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Determination of the relaxation and retardation spectra – a view from the up-to-date signal processing perspective

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Mechanics of Composite Materials Aims and scope

The problem of determination of relaxation and retardation spectra (RRS) is considered from the viewpoint of up-to-date signal processing. It is shown that the recovery of RRS represents the Mellin deconvolution problem, which transforms into the Fourier deconvolution problem for data on a logarithmic time or frequency scale, where it can also be treated as the inverse filtering problem. On this basis, discrete deconvolution (inverse) filters operating with geometrically sampled data are proposed to use as RRS estimators. Appropriate frequency responses and algorithms are derived for estimating RRS from eight different material functions. The noise amplification coefficient is suggested to use as a measure for quantifying the degree of ill-posedness and illconditioness of the RRS recovery problem and algorithms. A methodology is developed for designing RRS estimators with a desired noise amplification, producing maximum accurate spectra for available limited input data. Practical algorithms for determining RRS are proposed, and their performance is studied. The algorithms suggested are compared with the so-called moving-average formulae. It is demonstrated that the minimum frequency range for recovering the point estimate of a relaxation spectrum depends on the allowable noise amplification (the degree of ill-conditioness) and is in no way limited by 1.36 decades, as it is stated by the sampling localization theorem.

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References

  1. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd. ed., J. Wiley and Sons, NY (1980).

    Google Scholar 

  2. N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior, Springer-Verlag, Berlin (1989).

    Book  Google Scholar 

  3. R. G. Owens and T. N. Phillips, Computational Rheology, Imperial College Press, London (2002).

    Book  Google Scholar 

  4. N. G. McCrum, B. E. Read, and G. Williams, Anelastic and Dielectric Effects in Polymer Solids, J. Wiley and Sons, London (1967).

    Google Scholar 

  5. A. K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectric, London (1983).

    Google Scholar 

  6. C. P. Slichter, Principles of Magnetic Resonance, 3rd. enlarged and updated ed., Springer-Verlag, NY (1996).

  7. C. Friedrich, R. J. Loy, and R. S. Anderssen, “Relaxation time spectrum–molecular weight distribution relationships,” Rheol. Acta, 48, 151–162 (2009).

    Article  CAS  Google Scholar 

  8. M. R. Nobile and F. Cocchini, “A general relation between MWD and relaxation time spectrum,” Rheol Acta, 47, 509–519 (2008).

    Article  CAS  Google Scholar 

  9. W. Thimm, C. Friedrich, M. Marth, and J. Honerkamp, “An analytical relation between relaxation time spectrum and molecular weight distribution,” J. Rheol., 43, 1663–1672 (1999).

    Article  CAS  Google Scholar 

  10. A. R. Davies and R. S. Anderssen, “Sampling localization in determining the relaxation spectrum,” J. Non-Newtonian Fluid Mech., 73, 163–179 (1997).

    Article  CAS  Google Scholar 

  11. A. R. Davies and R. S. Anderssen, “Sampling localization and duality algorithms in practice,” J. Non-Newtonian Fluid Mech., 79, 235–252 (1998).

    Article  CAS  Google Scholar 

  12. J. R. Macdonald, “On relaxation-spectrum estimation for decades of data: accuracy and sampling-localization considerations,” Inverse Problems, 16, 1561–1583 (2000).

    Article  Google Scholar 

  13. M. Renardy, “On the use of Laplace transform inversion for reconstruction of relaxation spectra,” J. Non-Newtonian Fluid Mech., 154, 47–51 (2008).

    Article  CAS  Google Scholar 

  14. J. R. Macdonald, “Comparison of parametric and nonparametric methods for the analysis and inversion of immittance data: Critique of earlier work,” J. Comp. Phys., 157, 280–301 (2000).

    Article  Google Scholar 

  15. J. R. Macdonald and E. Tuncer, “Deconvolution of immittance data: Some old and new methods,” J. Electroanalytical Chem., 602, 255–262 (2007).

    Article  CAS  Google Scholar 

  16. F. Aslani and L. Sjögren, “Relaxation rate distribution from frequency or time dependent data,” Chem. Phys., 325, 299–312 (2006).

    Article  CAS  Google Scholar 

  17. S. M. F. D. S. Mustapha and T. N. Phillips, “A dynamic nonlinear regression method for the determination of the discrete relaxation spectrum,” J. Phys. D: Appl. Phys., 33, 1219–1229 (2000).

    Article  CAS  Google Scholar 

  18. E. A. Jensen, “Determination of discrete relaxation spectra using simulated annealing,” J. Non-Newtonian Fluid Mech., 107, 1–11 (2002).

    Article  CAS  Google Scholar 

  19. S. Hansen, “Estimation of the relaxation spectrum from dynamic experiments using Bayesian analysis and a new regularization constraint,” Rheol. Acta, 47, 169–178 (2008).

