Correction to: Comparison between thick level set (TLS) and cohesive zone models

Correction

In the publication of this article [1], there was an error in some equations.

• Equation (37) should instead read:

  $$\begin{aligned} \hat{\sigma } = F(\lambda I \hat{\sigma }) \; \; \text {and} \; \; \int _{\tilde{\phi } = 0}^{\hat{\ell }} h(\tilde{\phi }) \text {d}\tilde{\phi } = -g_{\text {dam}}'(0) \frac{1-g_{\text {dam}}(d(\hat{\ell }))}{g_{\text {dam}}(d(\hat{\ell }))} \hat{\sigma }^2 \end{aligned}$$

  (1)

• Equation (38) should instead read:

  $$\begin{aligned} \hat{\sigma } = F(\lambda I \hat{\sigma }) \; \; \text {and} \; \; \int _{0}^{\hat{\phi }} h(\tilde{\phi }) \text {d}\tilde{\phi } = -g_{\text {dam}}'(0) \frac{1-g_{\text {dam}}(d(\hat{\phi }))}{g_{\text {dam}}(d(\hat{\phi }))} \hat{\sigma }^2 \end{aligned}$$

  (2)

• Equation (45) should instead read:

  $$\begin{aligned} \hat{\sigma } = F(\lambda I \hat{\sigma }) \; \; \text {and} \; \; \int _{0}^{\hat{\phi }} h(\tilde{\phi }) \text {d}\tilde{\phi } = \frac{d(\hat{\phi })}{1-d(\hat{\phi })} \hat{\sigma }^2 \end{aligned}$$

  (3)

• Equation (49) should instead read:

  $$\begin{aligned} \hat{\sigma } = F \left( \lambda \frac{\hat{\phi }^2}{1-\hat{\phi }} \hat{\sigma }\right) \; \; \text {and} \; \; \int _{0}^{\hat{\phi }} h(\tilde{\phi }) \text {d}\tilde{\phi } = \frac{2\hat{\phi }-\hat{\phi }^2}{(\hat{\phi }-1)^2} \hat{\sigma }^2 \end{aligned}$$

  (4)

• Equation (50) should instead read:

  $$\begin{aligned} \hat{\sigma }(\hat{\phi }) = \frac{1-\hat{\phi }}{1-\hat{\phi }+ \lambda \hat{\phi }^2} \; \; \text {and} \; \; \int _{0}^{\hat{\phi }} h(\tilde{\phi }) \text {d}\tilde{\phi } = \frac{2\hat{\phi }-\hat{\phi }^2}{(1-\hat{\phi }+ \lambda \hat{\phi }^2)^2} \end{aligned}$$

  (5)

• Equation (63) should instead read:

  $$\begin{aligned} H(\hat{\phi }) = \int _{0}^{\hat{\phi }} h(\tilde{\phi }) \text {d}\tilde{\phi } \end{aligned}$$

  (6)

• Equation (65) should instead read:

  $$\begin{aligned} \lambda \le \frac{1}{2} \end{aligned}$$

  (7)

• Equation (70) should instead read:

  $$\begin{aligned} \mu _n = \min _{\Gamma } \sqrt{ \frac{(\overline{h(d) Y_c})_n}{\overline{Y}_n}} \end{aligned}$$

  (8)

• In Eq. (75), the expression of $\beta _n$ should instead read:

  $$\begin{aligned} \beta _n = Y_c \frac{\int _{\phi = 0}^{\ell } h'(d) \left( \frac{d'(\phi / \ell _c)}{\ell _c} \right) ^2 \left( 1- \frac{\phi }{\rho (s)} \right) \text {d} \phi }{\int _{\phi = 0}^{\ell } \frac{d'(\phi / \ell _c)}{\ell _c} \left( 1- \frac{\phi }{\rho (s)} \right) \text {d} \phi } \end{aligned}$$

  (9)

• Equation (76) should instead read:

  $$\begin{aligned} \Delta \mu _n^{\text {pred}} < \frac{\Delta \phi _{\text {max}} \beta _n - f_n }{\alpha _n} \end{aligned}$$

  (10)

• Equation (77) should instead read:

  $$\begin{aligned} \Delta \mu _n^{\text {pred}} = \min _{\Gamma } \frac{\Delta \phi _{\text {max}} \beta _n - f_n }{\alpha _n} \end{aligned}$$

  (11)

• Equation (78) should instead read:

  $$\begin{aligned} \mu ^{\text {corr}} = \sqrt{ \frac{(\overline{h(d) Y_c})_n}{\overline{Y}_n}} \Bigr |_{\arg \min _{\Gamma } \frac{\Delta \phi _{\text {max}} \beta _n - f_n }{\alpha _n}} \end{aligned}$$

  (12)

This has now been included in this erratum.