*4.1. Linear Softening Model*

For the cases with a linear softening model, the cohesive force can be expressed as

$$\begin{cases} f^c = f\_t \left( 1 - \frac{\omega}{\omega\_c} \right) & 0 \le \omega \le \omega\_c \\ f^c = 0 & \omega > \omega\_c \end{cases} \tag{25}$$

In this paper, the material properties of common concrete were used, where the tensile strength *ft* = 3.18 MPa and the critical crack opening *ω<sup>c</sup>* = 0.0314 mm. The fracture energy was *Ef* = 0.5 *ftω<sup>c</sup>* = 50 N/m. The plot of this softening model is provided in Figure 8.

**Figure 8.** Linear and exponential softening models in terms of the traction–separation relation.

Due to the nonlinearity of the problem, an explicit analytical solution for the displacement field around the crack tip is not available. In order to evaluate the relative error level for different mesh densities, we took the results obtained by the finest meshes with each enrichment scheme as reference exact solutions. The h-version convergence rate of the finite element method was quantified by means of the relative error in the *L*2-norm, which was calculated by the following equations.

$$E\_u = \frac{\left\| u - u^{ref} \right\|\_{L^2}}{\left\| u^{ref} \right\|\_{L^2}} \tag{26}$$

$$\|u\|\_{L2} = \sqrt{\int\_{\Omega \backslash \Gamma\_c} (u\_1)^2 + (u\_2)^2 d\Omega} \tag{27}$$

where the superscript *ref* denotes a reference solution.

In Figure 9, the deformed geometry of the cohesive crack problem is compared with that of the traction-free crack problem when the same load factor is applied. When a cohesive force exists between crack faces, the crack closes smoothly from the physical tip to the fictitious tip. The Contour plots of *σyy* for the cohesive crack problem and the tractionfree crack problem are provided in Figure 10. It can be seen that, in the cohesive crack model, a stress concentration appears ahead of the crack tip, which is the fracture process zone (FPZ), rather than at the back of the crack tip, which is the case for the traction-free crack problem. The stress gradient at the crack tip is much smaller compared with the case with the traction-free crack. The stress at the crack tip in Figure 10b is finite, and equal to material tensile strength. This means that the cohesive crack model abandons the unrealistic assumption in the LEFM that the stress at the crack tip is infinite, which can be seen in Figure 10a.

**Figure 9.** Deformed geometries of the beam with (**a**) a traction-free crack and (**b**) a cohesive crack.

**Figure 10.** Contour plot of *σyy* for (**a**) the traction-free crack problem and (**b**) the cohesive crack problem (unit: Mpa).

The stress *σyy* along the axis of symmetry is plotted in Figure 11. The stress profiles obtained by different enrichment schemes are difficult to distinguish from one another. They also show an obvious FPZ ahead of the crack tip, and the stress *σyy* is equal to the tensile strength at the crack tip. In contrast, a stress singularity appears around the crack tip in the traction-free crack problem.

**Figure 11.** Stress profiles of *σyy* obtained by different enrichment schemes for the linear softening model.

Figure 12 shows the relative error in the *L*2-norm plotted against the inverse of the element size h, which is taken as the square root of the area of the element. The rates of convergence were obtained by means of polynomial curve fitting of those data points. It is interesting that, as the linear softening model is considered, with the employment of the branch function *r* sin *<sup>θ</sup>* <sup>2</sup> , XFEM-s, XFEM-c1, and XFEM-c2 achieve a better convergence rate of more than 1, compared with the 0.7 obtained by XFEM-h. The employment of the additional branch function *r* cos *<sup>θ</sup>* <sup>2</sup> results in a similar convergence rate but a higher accuracy.

**Figure 12.** Convergence rate plot for the cohesive crack problem with the linear softening model (m is the convergence rate).

As far as accuracy is concerned, the XFEM-s scheme provides less accuracy for cohesive crack problems, especially when coarse meshes are used. It can be seen from the above stress contour plot in Figure 10 that the singularity vanishes at the crack tip, and only a finite stress gradient exists. If the parasitic terms resulting from the branch function are not corrected, it can reduce the accuracy severely for the case of cohesive cracks. With the corrected approximation for blending elements as in XFEM-c1 and XFEM-c2, the error level is improved by around 2 times, while the convergence rate remains almost the same, which is similar to the case of strong discontinuities.
