CONTINUUM MECHANICS for ENGINEERS

FIGURE 3.12
Normal and shear components at P to plane referred to principal axes.
In seeking the maximum and minimum (the so-called extremal) values of
the above components, let us consider first σN. As the normal ni assumes all
possible orientations at P, the values of σN will be prescribed by the functional
relation in Eq 3.7-1 subject to the condition that nini = 1. Accordingly, we may
use to advantage the Lagrangian multiplier method to obtain extremal values
of σN. To do so we construct the function fn ( , where
i)=
σijnn i j −σ( nn i i −1)
the scalar σ is called the Lagrangian multiplier. The method requires the
derivative of f(ni) with respect to nk to vanish; and, noting that ∂n ,
i/ ∂nk = δik
we have
∂ f
∂ n
k
= σij( δiknj+ δ jkni)− σ( 2niδik )= 0
But = , and δ kjn j = n k, so that this equation reduces to
σ ij
σ ji
σ σδ kj j n − ( kj)
= 0
(3.7-3)
which is identical to Eq 3.6-2, the eigenvalue formulation for principal
stresses. Therefore, we conclude that the Lagrangian multiplier σ assumes
the role of a principal stress and, furthermore, that the principal stresses
include both the maximum and minimum normal stress values.
With regard to the maximum and minimum values of the shear component
* * *
σS, it is useful to refer the state of stress to principal axes Px , as shown
1x2
x3
in Figure 3.12. Let the principal stresses be ordered in the sequence σI > σII ( ˆn )
> σIII so that is expressed in vector form by
ti t
( )
= ⋅ n= ˆ σ neˆ + σ n eˆ + σ n eˆ
nˆ * * *
I 1 1 II 2 2 III 3 3
(3.7-4)