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Investigating the behavior of fractional viscoelastic materials

Fractional calculus is an old mathematical topic; yet, its application in physics and engineering is more recent. In viscoelasticity and hereditary solid mechanics fractional order derivatives have been used to describe the system behavior. From a mathematical perspective, fractional derivatives are an extension of ordinary derivatives, and possess mathematical definitions and properties that stem from ordinary derivatives. In this particular project, we focus on the constitutive differential equations governing the time-dependent indentation response for axisymetric indenters into a fractional viscoelastic half-space. The indentation creep and relaxation functions suitable for the back analysis of fractional viscoelastic properties from indentation data are studied. Using the correspondence principle of viscoelasticty, we can find and compare the differential order of the governing equations of the indentation response with the one governing the material level. The 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 the well known fact that an indentation test is rather a shear test than a hydrostatic test. Additionally, 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.