Sheet Metal Forming -2

Modern continuous rolling mills produce large quantities of thin sheet metal at low cost.
A substantial fraction of all metals are produced as thin hot-rolled strip or cold-rolled
sheet; this is then formed in secondary processes into automobiles, domestic appliances,
building products, aircraft, food and drink cans and a host of other familiar products. Sheet
metals parts have the advantage that the material has a high elastic modulus and high yield
strength so that the parts produced can be stiff and have a good strength-to-weight ratio.
A large number of techniques are used to make sheet metal parts. This book is concerned
mainly with the basic mechanics that underlie all of these methods, rather than with a
detailed description of the overall processes, but it is useful at this stage to review briefly
the most common sheet forming techniques.
Common forming processes
Blanking and piercing. As sheet is usually delivered in large coils, the first operation
is to cut the blanks that will be fed into the presses; subsequently there may be further
blanking to trim off excess material and pierce holes. The basic cutting process is shown
in Figure I.1. When examined in detail, it is seen that blanking is a complicated process of
plastic shearing and fracture and that the material at the edge is likely to become hardened
locally. These effects may cause difficulty in subsequent operations and information on
tooling design to reduce problems can be found in the appropriate texts.

Figure I.1 Magnified section of blanking a sheet showing plastic deformation and cracking.
Bending. The simplest forming process is making a straight line bend as shown in
Figure I.2. Plastic deformation occurs only in the bend region and the material away
from the bend is not deformed. If the material lacks ductility, cracking may appear on

the outside bend surface, but the greatest difficulty is usually to obtain an accurate and
repeatable bend angle. Elastic springback is appreciable.
Various ways of bending along a straight line are shown in Figure I.3. In folding (a), the
part is held stationary on the left-hand side and the edge is gripped between movable tools
that rotate. In press-brake forming (b), a punch moves down and forces the sheet into
a vee-die. Bends can be formed continuously in long strip by roll forming (c). In roll
forming machines, there are a number of sets of rolls that incrementally bend the sheet,
and wide panels such as roofing sheet or complicated channel sections can be made in
this process. A technique for bending at the edge of a stamped part is flanging or wiping
as shown in Figure I.3(d). The part is clamped on the left-hand side and the flanging tool
moves downwards to form the bend. Similar tooling is used is successive processes to
bend the sheet back on itself to form a hem.

If the bend is not along a straight line, or the sheet is not flat, plastic deformation occurs
not only at the bend, but also in the adjoining sheet. Figure I.4 gives examples. In shrink
flanging (a), the edge is shortened and the flange may buckle. In stretch flanging (b), the

length of the edge must increase and splitting could be a problem. If the part is curved
near the flange or if both the flange and the part are curved, as in Figure I.4(c), the flange
may be either stretched or compressed and some geometric analysis is needed to determine
this. All these flanges are usually formed with the kind of tooling shown in Figure I.3(d).

Figure I.4 (a) A shrink flange showing possible buckling. (b) A stretch flange with edge cracking.
(c) Flanging a curved sheet.
Section bending. In Figure I.5, a more complicated shape is bent. At the left-hand end
of the part, the flange of the channel is stretched and may split, and the height of the leg,
h, will decrease. When the flange is on the inside, as on the right, wrinkling is possible
and the flange height will increase.

Stretching. The simplest stretching process is shown in Figure I.6. As the punch is pushed
into the sheet, tensile forces are generated at the centre. These are the forces that cause the
deformation and the contact stress between the punch and the sheet is very much lower
than the yield stress of the sheet.
The tensile forces are resisted by the material at the edge of the sheet and compressive
hoop stresses will develop in this region. As there will be a tendency for the outer region to
buckle, it will be held by a blank-holder as shown in Figure I.6(b). The features mentioned
are common in many sheet processes, namely that forming is not caused by the direct

contact stresses, but by forces transmitted through the sheet and there will be a balance
between tensile forces over the punch and compressive forces in the outer flange material.
Hole extrusion. If a hole smaller than the punch diameter is first pierced in the sheet, the
punch can be pushed through the sheet to raise a lip as in the hole extrusion in Figure I.7.
It will be appreciated that the edge of the hole will be stretched and splitting will limit
the height of the extrusion.

Stamping or draw die forming. The part shown in Figure I.8(a) is formed by stretching
over a punch of more complicated shape in a draw die. This consists of a punch, and draw
ring and blank-holder assembly, or binder. The principle is similar to punch stretching
described above, but the outer edge or flange is allowed to draw inwards under restraint to
supply material for the part shape. This process is widely used to form auto-body panels
and a variety of appliance parts. Much of the outer flange is trimmed off after forming
so that it is not a highly efficient process, but with well-designed tooling, vast quantities
of parts can be made quickly and with good dimensional control. Die design requires
the combination of skill and extensive computer-aided engineering systems, but for the
purpose of conceptual design and problem solving, the complicated deformation system
can be broken down into basic elements that are readily analysed. In this book, the analysis
of these macroscopic elements is studied and explained, so that the reader can understand
those factors that govern the overall process.
Deep drawing. In stamping, most of the final part is formed by stretching over the punch
although some material around the sides may have been drawn inwards from the flange. As

there is a limit to the stretching that is possible before tearing, stamped parts are typically
shallow. To form deeper parts, much more material must be drawn inwards to form the
sides and such a process is termed deep drawing. Forming a simple cylindrical cup is
shown in Figure I.9. To prevent the flange from buckling, a blankholder is used and the
clamping force will be of the same order as the punch force. Lubrication is important
as the sheet must slide between the die and the blankholder. Stretching over the punch
is small and most of the deformation is in the flange; as this occurs under compressive
stresses, large strains are possible and it is possible to draw a cup whose height is equal to
or possibly a little larger than the cup diameter. Deeper cups can be made by redrawing
as shown in Figure I.10.

Tube forming. There are a number of processes for forming tubes such as flaring and
sinking as shown in Figure I.11. Again, these operations can be broken down into a few
elements, and analysed as steady-state processes.
Fluid forming. Some parts can be formed by fluid pressure rather than by rigid tools.
Quite high fluid pressures are required to form sheet metal parts so that equipment can

be expensive, but savings in tooling costs are possible and the technique is suitable where
limited numbers of parts are required. For forming flat parts, a diaphragm is usually placed
over the sheet and pressurized in a container as in Figure I.12. As the pressure to form the
sheet into sharp corners can be very high, the forces needed to keep the container closed
are much greater than those acting on a punch in a draw die, and special presses are
required. Complicated tubular parts for plumbing fittings and bicycle frame brackets are
made by a combination of fluid pressure and axial force as in Figure I.13. Tubular parts,
for example frame structures for larger vehicles, are made by bending a circular tube,
placing it in a closed die and forming it to a square section as illustrated in Figure I.14.
Coining and ironing. In all of the processes above, the contact stress between the sheet
and the tooling is small and, as mentioned, deformation results from membrane forces in
the sheet. In a few instances, through-thickness compression is the principal deformation
force. Coining, Figure I.15, is a local forging operation used, for example, to produce a
groove in the lid of a beverage can or to thin a small area of sheet. Ironing, Figure I.16, is
a continuous process and often accompanies deep drawing. The cylindrical cup is forced
through an ironing die that is slightly smaller than the punch plus metal thickness dimension.
Using several dies in tandem, the wall thickness can be reduced by more than one-half
in a single stroke.

Summary. Only very simple examples of industrial sheet forming processes have been
shown here. An industrial plant will contain many variants of these techniques and numerous
presses and machines of great complexity. It would be an overwhelming task to deal
with all the details of tool and process design, but fortunately these processes are all made
up of relatively few elemental operations such as stretching, drawing, bending, bending

under tension and sliding over a tool surface. Each of the basic deformation processes can
be analysed and described by a ‘mechanics model’, i.e. a model similar to the familiar ones
in elastic deformation for tension in a bar, bending of a beam or torsion of a shaft; these
models form the basis for mechanical design in the elastic regime. This book presents
similar models for the deformation of sheet. In this way, the engineer can apply a familiar
approach to problem solving in sheet metal engineering.
Application to design
The objective in studying the basic mechanics of sheet metal forming is to apply this to
part and tool design and the diagnosis of plant problems. It is important to appreciate
that analysis is only one part of the design process. The first step in design is always to
determine what is required of the part or process, i.e. its function. Determining how to
achieve this comes later. When the function is described completely and in quantitative
terms, the designer can then address the ‘how’. This is typically an iterative process in
which the designer makes some decision and then determines the consequences. A good
designer will have a feeling for the consequences before any calculations are made and this
ability is derived from an understanding of the basic principles governing each operation.
Once the decision is made, simple and approximate calculations are usually sufficient to
justify the decision. There will be a point when an extensive and detailed analysis is needed
to confirm and prove the design, but this book is aimed at the initial but important stage
of the process, namely being able to understand the mechanics of sheet forming processes
and then analysing these in a quick and approximate manner.

