# Numerical Analysis of Micro-Residual Stresses in a Carbon/Epoxy Polymer Matrix Composite during Curing Process

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}and user-subroutines are used to simulate the thermo-curing process coupled with the mechanical constitutive model. Experimental characterization of the bulk resin properties and curing behavior was made to setup the models. The higher micro-residual stresses occur at the thinner fiber gaps, acting as triggers to failure propagation during mechanical loading. These micro-residual stresses achieve peak values above the yield stress of the resin 55 MPa, but without achieving damage. These micro-residual stresses reduce the transverse strength by at least 10%, while the elastic properties remain almost unaffected. The numerical results of the effective properties show a good agreement with the macro-scale experimentally measured properties at coupon level, including transverse tensile, longitudinal shear and transverse shear moduli and strengths, and minor in-plane and transverse Poisson’s ratios. A sensitivity analysis was performed on the thermal expansion coefficient, chemical shrinkage, resin elastic modulus and cure temperature. All these parameters change the micro-residual stress levels and reduce the strength properties.

## 1. Introduction

^{®}and user-material (UMAT) subroutines to simulate the curing process and estimate the micro-residual stresses and its influence on the mechanical performance of an aerospace grade epoxy resin. A distinction between residual (macro-scale) and micro-residual (micro-scale) stresses is made because the micro-residual stresses are developed inside the RVE with traction-free boundary conditions.

## 2. Materials and Methods

#### 2.1. Materials Selection

#### 2.2. Constitutive Models

#### 2.2.1. Carbon Fibers

#### 2.2.2. Polymer Matrix

- ${A}_{1}$; ${A}_{2}$: Reaction velocities [1/s].
- $\Delta {E}_{1}$; $\Delta {E}_{2}$: Activation energies [J/mol]
- $m$; $n:$ Fitting exponents
- $R$: Universal gas constant [8.31432 J/mol.K]
- $T\left(t\right)$: Temperature

- $v\left(0\right)=0.5$ Poisson’s ratio at un-cured state
- $v\left(1\right)={v}_{m}$ Poisson’s ratio at cured state
- $E\left(0\right)=0$ Elastic modulus at un-cured state
- $E\left(1\right)={E}_{m}$ Elastic modulus at cured state
- $\alpha \left(0\right)=0$ Chemical shrinkage at un-cured state
- $\alpha \left(1\right)={\alpha}_{m}$ Chemical shrinkage at cured state.

#### 2.2.3. Thermo-Curing Coupling Strategy

#### 2.3. Finite Element Model

#### 2.4. Micro-Residual Stress Analysis

#### 2.5. Experimental Test Procedures

- Transverse tensile tests following ASTM D3039.
- Longitudinal shear tests following ASTM D3518.
- Transverse shear tests following ASTM D5379.

## 3. Results and Discussion

#### 3.1. Experimental Results

#### 3.2. Residual Stress Analysis

#### 3.3. Effective Mechanical Properties

#### 3.4. Sensitivity Analysis Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Randomly generated RVE geometries used for the micro-residual stress analysis. (

**a**) RVE1, (

**b**) RVE2 and (

**c**) RVE3.

**Figure 4.**Results from DMA analysis. (

**a**) Storage and loss modulus (

**b**) Curing state and shift function.

**Figure 5.**(

**a**) Shift function for storage modulus and (

**b**) Elastic properties evolution with curing level.

**Figure 6.**Coupons after testing. (

**a**) Transverse tensile tests, (

**b**) Longitudinal shear tests and (

**c**) Transverse shear tests.

**Figure 14.**Shear stress-strain response comparison. (

**a**) Transverse direction and (

**b**) Longitudinal direction.

**Figure 15.**Damage patterns at failure for the different loading cases (SDV14 refers to the damage state variable).

**Figure 16.**Micro-residual stresses and failure index/cure level sensitivity to material and process parameters. (

**a**) Thermal expansion sensitivity results, (

**b**) Chemical shrinkage sensitivity results, (

**c**) Cure temperature sensitivity results, (

**d**) Matrix elastic modulus sensitivity results.

**Figure 17.**Tensile stress–strain response sensitivity to material and process parameters. (

**a**) Thermal expansion sensitivity results, (

**b**) Chemical shrinkage sensitivity results, (

**c**) Cure temperature sensitivity results and (

**d**) Elastic modulus sensitivity results.

Properties | Value |
---|---|

Fiber diameter (mm) | 7 × 10^{−3} |

Density (kg/m^{3}) | 1800 |

Specific Heat (kJ/kg·K) | 0.752 |

Thermal Conductivity (W/m·K) | 9.38 |

$CT{E}_{L}$ (°C^{−1}) | −0.38 × 10^{−6} |

$CT{E}_{T}$ (°C^{−1}) | 6.94 × 10^{−6} |

${E}_{L}$ (GPa) | 230 |

${E}_{T}$ (GPa) | 15 |

${v}_{L}$ (n.d) | 0.2 |

${G}_{L}$ (GPa) | 15 |

${G}_{T}$ (GPa) | 7 |

Properties | Value |
---|---|

Density (kg/m^{3}) | 1310 |

Specific Heat (kJ/kg·K) | 0.679 |

Thermal Cond. (W/m·K) | 0.15 ^{1} |

$CTE$ (10^{−6} °C^{−1}) | 61.0 ^{1} |

Shrinkage (%) | 2.0 ^{1} |

${T}_{g}$ (°C) | 135 ^{2} |

${E}_{m}$ (MPa) | 2850 |

${v}_{m}$ (n.d) | 0.33 |

${v}_{p}$ (n.d) | 0.30 ^{1} |

${S}_{y+}$ (MPa) | 55 |

${S}_{y-}$ (MPa) | 81 ^{3} |

${S}_{u+}$ (MPa) | 65 |

${S}_{u-}$ (MPa) | 93 ^{3} |

${\epsilon}_{f}$ (%) | 2.6 |

${G}_{flm}$ (N/mm) | 0.12 ^{3} |

^{1}Estimated from literature for similar epoxies.

