# 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 |

## References

- Das, T.K.; Ghosh, P.; Das, N.C. Preparation, development, outcomes, and application versatility of carbon fiber-based polymer composites: A review. Adv. Compos. Hybrid Mater.
**2019**, 2, 214–233. [Google Scholar] [CrossRef] - Zhou, J.; Liao, B.; Shi, Y.; Zuo, Y.; Tuo, H.; Jia, L. Low-velocity impact behavior and residual tensile strength of CFRP laminates. Compos. Part B Eng.
**2019**, 161, 300–313. [Google Scholar] [CrossRef] - Delaney, M.; Fung, S.; Kim, H. Dent depth visibility versus delamination damage for impact of composite panels by tips of varying radius. J. Compos. Mater.
**2018**, 52, 2691–2705. [Google Scholar] [CrossRef] - Hongliang, T.; Zhixian, L.; Xiaoping, M.; Chao, Z.; Chen, S. An experimental and numerical investigation on low-velocity impact damage and compression-after-impact behavior of composite laminates. Compos. Part B Eng.
**2019**, 167, 329–341. [Google Scholar] - Kadlec, M.; Kafka, V. Strain Concentration during the compression of a Carbon/Epoxy Composite after impact. Int. J. Struct. Integr.
**2015**, 6, 279–289. [Google Scholar] [CrossRef] - Bathe, K. Finite Element Procedures; PHI Learning Private Limited: Delhi, India, 1996; Volume 1. [Google Scholar]
- ABAQUS-6.14. Theory Manual. Available online: http://130.149.89.49:2080/v6.14/ (accessed on 8 December 2021).
- Lahey, S.R.; Miller, P.M.; Reymond, M. MSC.NASTRAN 2004 Reference Manual, Version 68; MSC.Software Corporation: Santa Ana, CA, USA, 2004; Volume 2, pp. 583–584. [Google Scholar]
- Vlach, J.; Raška, J.; Horňas, J.; Petrusová, L. Impacted area description effect on strength of laminate determined by calculation. Procedia Struct. Integr.
**2022**, 35, 132–140. [Google Scholar] [CrossRef] - Balasubramaniam, K.; Ziaja, D.; Jurek, M.; Fiborek, P.; Malinowski, P. Experimental and Numerical Analysis of Multiple Low-Velocity Impact Damages in a Glass Fibered Composite Structure. Materials
**2021**, 14, 7268. [Google Scholar] [CrossRef] - Pan, B. Digital image correlation for surface deformation measurement: Historical developments, recent advances and future goals. Meas. Sci. Technol.
**2018**, 29, 082001. [Google Scholar] [CrossRef] - Berthelot, J.M. Matériaux Composites, Comportement Mécanique et Analyse Des Structures, 1st ed.; Masson: Paris, France, 1992; pp. 154–196. [Google Scholar]
- Ning, Z.H.; Huo, G.L.; Liu, R.H.; Wu, W.L.; Xie, J.M. Progressive Failure Analysis of Laminates with Embedded Wrinkle Defects Based on an Elastoplastic Damage Model. Materials
**2020**, 13, 2422. [Google Scholar] [CrossRef] - Zhang, S.; Xing, T.; Zhu, H.; Chen, X. Experimental Identification of Statistical Correlation between Mechanical Properties of FRP Composite. Materials
**2020**, 13, 674. [Google Scholar] [CrossRef][Green Version] - ASTM standard. ASTM D3039/D3039M-14; Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials. ASTM International: Philadelphia, PA, USA, 2014.
- ASTM standard. ASTM D7136/D7136M-12; Standard Test Method for Measuring the Damage Resistance of a Fiber-Reinforced Polymer Matrix Composite to a Drop-Weight Impact Event. ASTM International: Philadelphia, PA, USA, 2012.
- T700G Technical Data Sheet. Available online: https://www.toraycma.com/wp-content/uploads/T700G-Technical-Data-Sheet-1.pdf. (accessed on 5 May 2022).
- Daniel, I.M.; Ishai, O. Engineering Mechanics of Composite Materials, 2nd ed.; Oxford University Press: New York, NY, USA, 2006; pp. 268–272. [Google Scholar]
- Kadlec, M.; Vích, O.; Novotný, D. Composite laminate deflection during low-velocity impact. In Book of Full Papers, Proceedings of the Experimental Stress Analysis Conference, Litomyšl, Czech Republic, 29 September 2021; Czech Technical University: Prague, Czech Republic; pp. 86–90.
