Surface effects on the spherical indentation of biological film/substrate structures

Micro-/nano-indentation has been the most popular technique to extract the mechanical characteristics of biological cells and tissues. However, due to surface effects and the existence of substrates, conventional contact models are unable to determine the accurate elastic modulus of biological samples by analyzing the measured load-indent depth data. In this study, the spherical indentation of the film/substrate structure considering the surface energy and large deformations is investigated. The hyperelasticity of biological films and substrates is considered through neo-Hookean constitutive model, and the surface effect is incorporated using the finite element method. The explicit formulas for the relations between load and indent depth are presented for films with two orders of magnitude modulus mismatch to the substrate. It is found that the modulus mismatch between film and underlying substrate would lead to an overestimation of modulus for the film on a stiffer substrate, but an underestimated modulus for that on a softer substrate if the conventional Hertzian theory is directly adopted in the analysis. Moreover, for indentation at micro-/nano-scale, the surface energy would pronouncedly reduce the indent depth under a given load and lead to a seemingly stiffer film. Our results provide the explicit expressions to accurately predict the spherical indentation response of biological film/substrate structures.