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

Andrés Parrilla Gómez, Nicolas Moes, Claude Stolz

► To cite this version:

Andrés Parrilla Gómez, Nicolas Moes, Claude Stolz. Correction to: Comparison between thick level set (TLS) and cohesive zone models. Advanced Modeling and Simulation in Engineering Sciences, 2017, 4 (1), 10.1186/s40323-017-0091-2. hal-03284752

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Submitted on 30 May 2024

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Correction to: Comparison between thick level set (TLS) and cohesive zone models

Andrés Parrilla Gómez, Nicolas Moës, Claude Stolz

► To cite this version:

Andrés Parrilla Gómez, Nicolas Moës, Claude Stolz. Correction to: Comparison between thick level set (TLS) and cohesive zone models. Advanced Modeling and Simulation in Engineering Sciences, 2017, 4 (1), 10.1186/s40323-017-0091-2 . hal-03284752

HAL Id: hal-03284752

https://hal.science/hal-03284752v1

Submitted on 30 May 2024

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

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

Andrés Parrilla Gómez¹, Nicolas Moës¹* and Claude Stolz¹,²

## Correction

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

* Equation (37) should instead read:

$$ \hat{\sigma} = F(\lambda I \hat{\sigma}) \quad \text{and} \quad \int_{\tilde{\phi}=0}^{\hat{\ell}} h(\tilde{\phi}) d\tilde{\phi} = -g'_{\text{dam}}(0) \frac{1 - g_{\text{dam}}(d(\hat{\ell}))}{g_{\text{dam}}(d(\hat{\ell}))} \hat{\sigma}^2 \quad (1) $$

* Equation (38) should instead read:

$$ \hat{\sigma} = F(\lambda I \hat{\sigma}) \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = -g'_{\text{dam}}(0) \frac{1 - g_{\text{dam}}(d(\hat{\phi}))}{g_{\text{dam}}(d(\hat{\phi}))} \hat{\sigma}^2 \quad (2) $$

* Equation (45) should instead read:

$$ \hat{\sigma} = F(\lambda I \hat{\sigma}) \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = \frac{d(\hat{\phi})}{1 - d(\hat{\phi})} \hat{\sigma}^2 \quad (3) $$

* Equation (49) should instead read:

$$ \hat{\sigma} = F \left( \lambda \frac{\hat{\phi}^2}{1 - \hat{\phi}} \hat{\sigma} \right) \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = \frac{2\hat{\phi} - \hat{\phi}^2}{(\hat{\phi} - 1)^2} \hat{\sigma}^2 \quad (4) $$

* Equation (50) should instead read:

$$ \hat{\sigma}(\hat{\phi}) = \frac{1 - \hat{\phi}}{1 - \hat{\phi} + \lambda \hat{\phi}^2} \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = \frac{2\hat{\phi} - \hat{\phi}^2}{(1 - \hat{\phi} + \lambda \hat{\phi}^2)^2} \quad (5) $$

* Equation (63) should instead read:

$$ H(\hat{\phi}) = \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} \quad (6) $$

* Equation (65) should instead read:

$$ \lambda \le \frac{1}{2} \quad (7) $$

<figure><img src="image_3.png" /></figure>

<footnote>*Correspondence:
nicolas.moes@ec-nantes.fr
¹GeM, UMR CNRS 6183, Ecole
Centrale de Nantes, 1 rue de la
Noë, 44321 Nantes, France
Full list of author information is
available at the end of the article</footnote>
<figure><img src="image_2.png" /></figure>

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

Andrés Parrilla Gómez¹, Nicolas Moës¹* and Claude Stolz¹,²

## Correction

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

* Equation (37) should instead read:

$$ \hat{\sigma} = F(\lambda I \hat{\sigma}) \quad \text{and} \quad \int_{\tilde{\phi}=0}^{\hat{\ell}} h(\tilde{\phi}) d\tilde{\phi} = -g'_{\text{dam}}(0) \frac{1 - g_{\text{dam}}(d(\hat{\ell}))}{g_{\text{dam}}(d(\hat{\ell}))} \hat{\sigma}^2 \quad (1) $$

* Equation (38) should instead read:

$$ \hat{\sigma} = F(\lambda I \hat{\sigma}) \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = -g'_{\text{dam}}(0) \frac{1 - g_{\text{dam}}(d(\hat{\phi}))}{g_{\text{dam}}(d(\hat{\phi}))} \hat{\sigma}^2 \quad (2) $$

* Equation (45) should instead read:

$$ \hat{\sigma} = F(\lambda I \hat{\sigma}) \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = \frac{d(\hat{\phi})}{1 - d(\hat{\phi})} \hat{\sigma}^2 \quad (3) $$

