Shell theory comparison
| Linear KL | Koiter | Reissner–Mindlin | Naghdi | |
|---|---|---|---|---|
| DOFs/node | 3 ($u_1,u_2,u_3$) | 3 ($u_1,u_2,u_3$) | 5 ($u_1,u_2,u_3,\varphi_1,\varphi_2$) | 5 ($u_1,u_2,u_3,\varphi_1,\varphi_2$) |
| Director | implicit: $\mathbf{n} = \mathbf{a}_1\times\mathbf{a}_2/|\cdot|$ | implicit: $\mathbf{n} = \mathbf{a}_1\times\mathbf{a}_2/|\cdot|$ | additive: $\mathbf{d} = \mathbf{G}_3+\varphi_1\mathbf{T}_1+\varphi_2\mathbf{T}_2$, $|\mathbf{d}|\neq 1$ | Rodrigues: $\mathbf{d} = \cos|\varphi|\,\mathbf{G}_3+\mathrm{sinc}|\varphi|(\varphi_1\mathbf{T}_1+\varphi_2\mathbf{T}_2)$, $|\mathbf{d}|=1$ |
| Membrane strain | linear: $\tfrac{1}{2}(\mathbf{A}_\alpha\cdot\mathbf{u}_{,\beta}+\mathbf{A}_\beta\cdot\mathbf{u}_{,\alpha})$ | Green–Lagrange: $\tfrac{1}{2}(a_{\alpha\beta}-A_{\alpha\beta})$ | linear: $\tfrac{1}{2}(\mathbf{A}_\alpha\cdot\mathbf{u}_{,\beta}+\mathbf{A}_\beta\cdot\mathbf{u}_{,\alpha})$ | Green–Lagrange: $\tfrac{1}{2}(a_{\alpha\beta}-A_{\alpha\beta})$ |
| Bending strain | $\kappa_{\alpha\beta} = -u_{3,\alpha\beta}$ (flat ref.) | $\kappa_{\alpha\beta} = b_{\alpha\beta}-B_{\alpha\beta}$ | $\tfrac{1}{2}(\mathbf{A}_\alpha\cdot\mathbf{d}_{,\beta}+\mathbf{A}_\beta\cdot\mathbf{d}_{,\alpha})-B_{\alpha\beta}$ | $\tfrac{1}{2}(\mathbf{a}_\alpha\cdot\mathbf{d}_{,\beta}+\mathbf{a}_\beta\cdot\mathbf{d}_{,\alpha})-B_{\alpha\beta}$ |
| Transverse shear | $\gamma=0$ (Kirchhoff) | $\gamma=0$ (Kirchhoff) | $\gamma_\alpha = \mathbf{A}_\alpha\cdot\mathbf{d}$ | $\gamma_\alpha = \mathbf{a}_\alpha\cdot\mathbf{d}$ |
| Finite rotations | no | yes | small only ($|\varphi|\ll 1$) | yes |
| C¹ for bending | yes | yes | no | no |
| In FerriteShells | — | ✓ _KL functions | — | ✓ _RM functions |
The key distinction between RM and Naghdi is which base vectors appear in the strain measures: RM uses the reference base vectors $A_\alpha$ (linearised around the reference configuration), while Naghdi replaces them with the current $a_\alpha$ everywhere, giving fully nonlinear strains. The director parametrisation (non-unit additive vs unit Rodrigues) is a separate but related choice — in practice the two always appear together.
Koiter has no director DOFs; the normal is always implicit from the surface geometry, so the Kirchhoff constraint (zero shear) is built in and C¹ continuity is required for bending.
Classical RM (additive director, $\|\mathbf{d}\|\neq 1$) is not implemented; the _RM functions go directly to the geometrically exact Naghdi form via Rodrigues parametrisation.