The mode II delamination toughness of arc unidirectional fiber-reinforced composites with ENF test

To obtain the mode II delamination toughness (the energy release rate GIIC) of arc unidirectional fiber-reinforced composites, the end notched flexure (ENF) test with arc specimen was used to study. In terms of theory, the formulas of compliance and GIIC for Arc-ENF specimen were derived by using energy method and Irwin-Kies formula, and the crack length correction and friction coefficient were respectively introduced to correct these formulas, with the verification by the numerical simulation. As the ENF test, the compliance curve of the Arc-ENF specimen was recorded under the condition of the constant loading speed, and the crack tip position was determined by the strain diagram of the specimen in 1-3 plane with the optical strain gauge. Finally, the friction coefficient and the GIIC of the Arc-ENF specimen in the ENF test were determined using the corrected formulas, the numerical simulation method and the ENF test data.


Introduction
The delamination is one of the main damage forms of fiber-reinforced composite structures. In order to prevent the occurrence of the delamination, it is necessary to evaluate the interlaminar properties of material during the design for material structures, and researchers have designed various test methods for analysis. For mode II delamination toughness test, the ENF specimen is common, with the test methods including the ENF test, four-point bending end notched flexure (4ENF) test and End load separation (ELS) test, etc. Brunner et al. [1] and Tabiei et al. [2] made review of these, and pointed out that the friction effect in the 4ENF test was obvious, which lead to results higher, and the clamp in ELS test will introduce many other factors, in which the test data are difficult to process, and the unstable crack propagation process in ENF test leads to a large error in the measurement results. Nevertheless, considering the advantages of ENF test in terms of test methods and data processing, standard D7905/D7905M-19e1 [3] still proposes it as the main test method for mode II delamination toughness test of unidirectional fiber-reinforced composite materials, for which the ENF test is adopted on the Arc-ENF specimen.
For the analytical solution of the ENF test, Russell and Street [4] firstly obtained the compliance and GIIC formulas based of the simple beam theory (SBT):   (2) where 2L, B and 2h are respectively the length, width and thickness of the ENF specimen, and a is the crack length, E1 is the elastic modulus in the fiber direction. Then, the compliance formula are modified based on the Timoshenko beam theory (TBT) by considering the effect of shear deformation [5], where G13 is the shear modulus of the 1-3 plane, and for GIIC formulas, SBT The reason that the expressions of GIIC are same is that the TBT is the first-order shear-deformation beam theory. And the effect of shear deformation on GIIC can be seen with the application of elasticity or higher-order shear-deformation beam theories [6][7][8], alternatively, enhanced beam theory (EBT) models with elastic or cohesive interfaces [9] are an another effective way, with the same expressions, where  is the crack-length correction parameter related to the modulus of materials, and △a is a constant. Besides, the effect of friction on the GIIC formula is also significant. Zebar et al. [10] studied the mixed 4ENF test of two types of composites, and obtained the GIIC formula considering the friction effect. Based on the point friction hypothesis and the beam theory, Wang et al. [11] used the energy method to derive a GIIC formula considering the friction effect, and the calculation results were consistent with the results of the finite element analysis. Liu et al. [12] obtained a similar formula under the same assumption, and proposed a new cohesion model for the simulation of 4ENF test, and the simulation results were in good agreement with the test results. Mencattelli et al. [13] studied the effect of friction in 4ENF test and found that the friction effect between the loading clamp and the specimen was the main factor influencing the test, and suggested the rolling supporting to reduce the influence of friction. Parrinello et al. [14][15][16] used the mechanical behavior of ideal rigid plastic materials to describe the friction behavior, and combined with Euler-Bernoulli beam theory to establish an analytical method for analysing four-point bending test, and the final theoretical results were consistent with the numerical simulation results.
With the development of computer technology, the numerical analysis has been widely used in the study of delamination of composites, in which cohesion zone model (CZM) [17,18] is widely concerned by researchers. Dourado [19] used a bilinear CZM to analyse the mode II delamination in composites with considering the fiber bridging effect in the process of test, and the CZM parameters were obtained by using a genetic algorithm to compare the result of simulation and test, and then De Morais [20] used this method on the mixed-mode (I+II) delamination. Nguyen, et al. [21] proposed a new mixed CZM for numerical analysis, and the simulation result had a good agree with the test of double cantilever beam (DCB), ENF and mixed-mode bending (MMB), which show the applicability of this model. Xie et al. [22] obtained the cohesion model by using the closed form solution method on the classical lamination theory (CLT), and applied it to the numerical analysis of DCB, ENF and MMB to study the influence of the crack length, interface strength, fracture energy and cohesive model form on the calculation results. In general, researchers can use the mechanical properties of matrix of composites as the CZM parameters in the numerical analysis.
The geometrical shape in the longitudinal direction of ENF specimen in different test introduced above is straight, however, in engineering applications, composites will have different geometric shapes according to design requirements or forming processes, such as the cylinder structure formed by winding forming process. For those structure with curvature, people want to get performance parameters as far as possible under the same processing conditions. Thus, based on the content introduced above, the GIIC of Arc-ENF specimen will be obtained, and the content is divided into three parts. The first part introduces the derivation, correction and verification of the formulas of compliance and GIIC for Arc-ENF specimen, and the second part introduces ENF test, finally, in the third part, GIIC for Arc-ENF specimen will be obtained by analysing the ENF test data with the modified formulas and CZM numerical simulation method.

