INPUT RUBBER COMPOUND:
rubber compound for shifting = ViscoelasticShifting_Example
rubber compound in strain sweep = ViscoelasticShifting_Example
Ntemp (number of temperature points) =35
Tstart, Tstop (temperature points between the limits, in Celcius) =-0.5011E+02 0.1200E+03
temperature values increses
Ndata (number of frequency points) = 8
frequency.start, frequency.stop (frequency points between the limits, in Hz) = 0.5000E+00 0.2800E+02
frequency values decreses
QUALITY OF INPUT DATA:
If any of the ratios below (first column) differ strongly from unity (say > 10 or < 0.1) this indicate bad input data!
Quality of ReE-data: maximum ratio ReE(i,j+1)/ReE(i,j) (j=frequency number) at temperature T(i) = 0.9963E+00 -0.5011E+02
minimum ratio ReE(i,j+1)/ReE(i,j) (j=frequency number) at temperature T(i) = 0.6606E+00 -0.1511E+02
Quality of ImE-data: maximum ratio ImE(i,j+1)/ImE(i,j) (j=frequency number) at temperature T(i) = 0.1133E+01 0.1000E+03
minimum ratio ImE(i,j+1)/ImE(i,j) (j=frequency number) at temperature T(i) = 0.6550E+00 -0.2011E+02
INPUT MASTER CURVE:
iflag (=0 read-in shift factor, =1 shift ReE, =2 shift ImE, =3 shift ImE/ReE=tand) = 1
(=-1 only RELAXATION MODULUS is calculated; require that the two IN_TREADCOMPOUND... files exist)
(Our experience is that in general =1 shift ReE gives best result)
(If iflag=0 then shift factor must be given in input file 'IN_TREADCOMPOUND.1temperature.C.2aT.3bT')
(In this case the data file 'IN_TREADCOMPOUND.1temperature.C.2aT.3bT' must give aT for the same NT different
temperatures as in the input file 'IN.TREADCOMPOUND.1NT.1Nf.1Nrow.1iT.1if.1istrain.1iReE.1iImE.Next.all.data')
ivert (=0 vertical shift factor b=1, =1 vertical shift factor b=T/Tref, =2 interpolation) = 2
Tref, Trefused (reference temperature for mastercurve; should be one of the temperatures for which E(omega) was measured) = 0.2000E+02 0.1994E+02
omega0 (the tand(T)-curve for this angular frequency (in 1/s); the maximum of tand defines glass temp Tg) = 0.1000E-01
(note: input frequencies in measured data are assumed to be in Hz but omega0 (angular frequency) in 1/s)
iremove0 (=1 (or 2) replace lowest (and next lowest) frequency data point with extrapolated value; useful e.g. if ImE<0) = 0
(if iremove0 =-1 or -2 then replace with extrapolated value only Im E - data)
(if iremove0 =11 (or 12) then cut away lowest (and next lowest) frequency point)
iremove1 (=1 (or 2) replace highest (and next highest) data point with extrapolated value; useful e.g. if ImE<0) = 0
(if iremove1 =-1 or -2 then replace with extrapolated value only Im E - data)
(if iremove1 =11 (or 12) then cut away highest (and next highest) frequency point)
ispline (=1 or 0 gives linear or cubic spline interpolation) = 1
npolySegment (integer order of polynomial smoothening of frequency segment, e.g. 2 or 3) = 3
npolyTotal (integer order of polynomial smoothening of master curve, e.g. 10) =10
iorder (=0 --> shift start at low temperature, =1 start at high temperature) = 0
(both procedures gives the same result if frequency strings overlap)
improve (=0 or 1; normally use 0 but when shifting ImE or tand you may need =1 to get smooth curve) = 0
xuse (a number < 1.0 but close to 1.0, e.g., 0.9; only used when improve=1) = 0.9500E+00
ilog (=0 or 1; normally use 0, an integer related to shifting procedure) = 0
X (if ratio ReE(j+1)/ReE(j) or ImE(j+1)/ImE(j) larger than X or smaller than 1/X then write WARNING) = 0.3000E+02
INPUT STRAIN SWEEP:
iStrainSweep (=1 or 2 viscoelastic E-modulus is calculated from strain sweeps at different temperatures) = 1
(the E-modulus is given at the reference temperature at the fixed common strain and stress or self-consistently (see below))
(iStrainSweep=1 --> the relevant shift factor aT used is assumed to be given by the earlier part of the code)
(iStrainSweep=2 --> first part of code not performed; shift factor aT is read-in from file: IN_TREADCOMPOUND.1temperature.C.2aT.3bT)
reduction (the E-modul resulting from strain sweep analyzis is presented for (max common stress or strain)x(reduction)) = 0.7000E+00
xkappa (viscoelastic modulus calculated self-consistently with stress sigma=sig00+0.67*xkappa*|E|, sig00=0.3MPa) = 0.7000E+00
(this relation between stress and E-modulus is expected from contact mechanics theory where xkappa = (average surface slope))
npolyEred, order of polynom for smoothning of Ereduction= 5
prestrain, if oscilatory strain amplitude (elongation mode) is epsilon1, then prestrain = prestrainfactor*strain1= 0.1250E+01
(in shearmode no prestain so put prestrainfactor=0.0 in shear mode experiments)
INPUT RELAXATION MODULUS:
(NOTE: it is useful to calculate the relaxation spectra as it is a good test of the accuracy on the
meassured ReE and ImE. The reason is that the relaxation spectra is a single real function and in general
it is not possible to obtain two functions, ReE and ImE, as the real and imaginary part of a complex function
which depend on a single real fit-function. However, if E(omega) is a causal linear response function then
ReE and ImE are related via a Kramers-Kronig relation and a complex function depending on a single real
fit-function (relaxation spectra) can exactly reproduce both ReE and ImE. However, if there is inaccuracy in the
measured E(omega) then ReE and ImE cannot be obtained from a complex function which depend on a single real function.