    Article  CAS  Google Scholar 

  20. F. J. Stadler and Ch. Bailly, “A new method for the calculation of continuous relaxation spectra from dynamic-mechanical data,” Rheol. Acta, 48, 33–49 (2009).

    Article  CAS  Google Scholar 

  21. A. V. Oppenheim and R. V. Schafer, Discrete-Time Signal Processing, 2nd. Ed., Prentice-Hall International, New Jersey (1999).

    Google Scholar 

  22. S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, 2nd. Ed., California Technical Publishing, San Diego (1999).

    Google Scholar 

  23. T. W. Parks and C. S. Burrus, Digital Filter Design, Wiley-Interscience, NY (1987).

    Google Scholar 

  24. V. Shtrauss, “Functional conversion of signals in the study of relaxation phenomena,” Signal Processing, 45, 293–312 (1995).

    Article  Google Scholar 

  25. V. Shtrauss, “Unified approach and technique to performance of Mellin convolution and related integral transform in relaxation experiments,” Meccanica, 32, 251–258 (1997).

    Article  Google Scholar 

  26. V. Shtrauss, “Signal processing in relaxation experiments,” Mech. Compos. Mater., 38, No. 1, 73–88 (2002).

    Article  Google Scholar 

  27. V. Shtrauss, “Digital signal processing for relaxation data conversion,” J. Non-Crystal. Solids, 351, 2911–2916 (2005).

    Article  CAS  Google Scholar 

  28. V. Shtrauss, “Digital estimators of relaxation spectra,” J. Non-Crystal Solids, –– 353, 4581–4585 (2007).

    Article  CAS  Google Scholar 

  29. V. Shtrauss, “Determination of relaxation and retardation spectrum by inverse functional filtering,” J. Non-Newtonian Fluid Mech.,165, 453–465 (2010).

    Article  CAS  Google Scholar 

  30. J. Bertrand, P. Bertrand, and J.-P. Ovarlez, Chapter 11. The Mellin Transform. In: The Transform and Applications Handbook, 2. ed. Ed. A. D. Poularikas, CRC Press, Boca Raton (2000).

  31. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, N.Y. (1968).

    Google Scholar 

  32. Shtrauss V. Solving inverse problems in relaxation experiments. In: Advances in Composite Materials and Structures VII. Eds. W. P. de Wilde, W. R. Blain, C. A. Brebbia, WIT Press, Southampton, Boston (2000), P. 239–248.

  33. V. D. Shtraus, “Nondestructive relaxation spectrometry of dielectrics,” Mech. Compos. Mater., 25, No. 4, 521–534 (1989).

    Article  Google Scholar 

  34. J. R. Macdonald and V. I. Piterbarg, “On the transformation of colored random noise by the Kronig–Kramers integral transform,” J. Electroanalytical Chem., 428, 1–9 (1997).

    Article  CAS  Google Scholar 

  35. R. S. Anderseen, A. R. Davies, and F. R. de Hoog, “On the interconversion integral equation for relaxation and creep,” Anzian J. (CTAC2006) (E), 48, C346–C363 (2007).

    Google Scholar 

  36. . S. Anderseen, A. R. Davies, and F. R. de Hoog, “On the sensitivity of interconversion between relaxation and creep,” Rheol. Acta, 47, 159–167 (2008).

    Article  Google Scholar 

  37. V. Shtrauss, “Sampling and algorithm design for relaxation data conversion,” WSEAS Trans. Signal Proc., 2, 984–990 (2006).

    Google Scholar 

  38. R. J. Loy, C. Newbury, R. S. Anderssen, and A. R. Davies, “A duality proof of sampling localisation in relaxation spectrum recovery,” Bull. Austral. Math. Soc., 64, 265–269 (2001).

    Article  Google Scholar 

  39. R. J. Loy, A. R. Davies, and R. S. Anderssen, “A revised duality proof of sampling localization in relaxation spectrum recovery,” Bull. Austral. Math. Soc., 78, 79–83 (2009).

    Article  Google Scholar 

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Acknowledgements

This work was supported by the European Regional Development Fund (ERDF) under project No. 2010/0213/2DP/2.1.1.1.0/10/APIA/VIAA/017.

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  1. Institute of Polymer Mechanics, University of Latvia, Riga, LV-1006, Latvia

    V. Shtrauss

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  1. V. Shtrauss
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Correspondence to V. Shtrauss.

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Translated from Mekhanika Kompozitnykh Materialov, Vol. 48, No. 1, pp. 37–66, January-February, 2012.

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Shtrauss, V. Determination of the relaxation and retardation spectra – a view from the up-to-date signal processing perspective. Mech Compos Mater 48, 27–46 (2012). https://doi.org/10.1007/s11029-012-9249-7

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