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Correction to: Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials

The Original Article was published on 11 April 2020

Correction to: Adv. Model. and Simul. (2020) 7:19 https://doi.org/10.1186/s40323-020-00152-7

Following publication of the original article [1], the authors reported the errors in the equations. In Eqs. (2), (3) and (22), at the end of all integrals, \(\hbox {d}\vec {X}_{\mathrm{md}}\vec {X}\) has been changed to \(\hbox {d}\vec {X}_\mathrm{m}\hbox { d}\vec {X}\). The corrected equations are given below:

$$\begin{aligned} \mathcal {E}(\vec {u}) = \frac{1}{|\Omega _\mathrm {m}|} \int _\Omega \int _{\Omega _\mathrm {m}} W(\vec {X},{{\varvec{F}}}) \, \mathrm {d}\vec {X}_\mathrm {m}\hbox { d}\vec {X}. \end{aligned}$$
(2)
$$\begin{aligned} \delta {\mathcal {E}}(\vec {u};\delta \vec {u}) = \frac{1}{|\Omega _\mathrm {m}|} \int _\Omega \int _{\Omega _\mathrm {m}} {{\varvec{P}}} : \vec {\nabla }_\mathrm {m}\delta \vec {u}(\vec {X},\vec {X}_\mathrm {m}) \, \mathrm {d}\vec {X}_\mathrm {m}\hbox { d}\vec {X}, \end{aligned}$$
(3)
$$\begin{aligned} \delta ^2{\mathcal {E}}(\vec {u};\delta \vec {u}) = \frac{1}{|\Omega _\mathrm {m}|} \int _\Omega \int _{\Omega _\mathrm {m}} \vec {\nabla }_\mathrm {m}\delta \vec {u}(\vec {X},\vec {X}_\mathrm {m}) : \mathbb {C} : \vec {\nabla }_\mathrm {m}\delta \vec {u}(\vec {X},\vec {X}_\mathrm {m}) \, \mathrm {d}\vec {X}_\mathrm {m}\hbox { d}\vec {X}, \end{aligned}$$
(22)

The original article [1] has been updated.

Reference

  1. Rokoš O, Zeman J, Doškář M, Krysl P. Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials. Adv Model Simul Eng Sci. 2020;7:19. https://doi.org/10.1186/s40323-020-00152-7.

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Correspondence to Ondřej Rokoš.

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Rokoš, O., Zeman, J., Doškář, M. et al. Correction to: Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials. Adv. Model. and Simul. in Eng. Sci. 7, 25 (2020). https://doi.org/10.1186/s40323-020-00160-7

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  • DOI: https://doi.org/10.1186/s40323-020-00160-7