# Mesoscale Process Modeling of a Thick Pultruded Composite with Variability in Fiber Volume Fraction

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Numerical Modeling Framework

#### 3.1. Chemical and Thermo-Mechanical Material Models of Polyester Resin

#### 3.2. Thermal and Thermo-Mechanical Constitutive Material Properties

#### 3.3. Coupled Temperature Displacement Model

## 4. Results and Discussion

#### 4.1. Fiber Distribution through the Cross-Section

#### 4.2. Effect of Nonuniformity on the Temperature and Cure Degree Evolution

#### 4.3. Effect of Nonuniformity on Global Residual Stress Distribution

#### 4.4. Local Residual Stresses

#### 4.5. Processing–Structure–Property Relationship

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${V}_{f}$ | Fiber volume fraction |

UD | Unidirectional |

RVE | Representative volume element |

FRPC | Fiber-reinforced polymer composite |

CHILE | Cure hardening instantaneous linear elastic |

CTE | Coefficient of thermal expansion |

SCFM | Self-consistent field micromechanics |

${T}_{g}$ | Glass transition temperature |

1D, 2D, 3D | 1-dimensional, 2-dimensional, 3-dimensional |

$\Delta {\u03f5}_{ij}^{mech}$ | Mechanical strain increment |

$\Delta {\u03f5}_{ij}^{tot}$ | Total strain increment |

$\Delta {\u03f5}_{ij}^{therm}$ | Thermal strain increment |

$\Delta {\u03f5}_{ij}^{chem}$ | Chemical strain increments |

$\Delta {\sigma}_{ij}$ | Incremental stress component |

J | Jacobian matrix |

$\rho $ | Density |

${C}_{p}$ | Specific heat capacity |

k | Thermal conductivity |

E | Elastic modulus |

$\nu $ | Poisson’s ratio |

${A}_{0}$ | Pre-exponential constant |

${E}_{a}$ | Activation energy |

m, n | Reaction orders |

$\alpha $ | Instantaneous cure degree |

R | Gas constant |

T | Absolute temperature |

${k}_{X}$, ${k}_{Y}$ | Thermal conductivity of composite in X and Y directions |

${k}_{f}$ | Thermal conductivity of fiber |

${k}_{r}$ | Thermal conductivity of resin |

${\omega}_{f}$ | Weight fraction of fiber |

${\omega}_{r}$ | Weight fraction of resin |

${T}_{C1}$, ${T}_{C2}$, ${T}_{C3}$ | Critical temperatures for elastic modulus model |

${E}_{r}^{0}$, ${E}_{r}^{1}$, ${E}_{r}^{\infty}$ | Critical elastic moduli for elastic modulus model |

${A}_{e}$, ${K}_{e}$ | Exponential rise constants in elastic modulus model |

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**Figure 2.**One of the analyzed cross sections (

**a**) with examples of different RVE sizes for 4 × 4 patches (Case A), (

**b**) 8 × 8 patches (Case B), (

**c**) 16 × 16 patches (Case C), (

**d**) 32 × 32 patches (Case D) (

**e**) and 64 × 64 patches (Case E) (

**f**). The grid used in (

**a**) reflects 32 × 32 patches.

**Figure 3.**(

**a**) Schematic representation of a pultrusion line and the modeling domain. (

**b**) Schematic representation of the modeling framework.

**Figure 4.**(

**a**) 2D modeling domain, (

**b**) a closer look at the mesh structure, (

**c**) temperature profile defined on the boundary (the outer surfaces of the profile).

**Figure 5.**${V}_{f}$ distribution through the cross-section for various RVE sizes; (

**a**) uniform, (

**b**) ‘Case A’ ($4\times 4$), (

**c**) ‘Case B’ ($8\times 8$), (

**d**) ‘Case C’ ($16\times 16$), (

**e**) ‘Case D’ ($32\times 32$), (

**f**) ‘Case E’ ($64\times 64$).

**Figure 6.**(

**a**) ${V}_{f}$ values distribution, and (

**b**) coefficient of variation with respect to the RVE edge length.

**Figure 7.**Temperature (

**top row**) and cure degree (

**bottom row**) distributions throughout the cross-section at the exit of the die.

**Figure 8.**Residual stress distribution (

**top row**; normal stress in X-direction,

**middle row**; normal stress in Y-direction,

**bottom**row; shear stress (XY)) after cooling down.

**Figure 9.**Local residual stress evolution within the selected RVEs. Schematic representations of the corresponding locations in the cross-section are on the left column (the grid is representative for ‘Case D’). Stress evolution curves in the X-direction are on the middle column. Stress evolution curves in the Y-direction are on the right column.

