Principal normal stress difference

In the example given, the constitutive equation used is a multimode Phan Tien Tanner (PTT). It requires the evaluation of both linear and nonlinear material-response parameters. The linear parameters are a sufficient number of the discrete relaxation spectrum 2, and r]i pairs, which are evaluated from small-strain dynamic experiments. The values of the two nonlinear material-response parameters are evaluated as follows. Three semiarbitrary initial values of the two nonlinear parameters are chosen and the principal normal stress difference field is calculated for each of them using the equation of motion and the multimode PTT. They are compared at each field point (i, j) to the experimentally obtained normal stress difference and used in the following cost function F... 

A systematic study of the basic rheological properties for a wide variety of polypropylene melts has been made by Minoshima et al. [89]. These authors measured shear viscosities at low shear rates in a Rheomatrics mechanical spectrophotometer and at high rates in an Instron capillary rheometer. The principal normal stress difference, Ni, was measured in the mechanical spectrophotometer with a cone and plate device. The elongational viscosity, of special importance to fiber formation, was measured in an apparatus built by Ide and White [90],... 

The principal normal stress differences, Ni, are plotted in Figure 3.7. The normal stress difference increases with molecular weight, molecular weight distribution, and shear rate. Because of instabilities existing in cone and plate geometry, data at high shear rates are unobtainable. 

FIGURE 3.7 Principal normal stress difference as a function of shear rate for polypropylene at 180°C. (From Minoshima, W. White, J.L. Spruiell, J.E. Polym. Eng. Sci., 1980, 20, 1166. With permission.)... 

Figure 2.3 Principal normal stress difference A/, vs. shear stress with 0 (O), 10 (A), and 22 ( ) vol% glass fibers (Chan et al. [70])...

Figure 2.4 Principal normal stress difference N, as function of shear stress a,2 polystyrene...

Polymer solutions are viscoelastic. This was shown by Eisenschitz and Philippoff [67] as early as 1933 using oscillatory experiments. Polymer solutions exhibit normal stresses and complex viscoelastic properties [66,68 to 70] (see Section 1.3.5). Tanner [70] and Ide and White [66] have sought to determine the dependence of normal stresses upon concentration in polymer solutions (see Section 1.3.5). They considered at the principal normal stress difference Nj at low shear rates. 

As will be shown below, the experiment in the sliding plate rheometer does not allow one to determine Nl, since the normal force is in fact related to the second normal stress difference. For this reason, we studied the stress-optical law in shear by assuming that the principal directions of shear and refractive index are close to each other in the x-y plane. It is then straightforward to express the difference of principal stresses in the x-y plane... 

When the material behavior is brittle rather than ductile, the mechanics of the failure process are much different. Instead of the slow coalescence of voids associated with ductile rupture, brittle fracture proceeds by the high-velocity propagation of a crack across the loaded member. If the material behavior is clearly brittle, fracture may be predicted with reasonable accuracy through use of the maximum normal stress theory of failure. Thus failure is predicted to occur in the multi-axial state of stress when the maximum principal normal stress becomes equal to or exceeds the maximum normal stress at the time of failure in a simple uniaxial stress test using a specimen of the sane material. 

Major consolidation stress a ) This is the principal normal stress (cti) under which the sample has been consolidated in the principal stress plane. The major consolidation stress should not be confused with the initial compaction stress, which is the stress that compacts the powder bed. Each different compaction stress, (7c, leads to a different yield locus and becomes one of a family of yield loci at different densifications. The major consolidation stress is obtained by drawing a Mohr semi-circle through the equilibrium or end point of the yield locus and tangential to the yield locus. 

All parameters depend on the time and the shear rate. Steady-state conditions are obtained for t — CO. Variable (°, ) denotes the steady state values of the shear stress. The anisotropic character of the flowing solutions give rise to additional stress components, which are different in all three principal directions. This phenomenon is called the Weissenberg effect, or the normal stress phenomenon. From a physical point of view, it means that all diagonal elements of the stress tensor deviate from zero. It is convenient to express the mechanical anisotropy of the flowing solutions by the first and second normal stress difference ... 

All fractures generated by internal fluid overpressure are here referred to as hydrofractures. The fracture-generating fluid may be oil, gas, magma, groundwater, or geothermal water. Hydrofractures include dykes, inclined sheets, mineral veins, many joints, and the man-made hydraulic fractures that are used in the petroleum industry to increase the permeability of reservoir rocks. Hydrofractures are primarily extension fractures (Gudmundsson et al. 2001). The difference between the total fluid pressure in a hydrofracture and the normal stress, which for extension fractures is the minimum compressive principal stress, oj, is referred to as the fluid overpressure. 

Each of these local principal stresses makes a contribution to the mean normal stress in the PB domain that has different but known elastic properties. These contributions can be determined through a concentration factor q given by Good-ier (1933) as... 

Normal stress n. (1) A stress directed at right angles to the area upon which it acts. (2) In a flowing viscoelastic liquid, tensile/com-pressive stresses at any point in the fluid in the principal coordinate directions, one of which will be the main direction of flow. Rheologists are usually concerned with differences between normal stresses acting in the flow direction and directions perpendicular to the flow. 

There are conditions of loading a product that is subjected to a combination of tensile, compressive, and/or shear stresses. For example, a shaft that is simultaneously bent and twisted is subjected to combined stresses, namely, longitudinal tension and compression, and torsional shear. For the purposes of analysis it is convenient to reduce such systems of combined stresses to a basic system of stress coordinates known as principal stresses. These stresses act on axes that differ in general from the axes along which the applied stresses are acting and represent the maximum and minimum values of the normal stresses for the particular point considered. There are different theories that relate to these stresses. They include Mohr s Circle, Rankine s, Saint Venant, Guest, Hencky-Von Mises, and Strain-Energy. 

In an incompressible medium, the rheological state at a point is, as far as the stress is concerned, completely described by the shear stresses (three in a general flow) and in the differences of the normal or direct stresses. We shall denote components of the stress tensor by (TijiiJ= 1,2, 3) and suppose the stress tensor is symmetric, so that Gij=Oji. (If Cij is a shear stress if i=j, Gij is a direct or normal stress.) We can change axes by rotation to reduce Gij to principal form—in these (mutually orthogonal) principal axes the principal stresses are g, G2 and 0-3 and all shear stresses vanish. The simplest stress-optical relation is to suppose that the dielectric tensor (Kij) and the stress tensor are coaxial, i.e. have the same principal axes, and that the differences in principal stresses are proportional to the corresponding differences in (principal) refractive indices. Hence if 2( 3) is a principal axis, and g and G2 lie in the xy plane, we have the simplest stress-optic relation as... 

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