Solutions to ramp-hold dynamic oscillation indentation tests for assessing the viscoelasticity of hydrogel by Kelvin-Voigt fractional derivative modeling

Hongmei Zhang, Tongqing Lu, Qingzhe Zhang, Yan Zhou, Hongrui Zhu, Joseph Harms, Xiaofeng Yang, Mingxi Wan, Michael F. Insana

Research output: Contribution to journalArticlepeer-review


Indentation type dynamic mechanical analysis (IT-DMA) is an attractive approach to characterizing viscoelastic properties of soft matter under both quasi-static and dynamic loadings. Frequency-dependent viscoelastic properties are usually characterized by storage and loss moduli whereas extracting inherent viscoelastic model parameters from dynamic loadings remains unsolved. In this study, closed-form solutions to dynamic oscillation loadings were first derived and adapted to a variety of excitations for arbitrary shape indenters. A uniform theoretical framework for ramp-hold-oscillation-relaxation/creep indentation tests are established and Kelvin-Voigt fractional derivative (KVFD) solutions are presented to predict the viscoelastic behavior for both quasi-static loadings (creep, relaxation) and dynamic loadings (sinusoidal displacement/force cycles at 5,10,15,20,30 Hz). Four gelatin-cream hydrogel samples were investigated under ramp-hold-oscillation-relaxation and ramp-hold-oscillation-creep testing conditions. Viscoelastic quantities of elasticity E0, fluidity α, and viscosity τ were extracted by fitting the experimental curves to KVFD predictions. The results show that KVFD models can accurately depict the viscoelastic behavior under both dynamic and quasi-static loadings with high goodness of fit (R2>0.936). To assess the proposed method, comparison of the viscoelastic quantities from different experimental protocols were conducted. Statistical results revealed viscoelastic parameters from different loading are in close agreement. For dynamic loadings, significant differences were found for [E0, α] estimates at different frequencies. For different experimental protocols, estimations were consistent in that E0 showed maximum value on Gel5Cream15 sample, and α was increased with increasing cream percentage. Moreover, viscoelastic quantities extracted from dynamic relaxation were correlated with those of the dynamic creep (r = 0.977, p < 0.001 for E0, and r = 0.969, p < 0.001 for α). Viscoelastic quantities from quasi-static relaxation were highly correlated with those from quasi-static creep (r = 0. 910, p < 0.001 for E0, and r = 0.979, p < 0.001 for α). The complex modulus from KVFD estimations was found to be correlated with that of the DMA measurement (for dynamic relaxation, r = 0.603, p < 0.001 for E′ and r = 0.818, p < 0.001 for E″; for dynamic creep, r = 0.756, p < 0.001 for E′ and r = 0.762, p < 0.001 for E″). Cross validation and assessment by alternative DMA method confirmed that KVFD solutions provided a method for relating viscoelastic measurements at load frequencies from quasi-static to dynamic for soft viscoelastic tissue-like materials.

Original languageEnglish (US)
Article number103431
JournalMechanics of Materials
StatePublished - Sep 2020


  • Hydrogel
  • Indentation
  • Kelvin-Voigt Fractional Derivative (KVFD) modeling
  • Ramp-hold-oscillation-relaxation/creep
  • Viscoelasticity

ASJC Scopus subject areas

  • Materials Science(all)
  • Instrumentation
  • Mechanics of Materials

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