Normal and shear stresses
Let the traction vector t(n) for a given current position x at time t act on an arbitrarily oriented surface element characterized by an outward unit normal vector n.
The traction vector t(n) may be resolved into the sum of a vector along the normal n to the plane, denoted by and a vector perpendicular to n denoted by From the results in section 2.3.3, it could be seen that
It could easily be verified that n·m = 0. This means m is a vector embedded in the surface. Then, σn is called the normal traction and τn is called the shear traction. As the names suggest, normal traction acts perpendicular to the surface and shear traction acts tangential or parallel to the surface.
Since, we obtain the useful relation
where we have made use of the property that n · m = 0.
It could be seen from figure 4.3 that the stress components corresponding to Cartesian basis, σxx, σyy and σzz act normal to their respective surfaces and hence are called normal stresses and the remaining independent components, σxy, σyz and σzx act parallel to the surface and hence are called shear stresses. (Remember that Cauchy stress is a symmetric tensor.) When the stress components are determined with respect to any coordinate basis vectors, there will be three normal stresses corresponding to σii and three shear stresses, σij , i ≠ j. Here to call certain components of the stresses as normal and others as shear, we have exploited the relationship between the components of the stress tensor and the traction vector.