# Strain-Field Modifications in the Surroundings of Impact Damage of Carbon/Epoxy Laminate

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Material Description

_{S}with total thickness of 1.584 mm. The laminate was made of a unidirectional carbon fabric (Toray, Tacoma, WA, USA) in an epoxy matrix (Hexcel, Saronno, Italy) with a volume fraction of 53.3%. The plies were hand laminated, vacuum bagging was performed, and the laminate was cured in an autoclave at 180 °C and 6 bar pressure. Fibers were of a non-commercial nature provided only for a specific scientific research purpose; the mechanical properties of the fibers can be considered equivalent to the T700G standard [17]. The properties of the matrix correspond to epoxy 3501-6 [18]. The scheme of the laminate composition is shown in Figure 2.

#### 2.2. Tensile Test

^{2}. During the experiment, the dependence of the force on the displacement of the sample was checked on an Instron 55R1185 testing machine (Instron, Norwood, MA, USA). Longitudinal elongation and transverse narrowing were measured with an Epsilon biaxial extensometer (Epsilon Technology Corp, Jackson, WY, USA). The sample was loaded to break in the absence of an extensometer. The experiment and the dimensions of the test specimen are shown in Figure 3a,b.

#### 2.3. Impact Test

#### 2.4. Post-Impact Tensile Test

#### 2.5. Tensile Test Simulation

#### 2.6. Impact Test Simulation

#### 2.7. Post-Impact Simulation Test

## 3. Results and Discussion

#### 3.1. Tensile Test Simulation Results

#### 3.2. Impact Test Simulation Results

#### 3.3. Post-Impact Simulation Results

#### 3.4. Results Extrapolation

## 4. Conclusions

- The basic experiments and their relation to the parameters of the FE model were shown.
- A high correlation with partial results of reference calculations with experiments was shown.
- The results of partial experiments of tensile and impact tests fall within the confidence intervals with a probability of 95%.
- The simulation showed that the dependence of the increment of the area affected by the impact is nonlinear.
- A method of identifying the extent of damage by the DIC method in relation to the impact energy of the test specimen and changes in relative deformations around the damage was successfully presented.
- Simulations have confirmed that areas near the top and bottom layers are most effective for collecting strain-field data changes. The possible application of SHM should be focused on these cross-sectional areas.
- Correlated models will be used to generate strain-field data in relation to the presented load cases.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$\left\{\mathsf{\sigma}\right\}$ | (MPa) | Stress tensor |

$\left\{\mathsf{\epsilon}\right\}$ | (1) | Strain tensor |

${\mathrm{E}}_{1}^{0}$ | (MPa) | Elastic modulus of non-damaged material in direction 1 |

${\mathrm{E}}_{2}^{0}$ | (MPa) | Elastic modulus of non-damaged material in direction 2 |

${\mathrm{G}}_{12}^{0}$ | (MPa) | Shear modulus of non-damaged material in plane 12 |

${\mathrm{G}}_{13}^{0}$ | (MPa) | Transverse shear modulus of non-damaged material in plane 13 |

${\mathrm{G}}_{23}^{0}$ | (MPa) | Transverse shear modulus of non-damaged material in plane 23 |

${\mathsf{\nu}}_{12}^{0}$ | (1) | Poisson’s ratio of undamaged material in plane 12 |

${\mathrm{E}}_{1}^{\mathrm{d}}$ | (MPa) | Consolidated elastic modulus of damaged material in direction 1 |

${\mathrm{E}}_{2}^{\mathrm{d}}$ | (MPa) | Consolidated elastic of damaged material in direction 2 |

${\mathrm{G}}_{12}^{d}$ | (MPa) | Consolidated shear modulus of damaged material in plane 12 |

${\mathrm{G}}_{13}^{d}$ | (MPa) | Consolidated transverse shear modulus of damaged material in plane 13 |

${\mathrm{G}}_{23}^{d}$ | (MPa) | Consolidated transverse shear modulus of damaged material in plane 23 |

${\mathsf{\nu}}_{12}^{0}$ | (1) | Poisson’s ratio of undamaged material in plane 12 |