Material properties

The most important criteria in selecting a material are related to the function of the
part – qualities such as strength, density, stiffness and corrosion resistance. For sheet material,
the ability to be shaped in a given process, often called its formability, should also be
considered. To assess formability, we must be able to describe the behaviour of the sheet
in a precise way and express properties in a mathematical form; we also need to know
how properties can be derived from mechanical tests. As far as possible, each property
should be expressed in a fundamental form that is independent of the test used to measure
it. The information can then be used in a more general way in the models of various metal
forming processes that are introduced in subsequent chapters.
In sheet metal forming, there are two regimes of interest – elastic and plastic
deformation. Forming a sheet to some shape obviously involves permanent ‘plastic’ flow
and the strains in the sheet could be quite large. Whenever there is a stress on a sheet
element, there will also be some elastic strain. This will be small, typically less than one
part in one thousand. It is often neglected, but it can have an important effect, for example
when a panel is removed from a die and the forming forces are unloaded giving rise to
elastic shape changes, or ‘springback’.
1.1 Tensile test
For historical reasons and because the test is easy to perform, many familiar material
properties are based on measurements made in the tensile test. Some are specific to the
test and cannot be used mathematically in the study of forming processes, while others
are fundamental properties of more general application. As many of the specific, or nonfundamental
tensile test properties are widely used, they will be described at this stage and
some description given of their effect on processes, even though this can only be done in
a qualitative fashion.

1.1.1 The load–extension diagram
Figure 1.2 shows a typical load–extension diagram for a test on a sample of drawing
quality steel. The elastic extension is so small that it cannot be seen. The diagram does
not represent basic material behaviour as it describes the response of the material to a
particular process, namely the extension of a tensile strip of given width and thickness.
Nevertheless it does give important information. One feature is the initial yielding load,
Py, at which plastic deformation commences. Initial yielding is followed by a region in
which the deformation in the strip is uniform and the load increases. The increase is
due to strain-hardening, which is a phenomenon exhibited by most metals and alloys in
the soft condition whereby the strength or hardness of the material increases with plastic
deformation. During this part of the test, the cross-sectional area of the strip decreases
while the length increases; a point is reached when the strain-hardening effect is just
balanced by the rate of decrease in area and the load reaches a maximum Pmax .. Beyond
this, deformation in the strip ceases to be uniform and a diffuse neck develops in the
reduced section; non-uniform extension continues within the neck until the strip fails.

1.1.2 The engineering stress–strain curve
Prior to the development of modern data processing systems, it was customary to scale the
load–extension diagram by dividing load by the initial cross-sectional area, A0 = w0t0, and
the extension by l0, to obtain the engineering stress–strain curve. This had the advantage
that a curve was obtained which was independent of the initial dimensions of the test-piece,
but it was still not a true material property curve. During the test, the cross-sectional area
will diminish so that the true stress on the material will be greater than the engineering
stress. The engineering stress–strain curve is still widely used and a number of properties
are derived from it. Figure 1.3(a) shows the engineering stress strain curve calculated from
the load, extension diagram in Figure 1.2.

As already indicated, this is not the true stress at maximum load as the cross-sectional
area is no longer A0. The elongation at maximum load is called the maximum uniform
elongation, Eu.
If the strain scale near the origin is greatly increased, the elastic part of the curve
would be seen, as shown in Figure 1.3(b). The strain at initial yield, ey, as mentioned, is
very small, typically about 0.1%. The slope of the elastic part of the curve is the elastic
modulus, also called Youngs modulus:

If the strip is extended beyond the elastic limit, permanent plastic deformation takes place;
upon unloading, the elastic strain will be recovered and the unloading line is parallel to
the initial elastic loading line. There is a residual plastic strain when the load has been
removed as shown in Figure 1.3(b).

In some materials, the transition from elastic to plastic deformation is not sharp and
it is difficult to establish a precise yield stress. If this is the case, a proof stress may be
quoted. This is the stress to produce a specified small plastic strain – often 0.2%, i.e. about
twice the elastic strain at yield. Proof stress is determined by drawing a line parallel to the
elastic loading line which is offset by the specified amount, as shown in Figure 1.3(c).
Certain steels are susceptible to strain ageing and will display the yield phenomena
illustrated in Figure 1.4. This may be seen in some hot-dipped galvanized steels and
in bake-hardenable steels used in autobody panels. Ageing has the effect of increasing
the initial yielding stress to the upper yield stress σU; beyond this, yielding occurs in a
discontinuous form. In the tensile test-piece, discrete bands of deformation called L¨uder’s
lines will traverse the strip under a constant stress that is lower than the upper yield stress;
this is known as the lower yield stress σL. At the end of this discontinuous flow, uniform
deformation associated with strain-hardening takes place. The amount of discontinuous
strain is called the yield point elongation (YPE). Steels that have significant yield point
elongation, more than about 1%, are usually unsuitable for forming as they do not deform
smoothly and visible markings, called stretcher strains can appear on the part.

