# Analytical Micromechanics Models for Elastoplastic Behavior of Long Fibrous Composites: A Critical Review and Comparative Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review on Micromechanics Models

#### 2.1. Numerical Micromechanics Models

#### 2.2. Analytical Micromechanics Models

#### 2.3. Micromechanics Models with Imperfect Interface

## 3. Comparison on Elastic Theories

#### 3.1. General Framework

#### 3.2. Summary of Elastic Models

#### 3.2.1. Eshelby Model

#### 3.2.2. SCM

#### 3.2.3. Mori–Tanaka Model

#### 3.2.4. GSCM

_{11}, ${\upsilon}_{12}$, G

_{12}, and E

_{22}, as Equations (34)–(38)

#### 3.2.5. Rule of Mixture

#### 3.2.6. Chamis Model

#### 3.2.7. Halpin–Tsai Equations

#### 3.2.8. Bridging Model

_{22}and G

_{12}with experiments. If no experiments are available, $\beta =\alpha =0.3$ are mostly recommended. The Equation (61) for ${a}_{12}$ and ${a}_{13}$ are solved from the symmetric condition of the composite compliance, i.e., ${M}_{ij}={M}_{ji}$.

#### 3.3. Quantitative Comparison

## 4. Comparison on Elastoplastic Behavior

#### 4.1. Comparison on Micromechanics Models

#### 4.2. Comparison on Modifications on Yield Stress

#### 4.2.1. First Moment Approach

#### 4.2.2. Second-Moment Approach

- (a)
- the inclusion aspect ratio is larger than 1,
- (b)
- the elastic stiffness contrast between a fiber and a matrix is high,
- (c)
- work-hardening phenomenon of a matrix is not significant.

#### 4.2.3. SCFs in Matrix

#### 4.3. Comparison on Linearization

#### 4.3.1. Secant Linearization

#### 4.3.2. Tangent Linearization

#### 4.3.3. Transformation Field Analysis (TFA)

#### 4.3.4. Affine Formulation

#### 4.3.5. Incremental-Secant Scheme

#### 4.3.6. Quantitative Comparison on the Linearizations

#### 4.4. Comparison on Modifications of a Plastic Eshelby Tensor

#### 4.4.1. Anisotropic Eshelby Tensor Approach

#### 4.4.2. Isotropic Matrix Method

#### 4.4.3. Isotropic Eshelby Tensor Method

#### 4.4.4. Peng’s Approach

#### 4.4.5. Quantitative Comparison on Modifications of the Eshelby Tensor

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ | |
---|---|---|---|---|---|---|

