Stress components

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Thus, tomographic photoelastic measurements enable only calculation of the distribution of (Tzz and - (Tzz- For determining the other stress components one needs additional information. 

The axial stress is the only stress component which can be determined directly from measurement data. Hence, we have the boundary-value problem with equations (27), (29)-(31) and the boundary conditions (34)-(36). 

Thus, the harmonic function >P(2 ,y) is a function of two variables which can be determined from the boundary conditions. This follows also from the fact that If the distribution of is described only by harmonic functions, the other stress components do not develop in cylinders [2]. 

Let us assume that stress gradient in axial direction is present but smooth. Then we can use a perturbation method and expand the solution of equation (30) in a series. The first term of this expansion will be a solution of the plane strain problem and potential N will be equal to zero. The next terms of the stress components will contain potential N also. 

Equations (1.6) and (1.7) are used to formulate explicit relationships between the extra stress components and the velocity gradients. Using these relationships the extra stress, t, can be eliminated from the governing equations. This is the basis for the derivation of the well-known Navier-Stokes equations which represent the Newtonian flow (Aris, 1989). 

In the differential models stress components, and their material derivatives, arc related to the rate of strain components and their material derivatives. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Using a known solution at the inlet. To provide an example for tins option, let us consider the finite element scheme described in Section 2.1. Assuming a fully developed flow at the inlet to the domain shown in Figure 3.3, v, (dvy/dy) = 0 and by the incompressibility condition (dvx/dx) - 0, x derivatives of all stress components are also zero. Therefore at the inlet the components of the equation of motion (3.25) are reduced to... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

Step 5 - using updated values of viscosity and calculated vg calculate v,., 17 and p. For viscoelastic fluids also calculate the additional stress components at this step. 

CALCULATES PRESSURE AND STRESS COMPONENTS AT REDUCED, INTEGRATION POINTS AND WRITES INTO OUTPUT FILE. 

VAKIA.TIONAL RECOVERY OF PRESSURE, AND STRESS COMPONENTS AT NODES... 

The shear stresses are proportional to the viscosity, in accordance with experience and intuition. However, the normal stresses also have viscosity-dependent components, not an intuitively obvious result. For flow problems in which the viscosity is vanishingly small, the normal stress component is negligible, but for fluid of high viscosity, eg, polymer melts, it can be significant and even dominant. 

Figure 9.2 Longitudinal stress-corrosion cracks in a heat exchanger tnbe the broad gap between the crack faces reveals that high-level residual hoop (circumferential) stresses from the tube-forming operation provided the stress component required for SCC.

In the last chapter we said that one of the requirements of a high-temperature material - in a turbine blade, or a super-heater tube, for example - was that it should resist attack by gases at high temperatures and, in particular, that it should resist oxidation. Turbine blades do oxidise in service, and react with H2S, SO2 and other combustion products. Excessive attack of this sort is obviously undesirable in such a highly stressed component. Which materials best resist oxidation, and how can the resistance to gas attack be improved ... 

The main causes of failure in gear couplings are wear or surface fatigue caused by lack of lubricant, incorrect lubrication, or excessive surface stresses. Component fracture caused by overload or fatigue is generally of secondary importance. 

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

The stress singularity for this and the other stress components Oy and... 

The stress-intensity factors are quite different from stress concentration factors. For the same circular hole, the stress concentration factor is 3 under uniaxial tension, 2 under biaxiai tension, and 4 under pure shear. Thus, the stress concentration factor, which is a single scalar parameter, cannot characterize the stress state, a second-order tensor. However, the stress-intensity factor exists in all stress components, so is a useful concept in stress-type fracture processes. For example. 

---