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
1 O
s
2
s
1
a s
f
s
f
Figure 2.8 Yield locus for plane stress for a von Mises yield condition.
2.5 The flow rule
A yield theory allows one to predict the values of stress at which a material element will deform plastically in plane stress, provided the ratio of the stresses in the plane of
the sheet and the flow stress of the material are known. In the study of metal forming processes, we will also need to be able to determine what strains will be associated with
the stress state when the element deforms. In elastic deformation, there is a one-to-one relation between stress and strain; i.e. if we know the stress state we can determine the
strain state and vice versa. We are already aware of this, because in the experimental study of elastic structures, stresses are determined by strain gauges. This is not possible in the
plastic regime. A material element may be at a yielding stress state, i.e. the stresses satisfy the yield condition, but there may be no change in shape. Alternatively, an element in
which the stress state is a yielding one, may undergo some small increment of strain that is determined by the displacements of the boundaries; i.e. the magnitude of the deformation
increment is determined by the movement of the boundaries and not by the stresses. However, what can be predicted if deformation occurs is the ratio of the strain increments;
this does depend on the stress state.
Again, with hindsight, the relationship between the stress and strain ratios can be antic- ipated by considering the nature of flow. In the tensile test, the stresses are in the ratio
1 : 0 : 0 and the strains in the ratio
1 : −12 : −12
so it is not simply a matter of the stresses and strains being in the same ratio. The appropriate relation for general deformation is not, therefore, an obvious one, but can be
found by resolving the stress state into the two components, namely the hydrostatic stress and the reduced or deviatoric stresses that have been defined above.
2.5.1 The Levy–Mises flow rule
As shown in Figure 2.6, the deviatoric or reduced stress components, together with the hydrostatic components, make up the actual stress state. As the hydrostatic stress is unlikely
to influence deformation in a solid that deforms at constant volume, it may be surmised
22 Mechanics of Sheet Metal Forming
that it is the deviatoric components that will be the ones associated with the shape change. This is the hypothesis of the Levy–Mises Flow Rule. This states that the ratio of the strain
increments will be the same as the ratio of the deviatoric stresses, i.e. dε
1
σ
′ 1
= dε
2
σ
′ 2
= dε
3
σ
′ 3
2.13a or
dε
1
2 − α
= dε
2
2α − 1
= dε
3
−1 + α 2.13b
If a material element is deforming in a plane stress, proportional process, as described by Equation 2.6, then Equation 2.13b can be integrated and expressed in terms of the
natural or true strains, i.e. ε
1
2 − α
= ε
2
2α − 1
= βε
1
2α − 1
= ε
3
−1 + α =
−1 + βε
1
−1 + α 2.13c
2.5.2 Relation between the stress and strain ratios
From the above, we obtain the relation between the stress and strain ratios: α
= 2β
+ 1 2
+ β and
β =
2α − 1
2 − α
2.14 It may be seen that while the flow rule gives the relation between the stress and strain
ratios, it does not indicate the magnitude of the strains. If the element deforms under a given stress state i.e. α is known the ratio of the strains can be found from Equation 2.13,
or 2.14. The relationship can be illustrated for different load paths as shown in Figure 2.9; the small arrows show the ratio of the principal strain increments and the lines radiating
from the origin indicate the loading path on an element. It may be seen that each of these strain increment vectors is perpendicular to the von Mises yield locus. It is possible to
predict this from considerations of energy or work.
1 12
1 1
−1 1
de
2
= −2de
1
de
2
= −de
1
de
2
= 0 s
1
, de
1
s
2
, de
2
de
2
= −de
1
2 de
2
= de
1
−s
f
Figure 2.9 Diagram showing the strain increment components for different stress states around the
von Mises yield locus.
Sheet deformation processes 23
