Nonlinear Viscoelastic Modelling
Macro-modelling of polyethylene does not separate the crystalline and amorphous phase behaviour but describes the overall material behaviour in bulk. A nonlinear viscoelastic model is generated by adopting a linear interpolation approach to include stress influence on the material parameters. These parameters are taken to be piecewise linear functions instead of being prescribed functions. The model is presented as a matrix of material constants corresponding to the selected stresses. The parameters corresponding to stress levels between the stresses used for defining the model are found from a linear interpolation.
The proposed model is formulated from a multiple Kelvin element model, shown in Figure 7. The number of Kelvin elements for a specific material is determined in the calibration process. This makes the model more adaptable to material variations.
E1 E2 En
Ц1 Ці Ци
Figure 7. Multiple Kelvin element viscoelastic solid.
where E0 and E; are functions of stress, and x; = , known as relaxation times, remain constant.
are the Kelvin dashpot viscosities. The creep function is given by:
1 N 1
[!2] V(t) = ^—r + Z ^—r
Then, the strain response under a constant stress oi becomes:
For a constant oi, the material parameters E0 and Ei are calibrated from a given test and reduced to constants. For the development of the proposed model, several creep tests are performed at different stress levels. A linear least-squares fitting is used in the parameter calibration at each stress, and the number of Kelvin elements N is automatically chosen by the fitting criterion. A series of constant E0(oi) and Ei(oi) values are generated corresponding to the creep stresses. Thus, the model can be presented as a table of E0 and Ei values, as listed in Table 4.
In order to define the material behaviour at an arbitrary stress, the parameters are linearly interpolated from the parameters corresponding to stresses below and above the given value. Thus, E0 and Ei are modelled as linear piecewise functions of stress. For modelling purposes, the instantaneous modulus E0(oi) and the coefficients of the exponential terms xi(oi) = ^Ei(oi) (instead of Ei(oi) ) are linearly interpolated for an arbitrary stress, a. The interpolations are given by the following equations:
 E0(o) = E0(am) + a~°m ^(On) – E0(Om)]
 xi (a) = xi (Bm) + ° °m [xi (On) – xi (Cm)]
On – Om
where om and on are the bounding stresses that define the interval in which о is located. When the number of Kelvin elements is not equal for all the stress levels used to define the model, a zero value for the absent coefficients xi can be used.
For numerical implementation, Eq. 11 can be discretized as:
where tj and tk are times at discrete steps, and оj and £k are the corresponding stress and strain.