K-BKZ model

where is the stress tensor, m (t-t’) is a time-memory function, h(I₁,I₂) is a damping function of the two strain invariants I₁,I₂ and is the Finger strain tensor. Temperature effects are included via an Arrhenius temperature dependency of material parameters. Furthermore, we have:

Time-memory function:

(2)

where a_i are the relaxation modulus, t_i are the relaxation times and N is the number of the pairs modul / time.

The damping function h(I₁,I₂) can be one of the following:

(3)

(4)

(5)

where a, b are adjustable parameters, I₁, I₂ are the first and the second invariants of the Finger strain tensor given by:

(6)

(7)

where l is the extension ratio.

Table 1 shows values of the m parameter for various types of deformation:

Table 1: Values of m for some types of deformation:

Type of deformation	m
Uniaxial elongation	-1/2
Planar extension (pure shear)	0
Ellipsoidal extension	1/2
Equibiaxial extension	1

 

The damping functions (eq. 3, 4) were suggested by Wagner [3,4]. The damping function (eq. 5) was proposed by Papanastasiou et al. [5].

Invariants of Finger strain tensor:

(8)

(9)

Finger strain tensor for uniaxial elongation:

(10)

Finger strain tensor biaxial extension ( l (t) is the same for both directions):

(11)

Equations for stresses in principal directions:

(12)

(13)

[1] Kaye, A.: ”Non-Newtonian Flow in Incompressible Fluids”, College of Aeronautics, Cranfield, CoA Note No. 134 (1962)

[2] Bernstein, B., Kearsley, E.A., Zapas, L.J.: Trans. Soc. Rheol. 7, p.391/410 (1963)

[3] Wagner, M.H.: J. Rheol. 34(6), p.943/958 (1990)

[4] Wagner, M.H.: Rheol. Acta 15, p.136/142, (1976)

[5] Papanastasiou, A.C., Scriven, L.E., Macosko, C.W.: J. Rheol. 27(4), p.387/410 (1983)