Basic formulas of compliance and GIIC for Arc-ENF specimen
Compared with the ENF test of standard specimen, the biggest difference of arc specimen is that their geometrical shape in the longitudinal direction is arc (Figure 1), and there is no difference in test methods. And there are two normal ways to derive the compliance formula. The first one is that, based on the beam theory, the displacement of the loading point can be determined by calculating the deformation of the specimen in the loading and boundary condition, and then the compliance expression will be obtained. And the another one is to determine the strain energy of the specimen based on the energy method, and then calculate the loading point displacement by using the Castigliano theorem, and finally obtain the expression of the compliance. However, since the bending deformation equation of the arc specimen based on the beam theory is a nonlinear equation (the first derivative of deflection cannot be ignored), the analytic solution of deformation cannot be obtained. Thus, the second way based on the energy method will be used to give the compliance formula. The strain energy U of the Arc-ENF specimen ( Figure 1) in ENF test is combined with bending deformation energy U1 and shear deformation energy U2, 22 3 where M is flexure moment and  is shear force, and their expressions are described in Appendix A; E1 is the elastic modulus in the 1 direction, I is the second moment of area of the specimen section, R ≈ R0+h, B is the width of the specimen, G13 the shear modulus of the 1-3 plane. Then, considering the Castigliano theorem =/ dU dF  and Irwin-Kies formula, the expression of compliance and GIIC can be written as

The correction of the basic formulas of compliance and GIIC for Arc-ENF specimen
The correction of the basic formulas can be realized in two aspects based on the research in the introduction section. The first one is the crack length correction in Eq. (4), reflecting the effect of shear deformation, and observing the form of 0 () i f  , the corrected term with a assumption that correction is related to the crack length can be written as where   is constant. And the corrected formulas are Another one is to consider the effect of friction. Due to the friction occurring between the upper and lower layers after delamination during the ENF test, the correction expression need to be derived from the energy of friction, '( ) where i=1,

