# Numerical Analysis of the Main Wave Propagation Characteristics in a Steel-CFRP Laminate Including Model Order Reduction

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Dispersion Relation in Fiber Metal Laminates

#### 2.1. Dispersion Relation of a Single Layer

**b**are the volume forces,

**u**represents the displacement field, and the dots indicate the second derivative with respect to the time. Furthermore, to derive the equation of motion, linear strains are considered

#### 2.2. Wave Propagation in Layered Structures

#### 2.3. Displacement Fields

## 3. Material and Model Definition

#### 3.1. Material Layout

#### 3.2. Model Definition

## 4. Analysis of the Dispersion Diagram

#### 4.1. Analytical Treatment

#### 4.2. Numerical Simulations

^{®}. The two-dimensional numerical model and the corresponding boundary conditions used are depicted in Figure 2. The total length of the model is 1 $\mathrm{m}$ and the discretization is realized by second-order Lagrangian elements under plane strain assumption. Subsequently, a structured mesh of rectangular elements with quadratic shape functions is generated. For the excitation of the wave field, a multifrequency signal is used, see Ref. [31]. It is tuned to be able to generate dispersion diagrams over a predefined frequency range and hence, to avoid multiple simulations. The same signal is used for both directions, covering a frequency range from 25 $\mathrm{k}$$\mathrm{Hz}$ to 245 $\mathrm{k}$$\mathrm{Hz}$ in the steps of 10 $\mathrm{k}$$\mathrm{Hz}$. The signal is applied by a force boundary condition at a distance of 4 $\mathrm{m}$$\mathrm{m}$ from the left edge at the upper surface of the plate. Subsequently, both fundamental ${A}_{0}$- and ${S}_{0}$-modes are excited. A mode-selective excitation is not necessary, because the evaluation is done at the center line of the plate. Due to the theoretical displacement field characteristics, the ${S}_{0}$-mode shows only an in-plane displacement component at the center line, whereas the ${A}_{0}$-mode is limited to an out-of-plane motion. Therefore, with the help of the displacement component ${u}_{1}$ the ${S}_{0}$-mode and with the help of the component ${u}_{3}$, the ${A}_{0}$-mode can be evaluated separately. Details about the model size and the discretization are provided in Table 2.

#### 4.3. Results and Comparison

## 5. Displacement Field Analysis

#### 5.1. Analytical Treatment

#### 5.2. Numerical Simulations

#### 5.3. Results and Comparison

## 6. Parametric Model Order Reduction of Numerical Model

#### 6.1. Fundamentals

#### 6.2. Application

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

2D | two dimensional |

2D-DFT | discrete two dimensional Fourier transformation |

3D | three dimensional |

CFRP | carbon fiber-reinforced plastic |

FML | fiber metal laminate |

FRP | fiber-reinforced polymer |

GFRP | glass fiber-reinforced plastic |

GUW | guided ultrasonic wave |

HiFi | high fidelity |

MOR | model order reduction |

PMOR | parametric model order reduction |

POD | proper orthogonal decomposition |

POM | proper orthogonal modes |

SHM | structural health monitoring |

SVD | singular value decomposition |

UD | unidirectional |

## Appendix A

**Figure A1.**In-plane (

**left**) and out-of-plane (

**right**) components of a normalized displacement field for ${A}_{0}$ wave mode perpendicular to the fiber orientation at 80 $\mathrm{k}$$\mathrm{Hz}$ derived from the analytical framework and numerical 2D- and 3D-simulations.

**Figure A2.**In-plane (

**left**) and out-of-plane (

**right**) components of a normalized displacement field for ${S}_{0}$ (

**top**) and ${A}_{0}$ (

**bottom**) wave modes in the fiber orientation at 80 $\mathrm{k}$$\mathrm{Hz}$ derived from the analytical framework and numerical 2D- and 3D-simulations.

