# Effect of Impact Angle on the Impact Mechanical Properties of Bionic Foamed Silicone Rubber Sandwich Structure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Bionic Sandwich Structure Design

^{2}, thickness of foam silicone rubber is 5 mm, and density is 950 kg/m

^{3}. Figure 1c represents the stacking order of sandwich structure, respectively, coating (Co)

_{60}/(0°)

_{6}/(90°)

_{6}/Core/(0°)

_{6}/(90°)

_{6}, the stacking order is top-down, and the subscript indicates the number of repetitions.

#### 2.2. Ontological Model and Numerical Simulation Method

#### 2.2.1. Composite Laminate Damage Criterion

_{T}, X

_{C}, Y

_{T}, and Y

_{C}are the tensile and compressive strengths in the longitudinal and transverse directions, respectively, and ${\sigma}_{ij}\left(i,j=1,2,3\right)$ are the Corsi stress tensor components. S

_{12}is the shear strength in the fiber and transverse directions, S

_{13}is the shear strength in the fiber and thickness directions, and S

_{23}is the shear strength in the transverse and thickness directions.

#### 2.2.2. Rubber Intrinsic Structure Model and Parameters

_{0}to represent the pore size of the foamed silicone rubber, the strain energy density W of the foamed silicone rubber can then be determined by the material parameters of the base rubber, i.e., the porosity f

_{0}at the initial time and the invariants I

_{1}, I

_{2}, I

_{3}of the deformation tensor B.

^{3}, take ${\rho}_{S}=1150\mathrm{kg}/{\mathrm{m}}^{3}$.

_{1}, λ

_{2}, and λ

_{3}for each point on the solid part, there is a corresponding strain energy density W and corresponding invariants I

_{1}, I

_{2}, and I

_{3}.

^{T}, then I

_{3}= det(B) = 1 holds at any point within the spherical shell. The strain energy density is simplified as W = W (I

_{1}, I

_{2}) and is related only to the first and second invariants I

_{1}and I

_{2}. The invariants I

_{1}and I

_{2}at any point in the spherical shell can be expressed as a function of the macroscopic principal elongation ${\widehat{\lambda}}_{1},{\widehat{\lambda}}_{2},{\widehat{\lambda}}_{3}$; the macroscopic invariants ${\widehat{I}}_{1},{\widehat{I}}_{2},\widehat{j}$ (Jacobi determinant of the macroscopic deformation gradient); and the coordinates X, Y, and Z of the reference configuration, i.e.,

^{2}+ Y

^{2}+ Z

^{2})

^{1/2}. Using the above equation, the integration of the strain energy density over the entire spherical shell representative unit can be obtained and divided by the volume of the ball V

_{0}= 4πb

^{3}/3, b being the outer radius of the ball. The average strain energy density of the foamed silicone rubber as a whole is obtained $\widehat{W}$ as follows:

_{2}in the strain energy density can be discarded, and the expression of the strain energy density W

_{Y}is

_{10}, C

_{20}, C

_{30}, are material parameters and Wv is zero if the material is incompressible and no volume deformation occurs. According to the uniaxial tensile test data in the published literature, C

_{10}= 0.6, C

_{20}= −0.21, and C

_{30}= 0.08.

#### 2.2.3. Finite Element Modeling

^{3}and Young’s modulus of 6 GPa in this study species, using eight-node linear hexahedral cells, shrinkage integration, and hourglass control. The thickness of the unidirectional carbon fiber layer is 0.15 mm, the density is 300 g/m

^{2}, and the fiber plate uses a universal continuous shell grid within the eight-node quadrilateral face, reduction integration, hourglass control, and finite film strain (SC8R). The foamed silicone rubber thickness is 5 mm, the density is 950 kg/m

^{3}, and the rubber sandwich uses four-node linear tetrahedral cells (C3D4). A finite element model of the rubber sandwich structure is generated and analyzed using ABAQUS/Explicit software with fixed boundary conditions and 48 fully constrained supports, applying symmetric edge specimens on the upper and lower surfaces with displacements and rotation angles set to zero in the x, y, and z directions. In this paper, it is assumed that the pores are initially spherical and uniformly distributed, making the porous material initially isotropic. Figure 2 shows the finite element model of the sandwich structure under impact loading.

_{n}and tangential velocity V

_{t}. In this paper, the impact velocities are set as 4.970 m/s, 5.495 m/s, 5.973 m/s, 6.419 m/s, 6.830 m/s, and 8 m/s, and the impact angles are 30°, 45°, 60°, 75°, and 90°, respectively.

#### 2.3. Model Validation

_{3}/Core/W (woven fiber)

_{3}, the porosity is f

_{0}= 0.17, the impact velocity is 2.97 m/s, and the impact angle is 90° is shown in Figure 4. It can be seen from the figure that the curves have the same variation trend and the relative error between the finite element simulation and the experimental results is within 15%, indicating that the experimental results have good correlation with the simulation results. The reason for the error may be that the internal pores of rubber are assumed to be uniformly distributed, have uniform pore size, and have a certain number of spherical pores in the calculation process, while the actual internal pore structure of rubber is more complex.

## 3. Results and Discussion

#### 3.1. Energy Change

_{r}is the residual velocity. When v

_{r}is negative, it means the falling hammer bounces off the sandwich structure, and a positive value means the falling hammer penetrates the sandwich structure.

^{2}> 0.99) as shown in Figure 5, with the expression valid only for v

_{i}> v

_{threshold}.

_{threshold}are the fitting parameters; v

_{i}is the initial velocity of the impact; and v

_{threshold}is defined as the velocity penetration threshold at a given angle.

