The frequency is a function of the dimensions of the bar and its Young's modulus.
You need to know what mode of oscillation you are exciting in your bar - there is a hug difference between the flexural and longitudinal modes.
If the rod is bending, you can find the formulas here. The derivation goes on and on... but you should be able to use the formula on the first page (for free-free):
$$f = \frac{1}{2\pi}\left(\frac{22.373}{L^2}\right)\sqrt{\frac{EI}{\rho}}$$
In this formula, $I$ is the second moment of area of the rod - see the wiki article for an explanation and to find the appropriate value for the shape of your rod.
If you have a higher mode, you can find the position of two fixed nodes and use the fixed-fixed equation instead.
And if you have longitudinal vibration, the answer is much simpler - you just have to look at the transit time of the sound wave from end to end. One round trip corresponds to the fundamental frequency, so
$$f = \frac{v}{2L} = \sqrt{\frac{E}{4L^2\rho}}\\
E = \rho\; \left(\;2\;L\;f\;\right)^2$$
$A \approx \frac{W_0*L_0}{L}*\frac{T_0*L_0}{L}=\frac{W_0*T_0*L_0^2}{L^2}$
If my interpretation is correct, you are assuming that $W*L=W_0*L_0$ and $T*L=T_0*L_0$
That would make the volume: $W_0*T_0*L_0^2/L$, which decreases when you stretch the material. For small strain, the Poisson ratio would approach 1. Poisson ratio should be between -1 and 0.5 for a stable, isotropic, linear elastic material. I don't know what the material is you're testing, but..
$A \approx \frac{A_0*L_0}{L}$ seems more suitable imo.
Best Answer
Hydrogels can have a Young's modulus as low as 5-10kPa easily (see for example gels of 1-2 percent agarose or alginate). You can probably get close to 1 kPa or less by lowering the percentage of agarose or alginate, but the gel may be prone to tearing and difficult to handle. You would probably need to experiment as these hydrogels are often used in biomedical applications, in which a minimum of physical integrity is required. Therefore, people usually use gels with larger Young's modulus than 1 kPa.