Verification of the formulas of compliance and GIIC for Arc-ENF specimen
In this section, the numerical simulation will be used with CZM based on the ABAQUS for the verification of the formulas in section 2.2. The model is 2D with geometric parameters listed in table 1, and the materials of Arc-ENF specimen have two different properties [23] (table 2). As for the interface model, the bilinear CZM is used with its parameters (table 3) referred to the matrix of composites, and the interface failure adopts the maximum energy criterion. Besides, when the initial crack length is small, a larger load is required to cause the initial crack propagation, and at the same time, the load will suddenly drop, so the initial crack length set during simulation is large. This phenomenon can occur referred to the maximum load-displacement curve in Figure 2 or Figure 3.
Firstly, the influence of the crack length correction on the compliance and GIIC is studied. In this part, two different types of composite materials are used under two different energy release rates (GIIC =0.7, 1.4 kJ/m 2 ), and finally the load F-displacement  curve is given. For the theoretical calculation results, the load force value at each point on the curve represents the maximum force value under the corresponding displacement, and the crack propagation occurs when the load reaches this value. So the simulation results should be consistent with the theoretical during the crack propagation. By observing  Figure 2 and Figure 3, it can be seen that there is a big difference between the theory and the simulation results before correction, but the corrected results are in good agreement, which indicates the reasonability of the correction. However, the value of   is related not only to the energy release rate of the interface, but also to the mechanical properties of the material. And when the difference between the longitudinal direction modulus and the transverse modulus of the material is small, the value of   is larger. Besides, with the increase of GIIC,   becomes lager. So referring to the values of   showing in Figure 2 and Figure 3,   =3.0° is suggested for M1 material, and   =4.0° is suggested for M2 material, or revising the value with the numerical simulation results.

The ENF test
The ENF test was used to obtain the GIIC of the Arc-ENF specimen, seen in Figure 5. And the geometric parameters are given in table 4 and the mechanical properties are given in table 5. To ensure the quasi-static crack propagation, the loading speed was set at 4×10 -6 m/s, and the radius of the cylinders used for bearing and loading was 1.5 mm. As for the position of the crack tip, it's difficult to observe with eyes. So the optical strain gauge was used to record the deformation in 1-3 plane of the specimen, and after processing, the strain diagram in any moment could be obtained, such as Figure 5.
And the position of the crack tip (Point A in Figure 5) was determined by 5% strain, empirically, in the longitudinal direction. Finally, the value of load and displacement of the specimen were recorded.

Result analysis
Although the corrected formulas of the compliance and GIIC for the Arc-ENF specimen verified above were effective, and the necessary data in the ENF test had been recorded, there are still two problems to be answered. The first one is that whether the crack tip position determined by the 5% strain is near or not to the 0  in the formulas, and another one is that the value of  for the specimen is not sure.
The following content will give answers and finally, the GIIC value and friction coefficient will be determined.

Discussion
The method to obtain the corrected formulas of the compliance and GIIC for the Arc-ENF specimen is similar to the normal ENF specimen in the ENF test, which contains two terms, the crack length correction and the effect of friction correction. For the first one, the correction reflecting the effect of shear deformation is expressed with an additional expression (Eq. 10) for the Arc-ENF specimen, which is different from that for the normal ENF specimen, expressed by the change of the crack length (Eq.4). It's notable that both of the corrections are small, compared with the bigger crack length correction found by comparing the strain diagrams from the optical strain gauge ( Figure 5) and the numerical simulation results ( Figure 6). The bigger crack length correction drives from the deformation of interlamination at the crack tip, and as the fracture extensibility of the interlamination increasing, the value of the correction will be bigger. To obtain the expression of this correction, it should be derived from the basic elastic theory, and the finding is also valuable for the further correction on the formulas for the normal ENF specimen.
For the effect of friction correction, there are many research results for the normal ENF specimen introduced in the section 1, which are based on the point-friction assumption. However, due to the smooth surface of the cylinders of bearing and loading, this assumption can be ignored in the ENF test for the Arc-ENF specimen, instead the friction occurring between the upper and lower layers after delamination is considered. And comparing the results of the numerical simulation and the ENF test, the value of GIIC calculated by the corrected formulas is reasonable, which has reference value on the correction of formulas for the normal ENF specimens.

Conclusion
In this paper, a corrected formulas of the compliance and GIIC for Arc-ENF specimen in the ENF test are introduced, with the verification by comparing the theoretical results with numerical simulation results. And for the test method, the optical strain gauge is used to determine the position of the crack tip. Finally, GIIC = 0.9878 kJ/m 2 of the Arc-ENF specimen is obtained from the use of the corrected formulas, the numerical simulation method and the ENF test data.