The inner radius of the spherical container is $a$, and the outer radius is $b$. The internal pressure is $P_1$, while the external pressure is $P_{\mathbf{e}}$. Use the displacement method to determine its stress components is ().(neglecting body forces)

Options:
A. $$
\sigma_{r}=\frac{r^{3} \left( b^{3} - a^{3} \right)} {b^{3} \left( r^{3} - a^{3} \right)} P_{1}-\frac{a^{3} \left( r^{3} - b^{3} \right)} {b^{3} \left( a^{3} - r^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=-\frac{b^{3} \left( r^{3}+2 a^{3} \right)} {a^{3} \left( r^{3} - b^{3} \right)} P_{1}+\frac{a^{3} \left( 2 r^{3} +b^{3} \right)} {a^{3} \left( b^{3} - r^{3} \right)} P_{e}
$$
B. $$
\sigma_{r}=-\frac{b^{3} \left( r^{3}-a^{3} \right)} {r^{3} \left( a^{3}-b^{3} \right)} P_{1}+\frac{a^{3} \left( b^{3}-r^{3} \right)} {r^{3} \left( a^{3}-b^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=\frac{b^{3} \left( 2 r^{3}+a^{3} \right)} {2 r^{3} \left( a^{3}-b^{3} \right)} P_{1}-\frac{a^{3} \left( b^{3}+2 r^{3} \right)} {2 r^{3} \left( a^{3}-b^{3} \right)} P_{e}
$$
C. $$
\sigma_{r}=\frac{b^{3} \left( a^{3}-r^{3} \right)} {r^{3} \left( b^{3}-a^{3} \right)} P_{1} -\frac{a^{3} \left( r^{3}-b^{3} \right)} {r^{3} \left( b^{3}-a^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=-\frac{a^{3} \left( r^{3}+2 b^{3} \right)} {2 r^{3} \left( b^{3}-a^{3} \right)} P_{1}+\frac{b^{3} \left( 2 a^{3}+r^{3} \right)} {2 r^{3} \left( b^{3}-a^{3} \right)} P_{e}
$$
D. $$
\sigma_{r}=\frac{b^{3} \left( a^{3} - r^{3} \right)} {r^{3} \left( b^{3} - a^{3} \right)} P_{1} -\frac{a^{3} \left( r^{3} - b^{3} \right)} {r^{3} \left( b^{3} - a^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=\frac{a^{3} \left( b^{3}-2 r^{3} \right)} {2 r^{3} \left( b^{3}-a^{3} \right)} P_{1}-\frac{b^{3} \left( 2 r^{3}-a^{3} \right)} {2 r^{3} \left( b^{3}-a^{3} \right)} P_{e}
$$
E. $$
\sigma_{r}=\frac{a^{3} \left( r^{3} - b^{3} \right)} {r^{3} \left( a^{3} - b^{3} \right)} P_{1}+\frac{b^{3} \left( a^{3} - r^{3} \right)} {r^{3} \left( a^{3} - b^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=-\frac{r^{3} \left( b^{3}+2 a^{3} \right)} {2 a^{3} \left( b^{3} - r^{3} \right)} P_{1}+\frac{b^{3} \left( 2 r^{3} + a^{3} \right)} {2 a^{3} \left( b^{3} - r^{3} \right)} P_{e}
$$
F. $$
\sigma_{r}=-\frac{r^{3} \left( b^{3}-a^{3} \right)} {a^{3} \left( b^{3} - r^{3} \right)} P_{1}+\frac{2 b^{3} \left( a^{3} - r^{3} \right)} {a^{3} \left( b^{3} - r^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=\frac{a^{3} \left( r^{3}+2 b^{3} \right)} {b^{3} \left( r^{3} - a^{3} \right)} P_{1}-\frac{r^{3} \left( 2 a^{3}+b^{3} \right)} {b^{3} \left( r^{3} - a^{3} \right)} P_{e}
$$
G. $$
\sigma_{r}=\frac{a^{3} \left( b^{3} - r^{3} \right)} {2 b^{3} \left( a^{3} - r^{3} \right)} P_{1}+\frac{b^{3} \left( r^{3} - a^{3} \right)} {2 a^{3} \left( r^{3} - b^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=-\frac{r^{3} \left( b^{3}+2 a^{3} \right)} {2 a^{3} \left( r^{3} - b^{3} \right)} P_{1}+\frac{a^{3} \left( b^{3} + 2 r^{3} \right)} {2 a^{3} \left( r^{3} - b^{3} \right)} P_{e}
$$
H. $$
\sigma_{r}=-\frac{b^{3} \left( a^{3} - r^{3} \right)} {2 r^{3} \left( b^{3} - a^{3} \right)} P_{1} +\frac{a^{3} \left( b^{3} - r^{3} \right)} {2 r^{3} \left( b^{3} - a^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=\frac{r^{3} \left( b^{3}+2 a^{3} \right)} {a^{3} \left( r^{3} - b^{3} \right)} P_{1}-\frac{a^{3} \left( b^{3} + 2 r^{3} \right)} {b^{3} \left( r^{3} - a^{3} \right)} P_{e}
$$
I. $$
\sigma_{r}=-\frac{a^{3} \left( b^{3}-r^{3} \right)} {r^{3} \left( b^{3}-a^{3} \right)} P_{1}-\frac{b^{3} \left( r^{3}-a^{3} \right)} {r^{3} \left( b^{3}-a^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=\frac{a^{3} \left( b^{3}+2 r^{3} \right)} {2 r^{3} \left( b^{3}-a^{3} \right)} P_{1}-\frac{b^{3} \left( 2 r^{3}+a^{3} \right)} {2 r^{3} \left( b^{3}-a^{3} \right)} P_{e}
$$
J. $$
\sigma_{r}=-\frac{b^{3} \left( r^{3} - a^{3} \right)} {a^{3} \left( b^{3}-r^{3} \right)} P_{1}+\frac{a^{3} \left( r^{3} - b^{3} \right)} {a^{3} \left( b^{3} - r^{3} \right)} P_{e}, \quad\sigma_{\theta}=\sigma_{\varphi}=\frac{b^{3} \left( 2 a^{3}-r^{3} \right)} {2 a^{3} \left( b^{3} - r^{3} \right)} P_{1}-\frac{a^{3} \left( r^{3}+2 b^{3} \right)} {2 a^{3} \left( b^{3} - r^{3} \right)} P_{e}
$$
Answer: I