1.1.3 The true stress–strain curve
There are several reasons why the engineering stress–strain curve is unsuitable for use in
the analysis of forming processes. The ‘stress’ is based on the initial cross-sectional area
of the test-piece, rather than the current value. Also engineering strain is not a satisfactory
measure of strain because it is based on the original gauge length. To overcome these
disadvantages, the study of forming processes is based on true stress and true strain;
these are defined below.
True stress is defined as

where A is the current cross-sectional area. True stress can be determined from the
load–extension diagram during the rising part of the curve, between initial yielding and

It can be seen that the true stress–strain curve does not reach a maximum as strainhardening
is continuous although it occurs at a diminishing rate with deformation. When
necking starts, deformation in the gauge length is no longer uniform so that Equation 1.11
is no longer valid. The curve in Figure 1.5 cannot be calculated beyond a strain corresponding
to maximum load; this strain is called the maximum uniform strain:

If the true stress and strain are plotted on logarithmic scales, as in Figure 1.6, many
samples of sheet metal in the soft, annealed condition will show the characteristics of this
diagram. At low strains in the elastic range, the curve is approximately linear with a slope
of unity; this corresponds to an equation for the elastic regime of

The fitted curve has a slope of n, which is known as the strain-hardening index, and an
intercept of logK at a strain of unity, i.e. when ε = 1, or log ε = 0; K is the strength
coefficient. The empirical equation or power law Equation 1.16(a) is often used to describe
the plastic properties of annealed low carbon steel sheet. As may be seen from Figure 1.6,

it provides an accurate description, except for the elastic regime and during the first few
per cent of plastic strain. Empirical equations of this form are often used to extrapolate
the material property description to strains greater than those that can be obtained in the
tensile test; this may or may not be valid, depending on the nature of the material.
1.1.4 (Worked example) tensile test properties
The initial gauge length, width and thickness of a tensile test-piece are, 50, 12.5 and
0.80mm respectively. The initial yield load is 1.791 kN. At a point, A, the load is 2.059 kN
and the extension is 1.22 mm. The maximum load is 2.94 kN and this occurs at an extension
of 13.55 mm. The test-piece fails at an extension of 22.69 mm.
Determine the following:

Anisotropy
Material in which the same properties are measured in any direction is termed isotropic, but
most industrial sheet will show a difference in properties measured in test-pieces aligned,
for example, with the rolling, transverse and 45◦ directions of the coil. This variation is
known as planar anisotropy. In addition, there can be a difference between the average of
properties in the plane of the sheet and those in the through-thickness direction. In tensile
tests of a material in which the properties are the same in all directions, one would expect,
by symmetry, that the width and thickness strains would be equal; if they are different,
this suggests that some anisotropy exists.

In materials in which the properties depend on direction, the state of anisotropy is usually
indicated by the R-value. This is defined as the ratio of width strain, εw = ln(w/w0), to
thickness strain, εt = ln(t/t0). In some cases, the thickness strain is measured directly,
but it may be calculated also from the length and width measurements using the constant
volume assumption, i.e.

If the change in width is measured during the test, the R-value can be determined continuously
and some variation with strain may be observed. Often measurements are taken
at a particular value of strain, e.g. at eeng. = 15%. The direction in which the R-value is
measured is indicated by a suffix, i.e. R0, R45 and R90 for tests in the rolling, diagonal
and transverse directions respectively. If, for a given material, these values are different,
the sheet is said to display planar anisotropy and the most common description of this is

which may be positive or negative, although in steels it is usually positive.
If the measured R-value differs from unity, this shows a difference between average
in-plane and through-thickness properties which is usually characterized by the normal
plastic anisotropy ratio, defined as

The term ‘normal’ is used here in the sense of properties ‘perpendicular’ to the plane of
the sheet.

Rate sensitivity
For many materials at room temperature, the properties measured will not vary greatly
with small changes in the speed at which the test is performed. The property most sensitive
to rate of deformation is the lower yield stress and therefore it is customary to specify the
cross-head speed of the testing machine – typically about 25 mm/minute.
If the cross-head speed, v, is suddenly changed by a factor of 10 or more during the
uniform deformation region of a tensile test, a small jump in the load may be observed
as shown in Figure 1.7. This indicates some strain-rate sensitivity in the material that can
be described by the exponent, m, in the equation

1.2 Effect of properties on forming
It is found that the way in which a given sheet behaves in a forming process will depend
on one or more general characteristics. Which of these is important will depend on the
particular process and by studying the mechanics equations governing the process it is
often possible to predict those properties that will be important. This assumes that the
property has a fundamental significance, but as mentioned above, not all the properties
obtained from the tensile test will fall into this category.
The general attributes of material behaviour that affect sheet metal forming are as
follows.
1.2.1 Shape of the true stress–strain curve
The important aspect is strain-hardening. The greater the strain-hardening of the sheet,
the better it will perform in processes where there is considerable stretching; the straining
will be more uniformly distributed and the sheet will resist tearing when strain-hardening
is high. There are a number of indicators of strain-hardening and the strain-hardening
index, n, is the most precise. Other measures are the tensile/yield ratio, T S/(σf)0, the total
elongation, ETot. and the maximum uniform strain, εu; the higher these are, the greater is
the strain-hardening.