Thus comparing the two curves in CHECK...ORIGINAL.and.FITTED will give information about the accuracy of measured E(omega).)
iEtime (=1, or 2, or 3 gives the relaxation spectra of E(omega), or 1/E(omega), or both) = 3
(if iEtime=1, E(omega)=E(infty)-\sum_n H1(n)/(1-i tau_n omega))
(if iEtime=2, 1/E(omega)=1/E(infty)+\sum_n H2(n)/(1-i tau_n omega))
nterma (= number of relaxation times in pole-expansion of E(omega) or 1/E(omega) = 30
nMontestep (= number of Monte-Carlo steps in optimization, use typically 500000 = 500000
The calculated error in the fit of pole-expansion of E(omega) = 0.1504E-01
The calculated error in the fit of pole-expansion of 1/E(omega) = 0.8397E-02
(For good pole-expansion fit of E or 1/E the error should be of order 0.01 or smaller)
CALCULATED DATA:
maximum of (ImE/ReE)(T) [or tand(T)] curve (for frequency omega0), and the temperature Tg (in C) at maximum = 0.6874E+00 -0.3553E+02
Tg is the glass transition temperature at the frequency omega0; usually Tg is defined with omega0 = 0.01 s^{-1}
left side minimum, maximum, right side minimum of tand(T) curve (for angular frequency omega0) tand = 0.1638E+00 0.6874E+00 0.2536E-01
in order to be able define the glass transition temperature Tg one must have tandMAX > tandMINleft, tandMINright
average effective activation energy Ea=-k_B T ln(aT) for segmental motion of polymer chain (in eV) = 0.1570E+01
Note: Ea cannot be interpreted as a simple activation energy since an additional temperature dependency influencing
the shift factor comes from the increase in the free volume (thermal expansion) with increasing temperature
A NOTE:
The WLF equation (which is one output of the code) holds only in the temperature range Tg < T < Tg + 100 C
The WLF equation follows if it is assumed that the free volume (related to thermal expansion) increases linearly
with temperature thus facilitating thermally activated chain segment motion; for T < Tg this effect is less
important and then an Arhenius type of shift factor aT=exp(-Ea/k_BT) may be more accurate than the
WLF formula log_{10}(aT) = - A (T-Tg)/(B+T-Tg) with A=17.44 and B=51.6 C
If T0 reference temperature WLF-output file gives aT/aT0 with the canonical A=17.44 and B=51.6 C
However, the coefficients A and B are in not universal and may have to be optimized for each rubber compund
STRAIN-SWEEP INFO:
the strain sweeps are for the frequency (Hz), and for the number of temperatures = 0.1000E+01 10
the minimum common strain (in %) and minimum common stress (in Pa) between all strain sweeps= 0.4295E-01 0.6276E+04
the maximum common strain (in %) and maximum common stress (in Pa) between all strain sweeps= 0.7557E+02 0.4152E+07
E-modulus is calculated for the (maximum common strain)x(reduction) (in %) = 0.5290E+02
E-modulus is calculated for the (maximum common stress)x(reduction) (in Pa)= 0.2907E+07
Note: if iStrainSweep=1 the non-linear strain data has been shifted assuming the same shifting factor as obtained in the first part
of the code, otherwise if iStrainSweep=2 the shift factor aT is read-in from file: IN_TREADCOMPOUND.1temperature.C.2aT.3bT)
ratioReE = ReE(large strain)/ReE(small strain), ratioImE = ImE(large strain)/ImE(small strain), = 0.3413E+00 0.4970E+00