Abstract
We propose a notion of internal work for a generalized hyperelastic material, taking into account the topological structure of its singular locus via the Maslov class. This is based on the interpretation of the crossing of the singular locus as a phase transition.
Sommario
In questa nota proponiamo una nozione di lavoro delle forze interne per un materiale iperelastico generalizzato che tiene conto della struttura topologica del suo insieme singolare per mezzo della classe di Maslov. Questa si ottiene interpretando l'attraversamento di tale insieme singolare come una transizione di fase.
Similar content being viewed by others
References
Cardin, F., ‘Morse families and constrained mechanical systems. Generalized hyperelastic materials’,Meccanica,26 (1991) 161–167.
Müller, I., ‘On the size of hysteresis in pseudo-elasticity’,Continuum. Mech. Thermodyn., (1989) 125–142.
Abraham, R., Marsden, J.E.Foundations of Mechanics, Benjamin-Cummings 2nd ed. 1978.
Arnol'd, V.I.,Methodes Mathematiques de la Mécanique classique, MIR, Moscou, 1978.
Benenti, S., ‘Symplectic relations in analytical mechanics’,Proceedings of the IUTAMM-ISIMM Symposium on Modern Developments in Analytical Mechanics Torino, 1982.
Maslov, V.P.,Théorie des Perturbations et Méthodes Asymptotiques, Editions de l'Université de Moscou, (1965), en russe. Traduction française: Dunod-Gauthier-Villars, Paris, 1971.
Hörmander, L., ‘Fourier integral operators I’,Acta Math. 127, (1971) 79.
Weinstein, A.,Lectures on Symplectic Manifolds, CBMS Conf. Series, AMS 29, 1977.
Tulczyjew, W.M., ‘Control of static mechanical systems’,Proceedings of Dynamical Systems and Microphysics, CISM, Udine, Italy, 1984, p. 359.
Ericksen, J.L., ‘Nonlinear elasticity of diatomic crystals’,Int. J. Solids Structures,6, (1970), 951.
Ericksen, J.L., ‘Multivalued strain energy functions for crystals’,Int. J. Solids and Structures 18 (1982) 913.
Pitteri, M., ‘Onv + 1 lattices’,J. Elasticity,15 (1985) 3.
James, R.D., ‘The stability and metastability of quartz’, inS. Antman, and others (eds.) Metastability and Incompletely Posed Problems IMA, Volumes in Mathematics and Applications, 3, Springer-Verlag, 1987.
Grioli, G., ‘On the stress in rigid bodies’,Meccanica,18 (1983) 3–7.
Marzano S., Podio Guidugli P., ‘Materiali elastici approssimativamente vincolati’,Rend. Sem. Mat. Univ. Padova,73 (1985) 99–117.
Truesdell, C., Noll, W.,The Non-Linear Field Theories of Mechanics, Handbuch der Physik, Springer-Verlag, Berlin, Heidelberg, New York, 1965.
Cohen, H., Pastrone, F., ‘Axisymmetric equilibrium states of non-linear elastic cylindrical shells’,Int. J. Non-Linear Mechanics,21 (1986) 37–50.
Bott, R., Tu, L.W.,Differential Forms in Algebraic Topology, Springer GTM 82 Berlin, New York, 1982.
Doubrovine, B., Novikov, S., Fomenko A.,Géométrie Contemporaine. Méthodes et Applications, vol. 2, MIR Moscou, 1983.
Guillemin, V., Sternberg, S.,Geometric Asymptotics, American Mathematical Society Providence Rhode Island, 1977.
Arnol'd, V.I., ‘Une classe caractéristique intervenant dans le conditions de quantification’,Functional Anal. Appl. 1 (1967), 1, also reprinted in [6].
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cardin, F., Spera, M. On the internal work in generalized hyperelastic materials. Meccanica 30, 727–734 (1995). https://doi.org/10.1007/BF00986577
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00986577