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Representation formulas for L∞norms of weakly convergent sequences of gradient fields in homogenization

Published online by Cambridge University Press:  13 February 2012

Robert Lipton  and
Tadele Mengesha
Robert Lipton
Affiliation:
Department of Mathematics, Louisiana State University, 384 Lockett Hall, Baton Rouge, LA 70803-4918, USA. [email protected]
Tadele Mengesha
Affiliation:
Department of Mathematics, Louisiana State University, 384 Lockett Hall, Baton Rouge, LA 70803-4918, USA. [email protected]
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Abstract

We examine the composition of the L∞ norm with weaklyconvergent sequences of gradient fields associated with the homogenization of second orderdivergence form partial differential equations with measurable coefficients. Here thesequences of coefficients are chosen to model heterogeneous media and are piecewiseconstant and highly oscillatory. We identify local representation formulas that in thefine phase limit provide upper bounds on the limit superior of theL∞ norms of gradient fields. The local representationformulas are expressed in terms of the weak limit of the gradient fields and localcorrector problems. The upper bounds may diverge according to the presence of roughinterfaces. We also consider the fine phase limits for layered microstructures and forsufficiently smooth periodic microstructures. For these cases we are able to provideexplicit local formulas for the limit of the L∞ norms of theassociated sequence of gradient fields. Local representation formulas for lower bounds areobtained for fields corresponding to continuously graded periodic microstructures as wellas for general sequences of oscillatory coefficients. The representation formulas areapplied to problems of optimal material design.

Keywords

L∞normsnonlinear compositionweak limitsmaterial designhomogenization
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1121 - 1146
Copyright
© EDP Sciences, SMAI, 2012

