Abstract
The basic results in homogenization theory are revisited in the abstract context of continuum mechanics in which the constitutive behaviour and the kinematic constraints are governed by pairs of conjugate convex potentials. The theory and the methods of this generalized elastic model are briefly recalled and applied to extend the classical linear theory of homogenization to the non-linear and possibly multivalued constitutive framework.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hashin, Z., Shtrikman, S. (1962): On some variational principles in anisotropic and non-homogeneous elasticity. J. Mech. Phys. Solids 10, 335–342
Hashin, Z., Shtrikman, S. (1962): A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10, 343–352
Rockafellar, R.T. (1970): Convex analysis. Princeton University Press, Princeton
Yosida, K. (1974): Functional analysis. 4th edition. Springer, New York
Ekeland, I., Temam, R. (1976): Convex analysis and variational problems. North-Holland, Amsterdam
Ioffe, A.D., Tihomirov, V.M. (1974): Theory of extremal problems. (Russian). Nauka, Moscow. Translation (1979): North-Holland, Amsterdam
Talbot, D.R.S., Willis, J.R. (1985): Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Math. 35, 39–54
Willis, J.R. (1989): The structure of overall constitutive relations for a class of nonlinear composites. IMA J. Appl. Math. 43, 231–242
Toland, J.F., Willis, J.R. (1989): Duality for families of natural variational principles in nonlinear electrostatics. SIAM J. Math. Anal. 20, 1283–1292
Romano, G., Rosati, L., Marotti de Sciarra, F., Bisegna, P. (1993): A potential theory for monotone multivalued operators. Quart. Appl. Math. 51, 613–631
Hiriart-Urruty, J.B., Lemaréchal, C. (1993): Convex analysis and minimization algorithms. Vols. I, II. Springer, Berlin
Romano, G. (1995): New results in subdifferential calculus with applications to convex optimization. Appl. Math. Optim. 32, 213–234
Romano, G. (2000): Structural mechanics. II. Continuous models. Libero, Napoli
Romano, G. (2000): On the necessity of Korn’s inequality. Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2000). Galway, Ireland, July 9–14, 2000
Romano, G. (2001): Scienza delle costruzioni. Tomo Zero. Hevelius, Benevento
Romano, G. (2002): Scienza delle costruzioni. Tomo I. Hevelius, Benevento
Romano, G. (2003): Scienza delle costruzioni. Tomo II. Hevelius, Benevento
Romano, G., Diaco, M., Sellitto, C. (2004): Tangent stiffness of elastic continua on manifolds. In: Romano, G., Rionero, S. (eds.): Recent trends in the applications of mathematics to mechanics. Springer, Berlin, pp. 155–184
Romano, G., Diaco, M. (2004): A functional framework for applied continuum mechanics. In: Fergola, P., Capone, F. (eds.): New Trends in Mathematical Physics. World Scientific, Singapore, to appear
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Italia
About this paper
Cite this paper
Romano, A., Romano, G. (2005). Basic issues in convex homogenization. In: Rionero, S., Romano, G. (eds) Trends and Applications of Mathematics to Mechanics. Springer, Milano . https://doi.org/10.1007/88-470-0354-7_15
Download citation
DOI: https://doi.org/10.1007/88-470-0354-7_15
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-0269-2
Online ISBN: 978-88-470-0354-5
eBook Packages: EngineeringEngineering (R0)