Yes, the area enclosed within the hysteresis curve is a measure of the damping coefficient. That will be a fairly accurate way to determine damping if damping is the dominant dissipative mechanism (which, for the kind of experiment you have in mind, it should be). I can not off the top of my head recall the relationship between the damping coefficient and the energy loss per cycle, though i suspect it will be strain rate dependant.
If you have a strain gauge and an accelerometer, you can plot the force vs velocity. The slope, in the linear region is the damping coefficient. There is another possibility, but is quite instrumentation intensive.
If you drive (harmonically) the isolator at some frequency \omega with a mass M>>m, (m is the mass of the isolator) supported by the isolator, then the ratio of the response amplitude to the driving amplitude, called the transmissibility, is given by :
K(\omega ) = \frac {x}{X} = \sqrt { \frac {\omega _0 ^4 + 4 \gamma ^2 \omega ^2}{(\omega _0 ^2 - \omega ^2)^2 + 4 \gamma ^2 \omega ^2}
where \omega _0 = \sqrt {k/M} is the natural frequency of the isolator panel (if k - the spring constant - is not known, a simple load-extension test will tell you what it is) amd \gamma = c/2M, where c is the damping coefficient.
So, by measuring K(w) at a range of frequencies, you can get a pretty good value for c. However, this requires a function generator, some appropriate kind of transducer to induce harmonic oscillations, an accelerometer, and a spectrum analyzer (and very likely a diff amp).
While I'm not certain what the ASTM recommended procedure involves, I know that the standard test is : E756-04 (Standard Test Method for Measuring Vibration-Damping Properties of Materials - developed by Committee E33.03 on Sound Transmission). However, this test only measures the loss factor \eta, which is in fact, the area under the hysteresis curve (I believe). I don't know if it actually tells you the damping coefficient. Would it be sufficient for you to know the loss factor ? The description of the test is not available online unless you choose to buy it from ASTM for about $40 or so. Perhaps someone has the ASTM Testing Handbook nearby and can look it up.
