Methods for estimation of the complex modulus generally produce data from which discrete results can be obtained for a set of frequencies. As these results are normally afflicted by noise, they are not necessarily consistent with the principle of causality and requirements of thermodynamics. A method is established for noise-corrected estimation of the complex modulus, subject to the constraints of causality, positivity of dissipation rate and reality of relaxation function, given a finite set of angular frequencies and corresponding complex moduli obtained experimentally. Noise reduction is achieved by requiring that two self-adjoint matrices formed from the experimental data should be positive semidefinite. The method provides a rheological model that corresponds to a specific configuration of springs and dashpots. The poles of the complex modulus on the positive imaginary frequency axis are determined by a subset of parameters obtained as the common positive zeros of certain rational functions, while the remaining parameters are obtained from a least squares fit. If the set of experimental data is sufficiently large, the level of refinement of the rheological model is in accordance with the material behavior and the quality of the experimental data. The method was applied to an impact test with a Nylon bar specimen. In this case, data at the 29 lowest resonance frequencies resulted in a rheological model with 14 parameters. The method has added improvements to the identification of rheological models as follows: (1) Noise reduction is fully integrated. (2) A rheological model is provided with a number of elements in accordance with the complexity of the material behavior and the quality of the experimental data. (3) Parameters determining poles of the complex modulus are obtained without use of a least squares fit.
Noise-Corrected Estimation of Complex Modulus in Accord With Causality and Thermodynamics: Application to an Impact Test
Manuscript received June 29, 2011; final manuscript received June 27, 2012; accepted manuscript posted July 6, 2012; published online November 19, 2012. Assoc. Editor: Krishna Garikipati.
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Collet, P., Gary, G., and Lundberg, B. (November 19, 2012). "Noise-Corrected Estimation of Complex Modulus in Accord With Causality and Thermodynamics: Application to an Impact Test." ASME. J. Appl. Mech. January 2013; 80(1): 011018. https://doi.org/10.1115/1.4007081
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