Research paperThe “rigorous” Maxwell homogenization scheme in 2D elasticity: Effective stiffness tensor of composite with elliptic inhomogeneities

- “Rigorous” Maxwell's scheme is formulated in terms of the elastic dipole moments.
- The complete solution for a finite cluster model of composite has been found.
- The elastic moduli of composite with elliptical inclusions have been evaluated.
- An accuracy of the Maxwell's scheme is comparable with that of Rayleigh's method.

Maxwell's homogenization scheme is formulated in terms of the induced dipole moments of the representative volume element (RVE) of actual composite and properly defined equivalent inclusion. Taking interactions between the inhomogeneities into account makes this scheme rigorous in the sense that the effective properties converge to their exact values with increasing RVE size. To demonstrate potential of the “rigorous” Maxwell's scheme, the 2D elasticity problem for a composite with elliptical inhomogeneities is considered. The complete series solutions for a finite cluster of inhomogeneities and equivalent anisotropic inclusion have been obtained by the method of complex potentials. The effective elastic moduli of composite have been found by equating the dipole strengths of RVE and equivalent inclusion. The convergence of solution is verified and an effect of structural parameters on the effective stiffness tensor of the composite has been evaluated. Numerical study shows that the proposed version of Maxwell's scheme enables evaluation of the effective properties of both periodic and random structure composites with accuracy, comparable with that of Rayleigh's method.