# The Effect of Fiber Waviness on the Residual Stress State and Its Prediction by the Hole Drilling Method in Fiber Metal Laminates: A Global-Local Finite Element Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Simulation of Residual Stress States in Metal/CFRP Hybrids and Its Prediction by the Simulated Hole Drilling Method

#### 2.1. Global-Local Finite Element Analysis

#### 2.2. Comparison between Experimental Data and Numerical Modeling

## 3. Analysis of Fiber Waviness on the Residual Stress State and the Reliability of the HDM

#### 3.1. Definition of Fiber Waviness

#### 3.2. Analysis Results

#### 3.2.1. In-Plane Fiber Waviness

#### 3.2.2. Out-of-Plane Fiber Waviness

## 4. Conclusions

- (1)
- In-plane fiber waviness leads to a rather low variance of residual stresses over thickness. Furthermore, oscillation of residual stresses occurs only at the boundaries of the samples section, which, however, quickly fades away. The simulated hole drilling method has proven that the residual stresses along the thickness direction can still be determined precisely in the case of such an in-plane waviness. The results indicated that the residual stress is not sensitive to the wave length and amplitude, if the waviness is in-plane.
- (2)
- Out-of-plane fiber waviness is much more critical with regard to the distribution and variance of residual stresses. The oscillation of residual stresses occurs over the entire sample width and is not limited to the boundary layers, as in the case of the in-plane fiber waviness described above. In addition, the investigation of two kinds of sinusoidal fiber waviness indicated that the long-wave profile with higher amplitude leads to significantly higher scattering of residual stresses, especially at the metal-CFRP interface. Applying the simulated hole drilling method, it was shown that the fiber waviness in thickness direction is a challenge for the determination of residual stress states such that the approach could not provide satisfactory results in the form of correct residual stress predictions.
- (3)
- Additionally it could be shown that the waviness in loading direction leads to a varying shear stiffness. This can be related to the kinematic of the fiber orientation, which can be with or against the overall loading direction. If shear stresses are present (e.g., in Timoshenko-Theory, multimaterial layup), this kind of fiber waviness leads to the oscillation of deformation fields.
- (4)
- The current limits of the incremental hole drilling method could be pointed out by the presented investigations. Without further adjustment of the calibration coefficients the oscillating stress and strain fields lead, in particular fiber waviness in thickness direction, to unreliable predictions. For the experimental application it can be concluded that the specimens have to be carefully examined with regard to fiber waviness, which, however, in most cases will be difficult to determine.

## Author Contributions

## Funding

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## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AHSS | Advanced high strength steels |

BIW | Body in white |

CFRP | Carbon fiber reinforced plastics |

FML | Fiber metal laminates |

CTE | Coefficient of thermal expansion |

CCS | Coefficient of chemical shrinkage |

HDM | Incremental hole drilling method |

RVE | Representative volume element |

DOC | Degree of cure |

FEA | Finite element analysis |

SHDM | Simulated hole drilling method |

${S}_{11},\phantom{\rule{0.277778em}{0ex}}{\sigma}_{11}$ | Normal stress in x direction [MPa] |

${S}_{13},\phantom{\rule{0.277778em}{0ex}}{\sigma}_{13}$ | Shear stress in x–z [MPa] |

${C}_{ii}$ | Entry of fourth order linear elasticity tensor [MPa] |

${\alpha}_{ii}$ | Coefficient of thermal expansion (CTE) [${\mathrm{K}}^{-1}$] |

${\beta}_{ii}$ | Coefficient of chemical shrinkage (CCS) [-] |

$\alpha $ | Angle of slope |

a | Amplitude of fiber waviness [mm] |

L | Wavelength of fiber waviness [mm] |

ORIENT | User subroutine to prescribe an orientation for defining local material directions |

USDFLD | User subroutine to define field variables |

GLARE | Glass fiber aluminum reinforced epoxy |

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**Figure 1.**Evolution of the direct elastic coefficients as a function of the degree of cure (

**a**–

**c**). Due to the assumption of the transversal isotropy, the terms ${C}_{22}$ and ${C}_{33}$ are equal and ${C}_{12}$ and ${C}_{13}$ are equal. (

**d**,

**e**) Evolution of the effective transversal isotropic coefficient of chemical shrinkage (CCS) and coefficient of thermal expansion (CTE) as a function of degree of cure [10].

**Figure 2.**On the left side, the geometry and boundary conditions of the considered metal/CFRP hybrid are shown. Due to the symmetry within the 2–3 plane, the displacement is freezed. The derived submodel, as well as the arrangement of the drilled hole and the strain gauges, are shown on the right side.