#### 3.2. Mechanical Response Analysis

#### 3.3. Failure Modes

## 4. Conclusions

- Numerical methods for calculating the structure of the foamed silicone rubber sandwich using a rubber intrinsic model with porosity and a three-dimensional Hashin criterion are effective;
- Based on the simulation data, the curve relationship between the initial velocity and the residual velocity was fitted using the Levenberg–Marquardt optimization algorithm, and the penetration thresholds for impact angles of 30°, 45°, 60°, 75°, and 90° were 6.747 m/s, 5.968 m/s, 5.640 m/s, and 5.482 m/s, and the impact resistance decreased by 11.5%, 16.4%, 18.7%, and 18.9% with the increase of impact angle;
- When the impact angle is greater than 45°, with the impact angle increases, the difference between the impact threshold is smaller and smaller. When the impact angle is greater than 75°, compared with 90°, the impact resistance difference is only 2.9%; at this time, the impact angle has less impact resistance performance;
- The impact angle has an obvious effect on the energy absorption characteristics of the rubber sandwich structure. At a certain impact speed, the smaller the impact angle, the longer the path of the falling hammer along the plane of the sandwich structure, the larger the contact area, the lighter the degree of damage, and the greater the energy absorbed by the sandwich structure; therefore, 90° is the most unfavorable impact angle for structural deformation, and avoiding the impact from the front of the sandwich structure can effectively reduce the degree of damage to the sandwich structure;
- The damage patterns of positive impact and oblique impact on the upper panel are different. For positive impact, the upper panel of the sandwich structure had fiber fracture caused by the shearing process. For oblique impact, fiber fracture and multiple cracks were produced at the edge of the falling hammer due to the larger contact area that the falling hammer passed through and then removed a large amount of material from the upper panel.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Bionic red-eared slider turtle sandwich structure design: (

**a**) A macroscopic morphology of a turtle shell. (

**b**) [6] A cross-sectional view of the turtle shell carapace showing composite layers. (

**c**) [6] A μCT reconstruction modified using an SEM fractography showing the sagittal surface of the carapace rib. The fractography reveals the dorsal and ventral cortices and the cancellous interior of the rib. (

**d**) Schematic diagram of stacking sequence of bionic sandwich structure and impact region.

**Figure 2.**Finite element model of composite laminate under impact loading. (

**a**) The whole model. (

**b**) The rubber model: there are 400 same-sized spherical voids uniformly distributed in the model. (

**c**) Half of void-containing cube.

**Figure 4.**Comparison of experimental results and simulation results for sandwich structures with porosity f

_{0}= 0.17 at 30 J impact energy, (

**a**) Contact Force-deflection; (

**b**) Energy-time.

**Figure 9.**Damage of the coating, upper panel, core, and bottom panel of sandwich structure at different impact angles when v = 4.970 m/s.

**Figure 10.**Damage of the coating, upper panel, core, and bottom panel of sandwich structure at different impact angles when v = 6.830 m/s.

Parameters | Symbol | Value | Units |
---|---|---|---|

Young’s modulus | E_{11}, E_{22}, E_{33} | 135, 8.8, 8.8 | GPa |

Poisson’s ratio | v_{12}, v_{13}, v_{23} | 0.33, 0.33, 0.35 | _ |

Shear modulus | G_{12}, G_{13}, G_{23} | 4.47, 4.47, 4.0 | GPa |

Ultimate tensile stress | X_{T}, Y_{T}, Z_{T} | 1548, 55.8, 55.8 | MPa |

Ultimate compressive stress | X_{C}, Y_{C}, Z_{C} | 1226, 131, 131 | MPa |

Ultimate shear stress | S_{12}, S_{13}, S_{23} | 89.9, 89.9, 51.2 | MPa |

θ V _{i} (m/s) | 30 v _{r} (m/s) | 45 v _{r} (m/s) | 60 v _{r} (m/s) | 75 v _{r} (m/s) | 90 v _{r} (m/s) |
---|---|---|---|---|---|

4.970 | −1.445 | −1.365 | −1.308 | −0.921 | −0.433 |

5.495 | −0.854 | −0.738 | −0.153 | 0.321 | 0.431 |

5.973 | −0.766 | 0.190 | 1.66411 | 2.185 | 2.170 |

6.419 | −0.143 | 2.255 | 2.814 | 3.176 | 3.083 |

6.830 | 1.082 | 3.089 | 3.603 | 3.910 | 3.959 |

8.000 | 4.327 | 5.119 | 5.518 | 5.634 | 5.696 |

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**MDPI and ACS Style**

Zhang, D.; Dong, H.; Zhao, S.; Yan, W.; Wang, Z.
Effect of Impact Angle on the Impact Mechanical Properties of Bionic Foamed Silicone Rubber Sandwich Structure. *Polymers* **2023**, *15*, 688.
https://doi.org/10.3390/polym15030688

**AMA Style**

Zhang D, Dong H, Zhao S, Yan W, Wang Z.
Effect of Impact Angle on the Impact Mechanical Properties of Bionic Foamed Silicone Rubber Sandwich Structure. *Polymers*. 2023; 15(3):688.
https://doi.org/10.3390/polym15030688

**Chicago/Turabian Style**

Zhang, Di, Hui Dong, Shouji Zhao, Wu Yan, and Zhenqing Wang.
2023. "Effect of Impact Angle on the Impact Mechanical Properties of Bionic Foamed Silicone Rubber Sandwich Structure" *Polymers* 15, no. 3: 688.
https://doi.org/10.3390/polym15030688