ELASTIC MODULI OF TWO DIMENSIONAL ELASTIC CONTINUA WITH ELLIPTICAL HOLES.

The effective moduli of a composite medium containing many randomly positioned and oriented elliptical objects is calculated and the results are applied to the case of voids. Two different self-consistent effective medium approximations are described. The asymmetric self-consistent method shows that the Young's modulus E* goes to zero as E* equals E//1 (p minus p//c)/(1 minus p//c), where phi equals 1 minus p is the concentration of the voids. The subscript 1 denotes the void free material. The Poisson ratio sigma is also linear in p and goes to a value sigma //c at p//c that is independent of the initial value of Poisson's ratio sigma . The two elastic moduli decouple in this case. The corresponding elastic threshold is p//c equals left bracket 1 plus ab/(a**2 plus b**2) right bracket ** minus **1, where a and b are the major semi-axes. The symmetric self-consistent method yields a different p//c equals 2 left bracket 1 plus ROOT 2(a plus b)**2/(a**2 plus b**2) right bracket ** minus **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 plus sigma //c equals 1 for all aspect ratios.