Abstract

In this work, the asymptotic homogenization method (AHM) is applied to a Dirichlet boundary problem for a non-homogeneous, one-dimensional, elliptic, second-order differential equation with continuously differentiable coefficient and continuous non-homogeneity. Both the coefficient and non-homogeneity exhibit locally periodic and rapidly oscillating behaviors. As alternatives to the exact solution, three formal asymptotic solutions (FASs) are presented in the form of two-scale series in terms of powers of the small geometric parameter characterizing the separation of structural scales of the locally microperiodic medium modeled by the described problem. Such FASs are constructed from the solutions of the recurrent sequence of problems for the coefficients of the powers of the small parameter, which is form by the so-called homogenized problem and local problems over the periodicity cell. Finally, an example is presented in order to illustrate the fact that the three FASs obtained by applying the AHM are good approximations of the exact solution.

Keywords

• Asymptotic homogenization method
• Formal asymptotic solution
• Homogenized problem
• Local problems
• Effective coefficient

References

• BAKHVALOV, N. S.; PANASENKO, G. P. Homogenisation: Averaging Processes in Periodic
• Media. Dordrecht: Kluwer Academic Publishers, 1989.
• BRAVO-CASTILLERO, J.; GUINOVART-D ́IAZ, R.; SABINA, F. J.; RODR ́ 350 IGUEZ-RAMOS,
• R. Closed-form expressions for the effective coefficients of fiber-reinforced composites with transversely isotropic constituents - II. Piezoelectric and square symmetry. Mechanics of Materials, v.33, n.1, p.237-248, 2001.
• BRAVO-CASTILLERO, J.; GUINOVART-D ́IAZ, R.; SABINA, F. J.; RODR ́ 354 IGUEZ-RAMOS,
• R.;SABINA DE LIS, J. C.; VALDIVIEZO-MIJANGOS, O. C. Effective elastic properties of periodic fibrous composites. Limit cases. Applications to porous and nonlinear materials.Computer Assisted Mechanics and Engineering Sciences, v.13, n.2, p.305-322, 2006.
• BRUNO, O. P. The effective conductivity of strongly heterogeneous composites. Proceedings of the Royal Society of London A, v.433, n.1, p.353-381, 1991.
• DECIO JR, R. M. S.; PÉREZ-FERNANDEZ, L. D.; BRAVO-CASTILLERO, J. Exactness of formal asymptotic solutions of a Dirichlet problem modeling the steady state of functionally graded microperiodic nonlinear rods. Journal of Applied Mathematics and Computational Mechanics, v.18, n.3, p.45-56, 2019.
• GUINOVART-D ́IAZ, R.; BRAVO-CASTILLERO, J.; RODR ́ 364 IGUEZ-RAMOS, R.; SABINA, F.J. Closed-form expressions for the effective coefficients of fiber-reinforced composites with transversely isotropic constituents. I: Elastic and hexagonal symmetry. Journal of the Mechanics and Physics of Solids, v.49, n.1, p.1445-1462, 2001.
• LARSSON, S.; THOMEE, V. Partial Differential Equations with Numerical Methods. Berlin Heidelberg: Springer-Verlag, 2009.
• LIMA, E. L. Curso de Análise, v.1. Rio de Janeiro: IMPA, 2010, 12a ed.
• LIMA, M. P. de; LAZZARI, L.; FERNANDEZ, L. dos S.; PEREZ-FERNÁNDEZ, L. D.; BRAVO-CASTILLERO, J. Homogeneização assintótica da equação do calor para meios unidimensionais periódicos continuamente heterogˆeneos. Vetor, v.26, n.2, p.73-83, 2016.
• LIMA, M. P. de; PEREZ-FERNÁNDEZ, L. D.; BRAVO-CASTILLERO, J. Homogenization of a continuously microperiodic multidimensional medium. Tendˆencias em Matemática Aplicada e Computacional, v.19, n.1, p.15-32, 2018.
• LUZ, L. N. M., PEREZ-FERNÁNDEZ, L. D., BRAVO-CASTILLERO, J. Comparação do caso contínuo com o caso contínuo por partes com contato perfeito para a equação elíptica via método de homogeneização assintótica. Revista Interdisciplinar De Pesquisa Em Engenharia, 380 v.7, n.2, p.17–29, 2022.
• ROCHA, F. C. da. Introdução `a técnica de camada-limite. Cadernos de Engenharia de Estruturas, v.12, n.55, p.37-50, 2010.
• PANASENKO, G. P. Homogenization for periodic media: from microscale to macroscale. Physics of Atomic Nuclei, v.71, n.4, p.681-694, 2008.
• RODRÍGUEZ-RAMOS, R.; SABINA, F. J.; BRAVO-CASTILLERO, J.; GUINOVART-D ́ 385 IAZ,R. Closed-form expressions for the effective coefficients of fiber-reinforced composites with transversely isotropic constituents - I. Elastic and square symmetry. Mechanics of Materials, v.33, n.1, p.223-235, 2001.
• SABINA, F. J.; RODRÍGUEZ-RAMOS, R.; BRAVO-CASTILLERO, J.; GUINOVART-D 389 IAZ, R. Closed-form expressions for the effective coefficients of fiber-reinforced composites with transversely isotropic constituents. II: Piezoelectric and hexagonal symmetry. Journal of the Mechanics and Physics of Solids, v.49, n.1, p.1463-1479, 2001.
• SABINA, F. J.; RODRÍGUEZ-RAMOS, R.; BRAVO-CASTILLERO, J.; GUINOVART-D ́ 393 IAZ, R.; VALDIVIEZO-MIJANGOS, O. C. Overall behavior of two-dimensional periodic composites. International Journal of Solids and Structures, v.39, n.2, p.483-497, 2002.
• SADD, M. H. Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier Academic Press,
• , 4th 397 ed.
• PEREZ-FERNÁNDEZ, L. D. Homogeneização assintótica de laminados microperiódicos el ́asticos não lineares. SigmaE, v.10, n.2, p.1-21, 2021.
• TORQUATO, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. New York: Springer-Verlag, 2002.

Paper information