The importance of the initial yield strength, as already mentioned, is related to the
strength of the formed part and particularly where lightweight construction is desired, the
higher the yield strength, the more efficient is the material. Yield strength does not directly
affect forming behaviour, although usually higher strength sheet is more difficult to form;
this is because other properties change in an adverse manner as the strength increases.
The elastic modulus also affects the performance of the formed part and a higher
modulus will give a stiffer component, which is usually an advantage. In terms of forming,
the modulus will affect the springback. A lower modulus gives a larger springback and
usually more difficulty in controlling the final dimensions. In many cases, the springback
will increase with the ratio of yield stress to modulus, (σf)0/E, and higher strength sheet
will also have greater springback.
1.2.2 Anisotropy
If the magnitude of the planar anisotropy parameter, R, is large, either, positive or
negative, the orientation of the sheet with respect to the die or the part to be formed will
be important; in circular parts, asymmetric forming and earing will be observed. If the
normal anisotropy ratio R is greater than unity it indicates that in the tensile test the width
strain is greater than the thickness strain; this may be associated with a greater strength in
the through-thickness direction and, generally, a resistance to thinning. Normal anisotropy
R also has more subtle effects. In drawing deep parts, a high value allows deeper parts to
be drawn. In shallow, smoothly-contoured parts such as autobody outer panels, a higher
value of R may reduce the chance of wrinkling or ripples in the part. Other factors such
as inclusions, surface topography, or fracture properties may also vary with orientation;
these would not be indicated by the R-value which is determined from plastic properties.
1.2.3 Fracture
Even in ductile materials, tensile processes can be limited by sudden fracture. The fracture
characteristic is not given by total elongation but is indicated by the cross-sectional area of
the fracture surface after the test-piece has necked and failed. This is difficult to measure
in thin sheet and consequently problems due to fracture may not be properly recognized.
1.2.4 Homogeneity
Industrial sheet metal is never entirely homogeneous, nor free from local defects. Defects
may be due to variations in composition, texture or thickness, or exist as point defects such
as inclusions. These are difficult to characterize precisely. Inhomogeneity is not indicated
by a single tensile test and even with repeated tests, the actual volume of material being
tested is small, and non-uniformities may not be adequately identified.
1.2.5 Surface effects
The roughness of sheet and its interaction with lubricants and tooling surfaces will affect
performance in a forming operation, but will not be measured in the tensile test. Special
tests exist to explore surface properties.

1.2.6 Damage
During tensile plastic deformation, many materials suffer damage at the microstructural
level. The rate at which this damage progresses varies greatly with different materials. It
may be indicated by a diminution in strain-hardening in the tensile test, but as the rate of
damage accumulation depends on the stress state in the process, tensile data may not be
indicative of damage in other stress states.
1.2.7 Rate sensitivity
As mentioned, the rate sensitivity of most sheet is small at room temperature; for steel it
is slightly positive and for aluminium, zero or slightly negative. Positive rate sensitivity
usually improves forming and has an effect similar to strain-hardening. As well as being
indicated by the exponent m, it is also shown by the amount of extension in the tensile
test-piece after maximum load and necking and before failure, i.e. ETotal − Eu, increases
with increasing rate sensitivity.
1.2.8 Comment
It will be seen that the properties that affect material performance are not limited to those
that can be measured in the tensile test or characterized by a single value. Measurement of
homogeneity and defects may require information on population, orientation and spatial
distribution.
Many industrial forming operations run very close to a critical limit so that small
changes in material behaviour give large changes in failure rates. When one sample of
material will run in a press and another will not, it is frequently observed that the materials
cannot be distinguished in terms of tensile test properties. This may mean that one or two
tensile tests are insufficient to characterize the sheet or that the properties governing the
performance are only indicated by some other test.
1.3 Other mechanical tests
As mentioned, the tensile test is the most widely used mechanical test, but there are many
other mechanical tests in use. For example, in the study of bulk forming processes such as
forging and extrusion, compression tests are common, but these are not suitable for sheet.
Some tests appropriate for sheet are briefly mentioned below:
• Springback. The elastic properties of sheet are not easily measured in routine tensile
tests, but they do affect springback in parts. For this reason a variety of springback
tests have been devised where the sheet is bent over a former and then released.
• Hardness tests. An indenter is pressed into the sheet under a controlled load and the size
of the impression measured. This will give an approximate measure of the hardness
of the sheet – the smaller the impression, the greater the hardness. Empirical relations
allow hardness readings to be converted to ‘yield strength’. For strain-hardening
materials, this yield strength will be roughly the average of initial yield and ultimate
tensile strength. The correlation is only approximate, but hardness tests can usefully
distinguish one grade of sheet from another.
12 Mechanics of Sheet Metal Forming

