# Elastodynamic Behaviour of Laminate Structures with Soft Thin Interlayers: Theory and Experiment

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## Abstract

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## 1. Introduction

## 2. Mathematical Modelling

#### 2.1. Exact Statement of Boundary Value Problem

#### 2.2. Modeling of the Film via EBCs

#### 2.3. Thickness Resonance Frequencies

## 3. Properties of Lamb Waves in Laminates with Soft Interlayer

#### 3.1. Main Properties of Dispersion Curves and Vibration Forms

#### 3.2. Influence of the Mechanical Properties of Interlayer

#### 3.3. Influence of the Thickness of Interlayer

#### 3.4. Influence of the Adhesive Bonding or Imperfect Contact

#### 3.5. Analysis of the Influence of the Film Parameters on the Basis of EBCs

## 4. Properties of Other Guided Waves in Laminates with Soft Interlayer

## 5. Comparison: Theory vs. Experiment

#### 5.1. Experimental Setup

#### 5.2. Analysis of the Experimental Data

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NDT | non-destructive testing |

SHM | stuctural health monitoring |

SCs | slowness curves |

EGWs | elastic guided waves |

GWs | guided waves |

LWs | Lamb waves |

ALW | antisymmetric Lamb wave |

SLW | symmetric Lamb wave |

EWs | edge waves |

BCs | boundary conditions |

EBCs | effective boundary conditions |

SBCs | spring-type boundary conditions |

LDV | laser Doppler vibrometer |

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**Figure 2.**Slownesses of LWs propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) and 2 mm thickness aluminium plate.

**Figure 3.**Displacement distribution ${u}_{k}({x}_{3},f)$ of LWs ${\mathrm{A}}_{0}$ and ${\mathrm{S}}_{0}$ propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) and LW ${\mathrm{a}}_{0}$ propagating in 2 mm thickness aluminium plate.

**Figure 4.**Displacement distribution ${u}_{k}({x}_{3},f)$ of LWs ${\mathrm{A}}_{1}$ and ${\mathrm{S}}_{1}$ propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) and LW ${\mathrm{s}}_{0}$ propagating in 2 mm thickness aluminium plate.

**Figure 5.**Displacement distribution ${u}_{k}({x}_{3},f)$ of LWs ${\mathrm{A}}_{2}$ and ${\mathrm{S}}_{2}$ propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) and LW ${\mathrm{a}}_{1}$ propagating in 2 mm thickness aluminium plate.

**Figure 6.**Classification of GWs propagating in homogeneous elastic waveguide and symmetric three-layered waveguide with thin soft mid-layer.

**Figure 7.**Slownesses of antisymmetric (

**a**) and symmetric (

**b**) LWs propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m interlayer/2 mm aluminium) for four materials: two-sided epoxy tape (dashed thick lines), two-component epoxy adhesive (dash-dotted lines), cyanoacrylate adhesive (dashed thin lines), silicone rubber (thick dotted lines).

**Figure 8.**SCs of LWs propagating in the laminate (2 mm aluminium/${h}_{2}$ thickness film/2 mm aluminium) at ${h}_{2}=10,40,50,100\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m (

**a**) and in 4.1 mm thickness plate (${h}_{2}=100\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m) calculated using EBCs (21) and (22) of zero order (thin dashed lines), second order (solid lines), three layer model (thick dashed lines) (

**b**).

**Figure 9.**SCs of symmetric LWs propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) with the imperfect contact for four different combinations of normal and tangential stiffnesses ${\varkappa}_{i}$ (

**a**) and symmetric LWs for two different values of the interface tangential stiffness if ${\varkappa}_{3}=\infty $ (

**b**).

**Figure 10.**Displacement distribution ${u}_{k}({x}_{3},f)$ of SLWs propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) with imperfect contact (${\kappa}_{1}=2.5$ GPa/mm, ${\kappa}_{3}=\infty $).

