2.4 Couple Stresses
Couple stress at a point can exist due to external loads, like a magnetic field, due to
materials’ microstructure or due to small size of the structure which can cause large
deformation gradients.
If we take moment with respect to Z axis
σ XY ΔYΔZ
ð
Þ ΔX þ m XZ À σ YX ΔXΔZ
ð
Þþm YZ ¼ 0
where m XZ , m YZ are distributed moments.
Therefore, σ XY 6 ¼ σ YX ; similarly σ XZ 6 ¼ σ ZX , σ YZ 6 ¼ σ ZY .
The skew symmetric second-order couple stress tensor can be given by
m
½ Š ¼
0
1
2
m XZ
À
1
2
m XY
À
1
2
m YZ
0
1
2
m YX
1
2
m ZY
À
1
2
m ZX
0
2
6
6
6
6
6
4
3
7
7
7
7
7
5
It is important to point out that 1/2 multiplier in front of the moment is due to the
relation between shear stresses and normal stresses [2 ε xy ¼ γ xy ] (Fig. 2.8).
2.4.1 Example 2.1
From equilibrium of forces along x axis, we can write the following relations, where
A represents the area along the inclined surface (Fig. 2.9):
(a)
(b)
dx
x
Y
σ yx
σ xy
σ xy
σ yx
dy
Y
X
Z
σ yz
σ xz
σ xy
dy
dz
dx
σ zy
σ yx
σ zx
Fig. 2.7 (a) Linear shear stresses acting at point Q. (b) x-y plane view of point Q
20
2 Stress and Strain in Continuum