# Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis

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## Abstract

**:**

## 1. Introduction

#### 1.1. Out-of-Plane Waviness

#### 1.2. Spatial Stochastic Analysis

## 2. Multiscale Analysis for Computation of Knock-Down-Factors

## 3. Random Field Analysis

#### 3.1. Covariance Function

#### 3.2. Numerical Discretization

## 4. Probabilistic Analysis

#### 4.1. Process Description

#### 4.2. FEM Modeling

## 5. Results

#### 5.1. Parameters

#### 5.2. Deterministic FE Analysis

#### 5.3. Probabilistic Analysis

#### 5.3.1. Statistical Distribution of KDF across Layers

#### 5.3.2. Relation between KDF and Waviness Parameters

- (1)
- Ratio of waviness amplitude to laminate thickness;
- (2)
- Maximum curvature in the fiber orientation direction.

#### Ratio of Waviness Amplitude to Laminate Thickness

- Only a minor ply fraction of the full laminate is subjected to a large decrease (>30 percent) of the KDF for higher waviness ratios.
- A large decrease of the KDF occurs only on plies with orientation angles of zero degrees and 45 degrees.
- For all different ply orientations except zero degrees, a similar result pattern (decreasing, constant within some range) can be observed.
- A large reduction due to a low KDF can already be identified for small to moderate waviness ratios ($0.2<r<0.3$).
- Even for moderate waviness ratios ($r\approx 0.4$) there is a large remaining scatter in the KDF for different random field samples between an unaffected resulting material behavior (KDF $\approx 1.0$) and a large degradation of the tensile strength property.

#### Maximum Curvature in Fiber Orientation Direction

#### Summary of the Metric Assessment

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Two-dimensional examples of out-of-plane fiber waviness in a composite laminate. (

**a**) Embedded single graded waviness, reprinted with permission from Ref. [5], 2013, S. Mukhopadhyay; (

**b**) hump, reprinted with permission from Ref. [6], 2012, P. Davidson; (

**c**) complex stochastic distributed waviness, reprinted with permission from Ref. [7], 2010, R. Hinterhölzl.

**Figure 2.**Illustration of homogenization approach for KDF parameter estimation based on pristine (left) and defective model configuration (right), reprinted with permission from Ref. [29], 2020, F. Heinecke).

**Figure 4.**KLE-based waviness modeling of a generic CFRP cross-section made up of 8 layers for three different numbers of eigenvalues. The numerical approximated random field is highlighted in red. (

**a**) 6 modes; (

**b**) 12 modes; (

**c**) 20 modes.

**Figure 6.**Two examples of random field realizations embedded in a reference FE model for a laminate of the thickness ${t}_{lam}$ with about 169,000 degrees of freedom. The upper half layers of the laminate are made transparent for visibility. The color indicates the relative displacements of the ply with respect to a nominal pristine position.

**Figure 7.**Node sets at model boundary and applied loads for tensile loadcase. (Reprinted with permission from Ref. [29], 2020, F. Heinecke).

**Figure 8.**(

**a**) KDF plot of deterministic analysis for ${R}_{x}$ loadcase; (

**b**) random field example used for deterministic analysis.

**Figure 10.**Convergence of KDF according to mesh refinement in the z-direction (layer thickness direction).

**Figure 11.**Distribution of mean of KDF for ${R}_{x}$ within the selected laminate configuration with related variances as second bar element of the stochastic results. For each layer, the ply angle is additionally printed.

**Figure 12.**Distribution of KDF histograms for ${R}_{x}$ at specific layers within selected laminate configuration.

**Figure 13.**Scatter plot of waviness ratio compared to resulting KDF for property ${R}_{x}$. Plies that share the same fiber orientation angle are colored equally.

**Figure 14.**Selection of three exemplary random fields with a similar waviness ratio at the midsurface of the laminate.

**Figure 16.**Scatter plot of maximum curvature in fiber orientation direction compared to resulting KDF for ${R}_{x}$. Plies that share the same fiber orientation angle are colored equally.

**Figure 17.**Correlation coefficient for tensile KDF related to fiber orientation aligned maximum curvature.

x-Direction | y-Direction |
---|---|

${u}^{yzP{I}^{-}}=0$ | ${u}^{xzP{I}^{-}}=0$ |

${u}^{yzP{I}^{+}}={u}^{{N}^{ref}}$ | ${u}^{xzP{I}^{+}}=0$ |

${F}_{x}=\sigma \xb7{A}^{yzP{I}^{+}}$ $\sigma =1$ |

Computation Step | Elapsed Time | |
---|---|---|

FE model generation | 31.75 s | |

Defective | FE analysis | 120.09 s |

Result postprocessing | 5.44 s | |

FE model generation | 25.75 s | |

Pristine | FE analysis | 112.83 s |

Result postprocessing | 5.30 s | |

KDF estimation | 10.53 s | |

Total runtime | 311.69 s |

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**MDPI and ACS Style**

Schuster, A.; Degenhardt, R.; Willberg, C.; Wille, T.
Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis. *J. Compos. Sci.* **2022**, *6*, 353.
https://doi.org/10.3390/jcs6120353

**AMA Style**

Schuster A, Degenhardt R, Willberg C, Wille T.
Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis. *Journal of Composites Science*. 2022; 6(12):353.
https://doi.org/10.3390/jcs6120353

**Chicago/Turabian Style**

Schuster, Andreas, Richard Degenhardt, Christian Willberg, and Tobias Wille.
2022. "Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis" *Journal of Composites Science* 6, no. 12: 353.
https://doi.org/10.3390/jcs6120353