Improved bounds for the effective energy of nonlinear 3D conducting composites
A. León–Mecíasª, J. Bravo–Castilleroª, A. Mesejo–Chiongª, L.D. Pérez–Fernándezb, and F.J. Sabinac
ª Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Habana 4, CP–10400, Cuba.
b Instituto de Cibernética, Matemática y Física, Calle 15 e/t CyD, Vedado 10400, Habana 4, Cuba.
c Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20–726, Delegación de Álvaro Obregón, 01000 México D.F, México.
]]> * Corresponding author:
Recibido el 6 de abril de 2006
Aceptado el 26 de abril de 2007
Abstract
Recent variational inequalities of Talbot are used to improve the lower and upper bounds for the effective energy of nonlinear 3–D two–phase conducting composites. The effective conductivity of the linear isotropic two–phase periodic conducting composite used as comparison material in the inequalities is computed through an asymptotic homogenization model by finite element analysis of the local problem on the three–dimensional cubic unit cell with one spherical inclusion. A brief mathematical description of the numerical method is included. Numerical calculations of the effective conducting linear property are compared with Bruno's bounds. It shows that the numerical solution for the limit cases of superconducting and empty inclusions improves the bounds when the inclusion volume fraction is greater than about 0.4. It is natural to expect an improvement in the whole volume fraction of Talbot's bounds for nonlinear conducting composites when the numerical calculation is used instead of bounds for the linear comparison problem, as is the case here.
]]> Keywords: Variational bounds; effective properties; conducting composites; asymptotic homogenization method; finite element method.
Resumen
Las cotas inferior y superior para la energía efectiva de un compuesto conductor no lineal, tridimensional bifásico son mejoradas usando desigualdades variacionales de Talbot. La conductividad efectiva del compuesto de comparación periódico, bifásico, lineal, usado en las desigualdades, se obtiene resolviendo los problemas locales que aparecen al aplicar el método de homogeneización asintótica, mediante un analisis de elemento finito, tomando como celda unitaria un cubo con una inclusión esférica. Los métodos numéricos empleados se describen brevemente. Los resultados numéricos para la propiedad efectiva del compuesto de comparación se comparan con las cotas de Bruno. Se observa que una de las cotas está muy cerca de la solución numérica, para los casos límites de inclusión superconductora y vacía, y cuando la fracción volumétrica de la inclusión es mayor que 0.4, esta se aleja. Es natural entonces esperar una mejora, en todo el rango de fracciones volumétricas de las cotas de Talbot para compuestos conductores no lineales, cuando se usan los cálculos numéricos, en lugar de las cotas para el compuesto de comparación lineal, como es aquí el caso.
Descriptores: Cotas variacionales; propiedades efectivas; compuestos conductores; método de homogeneización asintótica; método de elemento finito.
PACS: 02.70.Dc; 46.15.Cc; 66.70.+f
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]]> Acknowledgements
The authors are grateful to professor Harald Berger from Otto von Guericke University for useful comments during his visit to Havana University and to Lino E. Borroto from ICIMAF, Cuban Academy of Science, who taught us the first steps using ANSYS. Suggestions and recommendations of members of the Group of Solids Mechanics (R. Rodríguez Ramos and R. Guinovart Díaz) from Havana University, were also very important in this work. The provisions of the Projects PNCB CITMA No. 9/2004 and PAPIIT, DGAPA, UNAMIN101705 are also acknowledged. We wish to express out thanks to the Department of Mathematics and Mechanics of the IIMAS–UNAM and in particular to Ana Pérez Arteaga and Ramiro Chávez for computational support. AL is very grateful for the grant of a scholarship from Red de Macro–Universidades de América Latina y el Caribe. JB recognizes the support of a scholarship from Foundation CIMAC. The authors gratefully acknowledge the Referee's recommendations.