• Hydrostatic bulging test. In this test a circular disc is clamped around the edge and
bulged to a domed shape by fluid pressure. From measurement of pressure, curvature
and membrane or thickness strain at the pole, a true stress–strain curve under equal
biaxial tension can be obtained. The advantage of this test is that for materials that
have little strain-hardening, it is possible to obtain stress–strain data over a much larger
strain range than is possible in the tensile test.
• Simulative tests. A number of tests have been devised in which sheet is deformed in a
particular process using standard tooling. Examples include drawing a cup, stretching
over a punch and expanding a punched hole. The principles of these tests are covered
in later chapters.
1.4 Exercises
Ex. 1.1 A tensile specimen is cut from a sheet of steel of 1mm thickness. The initial
width is 12.5mm and the gauge length is 50 mm.
(a) The initial yield load is 2.89 kN and the extension at this point is 0.0563 mm.
Determine the initial yield stress and the elastic modulus.
(b) When the extension is 15%, the width of the test-piece is 11.41. Determine the
R-value.
[Ans: (a): 231 MPa, 205 GPa; (b) 1.88]
Ex. 1.2 At 4% and 8% elongation, the loads on a tensile test-piece of half-hard aluminium
alloy are 1.59 kN and 1.66 kN respectively. The test-piece has an initial width of 10 mm,
thickness of 1.4mm and gauge length of 50 mm. Determine the K and n values.
[Ans: 174 MPa, 0.12]
Ex. 1.3 The K, n and m values for a stainless steel sheet are 1140 MPa, 0.35 and 0.01
respectively. A test-piece has initial width, thickness and gauge length of 12.5, 0.45 and
50mm respectively. Determine the increase in load when the extension is 10% and the
extension rate of the gauge length is increased from 0.5 to 50 mm/minute.
[Ans: 0.27 kN]
Ex. 1.4 The following data pairs (load kN; extension mm) were obtained from the plastic
part of a load-extension file for a tensile test on an extra deep drawing quality steel sheet of
0.8mm thickness. The initial test-piece width was 12.5mm and the gauge length 50 mm.
1.57, 0.080; 1.90, 0.760; 2.24, 1.85; 2.57, 3.66; 2.78, 5.84; 2.90, 8.92
2.93,11.06; 2.94,13.49; 2.92, 16.59; 2.86, 19.48; 2.61, 21.82; 2.18, 22.69
Obtain engineering stress–strain, true stress, strain and log stress, log strain curves. From
these determine; initial yield stress, ultimate tensile strength, true strain at maximum load,
total elongation and the strength coefficient, K, and strain-hardening index, n.
[Ans: 156 MPa, 294 MPa, 0.24, 45%, 530 MPa, 0.24]

Sheet deformation processes:

2.1 Introduction
In Chapter 1, the appropriate definitions for stress and strain in tensile deformation were
introduced. The purpose now is to indicate how the true stress–strain curve derived from
a tensile test can be applied to other deformation processes that may occur in typical sheet
forming operations.
A common feature of many sheet forming processes is that the stress perpendicular to
the surface of the sheet is small, compared with the stresses in the plane of the sheet (the
membrane stresses). If we assume that this normal stress is zero, a major simplification is
possible. Such a process is called plane stress deformation and the theory of yielding for
this process is described in this chapter. There are cases in which the through-thickness or
normal stress cannot be neglected and the theory of yielding in a three-dimensional stress
state is described in an appendix.
The tensile test is of course a plane stress process, uniaxial tension, and this is now
reviewed as an example of plane stress deformation.
2.2 Uniaxial tension
We consider an element in a tensile test-piece in uniaxial deformation and follow the
process from an initial small change in shape. Up to the maximum load, the deformation
is uniform and the element chosen can be large and, in Figure 2.1, we consider the whole
gauge section. During deformation, the faces of the element will remain perpendicular to
each other as it is, by inspection, a principal element, i.e. there is no shear strain associated
with the principal directions, 1, 2 and 3, along the axis, across the width and through the
thickness, respectively.

Principal strain increments
During any small part of the process, the principal strain increment along the tensile axis
is given by Equation 1.10 and is

Constant volume (incompressibility) condition
It has been mentioned that plastic deformation occurs at constant volume so that these
strain increments are related in the following manner. With no change in volume, the
differential of the volume of the gauge region will be zero, i.e.

