Comparison of continuous case with continuous case piecewise continuous with perfect contact for the elliptical equation via asymptotic homogenization method

Authors

  • Larissa N. Meirelles Luz Universidade Federal de Pelotas
  • Leslie Pérez-Fernández Universidade Federal de Pelotas
  • Julian Bravo-Castillero Universidad Nacional Autónoma de México

Keywords:

Conductive means., Continuous and piecewise constant microperiodic heterogeneity., Effective behavior., Asymptotic homogenization., Maximum principle.

Abstract

The methods of mathematical homogenization allow the effective properties of heterogeneous media to be found with great precision and rigor based on the physical and geometric properties of their components. In particular, the asymptotic homogenization method is used to find the coefficients that represent the effective properties of a medium with a periodic structure. The present work aims to study this mathematical homogenization technique to obtain the effective behavior of micro-heterogeneous media, and to apply mathematical formalism to build a formal asymptotic solution of a one-dimensional linear problem with continuous and constant coefficients by parts. Still, the proximity between the solutions of the original and homogenized problems will be mathematically justified. In order to illustrate the theoretical results, an example is presented considering both types of heterogeneity in a case that presents the same effective behavior. 

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References

Bakhvalov, N. S.; Panasenko, G. P. Homogenisation: Averaging processes in periodic media. Dordrecht: Kluwer Academic Publishers, 1989. https://doi.org/10.1007/978-94-009-2247-1

Einstein, A. Eine neue Bestimmung der Moleküldimensionen. Annalen der Physik, v.324, n.2, p.289–306, 1906. https://doi.org/10.1002/andp.19063240204

Kudriavtsev, L. D. Curso de Análisis Matemático: Tomo I. Moscou: Mir, 1983.

Maxwell, J. C. Treatise on Electricity and Magnetism. Oxford: Clarendon Press, 1873.

Rayleigh, L. On the influence of obstacles arranged in a rectangular order upon the properties of medium. Philosophical Magazine, v.34, p.481–502, 1892. https://doi.org/10.1080/14786449208620364

Sadd, M. H. Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier Academic Press, 4ta. Ed., 2020. https://doi.org/10.1016/C2017-0-03720-5

Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties. New York: Springer-Verlag, 2002. https://doi.org/10.1007/978-1-4757-6355-3

Caballero-Pérez, R. O.; Bravo-Castillero, J.; Pérez-Fernández, L. D. A simple scheme for calculating the energy harvesting figures of merit of porous ceramics. Energy Harvesting and Systems, v. 7, n. 1, p. 25-32, 2020a. https://doi.org/10.1515/ehs-2021-0001

Hashin, Z. Analysis of composite materials—a survey. Journal of Applied Mechanics, v. 50, p. 481-505, 1983. https://doi.org/10.1115/1.3167081

Caballero-Pérez, R. O., Bravo-Castillero, J., Pérez-Fernández, L. D., Rodríguez-Ramos, R., & Sabina, F. J. Computation of effective thermo-piezoelectric properties of porous ceramics via asymptotic homogenization and finite element methods for energy-harvesting applications. Archive of Applied Mechanics, v. 90, p. 1415–1429, 2020b. https://doi.org/10.1007/s00419-020-01675-6

Sabina, F. J., Guinovart-Díaz, R., Espinosa-Almeyda, Y., Rodríguez-Ramos, R., Bravo-Castillero, J., López-Realpozo, J. C., ... & Sánchez-Dehesa, J. Effective transport properties for periodic multiphase fiber-reinforced composites with complex constituents and parallelogram unit cells. International Journal of Solids and Structures, v. 204, p. 96-113, 2020. https://doi.org/10.1016/j.ijsolstr.2020.08.001

Bravo-Castillero, J., Ramírez-Torres, A., Sabina, F. J., García-Reimbert, C., Guinovart-Díaz, R., & Rodríguez-Ramos, R. Analytical formulas for complex permittivity of periodic composites. estimation of gain and loss enhancement in active and passive composites. Waves in Random and Complex Media, v. 30, n. 4, p. 593-613, 2020. https://doi.org/10.1080/17455030.2018.1546063

Yañez-Olmos, D., Bravo-Castillero, J., Ramírez-Torres, A., Rodríguez-Ramos, R., & Sabina, F. J. Effective coefficients of isotropic complex dielectric composites in a hexagonal array. Technische Mechanik, v. 39, n. 2, p. 220-228, 2019. https://doi.org/10.24352/UB.OVGU-2019-020

Caballero-Pérez, R. O., Bravo-Castillero, J., Pérez-Fernández, L. D., Rodríguez-Ramos, R., & Sabina, F. J. Homogenization of thermo-magneto-electro-elastic multilaminated composites with imperfect contact. Mechanics Research Communications, v. 97, p. 16-21, 2019. https://doi.org/10.1016/j.mechrescom.2019.04.005

Álvarez-Borges, F. E., Bravo-Castillero, J., Cruz, M. E., Guinovart-Díaz, R., Pérez-Fernández, L. D., Rodríguez-Ramos, R., & Sabina, F. J. Reiterated homogenization of a laminate with imperfect contact: gain-enhancement of effective properties. Applied Mathematics and Mechanics, v. 39, n. 8, p. 1119-1146, 2018. https://doi.org/10.1007/s10483-018-2352-6

Guinovart-Sanjuán, D., Vajravelu, K., Rodríguez-Ramos, R., Guinovart-Díaz, R., Bravo-Castillero, J., Lebon, F., & Sabina, F. J. et al. Analysis of effective elastic properties for shell with complex geometrical shapes. Composite Structures, v. 203, p. 278-285, 2018. https://doi.org/10.1016/j.compstruct.2018.07.036

Mattos, Lucas Prado; cruz, Manuel Ernani; Bravo-Castillero, Julián. Finite element computation of the effective thermal conductivity of two-dimensional multi-scale heterogeneous media. Engineering Computations, v. 35 n. 5, p. 2107-2123, 2018. https://doi.org/10.1108/EC-11-2017-0444

Rodríguez-Ramos, R., Otero, J. A., Cruz-González, O. L., Guinovart-Díaz, R., Bravo-Castillero, J., Sabina, F. J., ... & Sevostianov, I. Computation of the relaxation effective moduli for fibrous viscoelastic composites using the asymptotic homogenization method. International Journal of Solids and Structures, v. 190, p. 281-290, 2020. https://doi.org/10.1016/j.ijsolstr.2019.11.014

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Published

2022-02-16

How to Cite

Luz, L. N. M., Pérez-Fernández, L., & Bravo-Castillero, J. (2022). Comparison of continuous case with continuous case piecewise continuous with perfect contact for the elliptical equation via asymptotic homogenization method. Revista Interdisciplinar De Pesquisa Em Engenharia, 7(2), 17–29. Retrieved from https://periodicos.unb.br/index.php/ripe/article/view/35015

Issue

Vol. 7 No. 2 (2021): Revista Interdisciplinar de Pesquisa em Engenharia

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Artigos

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