Dadymdroelliad y modwlws cymhlyg mewn glud-elastigedd llinol

Dadymdroelliad y modwlws cymhlyg mewn glud-elastigedd llinol
(Deconvolution of the complex modulus in linear viscoelasticity)

A. Russell Davies

The relaxation spectrum of a viscoelastic material holds the key to describing its relaxation mechanisms at a molecular level. It also plays a fundamental role in accessing the molecular weight distribution, and in modelling the dynamics of complex uids. The relaxation spectrum cannot be measured directly, but it may be locally determined from experimental measurements of viscoelastic response at a macroscopic level. In particular, the relaxation spectrum is a continuous distribution of relaxation times which may be recovered, at least locally, from measurements of the complex modulus of the material. Although mathematical expressions for the continuous spectrum have been known for over a century, these were inaccessible to numerical implementation for decades, since they involve inverse operators which are not continuous, resulting in severe instability. Progress was made when regularization methods for approximating discrete line spectra were introduced some two decades ago. It was not until 2012, however, that Davies and Goulding proposed a method of wavelet regularization for recovering continuous spectra in a mathematically rigorous framework. This work was further re ned in 2016 by introducing a mathematical form of high-order derivative spectroscopy involving sequences of derivatives of dynamic moduli, termed Maclaurin sequences. In this article, a rigorous justi cation for the use of Maclaurin sequences is presented. Furthermore, a new sequence is presented, which is termed a wavelet correction sequence, achieving the same accuracy as Maclaurin sequences, but with a reduced order of di erentiation.


  	A. Russell Davies, 'Dadymdroelliad y modwlws cymhlyg mewn glud-elastigedd llinol', Gwerddon, 24, August 2017, 22–37.


  1. Anderssen, R. S. et al. (2015), ‘Derivative based algorithms for continuous relaxation spectrum recovery’, Journal of Non-Newtonian Fluid Mechanics, 222, 132–40.
  2. Anderssen, R. S. et al. (2015), ‘Simple joint inversion localized formulae for relaxation spectrum recovery’, The ANZIAM Journal, 58(1), 1–9.
  3. Ankiewitz, S. et al. (2016), ‘On the use of continuous relaxation sbectra to characterize model polymers’, Journal of Rheology, 60(6), 1115–20.
  4. Bernstein, S. N. (1928), ‘Sur les fonctions absolument monotones’, Acta Mathematica, 52(1), 1–66.
  5. Davies, A. R., et al. (2016), ‘Derivative spectroscopy and the continuous relaxation spectrum’, Journal of Non-Newtonian Fluid Mechanics, 233, 107–18.
  6. Davies, A. R., a Goulding, N. J. (2012), ‘Wavelet regularization and the continuous relaxation spectrum’, Journal of Non-Newtonian Fluid Mechanics, 189, 19–30.
  7. Ferry, J. D. (1970), Viscoelastic properties of polymers (New York: Wiley).
  8. Fuoss, R., a Kirkwood, J. (1941), ‘Electrical Properties of Solids, VIII. Dipole Moments in Polyvinyl Chloride-Diphenyl Systems’, Journal of the American Chemical Society, 63(2), 385–94.
  9. Mallat, S. (2009), A Wavelet Tour of Signal Processing. The Sparse Way (San Diego: Academic Press).
  10. McDougall, I., Orbey, N., a Dealy, J. (2014), ‘Inferring meaningful relaxation sbectra from experimental data’, Journal of Rheology, 53(3), 770–97.
  11. Loy, R. J. et al. (2017), ‘Convergence in relaxation spectrum recovery’, Bulletin of the Australian Mathematical Society, 95(1), 121–32.
  12. Powell, M. J. D. (1981), Approximation theory and methods (Cambridge: CUP).
  13. Saut, J. C, a Joseph, D. D. (1983), ‘Fading memory’, Archive for Rational Mechanics and Analysis, 81(1), 53–95.
  14. Tanner, R. I., a Walters, K. (1998), Rheology: An Historical Perspective (Amsterdam: Elsevier).
  15. Tschoegl, N. W. (1989), The Phenomenological Theory of Linear Viscoelastic Behaviour (Berlin Heidelberg: Springer-Verlag).
  16. Walters, K. (1975), Rheometry (London: Chapman and Hall).