# Proximity Effects in Matrix-Inclusion Composites: Elastic Effective Behavior, Phase Moments, and Full-Field Computational Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Generation of Representative Volume Element of Matrix-Inclusion Composite with an Effective Minimal Distance between Inclusions

## 3. Elastic Properties and Homogenization

#### 3.1. Stiffness Contrasts between Phases

#### 3.2. Analytical Homogenization—Mori–Tanaka Estimates

#### 3.2.1. The First Moment

#### 3.2.2. The Second Moment

#### 3.3. Computational Homogenization—Full-Field Simulations

## 4. Numerical Results

#### 4.1. Results for the Effective Properties

#### 4.2. Results for Phase Mean Fields

#### 4.2.1. The First Moment

**Hydrostatic Stress**

**Equivalent Stress**

#### 4.2.2. The Second Moment

#### 4.3. Results for the Local Fields

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation Formulas for Second-Moment Calculation

## Appendix B. Complementary Results for the Effective Properties

**Figure A1.**Mean values and standard deviations on the mean value for the apparent moduli as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure A2.**Mean values and standard deviations on the mean value for the apparent moduli as a function of the packing parameter ${s}_{min}/r$ ($r=5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

## Appendix C. Complementary Results for Mean Fields

#### Appendix C.1. First Moment

**Figure A3.**Mean values and standard deviations on the mean value for the first moment of hydrostatic stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure A4.**Mean values and standard deviations on the mean value for the first moment of hydrostatic stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=25.1$ %.

**Figure A5.**Mean values and standard deviations on the mean value for the first moment of equivalent stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

#### Appendix C.2. Second Moment

**Figure A6.**Mean values and standard deviations on the mean value for the second moment of hydrostatic stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure A7.**Mean values and standard deviations on the mean value for the second moment of hydrostatic stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=25.1$ %.

**Figure A8.**Mean values and standard deviations on the mean value for the second moment of equivalent stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

## Appendix D. Complementary Results for Local Fields

**Figure A9.**Frequency histograms of equivalent stress by phase at $SR=10$, ${f}_{v}=13.4$ %, and the extreme ${s}_{min}$ values (${s}_{min}=0.1r$ and ${s}_{min}=1.2r$) for contrasts of 10 and 0.1.

**Figure A10.**Local equivalent stress at $SR=10$, ${f}_{v}=13.4$ % for extreme ${s}_{min}$ values and contrasts of 10 and 0.1—the cross-sectional views of an RVE.

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**Figure 1.**Cross-sectional views of RVE mesh for extreme ${s}_{min}$ values, with close-ups of inclusions at ${s}_{min}$, $SR=10$, $r=10\mathsf{\mu}$m, ${f}_{v}=13.4$ %.

**Figure 2.**Mean values and standard deviations on the mean value for the apparent moduli as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure 3.**Mean distance between two spheres ${s}_{mean}$ as a function of ${s}_{min}$ for different scaling ratios. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure 4.**Mean values and standard deviations on the mean value for the apparent moduli as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=25.1$ %.

**Figure 5.**Mean values and standard deviations on the mean value for the first moment of hydrostatic stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure 6.**Mean values and standard deviations on the mean value for the first moment of equivalent stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure 7.**Mean values and standard deviations on the mean value for the second moment of hydrostatic stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure 8.**Mean values and standard deviations on the mean value for the second moment of equivalent stress by phase as a function of the packing parameter ${s}_{min}/r$ ($r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m) for a volume fraction of inclusions ${f}_{v}=13.4$ %. For clarity, the error bars and dots are slightly shifted around for each studied minimum distance.

**Figure 9.**Frequency histograms of equivalent stress by phase at $SR=10$, ${f}_{v}=13.4$ %, and the extreme ${s}_{min}$ values (${s}_{min}=0.1r$ and ${s}_{min}=1.2r$) for contrasts of 0.01 and 100.

**Figure 10.**Local equivalent stress at $SR=10$, ${f}_{v}=13.4$ % for extreme ${s}_{min}$ values and contrasts of 100 and 0.01—the cross-sectional views of RVE.

${\mathit{n}}_{\mathit{min}}=2$ | $\mathit{g}=2$ | ||||
---|---|---|---|---|---|

$\mathit{g}=\mathbf{1}.\mathbf{15}$ | $\mathit{g}=\mathbf{1}.\mathbf{5}$ | $\mathit{g}=\mathbf{2}$ | ${\mathit{n}}_{\mathit{min}}=\mathbf{3}$ | ${\mathit{n}}_{\mathit{min}}=\mathbf{4}$ | |

${k}^{app}$ (MPa) | 20.916 | 20.918 | 20.920 | 20.914 | 20.919 |

Standard deviation of ${k}^{app}$ (MPa) | 0.461 | 0.462 | 0.463 | 0.468 | 0.460 |

Number of nodes | 279,000 | 65,300 | 32,500 | 88,900 | 158,700 |

**Table 2.**Average number of nodes in the mesh and number of samples with inclusion volume fraction ${f}_{v}=13.4$ % for different scaling ratios $SR$ with $r=10\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m.

$\mathit{SR}=2.5$ | $\mathit{SR}=5$ | $\mathit{SR}=10$ | |
---|---|---|---|

Number of nodes | 30,000 | 300,000 | 2,500,000 |

Number of samples | 350 | 70 | 35 |

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**MDPI and ACS Style**

Belgrand, L.; Ramière, I.; Largenton, R.; Lebon, F.
Proximity Effects in Matrix-Inclusion Composites: Elastic Effective Behavior, Phase Moments, and Full-Field Computational Analysis. *Mathematics* **2022**, *10*, 4437.
https://doi.org/10.3390/math10234437

**AMA Style**

Belgrand L, Ramière I, Largenton R, Lebon F.
Proximity Effects in Matrix-Inclusion Composites: Elastic Effective Behavior, Phase Moments, and Full-Field Computational Analysis. *Mathematics*. 2022; 10(23):4437.
https://doi.org/10.3390/math10234437

**Chicago/Turabian Style**

Belgrand, Louis, Isabelle Ramière, Rodrigue Largenton, and Frédéric Lebon.
2022. "Proximity Effects in Matrix-Inclusion Composites: Elastic Effective Behavior, Phase Moments, and Full-Field Computational Analysis" *Mathematics* 10, no. 23: 4437.
https://doi.org/10.3390/math10234437