Purely viscous liquid

Many of the new plastics, blends, and material systems require special, enhanced processing features or techniques to be successfully injection molded. The associated materials evolution has resulted in new plastics or grades, many of which are more viscoelastic. That is, they exhibit greater melt elasticity. The advanced molding technology has started to address the coupling of viscoelastic material responses with the process parameters. This requires an understanding of plastics as viscoelastic fluids, rather than as purely viscous liquids, as is commonly held... 

Thus, even when the elongation rate, as defined by equation 3.76, is constant, the separation of two material points increases exponentially with time. As stress relaxation occurs exponentially, it is clear that at high elongation rates the stress will increase very rapidly. In a purely viscous liquid the stress relaxes instantaneously and consequently this high resistance to stretching does not occur. 

The function iKt-t ) may be interpreted as a memory function having a form as shown in Figure 3.14. For an elastic solid, iff has the value unity at all times, while for a purely viscous liquid iff has the value unity at thfe current time but zero at all other times. Thus, a solid behaves as if it remembers the whole of its deformation history, while a purely viscous liquid responds only to its instantaneous deformation rate and is uninfluenced by its history. The viscoelastic fluid is intermediate, behaving as if it had a memory that fades exponentially with time. The purely elastic solid and the purely viscous fluid are just extreme cases of viscoelastic behaviour. 

The presence of a low-viscosity interfacial layer makes the determination of the boundary condition even more difficult because the location of a slip plane becomes blurred. Transitional layers have been discussed in the previous section, but this is an approximate picture, since it stiU requires the definition of boundary conditions between the interfacial layers. A more accurate picture, at least from a mesoscopic standpoint, would include a continuous gradient of material properties, in the form of a viscoelastic transition from the sohd surface to the purely viscous liquid. Due to limitations of time and space, models of transitional gradient layers will be left for a future article. 

For Hookean elastic solids the stress and strain are in phase, whereas for purely viscous liquids the strain lags 90° behind the applied stress. 

These materials exhibit both viscous and elastic properties. In a purely Hookean elastic solid, the stress corresponding to a given strain is independent of time, whereas for viscoelastic substances the stress will gradually dissipate. In contrast to purely viscous liquids, on the other hand, viscoelastic fluids flow when subjected to stress, but part of their deformation is gradually recovered upon removal of the stress. 

The Maxwell model comprises an elastic spring (of modulus G) in series with a dashpot containing a purely viscous liquid (of viscosity T ) (12,15,18). Following application of an external force to this system, the stress in the spring is equal to that in the dashpot and, furthermore, the total strain in the system (jtotal) is the sum of the strain in the spring (ys) and that in the dashpot (yd). 

Polymers are also unique in their viscoelastic nature, a behavior that is situated between that of a pure elastic solid and that of a pure viscous liquid-like material their mechanical properties present a strong dependence on time and temperature. Given all the factors that have to be taken into account to determine the mechanical properties of polymers, their measurement would appear to be very complex. However, there is a series of general principles that determine the different mechanical properties and that give a general idea of the expected results in different mechanical tests. These principles can be organized in a systematic manner to determine the interrelation of polymer structure and the observed mechanical properties, using equations and characteristic parameters of polymeric materials. 

Ata = -l,b = c = 0, equations [7.2.24] correspond to the Maxwell liquid with a discrete spectrum of relaxation times and the upper convective time derivative. For solution of polymer in a pure viscous liquid, it is convenient to represent this model in such a form that the solvent contribution into total stress tensor will be explicit ... 

Purely viscous constitutive equations, which account for some of the nonlinearity in shear but not for any of the history dependence, are commonly used in process models when the deformation is such that the history dependence is expected to be unimportant. The stress in an incompressible, purely viscous liquid is of the form given in equation 2, but the viscosity is a function of one or more invariant measures of the strength of the deformation rate tensor, [Vy - - (Vy) ]. [An invariant of a tensor is a quantity that has the same value regardless of the coordinate system that is used. The second invariant of the deformation rate tensor, often denoted IId, is a three-dimensional generalization of 2(dy/dy), where dy/dy is the strain rate in a one-dimensional shear flow, and so the viscosity is often taken to be a specific function-a power law, for example-of (illu). ]... 

The equations describing polymer processing operations are usually coupled, nonlinear partial-differential or partial-differential-integral equations in which two or three spatial directions and perhaps time appear as independent variables. Fully three-dimensional problems can usually be solved only for purely viscous liquids, and substantial simplification is usually required even for two-dimensional problems of viscoelastic liquids because of limitations of computer speed and memory. In some situations the geometry provides simplifications that lead to closed-form analytical solutions, but these are rare, and numerical methods are usually required in order to obtain process information. Numerical methods relevant to polymer processing flows are discussed in (3,4,16,17). The following three broad classes of numerical methods are in common use ... 

Another aspect of the situation is illuminated by describing the dissipation by the viscosity of the liquid. In a purely viscous liquid, l.e. in which absorption is due only to the shear viscosity, we have ... 

Filled polymer rheology is basically concerned with the description of the deformation of filled polymer systems under the influence of applied stresses. Softened or molten filled polymers are viscoelastic materials in the sense that their response to deformation lies in varying extent between that of viscous liquids and elastic solids. In purely viscous liquids, the mechanical energy is dissipated into the systems in the form of heat and cannot be recovered by releasing the stresses. Ideal solids, on the other hand, deform elastically such fliat the deformation is reversible and the energy of deformation is fully recoverable when the stresses are released. 

The density p enters because the inertia of the sample plays an essential role in the wave propagation. For a purely viscous liquid, 31m Xm and r]o = 231m jutp. The total mechanical impedance Rm + iXm is (3tM + i Xm) times the area of contact A plus the contributions from whatever inertia, elastance, and frictance must be attributed to the apparatus itself. 

Today we call this time-dependent response viscoelasticity. It is typical of all polymeric materials. Another common way to measure the phenomenon is by stress relaxation. As illustrated in Figure 3.1.2, when a polymeric liquid is subject to a step increase in strain, the stress relaxes in an exponential fashion. If a purely viscous liquid is subjected to the same deformation, the stress relaxes instantly to zero as soon as the strain becomes constant. An elastic solid would show no relaxation. 

Pa-s (pure viscous liquid with Newtonian viscosity of the solution), for curve 2, a=l, for curve 3, a = 0, that is, the latter two graphs correspond to viscoelastic solution with and without account for the rheological non-linearity, respectively. 

The situation considered here for semi-dilute solutions is comparable with the case of a chemical gel under swelling (or deswelling), but it is not equivalent. Indeed, whatever the time scale of the observation for a gel the situation remains identical, because crosslinks are permanent. For semi-dilute solutions, this is not the case they are pure viscous liquids at the observation time scale (e.g. 5 x 10 s) of CGD experiment. The elastic part of the longitudinal modulus (Eq. (D-16)) does not play any role because... 

---