The total differential of the second Piola–Kirchhoff stress tensor is
$$\begin{aligned} \mathrm {d}{\mathbf {S}}&= \frac{\partial {\mathbf {S}}}{\partial {\mathbf {E}}}: \mathrm {d}{\mathbf {E}}\end{aligned}$$
(61)
$$\begin{aligned}&= \frac{\partial S^{IJ}}{\partial E_{KL}} {\mathbf {G}}_{I}{\mathbf {G}}_{J} {\mathbf {G}}_{K} {\mathbf {G}}_{L} : \mathrm {d}E_{M N} {\mathbf {G}}^{M} {\mathbf {G}}^{N} \end{aligned}$$
(62)
$$\begin{aligned}&= {\mathbb {C}}^{IJKL} {\mathbf {G}}_{I}{\mathbf {G}}_{J} \mathrm {d}E_{K L}, \end{aligned}$$
(63)
where
\(I, J, K, L, M, N = 1, \ldots , {n_{\mathrm {sd}}}\). From Eq. (
4), the following expression can be used:
$$\begin{aligned} {\mathbb {C}}^{IJKL}&= 2 \frac{\partial S^{IJ}}{\partial C_{KL}}. \end{aligned}$$
(64)
For shells,
$$\begin{aligned} \mathrm {d}S^{IJ} = {\mathbb {C}}^{IJ\gamma \delta } \mathrm {d}E_{\gamma \delta } + {\mathbb {C}}^{IJ 33} \mathrm {d}E_{33}, \end{aligned}$$
(65)
because
\(\mathrm {d}E_{3\alpha } = \mathrm {d}E_{\alpha 3} = 0\). From Eq. (
65), we can write
$$\begin{aligned} \mathrm {d}S^{33} = {\mathbb {C}}^{33\gamma \delta } \mathrm {d}E_{\gamma \delta } + {\mathbb {C}}^{33 33} \mathrm {d}E_{33}, \end{aligned}$$
(66)
and from Eqs. (
4) and (
66), we can write
$$\begin{aligned} \mathrm {d}S^{33} = \frac{1}{2}{\mathbb {C}}^{33\gamma \delta } \mathrm {d}C_{\gamma \delta } + \frac{1}{2}{\mathbb {C}}^{33 33} \mathrm {d}C_{33}. \end{aligned}$$
(67)
From the plane stress condition
\(S^{33} = 0\),
\(\mathrm {d}S^{33} = 0\), and consequently
$$\begin{aligned} \mathrm {d}E_{33}&= - \frac{{\mathbb {C}}^{33\gamma \delta }}{{\mathbb {C}}^{33 33}} \mathrm {d}E_{\gamma \delta }, \end{aligned}$$
(68)
which makes
$$\begin{aligned} \mathrm {d}S^{\alpha \beta }&= {\mathbb {C}}^{\alpha \beta \gamma \delta } \mathrm {d}E_{\gamma \delta } - \frac{ {\mathbb {C}}^{\alpha \beta 33} {\mathbb {C}}^{33\gamma \delta } }{{\mathbb {C}}^{33 33}} \mathrm {d}E_{\gamma \delta }, \end{aligned}$$
(69)
and therefore we introduce
$$\begin{aligned} \hat{{\mathbb {C}}}^{\alpha \beta \gamma \delta }&= {\mathbb {C}}^{\alpha \beta \gamma \delta } - \frac{{\mathbb {C}}^{\alpha \beta 3 3 }{\mathbb {C}}^{33\gamma \delta }}{{\mathbb {C}}^{33 33}}. \end{aligned}$$
(70)
In computing
\(C_{33}\), we have different methods for incompressible and compressible materials. In the case of incompressible material, from Eq. (
49) we can write
\(C_{33} = \lambda _3^2\). Because
\(J = 1\),
$$\begin{aligned} \lambda _3&= \frac{1}{A / A_0}, \end{aligned}$$
(71)
and therefore
$$\begin{aligned} C_{33}&= \frac{A_0^2}{A^2}. \end{aligned}$$
(72)
In the case of compressible material, as can be found in [
32],
\(C_{33}\) can be calculated by Newton–Raphson iterations that would make
\(S^{33} = 0\). Because
\(C_{\gamma \delta }\) does not change during the iterations, the iteration increment is
$$\begin{aligned} \varDelta C_{33}^i&= - \frac{\left( S^{33}\right) ^i}{\left( \mathrm {d}S^{33}/ \mathrm {d}C_{33} \right) ^i}. \end{aligned}$$
(73)
From Eq. (
67) and remembering that
\(\mathrm {d}C_{\gamma \delta } = 0\) during the iterations,
$$\begin{aligned} \varDelta C_{33}^i&= - 2 \frac{\left( S^{33}\right) ^i}{\left( {\mathbb {C}}^{3333}\right) ^i}. \end{aligned}$$
(74)
The update takes place as
$$\begin{aligned} C_{33}^{i+1}&= C_{33}^i + \varDelta C_{33}^i, \end{aligned}$$
(75)
where superscript
i is the iteration counter, and as the initial guess we have the following three options:
$$\begin{aligned} C_{33}^{0}&= 1, \end{aligned}$$
(76)
$$\begin{aligned} C_{33}^{0}&= \frac{A^2_0}{A^2}, \end{aligned}$$
(77)
$$\begin{aligned} C_{33}^{0}&= \frac{1}{2} g_{\alpha \beta } G^{\alpha \beta }. \end{aligned}$$
(78)
The option given by Eq. (
78) comes from the constitutive law for zero bulk modulus. To preclude
\(C_{33}\) being negative, we introduce an alternative update method based on the logarithm of
\(C_{33}\):
$$\begin{aligned} \ln C_{33}^{i+1}&= \ln C_{33}^{i} + \frac{\mathrm {d}\ln C_{33}}{\mathrm {d}C_{33}} \varDelta C_{33}^{i}\end{aligned}$$
(79)
$$\begin{aligned}&= \ln C_{33}^{i} + \frac{\varDelta C_{33}^{i}}{C_{33}^i}. \end{aligned}$$
(80)