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

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The original article was published in Advanced Modeling and Simulation in Engineering Sciences 2015 2:18

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.

Reference

  1. 1.

    Gómez AP, Moës N, Stolz C. Comparison between thick level set (TLS) and cohesive zone models. Adv. Model. and Simul. in Eng. Sci. 2015;2:18. https://doi.org/10.1186/s40323-015-0041-9.

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Correspondence to Nicolas Moës.

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The original article can be found online at https://doi.org/10.1186/s40323-015-0041-9.

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