Moduli vs Temperature

The Tait model and parameters for Specific Volume are used to obtain the Bulk Modulus of the material over a range of temperatures taken from a table of either elastic or shear moduli vs temperature in order to compute the Poisson's Ratio at those temperatures and fit them to a generalized TANH function. This is only valid for isotropic materials such as unfilled thermoplastic polymers. If a material is fiber filled, this tab takes the Tait parameters from the Multiscale tab after extraction from the pvT of the composite.

Follow these steps to set up and review Poisson ratio fitting:
  1. To prepare the data, click the Moduli vs T button.
  2. Upload the Excel file with the data, particularly the tensile modulus versus temperature data. Tensile or shear modulus can be used based on the selection box to the right of the field on the Material Overview tab. This selection is mirrored into the Elastic Mechanics tab as a reminder.
  3. Verify that the temperature unit is in Celsius and the tensile modulus range units is in mega pascals (MPa) in this example.

    Once the data is loaded, there will be a plot of the tensile modulus versus temperature.

    Note: If shear modulus is selected during import, the calculation of the Poisson Ratio will use the appropriate equation with the bulk modulus, and the plot will be that of the shear moduli versus temperature. Shear moduli versus temperature can be obtained from test results containing the Dynamic shear moduli versus temperature in a separate operation.
    Figure 1. Moduli vs T - Poisson Ratio Function


  4. Fit Poisson Ratio:

    Identify the center of the sigmoid (T0 and nu0) and adjust the parameters if necessary to fit the Poisson ratio to the computed points. . The remaining parameters are the minimum Poisson Ratio (nu-min) and the temperature scaling factor, Theta.

    For example, one might set the midpoint (T0 and nu0) to 322.05 [K] and 0.427 respectively, and adjust other values like nu-min to 0.377 and Theta to 0.052.

  5. Analyze Cross Correlation:

    The plot will display blue dots representing the computed bulk modulus points, the elastic modulus data points, and the computed Poisson Ratio values.

    The red line is a plot of the function fit for the Poisson ratio.

    The purple line in the elastic modulus (or shear modulus) pane shows the function fit of the elastic moduli using the computed bulk moduli and the fit Poisson ratio function. The purple line in the Bulk Modulus pane shows the computed bulk modulus fit from the Poisson ratio function and the elastic or shear moduli test data.

    A good fit, in the temperature range of interest, will have a tight fit to both the elastic and bulk moduli. It is worth noting that the bulk modulus cross-correlation will not typically be ideal, but if performing compressible hyperelastic material development, when that becomes available, it will be more important and the use of dynamic shear moduli sometimes fits more easily.

  6. Check Bulk Modulus Behavior:

    Ensure the bulk modulus behavior is consistent. Ideally, the lines should be fairly close to the eye to indicate accurate fitting.

    Specific volume testing only goes to 20 [C]. As such, the bulk modulus may vary below this temperature. Semicrystalline polymers may have a glass transition below this temperature and the bulk modulus may shift below that point. If fitting is needed below this point, it is best to do additional testing. A combination of shear and tensile testing can be used to compute the Poisson Ratio for better fitting at low temperatures.

  7. Once fitting is finished, the Poisson Ratio for an unfilled Polymer and reviewed the plots, click Approve Model Fit to finalize the settings.

By following the steps described, one can accurately fit and verify the Poisson ratio for an unfilled isotropic material, ensuring . more accurate mechanical behavior prediction over a range of temperatures.