# Development and Evaluation of Crack Band Model Implemented Progressive Failure Analysis Method for Notched Composite Laminate

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## Abstract

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## Featured Application

**A model studied in this article, which allows for a good representation of behavior of notched geometry and less mesh dependence, is expected to benefit the design process of complex composite structures.**

## Abstract

## 1. Introduction

## 2. Development of Progressive Failure Analysis

#### 2.1. Progressive Failure Analysis

_{ij}” values include the damage variable, d, according to its directions.

_{f}, d

_{m}, and d

_{s}are damage variables for the fiber, matrix, and shear damage, respectively, which range from 0 to 1. Additionally, 0 and 1 imply undamaged status and complete damage status, respectively. As material failure occurs, the damage variables approach 1. When performing a PFA, the failure criteria are verified to detect failure in materials in terms of the effective stress that reflects the damaged stress status at each analysis increment. The effective stress, $\tilde{\sigma}$, is as follows:

#### 2.2. Damage Initiation and Evolution

_{c}is the characteristic length and “⟨⟩” refers to the Macaulay brackets. The definition of the Macaulay brackets is as follows:

_{I}), as shown in Equation (2). Consequently, an increase in the damage variables induced a decrease in the stiffness matrix.

## 3. Material and Test Procedures

#### 3.1. Composite Material and Test Specimen

#### 3.2. Test Procedures

#### 3.3. FEM Procedures

## 4. Results

#### 4.1. Mesh Dependency Study

#### 4.2. Evaluation of Load-Displacement Behavior

#### 4.3. Evaluation of Strain Behavior

#### 4.4. Failure Behavior Observation

## 5. Conclusions

- (1)
- The developed PFA model demonstrated results that were less-mesh-dependent in comparison to the MLT-PFA model that has been used in various studies. The lower mesh dependency occurred because each element failed when considering the constant fracture energy regardless of the element size when using the damage variable in the crack-band-model.
- (2)
- The analysis results were in good agreement with the experimental ones regardless of the stacking sequences, the number of notches, and the loading direction. This conclusion was determined by examining the load-displacement behavior and the strain distribution of the PFA results while doing a comparison to the experimental results.
- (3)
- Using the developed PFA model, the failure behavior of the composite laminate containing open holes was studied. Tensile failure behavior shows a final load-drop induced from the fiber failure damage while the compressive failure behavior shows that the final load-drop occurred by shear and matrix failure damage. The different fracture mechanism according to the loading direction was confirmed by comparing the fractured specimens.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 11.**Load-displacement curves for two-open-hole tests and PFA results under (

**a**) tensile, and (

**b**) compressive loading conditions.

**Figure 12.**Longitudinal strain (${\mathsf{\epsilon}}_{xx}$) contour at fracture moment obtained from PFA.

**Figure 13.**Longitudinal strain (${\mathsf{\epsilon}}_{xx}$) contour at fracture moment obtained from DIC.

**Figure 14.**Location of attached strain gauges and longitudinal strain (${\mathsf{\epsilon}}_{xx}$) value comparison for one-open-hole tensile tests: (

**a**) Location of strain gauges, (

**b**) strain comparison results for Type 1, and (

**c**) strain comparison results for Type 2 stacking sequences.

**Figure 15.**Location of attached strain gauges and longitudinal strain (${\mathsf{\epsilon}}_{xx}$) value comparison for one-open-hole compressive tests: (

**a**) Location of strain gauges, (

**b**) strain comparison results for location ① and ②, and (

**c**) strain comparison results for location ③ of the specimens.

**Figure 16.**Location of attached strain gauges and longitudinal strain (${\mathsf{\epsilon}}_{xx}$) value comparison for two-open-hole tensile tests: (

**a**) Location of strain gauges, (

**b**) strain comparison results for location ①, and (

**c**) strain comparison results for location ② and ③ of the specimens.

**Figure 17.**Location of attached strain gauges and longitudinal strain (${\mathsf{\epsilon}}_{xx}$) value comparison for two-open-hole compressive tests: (

**a**) Location of strain gauges, (

**b**) strain comparison results for location ①, and (

**c**) strain comparison results for location ② and ③ of the specimens.

**Figure 18.**Load-displacement curve for one-open-hole tensile analysis result with each damage points and the corresponding calculated damage contour when the (

**a**) shear failure, (

**b**) matrix tension failure, and (

**c**) fiber tension failure damage variable become 1 or greater, and (

**d**) final fracture occurs.

**Figure 19.**Load-displacement curve for two-open-hole compressive analysis result with each damage points and the corresponding calculated damage contour when the (

**a**) shear failure and (

**b**) matrix tension failure damage variable become 1 or greater and (

**c**) final fracture occurs.

**Figure 20.**Fractured test specimen for (

**a**) one-open-hole tensile test and (

**b**) two-open-hole compressive test.

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

Longitudinal modulus | ${E}_{1}$ | GPa | 147.7 |

Transverse modulus | ${E}_{2}$ | GPa | 8.52 |

Shear modulus | ${G}_{12}$ | GPa | 4.59 |

Poisson’s ratio | ${v}_{12}$ | - | 0.3 |

Longitudinal tensile strength | ${X}_{t}$ | MPa | 2737 |

Transverse tensile strength | ${X}_{c}$ | MPa | 1600 |

Longitudinal compressive strength | ${Y}_{t}$ | MPa | 51.32 |

Transverse compressive strength | ${Y}_{c}$ | MPa | 201.08 |

Shear strength | ${S}_{12}$ | MPa | 81.0 |

Fiber tensile fracture energy | ${G}_{ft}$ | kN/m^{2} | 180 |

Fiber compressive fracture energy | ${G}_{fc}$ | kN/m^{2} | 100 |

Matrix tensile fracture energy | ${G}_{mt}$ | kN/m^{2} | 0.30 |

Matrix compressive fracture energy | ${G}_{mc}$ | kN/m^{2} | 1.71 |

Specimen Type | Stacking Sequence |
---|---|

Type 1 | [0/+45/−45/90]_{s} |

Type 2 | [0/+45/0/−45]_{s} |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Yoon, D.; Kim, S.; Kim, J.; Doh, Y. Development and Evaluation of Crack Band Model Implemented Progressive Failure Analysis Method for Notched Composite Laminate. *Appl. Sci.* **2019**, *9*, 5572.
https://doi.org/10.3390/app9245572

**AMA Style**

Yoon D, Kim S, Kim J, Doh Y. Development and Evaluation of Crack Band Model Implemented Progressive Failure Analysis Method for Notched Composite Laminate. *Applied Sciences*. 2019; 9(24):5572.
https://doi.org/10.3390/app9245572

**Chicago/Turabian Style**

Yoon, Donghyun, Sangdeok Kim, Jaehoon Kim, and Youngdae Doh. 2019. "Development and Evaluation of Crack Band Model Implemented Progressive Failure Analysis Method for Notched Composite Laminate" *Applied Sciences* 9, no. 24: 5572.
https://doi.org/10.3390/app9245572