Correction: Kunc, O.; Fritzen, F. Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling. Math. Comput. Appl. 2019, 24, 56

The authors wish to make a correction to Formula (42) of the paper [1]. The correct formula reads

Correspondingly, a correction to Equations (A1)–(A4) of Appendix A of [1] is now provided. To this end, Green’s strain tensor $\overline{\mathit{E}}=\frac{1}{2}({\overline{\mathit{F}}}^{\mathsf{T}}\overline{\mathit{F}}-\mathit{I})$, the corresponding stored energy density function ${\overline{W}}^{\mathrm{E}}\left(\overline{\mathit{E}}\right)=\overline{W}\left(\overline{\mathit{F}}\right)$, the second Piola–Kirchhoff stress $\overline{\mathit{S}}=\partial {\overline{W}}^{\mathrm{E}}/\partial \overline{\mathit{E}}{|}_{\overline{E}}$, and the corresponding stiffness tensor ${\overline{\mathbb{C}}}^{\mathrm{E}}={\partial }^2{\overline{W}}^{\mathrm{E}}/{(\partial \overline{\mathit{E}})}^2{|}_{\overline{E}}$ are introduced. Starting from the well-known relationship $\overline{\mathit{P}}=\overline{\mathit{F}}\overline{\mathit{S}}$ between $\overline{\mathit{S}}$ and the first Piola–Kirchhoff stress $\overline{\mathit{P}}=\partial \overline{W}/\partial \overline{\mathit{F}}{|}_{\overline{F}}$ (see for instance [2]), we express the components of $\overline{\mathbb{C}}$ in terms of those of $\overline{\mathit{S}}$ and of ${\overline{\mathbb{C}}}^{\mathrm{E}}$: In the last step, the minor symmetry ${\overline{C}}_{mjno}^{\mathrm{E}}={\overline{C}}_{mjon}^{\mathrm{E}}$ has been exploited, and $i,j,k,l=1,2,3$ above and throughout. From this, the inverse relation can be derived. The fact that Green’s strain tensor is frame invariant, i.e., $\overline{\mathit{E}}\left(\overline{\mathit{R}}\overline{\mathit{U}}\right)=\overline{\mathit{E}}\left(\overline{\mathit{U}}\right)$, implies that both the left hand side ${\overline{C}}_{ijkl}^{\mathrm{E}}={\overline{C}}_{ijkl}^{\mathrm{E}}\left(\overline{\mathit{E}}\right)$ and the second Piola–Kirchhoff stress ${\overline{S}}_{lj}={\overline{S}}_{lj}\left(\overline{\mathit{E}}\right)$ are independent of $\overline{\mathit{R}}$. This is in contrast to ${\overline{C}}_{mjnl}={\overline{C}}_{mjnl}\left(\overline{\mathit{R}}\overline{\mathit{U}}\right)$ from which follows that By contraction of the indices i and k with the second index of $\overline{\mathit{F}}$ and the first index of ${\overline{\mathit{F}}}^{\mathsf{T}}$, respectively, Equation (1) follows.

The above changes do not affect the scientific results.