# An Experimental and Numerical Study of the Influence of Temperature on Mode II Fracture of a T800/Epoxy Unidirectional Laminate

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}Texipreg HS 160 RM) at temperatures of 20 °C, 80 °C and −30 °C. Mode II fracture toughness of AS4/3501-6 was also found to increase with temperatures in the range of −50 °C to 100 °C [18], owing to increasing matrix ductility. However, Davies and Charentenay [19] found that there was no temperature effect on the mode II fracture toughness of T300/914C laminates evaluated in the range from 30 °C to 120 °C. More researchers found that the mode II fracture toughness decreased with the increase in temperature [20,21,22]. The results of Sjögren and Asp [23,24] showed that the mode II fracture toughness of HTA/6376C laminates decreased with the increase in temperature from −50 °C to 100 °C. However, the fracture surface characteristics at different temperatures were similar to each other. It was observed that the crack jumped between the upper and lower boundaries of the fiber when the temperature was 100 °C. However, at room temperature, the delamination grew along the upper boundary of the fiber. Cowley and Beaumont [25] performed end-notched flexure (ENF) tests on composite laminates fabricated by IM8/APC and IM8/954-2 materials. They found that the mode II fracture toughness decreased more obviously with the increase in temperature, especially when close to the glass transition temperature of the material. Davidson et al. [26] found that the matrix ductility increased with the increase in temperature, resulting in the decrease in mode II fracture toughness of T800H/3900-2 laminates. Boni et al. [15] measured the mode II fracture toughness of Hexcel 913C-HTA laminates evaluated at room temperature and 70 °C and found that it decreased at a higher temperature. Zulkifli and Azari [27] studied the mode II delamination growth behavior of silk fiber/epoxy laminates at 20 °C, 23 °C, 26 °C, 38 °C, 50 °C and 75 °C. It was shown that the sliding cracking of matrix and fiber appeared during the experiment, and the mode II fracture toughness at 75 °C showed a 71% decrease compared to that at room temperature.

## 2. Delamination Tests on the ENF Specimen

#### 2.1. Specimen and Test Set-Up

_{1}, E

_{2}, and E

_{3}are the Young’s moduli in the 1-direction (main orthotropic material axe), 2-direction (perpendicular orthotropic material axe) and 3-direction, respectively; G

_{12}, G

_{13}and G

_{23}are the shear moduli in the 1-2 plane, 1-3 plane, and 2-3 plane, respectively; ν

_{12}, ν

_{13}and ν

_{23}are the Poisson’s ratios in the 1-2 plane, 1-3 plane, and 2-3 plane, respectively. X

_{T}and X

_{C}are the tensile and compressive strengths in the 1-direction, respectively; Y

_{T}and Y

_{C}are the tensile and compressive strengths in the 2-direction, respectively. Z

_{T}and Z

_{C}are the tensile and compressive strengths in the 3-direction, respectively; S

_{12}, S

_{13}, and S

_{23}are the shear strengths in the 1-2 plane, 1-3 plane, and 2-3 plane, respectively. The cured panel was cut into specimens with uniform geometrical dimensions of 180 mm length, 25 mm width and 4.8 mm thickness. A 40 mm long and 25 mm wide Teflon film was prearranged between the center plies to create an artificial pre-crack for the delamination. Unidirectional laminates with a stacking sequence of [0/0]

_{12}were studied here.

#### 2.2. Test Procedure

_{0}/L, was set at 0.7 [30]. In this case, the delamination could be initiated in a relatively stable manner, followed by continued and relatively stable delamination growth over a relatively long distance before reaching the maximum load.

#### 2.3. Date Reduction Method

#### 2.3.1. ECM Method

#### 2.3.2. SBT Method

_{1}is the flexural modulus. The limitation of the SBT method is that the contribution of shear deformation of the specimen to the energy release rate is neglected. Zero compliance at the crack root is assumed in Equation (4). However, in an actual case, there is some deflection and rotation at the crack tip.

#### 2.3.3. CBTE

_{e}is the effective delamination length and can be obtained by Equation (8).

## 3. Test Results

#### 3.1. Compliance Calibration Results

^{2}coefficient equal to 0.99) was obtained between the compliance and the cubic of crack length. The detailed fitting formulas are given in Figure 3. The values of the m at RT, 80 °C, and 130 °C were 0.006, 0.0089, and 0.0078, respectively. These values were used in Equation (3) for the calculation of fracture toughness.

