Mechanics of Composite Solids
Publication: Journal of Engineering Mechanics
Volume 128, Issue 8
Abstract
The mechanical properties of composite, elastic solids are determined using the method of volume averaging. Volume-averaged forms of Hooke’s law and Cauchy’s equation are developed for each phase. When the condition of local displacement equilibrium is valid, the volume-averaged displacement vectors are essentially equal and the spatially smoothed equations for each phase can be added to obtain a one-equation model of a two-phase system. This one-equation model contains area integrals of the spatial deviation displacement vector, and a closure problem is developed to determine this vector. The closed form for the stress-deformation relation contains effective coefficients to be determined by the solution of the closure problem.
Get full access to this article
View all available purchase options and get full access to this article.
References
Auriault, J.-L., and Bonnet, G.(1985). “Dynamique des composites elastiques periodiques.” Arch. Mech., 37, 269–284.
Beavers, G. S., and Joseph, D. D.(1967). “Boundary conditions at a naturally permeable wall.” J. Fluid Mech., 30, 197–207.
Bensoussan, A., Lions, J. L., and Papanicolaou, G. (1978). Asymptotic analysis for periodic structures, North-Holland, Amsterdam.
Bourgeat, A., Quintard, M., and Whitaker, S.(1988). “Eléments de comparaison entre la méthode d’homogénésation et la méthode prise de moyenne avec fermature.” C.R. Acad. Sci. Paris, t.306, Série II, 463–466.
Bowen, R. M.(1982). “Compressible porous media models by use of the theory of mixtures.” Int. J. Eng. Sci., 20, 697–735.
Ene, H. I., and Poliševski, D. (1987). Thermal flow in porous media, D. Reidel, Dordrecht.
Gray, W. G., Leijnse, A., Kolar, R. L., and Blain, C. A. (1993). Mathematical tools for changing spatial scales in the analysis of physical systems, CRC, Boca Raton, Fla.
Hashin, Z., and Shtrikman, S.(1963). “A variational approach to the theory of elastic behavior of multiphase materials.” J. Mech. Phys. Solids, 10, 127–140.
Jikov, V. V., Kozlov, S. M., and Oleinik, O. A. (1994). Homogenization of differential operators and integral functionals, Springer, New York.
Ochoa-Tapia, J. A., and Whitaker, S.(1995a). “Momentum transfer at the boundary between a porous medium and a homogeneous fluid. I: Theoretical development.” Int. J. Heat Mass Transf., 38, 2635–2646.
Ochoa-Tapia, J. A., and Whitaker, S.(1995b). “Momentum transfer at the boundary between a porous medium and a homogeneous fluid. II: Comparison with experiment.” Int. J. Heat Mass Transf., 38, 2647–2655.
Ochoa-Tapia, J. A., and Whitaker, S.(1997). “Heat transfer at the boundary between a porous medium and a homogeneous fluid.” Int. J. Heat Mass Transf., 40, 2691–2707.
Ochoa-Tapia, J. A., and Whitaker, S.(1998a). “Heat transfer at theboundary between a porous medium and a homogeneous fluid: The one-equation model.” J. Porous Media, 1, 31–46.
Ochoa-Tapia, J. A., and Whitaker, S.(1998b). “Momentum transfer at the boundary between a porous medium and a homogeneous fluid: Inertial effects.” J. Porous Media, 1, 201–217.
Quintard, M., and Whitaker, S.(1994a). “Transport in ordered and disordered porous media. I: The cellular average and the use of weighting functions.” Transp. Porous Media, 14, 163–177.
Quintard, M., and Whitaker, S.(1994b). “Transport in ordered and dis-ordered porous media. II: Generalized volume averaging.” Transp. Porous Media, 14, 179–206.
Quintard, M., and Whitaker, S.(1994c). “Transport in ordered and disordered porous media. III: Closure and comparison between theory and experiment.” Transp. Porous Media, 15, 31–49.
Quintard, M., and Whitaker, S.(1994d). “Transport in ordered and disordered porous media. IV: Computer-generated porous media.” Transp. Porous Media, 15, 51–70.
Quintard, M., and Whitaker, S.(1994e). “Transport in ordered and disordered porous media. V: Geometrical results for two-dimensional systems.” Transp. Porous Media, 15, 183–196.
Saffman, P.(1971). “On the boundary condition at the surface of a porous medium.” Stud. Appl. Math., 50, 93–101.
Sanchez-Palencia, E. (1980). “Nonhomogeneous media and vibration theory.” Lecture notes in physics, Springer, New York, 127.
Slattery, J. C. (1990). Interfacial transport phenomena, Springer, New York.
Torquato, S., and Hyun, S.(2001). “Effective-medium approximation for composite media: Realizable single-scale dispersions.” J. Appl. Phys., 89, 1725–1729.
Whitaker, S. (1999). The method of volume averaging, Kluwer Academic, Dordrecht.
Whitaker, S. (2002). “A detailed analysis of the mechanics of composite solids.” PDF file, available at [email protected]
Wood, B. D., Quintard, M., and Whitaker, S.(2000). “Jump conditions at nonuniform boundaries: The catalytic surface.” Chem. Eng. Sci., 55, 5231–5245.
Wood, B. D., Cherblanc, F., Quintard, M., and Whitaker, S. (2002). “Volume averaging for determining the effective dispersion tensor: Closure using periodic unit cells and comparisons with ensemble averaging theory.” Water Resour. Res., in press.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 128 • Issue 8 • August 2002
Pages: 823 - 828
Copyright
Copyright © 2002 American Society of Civil Engineers.
History
Received: Mar 25, 2002
Accepted: Mar 25, 2002
Published online: Jul 15, 2002
Published in print: Aug 2002
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.