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
3.2.5. Rule of Mixture
3.2.6. Chamis Model
3.2.7. Halpin–Tsai Equations
3.2.8. Bridging Model
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 | ||||||
---|---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 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 (Vf = 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 (Vf = 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 (Vf = 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 (Vf = 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 (Vf = 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 (Vf = 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 (Vf = 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 (Vf = 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 |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.8 | 7.15 | 2.67 | 2.42 | 0.28 |
E-Glass b/MY750 (Vf = 0.60) | 45.7 | 7.01 | 2.61 | 2.37 | 0.28 |
S2-Glass/Epoxy (Vf = 0.60) | 53.4 | 6.76 | 2.52 | 2.28 | 0.28 |
T300/BSL914C (Vf = 0.60) | 139.5 | 7.08 | 2.94 | 2.49 | 0.27 |
T300/PR319 (Vf = 0.60) | 138.4 | 1.98 | 0.75 | 0.67 | 0.28 |
AS carbon/Epoxy (Vf = 0.60) | 139.8 | 5.91 | 2.40 | 2.05 | 0.27 |
AS4/3501-6 (Vf = 0.60) | 136.6 | 7.30 | 3.09 | 2.60 | 0.27 |
IM7/8551-7 (Vf = 0.60) | 167.1 | 7.73 | 3.07 | 2.50 | 0.29 |
G40-800/5260 (Vf = 0.60) | 175.3 | 6.52 | 2.67 | 2.19 | 0.28 |
Average error | 3.1% | 44.3% | 53.5% | 45.5% | 7.3% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 18.91 | 11.34 | 6.96 | 0.23 |
E-Glass b/MY750 (Vf = 0.60) | 45.8 | 16.80 | 9.80 | 6.15 | 0.24 |
S2-Glass/Epoxy (Vf = 0.60) | 53.6 | 17.39 | 10.94 | 6.36 | 0.23 |
T300/BSL914C (Vf = 0.60) | 139.7 | 8.99 | 6.25 | 3.43 | 0.25 |
T300/PR319 (Vf = 0.60) | 138.4 | 4.19 | 4.18 | 1.55 | 0.24 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 8.06 | 5.82 | 3.06 | 0.25 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 9.14 | 6.37 | 3.53 | 0.25 |
IM7/8551-7 (Vf = 0.60) | 167.3 | 10.37 | 9.36 | 3.52 | 0.26 |
G40-800/5260 (Vf = 0.60) | 175.4 | 9.33 | 8.98 | 3.25 | 0.25 |
Average error | 3.1% | 14.3% | 61.9% | 11.5% | 18.3% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 11.7 | 4.60 | 4.06 | 0.25 |
E-Glass b/MY750 (Vf = 0.60) | 45.8 | 11.02 | 4.32 | 3.83 | 0.25 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 10.78 | 4.23 | 3.72 | 0.25 |
T300/BSL914C (Vf = 0.60) | 139.6 | 8.57 | 4.35 | 3.21 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 3.02 | 1.30 | 1.06 | 0.25 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 7.48 | 3.67 | 2.77 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 8.76 | 4.53 | 3.32 | 0.26 |
IM7/8551-7 (Vf = 0.60) | 167.3 | 9.67 | 4.92 | 3.23 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 8.47 | 4.36 | 2.92 | 0.25 |
Average error | 3.1% | 28.2% | 25.2% | 26.9% | 14.6% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 12.87 | 4.6 | 4.65 | 0.25 |
E-Glass b/MY750 (Vf = 0.60) | 45.8 | 12.03 | 4.32 | 4.33 | 0.25 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 11.8 | 4.23 | 4.25 | 0.25 |
T300/BSL914C (Vf = 0.60) | 139.6 | 8.77 | 4.35 | 3.32 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 3.27 | 1.29 | 1.19 | 0.25 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 7.72 | 3.67 | 2.9 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 8.93 | 4.54 | 3.42 | 0.25 |
IM7/8551-7 (Vf = 0.60) | 167.3 | 10.1 | 4.92 | 3.42 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 8.85 | 4.35 | 3.09 | 0.25 |
Average error | 3.1% | 25.1% | 25.2% | 22.4% | 14.9% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 8.252 | 3.076 | 3.076 | 0.26 |
E-Glass b/MY750 (Vf = 0.60) | 45.7 | 7.84 | 2.92 | 2.92 | 0.26 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 7.58 | 2.82 | 2.82 | 0.26 |
T300/BSL914C (Vf = 0.60) | 139.6 | 7.14 | 3.225 | 2.811 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 2.169 | 0.85 | 0.82 | 0.26 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 6.1 | 2.65 | 2.36 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 7.39 | 3.39 | 2.93 | 0.26 |
IM7/8551-7 (Vf = 0.60) | 167.2 | 7.72 | 3.42 | 2.81 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 6.78 | 2.99 | 2.51 | 0.26 |
