Multipole Expansion Method in Micromechanics of Composites

Abstract

In this Chapter, the development of Multipole Expansion Method and recent advances in its application to the micromechanics of composites are reviewed. Pioneered by J.C. Maxwell and Lord Rayleigh, the multipole expansion appears historically the first method of micromechanics. It employs a classical toolkit of mathematical physics and provides a complete solution for both the local fields and effective properties of composite. Rigorous analytical approach yields the simple and efficient computational algorithms whose results commonly serve as the benchmark data for other methods of micromechanics. The rational way of introducing the macro parameters of composite is discussed and the general formulas for the effective properties in terms of the mean gradients and dipole moments of disturbance fields are given. The technique of the method is demonstrated step-by-step starting with the problem for a single inclusion immersed in non-uniform far field, regarded as the generalized Eshelby’s model. Then, the multi-inclusion, finite cluster and representative unit cell models of composite are considered. These advanced models enable taking the microstructure of composite and interaction between the inclusions into account and can be viewed as the generalized Maxwell’s and Rayleigh’s models, respectively. The considered specific problems include conductivity and elasticity of particulate and fibrous composites with isotropic and anisotropic constituents as well as with perfect or imperfect interfaces. The perspectives of further development of the method in terms of theory and application area are outlined.