Stress cylindrical coordinates

In general, the velocity profile will be curved but as equation 1.33 contains only the local velocity gradient it can be applied in these cases also. An example is shown in Figure 1.13. Clearly, as the velocity profile is curved, the velocity gradient is different at different values of y and by equation 1.32 the shear stress r must vary withy. Flows generated by the application of a pressure difference, for example over the length of a pipe, have curved velocity profiles. In the case of flow in a pipe or tube it is natural to use a cylindrical coordinate system as shown in Figure 1.14. 

In cylindrical coordinates, the velocity gradient dvjdr generates the shear stress component rrx and Newton s law must be expressed in the two sign conventions as ... 

For the cylindrical coordinates of the shear-lag model shown in Fig. 4.6, the governing conditions adopted in this analysis are essentially the same as those described in Section 4.2.3, There is one exception in that the mechanical equilibrium condition between the external stress, internal stress components given by Eq. (4.11) is replaced by... 

For the cylindrical coordinates of the fiber push-out model shown in Fig. 4.36 where the external (compressive) stress is conveniently regarded as positive, the basic governing equations and the equilibrium equations are essentially the same as the fiber pull-out test. The only exceptions are the equilibrium condition of Eq. (4.15) and the relation between the IFSS and the resultant interfacial radial stress given by Eq. (4.29), which are now replaced by ... 

In the most general case, stresses on any of the six control-volume faces can potentially contribute to a force in any direction. In a cartesian coordinate system, only stresses in a certain direction can contribute to a force in that direction. In cylindrical coordinates and other noncartesian systems, the situation is more complex. As an example of this point, consider Fig. 2.15, which is a planar representation of the z face of the cylindrical differential element. Notice two important points that are revealed in this figure. One is that the the area of the 0 face varies from rdO on one side to (r + dr)dO on the other. Therefore, in computing net forces, the area s dependence on the r coordinate must be included. Specifically,... 

Fig. 2.16 In general, the stress state represented on a differential element in a cylindrical coordinate system has nine stress components. The same stress state can be represented as its principal components via a coordinate rotation.

Writing the strain-rate components in terms of the velocity field (Section 2.5) yields a general relationship between the flow field and the stress tensor in a particular coordinate system. For example, in a cylindrical coordinate system... 

Working in cylindrical coordinates, and substituting the force-per-unit-volume expressions that stem from the stress tensor (Eqs. 2.137, 2.138, and 2.139), the Navier-Stokes equations can be written as... 

The (stress or strain-rate) state at a point is a physical quantity that cannot depend on any particular coordinate-system representation. For example, the stress state is the same regardless of whether it is represented in cartesian or cylindrical coordinates. In other words, the state (as represented by a symmetric second-order tensor), is invariant to the particular coordinate-system representation. 

For convenience in the following discussion, Eq. (2.17) is expressed in cylindrical coordinates. The symmetric stress tensor T is given as... 

Therefore, with the substitution of Eq. (2.50) into Eq. (2.48), the stresses due to a normal force on the boundary of a semiinfinite solid are expressed in cylindrical coordinates by... 

If Te and Ts are respectively the transformation matrices for the strain and stress vectors the transformation matrix which allows writing the vectors in the cylindrical coordinate, we get ... 

Let us consider now the deformation and stresses of a cylindrical pipe under two different boundary conditions (Fig. 16.2). In both eases the length of the pipe is considered constant according to the requirements for a plane strain problem. The external and internal radii are R2 and R, respectively. If the applied forces and the displacements are also uniform, the deformation is purely radial, and in cylindrical coordinates = u r). According to the Navier equations, rot u = 0. Hence, Vdiv u = 0, which implies... 

According to these equations and using, as above, cylindrical coordinates in the stress-strain relationships, the zz component of the stress tensor can be written as... 

For solid walls, no-penetration and no-slip are typically applied to the momentum equation. Boundary conditions such as velocity, pressure and temperature at the inlet are usually known and specified, whereas their counterparts at the outlet are derived from assumptions of no-stress or fully developed and simulated flow. The thermal wall boundary conditions influence the flow significantly. Simple assumptions of constant wall temperature, insulated side walls and constant wall heat transfer flux have been used extensively for simple applications. More specifically, the following assumptions are normally made for a retort as shown in Figure 6.29 (the directions of the velocity in the following description refer to a cylindrical coordinate system in this figure). 

