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

Hongmei Zhang, Yue Wang, Michael F. Insana

Research output: Contribution to journalArticlepeer-review


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 E 0, fractional-order parameter α, and relaxation time constant . 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 R 2 > 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.

Original languageEnglish (US)
Article number025702
JournalMeasurement Science and Technology
Issue number2
StatePublished - Jan 11 2016


  • mechanical properties
  • plate compression
  • ramp-hold relaxation
  • soft polymer emulsion
  • spherical indentation

ASJC Scopus subject areas

  • Instrumentation
  • Engineering (miscellaneous)
  • Applied Mathematics


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