**Figure 3.**(

**a**) Prescribed temperature-time profile $T\left(x,t\right)$ for the whole domain. (

**b**) Corresponding residual stress state in thickness direction after completed curing and cooling down to ambient temperature. (

**c**) Validation of the predicted residual stress state in thickness direction based on results obtained by the incremental hole drilling method. For the residual stress analysis using the hole drilling method six combinations of strain gauges were used. The strain gauge combinations are depicted in (

**d**) [10]. In addition, the simulated hole drilling method (SHDM) is compared with experimental results in (

**c**).

**Figure 4.**Visualization of the fiber waviness under consideration. Each fiber waviness has been studied independently of each other. Furthermore, the fiber waviness has been assumed to be uniform within the entire cross-section.

**Figure 5.**(

**a**) Residual stress states with an in-plane fiber waviness compared to the reference residual stress state (without any imperfections). (

**b**) Contour plot of the residual stress distribution ${\sigma}_{11}$ shown for four section planes. The oscillation of the residual stress in thickness direction decreases rapidly from the boundary (y = 0 mm), after 2 mm the oscillation is almost absent.

**Figure 6.**Comparison of the theoretical residual stress and the predicted residual stress state using the simulated hole drilling method for the in-plane waviness. (

**a**) Sinus D1; (

**b**) Sinus D2.

**Figure 7.**Contour plot of the strain distribution ((

**a**)${\epsilon}_{11}$, (

**b**)${\epsilon}_{22})$ around the borehole at a drilling depth of $0.8$ mm. The placement of the strain gauges is shown in the figures. Due to the (in-plane) fiber waviness, the strain distribution around the borehole is not symmetrical.

**Figure 8.**(

**a**) Residual stress states with an out-of-plane fiber waviness (“Sinus D2”) compared to the reference residual stress state. The induced fiber waviness results in oscillating stress and strain fields in thickness direction along the x-axis. In addition, the waviness partially results in non-linear normal stress characteristics (see item 3). (

**b**) shows the normal stresses ${\sigma}_{11}$ across the thickness. The corresponding fiber waviness is shown redon top of the enlarged contourplot.

**Figure 9.**(

**a**) Residual stress states with an out-of-plane fiber waviness (“Sinus D1”) compared to the reference residual stress state. The induced fiber waviness results in oscillating stress and strain fields in thickness direction along the x-axis. (

**b**) The Normal stresses ${\sigma}_{11}$ across the thickness. On the top of the enlarged contour plot the corresponding fiber waviness is shown.

**Figure 10.**In the contour plot the distribution of the shear stress ${\sigma}_{13}$ in the CFRP-section with out-of-plane waviness (sinus D2) is illustrated. Below, the corresponding fiber waviness is plotted accordingly. Maximum shear stresses are located in areas with higher shear stiffness. The varying shear stiffness can be explained by the fact that the fiber orientation can oriented against (I) or with (II) the overall deformation direction.

**Figure 11.**Comparison of the theoretical residual stress and the predicted residual stress state using the simulated hole drilling method for the out-of-plane waviness “Sinus D2”. The comparison is done at the three characteristic positions (Figure 8). It can be clearly observed that the induced fiber imperfection lead to recognizable deviations. (

**a**) Pos. 1; (

**b**) Pos. 2; (

**c**) Pos. 3.

**Figure 12.**Comparison of the theoretical residual stress and the predicted residual stress state using the simulated hole drilling method for the out-of-plane waviness “Sinus D1”. The comparison is done at the two characteristic positions (Figure 9). It can be clearly observed that the induced fiber imperfection lead to recognizable deviations. (

**a**) Pos. 1; (

**b**) Pos. 2.

**Table 1.**Values of material parameters for the steel plate [10].

E | $\mathit{\nu}$ | ${\mathit{\alpha}}^{\mathbf{th}}$ |
---|---|---|

$210$ | $0.29$ | 1.2 × 10${}^{-5}$ |

**Table 2.**Values of wavelength and amplitude for the sine function used for defining fiber waviness (Equation 3.1 [38]).

Label | a [mm] | L [mm] |
---|---|---|

Sinus D1 | $0.0526$ | $2.29$ |

Sinus D2 | $1.19$ | $27.9$ |

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**MDPI and ACS Style**

Tinkloh, S.; Wu, T.; Tröster, T.; Niendorf, T.
The Effect of Fiber Waviness on the Residual Stress State and Its Prediction by the Hole Drilling Method in Fiber Metal Laminates: A Global-Local Finite Element Analysis. *Metals* **2021**, *11*, 156.
https://doi.org/10.3390/met11010156

**AMA Style**

Tinkloh S, Wu T, Tröster T, Niendorf T.
The Effect of Fiber Waviness on the Residual Stress State and Its Prediction by the Hole Drilling Method in Fiber Metal Laminates: A Global-Local Finite Element Analysis. *Metals*. 2021; 11(1):156.
https://doi.org/10.3390/met11010156

**Chicago/Turabian Style**

Tinkloh, Steffen, Tao Wu, Thomas Tröster, and Thomas Niendorf.
2021. "The Effect of Fiber Waviness on the Residual Stress State and Its Prediction by the Hole Drilling Method in Fiber Metal Laminates: A Global-Local Finite Element Analysis" *Metals* 11, no. 1: 156.
https://doi.org/10.3390/met11010156