### Abstract

Statistical N-point correlation functions are used for calculating properties of heterogeneous systems. The strength and the main advantage of the statistical continuum approach is the direct link to statistical information of microstructure. Two-point correlation functions are the lowest order of correlation functions that can describe the morphology and the microstructure-properties relationship. Experimentally, statistical pair correlation functions are obtained using SEM or small x-ray scattering techniques. Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim a new approximation methodology is utilized to obtain N-point correlation functions for non-FGM (functional graded materials) heterogeneous microstructures. Conditional probability functions are used to formulate the proposed theoretical approximation. In this approximation, weight functions are used to connect subsets of (N-1)-point correlation functions to estimate the full set of N-point correlation function. For the approximation of three and four point correlation functions, simple weight functions have been introduced. The results from this new approximation, for three-point probability functions, are compared to the real probability functions calculated from a computer generated three-phase reconstructed microstructure in three-dimensional space. This three-dimensional reconstruction was based on an experimental two-dimensional microstructure (SEM image) of a three-phase material. This comparison proves that our new comprehensive approximation is capable of describing higher order statistical correlation functions with the needed accuracy.

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

Pages (from-to) | 104-119 |

Number of pages | 16 |

Journal | Journal of the Mechanics and Physics of Solids |

Volume | 60 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Conditional probability
- Heterogeneous medium
- N-point correlation functions
- Statistical information of microstructure

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Condensed Matter Physics

### Cite this

*Journal of the Mechanics and Physics of Solids*,

*60*(1), 104-119. https://doi.org/10.1016/j.jmps.2011.09.009

**New approximate solution for N-point correlation functions for heterogeneous materials.** / Baniassadi, M.; Ahzi, Said; Garmestani, H.; Ruch, D.; Remond, Y.

Research output: Contribution to journal › Article

*Journal of the Mechanics and Physics of Solids*, vol. 60, no. 1, pp. 104-119. https://doi.org/10.1016/j.jmps.2011.09.009

}

TY - JOUR

T1 - New approximate solution for N-point correlation functions for heterogeneous materials

AU - Baniassadi, M.

AU - Ahzi, Said

AU - Garmestani, H.

AU - Ruch, D.

AU - Remond, Y.

PY - 2012/1

Y1 - 2012/1

N2 - Statistical N-point correlation functions are used for calculating properties of heterogeneous systems. The strength and the main advantage of the statistical continuum approach is the direct link to statistical information of microstructure. Two-point correlation functions are the lowest order of correlation functions that can describe the morphology and the microstructure-properties relationship. Experimentally, statistical pair correlation functions are obtained using SEM or small x-ray scattering techniques. Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim a new approximation methodology is utilized to obtain N-point correlation functions for non-FGM (functional graded materials) heterogeneous microstructures. Conditional probability functions are used to formulate the proposed theoretical approximation. In this approximation, weight functions are used to connect subsets of (N-1)-point correlation functions to estimate the full set of N-point correlation function. For the approximation of three and four point correlation functions, simple weight functions have been introduced. The results from this new approximation, for three-point probability functions, are compared to the real probability functions calculated from a computer generated three-phase reconstructed microstructure in three-dimensional space. This three-dimensional reconstruction was based on an experimental two-dimensional microstructure (SEM image) of a three-phase material. This comparison proves that our new comprehensive approximation is capable of describing higher order statistical correlation functions with the needed accuracy.

AB - Statistical N-point correlation functions are used for calculating properties of heterogeneous systems. The strength and the main advantage of the statistical continuum approach is the direct link to statistical information of microstructure. Two-point correlation functions are the lowest order of correlation functions that can describe the morphology and the microstructure-properties relationship. Experimentally, statistical pair correlation functions are obtained using SEM or small x-ray scattering techniques. Higher order correlation functions must be calculated or measured to increase the precision of the statistical continuum approach. To achieve this aim a new approximation methodology is utilized to obtain N-point correlation functions for non-FGM (functional graded materials) heterogeneous microstructures. Conditional probability functions are used to formulate the proposed theoretical approximation. In this approximation, weight functions are used to connect subsets of (N-1)-point correlation functions to estimate the full set of N-point correlation function. For the approximation of three and four point correlation functions, simple weight functions have been introduced. The results from this new approximation, for three-point probability functions, are compared to the real probability functions calculated from a computer generated three-phase reconstructed microstructure in three-dimensional space. This three-dimensional reconstruction was based on an experimental two-dimensional microstructure (SEM image) of a three-phase material. This comparison proves that our new comprehensive approximation is capable of describing higher order statistical correlation functions with the needed accuracy.

KW - Conditional probability

KW - Heterogeneous medium

KW - N-point correlation functions

KW - Statistical information of microstructure

UR - http://www.scopus.com/inward/record.url?scp=80755126111&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80755126111&partnerID=8YFLogxK

U2 - 10.1016/j.jmps.2011.09.009

DO - 10.1016/j.jmps.2011.09.009

M3 - Article

AN - SCOPUS:80755126111

VL - 60

SP - 104

EP - 119

JO - Journal of the Mechanics and Physics of Solids

JF - Journal of the Mechanics and Physics of Solids

SN - 0022-5096

IS - 1

ER -