Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

Abstract

A powerful but still easy to use technique is proposed for the processing and analysis of dynamic mechanical data. The experimentally determined dynamic moduli,G′(ω) andG″(ω), are converted into a discrete relaxation modulusG(t) and a discrete creep complianceJ(t). The discrete spectra are valid in a time window which corresponds to the frequency window of the input data. A nonlinear regression simultaneously adjust the parametersg _i ,λ _i ,i = 1,2, ⋯N, of the discrete spectrum to obtain a best fit ofG′, G″, and it was found to be essential that bothg _i andλ _i are freely adjustable. The number of relaxation times,N, adjusts during the iterative calculations depending on the needs for avoiding ill-posedness and for improved fit. The solution is insensitive to the choice of initial valuesg _i,0,λ _i,0,N ₀. The numerical program was calibrated with the gel equation which gives analytical expressions both in the time and the frequency domain. The sensitivity of the solution was tested with model data which, by definition, are free of experimental error. From the relaxation time spectrum, a corresponding discrete set of parametersJ ₀,η, J _d,i andΛ _i of the creep complianceJ(t) can then readily be calculated using the Laplace transform.