Shear flow stress

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

A solution to this problem requires a knowledge of the atom positions in the dislocation core. Using the atom positions in the Peierls dislocation, Pacheco and Mura (1969) estimated the force on a dislocation due to just a single interface. They obtained the increase in shear flow stress, Ate, due to an elastic modulus change across a sharp interface as... 

Substituting the relevant values for TiN and NbN from Table 8.1 gives a value for Q of only 0.06, one-half of that obtained above. Accounting for the observed concentration modulation, a total increment in hardness of only 12 GPa is predicted. If the uniaxial flow stress is taken as twice the shear flow stress, the predicted value of AH falls to only 9 GPa, much less than that observed here. 

Retardation time of Voigt element Shear flow stress Pressure activation volume Viscosity of Maxwell element Viscosity of Voigt element... 

The material speed at the interface may reach the local tool speed if sticking occurs, but slip will limit the speed to a lower value, and this aspect will be highly sensitive to the local temperature (and hence local viscosity or shear flow stress). 

In Fig. 2a, from data in Shaw (2004), the test conditions are at room temperature and low strain rate. Conditions are isothermal, and strain softening is due to the nucleation and growth of voids. The different curves result from applying compressive pressure p to the shear plane. The larger the ratio p/k, where k is the peak shear flow stress from the test, the larger is the strain y at which dx/dy = 0. 

Figure 2b considers the effect of heating on the stress-strain curve. The curve marked isothermal is that for p/k = 0.6 from Fig. 2a. That marked adiabatic is obtained from the isothermal curve assuming all of the plastic work is converted to heat and that the shear flow stress reduces with temperature rise at the rate of 30 MPa per 100 °C (a reasonable value). The two curves marked intermediate suppose one third and two thirds of the heat to be conducted away. In these cases, softening is the result of heating. The strain at which dx/dy = 0 increases from the adiabatic to the isothermal condition.

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

In addition to the apparent viscosity two other material parameters can be obtained using simple shear flow viscometry. These are primary and secondary nomial stress coefficients expressed, respectively, as... 

Kaye, A., Lodge, A. S. and Vale, D. G., 1968. Determination of normal stress difference in steady shear flow. Rheol. Acta 7, 368-379. 

Non-Newtonian Fluids Die Swell and Melt Fracture. Eor many fluids the Newtonian constitutive relation involving only a single, constant viscosity is inappHcable. Either stress depends in a more complex way on strain, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known coUectively as non-Newtonian and are usually subdivided further on the basis of behavior in simple shear flow. 

Consistent with this model, foams exhibit plug flow when forced through a channel or pipe. In the center of the channel the foam flows as a soHd plug, with a constant velocity. AH the shear flow occurs near the waHs, where the yield stress has been exceeded and the foam behaves like a viscous Hquid. At the waH, foams can exhibit waH sHp such that bubbles adjacent to the waH have nonzero velocity. The amount of waH sHp present has a significant influence on the overaH flow rate obtained for a given pressure gradient. 

Plastic Forming. A plastic ceramic body deforms iaelastically without mpture under a compressive load that produces a shear stress ia excess of the shear strength of the body. Plastic forming processes (38,40—42,54—57) iavolve elastic—plastic behavior, whereby measurable elastic respoase occurs before and after plastic yielding. At pressures above the shear strength, the body deforms plastically by shear flow. 

The force is direcdly proportional to the area of the plate the shear stress is T = F/A. Within the fluid, a linear velocity profile u = Uy/H is estabhshed due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient y = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. 

Bai [48] presents a linear stability analysis of plastic shear deformation. This involves the relationship between competing effects of work hardening, thermal softening, and thermal conduction. If the flow stress is given by Tq, and work hardening and thermal softening in the initial state are represented... 

Figure 10 shows that upon cessation of shear flow of the melt, shear stress relaxation of LLDPE is much faster than HP LDPE because of the faster reentangle-... 

Important for polymer processing is the fact that when the concentration of a hard filler is increased in the composite, the unsteady flow (in the sense of large-scale distortions) of the extrudate occurs at higher shear rates (stresses) than in the case of the base polymer [200, 201,206]. Moreover, the whirling of the melt flow is even suppressed by small additions of filler [207]. 

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],... 

At least, in absolute majority of cases, where the concentration dependence of viscosity is discussed, the case at hand is a shear flow. At the same time, it is by no means obvious (to be more exact the reverse is valid) that the values of the viscosity of dispersions determined during shear, will correlate with the values of the viscosity measured at other types of stressed state, for example at extension. Then a concept on the viscosity of suspensions (except ultimately diluted) loses its unambiguousness, and correspondingly the coefficients cn cease to be characteristics of the system, because they become dependent on the type of flow. 

Unfortunately, a few papers are known where normal stresses during shear flow of filled polymers were measured directly. Here an additional problem is connected with the solution of the problem what is considered a one-valued measure of elasticity of a material and under what conditions to compare the measured values of normal stresses. Moreover, the data at hand often represent rather a contradictory picture. 

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