# An Experimentally Based Micromechanical Framework Exploring Effects of Void Shape on Macromechanical Properties

^{1}

^{2}

^{2}Vehicle Design, SE-100 44 Stockholm, Sweden

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions, a lighter structure leaves room for more batteries, increasing the driving range of BEVs [1]. Optimized lightweighting can be achieved through the introduction of Carbon Fiber Reinforced Plastics (CFRP). CFRP has a high strength-to-weight ratio and superior mechanical properties. However, due to its composition of different constituents, a CFRP material exhibits a more complex failure behavior compared to traditional metallic materials and needs to be described by several different failure and damage mechanisms simultaneously. For a CFRP component, the failure events at the micromechanical scale advance and initiate failure events at higher scales (meso- and macro-scale), preferably modeled using multi-scale models [2].

## 2. Methodology

#### 2.1. Micrograph Data Extraction

#### 2.1.1. Image Processing

#### 2.1.2. Matrix, Fiber, and Void Data

#### 2.1.3. Fiber Diameter and Nearest Neighbor Distributions

#### 2.2. RVE Generation

#### 2.2.1. Characterization of RVEs

- if: ${r}_{void}>r$—remove the circumference within the void;
- if: ${r}_{void}<r$—remove the circumference of the void within the observation area.

#### 2.3. Numerical Analysis

#### 2.3.1. FE Modeling

#### 2.3.2. Periodic Boundary Conditions

#### 2.3.3. Computational Homogenization

## 3. Case Study

#### 3.1. Material System

#### 3.2. Micrograph Data Extraction

#### 3.2.1. Matrix, Fiber, and Void Data

^{®}, together with the micrographs with 5× magnification.

#### 3.2.2. Fiber Diameter and Nearest Neighbor Distributions

#### 3.3. RVE Generation

#### Statistical Characterization of RVEs

#### 3.4. FE Modeling

#### 3.5. Results of the Case Study

#### 3.5.1. Prediction of Macromechanical Properties

#### 3.5.2. Verification with Static Testing

## 4. Discussion and Conclusions

#### 4.1. RVE Generation

#### 4.2. Implementation of Voids

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BDA | Blob Detection Algorithm |

BEVs | Battery Electric Vehicles |

BVP | Boundary Value Problem |

CFRP | Carbon-Fiber-Reinforced Plastics |

CLAHE | Contrast-Limited Adaptive Histogram Equalization |

CLT | Classical Lamination Theory |

CoG | Center of Gravity |

CSR | Complete Spatial Randomness |

DIC | Digital Image Correlation |

FE | Finite Element |

FEA | Finite-Element Analysis |

LoG | Laplacian of Gaussian |

MSM | Multi-Scale Modeling |

NN | Nearest Neighbor |

PBC | Periodic Boundary Condition |

RVE | Representative Volume Element |

SMC | Sheet Molding Compounds |

SSE | Sum of Square Error |

UD | Unidirectional |

UTS | Ultimate Tensile Strength |

## Appendix A. Computational Homogenization

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**Figure 1.**The workflow for the micromechanical simulation approach in a multi-scale modeling framework, referred to as EMMMA.

**Figure 3.**A selection of distribution functions fit to the extracted micrograph data. (

**a**) First NN. (

**b**) Second NN.

**Figure 4.**Periodic RVEs generated with the algorithm (

**a**) without a void, (

**b**) with a circular void, (

**c**) with an elliptical void angled ${0}^{\circ}$, and (

**d**) with an elliptical void angled ${45}^{\circ}$.

**Figure 7.**Extracted void data for the manufactured plate looking at shape parameters for a fitted circular and elliptical void shape.

