 Correction
 Open Access
 Published:
Correction to: Comparison between thick level set (TLS) and cohesive zone models
Advanced Modeling and Simulation in Engineering Sciences volume 4, Article number: 4 (2017)
 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:
$$\begin{aligned} \hat{\sigma } = F(\lambda I \hat{\sigma }) \; \; \text {and} \; \; \int _{\tilde{\phi } = 0}^{\hat{\ell }} h(\tilde{\phi }) \text {d}\tilde{\phi } = g_{\text {dam}}'(0) \frac{1g_{\text {dam}}(d(\hat{\ell }))}{g_{\text {dam}}(d(\hat{\ell }))} \hat{\sigma }^2 \end{aligned}$$(1) 
Equation (38) should instead read:
$$\begin{aligned} \hat{\sigma } = F(\lambda I \hat{\sigma }) \; \; \text {and} \; \; \int _{0}^{\hat{\phi }} h(\tilde{\phi }) \text {d}\tilde{\phi } = g_{\text {dam}}'(0) \frac{1g_{\text {dam}}(d(\hat{\phi }))}{g_{\text {dam}}(d(\hat{\phi }))} \hat{\sigma }^2 \end{aligned}$$(2) 
Equation (45) should instead read:
$$\begin{aligned} \hat{\sigma } = F(\lambda I \hat{\sigma }) \; \; \text {and} \; \; \int _{0}^{\hat{\phi }} h(\tilde{\phi }) \text {d}\tilde{\phi } = \frac{d(\hat{\phi })}{1d(\hat{\phi })} \hat{\sigma }^2 \end{aligned}$$(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.
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/s4032301500419.
Open Access
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original article can be found online at https://doi.org/10.1186/s4032301500419.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Gómez, A.P., Moës, N. & Stolz, C. Correction to: Comparison between thick level set (TLS) and cohesive zone models. Adv. Model. and Simul. in Eng. Sci. 4, 4 (2017). https://doi.org/10.1186/s4032301700912
Received:
Accepted:
Published: