The pullout behavior of single fiber has been modelled in the literature (Lin and Li, 1997; Kanda and Li, 1999). In these existing models, when the fiber is subjected to a pullout force P, three stages are considered including elastic, debonding and pullout stage with different range of the slip S as shown in Figure 1. When the load is small, the local shear stress τ appears linear correlation with slip S and κ is the slope of the τ-S curve in elastic stage. When the shear stress exceeds the maximum shear strength τ m , the debonding of the fiber is initiated and the frictional strength is defined as a constant value τ d . After debonding reaches the embedded end of the fiber, the interface is purely governed by the frictional force and the fiber is assumed to be pulled out from the matrix, which is determined as pullout stage.

The model is established based on the static equilibrium requirement (Naaman et al., 1991; Li et al., 2003), and the tensile force in the fiber F is transferred to the matrix M through the interface. This relationship expressed in the differential form is d F d x = − d M d x = ψ τ (1) where ψ =   equivalent circumference of the fiber; and τ = shear stress at fiber-matrix interface.

When the fiber and matrix are fully bonded as shown in Figure 2A, the relationship between slip S 1 and the bonded load P 1   can be formulized as S 1 = P 1 Q η sinh ( η L ) [ cosh ( η L ) − 1 ] (2) where η 2 = ψ κ Q and slope κ   is defined from the shear strength diagram, Q = 1 A y E y + 1 A m E m . When shear stress at x = L reaches the maximum strength, the elastic stage is terminated and the maximum bonded load is calculated by P 1 m a x = τ m ψ η tanh ( η L ) (3)

When the shear stress exceeds the maximum shear strength τ m , debonding for a length of d in Figure 2B is initiated, resulting in two separate regions, bonded and debonded. The shear strength of debonded stage decreases from τ m to τ d . S 2 = P 2 m a x Q η cosh ( η ( L − d ) ) − 1 sinh ( η ( L − d ) ) + 1 2 Q d ( τ d ψ d + 2 P 2 m a x ) (4)   P 2 m a x = τ m ψ η t a n h ( η ( L − d ) ) (5)

It is assumed that at the time the fiber is completely debonded, the shear resistance still remains τ d , leading to a simplified rigid body motion after sliding begins ( Δ c > 0 ) as shown in Figure 2C. S 3 = − τ d ψ Q 2 ( L − Δ c ) 2 + P 3 , n t h Q ( L − Δ c ) (6) P 3 = τ d ψ ( L − Δ c ) (7)

The above section illustrates the stress transfer via interface between the fiber and matrix when the fiber is being pulled out. However, due to the great divergency in the interface properties between various fibers, the previous model mentioned above cannot apply to different types of fibers. The limitations will be explained in detail in this section.

To address the mentioned limitations of current model, a refined model for single fiber pullout is proposed in this section.

As mentioned before, the frictional strength varies in different fiber pullout stages, especially in the debonding and pullout stages. To account for this variation, a new parameter pullout frictional strength   τ p is adopted. During the fiber pullout process, fiber and matrix are fully bonded when the shear stress at interface is less than the maximum chemical strength τ m . Along the debonded zone the frictional strength is constant as τ d . Sliding begins right after the completion of debonding and the resisting frictional strength is assumed to drop to pullout strength τ p . By introducing this new parameter, the overall trend of pullout stage will be flatter and more consistent with the experiment than the previous model.

Moreover, considering the surface of some fibers bonded with the matrix is partly worn under pulley force which results in obvious slip hardening phenomenon, the new parameter blocking segment L b (as shown in Figure 6) and is introduced to describe different slipping phenomena.

The pulley force of fiber with slip hardening in the pullout stage can be expressed as   P 3 = τ p [ ( L b − Δ d + Δ c ) ( 1 + β ( Δ d − Δ c ) D ) + ( L + Δ c − L b ) ] (8) where, Δ d represents the fiber displacement, slip hardening coefficient β represents the rate of frictional change, Δ c represents the debonding length, and D represents fiber diameter.

By adding these new parameters, the fiber like PVA fiber which has obvious slip hardening phenomenon can be simulated more accurately, and the pullout phenomena of fibers with different properties can be unified through the processing of the pullout stage formula as shown in Figure 7.

Based on the analytical model established above, the curves obtained from the experiment can be simulated. The specific calculation process is summarized as follows in Figure 8.

In this section, the preparation and setup of single fiber pullout tests are introduced in detail. Moreover, parameters are fitted based on different experimental curves to obtain more accurate fiber bridging curve.