Boundary layer approximation. The Landau problem, which was described above, is an example of an exact solution of the Navier-Stokes equations. Schlichting [427] proposed another approach to the jet-source problem, which gives an approximate solution and is based on the boundary layer theory (see Section 1.7). The main idea of this method is to neglect the gradients of normal stresses in the equations of motion. In the cylindrical coordinates (71, ip, Z), with regard to the axial symmetry (Vv = 0) and in the absence of rotational motion in the flow (d/dip = 0), the system of boundary layer equations has the form... 

For mode 1 loading, the stresses near the tip of a crack in a plate are given from Eqn. (3.29) in polar-cylindrical coordinates, with the z-axis along the crack front, the x-axis in the direction of crack prolongation, and the y-axis perpendicular to the crack plane. 

When a well is drilled, the stress state of the rock around the wellbore is altered. The stresses around the wellbore are a function of position and, because of geometry, can be expressed in cylindrical coordinates (Figure 3b). The general solutions for the wellbore stresses can be quite complicated depending on the orientation of the well and the axes of... 

The stress is a rank two tensor. The relevant components here are the normal and tangential stress, namely, the components and T rs. They both enter the expression of forces exerted on the element of surface parallel to the 1/v interface. Let us first look at part. Its only contribution comes from the viscous stress. Its general expression in cylindrical coordinates is ... 

Shear-stress components for Newtonian fluids in cylindrical coordinates... 

The stress tensor in cylindrical coordinate system is given by... 

The discussion here follows directly from that in Section 2.2. A film of thickness h is bonded to a substrate of thickness hs, with no restrictions on the thickness ratio. The stress and deformation fields are referred to a cylindrical coordinate system with polar coordinates in the plane of the system and with the z—direction normal to the interface the origin of coordinates lies in the substrate midplane. The equi-biaxial stress components are referred to polar coordinates, but they could equally well be expressed in rectangular coordinates. As long as the response is in the range of geometrically linear behavior, the common shape of the film and substrate in plan view is immaterial. As in Section 2.2, the biaxial elastic moduli of the film and substrate are Mf and Mg, respectively, and the corresponding coefficients of linear thermal expansion are af and dg, respectively. 

Components of stress tensor in cylindrical coordinates Components of stress tensor in rectangular Cartesian coordinates Shear stress acting on mandrel surface [Eq. (46b)]... 

This is also calculated for the cylindrical coordinate system using equation [1.2] and the stress tensor (Table 1.2) for the velocity field [1.23] of the Poiseuille flow in a circular pipe. 

The height, H, of the apparatus is sufficiently large, and the gap width Rt R sufficiently small, to be able to neglect edge effects at the top and bottom in the force balance. The displacements and stresses can be regarded as axisyrmnetric with respect to the Oz axis and independent from z. Therefore, in the cylindrical coordinate system with the Oz axis, the different variables depend not on variables 6 and z, but only on r. Consequently, the stress tensor is written in the cylindrical coordinate system (Chapter 1, Table 1.2) as ... 

Returning to the problem of the blocked pipe, consider the flow in the cylindrical coordinate system whose Oz axis, which coincides with the axis of the pipe (see Figure 1.3 of Chapter 1), is oriented in the direction of the flow. By considering the non-zero component(s) in the velocity vector, verify that the only non-zero term in the strain rate tensor is Drz. Set out the form of this term explicitly and explain why ) 5 0. Explain also why the only non-zero shear stress in the pipe is Trz, and why Vrz <0. It will be deduced there from that the rheological relation for the Bingham fluid can be written, for the flow of that fluid in a pipe, as ... 

All the elasticity equations given by Eqs. 9.29 - 9.32 as well as the biharmonic stress function equation can be developed for cylindrical coordinates (see, Timoshenko and Goodier, (1970)). The biharmonic equation is written as,... 

Since one of the boundary conditions in Eki. 9.46 is now a displacement boundary condition, we also require the expressions for the displacements in terms of the constants A and C. These are found first by using the stress-strain-displacement relations and then by integrating the strain components to determine the displacements. The stress-strain-displacement equations for the condition of plane strain in cylindrical coordinates are,... 

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