Note: Equation 11–1 – Equation 11–4 are used to compute the extra-stress tensor for integral viscoelastic flow as well as differential viscoelastic flow. See Theory and Equations for details.
For an integral viscoelastic flow, Ansys Polyflow solves the constitutive
equations for the extra-stress tensor, the momentum equations, the
incompressibility equation, and (for nonisothermal flows) the energy equation.
For the constitutive equations for 
(Equation 11–1), 
is computed at time 
from the following equation:
 | (11–42) |
and 
are the scalar invariants of the Cauchy-Green strain tensor:
 | (11–43) |
and
 | (11–44) |
The various integral viscoelastic models are characterized by the form of the
functions 
, 
, and 
.
The momentum and incompressibility equations are provided in Equation 11–9 and Equation 11–10, respectively. See Theory and Equations for details.
For nonisothermal flows, the energy equation is presented as Equation 13–5 in Theory. To obtain
from Equation 11–1, 
can be computed from the isothermal constitutive equation
(Equation 11–42),
provided that a modified time scale 
is used for evaluating the strain history:
 | (11–45) |
The modified time scale is related to 
through the following equation:
 | (11–46) |
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. Equations for 
are presented later in this section.
The numerical technique used for solving integral viscoelastic flows implements an uncoupled scheme, where the computation of the viscoelastic extra stress is performed separately from that of the flow kinematics (and temperature). In this aspect, the technique differs from those used for differential viscoelastic flows where stresses, velocity, and pressure variables are all calculated simultaneously.
With an integral model, on the basis of known velocity and temperature fields, Ansys Polyflow first computes the viscoelastic extra stress by means of the constitutive equations (Equation 11–42 and Equation 11–45).The kinematics and temperature fields are then updated by solving Equation 11–9, Equation 11–10, and Equation 13–5, where the viscoelastic stresses act as a known pseudo-body-force term. (This means that stresses are computed at the Gauss point of the finite element.)
The Doi-Edwards model is characterized by shear thinning and a non-quadratic first normal-stress difference at high
shear rates. It also predicts a nonvanishing second normal-stress difference
and a finite steady extensional viscosity. In the Doi-Edwards model,
is computed from
 | (11–47) |
where
 | (11–48) |
and
 | (11–49) |
In Equation 11–47, 
represents the relaxation mode.
(optional, but strongly recommended) is computed from
Equation 11–2.
The KBKZ model provides additional accuracy by including a damping function
in its constitutive equations. 
is computed from
 | (11–50) |
and 
(optional, but strongly recommended) is computed from
Equation 11–2.
In Equation 11–50,
represents the relaxation mode and 
is a scalar parameter that controls the ratio of the
normal-stress differences:
 | (11–51) |
and 
is the damping function. The simplest case (Lodge-Maxwell model)
is for no damping: 
and 
.
The Papanastasiou-Scriven-Macosko (PSM)
model computes 
from
 | (11–52) |
where 
is a material parameter that primarily influences the
shear-thinning behavior.
The Wagner model
computes 
from
 | (11–53) |
where 
is a material parameter that influences both the shear
viscosity and the elongational behavior of the material.
The reversible PSM model uses Equation 11–52,
allowing 
to decrease and increase along a particle trajectory. The
irreversible PSM model allows 
only to decrease.
Similarly, the reversible Wagner model uses Equation 11–53,
allowing 
to decrease and increase along a particle trajectory. The
irreversible Wagner model allows 
only to decrease.
In both Equation 11–52 and Equation 11–53,
is computed from
 | (11–54) |
where 
and 
are given by Equation 11–43 and Equation 11–44. 
is a material parameter that influences only the
elongational behavior of the material.
The generalized Newtonian model that is equivalent to an integral model in simple steady shear (that is, the generalized Newtonian model that exhibits the same steady shear viscosity law as the integral model) is
 | (11–55) |
where
 | (11–56) |
is given by the viscoelastic constitutive equation (Equation 11–47 or
Equation 11–50) in a simple shear flow.
(optional, but recommended) is computed from Equation 11–2.
The solution for the equivalent generalized Newtonian model can be compared with the viscoelastic model results to determine the relative contribution of elasticity to the behavior of the flow being modeled.
Three models are available for the temperature shift function 
in Equation 11–46 the Arrhenius law, the Arrhenius approximate law, and
the WLF law, all described in Temperature-Dependent Viscosity Laws. It is also possible to eliminate the temperature
dependence using a temperature shift function equal to 1.
The numerical simulation requires the simultaneous solution of the constitutive equations (Equation 11–1, Equation 11–2, and Equation 11–42), together with the momentum equations (Equation 11–9) and the incompressibility condition (Equation 11–10).
In order to obtain convergence of the uncoupled scheme, an evolution scheme
that implements a progressive change from a Newtonian solution to the integral
viscoelastic solution is recommended. The scheme is based on the so-called
evolutive viscosity 
, which determines a Newtonian (or generalized Newtonian)
stress tensor 
. An extra-stress tensor 
is defined as follows:
 | (11–57) |
where 
and 
is a parameter that depends on the evolution parameter
.
Replacing 
by 
in Equation 11–9 yields
 | (11–58) |
When 
is zero, the problem is purely Newtonian (when 
is constant) or generalized Newtonian (when 
varies). When 
, Equation 11–58 defines the viscoelastic problem. In an evolution
scheme, 
at 
and 
at 
.
For every fixed value of 
, an iterative procedure is used to find velocity and pressure
(and temperature, for nonisothermal flows) fields that satisfy Equation 11–58. The
increments of the velocity and pressure, 
and 
, are calculated as follows:
 | (11–59) |
 | (11–60) |
When 
, Equation 11–60 becomes
 | (11–61) |
See Using Evolution to Compute Integral Viscoelastic Flow for details about using this numerical scheme.