#### 3.2. Load–Displacement Curve

#### 3.3. Fracture Toughness

#### 3.4. Analysis on the Fracture Surface

## 4. Numerical Simulation for the Mode II Delamination in ENF Specimens

#### 4.1. A model for the Thermal-Affected Material Properties

**ε**=

**S**•

**σ**, where

**ε**,

**S**, and

**σ**are the matrixes of engineering strain, compliance, and stress, respectively. The element S

_{ij}is a function of engineering constants E

_{ij}, ν

_{ij}, G

_{ij}(i, j = 1, 2, 3) of the composite material [48]. In order to include the temperature effect on the material mechanical properties, the compliance matrix

**S**is modified, as shown in Equation (9):

_{g}is the glass transition temperature, T

_{opr}is the operation temperature, and T

_{rm}is the room temperature. A united model [51], such as Equation (11), can be used to quantitatively characterize the temperature-dependent effect on the mechanical property of composite materials. The matrix and fiber stiffness and strength data are empirically fitted by exponential formulas as a function of T*. This model was first proposed for the hygrothermal environment. When there is no moisture absorption, this model degrades to the case with only temperature effect and is also applicable.

_{T}, X

_{C}, Y

_{T}, Y

_{C}and S

_{12}are the longitudinal tensile and compressive, transverse tensile and compressive, and in-planar shear strengths, respectively. The superscript “0” denotes the room temperature. Values of the exponents in Equation (11) can be determined by a micro-level based method [50,52,53]. As shown in Equation (12), mechanical properties E

_{33}, G

_{13}, ν

_{13}, G

_{23}, Z

_{T}, Z

_{C}and S

_{13}at the thermal condition are determined based on the transverse isotropic assumption. The ν

_{23}and S

_{23}are determined based on the method proposed by Christensen [54] and Pinho et al. [55]. The values of the exponents in Equation (11) are listed in Table 3 from which the basic mechanical properties of the composite material at 80 °C and 130 °C can be determined and are also listed in Table 3.

#### 4.2. Cohesive Zone Model with a Bilinear Constitutive Law

_{II}= (1 − d

_{II}) ${K}_{\mathrm{II}}^{0}$, where d

_{II}∈ [0,1] is the damage variable. The element completely fails and loses its load-bearing capacity when the interface stiffness reduces to zero (point 4). According to Griffith’s fracture theory, the area under the traction-relative displacement relationship is equal to the fracture toughness G

_{IIC}as shown in Equation (13):

#### 4.3. Numerical Model of the ENF Specimen

#### 4.4. Simulated Results

^{14}N/m

^{3}, 10

^{15}N/m

^{3}and 10

^{16}N/m

^{3}) were studied to reveal its influence on the initial slope of the predicted load–displacement curve. The predictions are shown in Figure 10a. ‘Test 1’ and ‘Test 2’ represented the experimental results of two tested specimens. For better comparisons, the initial nonlinear stage of the experimental results was removed. It indicated that the influence of interface stiffness was trivial and the predictions had good agreement with the experimental ones. Considering that a larger value of interface stiffness can ensure less structural stiffness loss, its value was reasonably chosen as 10

^{15}N/m

^{3}for the specimens evaluated at 80 °C. The influence of the viscosity coefficient (10

^{−6}, 10

^{−5}, and 10

^{−4}) on the predicted load–displacement curves is shown in Figure 10b and compared with the experimental ones. It can be seen that the viscosity coefficient had no effect on the linear and elastic stage, while it had an obvious effect on the subsequent load-drop stage. Consistent results were obtained when the viscosity coefficient was chosen as 10

^{−6}and 10

^{−5}because a higher viscosity coefficient usually results in a better convergence, which means fewer analysis steps and less computational time are required. Thus, the viscosity coefficient was chosen as 10

^{−5}for the specimens evaluated at 80 °C. The predicted load–displacement curves with different mesh sizes (0.25, 0.5, and 1 mm) are shown in Figure 10c. It can be seen that the effect of the studied mesh size on the predictions was also trivial. Because a finer mesh size will increase the computational cost, the mesh size was chosen as 0.5 mm. The final numerical results are compared with experimental results in Figure 10d, and good agreement was achieved between them.

^{14}, 10

^{15}, and 10

^{16}N/m

^{3}), viscosity coefficient (10

^{−5}, 10

^{−4}and 10

^{−3}) and mesh size (0.25 and 0.5 mm) on the numerical results. It can be seen that in the studied range, the interface stiffness and mesh size did not affect the predictions. For different viscosity coefficients, the predicted linear stages were the same. However, only when its value was 10

^{−5}was a satisfactory agreement between the predictions and the experimental results obtained for the load-drop stage. Therefore, the interfacial parameters of the cohesive elements suitable for ENF tests evaluated at 130 °C were as follows: interface stiffness of 10

^{15}N/m

^{3}, viscosity coefficient of 10

^{−5}, and mesh size of 0.5 mm. Based on the suitable interfacial parameter set, the numerical model for simulating the mode II delamination of composite laminates was established. The final numerical results were obtained and are presented in Figure 11d, and they agreed well with the experimental results.

