The Description of Stress and Material Functions

In dealing with the state of stresses of incompressible fluids under deformation or in flow, the total stress tensor T is divided into two parts  

Special types of states of stress are of particular importance. In a liquid that has been at rest (i.e., there is no deformation of a fluid) for a sufficiently long time, there is no tangential component of stress on any plane of a cube and the normal component of stress is the same for all three planes, each perpendicular to the others. This is the situation where only hydrostatic pressure, —p, exists. In such a situation, Eq. (2.113) reduces to 

Note that Eq. (2.116) can also be obtained from Eq. (2.113) with the assumption, ffii + 022 + 0 33 = 0. Since such an assumption is quite arbitrary, the definition of 

If we now consider the state of stress in an isotropic material, by definition the material has no preferred directions. In simple shear flow, we have 

Note that one cannot measure p and the components of the extra stress tensor a separately during flow of a liquid. Therefore, the absolute value of any one normal component of stress is of no rheological significance. The values of the differences of normal stress components are, however, not altered by the addition of any isotropic pressure (see Eq. (2.118)), and they presumably depend on the rheological properties of the material. It follows, therefore, that there are only three independent stress quantities of rheological significance, namely, one shear component and two differences of normal components  

---