# Derivation of the second Piola-Kirchhoff tensor

I tried many formulas to find the components of the second Piola Kirchhoff. I need help to derive equations 27-29 on reference 1.

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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 $$S_{11}=\mu \frac{C_{11}^{2}-\Gamma^{2}-\sqrt{C_{11}}}{C_{11}^{2}}, \quad S_{12}=\mu \frac{\Gamma}{C_{11}} \tag{29}$$ : https://i.stack.imgur.com/CwsMm.png

### References

1. Dadgar-Rad, F., & Sahraee, S. (2021). Large deformation analysis of fully incompressible hyperelastic curved beams. Applied Mathematical Modelling, 93, 89-100.
• Please, use Mathjax to type the relevant parts of the paper to your question. Jun 23, 2022 at 13:17
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Jun 23, 2022 at 15:58
• Does the following question help? scicomp.stackexchange.com/q/40706/36539
– NNN
Jun 24, 2022 at 1:53
• my question is clear how they find equations 27-29 i need the detail Jun 25, 2022 at 23:29