While the differential approach is well-suited for practical applications, the integral approach is generally used for advanced rheological research. Ansys Polymat provides several numerical models for viscoelastic flow, including Doi-Edwards and KBKZ. Appropriate choices for the viscoelastic model and related parameters can yield qualitatively and quantitatively accurate representations of viscoelastic behavior.
Note: The integral approach to modeling viscoelastic flow is limited to 2D and shell models and it cannot be applied to 3D models.
For an integral viscoelastic constitutive equation, the extra-stress tensor
is computed at time 
from the following equation:
 | (6–68) |
| where |
 = model-specific memory (kernel) function
|
 = model-specific function of  and 
| |
 = model-specific function of  and 
| |
 = current time | |
 = metric for time integrals |
and 
are the scalar invariants of the Cauchy-Green strain tensor:
 | (6–69) |
and
 | (6–70) |
The various integral viscoelastic models are characterized by the form of the
functions 
, 
, and 
.
For non-isothermal flows, 
can be computed from the isothermal constitutive equation
(Equation 6–68),
provided that a modified time scale 
is used for evaluating the strain history:
 | (6–71) |
The modified time scale is related to 
through the following equation:
 | (6–72) |
where 
is the shift function, which can be obtained from steady-state
shear-viscosity curves at different temperatures. This is the principle of
time-temperature equivalence.
To specify the viscosity model for an integral viscoelastic flow, you will click the Integral Viscoelastic models menu item in the Material Data menu.
Integral Viscoelastic models
If you want to choose a generalized Newtonian flow model with a shear-thinning behavior that is identical to the currently defined integral model, click the Switch to Generalized Newtonian Flow menu item in the Integral Viscoelastic models menu.
Switch to Generalized Newtonian Flow
If you want to use the Doi-Edwards model instead of the default KBKZ model, click the Switch to Doi - Edwards Model menu item in the Integral Viscoelastic models menu.
Switch to Doi - Edwards Model
If you want to define a spectrum of relaxation times, click the Modify the spectrum menu item in the Integral Viscoelastic models menu.
Modify the spectrum
The spectrum can be defined with (relaxation force, time) or (viscosity, time) data pairs.
If you are using the KBKZ model, you can click Modify the damping function to specify which damping function is to be used.
Modify the damping function
The default function is Lodge-Maxwell (that is, no damping).
For the KBKZ model, you can also click Modify N2 / N1 to define the ratio of the normal stress differences.
Modify N2 / N1
If you want to add a constant Newtonian viscosity component to the
viscoelastic stresses, you can click Modify add visc and
set a nonzero value for 
.
Modify add visc
To specify the temperature dependence of viscosity for an integral viscoelastic flow, you will click the Temperature dependence menu item in the Integral Viscoelastic models menu.
Temperature dependence
Note that the Management of the evolutive viscosity and Numerical integration menu items are not relevant for Ansys Polymat.
See Non-Automatic Fitting and Automatic Fitting for information about where and how the material data specification occurs in the non-automatic and automatic fitting procedures, respectively.
See Integral Viscoelastic Models and Temperature Dependence of Viscosity for details about the parameters and characteristics of each fluid model.