Extensional deformation function

In the strained state, the junction point vector deforms to a length and new orientation angles 0 and cp, Choosing the strained state orientation angles as the independent variables, the elemental strain is characterized by three quantities, the extensional deformation function... 

Fig. 12.19 Cold postextrusion micrographs as a function of the flow rate. The processing conditions were T = 177°C and no PPA. Each image is actually a composite of two micrographs in which the side and top are focused. The relative errors in throughputs are 0.05 Q = (a) 1.0, (b) 2.2, (c) 3.8, (d) 6.3, and (e) 11 g/min. The width of each image corresponds to 3 mm. [Reprinted by permission from K. B. Migler, Extensional Deformation, Cohesive Failure, and Boundary Conditions during Sharkskin Melt Fracture, J. Rheol., 46, 383 4-00 (2002).]...

De Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, Ithaca Dealy JM (1994) Official nomenclature for material functions describing the response of a viscoelastic fluid to various shearing and extensional deformations. J Rheol 38 179-191 Debbaut B, Crochet MJ (1988) Extensional effects in complex flows. J Non-Newtonian Fluid Mech 30 169-184... 

From Equation 10.21, the dimensions of viscosity are stress multiplied by time, and in the SI system viscosity is measured in units of pascal-seconds (Pas). For polymer melts and solutions, the fluid behavior is non-Newtonian and Equation 10.21 must be modified to allow the viscosity to become a material function of the shear rate. Similarly, material elements may be deformed by pulling on opposite sides of the cube with an equal force this constitutes an extensional deformation that may be characterized by an extensional viscosity. 

J. M. Dealy, Official nomenclature for material functions describing the response of a viscoelastic fluid to various shearing and extensional deformations, J. Rheol. 28, 181-195 (1984). 

Material functions must however be considered with respect to the mode of deformation and whether the applied strain is constant or not in time. Two simple modes of deformation can be considered simple shear and uniaxial extension. When the applied strain (or strain rate) is constant, then one considers steady material functions, e.g. q(y,T) or ri (e,T), respectively the shear and extensional viscosity functions. When the strain (purposely) varies with time, the only material functions that can realistically be considered from an experimental point of view are the so-called dynamic functions, e.g. G ((D,y,T) and ri (a), y,T) or E (o),y,T) and qg(o),y, T) where the complex modulus G (and its associated complex viscosity T] ) specifically refers to shear deformation, whilst E and stand for tensile deformation. It is worth noting here that shear and tensile dynamic deformations can be applied to solid systems with currently available instruments, whUst in the case of molten or fluid systems, only shear dynamic deformation can practically be experimented. There are indeed experimental and instrumental contingencies that severely limit the study of polymer materials in the conditions of nonlinear viscoelasticity, relevant to processing. 

Following Newton (1640), the viscosity is defined as the ratio of the stress over the deformation rate. Whether a shear or a simple extensional flow is considered, one has then the shear or the extensional viscosity, and if such quantities are rate dependent, one deals with shear or extensional viscosity functions. Experimentally,... 

Another strain measure that is closely related to the one defined by Eq. 10.14 was proposed by Wagner etal. [ 14]. This tensor involves a new scalar, which they call the molecular stress function. When used in an integral constitutive equation it was found to be able to describe the behavior of a high-density polyethylene in shear flow and several types of extensional deformation. 

Most experimental studies of melt behavior involve shearing flows, and we saw in Chapter 5 that linear viscoelastic behavior is a rich source of information about molecular structure. However, no matter how many material functions we determine in shear, outside the regime of linear viscoelasticity such information cannot be used to predict behavior in other types of deformation, ie., for any other flow kinematics. A class of flows that is of particular importance in commercial processing is extensional flow. In this type of flow, material elements are stretched very rapidly along streamlines. Nonlinear behavior in extensional deformations provides information about structural features of molecules that are not revealed by shear data. 

Fig. 14. Shear viscosity, Tj, and extensional viscosity, Tj as a function of deformation rate of a low density polyethylene (LDPE) at 150°C (111). To convert...

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

Figure 2.37 presents plots of elongational viscosities as a function of stress for various thermoplastics at common processing conditions. It should be emphasized that measuring elongational or extensional viscosity is an extremely difficult task. For example, in order to maintain a constant strain rate, the specimen must be deformed uniformly exponentially. In addition, a molten polymer must be tested completely submerged in a heated neutrally buoyant liquid at constant temperature. 

We first derive the kinematics of the deformation. The flow situation is shown in Fig. 14.14. Coordinate z is the vertical distance in the center of the axisymmetric bubble with the film emerging from the die at z = 0. The radius of the bubble R and its thickness 8 are a function of z. We chose a coordinate system C, embedded in the inner surface of the bubble. We discussed extensional flows in Section 3.1 where we defined the velocity field of extensional flows as... 

Figure 9.14 Droplet deformation as a function of the duration of deformation for different flow fields (ratio of shear and extensional flow) for a viscosity ratio of X = 3 the larger a, the larger the ratio of extensional flow a=0 corresponds to pure shear flow...

In these experiments, the tensile force is measured as a function of time, so that at a constant rate of deformation e it is possible to calculate the true tensile stress and the extensional viscosity r/c elastic properties of the deformation can be determined by measuring the elastic strain e. 

For correlating extensional viscosity data, it is obvious to attempt the same method as was used for non-steady state shear viscosity. Thus, the ratio r)Jrjeo is presumed to be determined by two dimensionless groups i0 and f/i0. As e is constant (i.e. qe), the ratio of these groups is equal to the tensile deformation e. Therefore, t/t0 will likewise be a function of t/xa and . 

The extensional stress growth functions in shear and in uniaxial extension were measured for neat PP and linear low density PE, LLDPE, as well as for their blends. Good agreement between the two types of deformation was obtained indicating that linear viscoelastic behavior was obtained strain hardening was not observed [8]. 

---