Homogenization of elastic plate equation∗
Abstract
We are interested in general homogenization theory for fourth-order elliptic equation describing the Kirchhoff model for pure bending of a thin solid symmetric plate under a transverse load. Such theory is well-developed for second-order elliptic problems, while some results for general elliptic equations were established by Zhikov, Kozlov, Oleinik and Ngoan (1979). We push forward an approach of Antoni´c and Balenovi´c (1999, 2000) by proving a number of properties of H-convergence for stationary plate equation.
Keyword : Kirchhoff model of elastic plate, homogenization, H-convergence, correctors
How to Cite
Burazin, K., Jankov, J., & Vrdoljak, M. (2018). Homogenization of elastic plate equation∗. Mathematical Modelling and Analysis, 23(2), 190-204. https://doi.org/10.3846/mma.2018.012
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
[1] G. Allaire. Shape optimization by the homogenization method. Springer, 2001.
[2] N. Antonić and N. Balenović. Optimal design for plates and relaxation. Mathematical Communications, 4:111–119, 1999.
[3] N. Antonić and N. Balenović. Homogenisation and optimal design for plates. Zeitschrift fur Angewandte Mathematik und Mechanik, 80:757–758, 2000.
[4] N. Antonić, N. Balenović and M. Vrdoljak. Optimal design for vibrating plates. Zeitschrift fur Angewandte Mathematik und Mechanik, 80:783–784, 2000.
[5] N. Antonić, A. Raguž and M. Vrdoljak. Homogenisation of nonlinear elliptic systems. In M. Rogina et al.(Ed.), Applied mathematics and computation, pp. 81–90, Department of Mathematics, University of Zagreb, 2001.
[6] D. Blanchard, A. Gaudiello and T. Mel’nyk. Boundary homogenization and reduction of dimension in a Kirchhoff-love plate. SIAM Journal Mathematical Analysis, 39(6):1764–1787, 2008. https://doi.org/10.1137/070685919
[7] M. Bukal and I. Velčić. On the simultaneous homogenization and dimension reduction in elasticity and locality of Γ-closure. Calculus of Variations and Partial Differential Equations, 56(3):59, 2017. https://doi.org/10.1007/s00526-017-1167-z
[8] A. Damlamian and M. Vogelius. Homogenization limits of the equations of elasticity in thin domains. SIAM Journal on Mathematical Analysis, 18(2):435–451, 1987. https://doi.org/10.1137/0518034
[9] L.V. Gibiansky and A.V. Cherkaev. Design of composite plates of extremal rigidity. In A. Cherkaev and R. Kohn(Eds.), Topics in the Mathematical Modelling of Composite Materials, pp. 95–137, Boston, MA, 1997. Birkh¨auser, Boston. https://doi.org/10.1007/978-1-4612-2032-9 5. The article is the translation of an article originally written in Russian and published as the report of Ioffe Physico Technical Institute, Academy of Sciences of USSR, publication 914, Leningrad, 1984
[10] K.A. Lurie, A.V. Cherkaev and A.V. Fedorov. Regularization of optimal design problems for bars and plates I-II. Journal of Optimization Theory and Applications, 37(4):499–543, 1982. https://doi.org/10.1007/BF00934953
[11] F. Murat and L. Tartar. H-Convergence. In Topics in the Mathematical Modelling of Composite Materials, volume 31 of Progress in Nonlinear Differential Equations and Their Applications, 1997. https://doi.org/10.1007/978-1-4612-2032-9 3. The article is the translation of notes originally written in French. Mimeographed notes. S´eminaire d’Analyse Fonctionnelle et Num´erique de l’Universit´e d’Alger, 1977/78
[12] R. Prakash. Optimal control problem for the time-dependent Kirchoff-Love plate in a domain with rough boundary. Asymptotic Analysis, 81(3-4):337–355, 2013. https://doi.org/10.3233/ASY-2012-1132
[13] L. Tartar. An introduction to the homogenization method in optimal design. In A. Cellina and A. Ornelas(Eds.), Optimal shape design, volume 1740 of Lecture Notes in Mathematics, pp. 47–156. Springer, Berlin, 2000. https://doi.org/10.1007/BFb0106742
[14] L. Tartar. The general theory of homogenization. Springer, 2009.
