Finite shear stress

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

The rheological properties of gum and carbon black compounds of an ethylene-propylene terpolymer elastomer have been investigated at very low shear stresses and shear rates, using a sandwich rheometer [50]. Emphasis was given to measurements of creep and strain recovery at low stresses, at carbon black flller contents ranging between 20 and 50% by volume. The EPDM-carbon black compounds did not exhibit a zero shear rate viscosity, which tended towards in-Anity at zero shear stress or at a finite shear stress (Fig. 13). This was explained... 

Bingham Plastic or Plastic Fluids. As shown in Fig. 2, this is the simplest of all non-Newtonian fluids in the sense that the relationship between shear stress and shear rate differs from that of a Newtonian fluid only by the fact that the linear relationship does not pass through the origin. Thus a finite shearing stress r is necessary to initiate move-... 

The symmetry of the stress tensor can be established using a relatively straightforward argument. The essence of the argument is that if the stress tensor were not symmetric, then finite shearing stresses would accelerate the angular velocity w of a differential fluid packet without bound—something that obviously cannot happen. 

Shear in an epoxy-bonded film. The results, in Figure 5.2, show that there is a linear relationship between compressive stress and shear stress, but that there is a finite shear stress in the uncompressed state. As a result, the coefficient of friction decreases as the contact pressure increases as shown in Figure 5.3. 

Fluids with a Yield Stress. Both pseudoplastic and dilatant fluids are characterized by the fact that no finite shear stress is required to make the fluids flow. A fluid with a yield stress is characterized by the property that a finite shear stress, To, is required to make the fluid flow. A fluid obeying... 

Generalized Plastic Fluid A fluid characterized by both of the following the existence of a finite shear stress that must be applied before flow begins (yield stress), and pseudoplastic flow at higher shear stresses. See also Bingham Plastic Fluid. 

For complicated fluids, such as toothpaste, paints, and jellies, Eq. 2.1 is not correct, because the fluids can sustain small but finite shear stresses without any motion. The equation simply is not applicable. To find its equivalent, it is necessary to make up a sum of forces which includes shear forces on the vertical sides of the cube. 

The mass averages of the molar mass and the chain-link number are used in these relationships since experiments have shown that the viscosities of polymers of various molar mass distributions at zero shear stress depend on the mass average molar mass. At finite shear stresses, polymers of the same molar mass average have higher viscosities when the distribution is narrower, since the lower-molar-mass component appears to function as a lubricant. But it has not been completely established which is the correct molar mass average to use. 

Nussdt himself derived the development of the film thickness and the heat transfer coefficient in case of laminar flow and neglected shear stress at the film surface [27]. Regarding a finite shear stress the film thickness of a condensating pure vapor phase at a distinct vertical position x reads as follows [42, 58] ... 

Above a critical packing fraction ( >c, foams and emulsions "jam" into a solid phase that can sustain finite shear stresses under shear strain. We will model the macroscopic mechanical response of foams and emulsions above, but still near, this rigidity transition within the context of Durian s "bubble model" for wet foams [8]. 

Rheology. The rheology of foam is striking it simultaneously shares the hallmark rheological properties of soHds, Hquids, and gases. Like an ordinary soHd, foams have a finite shear modulus and respond elastically to a small shear stress. However, if the appHed stress is increased beyond the yield stress, the foam flows like a viscous Hquid. In addition, because they contain a large volume fraction of gas, foams are quite compressible, like gases. Thus foams defy classification as soHd, Hquid, or vapor, and their mechanical response to external forces can be very complex. 

Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large paraU plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a sohd, which would undergo only a finite deformation. 

Fig. 2.4. Within the elastic range it is possible to relate uniaxial strain data obtained under shock loading to isotropic (hydrostatic) loading and shear stress. Such relationships can only be calculated if elastic constants are not changed with the finite amplitude stresses.

Strain gages may be applied to the test unit at all points where high stresses are anticipated, provided that the configuration of the units permits such techniques. The use of finite element analysis, models, brittle lacquer, etc., is recommended to confirm the proper location of strain gages. Three-element strain gages are recommended in critical areas to permit determination of the shear stresses and to eliminate the need for exact orientation of the gages. 

Some materials have the characteristics of both solids and liquids. For instance, tooth paste behaves as a solid in the tube, but when the tube is squeezed the paste flows as a plug. The essentia] characteristic of such a material is that it will not flow until a certain critical shear stress, known as the yield stress is exceeded. Thus, it behaves as a solid at low shear stresses and as a fluid at high shear stress. It is a further example of a shear-thinning fluid, with an infinite apparent viscosity at stress values below the yield value, and a falling finite value as the stress is progressively increased beyond this point. 

A fluid with a finite yield. stress is sheared between two concentric cylinders, 50 mm long. The inner cylinder is 30 mm diameter and the gap is 20 mm. The outer cylinder is held stationary while a torque is applied to the inner. The moment required just to produce motion was 0.01 N m. Calculate the force needed to ensure all the fluid is flowing under shear if the plastic viscosity is 0.1 Ns/ni2. 

Fig. 13. Shear stress t12 and first normal stress difference N1 during start-up of shear flow at constant rate, y0 = 0.5 s 1, for PDMS near the gel point [71]. The broken line with a slope of one is predicted by the gel equation for finite strain. The critical strain for network rupture is reached at the point at which the shear stress attains its maximum value...

Because the shear stress is always zero at the centerline in pipe flow and increases linearly with distance from the center toward the wall [Eq. (6-4)], there will be a finite distance from the center over which the stress is always less than the yield stress. In this region, the material has solid-like properties and does not yield but moves as a rigid plug. The radius of this plug (r0) is, from Eq. (6-4),... 

As an indenter creates an indentation it causes at least three types of finite deformation. It punches material downwards creating approximately circular prismatic dislocation loops. At the surface of the material it pushes material sideways. It causes shear on the planes of maximum shear stress under itself. Therefore, the overall pattern of deformation is very complex, and is reflected... 

Goel VK et al (1995) Interlaminar shear stresses and laminae separation in a disc finite element analysis of the L3-L4 motion segment subjected to axial compressive loads. Spine (Phila Pa 1976) 20(6) 689-698... 

The shear stress must remain finite at r = 0 so A = 0. Thus the shear stress distribution is given by... 

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