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SIMPLE JOINT INVERSION LOCALIZED FORMULAE FOR RELAXATION SPECTRUM RECOVERY

2016; Cambridge University Press; Volume: 58; Issue: 1 Linguagem: Inglês

10.1017/s1446181116000122

1446-8735

Robert Anderssen, A.R. Davies, F. R. de Hoog, R. J. Loy,

Material Dynamics and Properties

In oscillatory shear experiments, the values of the storage and loss moduli, $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ , respectively, are only measured and recorded for a number of values of the frequency $\unicode[STIX]{x1D714}$ in some well-defined finite range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ . In many practical situations, when the range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ is sufficiently large, information about the associated polymer dynamics can be assessed by simply comparing the interrelationship between the frequency dependence of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ . For other situations, the required rheological insight can only be obtained once explicit knowledge about the structure of the relaxation time spectrum $H(\unicode[STIX]{x1D70F})$ has been determined through the inversion of the measured storage and loss moduli $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ . For the recovery of an approximation to $H(\unicode[STIX]{x1D70F})$ , in order to cope with the limited range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ of the measurements, some form of localization algorithm is required. A popular strategy for achieving this is to assume that $H(\unicode[STIX]{x1D70F})$ has a separated discrete point mass (Dirac delta function) structure. However, this expedient overlooks the potential information contained in the structure of a possibly continuous $H(\unicode[STIX]{x1D70F})$ . In this paper, simple localization algorithms and, in particular, a joint inversion least squares procedure, are proposed for the rapid recovery of accurate approximations to continuous $H(\unicode[STIX]{x1D70F})$ from limited measurements of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ .