Elastic moduli of two-dimensional composite continua with elliptical inclusions

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Abstract
The elastic field around an e l l i p t i c a l i n c l u s i o n 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 a s y m m e t r i c self‐consistent method shows that the Young’s modulus E goes to zero as E=E 1(p−p c )/(1−p c ), where 1−p is the concentration of the v o i d s and subscript one denotes the void free material. The Poisson ratio σ is also linear in p 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 =[1+a b/(a 2+b 2)], where a and b are the major semiaxes. The s y m m e t r i c self‐consistent method yields a different p c =2[1+(2(a+b)2/(a 2+b 1))1/2]−1 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.

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