# Identification of Representative Equivalent Volumes on the Microstructure of 3D-Printed Fiber-Reinforced Thermoplastics Based on Statistical Characterization

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## Abstract

**:**

## 1. Introduction

#### Objective and Contributions

## 2. Adopted Spatial Descriptor Functions

#### 2.1. Nearest Neighbor Distance

#### 2.2. Second-Order Intensity Function

## 3. Algorithm for Image Processing

## 4. Applying Proposed Algorithm in the Analysis of Cross-Section Micrographs

## 5. Statistical Characterization of Cross-Section Micrographs

#### 5.1. Nearest Neighbor Distance Characterization

#### 5.2. Second-Order Intensity Function Characterization

#### 5.3. Statistical Characterization Discussion

## 6. Homogenized Properties

#### 6.1. Numerical Modeling

^{®}software with the application of periodic boundary conditions, assuming that the fiber distribution determined in the previous section are equivalent to those found in the cross-section micrographs. It is worth noting that the application of periodic boundary conditions on ABAQUS

^{®}are not directly available. In this case, it requires the definition of linear constraints using equations to determine the relative motion between the degrees of freedom of two or more nodes, as can be seen in [63]. In addition, the fibers were assumed as continuous in the finite element discretization, which leads the solution to be independent of the coordinate along the fiber direction. Therefore, it was not necessary to refine the mesh along this direction. However, it is worth remarking that in other different cases, e.g., modeling short fiber-reinforced composite materials and/or composite materials with inclusions, the mesh refinement is important in all directions. Table 5 presents the mechanical properties of the carbon fiber and thermoplastic matrix [7,8] adopted in the finite element discretization.

^{®}. More details about the element formulation is displayed in Figure 15 and additional information can be found in [64]. Table 6 summarizes details of the finite element discretization used in the Asymptotic Homogenization models.

#### 6.2. Numerical Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Response of the second-order intensity function $K\left(h\right)$, or Ripley’s K-Function, expected for arrangements with same number of fibers and same area of study (

**a**), distributed in random, hexagonal, and square patterns (

**b**). Adapted from [52].

**Figure 3.**Schematic illustration of a cropped region and its size in function of the parameter $\delta $.

**Figure 4.**Representation of a fiber/matrix distribution with deviations between the expected area for the whole circumference (

**a**) and the area of a set of pixels within a circumference (

**b**). The white squares represent the pixels corresponding to the fibers and the gray squares represent the pixels corresponding to the matrix. The red dashed lines represent the circles obtained with the support of MATLAB function imfindcircles().

**Figure 5.**Cross-section micrograph of sample TE-90-4-1 with random cropped regions for $\delta =50$, $\delta =100$, and $\delta =150$, and target fiber volume fraction ${V}_{f}^{*}=\{{V}_{f}:0.315\le {V}_{f}\le 0.320\}$.

**Figure 6.**Application of the proposed algorithm in a cropped region of specimen TE-0-1-1: (

**a**) reproduction of a cropped region from original micrograph used for computing the microstructure and (

**b**) computed microstructure using the proposed algorithm. The computed fiber volume fraction is ${V}_{f}=31.85\%$.

**Figure 7.**Application of the proposed algorithm in a cropped region of specimen TE-90-4-1: (

**a**) reproduction of a cropped region from original micrograph used for computing the microstructure and (

**b**) computed microstructure using the proposed algorithm. The computed fiber volume fraction is ${V}_{f}=31.96\%$.

**Figure 8.**Application of the proposed algorithm in a cropped region of specimen TE-90-4-2: (

**a**) reproduction of a cropped region from original micrograph used for computing the microstructure and (

**b**) computed microstructure using the proposed algorithm. The computed fiber volume fraction is ${V}_{f}=31.71\%$.

**Figure 10.**Fiber volume fraction computed for different sizes of cropped regions based on pixels counting and circumference areas.

**Figure 11.**Standard deviation of nearest neighbor distance $\hat{d}$ computed for different sizes of cropped regions.

**Figure 13.**Second-order intensity function obtained for cropped regions of size $\delta =[10,90]$, as well as for the main region of the cross-section micrograph (dashed lines) and for a complete random distribution (dash-dotted lines).

