# Statistical Analysis of Mechanical Stressing in Short Fiber Reinforced Composites by Means of Statistical and Representative Volume Elements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Influence of Finite Volume Approaches

#### 3.2. Influence of Finite Volume Size

- Sampling stress values from a statistical population;
- Calculating mean value of samples;
- Calculating stiffness out of mean of samples with global strain;
- Repeating steps 1–3 several times to achieve stiffness distribution;
- Repeating steps 1–4 for different volumes.

^{3}in the legend. The identical mean value of the three stiffness distributions and the different scattering are clearly visible. The largest scattering is the sampling with the smallest volume and vice versa.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Procedure of the creation of a model with a unique fiber ensemble. (

**a**): Considering the number of fibers. (

**b**): Fiber length according to LDF. (

**c**): Spatial orientation of fibers according to ODF. (

**d**): Determining volume. (

**e**): Placing fibers to achieve periodic microstructure.

**Figure 2.**Two different approaches to define the finite volume: Statistical Volume Element (SVE) with constant dimensions and Representative Volume Element (RVE) with constant fiber volume fraction.

**Figure 3.**Method of homogenization: creation of the microstructure (1), use of FEM to determine local stresses and strains (2), volume averaging (3) and calculation of effective modulus (4).

**Figure 4.**Mechanical characterization of PBT-GF20 and PBT by tensile testing of specimens milled out of injection molded plates. The tensile direction with the PBT-GF20 corresponds to the flow direction of the injection molding process (2-direction).

**Figure 6.**Reconstructed ODF by the maximum entropy method of the shown fiber orientation tensor of second order. The color shows the probability that a fiber points in this direction.

**Figure 7.**Normalized histogram of experimentally measured fiber lengths. Mean aspect ratio of fiber length distribution is 15.4.

**Figure 8.**Histograms, normal distributions and mean values of the effective stiffnesses, calculated with the RVE and SVE approach as well as mean value of experimentally measured stiffness. We used 100 fibers in 60 different fiber ensembles for both approaches.

**Figure 9.**Histogram and normal distribution of the effective stiffness, calculated with 50 and 100 fibers using the SVE approach.

**Figure 10.**(

**a**) Stress distribution of three unique fiber ensembles (top), (

**b**) Stress distribution of the matrix phase of the same ensembles (bottom).

**Figure 11.**(

**a**) Average stress distribution of the composite with 50 and 100 fibers (top). (

**b**) Average stress distribution in the matrix phase with 50 and 100 fibers (bottom).

**Figure 14.**Relative standard deviation of drawn effective stiffness in dependency of considered volume and relative standard deviation of Monte-Carlo simulations with 50 and 100 fibers.

**Figure 15.**FEA with two different mesh sizes (top: 1.0 mm, bottom: 0.25 mm) of a test specimen with uniform fiber orientation under consideration of the statistical deviation of the effective Young’s modulus due to the effect of different fiber ensembles.

Parameter | SVE | RVE |
---|---|---|

number of fibers | constant | constant |

fiber geometry | constant | constant |

fiber length | defined by LDF | defined by LDF |

fiber volume fraction | defined by fibers and total volume | constant |

total volume | constant | defined by fibers and fiber volume fraction |

fiber orientation | defined by ODF | defined by ODF |

fiber arrangement | random | random |

phase properties | constant | constant |

**Table 2.**Parameters of the isotropic, linear–elastic material model used for the individual phases and parameter of linear elastic fit of the PBT-GF20 composite for comparison.

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**MDPI and ACS Style**

Breuer, K.; Spickenheuer, A.; Stommel, M.
Statistical Analysis of Mechanical Stressing in Short Fiber Reinforced Composites by Means of Statistical and Representative Volume Elements. *Fibers* **2021**, *9*, 32.
https://doi.org/10.3390/fib9050032

**AMA Style**

Breuer K, Spickenheuer A, Stommel M.
Statistical Analysis of Mechanical Stressing in Short Fiber Reinforced Composites by Means of Statistical and Representative Volume Elements. *Fibers*. 2021; 9(5):32.
https://doi.org/10.3390/fib9050032

**Chicago/Turabian Style**

Breuer, Kevin, Axel Spickenheuer, and Markus Stommel.
2021. "Statistical Analysis of Mechanical Stressing in Short Fiber Reinforced Composites by Means of Statistical and Representative Volume Elements" *Fibers* 9, no. 5: 32.
https://doi.org/10.3390/fib9050032