We are developing dynamic atomic force microscopy (AFM) techniques to determine nanoscale elastic properties. Atomic force acoustic microscopy (AFAM) makes use of the resonant frequencies of an AFM cantilever while its tip contacts the sample surface at a given static load. Our methods involve nanosized silicon probes with tip radius R ranging from approximately 10 nm to 150 nm. The resulting radius of contact between the tip and the sample is less than 20 nm. However, the contact stress can be greater than a few tens of GPa, exceeding the theoretical yield strength of silicon by a factor of two to four. Our AFAM experiments indicate that, contrary to expectation, tips can sometimes withstand such stresses without fracture. We subjected ten tips to the same sequence of AFAM experiments. Each tip was brought into contact with a fused quartz sample at different static loads. The load was systematically increased from about 0.4 μN to 6 μN. Changes in tip geometry were observed in images acquired in a scanning electron microscope (SEM) between the individual AFAM experiments. All of the tips with R < 10 nm broke during the first AFAM experiments at static loads less than 1.6 μN. Tips with R > 40 nm plastically deformed under such loads. However, a group of tips with R from 25 nm to 30 nm neither broke nor deformed during the tests. In order to reach higher contact stresses, two additional tips with similar values of R were used in identical experiments on nickel and sapphire samples. Although the estimated stresses exceeded 40 GPa, we did not observe any tip fracture events. Our qualitative observations agree with more systematic studies performed by other groups on various nanostructures. The results emphasize the necessity of understanding the mechanics of nanometer-scaled bodies and the impact of size effects on measurements of mechanical properties on such scales.