# Numerical Analysis of Curing Residual Stress and Deformation in Thermosetting Composite Laminates with Comparison between Different Constitutive Models

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## Abstract

**:**

## 1. Introduction

## 2. Control Equation

#### 2.1. Heat Transfer Equation

#### 2.2. Chemical Reaction

#### 2.3. Thermo-Physical Properties

#### 2.4. Modified CHILE Model

## 3. Numerical Formulation

## 4. Material Model Verification

## 5. Results and Discussion

#### 5.1. Thermo-Chemical Analysis

#### 5.2. Residual Stress Analysis

#### 5.3. Curing Deformation Analysis

#### 5.4. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{p}, and the longitudinal thermal conductivity, k

_{L}, (x-direction in Equation (1)), can be respectively acquired by the following rule of mixtures [37]:

_{T}, (y and z-direction in Equation (1)) is able to be acquired in light of the E-S model [38]. It is established below:

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**Figure 1.**Put forward process to simulate residual stress and deformation under the viscoelastic model.

**Figure 4.**Temperature development in central point (5.08, 5.08, 1.27) of the composite laminate under from the modified cure hardening instantaneously linear elastic (CHILE) and viscoelastic models.

**Figure 5.**Degree of cure (DOC) development in central point (5.08, 5.08, 1.27) of the composite laminate under manufacture recommended cure cycle from the modified CHILE and viscoelastic models.

**Figure 6.**Distribution of temperature on the cross section at x = 5.08 in the course of curing from the viscoelastic model. (

**a**) 32 min; (

**b**) 46 min; (

**c**) 118 min; (

**d**) 133 min; (

**e**) 290 min; (

**f**) 301 min.

**Figure 7.**Comparison of thermal strain in various orientations at the point (5.08, 0, 1.27) from the modified CHILE and viscoelastic models.

**Figure 8.**Comparison of cure shrinkage in the through-thickness orientation from the modified CHILE and viscoelastic models.

**Figure 9.**Comparison of elastic modulus in through-thickness orientation in the course of cure cycle from the linear elastic model and the modified CHILE model at constant frequency 0.1 Hz.

**Figure 10.**Interlaminar normal stress ${\sigma}_{3}$ development at the point (5.08, 0, 1.27) of the composite laminate under manufacture recommended cure cycle from the modified CHILE and viscoelastic models.

**Figure 11.**Stress ${\sigma}_{2}$ development in the ${0}^{\xb0}$ ply at x = 5.08 and y = 5.08 of the composite laminate under manufacture recommended cure cycle from the modified CHILE and viscoelastic models.

**Figure 12.**Contours of the von Mises stresses from (

**a**) Viscoelastic model, (

**b**) Modified CHILE model.

**Figure 13.**Contours of the deformation after cool-down forecasted by (

**a**) Viscoelastic model, (

**b**) Modified CHILE model.

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

${A}_{1}$ (min^{−1}) | 2.102 × 10^{9} | $\u2206{E}_{1}$ (J/mol) | 8.07 × 10^{4} |

${A}_{2}$ (min^{−1}) | −2.014 × 10^{9} | $\u2206{E}_{2}$ (J/mol) | 7.78 × 10^{4} |

${A}_{3}$ (min^{−1}) | 1.96 × 10^{5} | $\u2206{E}_{3}$ (J/mol) | 5.66 × 10^{4} |

${H}_{R}$ (J/Kg) | 1.989 × 10^{5} | $R$ (J·mol^{−1}·K^{−1}) | 8.3143 |

Property | Value |
---|---|

Resin density ${\rho}_{r}$ (mg·m^{−3}) | $\begin{array}{c}90\alpha +1232\left(\alpha \le 0.45\right)\\ 1272\hspace{1em}\hspace{1em}\left(\alpha \ge 0.45\right)\end{array}$ |

Fiber density ${\rho}_{f}$ (mg·m^{−3}) | 1790 |

Resin specific heat capacity ${C}_{pr}$ (J·mol^{−1}·K^{−1}) | 4184[0.468 + 5.975 × 10^{−4}T − 0.141$\alpha $] |

Fiber specific heat capacity ${C}_{pf}$ (J·mol^{−1}·K^{−1}) | 1390 + 4.50T |

Thermal conductivity of resin ${k}_{r}$ (W·m^{−1}·K^{−1}) | 0.04184[3.85 + (0.035T − 0.141)$\alpha $] |

Thermal conductivity of fiber ${k}_{f}$ (W·m^{−1}·K^{−1}) | 0.742 + 9.02 × 10^{−4}T |

Property | AS4 Carbon Fiber | 3501-6 Epoxy Resin |
---|---|---|

Longitudinal elastic modulus ${E}_{1}$ (Gpa) | 206.8 | 3.2 |

Transverse elastic modulus ${E}_{2}={E}_{3}$ (Gpa) | 17.2 | 3.2 |

In-plane shear modulus ${G}_{12}={G}_{13}$ (Gpa) | 27.58 | 1.185 |

Transverse shear modulus ${G}_{23}$ (Gpa) | 6.894 | 1.185 |

In-plane Poisson’s ratio ${\upsilon}_{12}={\upsilon}_{13}$ | 0.2 | 0.35 |

Transverse Poisson’s ratio ${\upsilon}_{23}$ | 0.3 | 0.35 |

Longitudinal CTE ${\varphi}_{1}$() | −9 × 10^{−7} | 5.76 × 10^{−5} |

Transverse CTE ${\varphi}_{2}={\varphi}_{3}$() | 7.2 × 10^{−6} | 5.76 × 10^{−5} |

Longitudinal CSE ${\phi}_{1}$ | 0 | −0.01695 |

Transverse CSE ${\phi}_{2}={\phi}_{3}$ | 0 | −0.01695 |

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**MDPI and ACS Style**

Dai, J.; Xi, S.; Li, D.
Numerical Analysis of Curing Residual Stress and Deformation in Thermosetting Composite Laminates with Comparison between Different Constitutive Models. *Materials* **2019**, *12*, 572.
https://doi.org/10.3390/ma12040572

**AMA Style**

Dai J, Xi S, Li D.
Numerical Analysis of Curing Residual Stress and Deformation in Thermosetting Composite Laminates with Comparison between Different Constitutive Models. *Materials*. 2019; 12(4):572.
https://doi.org/10.3390/ma12040572

**Chicago/Turabian Style**

Dai, Jianfeng, Shangbin Xi, and Dongna Li.
2019. "Numerical Analysis of Curing Residual Stress and Deformation in Thermosetting Composite Laminates with Comparison between Different Constitutive Models" *Materials* 12, no. 4: 572.
https://doi.org/10.3390/ma12040572