Page 289 - Intro to Tensor Calculus

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(n)
We make the assumption that t is colinear with the normal vector to the surface passing through
i
the selected point. It is also assumed that for fluid elements at rest, there are no shear forces acting on the
planar element through an arbitrary point and therefore the stress tensor σ ij should be independent of the
orientation of the plane. That is, we desire for the stress vector σ ij to be an isotropic tensor. This requires
σ ij to have a specific form. To find this specific form we let σ ij denote the stress components in a general
i
coordinate system x , i =1, 2, 3 and let σ ij denote the components of stress in a barred coordinate system
i
x ,i =1, 2, 3. Since σ ij is a tensor, it must satisfy the transformation law
i
∂x ∂x j
σ mn = σ ij m n , i,j,m,n =1, 2, 3. (2.5.7)
∂x ∂x
We desire for the stress tensor σ ij to be an invariant under an arbitrary rotation of axes. Consider
therefore the special coordinate transformations illustrated in the figures 2.5-1(a) and (b).

Figure 2.5-1. Coordinate transformations due to rotations

For the transformation equations given in figure 2.5-1(a), the stress tensor in the barred system of
coordinates is
σ 11 = σ 22 σ 21 = σ 32 σ 31 = σ 12
(2.5.8)
σ 12 = σ 23 σ 22 = σ 33 σ 32 = σ 13
σ 33 = σ 11 .
σ 13 = σ 21 σ 23 = σ 31
If σ ij is to be isotropic, we desire that σ 11 = σ 11 , σ 22 = σ 22 and σ 33 = σ 33 . If the equations (2.5.8) are
to produce these results, we require that σ 11 , σ 22 and σ 33 must be equal. We denote these common values
by (−p). In particular, the equations (2.5.8) show that if σ 11 = σ 11 , σ 22 = σ 22 and σ 33 = σ 33 , then we must
require that σ 11 = σ 22 = σ 33 = −p. If σ 12 = σ 12 and σ 23 = σ 23 , then we also require that σ 12 = σ 23 = σ 31 .
We note that if σ 13 = σ 13 and σ 32 = σ 32 , then we require that σ 21 = σ 32 = σ 13 . If the equations (2.5.7) are
expanded using the transformation given in figure 2.5-1(b), we obtain the additional requirements that

σ 11 = σ 22 σ 21 = −σ 12 σ 31 = σ 32
σ 12 = −σ 21 σ 22 = σ 11 σ 32 = −σ 31 (2.5.9)
σ 13 = σ 23 σ 23 = −σ 13 σ 33 = σ 33 .