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  • Correction
  • Open Access

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

Advanced Modeling and Simulation in Engineering Sciences20174:4

https://doi.org/10.1186/s40323-017-0091-2

  • Received: 25 October 2017
  • Accepted: 25 October 2017
  • Published:

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.

Notes

Declarations

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.

Authors’ Affiliations

(1)
GeM, UMR CNRS 6183, Ecole Centrale de Nantes, 1 rue de la Noë, 44321 Nantes, France
(2)
IMSIA, UMR CNRS 9219, EdF, Av. Charles de Gaulle, 92141 Clamart, France

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/s40323-015-0041-9.View ArticleGoogle Scholar

Copyright

© The Author(s) 2017

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