Normal stress in shear

There are three important phenomena seen is polymeric liquids that make them different from simple fluids a non-Newtonian viscosity, normal stresses in shear flow, and elastic effects. All these effect are a result of the complex molecular structure of polymer macromolecules. 

Normal Stresses in Shear Flow. The tendency of polymer molecules to curl-up while they are being stretched in shear flow results in normal stresses in the fluid. For example, shear flows exhibit a deviatoric stress defined by... 

The lubrication approximation as previously derived is valid for purely viscous Newtonian fluids. But polymer melts are viscoelastic and also exhibit normal stresses in shearing flows, as is discussed in Chapter 3 nevertheless, for many engineering calculations in processing machines, the approximation does provide useful models. 

The coefficients Ti and b2, like non-Newtonian viscosity, are also found to be shear rate dependent. The non-Newtonian property of exhibiting normal stresses in shear flows plays an important role in processing under situations in which shear stresses vanish, as in extrudate swell, discussed later in this section. 

FIGURE 4.2 Simple shear flow deflnition of shear stress, strain, and shear rate Xyy, and are the Cartesian coordinates and the direction of normal stresses in shear flow. 

A priori, this condition is satisfied only with intermeshing rotors. Moreover, with polymer melts, normal stresses in shear flow are very large, even at relatively low shear rate for example, in the range of 10 dyn/cm at Y = 1 sec for polystyrene at 180 C (15) (and probably higher with elastomers). Consequently the N./ o ratio is larger for viscoelastic fluids than for Newtonian fluids and therefore the validity condition for using the lubrication theory at the nip is surely not fulfilled. 

Polypropylene melts are viscoelastic fluids. As such, the melts exhibit non-Newtonian viscosity, normal stresses in shear flow, excessive entrance-and-exit pressure drop, die swell, secondary entrance flows, melt fracture, and draw resonance. (Newtonian fluids also exhibit draw resonance.) Polypropylene melts are more viscoelastic than melts of nylon and polyester. 

Viscous and elastic moduli with small-amphtude sinusoidal shear can be determined by using an orthogonal rheometer [2]. Small-amplitude sinusoidal shear, using cone-and-plate or paraUel-plate test methods, can determine rheological behavior for normal stresses in shear flow, as well as for shear strain. 

However, various combinations of eiastic and viscous elements have been used to approximate the material behavior of polymer melts. Some models are combinations of springs and dashpots to represent the elastic and viscous responses, respectively. The most common ones being the Maxwell model for a polymer melt and the Kelvin or Voight model for a solid. One model that represents shear thinning behavior, normal stresses in shear flow and elastic behavior of certain polymer melts is the K-BKZ model [28-29]. 

Fig. 13.25. Normal stresses in shearing polymeric fluids are responsible for their swelling on exit from a die.

If we had used the Green tensor C instead of B in writing the constitutive equation, we would have obtained different results for the normal stresses in shear, (note eq. 1.4.25). Using B gives results that agree with observations for rubber. 

The transport properties of polymeric materials which distinguish them most from other materials are their flow properties or rheological behavior. There are many differences between the flow properties of a polymeric fluid and typical low molecular weight fluids such as water, benzene, sulfuric acid, and other fluids, which we classify as Newtonian. Newtonian fluids can be characterized by a single flow property called viscosity (/r) and their density (p). Polymeric fluids, on the other hand, exhibit a viscosity function that depends on shear rate or shear stress, time-dependent rheological properties, viscoelastic behavior such as elastic recoil (memory), additional normal stresses in shear flow, and an extensional viscosity that is not simply related to the shear viscosity, to name a few differences. 

C2.1.8.2 SHEAR THINNING AND NORMAL STRESS IN POLYMER MELTS... 

Other dimensional systems have been developed for special appHcations which can be found in the technical Hterature. In fact, to increase the power of dimensional analysis, it is advantageous to differentiate between the lengths in radial and tangential directions (13). In doing so, ambiguities for the concepts of energy and torque, as well as for normal stress and shear stress, are eliminated (see Ref. 13). 

Note that the solution predicts that simple shear produces normal stresses. In fact, although simple shear occurs at constant volume, the normal stresses in general give rise to a hydrostatic pressure. Determination of the normal stresses in the case of a hypoelastic equation of grade one could, at least in principle, determine the coefficients a, Ug, and Ug individually. 

The inherent anisotropy (most often only orthotropy) of composite materials leads to mechanical behavior characteristics that are quite different from those of conventional isotropic materials. The behavior of isotropic, orthotropic, and anisotropic materials under loadings of normal stress and shear stress is shown in Figure 1-4 and discussed in the following paragraphs. 

Thus, in order that E and G always be positive, i.e., that a positive normal stress or shear stress times the respective positive normal strain or shear strain yield positive work,... 

A key element in the experimental determination of the stiffness and strength characteristics of a lamina is the imposition of a uniform stress state in the specimen. Such loading is relatively easy for isotropic materials. However, for composite materials, the orthotropy introduces coupling between normal stresses and shear strains and between shear stresses and normal and shear strains when loaded in non-principal material coordinates for which the stress-strain relations are given in Equation (2.88). Thus, special care must be taken to ensure obtaining... 

During dynamic measurements frequency dependences of the components of a complex modulus G or dynamic viscosity T (r = G"/es) are determined. Due to the existence of a well-known analogy between the functions r(y) or G"(co) as well as between G and normal stresses at shear flow a, seemingly, we may expect that dynamic measurements in principle will give the same information as measurements of the flow curve [1],... 

The state of stress at a point in a structural member under a complex system of loading is described by the magnitude and direction of the principal stresses. The principal stresses are the maximum values of the normal stresses at the point which act on planes on which the shear stress is zero. In a two-dimensional stress system, Figure 13.2, the principal stresses at any point are related to the normal stresses in the x and y directions ax and ay and the shear stress rxy at the point by the following equation ... 

Metzner AB, Houghton WT, Salior RA, White JL. A method for measurement of normal stresses in simple shearing flow. Trans Soc Rheol 1961 133-147. 

Endo,H., Fujimoto.T., Nagasawa,M. Normal stress and shear stress in a viscoelastic liquid under steady state flow Effect of molecular weight heterogeneity. J. Polymer Sci. Pt.A-2 9,345-362 (1971). 

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