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References

Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, (1992) 1482–1518. Google Scholar
Avellaneda, M. and Lin, F.-H., Compactness methods in the theory of homogenization. Comm. Pure Appl. Math. 40 (1987) 803–847. Google Scholar
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications 5. North-Holland, Amsterdam (1978)
Bonnetier, E. and Vogelius, M., An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section. SIAM J. Math. Anal. 31 (2000) 651–677. Google Scholar
Boyarsky, B.V., Generalized solutions of a system of differential equations of the first order of elliptic type with discontinuous coefficients. Mat. Sb. N. S. 43 (1957) 451–503. Google Scholar
Caffarelli, L.A. and Peral, I., On W 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1–21. Google Scholar
Carlos-Bellido, J., Donoso, A. and Pedregal, P., Optimal design in conductivity under locally constrained heat flux. Arch. Rational Mech. Anal. 195 (2010) 333–351. Google Scholar
Casado-Diaz, J., Couce-Calvo, J. and Martin-Gomez, J.D., Relaxation of a control problem in the coefficients with a functional of quadratic growth in the gradient. SIAM J. Control Optim. 47 (2008) 1428–1459. Google Scholar
Chipot, M., Kinderlehrer, D. and Vergara-Caffarelli, L., Smoothness of linear laminates. Arch. Rational Mech. Anal. 96 (1985) 81–96. Google Scholar
De Giorgi, E. and Spagnolo, S., Sulla convergenza degli integrali dell’ energia peroperatori ellittici del secondo ordine. Boll. UMI 8 (1973) 391–411. Google Scholar
Duysinx, P. and Bendsoe, M.P., Topology optimization of continuum structures with local stress constraints. Internat. J. Numer. Math. Engrg. 43 (1998) 1453–1478. Google Scholar
Faraco, D., Milton’s conjecture on the regularity of solutions to isotropic equations. Ann. Inst. Henri Poincare, Nonlinear Analysis 20 (2003) 889–909. Google Scholar
Fujii, D., Chen, B.C. and Kikuchi, N., Composite material design of two-dimensional structures using the homogenization design method. Internat. J. Numer. Methods Engrg. 50 (2001) 2031–2051. Google Scholar
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin, New York (2001).
J.H. Gosse and S. Christensen, Strain invariant failure criteria for polymers in composite materials. AIAA (2001) 1184.
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, New York (1994).
Jimenez, S. and Lipton, R., Correctors and field fluctuations for the p ϵ(x)-Laplacian with rough exponents. J. Math. Anal. Appl. 372 (2010) 448–469. Google Scholar
A. Kelly and N.H. Macmillan, Strong Solids. Monographs on the Physics and Chemistry of Materials. Clarendon Press, Oxford, (1986).
Leonetti, F. and Nesi, V., Quasiconformal solutions to certain first order systems and the proof of a conjecture of G.W. Milton. J. Math. Pures. Appl. 76 (1997) 109–124. Google Scholar
Li, Y.Y. and Nirenberg, L., Estimates for elliptic systems from composite material. Comm. Pure Appl. Math. LVI (2003) 892–925. Google Scholar
Li, Y.Y. and Vogelius, M., Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Rational Mech. Anal. 153 (2000) 91–151. Google Scholar
Lipton, R., Assessment of the local stress state through macroscopic variables. Phil. Trans. R. Soc. Lond. Ser. A 361 (2003) 921–946. Google ScholarPubMed
Lipton, R., Bounds on the distribution of extreme values for the stress in composite materials. J. Mech. Phys. Solids 52 (2004) 1053–1069. Google Scholar
R. Lipton, Homogenization and design of functionally graded composites for stiffness and strength, in Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, edited by P.P. Castaneda et al., Kluwer Academic Publishers, Netherlands (2004) 169–192.
Lipton, R., Homogenization and field concentrations in heterogeneous media. SIAM J. Math. Anal. 38 (2006) 1048–1059. Google Scholar
Lipton, R. and Stuebner, M., Inverse homogenization and design of microstructure for point wise stress control. Quart. J. Mech. Appl. Math. 59 (2006) 139–161. Google Scholar
Lipton, R. and Stuebner, M., Optimal design of composite structures for strength and stiffness : an inverse homogenization approach. Struct. Multidisc. Optim. 33 (2007) 351–362. Google Scholar
R. Lipton and M. Stuebner, A new method for design of composite structures for strength and stiffness, 12th AIAA/ISSMO Multidisciplinary Analysis & Optimization Conference. American Institute of Aeronautics and Astronautics Paper AIAA, Victoria British Columbia, Canada (2008) 5986.
Markworth, A.J., Ramesh, K.S. and Parks, W.P., Modelling studies applied to functionally graded materials. J. Mater. Sci. 30 (1995) 2183–2193. Google Scholar
Meyers, N., An L p-Estimate for the gradient of solutions of second order elliptic divergence equations. Annali della Scuola Norm. Sup. Pisa 17 (1963) 189–206. Google Scholar
G.W. Milton, Modeling the properties of composites by laminates, edited by J. Erickson, D. Kinderleher, R.V. Kohn and J.L. Lions. Homogenization and Effective Moduli of Materials and Media, IMA Volumes in Mathematics and Its Applications 1 (1986) 150–174.
F. Murat and L. Tartar, Calcul des Variations et Homogénéisation, Les Méthodes de l’Homogénéisation : Théorie et Applications en Physique, edited by D. Bergman et al. Collection de la Direction des Études et Recherches d’Electricité de France 57 (1985) 319–369.
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. Google Scholar
Nuismer, R.J. and Whitney, J.M., Uniaxial failure of composite laminates containing stress concentrations, in Fracture Mechanics of Composites, ASTM Special Technical Publication, American Society for Testing and Materials 593 (1975) 117–142. Google Scholar
Ootao, Y., Tanigawa, Y. and Ishimaru, O., Optimization of material composition of functionally graded plate for thermal stress relaxation using a genetic algorithim. J. Therm. Stress. 23 (2000) 257–271. Google Scholar
Papanicolaou, G. and Varadhan, S.R.S., Boundary value problems with rapidly oscillating random coefficients, Random fields, Rigorous results in statistical mechanics and quantum field theory, Esztergom 1979. Colloq. Math. Soc. Janos Bolyai 27 (1981) 835–873. Google Scholar
E. Sanchez-Palencia, Non Homogeneous Media and Vibration Theory. Springer, Heidelberg (1980).
S. Spagnolo, Convergence in Energy for Elliptic Operators, Proceedings of the Third Symposium on Numerical Solutions of Partial Differential Equations, edited by B. Hubbard. College Park (1975); Academic Press, New York (1976) 469–498.