Spherical indentation creep following ramp loading
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Elastic-viscoelastic correspondence, utilizing Boltzmann integral operators, was used to generate displacement–time solutions for spherical indentation testing of viscoelastic materials. Solutions were found for creep following loading at a constant loading rate and compared with step-loading solutions. Experimental creep tests were performed with different loading rate–peak load level combinations on glassy and rubbery polymeric materials. The experimental data were fit to the spherical indentation ramp–creep solutions to obtain values of shear modulus and time-constants; good agreement was found between the experimental results and known modulus values. A multiple ramp-and-hold protocol was examined for the measurement of creep responses at several loads (and depths) within the same test. Emphasis is given to the use of multiple experiments (or multiple levels within a single experiment) to test a priori assumptions made in the correspondence solutions regarding linear viscoelastic material behavior and the creep function.
I. INTRODUCTION
Depth-sensing indentation (DSI, “nanoindentation”) has become a common technique for measuring mechanical behavior of a large variety of materials, including polymers and biological tissues. However, standard elastic-plastic (Oliver–Pharr1) analysis of indentation data obtained with sharp (conical or pyramidal) indentation tips does not always result in accurate results for polymeric materials.2 In particular, elevated estimates of the elastic modulus have been observed due both to viscoelastic creep effects and other less-understood problems associated with the assumed contact mechanics.3 In many cases, direct measurement of the time-dependent viscoelastic behavior of a polymer or tissue is desired, as opposed to techniques that seek to eliminate timedependent effects and isolate the elastic modulus.4 Recent investigations have utilized the elasticviscoelastic correspondence principal within the context of depth-sensing indentation testing.5–8 Key solutions for the viscoelastic spherical indentation problem were presented by Lee and Radok in 19609 and summarized by Johnson.10 The majority of load-controlled indentation studies using viscoelastic correspondence have assumed step-loading creep conditions,6–8 which are analytically convenient but experimentally impossible to implement. The implementation of Boltzmann integral operators
within the context of the viscoelastic correspondence problem allows for analytical solutions to be found for complicated time-based multiple-stage indentation experimental loading conditions, as long as the contact area is non-decreasing.9 Boltzmann integral operator solutions were recently explored for displacement-controlled conical indentation.11 The current investigation was undertaken to establish simple experimental techniques for DSI analysis of timedependent mechanical behavior using load-controlled spherical indentation. Elastic-viscoelastic correspondence analysis9,10 was used to generate solutions for indentation creep at fixed load fol
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