**Figure 8.**Presenting the first, second, and third NN distances between voids for the manufactured plate.

**Figure 9.**The best distribution functions of (

**a**) first NN and (

**b**) second NN, comparing the data from micrographs with 20× and 50× magnification.

**Figure 12.**Comparison of actual microstructure and the generated RVE fiber distribution. The different images show (

**1**) how smaller-matrix-dominating areas are captured, (

**2**) how areas with tightly packed fibers are captured, and (

**3**) how sparsely scattered fibers are difficult to capture.

**Figure 13.**Second-order intensity function $K\left(r\right)$ and the radial distribution function $G\left(r\right)$ for RVEs with different ${V}_{A}$. (

**a**) $K\left(r\right)$ with the void as a part of the observation area for circular voids, (

**b**) $G\left(r\right)$ with the void as a part of the observation area for circular voids (

**c**) $K\left(r\right)$ with the void as a part of the observation area for elliptical voids angled ${0}^{\circ}$, (

**d**) $G\left(r\right)$ with the void as a part of the observation area for elliptical voids angled ${0}^{\circ}$, (

**e**) $K\left(r\right)$ with the void not as a part of the observation area and adjusting the correction factor ${w}_{k}$ accordingly for circular voids, and (

**f**) $G\left(r\right)$ with the void not as a part of the observation area for circular voids.

**Figure 14.**Second-order intensity function $K\left(r\right)$ and the radial distribution function $G\left(r\right)$ for different RVEs with side length L and no voids, (

**a**) $K\left(r\right)$ and (

**b**) $G\left(r\right)$.

**Figure 16.**The effective macromechanical properties: (

**a**) ${E}_{x}$, (

**b**) ${E}_{y}$, and (

**c**) ${E}_{z}$, as a function of void area, ${V}_{v}$, for a 200 $\mathsf{\mu}$m RVE.

**Figure 17.**The effective macromechanical properties: (

**a**) ${G}_{xy}$, (

**b**) ${G}_{yz}$, and (

**c**) ${G}_{zx}$, as a function of void area, ${V}_{v}$, for a 200 $\mathsf{\mu}$m RVE.

**Figure 18.**The effective macromechanical properties: (

**a**) ${\nu}_{yz}$ and (

**b**) ${\nu}_{zy}$, as a function of void area, ${V}_{v}$, for a 200 $\mathsf{\mu}$m RVE.

**Table 1.**Average volume fractions for fiber (${V}_{f}$), matrix (${V}_{m}$), and void (${V}_{v}$) of the manufactured plate.

Magnification | Avg. ${\mathit{V}}_{\mathit{f}}$ (-) | Avg. ${\mathit{V}}_{\mathit{m}}$ (-) | Avg. ${\mathit{V}}_{\mathit{v}}$ (-) |
---|---|---|---|

5 | N/A | N/A | 0.0062 |

20 | 0.5808 | 0.4192 | N/A |

50 | 0.5963 | 0.4037 | N/A |

${\mathit{V}}_{\mathit{v}}$ (-) | ${\mathit{\u2300}}_{\mathit{v}}$ ($\mathsf{\mu}$m) | Area ($\mathsf{\mu}$m${}^{2}$) | ${\mathit{V}}_{\mathit{f}}$ (-) | ${\mathit{V}}_{\mathit{f}}$ (-) (w.o. Void) |
---|---|---|---|---|

0.2498 | 112.8 | 10,000 | 0.4341 | 0.5787 |

0.2003 | 101.0 | 8000 | 0.4757 | 0.5949 |

0.1500 | 87.4 | 6000 | 0.4943 | 0.5815 |

0.1001 | 71.4 | 4000 | 0.5300 | 0.5890 |

0.0499 | 50.4 | 2000 | 0.5656 | 0.5953 |

0.0249 | 35.6 | 1000 | 0.5751 | 0.5898 |

0.0125 | 25.2 | 500 | 0.5852 | 0.5925 |

0.0062 | 17.8 | 250 | 0.5880 | 0.5917 |

0.0025 | 11.2 | 100 | 0.5958 | 0.5972 |

**Table 3.**${V}_{v}$ of the RVEs with an elliptical void angled ${0}^{\circ}$ and the resulting ${V}_{f}$.

${\mathit{V}}_{\mathit{v}}$ [-] | Major Axis, a ($\mathsf{\mu}$m) | Minor Axis, b ($\mathsf{\mu}$m) | Area ($\mathsf{\mu}$m${}^{2}$) | ${\mathit{V}}_{\mathit{f}}$ (-) | ${\mathit{V}}_{\mathit{f}}$ (-) (w.o. Void) |
---|---|---|---|---|---|

