×

On the effective properties of composite elastic plate. (English) Zbl 1458.74123

Summary: We consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. Following the results of N. Antonić and N. Balenović [Math. Commun. 4, No. 1, 111–119 (1999; Zbl 0944.49017); ZAMM, Z. Angew. Math. Mech. 80, S757–S758 (2000; Zbl 0962.74050)], and the authors et al. [“Homogenization of elastic plate equation”, Math. Model. Anal. 23, No. 2, 190–204 (2018)], we show the local character of the set of all possible composites, also called the G-closure, and prove that the set of composites obtained by periodic homogenization is dense in that set. Moreover, we derive expressions for elastic coefficients of composite plate obtained by mixing two materials in thin layers, also known as laminated materials, and for mixing two materials in the low-contrast regime. Additionally, we derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds. In the setting of two-phase isotropic materials in dimension \(d = 2\), explicit optimal Hashin-Shtrikman bounds are calculated.

MSC:

74Q15 Effective constitutive equations in solid mechanics
74Q20 Bounds on effective properties in solid mechanics
74K20 Plates
74E30 Composite and mixture properties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allaire, G., Shape Optimization by the Homogenization Method (2001), Springer
[2] Allaire, G.; Kelly, A., Optimal design of low-contrast two phase structures for the wave equation, Math. Models Methods Appl. Sci., 21, 7, 1499-1538 (2011) · Zbl 1219.74039
[3] Allaire, G.; Kohn, R. V., Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions, Q. Appl. Math., 675, 699 (1993) · Zbl 0805.73042
[4] Antonić, N.; Balenović, N., Homogenisation and optimal design for plates, Z. Angew. Math. Mech., 80, 757-758 (2000)
[5] Antonić, N.; Balenović, N., Optimal design for plates and relaxation, Math. Commun., 4 (1999) · Zbl 0944.49017
[6] Antonić, N.; Balenović, N.; Vrdoljak, M., Optimal design for vibrating plates, Z. Angew. Math. Mech., 80, 3, 783-784 (2000)
[7] Allaire, G.; Gutiérrez, S., Optimal design in small amplitude homogenization, ESAIM: Math. Model. Numer. Anal., 41, 3, 543-574 (2007) · Zbl 1148.65048
[8] Aubin, T., Some Nonlinear Problems in Riemannian Geometry (1998), Springer · Zbl 0896.53003
[9] Babadjian, J. F.; Barchiesi, M., A variational approach to the local character of G-closure: the convex case, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 26, 351-373 (2009) · Zbl 1173.35012
[10] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), Springer · Zbl 1220.46002
[11] Burazin, K.; Jankov, J., Small-amplitude homogenization of elastic plate equation, Appl. Anal. (2019)
[12] Burazin, K.; Jankov, J.; Vrdoljak, M., Homogenization of elastic plate equation, Math. Model. Anal., 23, 2, 190-204 (2018)
[13] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley: Wiley New York · Zbl 0582.49001
[14] Casado-Diaz, J., Some smoothness results for the optimal design of a two-composite material which minimizes the energy, Calc. Var. Partial Differ. Equ., 53, 649-673 (2015) · Zbl 1317.49045
[15] Dahlquist, G.; Björk, A., Numerical Methods (1974), Prentice-Hall: Prentice-Hall Englewood Cliffs, N. J., (translated by N. Anderson)
[16] Francfort, G.; Murat, F., Homogenization and optimal bounds in linear elasticity, Arch. Ration. Mech. Anal., 94, 307-334 (1986) · Zbl 0604.73013
[17] Folland, G. B., Fourier Analysis and Its Applications (1992), Wadsworth & Brooks/Cole Advanced Books & Software · Zbl 0786.42001
[18] Gérard, P., Microlocal defect measures, Commun. Partial Differ. Equ., 16, 1761-1794 (1991) · Zbl 0770.35001
[19] Gibiansky, L. V.; Cherkaev, A. V., Design of composite plates of extremal rigidity, (Topics in the Mathematical Modelling of Composite Materials. Topics in the Mathematical Modelling of Composite Materials, Ser. PNLDE, vol. 31 (1997), Birkhauser: Birkhauser Basel), 95-137 · Zbl 0920.35159
[20] Gibiansky, L. V.; Cherkaev, A. V., Microstructures of Composites of Extremal Rigidity and Exact Bounds on the Associated Energy Density, Topics in the Mathematical Modelling of Composite Materials, Ser. PNLDE, vol. 31, 273-317 (1997), Birkhauser: Birkhauser Basel · Zbl 0928.74077
[21] Grafakos, L., Classical Fourier Analysis (2008), Springer · Zbl 1220.42001
[22] Gutiérrez, S., An optimal design method based on small amplitude homogenization, Chin. Ann. Math., 35B, 5, 843-854 (2015) · Zbl 1328.49010
[23] Gutiérrez, S.; Mura, J., Small amplitude homogenization applied to inverse problems, J. Comput. Mech., 41, 699 (2008) · Zbl 1162.74359