[15] V.V. Zhikov, S.M. Kozlov, O.A. Oleinik and Kha T’en Ngoan. Averaging and G-convergence of differential operators. Russian Mathematical Surveys, 34(5):69–147, 1979. https://doi.org/10.1070/RM1979v034n05ABEH003898
[2] N. Antonić and N. Balenović. Optimal design for plates and relaxation. Mathematical Communications, 4:111–119, 1999.
[3] N. Antonić and N. Balenović. Homogenisation and optimal design for plates. Zeitschrift fur Angewandte Mathematik und Mechanik, 80:757–758, 2000.
[4] N. Antonić, N. Balenović and M. Vrdoljak. Optimal design for vibrating plates. Zeitschrift fur Angewandte Mathematik und Mechanik, 80:783–784, 2000.
[5] N. Antonić, A. Raguž and M. Vrdoljak. Homogenisation of nonlinear elliptic systems. In M. Rogina et al.(Ed.), Applied mathematics and computation, pp. 81–90, Department of Mathematics, University of Zagreb, 2001.
[6] D. Blanchard, A. Gaudiello and T. Mel’nyk. Boundary homogenization and reduction of dimension in a Kirchhoff-love plate. SIAM Journal Mathematical Analysis, 39(6):1764–1787, 2008. https://doi.org/10.1137/070685919
[7] M. Bukal and I. Velčić. On the simultaneous homogenization and dimension reduction in elasticity and locality of Γ-closure. Calculus of Variations and Partial Differential Equations, 56(3):59, 2017. https://doi.org/10.1007/s00526-017-1167-z
[8] A. Damlamian and M. Vogelius. Homogenization limits of the equations of elasticity in thin domains. SIAM Journal on Mathematical Analysis, 18(2):435–451, 1987. https://doi.org/10.1137/0518034
[9] L.V. Gibiansky and A.V. Cherkaev. Design of composite plates of extremal rigidity. In A. Cherkaev and R. Kohn(Eds.), Topics in the Mathematical Modelling of Composite Materials, pp. 95–137, Boston, MA, 1997. Birkh¨auser, Boston. https://doi.org/10.1007/978-1-4612-2032-9 5. The article is the translation of an article originally written in Russian and published as the report of Ioffe Physico Technical Institute, Academy of Sciences of USSR, publication 914, Leningrad, 1984
[10] K.A. Lurie, A.V. Cherkaev and A.V. Fedorov. Regularization of optimal design problems for bars and plates I-II. Journal of Optimization Theory and Applications, 37(4):499–543, 1982. https://doi.org/10.1007/BF00934953
[11] F. Murat and L. Tartar. H-Convergence. In Topics in the Mathematical Modelling of Composite Materials, volume 31 of Progress in Nonlinear Differential Equations and Their Applications, 1997. https://doi.org/10.1007/978-1-4612-2032-9 3. The article is the translation of notes originally written in French. Mimeographed notes. S´eminaire d’Analyse Fonctionnelle et Num´erique de l’Universit´e d’Alger, 1977/78
[12] R. Prakash. Optimal control problem for the time-dependent Kirchoff-Love plate in a domain with rough boundary. Asymptotic Analysis, 81(3-4):337–355, 2013. https://doi.org/10.3233/ASY-2012-1132
[13] L. Tartar. An introduction to the homogenization method in optimal design. In A. Cellina and A. Ornelas(Eds.), Optimal shape design, volume 1740 of Lecture Notes in Mathematics, pp. 47–156. Springer, Berlin, 2000. https://doi.org/10.1007/BFb0106742
[14] L. Tartar. The general theory of homogenization. Springer, 2009.
[15] V.V. Zhikov, S.M. Kozlov, O.A. Oleinik and Kha T’en Ngoan. Averaging and G-convergence of differential operators. Russian Mathematical Surveys, 34(5):69–147, 1979. https://doi.org/10.1070/RM1979v034n05ABEH003898