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Homogenization Theory

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Book cover Numerical Simulation of Mechanical Behavior of Composite Materials

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Abstract

The representation of the behavior of homogeneous materials is carried out through mathematical laws or equations formulated at macroscopic level in the context of the mechanics of continuous media. Most formulations extrapolate this concept and regard the behavior of composite materials from a macroscopic point of view but it is disregarded from the compounding materials’ perspective. However, during the last decades several formulations have been developed to obtain the global behavior of composite materials through the stress and strain fields generated in the compounding materials. Consequently, new “multiscale approaches” have been proposed for the representation of the behavior of heterogeneous materials which use a representative elemental volume for the composite material modeling. The sphere assembly model proposed by Hashin (1962), (1983), is among these representation approaches, in which a domain is filled up by spheres of different sizes respecting the volumetric relation among the phases. Another method called “self-consistent method” was proposed by Hill (1965), Budiansky (1965), Hashin (1970), and Christensen (1979), where the heterogeneities of a medium are represented by an ellipsoidal or cylindrical inclusion within an infinite matrix of unknown elastic properties. There are also models based on the “Mori-Tanaka method” by Mori-Tanaka (1973), which considers cylindrical, ellipsoidal or plane fibers or fractures embedded in an isotropic matrix transversely isotropic or orthotropic. These methods follow an “eigenstrain”-based formulation, which considers an elastic, linear, homogeneous and infinite solid and the eigenstrain is also admitted. This idea was originally proposed by Eshelby (1958).

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Oller, S. (2014). Homogenization Theory. In: Numerical Simulation of Mechanical Behavior of Composite Materials. Lecture Notes on Numerical Methods in Engineering and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-04933-5_5

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  • DOI: https://doi.org/10.1007/978-3-319-04933-5_5

  • Publisher Name: Springer, Cham

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