Stress shear components

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. 

Tyx shear stress component, force in x direction on y area component,... 

Now the eddies transport momentum and the corresponding momentum flux components are equivalent to (negative) shear stress components ... 

Tyx is the shear stress component associated with the velocity gradient dvjdy and corresponding equations hold for other components. 

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 ... 

Both sign conventions are used in the fluid flow literature and consequently the reader should be able to work in either, as appropriate. The negative sign convention is convenient for flow in pipes because the velocity gradient dvjdr is negative and therefore the shear stress components turn out to be positive indicating that they physically act in the directions assumed in the sign convention. This is illustrated in Example 1.9. 

Independent of the sign convention used, the stress components can be classified into two types those that act tangentially to the face of the element and those that act normal to the face. Tangential components such as rxy, ryx, Tyz tend to cause shearing and are called shear stress components (or simply shear stresses). In contrast, the stress components rxx, Tyy, jzz act normal to the face of the element and are therefore called normal stress components (or normal stresses). Although there are six shear stress components, it is easily shown that ri = t, for t = j for example, ryx = rxy. Thus there are three independent shear stress components and three independent normal stress components. 

By analogy, equation 1.68 shows that a shear stress component can be interpreted as a flux of momentum because it is proportional to the gradient of the concentration of momentum. In particular, ryx is the flux in the y-direction of the fluid s x-component of momentum. Furthermore, the kinematic viscosity v can be interpreted as the diffusivity of momentum of the fluid. 

Velocity fluctuations can also cause extra apparent shear stress components. An element of fluid with a non-zero velocity component in the x-direction possesses an x-component of momentum. If this element of fluid also has a non-zero velocity component in the y-direction then as it moves in the y-direction it carries with it the x-component of momentum. The mass flow rate across a plane of area 8x8z normal to the y-coordinate direction is pvy8x8z and the x-component of momentum per unit mass is vx, so the rate of transfer of x-momentum in they-direction is given by the expression... 

In the case of laminar flow in a pipe, work is done by the shear stress component rTX and the rate of doing work is the viscous dissipation rate, that is the conversion of kinetic energy into internal energy. The rate of viscous dissipation per unit volume at a point, is given by... 

The distribution of the shear stress components is shown schematically in... 

The shear stress component is ryx but the subscripts have been omitted for brevity. 

In this array, the stress components in the first row act on a plane perpendicular to the X axis, the stress components in the second row act on a plane perpendicular to the y axis, and the stress components in the third row act on a plane perpendicular to the z axis. The stress matrix of Eq. (5.6) is symmetric, that is, the complementary shear stress components are equivalent—for example, % = Xyx, and so on. 

Hence, only six independent components of the stress tensor are needed to fully define the state of the stress at point P, where n u are the normal stress components, and it (i / j) are the shear-stress components. 

With the shear rate at hand, we can calculate the local viscous dissipation per unit volume. From Table 2.3 we note that the only nonvanishing shear-stress component is zyz — which is given by... 

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

We therefore observe that unlike in the Power Law model solution with a single shear stress component, xn, in the case of a CEF model, we obtain, in addition, two nonvanishing normal stress components. Adopting the sign convention for viscometric flow, where the direction of flow z is denoted as 1, the direction into which the velocity changes r, is denoted as 2, and the neutral direction 8 is denoted as direction 3, we get the expressions for the shear stress in terms of the shear rate, the primary, and secondary normal stress differences (see Eqs. 3.1-10 and 3.1-11) ... 

Polanyi and Mark discovered the slip properties of single-crystal tin, while Schmid worked out a law for the shear stress component along the slip direction in a slip plane. [44]... 

When studying the stability of the steady-state, time-dependent calculations are needed (see [7]). It can also be used as a simple method to compute the steady-state solution. A time-dependent approach using the Lesaint-Ravian technique for the normal stress components and the Baba-Tabata scheme for the shear stress component is developed by Saramito and Piau ([34]). This method allows one to obtain rapidly stationary solutions of the PTT models. Convergence with mesh refinement is obtained as well as oscillation-free solutions. 

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