Stress and strain ratios (isotropic material)
If we now restrict the analysis to isotropic materials, where identical properties will be
measured in all directions, we may assume from symmetry that the strains in the width
and thickness directions will be equal in magnitude and hence, from Equation 2.3,

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. dε1
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

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 σ1 and σ2, which are
both non-zero. The third stress, σ3, 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 2.2(a) for the uniaxial tension and Figure 2.2(b) 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 2.2(b), 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

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 σf. 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 σf 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
σf the objective is to determine, for a given value of α, the values of σ1 and σ2 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.

With hindsight, common yielding theories can be anticipated from knowledge of the
nature of plastic deformations in metals. These materials are polycrystalline and plastic
flow occurs by slip on crystal lattice planes when the shear stress reaches a critical level.
To a first approximation, this slip which is associated with dislocations in the lattice is
insensitive to the normal stress on the slip planes. It may be anticipated then that yielding
will be associated with the shear stresses on the element and is not likely to be influenced
by the average stress or pressure. It is appropriate to define these terms more precisely.
2.4.1 Maximum shear stress
On the faces of the principal element on the left-hand side of Figure 2.4, there are no
shear stresses. On a face inclined at any other angle, both normal and shear stresses will
act. On faces of different orientation it is found that the shear stresses will locally reach a
maximum for three particular directions; these are the maximum shear stress planes and
are illustrated in Figure 2.4. They are inclined at 45◦ to the principal directions and the
maximum shear stresses can be found from the Mohr circle of stress, Figure 2.5. Normal
stresses also act on these maximum shear stress planes, but these have not been shown in
the diagram.
The three maximum shear stresses for the element are

From the discussion above, it might be anticipated that yielding would be dependent on
the shear stresses in an element and the current value of the flow stress; i.e. that a yielding
condition might be expressed as
f (τ1, τ2, τ3) = σf
We explore this idea below

2.4.2 The hydrostatic stress
The hydrostatic stress is the average of the principal stresses and is defined as

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

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.

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 σ2 = σ3 = 0, the greatest maximum
shear stress at yielding is τcrit. = σf/2. Thus in this theory, the Tresca yield criterion,
yielding would occur in any process when

In plane stress, using the notation here, σ1 will be the maximum stress and, σ3 = 0, the
through-thickness stress. The minimum stress will be either σ3 if σ2 is positive, or, if σ2 is
negative, it will be σ2. In all cases, the diameter a of the Mohr circle of stress in Figure 2.5
will be equal to σf.
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 σf 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.

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 anticipated
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 : −1/2 : −1/2
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

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.

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.)

2.5.3 (Worked example) stress state
The current flow stress of a material element is 300 MPa. In a deformation process, the
principal strain increments are 0.012 and 0.007 in the 1 and 2 directions respectively.
Determine the principal stresses associated with this in a plane stress process.

2.6 Work of plastic deformation
If we consider a unit principal element as shown in Figure 2.10, then for a small deformation,
each side of the unit cube will move by an amount,

2.7 Work hardening hypothesis
In Section 2.5 it was shown that at a particular instant in a plane stress process where the
flow stress, σf, was known, the stresses and the ratio of the strain increments for a small
deformation could be determined. To model a process we need to be able to follow the
deformation along the given loading path as the flow stress changes. Clearly we would
need to know the strain hardening characteristic of the material as determined, for example,
by the true stress–strain curve in the tensile test.
It has been found by experiment that the flow stress increases in any process according
to the amount of plastic work done during this process; i.e. in two different processes, if
the work done in each is the same, the flow stress at the end of each process will be the
same regardless of the stress path. This statement is only true for monotonic processes
that follow the conditions given in Section 2.2.4; if there is a reversal in the process, the
flow stress cannot be predicted by any simple theory.
In a plane stress, proportional process, we can plot the relation between each of the
non-zero stresses and its strain as shown in Figure 2.11. From Equation 2.15(b), the sum
of the shaded areas shown is the total work done per unit volume of material in the process.
According to this work-hardening hypothesis, the flow stress at the end of this process is
that given by the tensile test curve, Figure 2.3, when an equal amount of work has been
done, i.e. when the sum of the areas in Figure 2.11 is equal to the area under the curve
in Figure 2.3.

The way in which this work-hardening hypothesis is implemented in any analysis is
described in the next section.

2.8 Effective stress and strain functions
The plastic work done per unit volume in an increment in a process is given by
Equation 2.15(a). It would useful if this could be expressed in the form

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

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 σf 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 approximation
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.

