2.8 Effective stress and strain functions
The plastic work done per unit volume in an increment in a process is given by Equation 2.15a. It would useful if this could be expressed in the form
dW vol.
= f
1
σ
1
, σ
2
, σ
3
df
2
ε
1
, ε
2
, ε
3
2.17 As the element is yielding during deformation, a suitable stress function to choose is that
given by the von Mises yielding criterion, which has already been shown to have the value of the flow stress. For plane stress this function is,
f
1
σ
1
, σ
2
, σ
3
= 1
− α + α
2
σ
1
This function is called the representative, effective or equivalent stress, σ , and if the material is yielding, it will be equal to the flow stress. For a general state of stress in an
isotropic material the effective stress function is, from Equation 2.12b,
σ =
1 2
{σ
1
− σ
2 2
+ σ
2
− σ
3 2
+ σ
3
− σ
1 2
} 2.18a
In plane stress, the effective stress function is σ
= σ
2 1
− σ
1
σ
2
+ σ
2 2
= 1
− α + α
2
σ
1
2.18b As indicated, if the material element is at yield, this function will have the magnitude of
the flow stress, σ
f
. The required strain function in Equation 2.17 can be found by substitution of the stress
function. This function is known as the representative, effective or equivalent strain incre- ment dε and for plane stress, the function is
dε = df
2
ε
1
, ε
2
, ε
3
= 4
3 {1 + β + β
2
}dε
1
2.19a In a general state of stress it can be written as
dε =
2 3
{dε
2 1
+ dε
2 2
+ dε
2 3
} =
2 9
{dε
1
− dε
2 2
+ dε
2
− dε
3 2
+ dε
3
− dε
1 2
} 2.19b
In a monotonic, proportional process, Equations 2.19a and b can be written in the inte- grated form with the natural or true strains ε substituted for the incremental strains dε; i.e.
ε =
4 3
{1 + β + β
2
}ε
1
= 2
3 {ε
2 1
+ ε
2 2
+ ε
2 3
} =
2 9
{ε
1
− ε
2 2
+ ε
2
− ε
3 2
+ ε
3
− ε
1 2
} 2.19c
where, ε, is the representative, effective, or equivalent strain. 26 Mechanics of Sheet Metal Forming
Because of the way in which these relations have been derived, it can be seen that the work done per unit volume in any process is given by
W vol.
=
ε
σ ε 2.20
It is also evident that because the stress function has been chosen as the von Mises stress which is equal in magnitude to the flow stress when the material is deforming, the effective
strain function will be equal to the strain in uniaxial tension when equal amounts of work are done in the general process and in uniaxial tension. Thus we have identified a general
stress–strain relation for an isotropic material deforming plastically, namely the effective stress–strain curve, σ
= f ε; this is coincident with the tensile test true stress–strain curve for an isotropic material.
The key to this principle of the equivalence of plastic work done is illustrated in Figure 2.11. To reiterate, in a plane stress process, there are two stress strain curves.
These must continuously satisfy the yield criterion and the condition that the work done in the process, i.e. the area under both curves, is equal to the work done in uniaxial tension.
This work done determines the current yield stress in uniaxial tension which is also the flow stress. The effective stress and strain functions ensure that these conditions are met
and enable the current flow stress for a material element deformed in any process to be determined from an experimental stress–strain curve obtained in a tension test. Material
properties can also be obtained from other tests, provided that the test enables an effective stress–strain curve to be obtained.
2.9 Summary
In this chapter, it is shown that for simple monotonic, proportional, plane stress processes, it is possible to determine at any instant the principal membrane stresses required for
deformation provided the current flow stress σ
f
is known and also that either the stress ratio α or strain ratio β are known. The current flow stress can be determined from the
tensile test stress strain curve using the effective stress and strain functions that are based on the equivalence of work. In practice, a process is often defined by the strain ratio β
obtained from measurement of final strains. The assumption is that this point is reached by a proportional process, but if only the initial and final conditions are known, care should
be taken in assuming that the strain ratio is constant.
The theory given in this chapter applies only for an instantaneous state in which the strain increment is small and the flow stress constant. In the next chapter, entire loading
paths are studied using the theory established here. It cannot be emphasized too strongly that while the theory of deformation given here is useful and practical, it is a simple approx-
imation of a very complex process. It is useful in process design and failure diagnosis in industry, but in some studies, more elaborate theories may be necessary.
2.10 Exercises
Ex. 2.1
A square element 8 × 8 mm in an undeformed sheet of 0.8 mm thickness becomes
a rectangle, 6.5 × 9.4 mm after forming. Assume that the stress strain law is:
σ = 6000.008 + ε
0.22
MPa Sheet deformation processes 27
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