- Pierard, O.; Friebel, C.; Doghri, I. Mean-field homogenization of multi-phase thermos-elastic composites: A general framework and its validation. Compos. Sci. Technol.
**2004**, 64, 1587–1603. [Google Scholar] [CrossRef] - Ye, F.; Wang, H. A simple Python code for computing effective properties of 2D and 3D representative volume element under periodic boundary conditions. arXiv
**2017**, arXiv:1703.03930. [Google Scholar] [CrossRef] - Hashin, Z. Failure criteria for unidirectional fiber composites. J. Appl. Mech.
**1980**, 47, 329–334. [Google Scholar] [CrossRef] - Hashin, Z.; Rotem, A. A fatigue failure criterion for fiber-reinforced materials. J. Compos. Mater.
**1973**, 7, 448–464. [Google Scholar] [CrossRef][Green Version] - Koloor, S.; Karimzadeh, A.; Abdulah, M.; Petru, M.; Yidris, N.; Sapuan, S.; Tamin, M. Linear-Nonlinear Stiffness Responses of carbon Fiber-Reinforced Polymer Composite Materials and Structures. Polymers
**2021**, 13, 344. [Google Scholar] [CrossRef] - Koloor, S.; Karimzadeh, A.; Yidris, N.; Petru, M.; Ayatollahi, M.; Tamin, M. An Energy-Based Concept for Yielding of Multidirectional FRP Composite Structures Using a Mesoscale Lamina Damage Model. Polymers
**2020**, 12, 157. [Google Scholar] [CrossRef][Green Version] - Šedek, J.; Bělský, P. Numerical evaluation of barely visible impact damage in a carbon fibre-reinforced composite panel with shear loading. WIT Trans. Eng. Sci.
**2017**, 116, 73–85. [Google Scholar] - Jia, L.; Yu, L.; Zhang, K.; Li, M.; Jia, Y.; Blackman, B.; Dear, J. Combined modelling and experimental studies of failure in thick laminates under out-of-plane shear. Compos. Part B Eng.
**2016**, 105, 8–22. [Google Scholar] [CrossRef] - Matzenmiller, A.; Lubliner, J.; Taylor, R. A constitutive model for anisotropic damage in fiber-composites. Mech. Mater.
**1995**, 20, 125–152. [Google Scholar] [CrossRef] - Donghyun, Y.; Sangdeok, K.; Jaehoon, K.; Youngdae, D. Development and Evaluation of Crack Band Model Implemented Progressive Failure Analysis Method for Notched Composite Laminate. Appl. Sci.
**2019**, 9, 5572. [Google Scholar] - Murakami, S. Continuum Damage Mechanics: A Continuum Mechanics Approach to the Analysis of Damage and Fracture; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012; Volume 185. [Google Scholar]
- Liao, B.; Jia, L.; Zhou, J.; Lei, H.; Gao, R.; Lin, Y.; Fang, D. An explicit–implicit combined model for predicting residual strength of composite cylinders subjected to low velocity impact. Compos. Struct.
**2020**, 247, 112450. [Google Scholar] [CrossRef] - Kim, E.; Rim, M.; Lee, I.; Hwang, T. Composite damage model based on continuum damage mechanics and low velocity impact analysis of composite plates. Compos. Struct.
**2013**, 95, 123–134. [Google Scholar] [CrossRef] - Afshar, A.; Daneshyar, A.; Mohammadi, S. XFEM analysis of fiber bridging in mixed-mode crack propagation in composites. Compos. Struct.
**2015**, 125, 314–327. [Google Scholar] [CrossRef] - Vlach, J.; Doubrava, R.; Růžek, R.; Raška, J.; Horňas, J.; Kadlec, M. Linearization of Composite Material Damage Model Results and Its Impact on the Subsequent Stress–Strain Analysis. Polymers
**2022**, 14, 1123. [Google Scholar] [CrossRef] - Wei, L.; Zhu, W.; Yu, Z.; Liu, J.; Wei, X. A new three-dimensional progressive damage model for fiber-reinforced polymer laminates and its applications to large open-hole panels. Compos. Sci. Technol.
**2019**, 182, 107757. [Google Scholar] [CrossRef] - Rektorys, K. Přehled Užité Matematiky II (Review of Applied Mathematics II); Prometheus: Praha, Czech, 1995; ISBN 9788085849721. [Google Scholar]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 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