# A Data Driven Modelling Approach for the Strain Rate Dependent 3D Shear Deformation and Failure of Thermoplastic Fibre Reinforced Composites: Experimental Characterisation and Deriving Modelling Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methodology

#### 2.1. Material Configuration and Specimen Preparation

#### 2.2. Experimental Setup and Testing Program

## 3. Experimental Results

#### 3.1. Stress–Strain Behaviour and Experimental Classification

#### 3.2. Failure and Fracture Behaviour

## 4. Material Modelling and Property Identification

- data driven (DD): A closed formulation describing the entire experimental curves by coherent formulae with the determining parameters being the material’s characteristics.

#### 4.1. Modulus and Strength Based (MaS) Approach

#### 4.1.1. Determination of Engineering Constants

#### 4.1.2. Strain Rate Dependency and Model Parameters

^{−4}and 3$\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right.$.

#### 4.2. Data Driven (DD) Approach

#### 4.2.1. Determination of Material Characteristics

#### 4.2.2. Modelling of the Strain Rate Dependency

#### 4.3. Determining the Range of Validity of the Material Modelling Approaches

## 5. Discussion

#### 5.1. Strain Rate and Configuration Dependent Experimental Characterisation

#### 5.2. Presented Modelling Approaches

#### 5.2.1. Modulus and Strength Based (MaS) Approach

#### 5.2.2. Data Driven (DD) Approach

#### 5.3. Interpretability of the Nonlinearities

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DD | data driven |

DIC | digital image correlation |

FEA | finite element analysis |

FRP | fibre reinforced plastic |

GF/PP | E-glass fibre polypropylene |

MaS | modulus and strength based |

MKF | multi-layered weft knitted fabric |

PP | polypropylene |

ROI | region of interest |

SHPB | split Hopkinson pressure bar |

TT | through thickness |

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**Figure 1.**(

**a**) Specimen configurations with respect to the fibre reinforcement plane and (

**b**) the corresponding stress states during Iosipescu testing [15].

**Figure 2.**Iosipescu shear test setup: (

**a**) specimen with undeformed (before) and deformed (after the experiment) specimen; (

**b**) representative strain distribution from DIC with the chosen ROI for further analysis.

**Figure 4.**Post-experimental images of representative deformation and failure patterns for the investigated material configurations.

**Figure 5.**Strain rate dependency of exemplary shear modulus ${G}_{0.55\%}^{sec}$ and ${\tau}^{max}$ for different material configurations.

**Figure 6.**Mas model predictions for secant moduli and maximum shear stress, exemplary for the 12-configuration at lowest and highest investigated strain rates.

**Figure 8.**Juxtaposition of experimental results and corresponding model predictions with parameters based on standard and constrained regression.

**Figure 11.**Comparison of models predicting onset and saturation of significant nonlinearity for the 12-configuration.

**Figure 12.**Summary of the onset and saturation points of significant nonlinearity for the 12-configuration.

Loading | Strain Rate $\dot{\mathit{\gamma}}$ | Sampling | Tests Per | ||
---|---|---|---|---|---|

Velocity [$\raisebox{1ex}{$\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$] | 12-conf. [$\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$] | 13-conf. [$\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$] | 31-conf. [$\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$] | [$\mathbf{frames}/\mathbf{s}$] | Velocity |

0.00001 | 2.2 × 10^{−4} * | 2.4 × 10^{−4} | 3.5 × 10^{−4} | 1 | 12 |

0.0001 | 3.6 × 10^{−3} | 3.5 × 10^{−3} | 2.9 × 10^{−3} | 25 | 12 |

0.01 | 2.8 × 10^{−1} | 3.4 × 10^{−1} | 3.0 × 10^{−1} | 5000 | 12 |

0.1 | 3.0 | 3.4 | 2.4 | 50,000 | 9 |

Tests per conf. | 15 | 15 | 15 |

^{*}In one experiment, the strain measurement was not triggered. It is excluded from the further analyses.

