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Abstrakty
In their usual form, homogenization methods produce first-order approximations of the exact solutions of problems for differential equations with rapidly oscillating coefficients which model the physical behavior of microstructured media. However, there is need of approximations containing higher-order terms when the usual first-order approximations, which are formed by superposing a macroscopic trend and a local perturbation, are not capable of reproducing the local details of the exact solutions. Here, two-scale asymptotic solutions with second-order terms are provided for a Dirichlet problem modeling the steady state of functionally-graded microperiodic nonlinear rods. The need of considering higherorder terms is illustrated through numerical examples for various power-law nonlinearities.
Rocznik
Tom
Strony
45--56
Opis fizyczny
Bibliogr. 25 poz. rys., tab.
Twórcy
autor
- Programa de Pós-Graduação em Modelagem Matemática, Universidade Federal de Pelotas Pelotas-RS, Brazil
- Instituto de Física e Matemática, Universidade Federal de Pelotas Pelotas-RS, Brazil
autor
- Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de México Ciudad de México, Mexico
Bibliografia
- [1] Panasenko, G.P. (2008). Homogenization for periodic media: from microscale to macroscale. Physics of Atomic Nuclei, 71(4), 681-694.
- [2] Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer.
- [3] Christensen, R.M. (1979). Mechanics of Composite Materials. Wiley.
- [4] Bakhvalov, N.S., & Panasenko, G.P. (1989). Homogenisation: Averaging Processes in Periodic Media. Kluwer.
- [5] Bensoussan, A., Lions, J.-L., & Papanicolau, G. (1978). Asymptotic Analysis for Periodic Structures. North-Holland.
- [6] Allaire, G. (1992). Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, 23(6), 1482-1518.
- [7] Nguetseng, G. (1989). A general convergence result for a functional related to the theory of homogenization. SIAM Journal on Mathematical Analysis, 20(3), 608-623.
- [8] Nguetseng, G. (2003). Homogenization structures and applications I. Zeitschrift für Analysis und ihre Anwendungen, 22(1), 73-108.
- [9] Spagnolo, S. (1976). Convergence in energy for elliptic operators. In: Numerical Solution of Partial Differential Equations (Vol. III). Academic Press, 468-498.
- [10] De Giorgi, E. (1984). G-operators and G-convergence. In: Proceedings of the International Congress of Mathematicians (Vol. 2). North-Holland, 1175-1191.
- [11] Murat, F., & Tartar, L. (1995). H-convergence. In: Topics in the Mathematical Modeling of Composite Materials: Progress in Nonlinear Differential Equations and Their Applications (Vol. 31). Springer, 21-43.
- [12] Tartar, L. (2010). The General Theory of Homogenization: A Personalized Introduction. Springer.
- [13] Ponte Casta˜neda, P., & Suquet, P. (1998). Nonlinear composites. Advances in Applied Mechanics, 34, 171-302.
- [14] Ponte Casta˜neda, P. (2002). Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I-theory. Journal of the Mechanics and Physics of Solids, 50(4), 737-757.
- [15] Ponte Castañeda, P., & Tiberio, E. (2000). A second-order homogenization method in finite elasticity and applications to black-filled elastomers. Journal of the Mechanics and Physics of Solids, 48(6-7), 1389-1411.
- [16] Keller, J.B. (1977). Effective behavior of heterogeneous media. In: Statistical Mechanics and Statistical Methods in Theory and Applications, Plenum, 631-644.
- [17] Keller, J.B. (1980). Darcy’s law for flow in porous media and the two-space method. In: Nonlinear Partial Equations in Engineering and Applied Science: Lecture Notes in Pure and Applied Mathematics (Vol. 54), Dekker, 429-443.
- [18] Su, F., Xu, Z., Cui, J.Z., & Dong, Q.L. (2011). Multi-scale method for the quasi-periodic structures of composite materials. Applied Mathematics and Computation, 217(12), 5847-5852.
- [19] Su, F., Xu, Z., Dong, Q.L., & Jiang, H. (2011). Multiscale computation method for parabolic problems of composite materials. Applied Mathematics and Computation, 217(21), 8337-8342.
- [20] Dong, Q.L., & Cao, L.Q. (2014). Multiscale asymptotic expansions methods and numerical algorithms for the wave equations in perforated domains. Applied Mathematics and Computation, 232, 872-887.
- [21] Hashin, Z. (1983). Analysis of composite materials - a survey. Journal of Applied Mechanics, 50, 481-505.
- [22] Elsgolts, L. (1970). Differential Equations and the Calculus of Variations. Mir.
- [23] Talbot, D.R.S., & Willis, J.R. (1994). Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry. Proceedings of the Royal Society of London A, 447, 365-384.
- [24] Dai, G., Jarrar, F., Ozturk, F., & Sheikh-Ahmad, J. (2018). On the effect of the complexity of the constitutive model in simulating superplastic forming. Defect and Diffusion Forum, 385, 379-384.
- [25] Padmanabhan, K.A., Prabu, S.B., & Ali, A.A. (2018). On the nuances in the power law description and interpretation of high homologous temperature creep and superplasticity data. Defect and Diffusion Forum, 385, 27-32.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-de230ef1-68ad-47de-ac26-07482cab4784