Abstract
Abstract: A rotationally invariant function f defined on the set Mn × n of all real square matrices of order n has a representation f on ℝn through the signed singular values of the matrix argument A ∊ Mn×n Using necessary and sufficient conditions for the rank 1 convexity of f in terms of f, three iterative procedures are derived to construct the rank 1 convex hull of f.
This is a preview of subscription content, log in via an institution to check access.
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
Ball, J.M. (1977). Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63, 337–403.
Ball, J.M. and James, R.D. (1987). Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., 100, 13–52.
Ball, J.M. and James, R.D. (1993). Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. Royal Soc. Lond. 338, 389–450.
Bladon, P., Terentjev, E.M. and Warner, M. (1993). Transitions and instabilities in liquidcrystal elastomers. Phys. Rev. E 47, R3838–R3840.
Chipot, M. and Kinderlehrer, D. (1988). Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103, 237–277.
Dacorogna, B. (1990). Direct methods in the calculus of variations, Springer, Berlin.
DeSimone, A. and Dolzmann, G. (2001). Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies. Preprint No. 41 Max-Planck Institute, Leipzig.
Kohn, R.V. and Strang, G. (1986). Optimal design and relaxation of variational problems, I, II, III. Comm. Pure Appl. Math., 39, 113–137, 139–182, 353–377.
Le Dret, H. and Raoult, A. (1994). Envelope quasi-convexe de la densité d’énergie de Saint Venant-Kirchhoff. C. R. Acad. Sci. Paris, 318, 93–98.
Le Dret, H. and Raoult, A. (1995). The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function. Proc. Royal Soc. Edinburgh, 125A, 1179–1192.
Morrey Jr., C.B. (1966). Multiple integrals in the calculus of variations, Springer, New York.
Müller, S. (1999). Variational models for microstructure and phase transitions. In Calculus of variations and geometric evolution problems (Cetraro, 1996) Lecture notes in Math. 1713, Berlin, Springer, 85–210.
Pedregal, P. (1997). Parameterized measures and variational principles, Birkhäuser, Basel.
Rosakis, P. (1997). Characterization of convex isotropic functions. J. Elasticity, 49, 257–267.
Roubíéek, T. (1997). Relaxation in optimization theory and variational calculus, W. de Gruyter, Berlin.
Silhavý, M. (1997). The mechanics and thermodynamics of continuous media, Springer, Berlin.
Silhavý, M. (1999). Convexity conditions for rotationally invariant functions in two dimensions. In Applied Nonlinear Analysis, A. Sequeira et al. (ed.), Kluwer Academic, New York, 513–530.
Silhavý, M. (1999). On isotropic rank 1 convex functions. Roy Soc. Edinburgh, 129A, 1081–1105.
Silhavý, M. (2001). Rotationally invariant rank 1 convex functions. Applied Mathematics & Optimization, 44, 1–15.
Silhavý, M. (2001). Rank 1 Convex hulls of isotropic functions in dimension 2 by 2. Mathematica Bohemica, 126, 521–529.
Silhavý, M. (2001). Monotonicity of rotationally invariant convex and rank 1 convex functions. Proc. Royal Soc. Edinburgh. To appear.
Silhavý, M. (2001). On the semiconvexity properties of rotationally invariant functions in two dimensions. To be published.
Silhavý, M. (2001). Relaxation in a class of SO (n)-invariant energies related to nematic elastomers. To be published.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
šilhavý, M. (2003). Rank 1 Convex Hulls of SO(n)-Invariant Functions. In: Miehe, C. (eds) IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Solid Mechanics and Its Applications, vol 108. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0297-3_2
Download citation
DOI: https://doi.org/10.1007/978-94-017-0297-3_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6239-0
Online ISBN: 978-94-017-0297-3
eBook Packages: Springer Book Archive