0.2503 | 73.6 | 43.3 | 10,000 | 0.4420 | 0.5896 |

0.2000 | 65.8 | 38.7 | 8000 | 0.4675 | 0.5844 |

0.1500 | 57.0 | 33.5 | 6000 | 0.4945 | 0.5818 |

0.1001 | 46.5 | 27.4 | 4000 | 0.5273 | 0.5859 |

0.0501 | 32.9 | 19.4 | 2000 | 0.5657 | 0.5956 |

0.0251 | 23.3 | 13.7 | 1000 | 0.5767 | 0.5915 |

0.0125 | 16.4 | 9.7 | 500 | 0.5870 | 0.5945 |

0.0062 | 11.6 | 6.8 | 250 | 0.5912 | 0.5949 |

0.0025 | 7.4 | 4.3 | 100 | 0.5938 | 0.5953 |

**Table 4.**${V}_{v}$ of the RVEs with an elliptical void angled ${45}^{\circ}$ and the resulting ${V}_{f}$.

${\mathit{V}}_{\mathit{v}}$ (-) | Major Axis, a ($\mathsf{\mu}$m) | Minor Axis, b ($\mathsf{\mu}$m) | Area ($\mathsf{\mu}$m${}^{2}$) | ${\mathit{V}}_{\mathit{f}}$ (-) | ${\mathit{V}}_{\mathit{f}}$ (-) (w.o. Void) |
---|---|---|---|---|---|

0.2503 | 73.6 | 43.3 | 10,000 | 0.4383 | 0.5847 |

0.2000 | 65.8 | 38.7 | 8000 | 0.4583 | 0.5728 |

0.1500 | 57.0 | 33.5 | 6000 | 0.4899 | 0.5763 |

0.1001 | 46.5 | 27.4 | 4000 | 0.5305 | 0.5895 |

0.0501 | 32.9 | 19.4 | 2000 | 0.5692 | 0.5993 |

0.0251 | 23.3 | 13.7 | 1000 | 0.5850 | 0.6000 |

0.0125 | 16.4 | 9.7 | 500 | 0.5830 | 0.5903 |

0.0062 | 11.6 | 6.8 | 250 | 0.5959 | 0.5996 |

0.0025 | 7.4 | 4.3 | 100 | 0.5868 | 0.5882 |

Property | Avg. Value |
---|---|

${E}_{x}$ | 136,395 MPa |

${E}_{y}$ | 7900 MPa |

${E}_{z}$ | 7904 MPa |

${G}_{xy}$ | 4235 MPa |

${G}_{yz}$ | 2908 MPa |

${G}_{xz}$ | 4244 MPa |

${\nu}_{xy}$ | 0.22 |

${\nu}_{yx}$ | 0.01 |

${\nu}_{xz}$ | 0.22 |

${\nu}_{zx}$ | 0.01 |

${\nu}_{yz}$ | 0.36 |

${\nu}_{zy}$ | 0.36 |

**Table 6.**Static tensile test results of the manufactured cross-ply plate with ${V}_{f}=0.5963$ and ${V}_{v}=0.0062$.

UTS (MPa) | Stiffness (MPa) |
---|---|

1377 | 68,519 |

1268 | 67,368 |

${\mathit{V}}_{\mathit{v}}$ (-) | Circular Void | Circular Void | Elliptical Void ${0}^{\circ}$ | Elliptical Void ${0}^{\circ}$ | Elliptical Void ${45}^{\circ}$ | Elliptical Void ${45}^{\circ}$ |
---|---|---|---|---|---|---|

${\mathit{E}}_{\mathit{x}}$ (MPa) | ${\mathit{G}}_{\mathit{xy}}$ (MPa) | ${\mathit{E}}_{\mathit{x}}$ (MPa) | ${\mathit{G}}_{\mathit{xy}}$ (MPa) | ${\mathit{E}}_{\mathit{x}}$ (MPa) | ${\mathit{G}}_{\mathit{xy}}$ (MPa) | |

