Comparison between the Mori-Tanaka and generalized self-consistent methods in the framework of anti-plane strain inclusion problem in strain gradient elasticity

In the present paper, two different averaging schemes are compared in the framework of the anti-plane inclusion problem of the strain gradient elasticity. Effective longitudinal shear modulus of fiber-reinforced composites was evaluated based on the Mori-Tanaka method (MTM) within the direct approach and based on the generalized self-consistent method (GSCM) within an energy approach. It is known that, in gradient theories, the non-uniform fields are realized inside the inhomogeneity even in the presence of a uniform strain or stress field prescribed at the infinity. Therefore, in MTM, an averaged concentration tensor is utilized. GSCM is used together with Eshelby's formula, which evaluates the total strain energy of the composite representative fragment by a particular surface integral. Thus, in this energy-based approach, one does not need to use averaged field quantities. The two mentioned approaches were compared and validated by finite element modeling in the framework of the simplified strain gradient elasticity theory. It was shown that both methods capture the inclusion size effects, however the quantitative predictions of these methods are significantly different for high volume fraction of small inclusions. It was shown that MTM underestimates the effective modulus compared to numerical simulations, while GSCM provides a sufficiently good solution. Thus, it seems that energy-based averaging schemes are preferable in the framework of gradient elasticity. Nevertheless, the MTM can be useful for the effective properties approximate evaluation of composites with small volume fraction of inclusions.