119-131 (1996).
13.
[13] Epstein, M. and Maugin, G.A.: Thermomechanics of volumetric growth in uniform bodies. Int. J. Plasticity, 16, 951-978 (2000).
14.
[14] Eshelby, J.D.: The force on an elastic singularity. Phil. Trans. R. Soc. London, A244, 87-112 (1951).
15.
[15] Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics, Applied Mathematical Sciences Series 137, Springer-Verlag, Berlin, 2000.
16.
[16] Hoger, A.: The material time derivative of logarithmic strainInt. J. Solids Struct., 22, 1019-1032 (1986).
17.
[17] Kienzler, R. and Herrmann, G.: Mechanics in Material Space, Springer-Verlag, Berlin, 2000.
18.
[18] Maugin, G.A.: Material Inhomogeneities in Elasticity, Applied Mathematics and Mathematical Computation Series 3, Cleveland, OH, Chemical Rubber Company Press, 1993.
[24] Steigmann, D.J.: Tension-field theory. Proc. R. Soc. London, A429, 141-173 (1990).
25.
[25] Truesdell. C. and Noll, W.: The non-linear field theories of mechanics. Handbuch der Physik, Vol III/3, Springer-Verlag, Berlin, 1965.
26.
[26] Wang, C.C.: On the geometric structure of simple bodies: a mathematical foundation for the theory of continuous distributions of dislocations. Arch. Rational Mech. Anal., 27, 33-94 (1967).