**Figure 14.**Second-order intensity function obtained for cropped regions of size $\delta =[100,150]$, as well as for the main region of the cross-section micrograph (dashed lines) and for a complete random distribution (dash-dotted lines).

**Figure 15.**Eight-node linear hexahedral elements with reduced integration of type C3D8R available on ABAQUS

^{®}.

**Figure 16.**Discretized cropped regions with size $\delta =10$ (

**a**), $\delta =20$ (

**b**), $\delta =30$ (

**c**), $\delta =40$ (

**d**), $\delta =50$ (

**e**) and $\delta =60$ (

**f**). For all the finite element models, the element size corresponds to one pixel size according to their respective computed microstructures.

**Table 1.**Adopted criteria for quantitatively characterizing a fiber distribution using the nearest neighbor distances.

Characteristic | Criteria |
---|---|

Agglomerated | $\widehat{\mu}(\hat{d}/{r}_{f})\to 2$ |

Dispersed | $\widehat{\mu}(\hat{d}/{r}_{f})>2$ |

Periodic | $\widehat{\sigma}\left(\hat{d}\right)\to 0$ |

Non-periodic | $\widehat{\sigma}\left(\hat{d}\right)>0$ |

**Table 2.**Conditions for characterizing a fiber arrangement as agglomerated/dispersed and periodic/non-periodic based on the second-order intensity function.

Characteristic | Function $\mathit{K}\left(\mathit{h}\right)$ |
---|---|

Agglomerated | $K\left(h\right)>{K}_{p}$ |

Dispersed | $K\left(h\right)<{K}_{p}$ |

Periodic | Stair-shaped |

Non-periodic | Monotonic Positive |

Specimen | Nearest Neighbor Distance | 2nd Order Intensity Function | ||
---|---|---|---|---|

Periodicity | Agglomeration | Periodicity | Agglomeration | |

TE-0-1-1 | $\delta \ge 60$ | $\delta \ge 40$ | $\delta \ge 10$ | $\delta \ge 50$ |

TE-90-4-1 | $\delta \ge 50$ | $\delta \ge 50$ | $\delta \ge 10$ | $\delta \ge 50$ |

TE-90-4-2 | $\delta \ge 50$ | $\delta \ge 50$ | $\delta \ge 10$ | $\delta \ge 40$ |

Size $\mathit{\delta}$ | Cropped Region | ${\mathit{V}}_{\mathit{f}}$ | $\widehat{\mathit{\sigma}}(\hat{\mathit{d}})$ | $\widehat{\mathit{\mu}}(\hat{\mathit{d}}/{\mathit{r}}_{\mathit{f}})$ | $\mathit{K}{\overline{)\left(\mathit{h}\right)}}_{\mathit{h}=6}$ |
---|---|---|---|---|---|

10 | CR05 | 31.72% | 2.0065 | 2.2921 | 1222 |

20 | CR05 | 31.66% | 1.8620 | 2.0823 | 2145 |

30 | CR07 | 31.81% | 1.8499 | 2.1752 | 2668 |

40 | CR07 | 31.51% | 1.8445 | 2.0874 | 3072 |

50 | CR01 | 31.96% | 2.0606 | 2.1472 | 3035 |

60 | CR02 | 31.75% | 1.8470 | 2.1051 | 2926 |

Mechanical Properties | Carbon Fiber | Thermoplastic Matrix |
---|---|---|

Longitudinal Modulus [GPa] | 230 | 3.2 |

Transverse Modulus [GPa] | 15 | 3.2 |

Longitudinal Shear Modulus [GPa] | 15 | 1.2 |

Transverse Shear Modulus [GPa] | 15 | 1.2 |

Poisson ratio | 0.2 | 0.3 |

Size $\mathit{\delta}$ | Number of Elements | Number of Nodes |
---|---|---|

10 | 2601 | 5408 |

20 | 10,201 | 20,808 |

30 | 22,801 | 46,208 |

40 | 40,401 | 81,608 |

50 | 63,001 | 127,008 |

60 | 90,601 | 182,408 |

Size $\mathit{\delta}$ | ${\mathit{E}}_{1}$ [MPa] | ${\mathit{E}}_{2}$ [MPa] | ${\mathit{E}}_{3}$ [MPa] | ${\mathit{G}}_{12}$ [MPa] | ${\mathit{G}}_{13}$ [MPa] | ${\mathit{G}}_{23}$ [MPa] | ${\mathit{\nu}}_{12}$ | ${\mathit{\nu}}_{13}$ | ${\mathit{\nu}}_{23}$ |
---|---|---|---|---|---|---|---|---|---|

