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Published in:
19-11-2019 | Correction
Correction to: Isogeometric hyperelastic shell analysis with out-of-plane deformation mapping
Published in: Computational Mechanics | Issue 1/2020
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Excerpt
In the original publication [1], a term was missing in Eq. (99). The correct form is
$$\begin{aligned} \delta \overline{\mathbf {g}}^\gamma&= \left( { \overline{g}^{\gamma \delta } \mathbf {n}\mathbf {n}} - \overline{\mathbf {g}}^{\delta }\overline{\mathbf {g}}^{\gamma } \right) \cdot \delta \overline{\mathbf {g}}_\delta . \end{aligned}$$
(99)
$$\begin{aligned} \delta _a \delta _b \overline{\kappa }_{\alpha \beta }&= \left( \delta _a \overline{\pmb {\varGamma }}_{\alpha \beta } - \left( \overline{\pmb {\varGamma }}_{\alpha \beta } \cdot \overline{\mathbf {g}}^\gamma \right) \frac{\partial \delta _a \overline{\mathbf {x}}}{\partial \xi ^\gamma } \right) \cdot \overline{\mathbf {g}}^\gamma \left( \mathbf {n}\cdot \delta _b \overline{\mathbf {g}}_\gamma \right) \nonumber \\&\quad + \delta _a \overline{\mathbf {g}}_\gamma \cdot \mathbf {n}\left( \overline{\mathbf {g}}^\gamma \cdot \delta _b \overline{\pmb {\varGamma }}_{\alpha \beta } + \overline{\pmb {\varGamma }}_{\alpha \beta } \right. \nonumber \\&\quad \left. \cdot \left( { \overline{g}^{\gamma \delta } \mathbf {n}\mathbf {n}} - \overline{\mathbf {g}}^{\delta } \overline{\mathbf {g}}^{\gamma } \right) \cdot \delta _b \overline{\mathbf {g}}_\delta \right) \end{aligned}$$
(97)
$$\begin{aligned}&= \left( \delta _a \overline{\pmb {\varGamma }}_{\alpha \beta } - \left( \overline{\pmb {\varGamma }}_{\alpha \beta } \cdot \overline{\mathbf {g}}^\gamma \right) \frac{\partial \delta _a \overline{\mathbf {x}}}{\partial \xi ^\gamma } \right) \cdot \overline{\mathbf {g}}^\gamma \left( \mathbf {n}\cdot \delta _b \overline{\mathbf {g}}_\gamma \right) \nonumber \\&\quad + \left( \delta _b \overline{\pmb {\varGamma }}_{\alpha \beta } - \left( \overline{\pmb {\varGamma }}_{\alpha \beta } \cdot \overline{\mathbf {g}}^{\delta } \right) \frac{\partial \delta _b \overline{\mathbf {x}}}{\partial \xi ^\delta } \right) \cdot \overline{\mathbf {g}}^\gamma \left( \mathbf {n}\cdot \delta _a \overline{\mathbf {g}}_\gamma \right) \nonumber \\&\quad +\left( \mathbf {n}\cdot \delta _a \overline{\mathbf {g}}_\gamma \right) \overline{b}_{\alpha \beta } \overline{g}^{\gamma \delta } \left( \mathbf {n}\cdot \delta _b \overline{\mathbf {g}}_\delta \right) . \end{aligned}$$
(98)