**Table 2.** Mix proportion.

\* by volume fraction.

It is important to note that interface's properties between fibre and matrix significantly influence the performance of a composite. The interfacial properties are also important in the fracture mechanism and the fracture toughness of the composite. The failure process in a composite material when a crack propagates is complex and involves matrix cracking. The bonding strength between fibre and matrix is to be considered as a source of energy dissipation of HPFRCC. The single fibre pull out test is the most common method to understanding the interfacial strength. Generally, the fibre pull out has three stages during debonding [13–15], as shown in Figure 2. Each stage of a single fibre pull out test can be expressed by:


**Figure 2.** Idealised interface law in three stages of single fibre pull out (adopted after [16]).

A numerical study for the behaviour of single fibre pull out was carried using commercial finite element (FE) software package ANSYS [17]. A 2-D axisymmetric model was employed for the simulation of the single fibre pull out process. In the developed model, a PVA fibre with a radius *Rf* was embedded at the centre of the cylindrical matrix, and *Ld* was the total embedded length of the fibre. The bottom of the model was constrained in both radial and axial directions. The interfacial properties were modelled using the bilinear cohesive zone model (CZM) in mode II, which was established by fracture mechanic models, such as the interface traction and separation. The relationship between normal critical energy *Gcn* and tangential critical energy *Gct* can be expressed by the maximum normal contact stress *σmax*, the maximum tangential contact stress *τmax*, the complete normal displacement *δn*, and the complete tangential displacement *δ<sup>t</sup>* [17]. Figure 3 presents the model of the FE single fibre pull out test with the fibre and matrix model which were meshed with 122,406 six node quadrilateral elements. The model was analysed using a non-linear geometrical method with convergent displacement control. To confirm the validity of the FE analysis of the single fibre pull out, an analytical fibre pull out test was conducted. An interfacial friction law for the slip mechanism between the fibre and the matrix has been investigated by several authors [16,18,19]. For an analytical fibre pull out, a proposed model by Zhan et al. [16], which was based on the interfacial law that could capture the major mechanism involved in various situations, was used to obtain the fibre pull out force. The results of the analytical and the FE analyses of the single fibre pull out model were overall in good agreement, with around 2% difference, as illustrated in Figure 4. Thus, the FE simulation can be used for investigating the interfacial behaviour between the fibre and the AAFA matrix.

**Figure 3.** Configuration of single fibre pull out simulation without an inclined angle.

**Figure 4.** Validation of finite element (FE) model with the analytical model by Zhan et al. [16].

Taguchi's DOE approach with eight parameters and three levels of test variables were selected in accordance to the literature [16,18,20–23], as shown in Table 3. The standard L27 (313) orthogonal array was used in accordance to these parameters, and the detail of L27 orthogonal array is shown in Table 4.





#### **3. Results and Discussion**

#### *3.1. Single Fibre Pull Out*

Based on the Taguchi's DOE approach, a statistical signal to noise (S/N) ratio analysis was performed to determine the effect of these parameters on the maximum fibre pull out force *Pmax*, as illustrated in Table 5 and Figure 5. The S/N ratio shows that the diameter of the fibre has the most effect on the fibre pull out force. The elastic modulus of the fibre and the matrix has a minor effect on the pull out force. A further analysis of the single fibre pull out behaviour was done using analysis of variance (ANOVA) and the results indicate that the contribution of the fibre diameter on pull out force is 44.69% of the total contribution factors. The overall results are presented in Table 6. It can be observed

that increasing the elastic modulus of the matrix, the diameter of the fibre, the tangential traction, and the embedded length of the fibre results in increasing the pull out force. The contributions of the elastic modulus of the matrix, the tangential traction and the embedded length of the fibre on the pull out force are 14.48%, 8.92%, and 9.47%, respectively. At the same time, increasing the Poisson's ratio results in decreasing in the pull out force but the contribution is minor. The contributions of the diameter of the fibre, Poison's ratio, the elastic modulus of matrix, and complete tangential displacement on the pull out force are fairly similar at about 2.5%.


**Table 5.** Numerical studies of single fibre pull out with Taguchi's DOE.


**Figure 5.** Signal to noise ratio of single fibre pull out.

**Table 6.** Analysis of Variance of fibre pull out force.

*<sup>a</sup>* degree of freedom; *<sup>b</sup>* sum of squares; *<sup>c</sup>* mean of squares.

The failure process in a composite material when a crack propagates is complex and involves matrix cracking. The bonding strength between fibre and matrix is to be considered as a source of energy dissipation. Thus, a single fibre pull out test of PVA fibre conducted with AAFA paste matrix (l/s = 0.6) was also conducted with OPC paste matrix (w/c = 0.3) to compare the interfacial bonding strength between AAFA and OPC matrices. The embedded length (*Ld*) of the fibre was around 4 mm which is half of the total length of the fibre, and the diameter of fibre (*df*) was 38 μm. Assuming uniform bonding, the maximum interfacial bonding stre. The results of the single fibre pull out test of AAFA and OPC matrices show that the pull out force is similar. A comparison of the maximum pull out force between the numerical and experimental results are presented in Table 7. The input parameters such as elastic modulus and Poisson's ratio of the FE model were adopted from the authors' previous work [24]. The comparison of the maximum pull out force between the numerical and experimental results are in good agreement, thus, validating the numerical analysis of the single fibre pull out with Taguchi's DOE.

**Table 7.** Maximum pull out force in the finite element and the experimental results.

