Stress tensor sign convention

The finite control-volume dimensions as illustrated in Fig. 2.13 may be a potential source of confusion. While the stress tensor represents the stress state at a point, it is only when the differential control volume is shrunk to vanishingly small dimensions that it represents a point. Nevertheless, the control volume is central to our understanding of how the stress acts on the fluid and in establishing sign conventions for the stress state. For example, consider the normal stress xrr, which can be seen on the r + dr face in the left-hand panel and on the r face in the right-hand panel. Both are labeled rrr, although their values are only equal when the control volume has shrunk to a point. Since the stress state varies continuously and smoothly throughout the flow, the stress state is in fact a little different at the centers of the six control-volume faces as illustrated in Fig. 2.13 where the... 

Note that we differentiate the stress tensor n discussed in this section from the previously discussed stress tensor n because they are defined on the basis of different sign conventions, as discussed later in the chapter. 

Sign Convention of the Stress Tensor x Consider a linear shear flow and examine... 

The stress tensor may be represented as a 3 x 3 matrix, with components Oy, where i and j both go from 1 to 3. The diagonal elements represent normal stresses, whereas the off-diagonal ones represent shear stresses. Positive normal stresses are tensile, while negative ones are compressive (but an opposite sign convention is sometimes used, most notably in the soil mechanics literature). Finally, from the balance of angular momentum (or torque in the static case), it follows that the stress tensor and its matrix representation are symmetric (ay = aji), meaning that only six out of the nine components are in fact independent. 

Fig. 1.3. A sketch of a Cartesian fixed control volume showing the surface forces acting on a moving fluid eiement. This Fig. is based on the illustrations of the surface forces given in the textbooks on fluid mechanics by [184], pp. 65-72 [185], 225-230 [2], pp. 60-66. However, contrary to the usual sign conventions applied in these books, the sign convention used in this book for the total stress tensor follows the approach given by Bird et al [11] [13]. Therefore, a pre factor of —1 have been introduced for the forces in the Fig. Only the surface forces in the a -direction are shown.

The minus sign in this equation is a matter of convention t(n) is considered positive when it acts inward on a surface whereas n is the outwardly directed normal, andp is taken as always positive. The fact that the magnitude of the pressure (or surface force) is independent of n is self-evident from its molecular origin but also can be proven on purely continuum mechanical grounds, because otherwise the principle of stress equilibrium, (2 25), cannot be satisfied for an arbitrary material volume element in the fluid. The form for the stress tensor T in a stationary fluid follows immediately from (2 59) and the general relationship (2-29) between the stress vector and the stress tensor ... 

It should be noted that the sign convention adopted here for components of the stress tensor is consistent with that found in many fluid mechanics and heat transfer books however, it is opposite to that found in some books on transport phenomena, e.g., Refs. 10,11, and 14. 

For readers not familiar with this notation, a few words of explanation may be useful. The indices on the typical component of the stress tensor have the following meaning. The second index indicates that this component of the stress acts in the Xj direction, while the first index indicates that it acts on a plane normal to the X axis. A component is positive when it acts on a fluid element in the plus Xj direction on the face of that element having the larger value of Xj. Thus, a tensile stress has a positive value, while a compressive stress is negative. Note that this sign convention is not used universally. 

---