10 | 75,187 | 5189 | 5153 | 2399 | 2292 | 2024 | 0.267 | 0.185 | 0.339 |

20 | 75,067 | 5165 | 5194 | 2361 | 2390 | 2032 | 0.269 | 0.182 | 0.336 |

30 | 75,397 | 5147 | 5178 | 2314 | 2360 | 2048 | 0.269 | 0.182 | 0.338 |

40 | 74,718 | 5156 | 5333 | 2231 | 2531 | 1991 | 0.276 | 0.175 | 0.323 |

50 | 75,748 | 5209 | 5262 | 2340 | 2454 | 2026 | 0.271 | 0.179 | 0.330 |

60 | 74,945 | 5172 | 5188 | 2364 | 2387 | 2036 | 0.269 | 0.183 | 0.336 |

Avg | 75,177 | 5173 | 5218 | 2335 | 2402 | 2026 | 0.270 | 0.181 | 0.334 |

CoV | 0.48% | 0.44% | 1.28% | 2.49% | 3.40% | 0.95% | 1.1% | 2.00% | 1.85% |

**Table 8.**Comparison of homogenized elastic properties with experimental data where $\Delta =100\times \left|\left(\mathrm{EXP}\phantom{\rule{4.pt}{0ex}}-\phantom{\rule{4.pt}{0ex}}\mathrm{HM}\right)/\mathrm{EXP}\right|$ with EXP the experimental data and HM the homogenized properties computed for the respective representative equivalent volume size.

Mech. Prop. | Experimental Data | Perfect Square Unit Cell | Representative Equivalent Volume | ||||
---|---|---|---|---|---|---|---|

From [17] | From [17] | Δ [%] | $\mathit{\delta}=50$ | Δ [%] | $\mathit{\delta}=60$ | Δ [%] | |

${E}_{1}$ [MPa] | 74,970 | 76,401 | 1.91% | 75,748 | 1.04% | 74,945 | 0.03% |

${E}_{2}$ [MPa] | 5529 | 5257 | 4.92% | 5209 | 5.79% | 5172 | 6.46% |

${G}_{12}$ [MPa] | 2352 | 2153 | 8.46% | 2340 | 0.51% | 2364 | 0.51% |

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**MDPI and ACS Style**

Dutra, T.A.; Ferreira, R.T.L.; Resende, H.B.; Oliveira, L.M.; Blinzler, B.J.; Asp, L.E.
Identification of Representative Equivalent Volumes on the Microstructure of 3D-Printed Fiber-Reinforced Thermoplastics Based on Statistical Characterization. *Polymers* **2022**, *14*, 972.
https://doi.org/10.3390/polym14050972

**AMA Style**

Dutra TA, Ferreira RTL, Resende HB, Oliveira LM, Blinzler BJ, Asp LE.
Identification of Representative Equivalent Volumes on the Microstructure of 3D-Printed Fiber-Reinforced Thermoplastics Based on Statistical Characterization. *Polymers*. 2022; 14(5):972.
https://doi.org/10.3390/polym14050972

**Chicago/Turabian Style**

Dutra, Thiago Assis, Rafael Thiago Luiz Ferreira, Hugo Borelli Resende, Luís Miguel Oliveira, Brina Jane Blinzler, and Leif E. Asp.
2022. "Identification of Representative Equivalent Volumes on the Microstructure of 3D-Printed Fiber-Reinforced Thermoplastics Based on Statistical Characterization" *Polymers* 14, no. 5: 972.
https://doi.org/10.3390/polym14050972