Sign conventions for stress

In analysing flow problems it is usual to select one coordinate axis (the x-coordinate axis in the above examples) to be parallel to the flow with the coordinate value increasing in the direction of flow so that the fluid s velocity is positive. If the velocity component vx varies in they-direction, it is normal to define the shear rate in terms of the change of vx in the positive y-direction, so 

When analysing simple flow problems such as laminar flow in a pipe, where the form of the velocity profile and the directions in which the shear stresses act are already known, no formal sign convention for the stress components is required. In these cases, force balances can be written with the shear forces incorporated according to the directions in which the shear stresses physically act, as was done in Examples 1.7 and 1.8. However, in order to derive general equations for an arbitrary flow field it is necessary to adopt a formal sign convention for the stress components. 

Before describing the two sign conventions that may be used, it may be helpful to consider a loose analogy with elementary mechanics. It is required to calculate the acceleration of a car from its mass and the forces acting on the car, all of which are known. It is necessary to evaluate the net 

The way to remember the conventions is as follows. For the positive sign convention, the stress component acting on the element s face at the higher y-value (the upper face) is taken as positive in the positive x-direction for the negative sign convention the same component is taken as positive in the negative x-direction. In each convention, the stress component acting on the opposite face is taken as positive in the opposite direction. 

With the positive sign convention, Newton s law of viscosity is expressed as 

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The information presented in this work builds upon developments from several more established fields of science. This situation can cause confusion as to the use of established sign conventions for stress, pressure, strain and compression. In this book, those treatments involving higher-order, elastic, piezoelectric and dielectric behaviors use the established sign conventions of tension chosen to be positive. In other areas, compression is taken as positive, in accordance with high pressure practice. Although offensive to a well structured sense of theory, the various sign conventions used in different sections of the book are not expected to cause confusion in any particular situation. 

Using the negative sign convention for stress components, the shear stress acting on the outer surface of the element (the higher value of r) must be measured in the negative x-direction and that on the inner surface in the positive x-direction, as indicated in Figure 1.17. 

Laminar flow in a pipe showing a typical fluid element and the velocity profile. The negative sign convention for stress components is shown... 

Negative sign convention for stress components. The diagram shows the directions in which components are taken as positive. The components acting on the faces normal to the x-axis have been omitted for clarity... 

The term viscosity has no meaning for a non-Newtonian fluid unless it is related to a particular shear rate y. An apparent viscosity fia can be defined as follows (using the negative sign convention for stress) ... 

It should be noted that for shear thinning and shear thickening behaviour the shear stress-shear rate curve passes through the origin. This type of behaviour is often approximated by the power law and such materials are called power law fluids . Using the negative sign convention for stress components, the power law is usually written as... 

It has been assumed that the flow is incompressible so that there are no fluctuations of the density. Equation 1.91 shows that the momentum flux consists of a part due to the mean flow and a part due to the velocity fluctuation. The extra momentum flux is proportional to the square of the fluctuation because the momentum is the product of the mass flow rate and the velocity, and the velocity fluctuation contributes to both. The extra momentum flux is equivalent to an extra apparent stress perpendicular to the face, ie a normal stress component. As (v x)2 is always positive it produces a compressive stress, which is positive in the negative sign convention for stress. 

In equation 1.94, (Tyx)v is the viscous shear stress due to the mean velocity gradient dvjdy and pv yv x is the extra shear stress due to the velocity fluctuations v x and v y. These extra stress components arising from the velocity fluctuations are known as Reynolds stresses. (Note that if the positive sign convention for stresses were used, the sign of the Reynolds stress would be negative in equation 1.94.)... 

Consider a fixed element of space with unit area in the x-z plane and having its surfaces at distances y and y+5y from the plate. Using the negative sign convention for stress components (which coincides with the... 

Using the negative sign convention for stress components, the surface force Sx can be written as... 

All the stress components shown are taken as positive in the directions indicated. Each stress component is written with two subscripts, the first denoting the face on which it acts and the second the direction in which the stress acts. Thus the stress component ryx acts on a face normal to the y-axis and in the x-direction. There are two ryx terms, one acting on the left face (aty) and the other on the right face (aty+5y). With the negative sign convention for stress components, the components ryx, ry, at the... 

The negative sign in this equation arises from the sign convention for stress and pressure. The stress in the liquid is positive when the liquid is in tension the pressure follows the opposite sign convention, so tension is negative pressure. 

In this book we are considering porous materials. Therefore, the stress treated here must be an effective stress a = a + pi where a is the total stress, and p is the pore fluid pressure. Note that in this section we are using the sign convention for stresses adopted in continuum mechanics, therefore the tension stress/strain is considered positive, and the pore fluid pressure is positive, since a = a+pl (details are described in Chap. 6). In this Section we denote the stress as effective stress a for simplicity. Readers can see that all results in this section also work for the effective stress. It should be noted that in this section the deviatoric stress is denoted as s whereas in other expositions the deviatoric stress is written as a. Similarly, the deviatoric strain is denoted as e. 

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