s
3
s
1
s
2
s
1
− s
2
s
2
− s
3
s
3
− s
1
t
1
t
1
= t
2
= t
3
= t
2
t
3
2 2
2
Figure 2.4 A principal element and the three maximum shear planes and stresses.
t
2
t
1
t
3
s
1
s
s
2
t
a
Figure 2.5 The Mohr circle of stress showing the maximum shear stresses.
2.4.2 The hydrostatic stress
The hydrostatic stress is the average of the principal stresses and is defined as σ
h
= σ
1
+ σ
2
+ σ
3
3 2.9
It can be considered as three equal components acting in all directions on the element as shown in Figure 2.6.
s
1
s
1 ′
s
2 ′
s
3 ′
s
2
s
3
s
h
s
h
s
h
= +
Figure 2.6 A principal element showing how the principal stress state can be composed of hydrostatic
and deviatoric components.
Hydrostatic stress is similar to the hydrostatic pressure p in a fluid, except that, by convention in fluid mechanics, p is positive for compression, while in the mechanics of
solids, a compressive stress is negative, hence, σ
h
= −p Sheet deformation processes 19
As indicated above, it may be anticipated that this part of the stress system will not contribute to deformation in a material that deforms at constant volume.
2.4.3 The deviatoric or reduced component of stress
In Figure 2.6, the components of stress remaining after subtracting the hydrostatic stress have a special significance. They are called the deviatoric, or reduced stresses and are
defined by
σ
′ 1
= σ
1
− σ
h
; σ
′ 2
= σ
2
− σ
h
; σ
′ 3
= σ
3
− σ
h
2.10a In plane stress, this may also be written in terms of the stress ratio, i.e.
σ
′ 1
= 2
− α 3
σ
1
; σ
′ 2
= 2α
− 1 3
σ
1
; σ
′ 3
= − 1
+ α 3
σ
1
2.10b The reduced or deviatoric stress is the difference between the principal stress and the
hydrostatic stress. The theory of yielding and plastic deformation can be described simply in terms of
either of these components of the state of stress at a point, namely, the maximum shear stresses, or the deviatoric stresses.
2.4.4 The Tresca yield condition
One possible hypothesis is that yielding would occur when the greatest maximum shear stress reaches a critical value. In the tensile test where σ
2
= σ
3
= 0, the greatest maximum shear stress at yielding is τ
crit.
= σ
f
2. Thus in this theory, the Tresca yield criterion, yielding would occur in any process when
σ
max .
− σ
min .
2 =
σ
f
2 or, as usually stated,
|σ
max .
− σ
min .
| = σ
f
2.11 In plane stress, using the notation here, σ
1
will be the maximum stress and, σ
3
= 0, the through-thickness stress. The minimum stress will be either σ
3
if σ
2
is positive, or, if σ
2
is negative, it will be σ
2
. In all cases, the diameter a of the Mohr circle of stress in Figure 2.5 will be equal to σ
f
. The Tresca yield criterion in plane stress can be illustrated graphically by the hexagon
shown in Figure 2.7. The hexagon is the locus of a point P that indicates the stress state at yield as the stress ratio α changes. In a work-hardening material, this locus will expand
as σ
f
increases, but here we consider only the instantaneous conditions where the flow stress is constant.
2.4.5 The von Mises yield condition
The other widely used criterion is that yielding will occur when the root-mean-square value of the maximum shear stresses reaches a critical value. Several names have been
associated with this criterion and here we shall call it the von Mises yield theory.
20 Mechanics of Sheet Metal Forming
1 P
s
f
− s
f
s
f
a s
2
s
1
Figure 2.7 Yield locus for plane stress for the Tresca yield criterion.
Bearing in mind that in the tensile test at yield, two of the maximum shear stresses will have the value of σ
f
2, while the third is zero, this criterion can be expressed mathemati- cally as
τ
2 1
+ τ
2 2
+ τ
2 3
3 =
2 σ
f
2
2
3 or
2τ
2 1
+ τ
2 2
+ τ
2 3
= σ
f
2.12a Substituting the principal stresses for the maximum shear stresses from Equation 2.8, the
yielding condition can be expressed also as 1
2 {σ
1
− σ
2 2
+ σ
2
− σ
3 2
+ σ
3
− σ
1 2
} = σ
f
2.12b By substituting for the deviatoric stresses, i.e.
σ
′ 1
= 2σ
1
− σ
2
− σ
3
3 etc. the yield condition can be written as
3 2
σ
′2 1
+ σ
′2 2
+ σ
′2 3
= σ
f
2.12c For the plane stress state specified in Equation 2.6, the criterion is
σ
2 1
− σ
1
σ
2
+ σ
2 2
= 1
− α + α
2
σ
1
= σ
f
2.12d In the principal stress space, this is an ellipse as shown in Figure 2.8.
It is reiterated that both the above theories apply only to isotropic material and they are a reasonable approximation to experimental observations. Although there are major
differences in the mathematical form of these two criteria, the values of stress predicted for any given value of α will not differ by more than 15. In the Mohr circle of stress,
the diameter of the largest circle, a, in Figure 2.5 will be in the range
σ
f
≤ a ≤ 2
√ 3
σ
f
= 1.15σ
f
Sheet deformation processes 21