0.25 | 52,352 | 2485 | 52,677 | 1956 | 52,687 | 2374 |

0.20 | 57,087 | 2825 | 55,909 | 2339 | 55,490 | 2649 |

0.15 | 59,858 | 3085 | 59,624 | 2768 | 59,495 | 3007 |

0.10 | 64,753 | 3467 | 63,790 | 3224 | 64,265 | 3403 |

0.05 | 68,734 | 3835 | 68,528 | 3733 | 69,320 | 3850 |

0.025 | 70,567 | 4078 | 69,933 | 3948 | 71,309 | 4081 |

0.0125 | 71,573 | 4149 | 71,585 | 4118 | 71,200 | 4119 |

0.006 | 71,806 | 4191 | 72,287 | 4217 | 72,419 | 4236 |

0.0025 | 72,580 | 4239 | 72,960 | 4294 | 71,912 | 4207 |

0 | 72,308 | 4235 | 72,308 | 4235 | 72,308 | 4235 |

**Table 8.**Comparison of the average effective macromechanical properties maintaining the void volume fraction. The 200 $\mathsf{\mu}$m RVE without a void is considered as the baseline. Only Young’s modulus and the shear modulus are presented; due to a very low variation, Poisson’s ratio is not included.

Property | RVE 200 $\mathsf{\mu}$m No Void (Baseline) | RVE 200 $\mathsf{\mu}$m Circ. Void | RVE 400 $\mathsf{\mu}$m Circ. Void | RVE 200 $\mathsf{\mu}$m Ellip. Void | RVE 400 $\mathsf{\mu}$m Ellip. Void |
---|---|---|---|---|---|

${V}_{v}$ (-) | 0 | 0.0062 | 0.0062 | 0.0062 | 0.0062 |

Void size | - | ${\u2300}_{v}$ = 17.8 $\mathsf{\mu}$m | ${\u2300}_{v}$ = 35.6 $\mathsf{\mu}$m | a = 11.6 $\mathsf{\mu}$m b = 6.8 $\mathsf{\mu}$m | a = 23.2 $\mathsf{\mu}$m b = 13.6 $\mathsf{\mu}$m |

${V}_{f}$ (-) | 0.5902 | 0.5880 (incl. void) | 0.5965 (incl. void) | 0.5912 (incl. void) | 0.5920 (incl. void) |

${E}_{x}$ (MPa) | 136,395 | −0.61% | +0.83% | +0.10% | −0.48% |

${E}_{y}$ (MPa) | 7900 | −2.11% | −1.09% | −2.24% | −2.59% |

${E}_{z}$ (MPa) | 7904 | −2.02% | −1.23% | −1.13% | −1.35% |

${G}_{xy}$ (MPa) | 4235 | −1.04% | 0.50% | −0.43% | −1.02% |

${G}_{yz}$ (MPa) | 2908 | −1.65% | −0.89% | −1.51% | −1.62% |

${G}_{xz}$ (MPa) | 4244 | −1.32% | +0.02% | −0.57% | −0.94% |

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**MDPI and ACS Style**

Eliasson, S.; Karlsson Hagnell, M.; Wennhage, P.; Barsoum, Z.
An Experimentally Based Micromechanical Framework Exploring Effects of Void Shape on Macromechanical Properties. *Materials* **2022**, *15*, 4361.
https://doi.org/10.3390/ma15124361

**AMA Style**

Eliasson S, Karlsson Hagnell M, Wennhage P, Barsoum Z.
An Experimentally Based Micromechanical Framework Exploring Effects of Void Shape on Macromechanical Properties. *Materials*. 2022; 15(12):4361.
https://doi.org/10.3390/ma15124361

**Chicago/Turabian Style**

Eliasson, Sara, Mathilda Karlsson Hagnell, Per Wennhage, and Zuheir Barsoum.
2022. "An Experimentally Based Micromechanical Framework Exploring Effects of Void Shape on Macromechanical Properties" *Materials* 15, no. 12: 4361.
https://doi.org/10.3390/ma15124361