Thermorheologically Simple Temperature Effects

When you define TRS in a viscoelastic material, the creep law is modified and takes the following form:

\frac{ⅆ {\bar{ε}}^{c r}}{ⅆ τ} = \frac{1}{a_{T} (θ)} g^{c r} ({\bar{ε}}^{c r}, I_{1}^{c r}, {\bar{I}}_{1}, {\bar{I}}_{2}, J, p, \tilde{q}, τ),

where

τ

and

a_{T} (θ)

denote the reduced time and the shift function, respectively. The reduced time is related to the actual time through the integral differential equation

τ = \int_{0}^{t} \frac{ⅆ t'}{a_{T} (θ)}, \frac{ⅆ τ}{ⅆ t} = \frac{1}{a_{T} (θ)} .

Regardless of the method used to define the viscoelastic behavior, thermo-rheologically simple temperature effects can be included by specifying the method used to define the shift function. You can define the following forms of the shift function: the Williams-Landel-Ferry (WLF) form and the Arrhenius form.

Williams-Landel-Ferry (WLF) Form

The shift function can be defined by the Williams-Landel-Ferry (WLF) approximation, which takes the form:

\log_{10} (A) = \frac{C_{1} (θ - θ_{0})}{C_{2} + (θ - θ_{0})},

where

θ_{0}

is the reference temperature at which the relaxation data are given,

θ

is the temperature of interest,

C_{1}

and

C_{2}

are calibration constants obtained at this temperature. If

θ \leq {θ_{0} - C}_{2}

, deformation changes will be elastic, based on the instantaneous moduli.

Table 1. Definition=Williams-Landel-Ferry
Input Data	Description
Reference Temperature	Reference temperature, $θ_{0}$ .
Calibration Constant 1	Calibration Constant, $C_{1}$ .
Calibration Constant 2	Calibration Constant, $C_{2}$ .

Arrhenius form

The Arrhenius shift function is commonly used for semi-crystalline polymers. It takes the form

\ln (A) = \frac{E_{0}}{R} (\frac{1}{(θ - θ^{Z})} - \frac{1}{(θ_{0} - θ^{Z})}),

where

E_{0}

is the activation energy,

R

is the universal gas constant,

θ^{Z}

is the absolute zero in the temperature scale being used,

θ_{0}

is the reference temperature at which the relaxation data are given, and

θ

is the temperature of interest.

Table 2. Definition=Arrhenius
Input Data	Description
Reference Temperature	Reference temperature, $θ_{0}$ .
Activation Energy	Activation energy, $R$ .