# Microstructural Analysis of the Transverse and Shear Behavior of Additively Manufactured CFRP Composite RVEs Based on the Phase-Field Fracture Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{2}/0/90

_{2}]

_{s}, as shown in Figure 1, and additively manufactured using a Markforged X7 3D printer. This LSS was adopted after a preliminary microstructural examination of similar AM composites. A similar LSS is used in acquiring the multiaxial response through uniaxial tensile tests on rectangular specimens [83], which was also studied in [14]. This sequence also enables the measurements of unidirectionally reinforced layer composition and gives a better insight into laminar dimensions, layer-wise defect accumulation, and repeatability issues between single, double, and multiple adjacent layers. However, these specimens did not account for the lengthwise defect accumulation due to uneven material solidification.

_{1C}was adopted for polyamide according to [88] and recalculated to acquire the G

_{C}values. In contrast to the fiber and the matrix, the values for the interface properties were assumed and calibrated according to the experimentally observed material behavior.

## 3. Results

#### 3.1. Microstructural Inspection

#### 3.2. Experimental Acquisition of Lamina Properties

#### 3.3. RVE Design

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Reprints

## Conflicts of Interest

## Appendix A

**u**, under the assumption of small strain sand isothermal conditions, as shown in Equation (A1) [46,80,89].

**T**is defined for the outward unit normal

**n**at boundary $\partial \Omega $, and it is work-conjugate to the displacements

**u**[46,80,89]. The work-conjugate to the phase field $\varphi $ is given by $\omega $, while $\mathit{\xi}$ is the micro-stress vector, a work-conjugate to the $\nabla \varphi $, as the virtual quantity is given by $\delta $ [46,80,89].

- Define the user material with five material properties including the Young’s modulus $E$, Poisson’s ratio $\nu $, phase field length scale $l$, fracture toughness ${G}_{c}$, and the tensile strength ${f}_{t}$ which is applicable for the phase field cohesive zone models, while otherwise neglected
- Set a solution-dependent state variable (SDV)
- Define the material conductivity equal to one
- State the analysis step as coupled temperature-displacement with steady-state or transient options, a constant increment size, a separated solution technique, and symmetric equation solver matrix storage
- Change the values of the solution controls parameters ${I}_{0},{I}_{\mathrm{R}},{I}_{\mathrm{P}},{I}_{\mathrm{C}},{I}_{\mathrm{L}},\mathrm{and}{I}_{\mathrm{G}}$ to 5000, to avoid convergence problems due to large number of iterations
- Define the initial temperature condition equal to zero to describe the undamaged material in the initial step
- Adopt the element type as coupled temperature–displacement

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**Figure 2.**CAE dimensions: (

**a**) UD-90 specimen dimensions, and (

**b**) SH-45 specimen dimensions; recreated according to [81].

**Figure 3.**(

**a**) Layer stacking sequence for the UD-90 set, and (

**b**) layer stacking sequence for the SH-45 set; recreated according to [81].

**Figure 4.**CFRP SEM images: (

**a**) y-z cross-section 1600× magnification, (

**b**) x-z cross-section 800× magnification, and (

**c**) x-y cross-section 800× magnification.

**Figure 5.**CFRP laminate’s four middle unidirectional layers: (

**a**) SEM image, (

**b**) WEKA classification, (

**c**) the fiber vs. matrix probability map, and (

**d**) voids and debris probability map.

**Figure 7.**DIC images of the 50 mm gauge length during the experimental procedure: (

**a**) UD-90 pre-failure, (

**b**) UD-90 at failure, (

**c**) SH-45 at 5% of shear strain, and (

**d**) SH-45 at 5% of axial strain.

**Figure 8.**(

**a**) Single-fiber unit cell (RVE-1), (

**b**) RVE with rectangular fiber placement (RVE-2), and (

**c**) RVE with hexagonal fiber placement (RVE-3).

**Figure 9.**Comparison between the experimental results and the RVE-3 response for characteristic element length in the range from $2.5\times {10}^{-4}$ to $5\times {10}^{-4}$.

**Figure 10.**Comparison between the experimental and the numerical results: (

**a**) transversally loaded unidirectionally reinforced case, and (

**b**) the in-plane shear case.

**Figure 11.**Damage evolution in the transverse tensile and shear cases: (

**a**) damage index of 0.5 in transverse tension, (

**b**) damage index of 0.75 in transverse tension, (

**c**) failure in transverse tension, (

**d**) damage index of 0.5 in shear, (

**e**) damage index of 0.75 in shear, and (

**f**) failure in shear.

Fiber | Matrix | Interface | |
---|---|---|---|

Elastic modulus, E_{11} [MPa] | 191,000 | 3000 | 100 |

Poisson ratio, v [/] | 0.2 | 0.3 | 0.3 |

Toughness G_{c}, [kJ/m^{2}] | 0.763 | 1 | 0.3 |

Phase field length, l [mm] | $1\times {10}^{-4}$ | $1\times {10}^{-3}$ | $1\times {10}^{-3}$ |

Laminate [13] | Filament [85] | |
---|---|---|

Fiber volume ratio | 0.536 ± 0.026 | 0.34 ± 0.002 |

Matrix volume ratio | 0.41 ± 0.02 | 0.66 ± 0.002 |

Fiber local ratio | 0.568 ± 0.028 ^{1} | 0.90 ^{2} |

$\mathrm{Fiber}\mathrm{diameter},\mathsf{\mu}\mathrm{m}$ | 7.00 ± 0.41 | 7.2 ± 0.30 |

$\mathrm{Layer}\mathrm{height},\mathsf{\mu}\mathrm{m}$ | 138.22 ± 5.11 | / |

^{1}Measurements acquired in multiple randomly selected 100 µm × 100 µm zones;

^{2}measurements based on selected fiber agglomerations [85].

Length (L), mm | Width (W), mm | Thickness (t), mm | |
---|---|---|---|

UD-90 | 220 | 26.903 ± 0.154 | 1.33 ± 0.017 |

SH-45 | 220 | 27.250 ± 0.017 | 2.30 ± 0.017 |

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## Share and Cite

**MDPI and ACS Style**

Gljušćić, M.; Lanc, D.; Franulović, M.; Žerovnik, A.
Microstructural Analysis of the Transverse and Shear Behavior of Additively Manufactured CFRP Composite RVEs Based on the Phase-Field Fracture Theory. *J. Compos. Sci.* **2023**, *7*, 38.
https://doi.org/10.3390/jcs7010038

**AMA Style**

Gljušćić M, Lanc D, Franulović M, Žerovnik A.
Microstructural Analysis of the Transverse and Shear Behavior of Additively Manufactured CFRP Composite RVEs Based on the Phase-Field Fracture Theory. *Journal of Composites Science*. 2023; 7(1):38.
https://doi.org/10.3390/jcs7010038

**Chicago/Turabian Style**

Gljušćić, Matej, Domagoj Lanc, Marina Franulović, and Andrej Žerovnik.
2023. "Microstructural Analysis of the Transverse and Shear Behavior of Additively Manufactured CFRP Composite RVEs Based on the Phase-Field Fracture Theory" *Journal of Composites Science* 7, no. 1: 38.
https://doi.org/10.3390/jcs7010038