# Influence on Elastic Wave Propagation Behavior in Polymers Composites: An Analysis of Inflection Phenomena

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}/Polydimethylsiloxane (PDMS) composites [32]. As the concentration of TiO

_{2}increased from 0 to 60 vol. %, the velocity of elastic waves initially declined and then augmented. The study also drew on two theoretical models, the coherent potential approximation model and the Waterman–Truell multiple scattering model, to reinforce the discoveries. However, the exact cause of this phenomenon remains unknown. Therefore, there is a need for further research to better understand the behavior of composite materials beyond this inflection point.

_{2}/Polyurea composites and found that both particle number and diameter increased the attenuation of PPC. The results showed that particle scattering was the main cause of enhanced wave attenuation, which was verified by comparing the simulation and theoretical results. Mylavarapu [39] developed a model that considers factors such as the particle size, porosity, and radius ratio to determine the elastic wave attenuation coefficient. Nonetheless, this investigation did not consider the impact of particle interactions because of the small PPC content.

## 2. Materials and Methods

#### 2.1. Materials and Preparation

^{3}. The glass transition temperature (T

_{g}) of PMMA was 105.5 °C, which was measured using differential scanning calorimetry (DSC 2500, T.A. Instruments, New Castle, DE, USA).

#### 2.2. Characterization

#### 2.2.1. Content Testing

^{3}) and PMMA (1.18 g/cm

^{3}), respectively.

#### 2.2.2. Microstructure

#### 2.2.3. Elastic Wave Properties Measurements

#### 2.3. FEM Simulation

^{2}and filling the remaining space with PMMA, as depicted in Figure 2. A single period sinusoidal stress wave with a peak stress of 10 KPa and frequency of 5 MHz was loaded onto the left-hand boundary of the rectangle. To account for the attenuation of the Lamé waves, the attenuation coefficient of PMMA was added to the simulation according to the Rayleigh damping model. The experimentally derived attenuation coefficient of PMMA was 72 Np/m. The Cu particles were considered rigid owing to the substantial differences in the properties of Cu and PMMA. The parameters of the finite element simulation are summarized in Table 1.

## 3. Results and Discussion

#### 3.1. Composites Structures

#### 3.2. Elastic Wave Velocity and Attenuation

_{2}/PDMS [32]. The findings indicate that as the particle size of Cu powder increases, there was a growth in the wave velocity, in addition to an increase in the order of magnitude surge in the wave attenuation coefficient. For example, the wave velocity ranges from 1.89 km/s to 2.01 km/s in Cu/PMMA composites containing 30 vol. % of particles, and the wave attenuation coefficient ranges from 81.6 Np/m to 1678.8 Np/m as the particle size increases from 1 μm to 100 μm. With an increase in the particle size, the inflection point of the wave velocity and attenuation moves in the direction of the lower content.

#### 3.3. Analysis of Wave Velocity and Attenuation

_{s}is the number of particles.

_{1}) occurring in 0–0.4 μs and the second peak (A

_{2}) occurring in 0.8–1.4 μs. A

_{1}represents the initial input elastic wave intensity attenuated by matrix absorption and single-particle scattering ($-{\alpha}_{mat}-{\alpha}_{par})$. At the initial stage when the elastic wave has just entered the sample, the primary cause of its attenuation is absorption by the matrix and particle-independent scattering. The caused by particle interactions can be neglected at this point.

_{1}) and the $\left(-{\alpha}_{mat}-{\alpha}_{par}\right)$ decreased linearly with increasing particle content, as shown in Figure 9b. The second peak (A

_{2}) is an aftershock caused by independent particle scattering and particle interactions behind the wavefront [42]. Figure 9c indicates that the height of the second peak is consistent with the changing trend of single-particle scattering and particle interaction (${\alpha}_{par}+{\alpha}_{sca}$), with both initially increasing and then decreasing as the particle content increases.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Elastic wave Propagation Behavior of Cu/PMMA composites: (

**a**) Elastic wave velocity; (

**b**) Elastic wave attenuation.

**Figure 7.**Each component of attenuation: (

**a**) 1 μm Cu/PMMA; (

**b**) 10 μm Cu/PMMA; (

**c**) 100 μm Cu/PMMA; and (

**d**) particle interaction of 1, 10, 100 μm Cu/PMMA.

**Figure 8.**Stress distribution of elastic wave propagating behavior in Cu/PMMA composites: (

**a**) At 0.4 μs; (

**b**) At 1.2 μs.

**Figure 9.**Analysis of mechanical energy after wave of 10 μm Cu/PMMA: (

**a**) Mechanical energy intensity; (

**b**) Height of the first peak(A

_{1}) and $\left(-{\alpha}_{mat}-{\alpha}_{par}\right)$; and (

**c**) Height of the second peak(A

_{2}) and (${\alpha}_{par}+{\alpha}_{sca}$).

**Figure 10.**Stress distribution of elastic wave propagating behavior in Cu/PMMA composites: (

**a**–

**d**) particle size of 10 μm; (

**e**–

**h**) particle size of 100 μm.

**Figure 11.**Central line of the wavefront from simulations after elastic wave load for 40–60% volume fraction of 10 μm (

**left**) and 100 μm (

**right**) in Cu/PMMA.

c_{l} | c_{s} | |
---|---|---|

Cu | 4763 | 2318 |

PMMA | 2740 | 1372 |

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**MDPI and ACS Style**

Luo, G.; Cheng, P.; Yu, Y.; Geng, X.; Zhao, Y.; Xia, Y.; Zhang, R.; Shen, Q.
Influence on Elastic Wave Propagation Behavior in Polymers Composites: An Analysis of Inflection Phenomena. *Polymers* **2023**, *15*, 1680.
https://doi.org/10.3390/polym15071680

**AMA Style**

Luo G, Cheng P, Yu Y, Geng X, Zhao Y, Xia Y, Zhang R, Shen Q.
Influence on Elastic Wave Propagation Behavior in Polymers Composites: An Analysis of Inflection Phenomena. *Polymers*. 2023; 15(7):1680.
https://doi.org/10.3390/polym15071680

**Chicago/Turabian Style**

Luo, Guoqiang, Pu Cheng, Yin Yu, Xiangwei Geng, Yue Zhao, Yulong Xia, Ruizhi Zhang, and Qiang Shen.
2023. "Influence on Elastic Wave Propagation Behavior in Polymers Composites: An Analysis of Inflection Phenomena" *Polymers* 15, no. 7: 1680.
https://doi.org/10.3390/polym15071680