* Equation (49) should instead read:

$$ \hat{\sigma} = F \left( \lambda \frac{\hat{\phi}^2}{1 - \hat{\phi}} \hat{\sigma} \right) \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = \frac{2\hat{\phi} - \hat{\phi}^2}{(\hat{\phi} - 1)^2} \hat{\sigma}^2 \quad (4) $$

* Equation (50) should instead read:

$$ \hat{\sigma}(\hat{\phi}) = \frac{1 - \hat{\phi}}{1 - \hat{\phi} + \lambda \hat{\phi}^2} \quad \text{and} \quad \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} = \frac{2\hat{\phi} - \hat{\phi}^2}{(1 - \hat{\phi} + \lambda \hat{\phi}^2)^2} \quad (5) $$

* Equation (63) should instead read:

$$ H(\hat{\phi}) = \int_{0}^{\hat{\phi}} h(\tilde{\phi}) d\tilde{\phi} \quad (6) $$

* Equation (65) should instead read:

$$ \lambda \le \frac{1}{2} \quad (7) $$

<figure><img src="image_3.png" /></figure>

<footnote>*Correspondence:
nicolas.moes@ec-nantes.fr
¹GeM, UMR CNRS 6183, Ecole
Centrale de Nantes, 1 rue de la
Noë, 44321 Nantes, France
Full list of author information is
available at the end of the article</footnote>
* Equation (70) should instead read:

$$ \mu_n = \min_{\Gamma} \sqrt{\frac{(h(d)Y_c)_n}{\bar{Y}_n}} \qquad (8) $$

* In Eq. (75), the expression of $\beta_n$ should instead read:

$$ \beta_n = Y_c \frac{\int_{\phi=0}^{\ell} h'(d) \left( \frac{d'(\phi/\ell_c)}{\ell_c} \right)^2 \left( 1 - \frac{\phi}{\rho(s)} \right) d\phi}{\int_{\phi=0}^{\ell} \frac{d'(\phi/\ell_c)}{\ell_c} \left( 1 - \frac{\phi}{\rho(s)} \right) d\phi} \qquad (9) $$

* Equation (76) should instead read:

$$ \Delta\mu_n^{\text{pred}} < \frac{\Delta\phi_{\text{max}}\beta_n - f_n}{\alpha_n} \qquad (10) $$

* Equation (77) should instead read:

$$ \Delta\mu_n^{\text{pred}} = \min_{\Gamma} \frac{\Delta\phi_{\text{max}}\beta_n - f_n}{\alpha_n} \qquad (11) $$

* Equation (78) should instead read:

$$ \mu^{\text{corr}} = \sqrt{\frac{(h(d)Y_c)_n}{\bar{Y}_n}} \bigg|_{\text{arg min}_{\Gamma} \frac{\Delta\phi_{\text{max}}\beta_n - f_n}{\alpha_n}} \qquad (12) $$

This has now been included in this erratum.

Author details

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

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

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 25 October 2017 Accepted: 25 October 2017

Published online: 16 November 2017

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.
* Equation (70) should instead read:

$$ \mu_n = \min_{\Gamma} \sqrt{\frac{(h(d)Y_c)_n}{\bar{Y}_n}} \qquad (8) $$

* In Eq. (75), the expression of $\beta_n$ should instead read:

$$ \beta_n = Y_c \frac{\int_{\phi=0}^{\ell} h'(d) \left( \frac{d'(\phi/\ell_c)}{\ell_c} \right)^2 \left( 1 - \frac{\phi}{\rho(s)} \right) d\phi}{\int_{\phi=0}^{\ell} \frac{d'(\phi/\ell_c)}{\ell_c} \left( 1 - \frac{\phi}{\rho(s)} \right) d\phi} \qquad (9) $$

* Equation (76) should instead read:

$$ \Delta\mu_n^{\text{pred}} < \frac{\Delta\phi_{\text{max}}\beta_n - f_n}{\alpha_n} \qquad (10) $$

* Equation (77) should instead read:

$$ \Delta\mu_n^{\text{pred}} = \min_{\Gamma} \frac{\Delta\phi_{\text{max}}\beta_n - f_n}{\alpha_n} \qquad (11) $$

* Equation (78) should instead read:

$$ \mu^{\text{corr}} = \sqrt{\frac{(h(d)Y_c)_n}{\bar{Y}_n}} \bigg|_{\arg\min_{\Gamma} \frac{\Delta\phi_{\text{max}}\beta_n - f_n}{\alpha_n}} \qquad (12) $$

This has now been included in this erratum.

Author details

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

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

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 25 October 2017 Accepted: 25 October 2017

Published online: 16 November 2017

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.