Torque stress

For low-viscosily fluids, one manner of obtaining increasing the torque (stress) value is to increase the area of shear as shown in Figure 3-5. Compared to the geometry... 

Consider as a specific example the torque acting on the parallelepiped about the z-axis. The normal stresses Taa do not contribute to this torque. Stresses and do not contribute because they point in the z-direction (see Fig. 1.2). Similarly, r and Tyz cannot add to the net torque because thej are balanced by stresses that are equal in magnitude but point in the opposite direction on the bottom plane z = 0. This then leaves us with two remaining contributions to the net torque, namely Ty (dydz) d.T and T y dxdz) dy. 

Figure 2.11 Complex torque (stress) signal(---) obtained by superposition of the three waveforms shown in Figure 2.10 and a computer generated typical strain response( )...

Both controlled-angular displacement (strain-controlled) and controlled-torque (stress-controlled) rotational rheometers are used, with the former giving superior performance at high frequencies and the latter better precision at low frequencies. There have also recently appeared on the market instruments said to be capable of operating in both modes. Controlled torque instruments can also be used to make creep and creep recovery tests, which are described in the next section. In order to obtain a linear viscoelastic characterization that includes the terminal zone, it is sometimes useful to combine data from oscillatory flow with those from a creep experiment, and this is also discussed in the following section. Rheometrical methods are described in some detail in several books [3,8,9]... 

The components tij form the stress tensor and the components and belong to torque stresses. Recalling that dSj can be replaced by UjdS where i/ is the outward unit normal to the surface 5, we can use the constraints (6.53)2, (6.54)2 and (6.55)2 and equate the corresponding surface terms appearing as coefficients of Sxiy Ao and Aci in the surface integrals in equations (6.50) and (6.69). Doing so allows us to express the surface forces in terms of the corresponding stress tensors and the outward unit normal i/ as... 

Next, it is important to determine if the material requires pre-treatment (such as pre-shearing) before measurements. This can be determined from the pseudoviscosity profile of the material. Pre-shearing will determine a zero-time of shear, thereby eliminating any structure history prior to loading. This is done by performing a continnons flow test under the broad torque range. The data can be viewed as viscosity versus torque/stress and converted to viscosity versus shear rate. 

Figure 5,16. It is assumed that by using an exactly symmetric cone a shear rate distribution, which is very nearly uniform, within the equilibrium (i.e. steady state) flow held can be generated (Tanner, 1985). Therefore in this type of viscometry the applied torque required for the steady rotation of the cone is related to the uniform shearing stress on its surface by a simplihed theoretical equation given as...

The stress field corresponding to this regime is shown in Figure 6.18. As this figure shows the measuring surface of the cone is affected by these secondary stresses and hence not all of the measured torque is spent on generation of the primary (i.e, viscometric) flow in the circumferential direction. 

The equations and methods for determining viscosity vary greatly with the type of instmment, but in many cases calculations may be greatly simplified by calibration of the viscometer with a standard fluid, the viscosity of which is known for the conditions involved. General procedures for calibration measurement are given in ASTM D2196. The constant thus obtained is used with stress and shear rate terms to determine viscosity by equation 25, where the stress term may be torque, load, or deflection, and the shear rate may be in rpm, revolutions per second (rps), or s F... 

Therefore, the viscosity can be determined from the torque and angular velocity. However, the viscosity is usually calculated from the shear rate and shear stress, which can be obtained from the Margules equation. The shear rate is given by equation 27, where r is any given radius. 

The shear rate and shear stress can be calculated for any radius r from these equations. In most cases the radius used is R because the shear stress and shear rate of interest are at the inner, torque-sensing cylinder. Thus equations 27 and 28 become... 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

Fig. 29. Measurements of yield stress with a vane device and Rheometrics controlled stress rheometer. The torque required to cause yielding is between 1.88...

Specific Commercial Rotational Viscometers. Information on selected commercial rotational viscometers can be found ia Table 7. The ATS RheoSystems Stresstech rheometer is an iastmment that combines controlled stress as well as controlled strain (shear rate) and oscillatory measurements. It has a torque range of 10 to 50 mN-m, an angular velocity range of 0 to 300 rad/s, and a frequency range of seven decades. Operation and temperature programming (—30 to 150°C higher temperatures optional) are computer controlled. 

The stress resulting from the axial forces must be considered in analy2ing the parts which resist axial separation, such as bolts, nuts, rings, etc. On scroU centrifuges the axial force owing to the axial component of the conveyor torque must also be considered. 

When 4-(mercaptoacetamido)diphenylamine [60766-26-9] (39) is added to EPDM mbber and mixed in a torque rheometer for 15 minutes at 150°C, 87% of it chemically binds to the elastomer (24). The mechanical and thermal stress placed on the polymer during mixing mptures the polymer chain, producing radicals that initiate the grafting process. 

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). 

There are no thermal-expansion problems in spirals. Since the center of the unit is not fixed, it can torque to reheve stress. 

Fig. 19.7. A rotation viscometer. Rotating the inner cylinder shears the viscous glass. The torque (and thus the shear stress aj is measured for a given rotation rate (and thus shear strain rate y).

Figure 4.41 shows the Stress-Strength Interference (SSI) diagrams for the two assembly operation failure modes. The instantaneous stress on the relief section on first assembly is composed of two parts first the applied tensile stress,. v, due to the pre-load, F, and secondly, the torsional stress, t, due to the torque on assembly, M, and this is shown in Figure 4.41(a) (Edwards and McKee, 1991). This stress is at a maximum during the assembly operation. If the component survives this stress, it will not fail by stress rupture later in life. 

The shear stress, t, due to the assembly torque diminishes to zero with time, the preload, F, remaining constant, and so the stress on the solenoid section is only the direct stress,. v, as given in equation 4.75 (see Figure 4.41(b)) (Edwards and McKee, 1991). A second reliability can then be determined by considering the requirement that the pre-load stress remains above a minimum level to avoid loosening in service (0.5 S/)min from experiment) (Marbacher, 1999). The reliability, R, can then be determined from the probabilistic requirement, P, to avoid loosening ... 

It is required to find the torque without slippage that can be transmitted by a hub that is assembled by an interference fit to a powered shaft. The hub outside diameter D = 070 mm, and the shaft diameter d = 050 mm, as shown in Figure 4.55. The length of the hub is 100 mm. Both hub and shaft are machined from hot rolled steel SAE 1035 with a yield strength Sy A(342,26) MPa (see Table 4.6). Given that the hub is stopped suddenly in service due to a malfunction, and considering only the torsional stresses, what is the probability that the shaft will yield ... 

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