**Figure 10.**Scatter plots of the in-plane principal stresses with respect to ${V}_{f}$ for each patch size cases; (

**a**) maximum in-plane stress, (

**b**) minimum in-plane stress.

**Figure 11.**Maximum in-plane principal stresses vs. ${V}_{f}$ for the core (

**a**), the neutral (

**b**) and the outer regions (

**c**). The minimum in-plane principal stresses vs. ${V}_{f}$ for the core (

**d**), the neutral (

**e**) and the outer regions (

**f**). Best linear fits and confidence intervals of the maximum (

**g**) and the minimum (

**h**) in-plane principal stresses.

Label | Number of Patches | RVE Edge Length |
---|---|---|

Case A | $4\times 4$ | 5 mm |

Case B | $8\times 8$ | $2.5$ mm |

Case C | $16\times 16$ | $1.25$ mm |

Case D | $32\times 32$ | $0.625$ mm |

Case E | $64\times 64$ | $0.3125$ mm |

**Table 2.**Cure kinetics parameters (reproduced from [41] with the permission).

${\mathit{A}}_{0}$ [1/s] | ${\mathit{E}}_{\mathit{a}}$ [kJ/mol] | m | n |
---|---|---|---|

$7.558\times {10}^{9}$ | $82.727$ | $0.630$ | $1.847$ |

**Table 3.**Parameters used in the CHILE model to characterize the instantaneous elastic modulus of the resin system (reproduced from [41] with the permission).

${\mathit{E}}_{\mathit{r}}^{0}$ [GPa] | ${\mathit{E}}_{\mathit{r}}^{1}$ [GPa] | ${\mathit{E}}_{\mathit{r}}^{\mathit{\infty}}$ [GPa] | ${\mathit{T}}_{\mathit{C}0}[\xb0\mathrm{C}]$ | ${\mathit{T}}_{\mathit{C}1}[\xb0\mathrm{C}]$ | ${\mathit{T}}_{\mathit{C}2}[\xb0\mathrm{C}]$ | ${\mathit{A}}_{\mathit{e}}$ [GPa] | ${\mathit{K}}_{\mathit{e}}[1/\xb0\mathrm{C}]$ |
---|---|---|---|---|---|---|---|

$0.0195$ | $0.73$ | $3.76$ | $-60$ | 30 | 110 | $0.20$ | $0.043$ |

**Table 4.**Thermal and mechanical material properties of the glass fiber and resin system used in this study (reproduced from [7] with the permission).

$\mathit{\rho}$ [kg/m ${}^{3}$] | ${\mathit{C}}_{\mathit{p}}$ [J/ (kg · K)] | k [W/ (m · K)] | E [GPa] | $\mathit{\nu}$ | $\mathit{CTE}$ [ppm/$\xb0$C] | |
---|---|---|---|---|---|---|

Polyester resin | 1100 | 1830 | $0.17$ | Table | $0.40$ | 72–180 |

Glass fiber | 2560 | 670 | 11.4 (axial) , 1.04 (transverse) | 73 | $0.22$ | $5.04$ |

**Table 5.**Average stress and ${V}_{f}$ values in the core (tensile) and the outer regions (compressive).

Uniform | Case A | Case B | Case C | Case D | Case E | |
---|---|---|---|---|---|---|

Stress [MPa] in X @core (${V}_{f}$) | $2.53\left(0.58\right)$ | $3.83\left(0.55\right)$ | $3.84\left(0.55\right)$ | $3.89\left(0.55\right)$ | $3.86\left(0.55\right)$ | $3.87\left(0.55\right)$ |

Stress [MPa] in Y @core (${V}_{f}$) | $2.53\left(0.58\right)$ | $3.82\left(0.55\right)$ | $4.01\left(0.55\right)$ | $4.12\left(0.55\right)$ | $4.07\left(0.55\right)$ | $4.10\left(0.55\right)$ |

Stress [MPa] in X @outer (${V}_{f}$) | $-4.28\left(0.58\right)$ | $-5.51\left(0.59\right)$ | $-7.11\left(0.62\right)$ | $-7.29\left(0.62\right)$ | $-7.39\left(0.62\right)$ | $-7.48\left(0.61\right)$ |

Stress [MPa] in Y @outer (${V}_{f}$) | $-4.28\left(0.58\right)$ | $-5.52\left(0.59\right)$ | $-6.90\left(0.60\right)$ | $-7.16\left(0.60\right)$ | $-7.16\left(0.60\right)$ | $-7.29\left(0.60\right)$ |

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**MDPI and ACS Style**

Yuksel, O.; Sandberg, M.; Hattel, J.H.; Akkerman, R.; Baran, I. Mesoscale Process Modeling of a Thick Pultruded Composite with Variability in Fiber Volume Fraction. *Materials* **2021**, *14*, 3763.
https://doi.org/10.3390/ma14133763

**AMA Style**

Yuksel O, Sandberg M, Hattel JH, Akkerman R, Baran I. Mesoscale Process Modeling of a Thick Pultruded Composite with Variability in Fiber Volume Fraction. *Materials*. 2021; 14(13):3763.
https://doi.org/10.3390/ma14133763

**Chicago/Turabian Style**

Yuksel, Onur, Michael Sandberg, Jesper H. Hattel, Remko Akkerman, and Ismet Baran. 2021. "Mesoscale Process Modeling of a Thick Pultruded Composite with Variability in Fiber Volume Fraction" *Materials* 14, no. 13: 3763.
https://doi.org/10.3390/ma14133763