# Numerical Simulations of the Low-Velocity Impact Response of Semicylindrical Woven Composite Shells

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Experimental Procedure

^{2}) was used as reinforcement, and the composite was produced by hand lay-up with 9 woven fabric layers (corresponding to 1.6 mm of final thickness). In order to ensure a constant fiber volume fraction and uniform thickness, as well as to eliminate any air bubbles, the laminates were placed inside a vacuum bag immediately after impregnation. The manufacturing process culminated with curing at 40 °C for 24 h. More details about the materials and manufacturing process can be found in [16,17,18]. Figure 1 shows the specimens’ dimensions and the schematic view of the test conditions.

## 3. Damage Models

^{6}N/mm

^{3}, as suggested by Camanho et al. [30]. In addition, it is considered that its value is the same for all directions, that is, ${k}_{n}={k}_{s}={k}_{t}$, as used in [31,32,33] with satisfactory results. It is noteworthy to mention that considering high values for the cohesive stiffness potentially results in convergence problems. On the other hand, the use of low values may affect the global stiffness and thus compromise the validation of the FE model [32]. A value of $\eta =1.45$ was considered for the interaction parameter in the definition of the cohesive model [34,35,36,37,38]. Identically to the intralaminar properties, there is a wide range in the data for the interlaminar strength parameters and fracture toughness. Given this information, the values used are averages, and preliminary parametric analysis using a coarser FE mesh was completed to determine which values the best-matched experimental data.

## 4. Finite Element Model

## 5. Numerical Results

## 6. Numerical–Experimental Correlation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Description | Symbol | Description |

$\rho $ | Density | ${\tilde{\sigma}}_{0}$ | Shear hardening function |

${E}_{1}$ | Young’s modulus along fiber direction 1 | ${r}_{\alpha}$ | Axial damage thresholds |

${E}_{2}$ | Young’s modulus along fiber direction 2 | ${r}_{12}$ | Shear damage threshold |

${E}_{2}$ | Though-thickness Young’s modulus | ${X}_{\alpha}$ | Tensile/compressive strength along the fiber directions |

${G}_{12}$ | In-plane shear modulus | ${d}_{1}$ | Tensile/compressive damage variable along direction 1 |

${G}_{13}$ | Out of plane shear modulus | ${d}_{2}$ | Tensile/compressive damage variable along direction 2 |

${\nu}_{12}$ | In-plane Poisson’s ratio | ${d}_{12}$ | Shear damage variable |

${X}_{1+}$ | Tensile strength along direction 1 | ${g}_{0}^{\alpha}$ | Elastic energy density |

${X}_{1-}$ | Compressive strength along direction 1 | ${L}_{e}$ | Characteristic length of the element |

${X}_{2+}$ | Tensile strength along direction 2 | ${\overline{\epsilon}}^{pl}$ | Plastic strain due to shear deformation |

${X}_{2-}$ | Compressive strength along direction 2 | ${k}_{n}$ | Elastic normal interlaminar stiffness |

${S}_{12}$ | In-plane shear strength | ${k}_{s}$ | Elastic shear interlaminar stiffness |

${G}_{f}^{\alpha}$ | Intralaminar fracture toughness along direction 1 and 2 | ${k}_{t}$ | Elastic tangential interlaminar stiffness |

${\alpha}_{12}$ | Parameter in the equation of shear damage | ${\tau}_{n}^{0}$ | Maximum normal contact stress |

${d}_{12}^{max}$ | Maximum shear damage | ${\tau}_{s}^{0}$ | Maximum 1st shear contact stress |

$C$ | Coefficient in hardening equation | ${\tau}_{t}^{0}$ | Maximum 2nd shear contact stress |

$\mathrm{l}p$ | Power term in hardening equation | ${G}_{Ic}$ | Interlaminar normal fracture toughness |

${F}_{\alpha}$ | Axial damage activation function | ${G}_{IIc}$ | Interlaminar 1st shear fracture toughness |

${F}_{12}$ | Shear damage activation function | ${G}_{IIIc}$ | Interlaminar 2nd shear fracture toughness |

${F}_{pl}$ | Plasticity activation function | 𝜂 | Benzeggagh–Kenane exponent |

${\tilde{\sigma}}_{\alpha}$ | Effective tensile/compressive stress | $\mu $ | Friction coefficient |

${\tilde{\sigma}}_{y0}$ | Initial effective shear yield stress | M | Parameter defined for the cohesive zone model |

${\tilde{\sigma}}_{12}$ | Effective shear stress | ${N}_{e}$ | Number of elements in the cohesive zone |

