Relaxation and Retardation Time Spectra
all graphs were generated with IRIS and exported as bitmap.
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Relaxation and Retardation Time Spectra
all graphs were generated with IRIS and exported as bitmap.
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Objective: Calculation of the relaxation time spectrum H(
) from dynamic mechanical data G', G"(w).
The data file from the dynamic mechanical experiment may look
something like this:
Freq,G',G" rad/s,Pa,Pa 4.571,65.1,4815 8.127,229.9,8640 10,332.8,10880 14.45,622.2,15580 19.05,1083,20280 25.7,1914,27890 48.98,6428,51420 60.25,10040,64040 81.27,17890,85190 144.5,49990,137800 190.5,77120,166400 316.2,152900,217700 602.5,272800,235000 1071,371000,227100 1479,429500,227800 1905,455900,214100 3388,533700,204800 6025,619900,193200 11220,691400,177300 14790,724800,174800 25110,759500,166900 35480,805200,160300 79430,872500,163000 141300,921400,177900 199500,952100,186800 354800,1004000,239300 630900,1088000,312400 2511000,1322000,756400 7063000,1813000,1537000 |
We enter the FREQ,G',G" data as received into IRIS and calculate the parsimonious spectra. The result looks like this:
RELAXATION TIME SPECTRUM
Fit-Results Iterations : 1730 [-] Comput. Time : 126 [s] Average Deviat: 7.816D-03 [-] Spectrum SD: 1.487D-03 [-] (Variance) G inf : 0.000D+00 [Pa] Time min : 1.416E-07 [s] Time max : 2.188E-01 [s] Method of spectra determination by M.Baumgaertel and H.H. Winter ===================================================== No. g lambda - [Pa] [s] 1 6.077D+06 3.102D-08 2 4.533D+05 2.024D-07 3 2.392D+05 1.105D-06 4 1.293D+05 7.497D-06 5 1.779D+05 4.284D-05 6 2.538D+05 2.803D-04 7 3.631D+05 1.999D-03 8 5.797D+04 5.137D-03
RETARDATION TIME SPECTRUM
The retardation spectrum gets determined from the above relaxation time spectrum by Gross' method using the Laplace transformation, see Baumgaertel et al. 1989: Eta0 : 1.104E+03 Pa s Jg : 1.290E-07 1/Pa Time min : 1.416E-07 s Time max : 2.188E-01 s ================================================================================ No. j LAMBDA - 1/[Pa] [s] 1 2.224D-07 1.136D-07 2 3.771D-07 3.323D-07 3 2.611D-07 1.431D-06 4 1.554D-07 8.651D-06 5 2.703D-07 5.351D-05 6 6.099D-07 4.226D-04 7 4.374D-07 4.263D-03
ORIGINAL DATA, WRITTEN IN UNIVERSAL DATA STANDARD
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No. FREQ G' G"
[rad/s] [Pa] [Pa]
1 4.571E+00 6.510E+01 4.815E+03
2 8.127E+00 2.299E+02 8.640E+03
3 1.000E+01 3.328E+02 1.088E+04
4 1.445E+01 6.222E+02 1.558E+04
5 1.905E+01 1.083E+03 2.028E+04
6 2.570E+01 1.914E+03 2.789E+04
7 4.898E+01 6.428E+03 5.142E+04
8 6.025E+01 1.004E+04 6.404E+04
9 8.127E+01 1.789E+04 8.519E+04
10 1.445E+02 4.999E+04 1.378E+05
11 1.905E+02 7.712E+04 1.664E+05
12 3.162E+02 1.529E+05 2.177E+05
13 6.025E+02 2.728E+05 2.350E+05
14 1.071E+03 3.710E+05 2.271E+05
15 1.479E+03 4.295E+05 2.278E+05
16 1.905E+03 4.559E+05 2.141E+05
17 3.388E+03 5.337E+05 2.048E+05
18 6.025E+03 6.199E+05 1.932E+05
19 1.122E+04 6.914E+05 1.773E+05
20 1.479E+04 7.248E+05 1.748E+05
21 2.511E+04 7.595E+05 1.669E+05
22 3.548E+04 8.052E+05 1.603E+05
23 7.943E+04 8.725E+05 1.630E+05
24 1.413E+05 9.214E+05 1.779E+05
25 1.995E+05 9.521E+05 1.868E+05
26 3.548E+05 1.004E+06 2.393E+05
27 6.309E+05 1.088E+06 3.124E+05
28 2.511E+06 1.322E+06 7.564E+05
29 7.063E+06 1.813E+06 1.537E+06