Modified flat-punch model for hyperelastic polymeric and biological materials in nanoindentation

Nanoindentation can characterize in-situ elastic modulus E of an object by pressing an indenter into the sample surface and fitting the detected data to a contact equation. In this work, we found the conventional contact-mechanism theory resulted in a high uncertainty of E for the hyperelastic materials, including some polymers and biological cells. The evaluated E displayed an exponential decrease with increasing indent distance when fitting to Hertz model and caused high E variance as a function of indent depth. To obtain a reliable E of those specimens, a new equation for E computation directly adopting the mechanical behavior of the sample was proposed. Indenting on hyperelastic polydimethylsiloxane (PDMS), we observed linear force-displacement curves and used its power-law for the selection of the correct equation. The flat-punch model was thus chosen and showed constant E independent of the indent size, which meant the indent depth in this paper. After eliminating the depth effect on E, we referred the nanoindentation results to the bulk E of PDMS for the revision of the flat-punch model. A new equation was generated and displayed the improvement on not only the precision (remove depth effect) but also the accuracy (compare to compression test) of E for PDMS. The suitability of the modified flat-punch model for hyperelastic material implied the practical deformational mechanism different from the general idea. Applied on microbial samples, our new equation characterized two bacteria and showed consistent results with their membrane structures. In conclusion, we suggest the modified flat-punch model improves the description of mechanical behaviors and derived the correct E for hyperelastic materials.