CONTINUUM MECHANICS for ENGINEERS

Stress Tensor
The quantities σ ji ti are the components of a second-order tensor known
as the stress tensor. This is shown by considering the transformation of the
components of the stress vector between coordinate systems Px1 x2 x3 and
as given by the transformation matrix having elements (see Section 2.5)
j
≡ ( ê )
t
( ˆn )
i
Px′ x′ x′
1 2 3
(3.3-9)
Since t can be expressed in terms of its components in either coordinate
system,
(3.3-10a)
nˆ ( )
ˆ ˆ ˆ
t
( n) n n
=
( ) ′
t eˆ = t
( )
eˆ′
or, from Eq 3.3-8,
aij = e′ i ⋅ej
(3.3-10b)
But from Eq 2.5-2, eˆ′ ˆ i = aijej
and from Eq 2.5-9, n′ j = ajsns, so that now
Eq 3.3-10b becomes, after some manipulations of the summed indices,
(3.3-11)
Because the vectors are linearly independent and since Eq 3.3-11 must be
valid for all vectors ns, we see that
êr (3.3-12)
But this is the transformation equation for a second-order tensor, and thus by
Eq 2.5-12 the tensor character of the stress components is clearly established.
The Cauchy stress formula given by Eq 3.3-8 expresses the stress vector
associated with the element of area having an outward normal n i at point P
in terms of the stress tensor components σ ji at that point. And although the
state of stress at P has been described as the totality of pairs of the associated
normal and traction vectors at that point, we see from the analysis of the
tetrahedron element that if we know the stress vectors on the three coordinate
planes of any Cartesian system at P, or equivalently, the nine stress
tensor components σ ji at that point, we can determine the stress vector for
any plane at that point. For computational purposes it is often convenient
to express Eq 3.3-8 in the matrix form
ˆ ˆ
i i i i
t
( nˆ )
= σ neˆ = σ′
n′
eˆ
ji j i ji j i
( σsr − ajsairσ′ ji) nsêr=
0
σsr =ajsa irσ′ ji