**Figure 11.**Displacement distribution ${u}_{k}({x}_{3},f)$ of SLWs propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) with imperfect contact (${\kappa}_{1}=4.2$ GPa/mm, ${\kappa}_{3}=\infty $).

**Figure 13.**Slownesses of LWs propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) with an imperfect contact at the interfaces (${\varkappa}_{1}={\varkappa}_{3}=20\phantom{\rule{0.166667em}{0ex}}\mathrm{GPa}/\mathrm{mm}$) for different approximations of EBCs: (

**a**)—zero-order EBCs (48), (

**b**)—first order EBCs (47), (

**c**)—reduced EBCs (46), (

**d**)—simplified EBCs (44) and (45).

**Figure 14.**Slownesses of all GWs propagating in 4.05 mm thickness symmetric laminate with a soft thin interlayer (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium).

**Figure 15.**Slownesses of EWs and SH-waves propagating in 4.05 mm thickness symmetric laminate with a soft thin interlayer (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) and 2 mm aluminium plate.

**Figure 16.**Attenuation of EWs propagating in 4.05 mm thickness symmetric laminate with a soft thin interlayer (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) and 2 mm aluminium plate.

**Figure 17.**Sketch of the experimental setup (

**a**). Out-of-plane velocities (

**b**) and their spectrum (

**c**) measured by the LDV at a point located 70 mm away from the piezoelectric actuator center after its broadband excitation with 1 $\mathsf{\mu}$s rectangular pulse tone burst voltage.

**Figure 18.**Slownesses of LWs propagating in 4.05 mm thickness plate (2 mm aluminium/50 $\mathsf{\mu}$m film/2 mm aluminium) determined via the MPM (circles) and estimated theoretically (solid lines).

Material | Density | Young Modulus | Poisson’s Ratio |
---|---|---|---|

$\mathit{\rho},\mathbf{kg}/{\mathbf{m}}^{3}$ | E, GPa | $\mathit{\nu}$ | |

Aluminium | 2700 | 70 | 0.33 |

Cyanoacrylate adhesive [34] | 1248 | 1.7 | 0.4 |

Silicone rubber [35] | 1150 | 3.1 | 0.48 |

Two-component epoxy adhesive [36] | 1345 | 2.75 | 0.35 |

Two-sided epoxy tape [37] | 930 | 0.5 | 0.4 |

Material | Eff. Stiffness | Density | Young Modulus | Poisson’s Ratio | |
---|---|---|---|---|---|

GPa/mm | $\mathit{\rho}$, $\mathbf{kg}/{\mathbf{m}}^{3}$ | E, GPa | $\mathit{\nu}$ | ||

${\mathbf{\varkappa}}_{\mathbf{1}}^{\mathrm{eff}}$ | ${\mathbf{\varkappa}}_{\mathbf{3}}^{\mathrm{eff}}$ | ||||

Aluminium | – | – | 2715 | 72 | 0.345 |

Two-sided epoxy tape | 1.1 | 26 | 900 | 0.26–0.35 | $\left[0.505-{\displaystyle \frac{0.16\phantom{\rule{0.166667em}{0ex}}E}{\mathrm{GPa}}}\right]$–0.5 |

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**MDPI and ACS Style**

Wilde, M.V.; Golub, M.V.; Eremin, A.A. Elastodynamic Behaviour of Laminate Structures with Soft Thin Interlayers: Theory and Experiment. *Materials* **2022**, *15*, 1307.
https://doi.org/10.3390/ma15041307

**AMA Style**

Wilde MV, Golub MV, Eremin AA. Elastodynamic Behaviour of Laminate Structures with Soft Thin Interlayers: Theory and Experiment. *Materials*. 2022; 15(4):1307.
https://doi.org/10.3390/ma15041307

**Chicago/Turabian Style**

Wilde, Maria V., Mikhail V. Golub, and Artem A. Eremin. 2022. "Elastodynamic Behaviour of Laminate Structures with Soft Thin Interlayers: Theory and Experiment" *Materials* 15, no. 4: 1307.
https://doi.org/10.3390/ma15041307