### Abstract

Statistical continuum theory based approaches are commonly used for the computation of the effective properties of heterogeneous materials. Statistical distribution and morphology of the microstructure are represented by n-point probability function. One-point probability function statistical representation of the microstructure leads to volume fraction dependent homogenization. However, second and higher order probability functions include the information of phase distribution and morphology. Most statistical based homogenization methods are limited to two-point probability function due to the lack of simple approximation of higher order probability functions that can be easily exploited. In this paper, a new approximation of the three-point probability function is proposed and discussed. The new approximation results are compared to existing approximations from the literature and to the real probability functions calculated from a computer generated two-phase micrographs.

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

Pages (from-to) | 3782-3787 |

Number of pages | 6 |

Journal | International Journal of Solids and Structures |

Volume | 46 |

Issue number | 21 |

DOIs | |

Publication status | Published - 15 Oct 2009 |

Externally published | Yes |

### Fingerprint

### Keywords

- Heterogeneous media
- Probability functions
- Statistical continuum

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Materials Science(all)
- Condensed Matter Physics
- Applied Mathematics
- Modelling and Simulation

### Cite this

*International Journal of Solids and Structures*,

*46*(21), 3782-3787. https://doi.org/10.1016/j.ijsolstr.2009.07.004

**A new approximation for the three-point probability function.** / Mikdam, A.; Makradi, A.; Ahzi, Said; Garmestani, H.; Li, D. S.; Y. Remond, Remond.

Research output: Contribution to journal › Article

*International Journal of Solids and Structures*, vol. 46, no. 21, pp. 3782-3787. https://doi.org/10.1016/j.ijsolstr.2009.07.004

}

TY - JOUR

T1 - A new approximation for the three-point probability function

AU - Mikdam, A.

AU - Makradi, A.

AU - Ahzi, Said

AU - Garmestani, H.

AU - Li, D. S.

AU - Y. Remond, Remond

PY - 2009/10/15

Y1 - 2009/10/15

N2 - Statistical continuum theory based approaches are commonly used for the computation of the effective properties of heterogeneous materials. Statistical distribution and morphology of the microstructure are represented by n-point probability function. One-point probability function statistical representation of the microstructure leads to volume fraction dependent homogenization. However, second and higher order probability functions include the information of phase distribution and morphology. Most statistical based homogenization methods are limited to two-point probability function due to the lack of simple approximation of higher order probability functions that can be easily exploited. In this paper, a new approximation of the three-point probability function is proposed and discussed. The new approximation results are compared to existing approximations from the literature and to the real probability functions calculated from a computer generated two-phase micrographs.

AB - Statistical continuum theory based approaches are commonly used for the computation of the effective properties of heterogeneous materials. Statistical distribution and morphology of the microstructure are represented by n-point probability function. One-point probability function statistical representation of the microstructure leads to volume fraction dependent homogenization. However, second and higher order probability functions include the information of phase distribution and morphology. Most statistical based homogenization methods are limited to two-point probability function due to the lack of simple approximation of higher order probability functions that can be easily exploited. In this paper, a new approximation of the three-point probability function is proposed and discussed. The new approximation results are compared to existing approximations from the literature and to the real probability functions calculated from a computer generated two-phase micrographs.

KW - Heterogeneous media

KW - Probability functions

KW - Statistical continuum

UR - http://www.scopus.com/inward/record.url?scp=69249216511&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69249216511&partnerID=8YFLogxK

U2 - 10.1016/j.ijsolstr.2009.07.004

DO - 10.1016/j.ijsolstr.2009.07.004

M3 - Article

VL - 46

SP - 3782

EP - 3787

JO - International Journal of Solids and Structures

JF - International Journal of Solids and Structures

SN - 0020-7683

IS - 21

ER -