I'm development a physics model that require a expression for elongation of a elastic material, $\lambda=\frac{L}{L_o}$ [where $L$ is the thickness of the material and $L_o \equiv L(\sigma = 0)$] as a function of the stress, $\sigma$, for uniaxial compression. Mooney-Rivlin model fits very well my experimental data $ \sigma$ vs. $\lambda$:
$\sigma=\left( 2C_1 + \frac{2C_2}{\lambda} \right )\left( \lambda ^2- \frac{1}{\lambda } \right)$
But I need $\lambda=\lambda(\sigma)$.
I think that this require to solve de equation
$0= 2C_1\lambda^4 + 2C_2\lambda^3+\sigma\lambda^2-2C_1\lambda-2C_2$
This 4th grade polynomic equation has exact analytical solution, but that solution implies enormous expressions.
I only want a aproximate (analytical) solution for that equation.
Is a mathematica reason for wich the data $\lambda$ vs $\sigma$ is fitted very well with a approximate solution $\lambda(\sigma)\approx 1+ \alpha \sigma^{1/4}+\beta\sigma^{1/3}$ ?
Best regards!