# Thermoelastic Investigation of Carbon-Fiber-Reinforced Composites Using a Drop-Weight Impact Test

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## Abstract

**:**

^{®}code to analyze whether the coupled heat and wave equation phenomenon exists in a two-dimensional polar coordinate system by discretizing through a forward-time central-space (FTCS) finite-difference method (FDM). The results show the coupling has no significant impact as the waves generated due to impact disappears in 0.015 s. In contrast, heat diffusion happens for over a one-second period. This study demonstrates that the heat equation alone governs the CFRP heat flow process, and the thermoelastic effect is negligible for the specific drop-weight impact load.

## 1. Introduction

#### 1.1. Non-Destructive Testing (NDT) of Composites—Thermographic Evaluation Method

#### 1.2. Drop-Weight Impact Testing (DWIT)

#### 1.3. Thermoelastic and Thermoplastic Effects

#### 1.4. Numerical Analysis

## 2. Methodology

^{®}software. The derived numerical analysis solutions were validated using experimental findings to optimize the models.

#### 2.1. CFRP Sample

^{®}(Allred and Associates Inc., Elbridge, NY, USA) [26]. Each sample was a solid quasi-isotropic carbon-fiber sheet 1 mm thick (Figure 2) [27]. The CFRP sheets comprised five plies at 0°/+45°/90°/−45°/0° orientation laminates (Figure 2). The samples were composed of a tough and rigid carbon-reinforced epoxy matrix, with a textured finish on both sides. The properties of the samples are given in Table 1.

#### 2.2. Experimental Setup to Perform a Drop Test

^{®}T1030sc thermal camera [31], and were analyzed using Researcher IR Max software [32] to investigate the heat generation due to friction and diffusion associated with the thermoelastic effects. The physical process to collect IR thermographic images when excitation occurs through a mechanical impact can be explained in three steps (diagrammatic representation of the experiment is shown in Figure 6):

## 3. Finite Difference Method (MATLAB^{®})

#### 3.1. Mathematical Model

#### 3.2. Numerical Analysis

^{®}. The results were compared with the experiments, as discussed in Section 4.2.1. Afterward, the coupled heat and wave equation in the two-dimensional polar coordinate system was solved. As an initial condition, a constant temperature of 21.9 ${}^{\xb0}$C was specified throughout the domain, except at the source. It is vital for the stability and accuracy of the FDM to choose the correct timestep value. In this work, the Courant–Friedrichs–Lewy (CFL) condition [48] was used to decide the timestep size. The CFL condition is given in Equation (20):

## 4. Results and Discussion

^{®}. The strain waves experienced significant damping, and the amplitude dropped to 0.15 in 0.012 s. The dynamic strain reflected the behavior of the CFRP under drop impact for in-plane measurement.

#### 4.1. Thermographic Experimental Results

#### 4.2. Numerical Simulations Results

#### 4.2.1. Wave Equation Results

^{®}(time = 3000 timesteps, radius = 87 mm). The initial conditions given in the code were the quadratic profile and a maximum deformation of 0.5 V (recorded by high-frequency oscilloscope from experiments; this value is directly proportional to the strain). The damping term was also added, with a coefficient of damping = 0.006. After plotting the experimental and MATLAB

^{®}simulation results together (Figure 10), the wave equation predicted a profile and frequency of 205 Hz, as obtained from the experiments (code is attached in the Appendix).

#### 4.2.2. Coupled Heat and Wave Equation Results

#### 4.3. Summary

## 5. Conclusions

- The thermoelastic phenomenon generated upon impact in a quasi-isotropic CFRP sheet is high-speed and completes in almost 0.015 s.
- Heat diffusion from the hotspot, which developed due to friction between the steel ball and the CFRP surface, occurs for nearly 1.2 s.
- No significant effect of the coupling heat and wave equation was observed, as temperature change due to thermoelasticity (elastic waves) was minimal compared to temperature change due to heat conduction.
- We found that thermoelasticity has a negligible effect, and heat diffusion alone is responsible for the heat dissipation in the CFRP.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The carbon fiber quasi-isotropic sheets [28].

**Figure 3.**(

**a**) The spherical steel ball, which measured 20 mm in diameter and weighed 33 g. (

**b**) The steel ball wrapped in black tape.

**Figure 5.**(

**a**) The CFRP sample clamped in a circular flange with an inner diameter of 173 mm (top view). (

**b**) The CFRP sample clamped in a circular flange (side view).

**Figure 7.**Experimental strain vs. time data plotted in MATLAB

^{®}(natural response after the ball is removed).

**Figure 11.**The ΔT images of the CFRP specimen after the impact with a steel ball (experiments vs. simulations).

**Figure 12.**The temperature drop after impact during the DWIT (experiments vs. numerical simulations).

**Table 1.**Properties of the CFRP samples [27].

Carbon Fiber Modulus | 228 GPa |
---|---|

Weight of sheet | 72.1 g |

Density | 1540 kg/m^{3} |

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**MDPI and ACS Style**

Andleeb, Z.; Malik, S.; Abbas Khawaja, H.; Samuelsen Nordli, A.; Antonsen, S.; Hussain, G.; Moatamedi, M.
Thermoelastic Investigation of Carbon-Fiber-Reinforced Composites Using a Drop-Weight Impact Test. *Appl. Sci.* **2021**, *11*, 207.
https://doi.org/10.3390/app11010207

**AMA Style**

Andleeb Z, Malik S, Abbas Khawaja H, Samuelsen Nordli A, Antonsen S, Hussain G, Moatamedi M.
Thermoelastic Investigation of Carbon-Fiber-Reinforced Composites Using a Drop-Weight Impact Test. *Applied Sciences*. 2021; 11(1):207.
https://doi.org/10.3390/app11010207

**Chicago/Turabian Style**

Andleeb, Zahra, Sohail Malik, Hassan Abbas Khawaja, Anders Samuelsen Nordli, Ståle Antonsen, Ghulam Hussain, and Mojtaba Moatamedi.
2021. "Thermoelastic Investigation of Carbon-Fiber-Reinforced Composites Using a Drop-Weight Impact Test" *Applied Sciences* 11, no. 1: 207.
https://doi.org/10.3390/app11010207