*4.2. Exponential Softening Model*

Because of its simplicity, the linear softening model is frequently used; however, for certain brittle materials, a nonlinear softening model may be more accurate. When the cohesive traction–displacement relation changes, the stress gradient around the crack tip differs, and that may affect the convergence rates of enrichment schemes. The same DCB specimen was used for the numerical study of convergence rates. The cohesive force and displacement gap relation can be expressed as

$$f^c = f\_t \cdot \exp(-\frac{f\_t}{E\_f} \cdot \omega) \tag{28}$$

The tensile strength was the same as *ft* = 3.18 MPa, and the fracture energy was made to be smaller than *Ef* = 12.5 N/m to increase the gradient of the cohesive force. The linear and exponential softening models used in this study are depicted in Figure 8. It can be seen from the profile that they will result in a similar FPZ length, but different stress gradients.

The stress *σyy* along the symmetric line produced by different enrichment schemes is provided in Figure 13, as well as a comparison of the two cohesive constitutive models. Likewise, the stress profiles produced by these enrichment schemes are hard to distinguish from one another. It can be seen from Figure 13 b that, when the traction–separation relation changes, although the cohesive force remains equal to the tensile strength at the crack tip, the stress gradient differs in the FPZ, and does not make much difference at the back of the crack tip.

**Figure 13.** (**a**) Stress profiles of *σyy* obtained by different enrichment schemes for the exponential softening model and (**b**) a comparison of different cohesive constitutive models.

Figure 14 shows the convergence rates of these enrichment schemes when the exponential softening model is inserted. Likewise, the enrichment schemes with tip branch functions exhibit a higher convergence rate. The employment of the tip branch function *r* cos *<sup>θ</sup>* <sup>2</sup> increases both the convergence rate and accuracy substantially. However, especially when coarse meshes are used, these enrichment schemes achieve lower accuracy than XFEM-h in terms of error level. In comparison with the cases of the linear constitutive law, the difference between the convergence rates of these enrichment schemes is more pronounced for the cases of the exponential constitutive law.

**Figure 14.** Plot of the convergence rate for the cohesive crack problem with the exponential softening law (m is the convergence rate).

#### *4.3. Mixed-Mode Crack Problem*

In this case, a plate with an inclined cohesive crack was analyzed using all four enrichment schemes in order to investigate their convergence properites in depth. The boundary conditions are shown in the sketch in Figure 15. The dimensions of the plate are 200 by 400 mm, with a thickness of 20 mm. The inclined crack is located at [0 150; 100 200]. A uniformly distributed tensile force *fext* = 1 Mpa was applied on the top edge with the plane stress condition. All the material properties and softening laws were the same as in previous cases.

**Figure 15.** Boundary conditions of a plate containing an inclined crack.

The convergence rates of these enrichment schemes are provided in Figure 16. They follow similar tendencies. The enrichment schemes with tip branch functions have similar convergence properties, while the corrected approximation and additional tip branch functions can increase the accuracy.

**Figure 16.** Plots of the convergence rate for the mixed-mode crack problem with (**a**) the linear softening law and (**b**) the exponential softening law (m is the convergence rate).