## 5. Conclusions

^{15}N/m

^{3}, viscosity coefficient of 10

^{−5}, and mesh size of 0.5 mm. The temperature did not seem to affect the suitable value of the interfacial parameter. The predicted load–displacement responses showed satisfactory agreement with the experimental, which illustrated the applicability and accuracy of the established numerical model. It is worthwhile to point out that the aerospace composite structures are usually exposed to thermal cycles. Further studies are required on the delamination behavior of those structures exposed to thermal cycles, which should be an interesting topic.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**P-d curves from the compliance calibration of ENF specimens (

**a**) original curve and (

**b**) linear elastic section for the determination of compliance.

**Figure 10.**Influence of (

**a**) interface stiffness, (

**b**) viscosity coefficient, (

**c**) mesh size on the predicted load–displacement response and (

**d**) numerical results of the ENF test at 80 °C.

**Figure 11.**Influence of (

**a**) interface stiffness, (

**b**) viscosity coefficient, (

**c**) mesh size on the predicted load–displacement response, and (

**d**) numerical results of the ENF test at 130 °C.

E_{1} (Gpa) | E_{2}, E_{3} (Gpa) | G_{12}, G_{13} (Gpa) | G_{23} (Gpa) | ν_{12}, ν_{13} | ν_{23} |

163.5 | 9.0 | 4.14 | 3.08 | 0.32 | 0.46 |

X_{T} (MPa) | X_{C} (MPa) | Y_{T}, Z_{T} (MPa) | Y_{C}, Z_{C} (MPa) | S_{12}, S_{13} (MPa) | S_{23} (MPa) |

2992 | 1183 | 70.6 | 278 | 172 | 105 |

**Table 2.**Mode II fracture toughness of ENF specimens evaluated at different temperatures. (Unit: J/m

^{2}).

Temperature | Method | Test 1 | Test 2 | Mean Value | S.D. | CoV (%) |

RT | ECM | 2269.68 | 2259.09 | 2264.39 | 5.29 | 0.2 |

SBT | 2237.93 | 2110.97 | 2174.45 | 63.48 | 2.9 | |

CBTE | 2738.67 | 2684.17 | 2711.42 | 27.25 | 1.0 | |

80 °C | ECM | 1920.93 | 1989.62 | 1955.28 | 34.34 | 1.8 |

SBT | 1916.81 | 1953.56 | 1935.19 | 18.38 | 0.9 | |

CBTE | 2273.08 | 2300.07 | 2286.58 | 13.50 | 0.6 | |

130 °C | ECM | 1644.45 | 1558.94 | 1601.70 | 42.78 | 2.7 |

SBT | 1584.24 | 1671.48 | 1627.86 | 43.63 | 2.7 | |

CBTE | 1660.58 | 1595.40 | 1627.99 | 32.59 | 2.0 |

**Table 3.**Values of the exponents for the engineering constant calculations and the determined mechanical properties of T800/epoxy composites at high temperatures.

Item | a | b | c | d | e |

Value | 0.04 | 0.5 | 0.5 | 0.04 | 0.54 |

Item | f | g | h | T_{g}/°C | T_{rm}/°C |

Value | 0.50 | 0.50 | 0.50 | 185 | 23 |

Elastic property | 80 °C | 130 °C | Strength (MPa) | 80 °C | 130 °C |

E_{1} (GPa) | 160.7 | 156.6 | X_{T} | 2941 | 2866 |

E_{2}, E_{3} (GPa) | 7.245 | 5.24 | X_{C} | 936 | 661 |

G_{12}, G_{13} (GPa) | 3.33 | 2.41 | Y_{T}, Z_{T} | 56.8 | 41.2 |

G_{23} (GPa) | 2.44 | 1.76 | Y_{C}, Z_{C} | 224 | 162 |

ν_{12}, ν_{13} | 0.33 | 0.33 | S_{12}, S_{13} | 138 | 100 |

ν_{23} | 0.485 | 0.487 | S_{23} | 84 | 61 |

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**MDPI and ACS Style**

Gong, Y.; Jiang, L.; Li, L.; Zhao, J.
An Experimental and Numerical Study of the Influence of Temperature on Mode II Fracture of a T800/Epoxy Unidirectional Laminate. *Materials* **2022**, *15*, 8108.
https://doi.org/10.3390/ma15228108

**AMA Style**

Gong Y, Jiang L, Li L, Zhao J.
An Experimental and Numerical Study of the Influence of Temperature on Mode II Fracture of a T800/Epoxy Unidirectional Laminate. *Materials*. 2022; 15(22):8108.
https://doi.org/10.3390/ma15228108

**Chicago/Turabian Style**

Gong, Yu, Linfei Jiang, Linkang Li, and Jian Zhao.
2022. "An Experimental and Numerical Study of the Influence of Temperature on Mode II Fracture of a T800/Epoxy Unidirectional Laminate" *Materials* 15, no. 22: 8108.
https://doi.org/10.3390/ma15228108