An empirical measure of nonlinear strain for soft tissue indentation

Indentation is a primary tool in the investigation of the mechanical properties of very soft tissue such as the brain. However, the usual material characterization protocols are not applicable because the resulting deformation is inhomogeneous, with even the identification of the amount of strain ambiguous and uncertain. Focusing on spherical indentation only, a standard is needed to quantify the amount of strain in terms of the probe radius and displacement so that different indentation experiments can be compared and contrasted. It is shown here that the minimum axial value of the Eulerian logarithmic strain tensor has many desirable properties of such a standard, such as invariance under the choice of material model, and experimental conditions for a given probe displacement. The disadvantage of this measure is that sophisticated finite element techniques need to be used in its determination. An empirical relation is obtained between this strain and the probe radius and displacement to circumvent this problem, and it is shown that this relationship is an excellent predictor of the strain measure. Two essential features of this empirical measure for nonlinear strains are that the exact strain measure for the linear theory is recovered on restriction to infinitesimal deformations and that the simulations use models based on reliable and accurate indentation data obtained from freshly harvested murine brains using a bespoke micro-indentation device.