References
1. G. Allaire, SIAM J Math Anal 23 (1992) 1482. [ Links ]
2. N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media. Mathematics and Applications (Dordrecht: Kluwer Academic Publishers, 1989). [ Links ]
3. A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for periodic Structures (Amsterdam: North–Holland, 1978). [ Links ]
4. O.P Bruno, Proc. R. Soc. Lond A 433 (1991) 353. [ Links ]
5. J.B. Castillero, J.A. Otero, R.R. Ramos, and A. Bourgeat, Int. J. Sol. Struct. 35 (1998) 527. [ Links ]
6. J. Bravo–Castillero, R. Guinovart–Díaz, F.J. Sabina, and R. Rodríguez–Ramos, Mech. Mater. 33 (2001) 237. [ Links ]
7. Z. Fang, W. Sun, and J.T. Tzeng, J. of Composite Materials 38 (2004) 1371. [ Links ]
8. A. Galka, J.J. Telega, and R. Wojnar, Comp. Assisted Mech. Eng Sci. 3 (1996) 133. [ Links ]
9. L. Gibiansky and S. Torquato, J. Appl. Phys. 84 (1998) 301. [ Links ]
10. R. Guinovart–Díaz, J. Bravo–Castillero, R. Rodríguez–Ramos, and F.J. Sabina, J. Mech. Phys. Solids 49 (2001) 1445. [ Links ]
11. Z. Hashin and S. Shtrikman, J. Appl. Phys. 33 (1962) 3125. [ Links ]
12. P. Knabner and L. Angermann, Numerik partieller Differential–gleichungen (Berlin Heidelberg: Springer, 2000). [ Links ]
13. J. Llorca, M. Elices, and Y. Termona, Acta mater. 48 (2000) 4589. [ Links ]
14. G.W. Milton, The Theory of Composites (Cambridge University Press 2002). [ Links ]
15. O.A. Oleinik, A.S. Schmaev, and G.A. Yosifian, Mathematical problems in elasticity and homogenization (Amsterdam: North Holland Publishing Company, 1992). [ Links ]
16. J.A. Otero, J. Bravo–Castillero, R. Guinovart–Díaz, R. Rodríguez–Ramos, and G.A. Maugin, Arch. Mech. 55 (2003) 357. [ Links ]
17. F. Périn, European Journal of Mechanics A/Solids 23 (2004) 139. [ Links ]
18. B.E. Pobedrya, Mechanics of composites Materials (Moscow: Moscow University Press, Russian 1984). [ Links ]
19. P. Ponte Castañeda, Phil. Trans. R. Soc. Lond. A 340 (1992) 531. [ Links ]
20. R. Rodríguez–Ramos, F.J. Sabina, R. Guinovart–Díaz, and J. Bravo–Castillero, Mech. Mater. 33 (2001) 223. [ Links ]
21. F.J. Sabina, R. Rodríguez–Ramos, J. Bravo–Castillero, and R. Guinovart–Díaz, J. Mech. Phys. Solids 49 (2001) 1463. [ Links ]
22. F.J. Sabina, J. Bravo–Castillero, R. Guinovart–Díaz, R. Rodríguez–Ramos, and O.C. Valdiviezo–Mijangos, International Journal of Solids and Structures 39 (2002) 483. [ Links ]
23. E. Sánchez–Palencia, Non Homogeneous Media and Vibration Theory. Lecture Notes in Physics (Berlin: Springer Verlag, 1980). [ Links ]
24. J. Segurado and J. Llorca, J. Mech. Phys. Solids 50 (2002) 2107. [ Links ]
25. D.R.S. Talbot and J.R. Willis, IMA J. Appl. Math. 35 (1985) 39. [ Links ]
26. D.R.S. Talbot, Proc. R. Soc. Lond. A 455 (1999) 3617. [ Links ]
27. S. Torquato, J. Appl. Phys. 58 (1985) 3790. [ Links ]
28. S. Torquato, Random Heterogeneous Materials. Microstructure and Macroscopic Properties (Springer, 2002). [ Links ]
29. J.R. Willis, J. Appl. Mech. 50 (1983) 1202. [ Links ]
30. O.C. Zienkiewicz, The Finite Element Method in Engineering Science (McGraw–Hill Publishing Company, 1977). [ Links ] ]]>