Indentation analysis of fractional viscoelastic solids

Abstract

The constitutive differential equations governing the time-dependent indentation response for axisymetric indenters into a fractional viscoelastic half-space are derived, together with indentation creep and relaxation functions suitable for the backanalysis of fractional viscoelastic properties from indentation data. These novel fractional viscoelastic indentation relations include, as a subset, classical integer-type viscoelastic models such as the Maxwell model or Zener model. Using the correspondence principle of viscoelasticty, it is found that the differential order of the governing equations of the indentation response is higher than the one governing the material level. This difference in differential order between the material scale and indentation scale is more pronounced for the viscoelastic shear response than for the viscoelastic bulk response, which translates, into fractional derivatives, the well-known fact that an indentation test is rather a shear test than a hydrostatic test. By way of example, an original method for the inverse analysis of fractional viscoelastic properties is proposed and applied to experimental indentation creep data of polystyrene. The method is based on fitting the time-dependent indentation data (in the Laplace domain) to the fractional viscoelastic model response. Applied to polysterene, it is shown that the particular time-dependent response of this material is best captured by a bulk-and-deviator fractional viscoelastic model of the Zener type.