Abstract
We investigate the interplay and intertwining of the amplitude and frequency of highly oscillatory sequences of bounded weakly differentiable maps forming laminated microstructures and fine structures.
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. (1989). A version of the fundamental theorem for Young measures. (Partial Differential Equations and Continuum Models of Phase Transitions (M. Rascle, D. Serre and M. Slemrods, eds.), Lecture Notes in Physics) (344) (Springer-Verlag, New-York).
Ball, J.M., Holmes, P.J., James, R.D., Pego, R.L. and Swart, P.J. (1971). On the dynamics of fine structure. J. Nonlinear Sci., 1:17–70.
Ball, J.M. and James, R.D. (1987). Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., 100:13–52.
Ciarlet, P.G. (1991). Basic Error Estimates for Elliptic Problems. in: Handbook of Numerical Analysis, P.G. Ciarlet, J.L. Lions Editors, North-Holland.
Collins, C. and Luskin, M. (1991). Optimal order estimates for the finite element approximation of the solution of a non-convex variational problem. Math. Comp., 621–637
Collins, C., Luskin, M. and Kinderlehrer, D. (1991). Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal., 28.
Chu, C. (1993). Hysteresis and microstructures: a study of biaxial loading on compound twins of copper-aluminium-nickel single crystals. Ph.D. dissertation (University of Minnesota).
Ericksson, K., Estep, D., Hansbo, P. and Johnson, C. (1996). Computational Differential Equations. Cambridge University Press.
Friesecke, G. and McLeod, J.B. (1996). Dynamics as a mechanism preventing the formation of finer and finer microstructure. Arch. Rat. Mech. Analysis, 133.
Friesecke, G. and McLeod, J.B. (1997). Dynamic stability of nonminimizing phase mixtures. Preprint.
Gromov, M. (1986). Partial differential relations. Springer.
Klouček, P (1999). The Finite Element Approximations of Binomial Microstructures. CAAM Technical report TR98-14, Rice University, to appear in SIAM J. Numer. Anal.
Klouček, P (1998). The Relaxation of Non-Quasiconvex VariationalIntegrals. Num. Math., in press.
Klouček, P (1998). The computational modeling of nonequilibrium thermodynamics of the martensitic transformation. J. Comp. Mech., 22(3).
Klouček, P, Bo Li and Luskin, M. (1996). Nonconforming finite element approximation of the microstructure. Math. Comp., 65(215):1111–1135.
Kohn, R. and Müller, S.(1992). Branching of twins near an austenite/twinned-martensite interface. Philosophical Magazine, 6A:697–715.
Li, B. and Luskin, M.(1996). Finite element analysis of microstructure for the cubic to tetragonal transformation. To appear in SIAM J. Mumer. Math.
Luskin, M. (1996). Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math.
Luskin, M. (1996). On the computation of Crystalline Microstructure. Acta Numerica.
Müller, S. and Šverák, V. (1995). Attainment results for the two-well problem by convex integration. Manuscript.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Kluwer Academic Publishers
About this chapter
Cite this chapter
Klouček, P. (2002). Remarks on Compactness in the Formation of Fine Structures. In: Sequeira, A., da Veiga, H.B., Videman, J.H. (eds) Applied Nonlinear Analysis. Springer, Boston, MA. https://doi.org/10.1007/0-306-47096-9_18
Download citation
DOI: https://doi.org/10.1007/0-306-47096-9_18
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-306-46303-7
Online ISBN: 978-0-306-47096-7
eBook Packages: Springer Book Archive