By Qing-Hua Qin, Qing-Sheng Yang

This booklet features a accomplished therapy of heterogeneous fabrics lower than coupled thermal, magnetic, electrical, and mechanical lots. The easy-to-understand textual content clarifies the most complicated suggestions for analysing and fixing multifield difficulties of heterogeneous fabrics: micromechanics technique and homogenization strategy. Readers will enjoy the authors’ thorough insurance of the basics by way of specified mathematical derivation with labored examples.

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**Example text**

The self-consistent model or M-T model in conjunction with the FEM can be applied to dealing with composites with arbitrary shaped inclusions. It is assumed that a typical inclusion is embedded in an infinite effective medium subjected to a uniform strain "&. This boundary value problem can be solved by FEM and then the average strain in the inclusion can be obtained. 20a). In this process, the unknown effective 28 Chapter 2 Homogenization theory for heterogeneous materials properties must be used in FE calculation, requiring an iteration procedure [41].

7) is effective modulus along the axial or single direction. Ei (i = m, f) are the elastic moduli of the constituents. The equation above indicates that the upper bound of effective stiffness can be expressed by a mixture law. This result is referred to as the Voigt approximation [30]. (2) Constant stress: It is assumed that a homogeneous traction boundary condition is applied to a composite , and the stresses in the matrix and fiber of the composite are the same. 8) This is referred to as the Reuss approximation [31].

23), we can see that the first term in Eq. 23) is the well-known rule of mixture, while the second term is a correction term due to the heterogeneity of the microstructure. In the constant strain model, it is assumed that the strains undergone in each phase have the same values. 2) This is the known rule of mixture. A simple expression under uniaxial state is - -- -1 El1 V where Ell f El1 dV -- vlE (I) II y is the Young's modulus, Vj + v 2 E II(2) +... 3) are the volume fraction and the Young's modulus of phase i, respectively.