From my understanding, there is no set dimensions for mechanical characterization. In general, the dimensions you use should be similar to the size and loading conditions of the structure when deployed for application purposes.
As explained above, there are usually no limitations except for the clamp you may use. In this aspect, depending on the clamp and the analysis you want to perform, it will be the sample dimensions. Also, it should be noted that is recommended to respect some relations of Length, Width and Thickness. In this aspect, I recommend you a file from TA Instruments which can be downloaded from Google in which you can find a detailed explanation of what I mention. This can be found just by searching "DMTA TA Instruments PDF". Either way, I just attached here.
From a more general point of view, what you are asking for is generally called a "MRV", as in Minimum Representative Volume.
That is, the minimal volume you can work upon and have a representative result for bigger scales.
I can't help you with your specific issue but when MRV is not known, it should be researched : measure whatever you are measuring on different volumes of growing dimensions and look for when the result stabilizes. This would be your MRV for THAT measurement (not necessarily for others).