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**Figure 1.**(

**a**) Manufacturing process; (

**b**) Geometry and dimensions of the specimens (in mm); (

**c**) Schematic view of the test conditions.

**Figure 7.**Effect of the FE mesh discretization on the low-velocity impact response of the force–time impact curves.

**Figure 8.**Effect of the FE mesh discretization on the low-velocity impact response of the energy–time impact curves.

**Figure 9.**Effect of the FE mesh size on the low-velocity impact response of the force–displacement impact curves.

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

Density | $\rho $ | kg/m^{3} | 1900 |

Stiffness properties | ${E}_{1+}={E}_{1-}$ | GPa | 21.9 |

${E}_{2+}={E}_{2-}$ | GPa | 21.9 | |

${E}_{3}$ | GPa | 8.6 | |

${G}_{12}$ | GPa | 3.4 | |

${G}_{13}$ | GPa | 2.4 | |

${\nu}_{12}$ | - | 0.14 | |

Strength properties | ${X}_{1+}={X}_{2+}$ | MPa | 250 |

${X}_{1-}={X}_{2-}$ | MPa | 200 | |

$S$ | MPa | 40 | |

Fracture toughness | ${G}_{f}^{\alpha}$ | N/mm | 4500 |

Shear plasticity | ${d}_{12}^{max}$ | - | 1 |

${\tilde{\sigma}}_{y0}$ | MPa | 25 | |

$C$ | - | 800 | |

$p$ | - | 0.552 |

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

Stiffness properties | ${k}_{n}={k}_{s}={k}_{t}$ | N/mm^{3} | 10^{6} |

Strength properties | ${\tau}_{n}^{0}$ | MPa | 15 |

${\tau}_{s}^{0}={\tau}_{t}^{0}$ | MPa | 30 | |

Fracture toughness | ${G}_{Ic}$ | N/mm | 0.3 |

${G}_{IIc}={G}_{IIIc}$ | N/mm | 0.6 | |

$\eta $ | - | 1.45 |

**Table 3.**Effect of mass scaling on the numerical predictions of maximum load, maximum displacement, and contact time.

Max. Load (N) | Dif. ^{1}(%) | Max. Displacement (mm) | Dif. ^{1}(%) | Contact Time (ms) | Dif. ^{1}(%) | |
---|---|---|---|---|---|---|

With mass scaling | 797 | - | 11.5 | - | 23 | - |

Without mass scaling | 730 | 8.8 | 11.5 | 0 | 22.6 | 1.8 |

^{1}Dif. = Difference.

**Table 4.**Effect of FE mesh discretization on the numerical predictions of maximum load, displacement, and contact time.

Mesh Size (mm) | Max. Load (N) | Difference (%) | Max. Displacement (mm) | Difference (%) | Contact Time (ms) | Difference (%) |
---|---|---|---|---|---|---|

0.3 | 797 | - | 11.5 | - | 23 | - |

0.5 | 818 | 2.6 | 11.5 | 0 | 23 | 0 |

1 | 828 | 3.8 | 11.5 | 0 | 23.5 | 2.2 |

2 | 806 | 1.1 | 11.5 | 0 | 23.1 | 0.4 |

Mesh Size (mm) | Maximum Load (N) | Maximum Displacement (mm) | Contact Time (ms) | ||||||
---|---|---|---|---|---|---|---|---|---|

Num. | Exp. | Error (%) | Num. | Exp. | Error (%) | Num. | Exp. | Error (%) | |

0.3 | 797 | 757 | 5.0 | 11.5 | 11.2 | 2.7 | 23 | 20.6 | 10.4 |

0.5 | 818 | 7.5 | 11.5 | 2.7 | 23 | 10.4 | |||

1 | 828 | 8.6 | 11.5 | 2.7 | 23.5 | 12.3 | |||

2 | 806 | 6.1 | 11.5 | 2.7 | 23.1 | 10.8 |

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**MDPI and ACS Style**

Ferreira, L.M.; Coelho, C.A.C.P.; Reis, P.N.B.
Numerical Simulations of the Low-Velocity Impact Response of Semicylindrical Woven Composite Shells. *Materials* **2023**, *16*, 3442.
https://doi.org/10.3390/ma16093442

**AMA Style**

Ferreira LM, Coelho CACP, Reis PNB.
Numerical Simulations of the Low-Velocity Impact Response of Semicylindrical Woven Composite Shells. *Materials*. 2023; 16(9):3442.
https://doi.org/10.3390/ma16093442

**Chicago/Turabian Style**

Ferreira, Luis M., Carlos A. C. P. Coelho, and Paulo N. B. Reis.
2023. "Numerical Simulations of the Low-Velocity Impact Response of Semicylindrical Woven Composite Shells" *Materials* 16, no. 9: 3442.
https://doi.org/10.3390/ma16093442