# Interaction between a punch and an arbitrary crack or inclusion in a transversely isotropic half-space

## Abstract

Here, we consider the problem of an arbitrary shaped rigid punch pressed against the boundary of a transversely isotropic half-space and interacting with an arbitrary flat crack or inclusion, located in the plane parallel to the boundary. The set of governing integral equations is derived for the most general conditions, namely the presence of both normal and tangential stresses under the punch, as well as general loading of the crack faces. In order to verify correctness of the derivations, two different methods were used to obtain governing integral equations: generalized method of images and utilization of the reciprocal theorem. Both methods gave the same results. Axisymmetric coaxial case of interaction between a rigid inclusion and a flat circular punch both centered along the z-axis is considered as an illustrative example. Most of the final results are presented in terms of elementary functions.

- Authors:

- Fabrikant-Research, Montreal, QC (Canada)
- Suffolk Univ., Boston, MA (United States). Dept. of Mathematics and Computer Science
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Center for Nanophase Materials Science (CNMS)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1426573

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Zeitschrift fuer Angewandte Mathematik und Physik

- Additional Journal Information:
- Journal Volume: 69; Journal Issue: 1; Journal ID: ISSN 0044-2275

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Fabrikant, I., Karapetian, E., and Kalinin, S. V.
```*Interaction between a punch and an arbitrary crack or inclusion in a transversely isotropic half-space*. United States: N. p., 2017.
Web. doi:10.1007/s00033-017-0894-5.

```
Fabrikant, I., Karapetian, E., & Kalinin, S. V.
```*Interaction between a punch and an arbitrary crack or inclusion in a transversely isotropic half-space*. United States. doi:10.1007/s00033-017-0894-5.

```
Fabrikant, I., Karapetian, E., and Kalinin, S. V. Sat .
"Interaction between a punch and an arbitrary crack or inclusion in a transversely isotropic half-space". United States.
doi:10.1007/s00033-017-0894-5.
```

```
@article{osti_1426573,
```

title = {Interaction between a punch and an arbitrary crack or inclusion in a transversely isotropic half-space},

author = {Fabrikant, I. and Karapetian, E. and Kalinin, S. V.},

abstractNote = {Here, we consider the problem of an arbitrary shaped rigid punch pressed against the boundary of a transversely isotropic half-space and interacting with an arbitrary flat crack or inclusion, located in the plane parallel to the boundary. The set of governing integral equations is derived for the most general conditions, namely the presence of both normal and tangential stresses under the punch, as well as general loading of the crack faces. In order to verify correctness of the derivations, two different methods were used to obtain governing integral equations: generalized method of images and utilization of the reciprocal theorem. Both methods gave the same results. Axisymmetric coaxial case of interaction between a rigid inclusion and a flat circular punch both centered along the z-axis is considered as an illustrative example. Most of the final results are presented in terms of elementary functions.},

doi = {10.1007/s00033-017-0894-5},

journal = {Zeitschrift fuer Angewandte Mathematik und Physik},

number = 1,

volume = 69,

place = {United States},

year = {Sat Dec 09 00:00:00 EST 2017},

month = {Sat Dec 09 00:00:00 EST 2017}

}