E-Glass ^{a}/LY556(V _{f} = 0.62) | Fiber | 80 | 80 | 33.33 | 33.33 | 0.2 |

Matrix | 3.35 | 3.35 | 1.24 | 1.24 | 0.35 | |

Composite | 53.5 | 17.7 | 5.83 | 6.32 | 0.28 | |

E-Glass ^{b}/MY750(V _{f} = 0.60) | Fiber | 74 | 74 | 30.8 | 30.8 | 0.2 |

Matrix | 3.35 | 3.35 | 1.24 | 1.24 | 0.35 | |

Composite | 45.6 | 16.2 | 5.83 | 5.79 | 0.28 | |

S2-Glass/Epoxy (V _{f} = 0.60) | Fiber | 87 | 87 | 36.3 | 36.3 | 0.2 |

Matrix | 3.2 | 3.2 | 1.19 | 1.19 | 0.35 | |

Composite | 52 | 19 | 6.7 | 6.7 | 0.3 | |

T300/BSL914C (V _{f} = 0.60) | Fiber | 230 | 15 | 15 | 7 | 0.2 |

Matrix | 4 | 4 | 1.48 | 1.48 | 0.35 | |

Composite | 138 | 11 | 5.5 | 3.93 | 0.28 | |

T300/PR319 (V _{f} = 0.60) | Fiber | 230 | 15 | 15 | 7 | 0.2 |

Matrix | 0.95 | 0.95 | 0.35 | 0.35 | 0.35 | |

Composite | 129 | 5.6 | 1.33 | 1.86 | 0.32 | |

AS carbon/Epoxy (V _{f} = 0.60) | Fiber | 231 | 15 | 15 | 7 | 0.2 |

Matrix | 3.2 | 3.2 | 1.19 | 1.19 | 0.35 | |

Composite | 140 | 10 | 6 | 3.35 | 0.3 | |

AS4/3501-6 (V _{f} = 0.60) | Fiber | 225 | 15 | 15 | 7 | 0.2 |

Matrix | 4.2 | 4.2 | 1.57 | 1.57 | 0.34 | |

Composite | 126 | 11 | 6.6 | 3.93 | 0.28 | |

IM7/8551-7 (V _{f} = 0.60) | Fiber | 276 | 19 | 27 | 7 | 0.2 |

Matrix | 4.08 | 4.08 | 1.48 | 1.48 | 0.38 | |

Composite | 165 | 8.4 | 5.6 | 2.8 | 0.34 | |

G40-800/5260 (V _{f} = 0.60) | Fiber | 290 | 19 | 27 | 7 | 0.2 |

Matrix | 3.45 | 3.45 | 1.28 | 1.28 | 0.35 | |

Composite | 173 | 10 | 6.94 | 3.56 | 0.33 |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.8 | 7.15 | 2.67 | 2.42 | 0.28 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.7 | 7.01 | 2.61 | 2.37 | 0.28 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.4 | 6.76 | 2.52 | 2.28 | 0.28 |

T300/BSL914C (V_{f} = 0.60) | 139.5 | 7.08 | 2.94 | 2.49 | 0.27 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 1.98 | 0.75 | 0.67 | 0.28 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.8 | 5.91 | 2.40 | 2.05 | 0.27 |

AS4/3501-6 (V_{f} = 0.60) | 136.6 | 7.30 | 3.09 | 2.60 | 0.27 |

IM7/8551-7 (V_{f} = 0.60) | 167.1 | 7.73 | 3.07 | 2.50 | 0.29 |

G40-800/5260 (V_{f} = 0.60) | 175.3 | 6.52 | 2.67 | 2.19 | 0.28 |

Average error | 3.1% | 44.3% | 53.5% | 45.5% | 7.3% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 18.91 | 11.34 | 6.96 | 0.23 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.8 | 16.80 | 9.80 | 6.15 | 0.24 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.6 | 17.39 | 10.94 | 6.36 | 0.23 |

T300/BSL914C (V_{f} = 0.60) | 139.7 | 8.99 | 6.25 | 3.43 | 0.25 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 4.19 | 4.18 | 1.55 | 0.24 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 8.06 | 5.82 | 3.06 | 0.25 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 9.14 | 6.37 | 3.53 | 0.25 |

IM7/8551-7 (V_{f} = 0.60) | 167.3 | 10.37 | 9.36 | 3.52 | 0.26 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 9.33 | 8.98 | 3.25 | 0.25 |

Average error | 3.1% | 14.3% | 61.9% | 11.5% | 18.3% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 11.7 | 4.60 | 4.06 | 0.25 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.8 | 11.02 | 4.32 | 3.83 | 0.25 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 10.78 | 4.23 | 3.72 | 0.25 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 8.57 | 4.35 | 3.21 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 3.02 | 1.30 | 1.06 | 0.25 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 7.48 | 3.67 | 2.77 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 8.76 | 4.53 | 3.32 | 0.26 |

IM7/8551-7 (V_{f} = 0.60) | 167.3 | 9.67 | 4.92 | 3.23 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 8.47 | 4.36 | 2.92 | 0.25 |

Average error | 3.1% | 28.2% | 25.2% | 26.9% | 14.6% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 12.87 | 4.6 | 4.65 | 0.25 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.8 | 12.03 | 4.32 | 4.33 | 0.25 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 11.8 | 4.23 | 4.25 | 0.25 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 8.77 | 4.35 | 3.32 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 3.27 | 1.29 | 1.19 | 0.25 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 7.72 | 3.67 | 2.9 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 8.93 | 4.54 | 3.42 | 0.25 |

IM7/8551-7 (V_{f} = 0.60) | 167.3 | 10.1 | 4.92 | 3.42 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 8.85 | 4.35 | 3.09 | 0.25 |

Average error | 3.1% | 25.1% | 25.2% | 22.4% | 14.9% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 8.252 | 3.076 | 3.076 | 0.26 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.7 | 7.84 | 2.92 | 2.92 | 0.26 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 7.58 | 2.82 | 2.82 | 0.26 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 7.14 | 3.225 | 2.811 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 2.169 | 0.85 | 0.82 | 0.26 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 6.1 | 2.65 | 2.36 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 7.39 | 3.39 | 2.93 | 0.26 |

