Topics in Modal Analysis II, Volume 8

30 Estimating Frequency-Dependent Mechanical Properties of Materials 323 30.6 Summary A frequency-dependent framework was developed and implemented to describe one-dimensional stress wave propagation through a modified split Hopkinson bar. Experimental data from a modified split Hopkinson pressure bar is used in conjunction with this model to estimate the complex modulus of polyurethane. The estimates of the complex modulus are obtained for three different parameterizations of the complex modulus as a function of frequency, specifically constant, linear, and quadratic models. Results of the estimation are discussed and compared. Acknowledgements The authors would like to thank AFOSR (Program Manager: Dr. David Stargel) and the Air Force Research Laboratory for supporting this research effort. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Air Force. References 1. Doyle JF (1997) Wave propagation in structures: spectral analysis using fast discrete fourier transforms. Springer, New York 2. Nunziato JW, Sutherland HJ (1973) Acoustical determination of stress relaxation functions for polymers. J Appl Phys 44(1):184–187 3. Lundberg B, Ödeen S (1993) In situ determination of the complex modulus from strain measurements on an impacted structure. J Sound Vib 167(3):413–419 4. Ahonsi B, Harrigan JJ, Aleyaasin M (2012) On the propagation coefficient of longitudinal stress waves in viscoelastic bars. Int J Impact Eng 45:39–51 5. Ödeen S, Lundberg B (1993) Determination of complex modulus from measured end-point accelerations of an impacted rod specimen. J Sound Vib 165(1):1–8 6. Mousavi S, Nicolas DF, Lundberg B (2004) Identification of complex moduli and Poisson’s ratio from measured strains on an impacted bar. J Sound Vib 277(4–5):971–986 7. Mousavi S, Welch K, Valdek U, Lundberg B (2005) Non-equilibrium split Hopkinson pressure bar procedure for non-parametric identification of complex modulus. Int J Impact Eng 31(9):1133–1151 8. Welch K, Mousavi S, Lundberg B, Strømme M (2005) Viscoelastic characterization of compacted pharmaceutical excipient materials by analysis of frequency-dependent mechanical relaxation processes. Eur Phys J E Soft Matter 18(1):105–112 9. Collet P, Gary G, Lundberg B, Mohr D (2012) Complex modulus estimation respecting causality: application to viscoelastic bars. EPJ Web of Conferences 26:01012 10. Welch K, Strømme M (2007) Low frequency limitations of the split Hopkinson pressure bar method for identification of complex modulus. Int J Impact Eng 34(6):1036–1046 11. Mahata K, Mousavi S, Söderström T (2006) On the estimation of complex modulus and Poisson’s ratio using longitudinal wave experiments. Mech Syst Signal Process 20(8):2080–2094 12. Willis RL, Wu L, Berthelot YH (2001) Determination of the complex Young and shear dynamic moduli of viscoelastic materials. J Acoust Soc Am 109(2):611–621 13. Söderström T (2002) System identification techniques for estimating material functions from wave propagation experiments. Inverse Probl Eng 10(5):413–439 14. Rensfelt A, Söderström T (2011) Parametric identification of complex modulus. Automatica 47(4):813–818 15. Mahata K, Söderström T (2007) Bayesian approaches for identification of the complex modulus of viscoelastic materials. Automatica 43(8):1369–1376 16. Liu Q, Subhash G (2006) Characterization of viscoelastic properties of polymer bar using iterative deconvolution in the time domain. Mech Mater 38(12):1105–1117 17. Mousavi S (2007) Identification of viscoelastic materials by use of wave propagation methods. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 369:59 18. Bracewell RM (1965) The fourier transform and its applications. McGraw-Hill, New York, pp 6–7, 244–250 19. Ueda K, Umeda A (1998) Dynamic response of strain gages up to 300 khz. Exp Mech 38(2):93–98 20. 2001, Kulite Strain Gage Manual, Kulite Semiconductor Products 21. 2010, OFV-552 Laser Vibrometer, Polytec 22. 2009, Precision 28144 Quad-Channel Wideband Transducer Conditioner with Voltage and Current Excitation (Datasheet), Precision Filters, Ithaca 23. 2003, NI PXI-6133 Specifications, National Instruments, Austin 24. 2011, Matlab User’s Manual (version 2011b), Mathworks 25. Foley JR, Jordan JL, Siviour CR (2012) Probabilistic estimation of the constitutive parameters of polymers. EPJ Web of Conferences 26:04041

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