${\mathrm{d}}_{1}$ | (1) | Consolidated damage factor in direction 1 |

${\mathrm{d}}_{2}$ | (1) | Consolidated damage factor in direction 2 |

${\mathrm{d}}_{\mathrm{s}}$ | (1) | Consolidated shear damage factor |

${F}_{1,u}^{+}$ | (MPa) | Ultimate tensile stress in direction 1 |

${F}_{1,u}^{-}$ | (MPa) | Ultimate compressive stress in direction 1 |

${F}_{2,u}^{+}$ | (MPa) | Ultimate tensile stress in direction 2 |

${F}_{2,u}^{-}$ | (MPa) | Ultimate compressive stress in direction 2 |

${F}_{12,u}$ | (MPa) | Ultimate shear stress in plane 12 |

${F}_{13,u}$ | (MPa) | Ultimate shear stress in plane 13 |

${F}_{23,u}$ | (MPa) | Ultimate shear stress in plane 23 |

${\epsilon}_{1,u}^{+}$ | (1) | Ultimate tensile strain in direction 1 |

${\epsilon}_{1,u}^{-}$ | (1) | Ultimate compressive strain in direction 1 |

${\epsilon}_{2,u}^{+}$ | (1) | Ultimate tensile strain in direction 2 |

${\epsilon}_{2,u}^{-}$ | (1) | Ultimate compressive strain in direction 2 |

${U}_{1}^{+}$ | (mJ/mm^{2}) | Tensile fracture energy in direction 1 |

${U}_{1}^{-}$ | (mJ/mm^{2}) | Compressive fracture energy in direction 1 |

${U}_{2}^{+}$ | (mJ/mm^{2}) | Tensile fracture energy in direction 2 |

${U}_{2}^{-}$ | (mJ/mm^{2}) | Compressive fracture energy in direction 2 |

${R}_{3,u}$ | (MPa) | Normal cohesive strength in the direction 3 |

${R}_{13,u}$ | (MPa) | Shear cohesive strength in plane 13 |

${R}_{23,u}$ | (MPa) | Shear cohesive strength in plane 23 |

${D}_{1}$ | (mJ/mm^{2}) | Normal fracture energy in the direction 3 |

${D}_{13}$ | (mJ/mm^{2}) | Shear fracture energy in plane 13 |

${D}_{23}$ | (mJ/mm^{2}) | Shear fracture energy in plane 23 |

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**Figure 5.**Deflection under the impactor from the simulation results; Initial state of impact (

**a**); Maximum plate displacement under impactor (

**b**).

**Figure 11.**Damage level of the specimen; Crack initiation from the FE model (

**a**), Final damage from the FE model (

**b**), Damage from the experiment (

**c**).

**Figure 12.**Damaged area at layer no. 1; 0.5 J (

**a**); 1.0 J (

**b**); 2.5 J (

**c**); 5.0 J (

**d**); 7.5 J (

**e**); 10.0 J (

**f**).

**Figure 13.**Area damaged from impact; layer no. 12; 0.5 J (

**a**); 1.0 J (

**b**); 2.5 J (

**c**); 5.0 J (

**d**); 7.5 J (

**e**); 10.0 J (

**f**).

**Figure 14.**The cross-section profile of the damaged area; 0.5 J (

**a**); 1.0 J (

**b**); 2.5 J (

**c**); 5.0 J (

**d**); 7.5 J (

**e**); 10.0 J (

**f**).

**Figure 15.**Max. principal strain distribution (μm/m) at load 80 kN; (

**a**) Simulation; (

**b**) Experiment sample 1; (

**c**) Experiment sample 2.

**Figure 16.**Max. principal strain distribution (μm/m) at load 80 kN; (

**a**) Horizontal path—scale 1; (

**b**) Horizontal path—scale 2; (

**c**) Vertical path—scale 1; (

**d**) Vertical path—scale 2.

**Figure 17.**Max. principal strain distribution (μm/m) at load 110 kN; (

**a**) Simulation; (

**b**) Experiment sample 1; (

**c**) Experiment sample 2.

**Figure 18.**Max. principal strain distribution (μm/m) at load 110 kN; (

**a**) Horizontal path—scale 1; (

**b**) Horizontal path—scale 2; (

**c**) Vertical path—scale 1; (

**d**) Vertical path—scale 2.

**Figure 19.**Max. principal strain distribution (μm/m); Pristine 80 kN (

**a**); Damaged (10 J) 80 kN (

**b**); Pristine 110 kN (

**c**); Damaged (10 J) 110 kN (

**d**).

**Figure 20.**Comparison of max. principal strain distribution (μm/m); (

**a**) Horizontal path at load 80 kN; (

**b**) Horizontal path at load 110 kN, (

**c**) Vertical path at load 80 kN; (

**d**) Vertical path at load 110 kN.

${\mathbf{E}}_{1}^{0}\left(\mathbf{MPa}\right)$ | ${\mathbf{E}}_{2}^{0}\left(\mathbf{MPa}\right)$ | ${\mathbf{\nu}}_{12}^{0}\left(1\right)$ | ${\mathbf{G}}_{12}^{0}\left(\mathbf{MPa}\right)$ | ${\mathbf{G}}_{13}^{0}\left(\mathbf{MPa}\right)$ | ${\mathbf{G}}_{23}^{0}\left(\mathbf{MPa}\right)$ |
---|---|---|---|---|---|

129,840 | 13,340 | 0.26 | 4890 | 4890 | 4630 |

${\mathbf{F}}_{1,\mathbf{u}}^{+}\left(\mathbf{MPa}\right)$ | ${\mathbf{F}}_{1,\mathbf{u}}^{-}\left(\mathbf{MPa}\right)$ | ${\mathbf{F}}_{2,\mathbf{u}}^{+}\left(\mathbf{MPa}\right)$ | ${\mathbf{F}}_{2,\mathbf{u}}^{-}\left(\mathbf{MPa}\right)$ | ${\mathbf{F}}_{12,\mathbf{u}}\left({\mathbf{F}}_{13,\mathbf{u}}\right)\left(\mathbf{MPa}\right)$ | ${\mathbf{F}}_{23,\mathit{u}}\left(\mathbf{MPa}\right)$ |
---|---|---|---|---|---|

