Plastic materials, mathematical modeling

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

Mathematical Modeling of Forming and Performance of Plastic Materials... 

Cold flow studies have several advantages. Operation at ambient temperature allows construction of the experimental units with transparent plastic material that provides full visibility of the unit during operation. In addition, the experimental unit is much easier to instrument because of operating conditions less severe than those of a hot model. The cold model can also be constructed at a lower cost in a shorter time and requires less manpower to operate. Larger experimental units, closer to commercial size, can thus be constructed at a reasonable cost and within an affordable time frame. If the simulation criteria are known, the results of cold flow model studies can then be combined with the kinetic models and the intrinsic rate equations generated from the bench-scale hot models to construct a realistic mathematical model for scale-up. 

Research problems with one response undoubtedly have an advantage. In practice, however, we mostly meet research subjects with several responses, which often means a literally large number of responses. Thus, for example, when producing rubber, plastic and other composite materials one must take into account responses such as physical-chemical, technological, economic, mechanical (tensile strength, elongation, module, etc.) and others. One can define the mathematical model for each of the mentioned responses but simultaneous optimization of several functions is mathematically impossible. 

Another approach to obtain migration data particularly for some plastic materials is the use of modelling. Today this approach is only suitable for certain materials but is accepted by the EU Commission. Diffusion within, and migration from, food contact materials are predictable processes that can be described by mathematical equations. Mass transfer from a plastic material, for instance, into food simulants obeys Tick s laws of diffusion in most cases. Physico-mathematical diffusion models have been established, verified and validated for migration from many plastics into food simulants and are accepted in the USA and in the EU. 

Rheocalic V2.4. The Bingham mathematic model was used to determine viscosity. The Bingham equation is t = Tq + ly. Where t is the shear stress applied to the material, y is the shear strain rate (also called the strain gradient). To is the yield stress and p is the plastic viscosity. 

The mathematical description of the powder billet SSE cannot rely on the above conditions of plasticity, as they do not take the change in density of the extruded sample into account. Nor the conditions of plasticity used to model the compaction of metal powders are suitable for this case (48). They imply that with the increase in hydrostatic pressure the relative density of powder material tends to unity. However, in the case of hydrostatic compaction of polymer powders, the relative density of compacts tends to a value less than unity, which is typical of the polymer. The reason is in difference of compaction mechanisms for the metal and polymer powders. 

As mentioned earlier, there have been many attempts to develop mathematical models that would accurately represent the nonlinear stress-strain behavior of viscoelastic materials. This section will review a few of these but it is appropriate to note that those discussed are not all inclusive. For example, numerical approaches are most often the method of choice for all nonlinear problems involving viscoelastic materials but these are beyond the scope of this text. In addition, this chapter does not include circumstances of nonlinear behavior involving gross yielding such as the Luder s bands seen in polycarbonate in Fig. 3.7. An effort is made in Chapter 11 to discuss such cases in connection with viscoelastic-plasticity and/or viscoplasticity effects. The nonlinear models discussed here are restricted to a subset of small strain approaches, with an emphasis on the single integral approach developed by Schapery. 

The above mathematical models (and later derivatives) define constitutive relationships for the plastic strain regime and they all assume a linear elastic behavior terminated by a yield point that is rate dependent. Hence the yield surface of the material is rate dependent. Since the purpose of these models are to develop methods to calculate deformations which are rate dependent beyond the yield point of a material they are often referred to by the term viscoplasticity, (see Perzyna, (1980), Christescu, (1982)). This practice is analogous to referring to methods to calculate deformation beyond the yield point of an ideal rate independent elastic-plastic material as classical plasticity. However, more general theories of viscoplasticity have been developed in some of which no yield stress is necessary. See Bodner, (1975) and Lubliner, (1990) for examples. 

A compression molding press is usually a rugged piece of equipment that provides sufficient force to compress the material. The force necessary to mold a part must be smaller than the press capacity. The force required depends on the nature of the plastic and part geometry. Dependence on the nature of the material being compressed complicates the task of mathematical modeling of this process. There are complex approximate expressions that relate the force required to the geometry of the part and material properties. 

The plastic strain energy density can be physically interpreted as the energy of distortion associated with the change in shape of a volume elanent and is related to failure, particularly under conditions of ductile behavior, as it often occurs in many of the polymeric materials submitted to cyclic mechanical solicitation (Kanchanomai and Mutoh, 2004). According to the mathematical model proposed by Morrow (American Society for Testing and Materials, 1965), the strain energy density can be evaluated numerically as the inner area of the saturated hysteresis loop for the uniaxial fatigue experiments (Fig. 7.6), and is expressed mathematically as ... 

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