# Finite Element Modeling Approaches, Experimentally Assessed, for the Simulation of Guided Wave Propagation in Composites

^{*}

## Abstract

**:**

## 1. Introduction

^{®}CAE environment (Dassault Systems, Simulia Corp., Providence, RI, USA), the implicit method foresees a procedure available for piezoelectric analysis allowing using C3D8E finite elements for the modeling of the piezoelectric sensors, which are not available for dynamic explicit FEA. One limitation of this approach is that the Abaqus

^{®}code does not account for piezoelectric effects in the total energy balance equation, which can lead to an apparent imbalance of the total energy of the model in some situations.

## 2. Materials and Methods

#### 2.1. Test Cases Overview

#### 2.2. SHM Experimental Equipment and Tests

## 3. Numerical Approach

^{®}Xeon

^{®}Gold 6248R CPU with a total of 24 cores and 48 threads.

#### Explicit vs. Implicit FEA

## 4. Results and Analysis

#### 4.1. Numerical Analyses vs. Experiments

_{0}and A

_{0}modes.

^{®}code automatically extracted the time of flight (ToF) of the 0-order wave packets (${S}_{0}$ and ${A}_{0}$) from the HT envelopes. By knowing the transducers’ position, the wave group velocities (${\mathrm{c}}_{\mathrm{g}}$) of the ${S}_{0}$ and ${A}_{0}$ GW packets were easily calculated. Figure 14 and Figure 15 report the comparison of the two numerical approaches, in terms of both dispersion and slowness phenomena, against the experimental data for the flat and curved panels, respectively. The ${S}_{0}$ wave mode is represented by the solid lines, while the ${A}_{0}$ mode is represented by the dotted lines. The separate colors are part of the specific approach. For both plates, the two numerical approaches provided coherent results when compared with the experiments. Consequently, the explicit method turned out to be preferable, due to the generally lower computational costs. However, if the objective was to study the GW propagation in the loaded configuration of the panel, which was representative of an operating scenario, the implicit formulation can be adopted as well. As mentioned in Section 1, once validated, this approach can simplify the modeling of the load. In fact, for the modeling of a quasi-static load under the explicit scheme, several efforts are needed: an explicit quasi-static approach must be properly defined, to reduce the inertia forces and the raising load-induced kinetic energy of the model [28]. For all these reasons, an implicit scheme should be preferred for easier modeling, despite the higher computational costs.

#### 4.2. Curvature Effect of GW

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFRP | Carbon fiber-reinforced plastic |

CIEDA | Combined implicit-explicit dynamic analysis |

Cg | Group velocity |

DOF | Degrees of freedom |

EDA | Explicit dynamic analysis |

EOCs | Operating loading and environmental conditions |

FEA | Finite element analysis |

GW | Guided wave |

IDA | Implicit dynamic analysis method |

PTRT | Piezoelectric transmitter/receiver transducer |

SHM | Structural health monitoring |

ToF | Time of flight |

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**Figure 3.**Schematic of the geometry and PTRTs network for the curved panel. (

**a**) Front view. (

**b**) Top view.

**Figure 4.**(

**a**) PIC255 sensors, as used in the experimental tests. (

**b**) Relative welding, with the wires connected to the electrodes.

**Figure 5.**Experimental setup for the (

**a**) flat and (

**b**) curved panels, with five surface-mounted sensors (numbered from 1 to 5 in the red circles). The yellow circles refer to (1) the flat and curved plates, (2) the oscilloscope, (3) the computer, (4) the probes, and (5) the connectors.

**Figure 6.**An example of an experimental acquired data set, considering the actuator in position PTRT 1.

**Figure 7.**Actuator input for numerical investigations for the implicit (blue line) and explicit (orange line) formulations.

**Figure 8.**An explicit FEA–GW excitation signal, modeled by applying an equivalent radial displacement field on the upper actuator edge.

**Figure 10.**Comparison between the explicit FEA and the experimental acquisitions on the flat panel for the investigated propagation paths—200 kHz.

**Figure 11.**Comparison between the implicit FEA and the experimental acquisitions on the flat panel for the investigated propagation paths—200 kHz.

**Figure 12.**Comparison between the explicit FEA and the experimental acquisitions on the curved panel for the investigated propagation paths—200 kHz.

**Figure 13.**Comparison between the implicit FEA and the experimental acquisitions on the curved panel for the investigated propagation paths—200 kHz.

**Figure 14.**Dispersion curves of the ${S}_{0}$ and ${A}_{0}$ mode group velocities propagating in the flat panel, evaluated with the explicit approach (in green), the implicit approach (in blue), and the experiments (in red).

**Figure 15.**Dispersion curves of the ${S}_{0}$ and ${A}_{0}$ mode group velocities on the curved panel, evaluated with the explicit approach (in green), the implicit approach (in blue), and the experiments (in red).