${\mathit{A}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{S}}_{0}$ | ${\mathit{S}}_{0}$ | |
---|---|---|---|---|

in-Plane | Out-of-Plane | in-Plane | Out-of-Plane | |

In fiber direction | $2.01\%$ | $0.01\%$ | $0.007\%$ | $0.035\%$ |

Perpendicular | $1.02\%$ | $0.006\%$ | $0.023\%$ | $0.092\%$ |

${\mathit{A}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{S}}_{0}$ | ${\mathit{S}}_{0}$ | |
---|---|---|---|---|

in-Plane | Out-of-Plane | in-Plane | Out-of-Plane | |

In fiber direction | $0.33\%$ | $0.004\%$ | $0.008\%$ | $0.055\%$ |

Perpendicular | $1.06\%$ | $0.006\%$ | $0.036\%$ | $0.19\%$ |

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**Figure 2.**Numerical model for the computation of the dispersion diagrams including the boundary conditions.

**Figure 3.**Comparison of numerically and analytically determined dispersion diagrams in fiber direction (

**left**) and perpendicular (

**right**) to the fiber orientation.

**Figure 4.**Deviation of the numerically and analytically determined dispersion diagrams in fiber direction (

**left**) and perpendicular (

**right**) to the fiber orientation.

**Figure 5.**Numerical 2D model for the computation of the displacement fields including the boundary conditions.

**Figure 6.**Numerical 3D model for the computation of displacement fields including the boundary conditions.

**Figure 7.**In-plane (

**left**) and out-of-plane (

**right**) components of normalized displacement field for ${S}_{0}$ (

**top**) and ${A}_{0}$ (

**bottom**) wave modes in the fiber orientation at 120 $\mathrm{k}$$\mathrm{Hz}$ from analytical treatment and numerical 2D- and 3D-simulations.

**Figure 8.**In-plane (

**left**) and out-of-plane (

**right**) components of normalized displacement field for ${S}_{0}$ (

**top**) and ${A}_{0}$ (

**bottom**) wave modes perpendicular to the fiber orientation at 120 $\mathrm{k}$$\mathrm{Hz}$ from analytical treatment and numerical 2D- and 3D-simulations.

**Figure 9.**In-plane

**(left**) and out-of-plane (

**right**) components of normalized displacement field for ${S}_{0}$ wave modes perpendicular to the fiber orientation at 80 $\mathrm{k}$$\mathrm{Hz}$ from analytical treatment and numerical 2D- and 3D-simulations.

${\mathit{E}}_{11}$ | ${\mathit{E}}_{22}$ | ${\mathit{G}}_{12}$ | ${\mathit{\nu}}_{12}$ | ${\mathit{\nu}}_{23}$ | $\mathit{\rho}$ | |
---|---|---|---|---|---|---|

[GPa] | [GPa] | [GPa] | [-] | [-] | [kg/m^{3}] | |

Steel | 180 | 180 | 69.2 | 0.30 | 0.30 | 7900 |

CFRP | 127 | 9.24 | 4.83 | 0.30 | 0.37 | 1580 |

Parameter | Value | Unit |
---|---|---|

Thickness | 1.98 | $\mathrm{m}\mathrm{m}$ |

Length | 1000 | $\mathrm{m}\mathrm{m}$ |

Element size CFRP | $0.2\times 0.25$ | $\mathrm{m}{\mathrm{m}}^{2}$ |

Element size steel | $0.2\times 0.12$ | $\mathrm{m}{\mathrm{m}}^{2}$ |

Nodes per element | 9 | - |

Propagation length | 1000 | $\mathrm{m}\mathrm{m}$ |

Simulation time | 10 | $\mathrm{m}\mathrm{s}$ |

Sampling frequency | 6 | $\mathrm{MHz}$ |

Min. excitation frequency | 25 | $\mathrm{kHz}$ |

Max. excitation frequency | 245 | $\mathrm{kHz}$ |

**Table 3.**Parameters of the numerical models and simulations for the computation of displacement fields.

Parameter | Value | Unit | |
---|---|---|---|

Thickness | 1.98 | $\mathrm{m}\mathrm{m}$ | |

Length | 0.3 | $\mathrm{m}$ | |

2D model | Element size CFRP | $0.75\times 0.25$ | $\mathrm{m}{\mathrm{m}}^{2}$ |

