A Load-Controlled Indentation Framework for Characterizing Soft Viscoelastic Materials in Biomechanics

This work develops a load-control indentation method for creep and relaxation testing of soft viscoelastic materials, with direct application to biological tissues. The material is modeled using a modified power-law (MPL) viscoelastic law, which captures the broad-band relaxation behavior observed in tissues and gels, unlike the standard linear solid (SLS) model. The classical Greenwood–Ting auxiliary-time construction, originally derived for SLS materials, is extended to the MPL case to handle decreasing contact radius during unloading under arbitrary load histories. Spherical indentation under a step, triangular ramp, and half-sine load histories are analyzed within the Hertz framework. During intervals where the contact radius increases, the Lee–Radok correspondence principle with the MPL creep function is used. During intervals where the contact radius decreases, a generalized Greenwood–Ting auxiliary-time construction, extended from SLS to MPL materials, is applied. The model predicts the full-time evolution of contact radius, indentation depth, and load–indentation hysteresis. Analytical results are validated against axisymmetric finite-element simulations. The effects of loading rate, relaxation ratio, and power-law exponent are examined to show how the broad-band relaxation spectrum controls hysteresis and energy dissipation. The approach provides a practical and robust basis for load-controlled creep and relaxation indentation tests on soft biological materials where SLS-based models are insufficient.