According to the parameter results obtained from single fiber pullout simulation above, even the same kind of fibers vary a lot in discreteness. Considering such difference, average values were used in the previous study to apply these parameters to a larger scale like simulation of fiber bridging curve. However, the randomness of interface parameters cannot be expressed completely by using the average values, so the parameters used in fiber bridging curve are obtained from normal distribution in this paper.

As illustrated in Figure 13, all parameters were randomized for PVA and PET fibers, respectively. Then a modified single crack model will be established based on the parameters selected randomly from the probability distribution of these parameters.

Fiber-bridging constitutive law σ(δ) describes the relationship between the bridging stress σ transferred across a crack and the opening of this crack δ. On the one hand, this curve is related to the material microstructure. On the other hand, tensile strain-hardening behavior of composites is governed by this law. Therefore, to successfully design SHCC material properties, it is necessary to understand and simulate the σ(δ) curve with good accuracy.

For a fiber perpendicular to the interface between fiber and matrix, the parameters are selected randomly from the probability distribution of these parameters above in Figure 13. Therefore, a single fiber pullout force can be calculated when the orientation angle ϕ is 0. However, the bridging force is contributed by the fibers which are randomly distributed with different orientation angle ϕ and embedding length L. Hence, the fiber-bridging σ(δ) relation is available through assuming the force of fibers with different centroidal distance z and orientation angle ϕ on the crack plane. The position relation of a single fiber and matrix is shown in Figure 14.

Equation 9 shows the formulation of fiber-bridging σ(δ) relation. σ ( δ ) = 1 A f V f ∫ P ( z , ϕ ) p ( z , ϕ ) d z d ϕ (9) A f and V f are the cross-sectional area and volume fraction of fiber, respectively. P ( z , ϕ ) is the bridging force contributed by a single fiber with distance z and orientation angle ϕ. p ( z , ϕ ) is the probability density function of fiber with distance z and orientation angle ϕ.

The probability density functions of distance z and orientation angle ϕ are independent of each other. So p ( z , ϕ ) can be expressed as following p ( z , ϕ ) = p ( z ) p ( ϕ ) (10) p ( z ) = 2 l f , 0 ≤ z ≤ l f 2 (11) p ( ϕ ) = sin ( ϕ ) , 0 ≤ ϕ ≤ π 2 (12)

A discrete method was adopted to deal with such complex analytic expressions in previous studies (Lu and Leung, 2016), which divides the fibers according to the combination of their inclination and embedding length. Assuming that the fibers are randomly distributed, fibers with different embedment length are uniformly divided into 100 groups from 0 to L/2, and those at different inclination angles are divided into 100 groups from 0 to π / 2 . Hence, 10,000 different groups are considered totally. Meanwhile, assuming that fibers with longer embedment length (from L/2 to L) have the same distribution as fibers with shorter embedment length and their lengths sum up to L. Therefore, when calculating the friction transfer with a certain embedment length, the frictional stress from both the long and short embedment side should be calculated and averaged.

Through this way, the stress-crack opening curves are simulated and compared with the experimental results. The specific calculation process is summarized in Figure 15.

Single crack tensile test was conducted to compare with the simulation results. Dog-bone specimen is used according to the SHCC mechanical test specification, and the specific size is shown in Figure 16.

The test preparation is shown in Figure 17. The preparation of specimens mainly includes weighing, mixing, pouring, demolding and curing of raw materials. After curing, the middle section of the specimen was notched by a cutting machine to prevent the phenomenon of multi-crack cracking during tensile test and the dimension of the cross section of PVA and PET fiber specimens are shown in Figure 18. For the test setup, the edge of electronic extensometer was glued on the side of the specimen, and the force arm of the extensometer was finally clamped and fixed on the tool edge.

It can be seen in Figure 19 that the bridging stress reaches the peak stress at roughly 5 MPa in PVA fibers while PET fibers can only reach 3 MPa peak stress. Besides the peak bridging stress, the corresponding crack opening displacement δ for PVA fiber specimen is wider than that for PET fiber specimen, which means that PVA fibers may have better ductility. Meanwhile, due to the larger crack width, the durability of PVA fiber specimen may be not as good. Compared with the PVA fiber specimen, PET fibers can continue to bridge the matrix under a larger crack opening displacement more than 2 mm. It is likely due to the interfacial bond between PVA fiber and the matrix is stronger than that between PET fiber and the matrix. Due to the weaker bond with the matrix, PET fiber shows higher tendency to pull out, which accounts for the flatter decline trend of PET fiber in the pullout stage as shown in Figure 19B.

Overall, the single crack model which is built based on the proposed single fiber pullout model fits well with the test results so the validity is confirmed again.