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 − pc)/(1 − pc), 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 σc at pc 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 pc = √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 pc + σc = 1 for all aspect ratios. Similar results are presented for the case of rigid reinforcements.
ASJC Scopus subject areas
- Arts and Humanities (miscellaneous)
- Acoustics and Ultrasonics