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Home / Technical Notes / Stress Analysis / Analysis of Nonlinear Materials / Converting the Measurement Data of Viscoelastic Materials
Converting the Measurement Data of Viscoelastic Materials
Measuring Viscoelastic Materials
There are two measurement methods.
- Static measurement: Stress relaxation is measured under a constant strain, or creep strain is measured under a constant stress
- Dynamic measurement: Stress is measured under a periodic strain, and vice versa.
Relaxation modulus is obtained through static measurement. Dynamic modulus is obtained through dynamic measurement.
Dynamic measurement is more common and performed at various temperatures and frequencies.
Here is an example of dynamic measurement.
Temperature: 25 to 250[deg] at 5[deg] step
Frequency: 1, 2, 5, 10, 25[Hz]
See Viscoelast_Mat.csv for detail.
Graphs below are the frequency response of storage modulus and loss modulus for each temperature.
Graphs below are the temperature characteristics of storage modulus and loss modulus for each frequency.
These data can be entered as the material property.
The entered data are automatically converted to a shift function and Prony series and used for the simulation.
Converting Dynamic Measurement Data to Master Curve and Shift Factor
The temperature increase corresponds to the frequency decrease.
Therefore, high temperature data can be translated to lower frequency data, while matching the data to the one of initial temperature.
Through this data conversion, a master curve can be created for each modulus.
Shift function's temperature plot is referenced to the initial temperature.
The reference temperature doesn't need to be the initial temperature.
It can be set to any temperature.
The shift function graph above indicates that if the reference temperature is changed to 100[deg] the master curve will shift to higher frequency by 8 digits.
The shift function's temperature plot will also shift so that 100[deg] becomes the origin.
The simulation results will be the same no matter what the reference temperature is.
Converting Master Curve Frequency Response to Prony Series
The figures below show the relations between master curves, relaxation modulus and Prony series.
Dynamic modulus is converted to relaxation modulus as follows:
| Dynamic modulus |
 |
| Relaxation modulus |
 |
Equations to convert Prony series to dynamic modulus or relaxation modulus are listed in [Analysis of Viscoelastic Materials].
There are no equations to convert master curves or relaxation modulus to Prony series.
Femtet obtains the initial modulus and Prony series through optimization calculations:
The coefficient of modulus is adjusted little by little until the results become close enough to master curves or relaxation modulus.
The left graph below shows the master curves: the originals and the ones calculated from Prony series. The right graph below shows the relaxation modulus: one calculated from Prony series and the other approximated from master curves.
They are well matched.
There are two optimization parameters: the number of Maxwell elements per frequency digit, and the weight yielding to loss modulus.
The more Maxwell elements used, the more memories consumed.
Weight yielding to loss modulus is used to optimize the loss modulus in such a situation that the storage modulus is optimized but the loss modulus is not yet.
In the static analysis of the viscoelastic materials, the important thing is the matching of relaxation moduli by Prony series and by Ninomiya-Ferry equation.
If the storage modulus is optimized, the relaxation modulus by the Ninomiya-Ferry equation is also optimized at the same time.
It is, therefore, not needed to change the default setting of 0 for the Weight yielding to the loss modulus.
Converting Relaxation Modulus Data to Prony Series
Time dependency data of the relaxation modulus obtained by the static measurement can be converted to the Prony series. The static measurement means the stress relaxation measurement under a constant strain or creep strain measurement under a constant stress.
Optimization is needed. Its parameter is the number of Maxwell elements per time digit.