We may summarize the tensile test process for an isotropic material in terms of the strain increments and stresses in the following manner:
dε
1
= dl
l ;
dε
2
= − 1
2 dε
1
; dε
3
= − 1
2 dε
1
2.4a and
σ
1
= P
A ;
σ
2
= 0; σ
3
= 0 2.4b
2.2.4 True, natural or logarithmic strains
It may be noted that in the tensile test the following conditions apply: • the principal strain increments all increase smoothly in a constant direction, i.e. dε
1
always increases positively and does not reverse; this is termed a monotonic process; • during the uniform deformation phase of the tensile test, from the onset of yield to
the maximum load and the start of diffuse necking, the ratio of the principal strains remains constant, i.e. the process is proportional; and
• the principal directions are fixed in the material, i.e. the direction 1 is always along the axis of the test-piece and a material element does not rotate with respect to the
principal directions.
If, and only if,
these conditions apply, we may safely use the integrated or large strains defined in Chapter 1. For uniaxial deformation of an isotropic material, these strains are
ε
1
= ln l
l ;
ε
2
= ln w
w = −
1 2
ε
1
; ε
3
= ln t
t
o
= − 1
2 ε
1
2.5
2.3 General sheet processes plane stress
In contrast with the tensile test in which two of the principal stresses are zero, in a typical sheet process most elements will deform under membrane stresses σ
1
and σ
2
, which are both non-zero. The third stress, σ
3
, perpendicular to the surface of the sheet is usually quite small as the contact pressure between the sheet and the tooling is generally very much
lower than the yield stress of the material. As indicated above, we will make the simplifying assumption that it is zero and assume plane stress deformation, unless otherwise stated. If
we also assume that the same conditions of proportional, monotonic deformation apply as for the tensile test, then we can develop a simple theory of plastic deformation of sheet
that is reasonably accurate. We can illustrate these processes for an element as shown in Figure 2.2a for the uniaxial tension and Figure 2.2b for a general plane stress sheet
process.
2.3.1 Stress and strain ratios
It is convenient to describe the deformation of an element, as in Figure 2.2b, in terms of either the strain ratio β or the stress ratio α. For a proportional process, which is the
only kind we are considering, both will be constant. The usual convention is to define the
16 Mechanics of Sheet Metal Forming
a b
Tensile test Plane stress
s
3
= 0, e
3
= −e
1
2
s
1
, e
1
s
2
= 0, e
2
= −e
1
2 s
3
= 0, e
3
= − 1 + be
1
s
1
, e
1
s
2
= as
1
e
2
= be
1
Figure 2.2 Principal stresses and strains for elements deforming in a uniaxial tension and b a
general plane stress sheet process.
principal directions so that σ
1
σ
2
and the third direction is perpendicular to the surface where σ
3
= 0. The deformation mode is thus: ε
1
; ε
2
= βε
1
; ε
3
= −1 + βε
1
2.6 σ
1
; σ
2
= ασ
1
; σ
3
= 0 The constant volume condition is used to obtain the third principal strain. Integrating the
strain increments in Equation 2.3 shows that this condition can be expressed in terms of the true or natural strains:
ε
1
+ ε
2
+ ε
3
= 0 2.7
i.e. the sum of the natural strains is zero. For uniaxial tension, the strain and stress ratios are β
= −12 and α = 0.
2.4 Yielding in plane stress
The stresses required to yield a material element under plane stress will depend on the current hardness or strength of the sheet and the stress ratio α. The usual way to define the
strength of the sheet is in terms of the current flow stress σ
f
. The flow stress is the stress at which the material would yield in simple tension, i.e. if α
= 0. This is illustrated in the true stress–strain curve in Figure 2.3. Clearly σ
f
depends on the amount of deformation to which the element has been subjected and will change during the process. For the moment,
we shall consider only one instant during deformation and, knowing the current value of σ
f
the objective is to determine, for a given value of α, the values of σ
1
and σ
2
at which the element will yield, or at which plastic flow will continue for a small increment. We
consider here only the instantaneous conditions in which the strain increment is so small that the flow stress can be considered constant. In Chapter 3 we extend this theory for
continuous deformation.
There are a number of theories available for predicting the stresses under which a material element will deform plastically. Each theory is based on a different hypothesis
about material behaviour, but in this work we shall only consider two common models and apply them to the plane stress process described by Equations 2.6. Over the years,
many researchers have conducted experiments to determine how materials yield. While no single theory agrees exactly with experiment, for isotropic materials either of the models
presented here are sufficiently accurate for approximate models.
Sheet deformation processes 17