Element size steel | $0.75\times 0.12$ | $\mathrm{m}{\mathrm{m}}^{2}$ | |

Nodes per element | 9 | - | |

Thickness | 1.98 | $\mathrm{m}\mathrm{m}$ | |

Length | 0.3 | $\mathrm{m}$ | |

3D model | Width | 0.15 | $\mathrm{m}$ |

Element size CFRP | $0.75\times 0.75\times 0.25$ | $\mathrm{m}{\mathrm{m}}^{3}$ | |

Element size steel | $0.75\times 0.75\times 0.12$ | $\mathrm{m}{\mathrm{m}}^{3}$ | |

Nodes per element | 27 | - | |

Excitation frequency | 80 | $\mathrm{k}\mathrm{Hz}$ | |

Analysis at 80 $\mathrm{kHz}$ | Sampling frequency | 1.6 | $\mathrm{MHz}$ |

Simulation time | 0.3 | $\mathrm{m}\mathrm{s}$ | |

Excitation frequency | 120 | $\mathrm{kHz}$ | |

Analysis at 120 $\mathrm{kHz}$ | Sampling frequency | 2.4 | $\mathrm{MHz}$ |

Simulation time | 0.21 | $\mathrm{m}\mathrm{s}$ |

${\mathit{A}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{S}}_{0}$ | ${\mathit{S}}_{0}$ | |
---|---|---|---|---|

in-Plane | Out-of-Plane | in-Plane | Out-of-Plane | |

In fiber direction | $0.072\%$ | $0.019\%$ | $0.042\%$ | $0.22\%$ |

Perpendicular | $0.57\%$ | $0.016\%$ | $0.06\%$ | $0.28\%$ |

${\mathit{A}}_{0}$ | ${\mathit{A}}_{0}$ | ${\mathit{S}}_{0}$ | ${\mathit{S}}_{0}$ | |
---|---|---|---|---|

in-Plane | Out-of-Plane | in-Plane | Out-of-Plane | |

In fiber direction | $0.22\%$ | $0.055\%$ | $0.054\%$ | $1.53\%$ |

Perpendicular | $1.02\%$ | $0.01\%$ | $0.074\%$ | $0.31\%$ |

Model | Training Time | Computational Time |
---|---|---|

High-dimensional | - | 66.29 s |

Reduced-order | 17.6 h | 1.45 s |

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## Share and Cite

**MDPI and ACS Style**

Mikhaylenko, A.; Rauter, N.; Bellam Muralidhar, N.K.; Barth, T.; Lorenz, D.A.; Lammering, R.
Numerical Analysis of the Main Wave Propagation Characteristics in a Steel-CFRP Laminate Including Model Order Reduction. *Acoustics* **2022**, *4*, 517-537.
https://doi.org/10.3390/acoustics4030032

**AMA Style**

Mikhaylenko A, Rauter N, Bellam Muralidhar NK, Barth T, Lorenz DA, Lammering R.
Numerical Analysis of the Main Wave Propagation Characteristics in a Steel-CFRP Laminate Including Model Order Reduction. *Acoustics*. 2022; 4(3):517-537.
https://doi.org/10.3390/acoustics4030032

**Chicago/Turabian Style**

Mikhaylenko, Andrey, Natalie Rauter, Nanda Kishore Bellam Muralidhar, Tilmann Barth, Dirk A. Lorenz, and Rolf Lammering.
2022. "Numerical Analysis of the Main Wave Propagation Characteristics in a Steel-CFRP Laminate Including Model Order Reduction" *Acoustics* 4, no. 3: 517-537.
https://doi.org/10.3390/acoustics4030032