### Abstract

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 = E_{1}(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 in p and goes to a value σ(c) at p(c) that is independent of the initial value of Poisson's ratio σ_{1}. Unlike the corresponding three-dimensional case, the two elastic moduli decouple in this special case. The corresponding elastic threshold is p(c) = [1 + ab /(a^{2} + b^{2})], where a and b are the major semiaxes. The symmetric self-consistent method yields a different p(c) = 2[1 + √2(a + b^{2})^{2}/(a^{2} + b^{1})]^{-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.

Original language | English (US) |
---|---|

Pages (from-to) | 1674-1680 |

Number of pages | 7 |

Journal | Journal of the Acoustical Society of America |

Volume | 77 |

Issue number | 5 |

State | Published - 1985 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Acoustics and Ultrasonics

### Cite this

*Journal of the Acoustical Society of America*,

*77*(5), 1674-1680.

**Elastic moduli of two-dimensional composite continua with elliptical inclusions.** / Thorpe, Michael; Sen, P. N.

Research output: Contribution to journal › Article

*Journal of the Acoustical Society of America*, vol. 77, no. 5, pp. 1674-1680.

}

TY - JOUR

T1 - Elastic moduli of two-dimensional composite continua with elliptical inclusions

AU - Thorpe, Michael

AU - Sen, P. N.

PY - 1985

Y1 - 1985

N2 - 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 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 + ab /(a2 + b2)], where a and b are the major semiaxes. The symmetric self-consistent method yields a different p(c) = 2[1 + √2(a + b2)2/(a2 + b1)]-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.

AB - 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 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 + ab /(a2 + b2)], where a and b are the major semiaxes. The symmetric self-consistent method yields a different p(c) = 2[1 + √2(a + b2)2/(a2 + b1)]-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.

UR - http://www.scopus.com/inward/record.url?scp=0021807226&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0021807226&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0021807226

VL - 77

SP - 1674

EP - 1680

JO - Journal of the Acoustical Society of America

JF - Journal of the Acoustical Society of America

SN - 0001-4966

IS - 5

ER -