2965.41 | 2911.81 | 100.88 | 109.42 | 100.76 | 98.41 |

${\mathbf{U}}_{1}^{+}(\mathbf{mJ}/{\mathbf{mm}}^{2})$ | ${\mathbf{U}}_{1}^{-}(\mathbf{mJ}/{\mathbf{mm}}^{2})$ | ${\mathbf{U}}_{2}^{+}(\mathbf{mJ}/{\mathbf{mm}}^{2})$ | ${\mathbf{U}}_{2}^{-}(\mathbf{mJ}/{\mathbf{mm}}^{2})$ |
---|---|---|---|

35.56 | 34.28 | 0.92 | 1.08 |

$\mathbf{\Phi}$ | ${\mathbf{R}}_{1,\mathbf{u}}\left(\mathbf{MPa}\right)$ | ${\mathbf{R}}_{13,\mathbf{u}}\left(\mathbf{MPa}\right)$ | ${\mathbf{R}}_{23,\mathbf{u}}\left(\mathbf{MPa}\right)$ |
---|---|---|---|

>45 | 100.88 | 100.76 | 100.76 |

≤45 | 174.03 | 173.44 | 173.44 |

${\mathbf{D}}_{1}\text{}(\mathbf{mJ}/{\mathbf{mm}}^{2})$ | ${\mathbf{D}}_{13}\text{}(\mathbf{mJ}/{\mathbf{mm}}^{2})$ | ${\mathbf{D}}_{23}\text{}(\mathbf{mJ}/{\mathbf{mm}}^{2})$ |
---|---|---|

35.56 | 0.92 | 0.92 |

${\mathbf{E}}_{1}^{\mathbf{d}}\left(\mathbf{MPa}\right)$ | ${\mathbf{E}}_{2}^{\mathbf{d}}\left(\mathbf{MPa}\right)$ | ${\mathbf{\nu}}_{12}^{\mathbf{d}}\left(1\right)$ | ${\mathbf{G}}_{23}^{\mathbf{d}}\left(\mathbf{MPa}\right)$ | ${\mathbf{G}}_{13}^{\mathbf{d}}\left(\mathbf{MPa}\right)$ | ${\mathbf{G}}_{12}^{\mathbf{d}}\left(\mathbf{MPa}\right)$ |
---|---|---|---|---|---|

${\mathrm{E}}_{1}^{0}\left(1-{\mathrm{d}}_{1}\right)$ | ${\mathrm{E}}_{2}^{0}\left(1-{\mathrm{d}}_{2}\right)$ | ${\mathsf{\nu}}_{12}^{0}\left(1-{\mathrm{d}}_{1}\right)$ | ${\mathrm{G}}_{12}^{0}\left(1-{\mathrm{d}}_{\mathrm{s}}\right)$ | ${\mathrm{G}}_{12}^{0}\left(1-{\mathrm{d}}_{\mathrm{s}}\right)$ | ${\mathrm{G}}_{12}^{0}\left(1-{\mathrm{d}}_{\mathrm{s}}\right)$ |

Experiment | Simulation | |
---|---|---|

Tensile modulus | 76,965 ± 863 MPa | 76,964 MPa |

Poisson ratio | 0.30 ± 0.03 | 0.3 |

Ultimate tensile load | 49,820 ± 1075 N | 50,854 N |

Experiment | Simulation | |
---|---|---|

Energy | 10.45 ± 0.22 J | 10.45 J |

Penetration | 6.99 ± 0.21 mm | 6.71 mm |

Residual deformation | 1.12 ± 0.36 mm | 0.77 mm |

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## Share and Cite

**MDPI and ACS Style**

Vlach, J.; Doubrava, R.; Růžek, R.; Raška, J.; Horňas, J.; Kadlec, M.
Strain-Field Modifications in the Surroundings of Impact Damage of Carbon/Epoxy Laminate. *Polymers* **2022**, *14*, 3243.
https://doi.org/10.3390/polym14163243

**AMA Style**

Vlach J, Doubrava R, Růžek R, Raška J, Horňas J, Kadlec M.
Strain-Field Modifications in the Surroundings of Impact Damage of Carbon/Epoxy Laminate. *Polymers*. 2022; 14(16):3243.
https://doi.org/10.3390/polym14163243

**Chicago/Turabian Style**

Vlach, Jarmil, Radek Doubrava, Roman Růžek, Jan Raška, Jan Horňas, and Martin Kadlec.
2022. "Strain-Field Modifications in the Surroundings of Impact Damage of Carbon/Epoxy Laminate" *Polymers* 14, no. 16: 3243.
https://doi.org/10.3390/polym14163243