K-BKZ constitutive equation

Factorization of the function concentrated polystyrene solution, time-strain factorability is not valid at short times after the imposition of the step shear. An accurate K-BKZ constitutive equation for shearing flows of this material will be much more complex than that for melt I. Furthermore, in strain histories in which a strain reversal takes place, such as constrained recoil (Wagner and Laun, 1978) or double-step strains with the second strain of sign opposite the first (Doi, 1980 Larson and Valesano, 1986), good agreement... 

With viscoelastic models used by an increasing number of researchers, time and temperature dependence, as well as strain hardening and nonisotropic properties of the deformed parison can, in principle, be accounted for. Kouba and Vlachopoulos (97) used the K-BKZ viscoelastic constitutive equation to model both thermoforming and parison membrane stretching using two-dimensional plate elements in three-dimensional space. Debbaut et al. (98,99) performed nonisothermal simulations using the Giesekus constitutive equation. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

Birefringence measurements are often performed to compare theoretical and experimental stress distributions in an abrupt contraction (see Section III-l). Such comparisons have been already published for the White-Metzner [30], K-BKZ [27, 31] and Wagner [32, 33] constitutive equations. Generally speaking. 

See Tanner (1985) or Larson (1988) for more details about the K-BKZ category of constitutive equations. 

The development of molecular constitutive equations for commercial melts is still a challenging unsolved problem in polymer rheology. Nevertheless, it has been found that for many melts, especially those without long-chain branching, the rheological behavior can be described by empirical or semiempirical constitutive equations, such as the separable K-BKZ equation, Eq. (3-72), discussed in Section 3.7.4.4 (Larson 1988). To use the separable K-BKZ equation, the memory function m(t) and the strain-energy function U, or its strain derivatives dU/dli and W jdh, must be obtained empirically from rheological data. 

While these functions have been adjusted to describe shear and uniaxial extensional flows, they seem to work poorly for planar extension of LDPE (Samurkas et al. 1989). Planar extensional flow represents a particularly difficult test for K-BKZ-type constitutive equations, since fits to shear data fix all the model parameters required for planar extension, and there is therefore no wiggle room left to obtain a fit to the latter. (This is because I = I2 in both shear and planar extension.) A recent non-K-BKZ molecular constitutive equation derived from reptation-related ideas shows improved qualitative agreement with planar extensional data (McLeish and Larson 1998). 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

The framework for examining arbitrary deformation histories for the rep-tation fluid has now been established and one can obtain a constitutive law for the stress response to arbitrary deformation histories. While the DE model can provide a more general constitutive equation than that to be developed now, the more general form requires numerical solution. The approximation known as the independent alignment assumption (lA) results in a closed form solution that gives a special case of the K-BKZ theory developed previously. 

The K-BKZ and other integral constitutive equations discussed above can be regarded as generalizations of the Lodge integral, eq4.3.18. The upper-convected Maxwell (UCM) equation, which is the differential equivalent of the Lodge equation, can also be generalized to make possible more realistic predictions of nonlinear phenomena. 

As an example of a popular viscoelastic constitutive equation used in the past 25 years, vhich possesses enough degree of complexity so as to capture as accurately as possible the complex nature of polymeric liquids, we present here the K-BKZ integral constitutive equation with multiple relaxation times proposed by Papanastasiou et al. [27] and further modified by Luo and Tanner [28]. This is often referred to in the literature as K-BKZ/PSM model (from the initials of Papanastasiou, Scriven, Macosko) and is vritten as... 

Figure 4.4 Rheological data and their best fit for the lU PAC-LDPE melt-A using the K-BKZ/PSM integral constitutive equation with eight relaxation modes and the data ofTable 4.1 [32]. Symbols correspond to experimental data [29], solid lines correspond to their best fit.

Transient Response Creep. The creep behavior of the polymeric fluid in the nonlinear viscoelastic regime has some different features from what were foimd with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation behavior (or vice versa). For most cases, the material properties that appear in the constitutive equations are written in terms of the relaxation response. We discuss this subsequently in the context of the K-BKZ model. 

Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U(Ji, I2, i). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

The DE Constitutive Equations. The DE model (52-56) made a major breakthrough in polymer viscoelasticity in that it provided an important new molecular physics based constitutive relation (between the stress and the applied deformation history). This section outlines the DE approach that built on the reptation-tube model developed above and gave a nonlinear constitutive equation, which in one simplified form gives the K-BKZ equation (70,71). The model also inspired a significant amount of experimental work. One should begin by... 

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