Indentation test of soft tissues with curved substrates: a finite element study

Indentation is a commonly used approach to measure the mechanical properties of soft tissues, such as articular cartilage and limb tissues. The Young's modulus of tissue can be calculated from the indentation test using a mechanical model, where the soft tissue is normally assumed to have a flat substrate. In this study, a series of 2D finite element models were established to investigate the effects of bones with various curvatures embedded in the soft tissues during an indentation test. For each curvature of the hard substrate, the errors in the calculation of the Young's modulus were estimated for different indentation depths (0-10%) and aspect ratios a/h of the indentor diameter and the tissue thickness (0.2-2 in seven steps). The radius ratio a/R of the indentor and the curved substrate ranged from 0 to 0.38 in nine steps. Results showed that the error in calculation of the Young's modulus increased by 21.2% when the curvature a/R of the bone increased from 0 to 0.38 (under the condition of a/h=1.0, Poisson's ratio v=0.45). The error increased from 6.0 to 18.2% when the tissue thickness increased from 0.2 to 2 (a/R=0.18, v=0.45). It was found that the error in the Young's modulus calculation caused by the curved hard substrates could be corrected by a single factor for different indentation depths. This factor depends on the Poisson's ratio, the aspect ratio a/h and the radius ratio a/R.