^{2}From the storage modulus onset.

^{3}Estimated from experimental tests with glass fiber prepregs. The compression values were extrapolated from transverse compression tests and the fracture energy corresponds to the interlaminar fracture toughness measured with DCB tests according to ASTM 5528.

Test | Description | Elastic Tensor Parameters | Yield Surface Parameters | Failure Surface Parameters |
---|---|---|---|---|

Longitudinal uniaxial tension | ${E}_{L}$; ${v}_{L}$ | - | - | |

Transverse uniaxial tension ^{1} | ${E}_{T};{v}_{T}$ | ${Y}_{+UT}$ | ${S}_{+UT}$ | |

Transverse shear | ${G}_{T}$ | ${Y}_{ST}$ | ${S}_{ST}$ | |

Longitudinal shear | ${G}_{L}$ | ${Y}_{SL}$ | ${S}_{SL}$ |

^{1}Although ${v}_{T}$ is not required to identify the elastic tensor, it can be easily determined.

Time (min) | 0 | 23.25 | 38.25 | 50 | 110 | 130 |

Temperature (°C) | 25 | 85 | 85 | 120 | 120 | 20 |

Parameter | Value | ||||
---|---|---|---|---|---|

Nominal | Case 1 | Case 2 | Case 3 | Case 4 | |

Shrinkage (mm/mm) | −0.02 | −0.01 | −0.015 | −0.025 | −0.03 |

CTE (10^{−6} °C^{−1}) | 60 | 20 | 40 | 80 | 100 |

Elastic mod. (MPa) | 2280 | 2565 | 2850 | 3135 | 3420 |

T cure (°C) | 120 | 90 | 105 | 130 | - |

${\mathit{A}}_{1}\text{}(1/\mathbf{s})$ | $\Delta {\mathit{E}}_{1}\text{}(\mathbf{kJ}/\mathbf{mol})$ | ${\mathit{A}}_{2}\text{}(1/\mathbf{s})$ | $\Delta {\mathit{E}}_{2}\text{}(\mathbf{kJ}/\mathbf{mol})$ | $\mathit{n}$ | $\mathit{m}$ |
---|---|---|---|---|---|

3.24 × 10^{15} | 134,627 | 0 | 0 | 1.00 | 0 |

Test Standard | Property | Average | Standard Deviation |
---|---|---|---|

ASTM D3039 | ${E}_{T}$ (MPa) | 7313 | 192 |

ASTM D3039 | ${v}_{L}$ (n/d) | 0.0109 | 0.0044 |

ASTM D5379 | ${G}_{T}$ (MPa) | 2812.8 | 73.9 |

ASTM D3518 | ${G}_{L}$ (MPa) | 3215 | 187 |

ASTM D3039 | ${S}_{+UT}$ (MPa) | 42.91 | 4.15 |

ASTM D5379 | ${S}_{ST}$ (MPa) | 22.34 | 2.17 |

ASTM D3518 | ${S}_{SL}$ (MPa) | 21.41 | 1.577 |

Experimental Results | Numerical Predictions | Experiments vs. Cure Predictions (%) | ||
---|---|---|---|---|

Cure | No-Cure | |||

${E}_{22}$ (MPa) | 7313 ± 192 | 7151 ± 114 | 7235 ± 55 | 2.3 |

${v}_{21}$ (n/d) | 0.0109 ± 0.0044 | 0.0115 ± 0.0002 | 0.0115 ± 0.0002 | −5.1 |

${v}_{23}$ (n/d) | 0.34 ^{1} | 0.34 ± 0.01 | 0.34 ± 0.01 | −0.6 |

${G}_{23}$ (MPa) | 2729 ± 132 | 2674 ± 71 | 2733 ± 74 | 2.0 |

${G}_{12}$ (MPa) | 3215 ± 187 | 3353 ± 63 | 3460 ± 80 | −4.3 |

${S}_{+UT}$ (MPa) | 42.9 ± 4.2 | 46.0 ± 2.0 | 51.9 ± 1.8 | −7.3 |

${S}_{ST}$ (MPa) | 22.3 ± 2.2 | 25.0 ± 2.6 | 32.3 ± 0.7 | −12.2 |

${S}_{SL}$ (MPa) | 21.4 ± 1.6 | 23.5 ± 1.2 | 26.8 ± 0.8 | −9.6 |

^{1}Estimated from shear modulus.

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**MDPI and ACS Style**

Gonçalves, P.T.; Arteiro, A.; Rocha, N.; Pina, L.
Numerical Analysis of Micro-Residual Stresses in a Carbon/Epoxy Polymer Matrix Composite during Curing Process. *Polymers* **2022**, *14*, 2653.
https://doi.org/10.3390/polym14132653

**AMA Style**

Gonçalves PT, Arteiro A, Rocha N, Pina L.
Numerical Analysis of Micro-Residual Stresses in a Carbon/Epoxy Polymer Matrix Composite during Curing Process. *Polymers*. 2022; 14(13):2653.
https://doi.org/10.3390/polym14132653

**Chicago/Turabian Style**

Gonçalves, Paulo Teixeira, Albertino Arteiro, Nuno Rocha, and Luis Pina.
2022. "Numerical Analysis of Micro-Residual Stresses in a Carbon/Epoxy Polymer Matrix Composite during Curing Process" *Polymers* 14, no. 13: 2653.
https://doi.org/10.3390/polym14132653