### Abstract

The two-dimensional elliptic inclusion problems have been widely studied and many important results have been obtained. Nevertheless, explicit real-form solutions for the stress inside the inclusion and in the matrix around the inclusion are available only for special anisotropic elastic materials. Solutions for which the inclusion as well as the matrix are of general anisotropic elastic materials are presented in this paper using the Stroh formalism. New identities which relate Stroh's complex eigenvalues and eigenvectors to real quantities obtainable directly from the elastic constants are derived, through which several solutions are expressed in real form. Among the real-form solutions obtained are the stress inside the inclusion, the hoop stress in the matrix around the interface boundary and the rotation of the inclusion. The special cases in which the inclusion is rigid or a void are also studied. For the elliptic hole subject to a uniform tensile stress perpendicular to the major axis of the elliptic hole, the stress-concentration factor at the end points of the major axis of the hole has a simple expression resembling the well-known classical result 1+ 2(a/b) for isotropic materials where 2a, 2b are, respectively, the major and minor axes of the elliptic hole. The hoop stress at the end points of the minor axis is independent of the shape of the elliptic hole.

Original language | English |
---|---|

Pages (from-to) | 553-572 |

Number of pages | 20 |

Journal | Quarterly Journal of Mechanics and Applied Mathematics |

Volume | 42 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1989 Nov 1 |

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### All Science Journal Classification (ASJC) codes

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics

### Cite this

*Quarterly Journal of Mechanics and Applied Mathematics*,

*42*(4), 553-572. https://doi.org/10.1093/qjmam/42.4.553

}

*Quarterly Journal of Mechanics and Applied Mathematics*, vol. 42, no. 4, pp. 553-572. https://doi.org/10.1093/qjmam/42.4.553

**Two-dimensional problems of the anisotropic elastic solid with an elliptic inclusion.** / Hwu, Chyanbin; Ting, T. C.T.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Two-dimensional problems of the anisotropic elastic solid with an elliptic inclusion

AU - Hwu, Chyanbin

AU - Ting, T. C.T.

PY - 1989/11/1

Y1 - 1989/11/1

N2 - The two-dimensional elliptic inclusion problems have been widely studied and many important results have been obtained. Nevertheless, explicit real-form solutions for the stress inside the inclusion and in the matrix around the inclusion are available only for special anisotropic elastic materials. Solutions for which the inclusion as well as the matrix are of general anisotropic elastic materials are presented in this paper using the Stroh formalism. New identities which relate Stroh's complex eigenvalues and eigenvectors to real quantities obtainable directly from the elastic constants are derived, through which several solutions are expressed in real form. Among the real-form solutions obtained are the stress inside the inclusion, the hoop stress in the matrix around the interface boundary and the rotation of the inclusion. The special cases in which the inclusion is rigid or a void are also studied. For the elliptic hole subject to a uniform tensile stress perpendicular to the major axis of the elliptic hole, the stress-concentration factor at the end points of the major axis of the hole has a simple expression resembling the well-known classical result 1+ 2(a/b) for isotropic materials where 2a, 2b are, respectively, the major and minor axes of the elliptic hole. The hoop stress at the end points of the minor axis is independent of the shape of the elliptic hole.

AB - The two-dimensional elliptic inclusion problems have been widely studied and many important results have been obtained. Nevertheless, explicit real-form solutions for the stress inside the inclusion and in the matrix around the inclusion are available only for special anisotropic elastic materials. Solutions for which the inclusion as well as the matrix are of general anisotropic elastic materials are presented in this paper using the Stroh formalism. New identities which relate Stroh's complex eigenvalues and eigenvectors to real quantities obtainable directly from the elastic constants are derived, through which several solutions are expressed in real form. Among the real-form solutions obtained are the stress inside the inclusion, the hoop stress in the matrix around the interface boundary and the rotation of the inclusion. The special cases in which the inclusion is rigid or a void are also studied. For the elliptic hole subject to a uniform tensile stress perpendicular to the major axis of the elliptic hole, the stress-concentration factor at the end points of the major axis of the hole has a simple expression resembling the well-known classical result 1+ 2(a/b) for isotropic materials where 2a, 2b are, respectively, the major and minor axes of the elliptic hole. The hoop stress at the end points of the minor axis is independent of the shape of the elliptic hole.

UR - http://www.scopus.com/inward/record.url?scp=0024770098&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024770098&partnerID=8YFLogxK

U2 - 10.1093/qjmam/42.4.553

DO - 10.1093/qjmam/42.4.553

M3 - Article

VL - 42

SP - 553

EP - 572

JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

IS - 4

ER -