Other Viscoplastic Models

Casson (1959) proposed an alternate model to describe the flow of viscoplastic fluids. The three-dimensional form is left as an exercise to the reader (hint see eq. 2.5.9), but the one-dimensional form of the Casson model is given by. 

This model has a more gradual transition from the Newtonian to the yield region. For many materials, such as blood and food products, it provides a better fit. 

Comparison of Bingham and Casson fits to the iron oxide suspension data over the entire range of experimental data obtained parameters for the Bingham model are t) = 0.25 Pa-s and T,. 1.66 Pa, while for the Casson model they are O.IS Pa-s and 1.66 Pa, respectively. 

Papanastasiou (1987) proposed a modification to the viscoplastic fluid models that avoids the discontinuity in the flow curve due to the incorporation of the yield criterion. Papanastasiou s modification involves the incorporation of an exponential term, thereby permitting the use of one equation for the entire flow curve, before and after yield. The one-dimensional form of Papanastasiou s modification is 

The three-dimensional forms of the Herschel-Bulkley (eq. 2.5.6) and Casson (eq. 2.5.5) equations, with Papanastasiou s modiflcation, are as follows. 

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Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

The results of the latest research into helical flow of viscoplastic fluids (media characterized by ultimate stress or yield point ) have been systematized and reported most comprehensively in a recent preprint by Z. P. Schulman, V. N. Zad-vornyh, A. I. Litvinov 15). The authors have obtained a closed system of equations independent of a specific type of rheological model of the viscoplastic medium. The equations are represented in a criterion form and permit the calculation of the required characteristics of the helical flow of a specific fluid. For example, calculations have been performed with respect to generalized Schulman s model16) which represents adequately the behavior of various paint compoditions, drilling fluids, pulps, food masses, cement and clay suspensions and a number of other non-Newtonian media characterized by both pseudoplastic and dilatant properties. 

A number of more advanced and general models attempting to predict the yielding, viscoplastic flow, time-dependence, and large strain behavior of fluoropolymers and other thermoplastics have recently been developed.in this section, we discuss the Dual Network Fluoropolymer (DNF) model. 

The comparative performance of these three as well as several other models for viscoplastic behaviour has been thoroughly evaluated in an extensive review paper by Bird et al. [1983],... 

In general, SMPF is perceived as a two-phase composite material with a crystalline phase mixed with an amorphous phase. A multiscale viscoplasticity theory is developed. The amorphous phase is modeled using the Boyce model, while the crystalline phase is modeled using the Hutchinson model. Under an isostrain assumption, the micromechanics approach is used to assemble the microscale RVE. The kinematic relation is used to link the micro-mechanics constitutive relation to the macroscopic constitutive law. The proposed theory takes into account the stress induced crystallization process and the initial morphological texture, while the polymeric texture is updated based on the apphed stresses. The related computational issue is discussed. The predictabihty of the model is vahdated by comparison wifli test results. It is expected that more accurate measurement of the stress and strain in the SMPF with large deformation may further enhance the predictability of the developed model. It is also desired to reduce the number of material parameters in the model. In other words, a deeper understanding and physics based theoretical modeling are needed. 

Plasticity and Viscoplasticity and Other Models. As discussed above, the alternative representation of the nonlinear viscoelastic response of polymers is that of plasticity and viscoplasticity. In some respects, these models could be recast as viscoelastic models and they would be equivalent to some of the models discussed above. However, the perspective that glassy polymers are really fluids and do follow time-temperature superposition is lost with these models. Hence, the physical interpretation of material parameters, in this author s opinion, becomes very qnestionable. Therefore, only references to the major papers on polymer plasticity and viscoplasticity are given (174-177). 

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