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

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.