Elastic moduli of two-dimensional composite continua with elliptical inclusions

The elastic field around an elliptical inclusion in two dimensions is obtained. This result is then used to compute the effective moduli of a composite medium containing many randomly positioned and oriented elliptical objects. Two different self-consistent methods are described and the special cases of voids and rigid reinforcement are considered in detail. The asymmetric self-consistent method shows that the Young’s modulus E goes to zero as E = E₁(p − p_c)/(1 − p_c), where 1 − p is the concentration of the voids and subscript one denotes the void free material. The Poisson ratio σ is also linear inp and goes to a value σ_c at p_c that is independent of the initial value of Poisson’s ratio σ₁. Unlike the corresponding three-dimensional case, the two elastic moduli decouple in this special case. The corresponding elastic threshold is p_c = √2(a + b)²/(a²+b¹)]^-1, where a and b are the major semiaxes. The symmetric self-consistent method yields a different [1 + ab /(a² + b²)] and a different concentration dependence of the effective moduli. Both methods give the same result for circular inclusions and both methods give p_c + σ_c = 1 for all aspect ratios. Similar results are presented for the case of rigid reinforcements.