Ramp-hold relaxation solutions for the KVFD model applied to soft viscoelastic media

The standard step-hold load-relaxation profile can yield variable estimates of mechanical properties due to the difficulty in achieving a step strain experimentally. A ramp-hold profile overcomes this limitation if appropriate model functions can be derived. Utilizing Boltzmann hereditary integral operators for two indentation geometries, analytical ramp solutions for load-relaxation were developed based on the Kelvin-Voigt fractional derivative (KVFD) model. The results identify three model parameters for characterizing viscoelastic behavior from a single model curve fit to the data: the elastic modulus E₀, fractional-order parameter α, and relaxation time constant τ. The quantitative nature of the analysis was validated through measurements on gelatin emulsion samples exhibiting viscoelastic behavior. KVFD-model-based solutions provide mathematically simple and experimentally flexible descriptions of load-relaxation behavior for a range of viscoelastic properties and experimental conditions; e.g. one closed-form solution can fit the ramp and the hold phases of the relaxation time series. Experiments show that the solution for a spherical indenter and plate compressor each fit well to the corresponding experimental relaxation curves with a coefficient of determination R² > 0.98. Parameters obtained from the spherical-tip indentation and plate-compression geometries agree within one standard deviation, confirming that the ramp solution based KVFD model yields consistent measurements for characterizing viscoelastic materials.