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

$$\begin{array}{cc}\hfill {\overline{C}}_{ijkl}\left(\overline{\mathit{F}}\right)={\overline{C}}_{ijkl}\left(\overline{\mathit{R}}\overline{\mathit{U}}\right)& =\sum _{m,n=1}^3{\overline{R}}_{im}{\overline{C}}_{mjnl}\left(\overline{\mathit{U}}\right){\overline{R}}_{kn}(i,j,k,l=1,2,3).\hfill \end{array}$$

(1)

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 ${\overline{\mathbb{C}}}^{\mathrm{E}}$:

$$\begin{array}{cc}\hfill {\overline{C}}_{ijkl}& {\displaystyle =\frac{{\displaystyle {\partial }^2\overline{W}}}{{\displaystyle \partial {\overline{F}}_{ij}\partial {\overline{F}}_{kl}}}=\frac{{\displaystyle \partial {\overline{P}}_{ij}}}{{\displaystyle \partial {\overline{F}}_{kl}}}=\sum _{m=1}^3\frac{{\displaystyle \partial {\overline{F}}_{im}{\overline{S}}_{mj}}}{{\displaystyle \partial {\overline{F}}_{kl}}}=\sum _{m=1}^3\left({\displaystyle {\delta }_{ik}{\delta }_{lm}{\overline{S}}_{mj}+{\overline{F}}_{im}\frac{{\displaystyle \partial {\overline{S}}_{mj}}}{{\displaystyle \partial {\overline{F}}_{kl}}}}\right)}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & {\displaystyle ={\delta }_{ik}{\overline{S}}_{lj}+\sum _{m,n,o=1}^3{\overline{F}}_{im}\frac{{\displaystyle \partial {\overline{S}}_{mj}}}{{\displaystyle \partial {\overline{E}}_{no}}}\frac{{\displaystyle \partial {\overline{E}}_{no}}}{{\displaystyle \partial {\overline{F}}_{kl}}}}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\displaystyle ={\delta }_{ik}{\overline{S}}_{lj}+\sum _{m,n,o=1}^3{\overline{F}}_{im}{\overline{C}}_{mjno}^{\mathrm{E}}\frac{{\displaystyle \partial {\overline{E}}_{no}}}{{\displaystyle \partial {\overline{F}}_{kl}}}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & ={\delta }_{ik}{\overline{S}}_{lj}+\sum _{m,p=1}^3{\overline{F}}_{im}{\overline{C}}_{mjpl}^{\mathrm{E}}{\overline{F}}_{kp}.\hfill \end{array}$$

(5)

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

$$\begin{array}{cc}\hfill {\overline{C}}_{ijkl}^{\mathrm{E}}& =-{\left({\overline{U}}^{-2}\right)}_{ik}{\overline{S}}_{lj}+\sum _{m,n=1}^3{\left({\overline{F}}^{-1}\right)}_{im}{\overline{C}}_{mjnl}{\left({\overline{F}}^{-\mathsf{T}}\right)}_{nk}\hfill \end{array}$$

(6)

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

$$\begin{array}{cc}\hfill \sum _{m,n=1}^3{\left({\overline{F}}^{-1}\right)}_{im}{\overline{C}}_{mjnl}\left(\overline{\mathit{R}}\overline{\mathit{U}}\right){\left({\overline{F}}^{-\mathsf{T}}\right)}_{nk}& =\sum _{m,n=1}^3{\left({\overline{U}}^{-1}\right)}_{im}{\overline{C}}_{mjnl}\left(\overline{\mathit{U}}\right){\left({\overline{U}}^{-\mathsf{T}}\right)}_{nk},\hfill \end{array}$$

(7)

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.