**Figure 16.**Comparison of the ${S}_{0}$ and ${A}_{0}$ mode dispersion curves for the flat panel (in red) and the curved panel (in blue)—experiments.

**Figure 17.**Comparison of the ${S}_{0}$ and ${A}_{0}$ modes dispersion curves on the flat panel (in red) and the curved panel (in blue)—explicit FEA.

**Figure 18.**Comparison of the ${S}_{0}$ and ${A}_{0}$ dispersion curves on the flat panel (in red) and the curved panel (in blue)—implicit FEA.

Material Property | Symbol | Units | CFRP Lamina | PTRT |
---|---|---|---|---|

Mass density | $\mathsf{\rho}$ | ${\mathrm{kg}\mathrm{m}}^{-3}$ | $1534$ | $7800$ |

Longitudinal Young’s modulus | ${\mathrm{E}}_{11}$ | $\mathrm{GPa}$ | $123.182$ | $62.1$ |

Transversal Young’s modulus | ${\mathrm{E}}_{22}$ | $\mathrm{GPa}$ | $7.700$ | $62.1$ |

${\mathrm{E}}_{33}$ | $7.700$ | $48.3$ | ||

Shear modulus | ${\mathrm{G}}_{12}$ | $\mathrm{GPa}$ | $3.60$ | $23.5$ |

${\mathrm{G}}_{13}$ | $3.60$ | $21$ | ||

${\mathrm{G}}_{23}$ | $2.70$ | $21$ | ||

Poisson’s ratio | ${\mathsf{\nu}}_{12}$ | $-$ | $0.360$ | $0.32$ |

${\mathsf{\nu}}_{13}$ | $0.360$ | $0.44$ | ||

${\mathsf{\nu}}_{23}$ | $0.4$ | $0.44$ |

Case Studies | FEA Type | Actuator |
---|---|---|

Flat panel | Explicit, Implicit | PTRT 1, PTRT 2 |

Curved panel | Explicit, Implicit | PTRT 1, PTRT 2 |

Material Property | Symbol | Units | Value |
---|---|---|---|

Dielectric constant | ${\mathrm{K}}_{3}$ | $-$ | $1280$ |

Dielectric permittivity at constant strain | ${\mathsf{\epsilon}}_{11}={\mathsf{\epsilon}}_{22}$ | ${\mathrm{nF}\mathrm{m}}^{-1}$ | $8.245$ |

${\mathsf{\epsilon}}_{33}$ | $7.122$ | ||

Piezoelectric strain coefficients | ${\mathrm{d}}_{31}$ | ${\mathrm{pC}\mathrm{N}}^{-1}$ | $-180$ |

${\mathrm{d}}_{33}$ | $400$ | ||

${\mathrm{d}}_{15}$ | $550$ |

**Table 4.**FE model details for the implicit and explicit FEA formulations, with reference to the case studies—8 NPW under a 300 kHz carrier.

Approach | Panel Shape | Component | Element Type | Number of Nodes | Number of Elements | Average Element Size (mm) |
---|---|---|---|---|---|---|

Explicit | Flat | Plate | S4R | 25,291 | 25,600 | 2 |

PTRT | C3D8R | 1251 | 768 | 0.5 | ||

Curved | Plate | S4R | 41,184 | 40,765 | 2 | |

PTRT | C3D8R | 1278 | 796 | 0.5 | ||

Implicit | Flat | Plate | C3D8I | 48,672 | 24,025 | 2 |

PTRT | C3D8E | 1251 | 768 | 0.5 | ||

Curved | Plate | C3D8I | 82,368 | 40,765 | 2 | |

PTRT | C3D8E | 1278 | 796 | 0.5 |

Measurement Direction | Actuator | Receiver |
---|---|---|

45° | PTRT 1 | PTRT 2 |

−45° | PTRT 1 | PTRT 3 |

0° | PTRT 2 | PTRT 3 |

90° | PTRT 2 | PTRT 4 |

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**MDPI and ACS Style**

De Luca, A.; Perfetto, D.; Polverino, A.; Aversano, A.; Caputo, F.
Finite Element Modeling Approaches, Experimentally Assessed, for the Simulation of Guided Wave Propagation in Composites. *Sustainability* **2022**, *14*, 6924.
https://doi.org/10.3390/su14116924

**AMA Style**

De Luca A, Perfetto D, Polverino A, Aversano A, Caputo F.
Finite Element Modeling Approaches, Experimentally Assessed, for the Simulation of Guided Wave Propagation in Composites. *Sustainability*. 2022; 14(11):6924.
https://doi.org/10.3390/su14116924

**Chicago/Turabian Style**

De Luca, Alessandro, Donato Perfetto, Antonio Polverino, Antonio Aversano, and Francesco Caputo.
2022. "Finite Element Modeling Approaches, Experimentally Assessed, for the Simulation of Guided Wave Propagation in Composites" *Sustainability* 14, no. 11: 6924.
https://doi.org/10.3390/su14116924