IM7/8551-7 (V_{f} = 0.60) | 167.2 | 7.72 | 3.42 | 2.81 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 6.78 | 2.99 | 2.51 | 0.26 |

Average error | 3.1% | 41.5% | 48.1% | 36.5% | 12.9% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 13.64 | 5.13 | 5.13 | 0.26 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.7 | 12.86 | 4.83 | 4.83 | 0.26 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 12.60 | 4.73 | 4.73 | 0.26 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 9.26 | 4.91 | 3.81 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 3.46 | 1.45 | 1.33 | 0.27 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 8.19 | 4.14 | 3.33 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 9.50 | 5.12 | 3.93 | 0.26 |

IM7/8551-7 (V_{f} = 0.60) | 167.2 | 10.42 | 5.52 | 3.80 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 9.43 | 4.88 | 3.49 | 0.26 |

Average error | 3.1% | 21.4% | 18.1% | 15.0% | 12.9% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 11.7 | 4.6 | 4.06 | 0.26 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.7 | 11.02 | 4.32 | 3.83 | 0.26 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 10.78 | 4.23 | 3.72 | 0.26 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 8.57 | 4.35 | 3.21 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 3.02 | 1.29 | 1.06 | 0.27 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 7.48 | 3.67 | 2.77 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 8.76 | 4.54 | 3.32 | 0.26 |

IM7/8551-7 (V_{f} = 0.60) | 167.2 | 9.67 | 4.92 | 3.23 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 8.47 | 4.35 | 2.92 | 0.26 |

Average error | 3.1% | 28.2% | 25.2% | 26.9% | 12.9% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 18.1 | 6.28 | 6.24 | 0.26 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.7 | 16.8 | 5.84 | 5.8 | 0.26 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 16.9 | 5.81 | 5.77 | 0.26 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 9.6 | 5.35 | 3.66 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 4.41 | 1.82 | 1.55 | 0.27 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 8.7 | 4.64 | 3.29 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 9.7 | 5.54 | 3.76 | 0.26 |

IM7/8551-7 (V_{f} = 0.60) | 167.2 | 11.2 | 6.46 | 3.76 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 10.2 | 5.8 | 3.51 | 0.26 |

Average error | 3.1% | 12.4% | 14.6% | 9.0% | 12.9% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 16.26 | 4.96 | 6.49 | 0.25 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.8 | 14.9 | 4.58 | 5.89 | 0.25 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 14.86 | 4.5 | 5.89 | 0.25 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 9.42 | 4.5 | 3.71 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 3.98 | 1.38 | 1.58 | 0.25 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 8.45 | 3.82 | 3.32 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 9.54 | 4.68 | 3.79 | 0.25 |

IM7/8551-7 (V_{f} = 0.60) | 167.2 | 10.88 | 5.15 | 3.79 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 9.63 | 4.57 | 3.47 | 0.25 |

Average error | 3.1% | 15.9% | 22.1% | 8.8% | 15.4% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} =0.62) | 50.9 | 16.26 | 4.95 | 6.49 | 0.25 |

E-Glass ^{b}/MY750 (V_{f} =0.60) | 45.8 | 14.90 | 4.57 | 5.89 | 0.25 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 14.86 | 4.50 | 5.89 | 0.25 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 9.42 | 4.51 | 3.71 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 3.97 | 1.38 | 1.57 | 0.25 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 8.44 | 3.83 | 3.32 | 0.25 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 9.53 | 4.70 | 3.79 | 0.25 |

IM7/8551-7 (V_{f} = 0.60) | 167.2 | 10.87 | 5.18 | 3.78 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 9.62 | 4.60 | 3.47 | 0.25 |

Average error | 3.1% | 14.9% | 22% | 8.9% | 15.3% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

Composites | $\text{}{\mathit{E}}_{11}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{E}}_{22}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{12}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{G}}_{23}\left(\mathit{G}\mathit{P}\mathit{a}\right)\text{}$ | $\text{}{\mathit{\upsilon}}_{12}\text{}$ |
---|---|---|---|---|---|