[24] Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids, 11, 127-140 (1963) · Zbl 0108.36902
[25] Hebey, E., Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities (2000), American Mathematical Society · Zbl 0981.58006
[26] Kohn, R.; Strang, G., Optimal design and relaxation of variational problems I-III, Commun. Pure Appl. Math., 39, 113-137 (1986), 139-182, 353-377 · Zbl 0609.49008
[27] Lax, P., Linear Algebra (1997), John Wiley: John Wiley New York
[28] Lewis, A. S., Convex analysis on the Hermitian matrices, SIAM J. Optim., 6, 164-177 (1996) · Zbl 0849.15013
[29] Lurie, K. A., Applied Optimal Control Theory of Distributed Systems (1993), Springer: Springer New York · Zbl 0835.49015
[30] Lurie, K. A., On the optimal distribution of the resistivity tensor of the working substance in a magnetohydrodynamic channel, J. Appl. Math. Mech. (Prikladnaya Matematika i Mekhanika), 34, 2, 255-274 (1970), (in Russian) · Zbl 0257.76096
[31] Lurie, K. A.; Cherkaev, A. V., Exact estimates of conductivity of composites formed by two isotropically conducting media, taken in prescribed proportion, Proc. R. Soc. Edinb., 99A, 71-87 (1984) · Zbl 0564.73079
[32] Lurie, K. A.; Cherkaev, A. V., Exact estimates of the conductivity of a binary mixture of isotropic materials, Proc. R. Soc. Edinb., 104A, 21-38 (1986) · Zbl 0623.73011
[33] Lurie, K. A.; Cherkaev, A. V., Effective characteristics of composite materials and the optimal design of structural elements, Usp. Mekhaniki, 9, 3-81 (1986)
[34] Lurie, K. A.; Cherkaev, A. V.; Fedorov, A. V., Regularization of optimal design problems for bars and plates, part 2, J. Optim. Theory Appl., 37, 4 (1982) · Zbl 0464.73109
[35] Maurice, G.; Ganghoffer, J.-F.; Rahali, Y., Second gradient homogenization of multilayered composites based on the method of oscillating functions, Math. Mech. Solids, 24, 7, 2197-2230 (2019) · Zbl 07254350
[36] Milton, G. W., The Theory of Composites (2002), Cambridge University Press · Zbl 0993.74002
[37] Milton, G. W.; Briane, M.; Harutyunyan, D., On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials, Math. Mech. Complex Syst., 5, 41-94 (2017) · Zbl 1368.74053
[38] Mirsky, L., On the trace of a matrix product, Math. Nachr., 20, 171-174 (1959) · Zbl 0136.24901
[39] Muñoz, J.; Pedregal, P., On the relaxation of an optimal design problem for plates, Asymptot. Anal., 16, 125-140 (1998) · Zbl 0943.74048
[40] Murat, F.; Tartar, L., H-convergence, (Topics in the Mathematical Modelling of Composite Materials. Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., vol. 31 (1997), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 21-43 · Zbl 0920.35019
[41] Murat, F.; Tartar, L., Optimality conditions and homogenization, (Marino, A.; etal., Nonlinear Variational Problems (1985), Pitman: Pitman Boston), 1-8 · Zbl 0569.49015
[42] Phan-Thien, N., Understanding Viscoelasticity (2002), Springer · Zbl 1036.74001
[43] Raitum, U. E., The extension of extremal problems connected with a linear elliptic equation, Sov. Math., 19, 1342-1345 (1978) · Zbl 0428.49002
[44] Raitums, U., On the local representation of G-closure, Arch. Ration. Mech. Anal., 158, 213-234 (2001) · Zbl 1123.35320
[45] Tartar, L., H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. R. Soc. Edinb., 115A, 193-230 (1990) · Zbl 0774.35008
[46] Tartar, L., The General Theory of Homogenization (2009), Springer
[47] Tartar, L., Estimation de coefficients homogénéisés, (Glowinski, R.; Lions, J. L., Computing Methods in Applied Sciences and Engineering, Third International Symposium, December 1977. Computing Methods in Applied Sciences and Engineering, Third International Symposium, December 1977, Lecture Notes in Math., vol. 704 (1979), Springer Verlag), 364-373 · Zbl 0443.35061
[48] Tartar, L., Estimations Fines des Coefficients Homogénéisés, (Krée, P., Ennio de Giorgi Colloquium. Ennio de Giorgi Colloquium, Pitman Research Notes in Math., vol. 125 (1985)), 168-187 · Zbl 0586.35004
[49] Vrdoljak, M., Classical optimal design in two-phase conductivity problems, SIAM J. Control Optim., 54, 4, 2020-2035 (2016) · Zbl 1346.49029
[50] Vrdoljak, M., On some questions in relaxation theory for optimal design problems (2004), Doctoral thesis in Croatian, Zagreb
[51] Yamamoto, Y., From Vector Spaces to Function Spaces: Introduction to Functional Analysis with Applications (2012), SIAM · Zbl 1269.46001
[52] Zhikov, V. V.; Kozlov, S. M.; Oleinik, O. A.; T’en Ngoan, Kha, Averaging and G-convergence of differential operators, Russ. Math. Surv., 34, 5, 69-147 (1979) · Zbl 0445.35096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.