Modeling ramp-hold indentation measurements based on Kelvin-Voigt fractional derivative model

Hongmei Zhang, Qing Zhe Zhang, Litao Ruan, Junbo Duan, Mingxi Wan, Michael F. Insana

Research output: Contribution to journalArticlepeer-review


Interpretation of experimental data from micro- and nano-scale indentation testing is highly dependent on the constitutive model selected to relate measurements to mechanical properties. The Kelvin-Voigt fractional derivative model (KVFD) offers a compact set of viscoelastic features appropriate for characterizing soft biological materials. This paper provides a set of KVFD solutions for converting indentation testing data acquired for different geometries and scales into viscoelastic properties of soft materials. These solutions, which are mostly in closed-form, apply to ramp-hold relaxation, load-unload and ramp-load creep-testing protocols. We report on applications of these model solutions to macro- and nano-indentation testing of hydrogels, gastric cancer cells and ex vivo breast tissue samples using an atomic force microscope (AFM). We also applied KVFD models to clinical ultrasonic breast data using a compression plate as required for elasticity imaging. Together the results show that KVFD models fit a broad range of experimental data with a correlation coefficient typically R 2 > 0.99. For hydrogel samples, estimation of KVFD model parameters from test data using spherical indentation versus plate compression as well as ramp relaxation versus load-unload compression all agree within one standard deviation. Results from measurements made using macro- and nano-scale indentation agree in trend. For gastric cell and ex vivo breast tissue measurements, KVFD moduli are, respectively, 1/3-1/2 and 1/6 of the elasticity modulus found from the Sneddon model. In vivo breast tissue measurements yield model parameters consistent with literature results. The consistency of results found for a broad range of experimental parameters suggest the KVFD model is a reliable tool for exploring intrinsic features of the cell/tissue microenvironments.

Original languageEnglish (US)
Article number035701
JournalMeasurement Science and Technology
Issue number3
StatePublished - Feb 15 2018


  • AFM measurements
  • KVFD models
  • hydrogels
  • ramp-hold indentation
  • tissue samples

ASJC Scopus subject areas

  • Instrumentation
  • Engineering (miscellaneous)
  • Applied Mathematics


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