E-Glass ^{a}/LY556 (V_{f} = 0.62) | 50.9 | 15.40 | 4.57 | 6.02 | 0.25 |

E-Glass ^{b}/MY750 (V_{f} = 0.60) | 45.8 | 14.16 | 4.21 | 5.49 | 0.25 |

S2-Glass/Epoxy (V_{f} = 0.60) | 53.5 | 14.09 | 4.13 | 5.47 | 0.25 |

T300/BSL914C (V_{f} = 0.60) | 139.6 | 9.20 | 4.21 | 3.58 | 0.26 |

T300/PR319 (V_{f} = 0.60) | 138.4 | 3.78 | 1.27 | 1.47 | 0.25 |

AS carbon/Epoxy (V_{f} = 0.60) | 139.9 | 8.21 | 3.56 | 3.19 | 0.26 |

AS4/3501-6 (V_{f} = 0.60) | 136.7 | 9.33 | 4.39 | 3.67 | 0.25 |

IM7/8551-7 (V_{f} = 0.60) | 167.2 | 10.57 | 4.78 | 3.65 | 0.27 |

G40-800/5260 (V_{f} = 0.60) | 175.4 | 9.35 | 4.24 | 3.35 | 0.25 |

Average error | 3.1% | 18.4% | 27.0% | 11.8% | 15.0% |

^{a}E-Glass 21 × K43 Gevetex;

^{b}Silenka E-Glass 1200tex.

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**Figure 1.**Schematic of a multi-scale framework: (

**a**) macro-scale model; (

**b**) meso-scale model; (

**c**) micro-scale model.

**Figure 2.**Schematic of a representative volume element RVE and a repeating unit cell (RUC) (solid line-undeformed, dash line-deformed). (

**a**) An RVE for a unidirectional (UD) composite with randomly distributed fibers; (

**b**) an RUC for a UD composite with periodically distributed fibers.

**Figure 4.**Imperfect interface phenomenon in fibrous composites. (

**a**) Crack growth in a SiC/SiC woven composite under cyclic load [242]; (

**b**) Ear-hole formation in a SiC/Ti-6Al-4V composite [243]; (

**c**) Interphase produced by chemical reaction in a SiC/Ti-6Al-4V composite [244]; (

**d**) BN coated T300 fiber [245].

**Figure 6.**Schematic of a multi-fiber model and a single fiber model. (

**a**) Multi-fiber model; (

**b**) single fiber model.

**Figure 9.**Comparison between elastoplastic models (IM7/8551-7 UD composite). (

**a**) Transverse compression; (

**b**) In-plane shear.

**Figure 10.**Comparison between elastoplastic models (E-Glass/Epoxy UD composite). (

**a**) Transverse compression; (

**b**) In-plane shear.

**Figure 11.**Comparison between elastoplastic models (AS4/Peek UD composite). (

**a**) 30° off-axis tension; (

**b**) 45° off-axis tension; (

**c**) 60° off-axis tension.

**Figure 12.**Short fiber reinforced polyamide composite under uniaxial tension [280]. (

**a**) Longitudinal tension; (

**b**) Transverse tension.

**Figure 13.**Finite element analysis (FEA) strain contour for short fiber reinforced polyamide composite [280].

**Figure 14.**Comparison between the first and second-moment approach. (

**a**) Ceramic reinforced aluminum composite (aspect ratio = 1, plastic parameter n = 0.05); (

**b**) Ceramic reinforced aluminum composite (aspect ratio = 3, plastic parameter n = 0.05); (

**c**) Ceramic reinforced aluminum composite (aspect ratio = 3, plastic parameter n = 0.4); (

**d**) two-phase steel with martensite inclusions (aspect ratio = 3, plastic parameter n = 0.31).

**Figure 15.**Comparison of models with/without stress concentration factors (SCFs)—IM7/8551-7 UD composite. (

**a**) Transverse compression; (

**b**) In-plane shear.

**Figure 16.**Comparison of models with/without stress concentration factors (SCFs)—E-glass/Epoxy UD composite. (

**a**) Transverse compression; (

**b**) In-plane shear.

**Figure 17.**Comparison of models with/without SCFs—AS4/Peek UD composite. (

**a**) 30° Off-axis tension; (

**b**) 45° Off-axis tension; (

**c**) 60° Off-axis tension.

**Figure 19.**Schematic of the incremental-secant linearization [112].