I tried many formulas to find the components of the second Piola Kirchhoff. I need help to derive equations 27-29 on reference 1.
[![enter image description here][1]][1]
we have
$$
\mathbf{C}=C_{11} \mathbf{T} \otimes \mathbf{T}+C_{12}(\mathbf{T} \otimes \mathbf{N}+\mathbf{N} \otimes \mathbf{T})+C_{22} \mathbf{N} \otimes \mathbf{N}+C_{33} \mathbf{E}_{3} \otimes \mathbf{E}_{3} \tag{16}
$$
Here, the components $C_{11}$ and $C_{12}$ are expressed in terms of the kinematic quantities of deformation, namely $\left\{u_{1}, u_{2}, \psi, \theta, \kappa_{0}, \mathbf{T}, \mathbf{N}\right\}$, in the following form:
$$
C_{11} \approx \mathbf{t} \cdot \mathbf{t}+2 z\left(\mathbf{t} \cdot \mathbf{b}+\kappa_{0} \mathbf{t} \cdot \mathbf{t}\right), C_{12} \approx \Gamma=\mathbf{t} \cdot \mathbf{n} \tag{17}
$$
we have
.\begin{array}{r}
I_{1}=\operatorname{tr} \mathrm{C}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2} \\
I_{2}=\frac{1}{2}\left(I_{1}^{2}-\operatorname{tr} \mathbf{C}^{2}\right)=\lambda_{1}^{2} \lambda_{2}^{2}+\lambda_{2}^{2} \lambda_{3}^{2}+\lambda_{3}^{2} \lambda_{1}^{2} \\
I_{3}=\operatorname{det} \mathbf{C}=\left(\lambda_{1} \lambda_{2} \lambda_{3}\right)^{2} \tag{20}
\end{array}
consider the neo-Hookean material model with the strain energy density function
$$
W=\frac{\mu}{2}\left(I_{1}-3\right),
$$
where $\mu$ is the shear modulus. From Eq. (23), it follows that
$$
\mathbf{S}=\mu \mathbf{I}-P \mathbf{C}^{-1}
$$
With the aid of Eqs. (16), $(20)_{3}$ and (26), the plane stress assumptions $S_{22}=S_{33}=0$ and the incompressibility constraint $I_{3}=1$, one will have
$$
\left.\begin{array}{r}
S_{22}=\mu-\left(C_{11} C_{22}-\Gamma^{2}\right)^{-1} C_{11} P=0 \\
S_{33}=\mu-C_{33}^{-1} P=0 \\
I_{3}=\left(C_{41} C_{22}-\Gamma^{2}\right) C_{33}=1
\end{array}\right\} \tag{27}
$$
This provides a system of three nonlinear algebraic equations in $\left\{C_{22}, C_{33}, P\right\}$. The exact solution of Eqs. (27) is given by
$$
C_{22}=\frac{\Gamma^{2}+\sqrt{C_{11}}}{C_{11}}, C_{33}=\frac{1}{\sqrt{C_{11}}}, P=\frac{\mu}{\sqrt{C_{11}}} \tag{28}
$$
It is worthwhile to note that the solution obtained for $C_{33}$ satisfies the condition $C_{33}=\lambda_{3}^{2}>0$. Now, the nonzero components of the second Piola-Kirchhoff stress $S$ in the Frenet frame become
$$
S_{11}=\mu \frac{C_{11}^{2}-\Gamma^{2}-\sqrt{C_{11}}}{C_{11}^{2}}, \quad S_{12}=\mu \frac{\Gamma}{C_{11}} \tag{29}
$$
[1]: https://i.stack.imgur.com/CwsMm.png
References
- Dadgar-Rad, F., & Sahraee, S. (2021). Large deformation analysis of fully incompressible hyperelastic curved beams. Applied Mathematical Modelling, 93, 89-100.
