# Improved Voigt and Reuss Formulas with the Poisson Effect

## Abstract

**:**

## 1. Introduction

## 2. Methods

**FE Model I**: The sliding phase interface Figure 2a is combined with the IBC in Table 1. This model is expected to be consistent with Equations (1) and (2) and has no influence from the Poisson effect.

**FE Model II**: The bonded phase interface Figure 2b is combined with HBC in Table 1. This model is anticipated to agree with Equations (3a)–(3c) and (4a), (4b) and reflect the influence of the Poisson effect.

- The average phase stresses as calculated by Equation (8).
- The total strain energy in the RVE, which can be computed from the finite element stress and strain vectors, $\mathit{\sigma}$ and $\mathit{\epsilon}$, by $U=\frac{1}{2}{\int}_{V}\mathit{\sigma}\cdot \mathit{\epsilon}dV$.
- The effective Young’s modulus and Poisson’s ratio as determined by Equations (6) and (7).

## 3. Results

- There is an excellent agreement between Equations (1) and (2) and FE Model I, and between Equations (3a)–(3c) and (4a), (4b) and FE Model II, suggesting that FE Model I and II do have the ability to simulate, respectively, the scenarios without and with the Poisson effect.
- The Poisson effect has much greater influence over the effective Poisson’s ratio than over the effective Young’s modulus. The Voigt and the Reuss formulas generally have very low accuracy if they are applied to estimate the effective Poisson’s ratio.

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 3.**Average stresses in (

**a**) Phase 1 and (

**b**) Phase 2 of Composite #2 under the iso-strain condition (The volume fraction Phase 2 is 0.35).

**Figure 4.**Average stresses in (

**a**) Phase 1 and (

**b**) Phase 2 of Composite #2 under the iso-stress condition (The volume fraction Phase 2 is 0.35).

**Figure 5.**Increase (%) in strain energy in FE Model II compared with FE Model I to achieve the same deformation in the loading direction under (

**a**) the iso-strain, and (

**b**) the iso-stress conditions.

**Figure 6.**Iso-strain condition—Effective Young’s moduli predicted by analytical formulas and FEM. (

**a**) Composite #1. (

**b**) Composite #2. (

**c**) Composite #3. (

**d**) Composite #4.

**Figure 7.**Iso-strain condition—Effective Poisson’s ratio predicted by analytical formulas and FEM. (

**a**) Composite #1. (

**b**) Composite #2. (

**c**) Composite #3. (

**d**) Composite #4.

**Figure 8.**Iso-stress condition—Effective Young’s moduli predicted by analytical formulas and FEM. (

**a**) Composite #1. (

**b**) Composite #2. (

**c**) Composite #3. (

**d**) Composite #4.

**Figure 9.**Iso-stress condition—Effective Poisson’s ratios predicted by analytical formulas and FEM. (

**a**) Composite #1. (

**b**) Composite #2. (

**c**) Composite #3. (

**d**) Composite #4.

Symbol | Meaning |
---|---|

${E}_{i}$$\left(i=1,2\right)$ | Young’s modulus of Phase $i$ |

${\nu}_{i}$$\left(i=1,2\right)$ | Poisson’s ratio of Phase $i$ |

${f}_{i}$$\left(i=1,2\right)$ | Volume fraction of Phase $i$ |

${\epsilon}_{x}^{i}$$,\text{}{\epsilon}_{y}^{i}$$,\text{}{\epsilon}_{z}^{i}$$\left(i=1,2\right)$ | Strain components in Phase $i$ |

${\sigma}_{x}^{i}$$,\text{}{\sigma}_{y}^{i}$$,\text{}{\sigma}_{z}^{i}$$\left(i=1,2\right)$ | Stress components in Phase $i$ |

${\overline{\epsilon}}_{x}$$,\text{}{\overline{\epsilon}}_{y}$$,\text{}{\overline{\epsilon}}_{z}$ | Average strain components in the RVE |

${\overline{\sigma}}_{x}$$,\text{}{\overline{\sigma}}_{y}$$,\text{}{\overline{\sigma}}_{z}$ | Average stress components in the RVE |

${V}_{i}$$(i=1,2$) | The volume of Phase $i$ |

RVE Surface | Iso-Strain [Figure 1b] | Iso-Stress [Figure 1c] | ||
---|---|---|---|---|

${\overline{\mathit{E}}}_{\mathit{x}}$$,\text{}{\overline{\mathit{\nu}}}_{\mathit{x}\mathit{y}}$$,\text{}{\overline{\mathit{\nu}}}_{\mathit{x}\mathit{z}}$ | ${\overline{\mathit{E}}}_{\mathit{z}}$$,\text{}{\overline{\mathit{\nu}}}_{\mathit{z}\mathit{x}}={\overline{\mathit{\nu}}}_{\mathit{z}\mathit{y}}$ | |||

HBC | IBC | HBC | IBC | |

$x=0$ | ${u}_{x}=0$ | ${u}_{x}=0$ | ${u}_{x}=0$ | ${u}_{x}=0$ |

$y=0$ | ${u}_{y}=0$ | ${u}_{y}=0$ | ${u}_{y}=0$ | ${u}_{y}=0$ |

$z=0$ | ${u}_{z}=0$ | ${u}_{z}=0$ | ${u}_{z}=0$ | ${u}_{z}=0$ |

$x=L$ | ${u}_{x}=1$ | ${u}_{x}=1$ | $\mathrm{Homogeneous}\text{}{u}_{x}$ | Free |

$y=L$ | $\mathrm{Homogeneous}\text{}{u}_{y}$ * | Free | $\mathrm{Homogeneous}\text{}{u}_{y}$ | Free |

$z=L$ | $\mathrm{Homogeneous}\text{}{u}_{z}$ | Free | ${u}_{z}=1$ | ${u}_{z}=1$ |

Composite # | Softer Phase | Stiffer Phase | Phase Contrast of | |||
---|---|---|---|---|---|---|

Young’s Modulus (MPa) | Poisson’s Ratio | Young’s Modulus (MPa) | Poisson’s Ratio | Young’s Modulus | Poisson’s Ratio | |

1 | 80.0 | 0.20 | 120.0 | 0.15 | Small | Small |

2 | 80.0 | 0.45 | 120.0 | 0.15 | Small | Large |

3 | 80.0 | 0.20 | 12,000.0 | 0.15 | Large | Small |

4 | 80.0 | 0.45 | 12,000.0 | 0.15 | Large | Large |

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**MDPI and ACS Style**

Luo, Y.
Improved Voigt and Reuss Formulas with the Poisson Effect. *Materials* **2022**, *15*, 5656.
https://doi.org/10.3390/ma15165656

**AMA Style**

Luo Y.
Improved Voigt and Reuss Formulas with the Poisson Effect. *Materials*. 2022; 15(16):5656.
https://doi.org/10.3390/ma15165656

**Chicago/Turabian Style**

Luo, Yunhua.
2022. "Improved Voigt and Reuss Formulas with the Poisson Effect" *Materials* 15, no. 16: 5656.
https://doi.org/10.3390/ma15165656