Average error | 3.1% | 41.5% | 48.1% | 36.5% | 12.9% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 13.64 | 5.13 | 5.13 | 0.26 |
E-Glass b/MY750 (Vf = 0.60) | 45.7 | 12.86 | 4.83 | 4.83 | 0.26 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 12.60 | 4.73 | 4.73 | 0.26 |
T300/BSL914C (Vf = 0.60) | 139.6 | 9.26 | 4.91 | 3.81 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 3.46 | 1.45 | 1.33 | 0.27 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 8.19 | 4.14 | 3.33 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 9.50 | 5.12 | 3.93 | 0.26 |
IM7/8551-7 (Vf = 0.60) | 167.2 | 10.42 | 5.52 | 3.80 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 9.43 | 4.88 | 3.49 | 0.26 |
Average error | 3.1% | 21.4% | 18.1% | 15.0% | 12.9% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 11.7 | 4.6 | 4.06 | 0.26 |
E-Glass b/MY750 (Vf = 0.60) | 45.7 | 11.02 | 4.32 | 3.83 | 0.26 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 10.78 | 4.23 | 3.72 | 0.26 |
T300/BSL914C (Vf = 0.60) | 139.6 | 8.57 | 4.35 | 3.21 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 3.02 | 1.29 | 1.06 | 0.27 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 7.48 | 3.67 | 2.77 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 8.76 | 4.54 | 3.32 | 0.26 |
IM7/8551-7 (Vf = 0.60) | 167.2 | 9.67 | 4.92 | 3.23 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 8.47 | 4.35 | 2.92 | 0.26 |
Average error | 3.1% | 28.2% | 25.2% | 26.9% | 12.9% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 18.1 | 6.28 | 6.24 | 0.26 |
E-Glass b/MY750 (Vf = 0.60) | 45.7 | 16.8 | 5.84 | 5.8 | 0.26 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 16.9 | 5.81 | 5.77 | 0.26 |
T300/BSL914C (Vf = 0.60) | 139.6 | 9.6 | 5.35 | 3.66 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 4.41 | 1.82 | 1.55 | 0.27 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 8.7 | 4.64 | 3.29 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 9.7 | 5.54 | 3.76 | 0.26 |
IM7/8551-7 (Vf = 0.60) | 167.2 | 11.2 | 6.46 | 3.76 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 10.2 | 5.8 | 3.51 | 0.26 |
Average error | 3.1% | 12.4% | 14.6% | 9.0% | 12.9% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 16.26 | 4.96 | 6.49 | 0.25 |
E-Glass b/MY750 (Vf = 0.60) | 45.8 | 14.9 | 4.58 | 5.89 | 0.25 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 14.86 | 4.5 | 5.89 | 0.25 |
T300/BSL914C (Vf = 0.60) | 139.6 | 9.42 | 4.5 | 3.71 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 3.98 | 1.38 | 1.58 | 0.25 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 8.45 | 3.82 | 3.32 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 9.54 | 4.68 | 3.79 | 0.25 |
IM7/8551-7 (Vf = 0.60) | 167.2 | 10.88 | 5.15 | 3.79 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 9.63 | 4.57 | 3.47 | 0.25 |
Average error | 3.1% | 15.9% | 22.1% | 8.8% | 15.4% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf =0.62) | 50.9 | 16.26 | 4.95 | 6.49 | 0.25 |
E-Glass b/MY750 (Vf =0.60) | 45.8 | 14.90 | 4.57 | 5.89 | 0.25 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 14.86 | 4.50 | 5.89 | 0.25 |
T300/BSL914C (Vf = 0.60) | 139.6 | 9.42 | 4.51 | 3.71 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 3.97 | 1.38 | 1.57 | 0.25 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 8.44 | 3.83 | 3.32 | 0.25 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 9.53 | 4.70 | 3.79 | 0.25 |
IM7/8551-7 (Vf = 0.60) | 167.2 | 10.87 | 5.18 | 3.78 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 9.62 | 4.60 | 3.47 | 0.25 |
Average error | 3.1% | 14.9% | 22% | 8.9% | 15.3% |
Composites | |||||
---|---|---|---|---|---|
E-Glass a/LY556 (Vf = 0.62) | 50.9 | 15.40 | 4.57 | 6.02 | 0.25 |
E-Glass b/MY750 (Vf = 0.60) | 45.8 | 14.16 | 4.21 | 5.49 | 0.25 |
S2-Glass/Epoxy (Vf = 0.60) | 53.5 | 14.09 | 4.13 | 5.47 | 0.25 |
T300/BSL914C (Vf = 0.60) | 139.6 | 9.20 | 4.21 | 3.58 | 0.26 |
T300/PR319 (Vf = 0.60) | 138.4 | 3.78 | 1.27 | 1.47 | 0.25 |
AS carbon/Epoxy (Vf = 0.60) | 139.9 | 8.21 | 3.56 | 3.19 | 0.26 |
AS4/3501-6 (Vf = 0.60) | 136.7 | 9.33 | 4.39 | 3.67 | 0.25 |
IM7/8551-7 (Vf = 0.60) | 167.2 | 10.57 | 4.78 | 3.65 | 0.27 |
G40-800/5260 (Vf = 0.60) | 175.4 | 9.35 | 4.24 | 3.35 | 0.25 |
Average error | 3.1% | 18.4% | 27.0% | 11.8% | 15.0% |
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