${\mathit{G}}^{\mathit{sec}}$ [GPa] | ${\mathit{G}}^{\mathit{tan}}$ [GPa] | $\mathit{\tau}$ [MPa] | |||
---|---|---|---|---|---|

$\dot{\mathit{\gamma}}$ [$\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$] | 0.55% | 1% | 2% | 5% | max |

12-configuration | |||||

2.2 × 10^{−4} | 1.4 | 1.3 | 1.0 | 0.3 | 48.4 |

3.6 × 10^{−3} | 1.8 | 1.5 | 1.1 | 0.2 | 56.4 |

2.8 × 10^{−1} | 2.4 | 1.9 | 1.4 | 0.4 | 65.0 |

3.0 | 2.5 | 1.9 | 1.5 | 0.6 | 72.3 |

13-configuration | |||||

2.4 × 10^{−4} | 1.2 | 1.0 | 0.6 | 0.07 | 22.6 |

3.5 × 10^{−3} | 1.2 | 1.0 | 0.7 | 0.12 | 25.4 |

3.4 × 10^{−1} | 1.4 | 1.1 | 0.8 | 0.11 | 29.4 |

3.4 | 1.6 | 1.2 | 0.9 | 0.14 | 34.1 |

31-configuration | |||||

3.5 × 10^{−4} | 1.1 | 0.8 | 0.6 | -${}^{*}$ | 12.1 |

3.7 × 10^{−3} | 1.4 | 1.1 | 1.0 | -${}^{*}$ | 15.9 |

2.6 × 10^{−1} | 1.6 | 1.2 | 1.1 | -${}^{*}$ | 17.5 |

2.6 | 2.0 | 1.6 | 1.4 | -${}^{*}$ | 21.9 |

^{*}Strain of 5% has not been reached.

${\dot{\mathit{\gamma}}}_{\mathit{ref}}$ [$\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$] | ${\mathit{G}}_{0.55\%,\mathit{ref}}^{\mathit{sec}}$ [MPa] | ${\mathit{\tau}}_{\mathit{ref}}^{\mathit{max}}$ [MPa] | ${\mathit{A}}^{\mathit{G}}$$[-]$ | ${\mathit{A}}^{\mathit{\tau}}$$[-]$ | |
---|---|---|---|---|---|

12-conf. | 2.2 × 10^{−4} | 1.4 | 48.4 | 0.09 | 0.05 |

13-conf. | 3.0 × 10^{−4} | 1.2 | 22.6 | 0.045 | 0.05 |

31-conf. | 12.1 | 0.08 |

${\dot{\mathit{\gamma}}}_{\mathit{ref}}$ [$\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.$] | ${\mathit{G}}_{0,\mathit{ref}}^{\mathit{tan}}$ [MPa] | ${\mathit{a}}_{\mathit{ref}}$$[-]$ | ${\mathit{A}}^{\mathit{G}}$$[-]$ | ${\mathit{A}}^{\mathit{a}}$$[-]$ | |
---|---|---|---|---|---|

12-conf. | 2.2 × 10^{−4} | 1372.48 | 29.30 | 0.090 | 0.032 |

13-conf. | 2.4 × 10^{−4} | 1463.62 | 67.29 | 0.113 | 0.031 |

31-conf. | 3.4 × 10^{−4} | 2670.44 | 220.35 | −0.029 | −0.081 |

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**MDPI and ACS Style**

Gerritzen, J.; Hornig, A.; Gröger, B.; Gude, M.
A Data Driven Modelling Approach for the Strain Rate Dependent 3D Shear Deformation and Failure of Thermoplastic Fibre Reinforced Composites: Experimental Characterisation and Deriving Modelling Parameters. *J. Compos. Sci.* **2022**, *6*, 318.
https://doi.org/10.3390/jcs6100318

**AMA Style**

Gerritzen J, Hornig A, Gröger B, Gude M.
A Data Driven Modelling Approach for the Strain Rate Dependent 3D Shear Deformation and Failure of Thermoplastic Fibre Reinforced Composites: Experimental Characterisation and Deriving Modelling Parameters. *Journal of Composites Science*. 2022; 6(10):318.
https://doi.org/10.3390/jcs6100318

**Chicago/Turabian Style**

Gerritzen, Johannes, Andreas Hornig, Benjamin Gröger, and Maik Gude.
2022. "A Data Driven Modelling Approach for the Strain Rate Dependent 3D Shear Deformation and Failure of Thermoplastic Fibre Reinforced Composites: Experimental Characterisation and Deriving Modelling Parameters" *Journal of Composites Science* 6, no. 10: 318.
https://doi.org/10.3390/jcs6100318