Deformation of sheet
in plane stress
3.1 Uniform sheet deformation processes
In Chapter 2, an instant in the plane stress deformation of a work-hardening material was
considered. We now apply the theory to some region of a sheet undergoing uniform, proportional
deformation as shown in Figure 3.1. If the undeformed sheet, of initial thickness
t0 is marked with a grid of circles of diameter d0 or a square mesh of pitch d0 as shown
in Figure 3.1(a), then during uniform deformation, the circles will deform to ellipses of
major and minor axes d1 and d2 respectively. If the square grid is aligned with the principal
directions, it will become rectangular as shown in Figure 3.1(b). The thickness is denoted
by t . At the instant shown in Figure 3.1(b), the deformation stresses are σ1 and σ2.

3.1.1 Tension as a measure of force in sheet forming
In sheet processes, deformation occurs as the result of forces transmitted through the sheet.
The force per unit width of sheet is the product of stress and thickness and in Figure 3.1(c)
is represented by,
T = σt
where, T , is known as the tension, traction or stress resultant. Because this is the product
of the current thickness t as well as the current stress σ, it is the appropriate measure of
force and will be used throughout this work in modelling processes. The term, tension,
will be used even though this suffers from the disadvantage that the force is not always
a tensile force. If the tension is negative, it indicates a compressive force. This is not a
serious problem as in plane stress sheet forming, almost without exception, one tension
will be positive, i.e. the sheet is always pulled in one direction. It is impractical to form
sheet by pushing on the edge; the expression used by practical sheet formers is that ‘you
cannot push on the end of a rope’.
In the convention used here, the principal direction 1 is that in which the principal
stress has the greatest (most positive) value, and the major tension T1 = σ1t will always
be positive. In stretching processes, the minor tension T2 = σ2t is tensile or positive. In
other processes, the minor tension could be compressive and in some cases the thickness
will increase. If T2 is compressive and large in magnitude, wrinkling may be a problem.
In discussing true stress in Section 1.1.3, it was shown that for most real materials,
strain-hardening continues, although at a diminishing rate, and true stress does not reach a
maximum. As tension includes thickness, which in many processes will diminish, T may
reach a maximum; this limits the sheet’s ability to transmit load and is one of the reasons
for considering tension in any analysis.
3.2 Strain distributions
In the study of any process, we usually determine first the strain over the part. This can
be done by measuring a grid as in Figure 3.1, or by analysis of the geometric constraint
exerted on the part. An example is the deep drawing process in Figure 3.2(a) and in
the Introduction, Figure I.9. As the process is symmetric about the axis, we need only
consider the strain at points on a line as shown in Figure 3.2(b). Plotting these strains in
the principal strain space, Figure 3.2(c), gives the locus of strains for a particular stage in
the process. As the process continues, this locus will expand, but not necessarily uniformly;
some points may stop straining, while others go on to reach a process limit.
For any process, there will be a characteristic strain pattern, as shown in Figure 3.2(c).
This is sometimes known as the ‘strain signature’. Considerable information can be
obtained from such a diagram and the way it is analysed is outlined in the following
section.
3.3 Strain diagram
The individual points on the strain locus in Figure 3.2(c) can be obtained from measurements
of a grid circles as shown in Figure 3.1. (If a square grid is used, the analysis

method is outlined in Appendix A.2.) If the major and minor axes are measured and the
current thickness determined, the analysis is as follows.

Thickness strain and thickness
From Equation 3.1(a), the thickness strain is determined by measurement of thickness, or
alternatively from the major and minor strains assuming constant volume deformation, i.e.

3.3.4 Summary of the deformation at a point
From the above, the principal strains and the strain ratio can be determined. The straining
process is conveniently described by the principal strains, i.e.

where β is constant.
Each point in the strain diagram in Figure 3.2(c) indicates the magnitude of the final
major and minor strain and the assumed linear path to reach this point. Referring to
Figure 3.3(a), we examine in more detail the character of different strain paths. This
diagram, Figure 3.3(a) does not represent any particular process, but will be used to discuss
the different deformation processes. The ellipse shown is a contour of equal effective
strain, ε; each point on the ellipse will represent strain in a material element that, from
the work-hardening hypothesis in Section 2.7, has the same flow stress, σf.

Modes of deformation
If, by convention, we assign the major principal direction 1 to the direction of the greatest
(most positive) principal stress and consequently greatest principal strain, then all points
will be to the left of the right-hand diagonal in Figure 3.3(a), i.e. left of the strain path in

Figure 3.3 (a) The strain diagram showing the different deformation modes corresponding to different
strain ratios. (b) Equibiaxial stretching at the pole of a stretched dome. (c) Deformation in plane strain
in the side-wall of a long part. (d) Uniaxial extension of the edge of an extruded hole. (e) Drawing
or pure shear in the flange of a deep-drawn cup, showing a grid circle expanding in one direction and
contracting in the other. (f) Uniaxial compression at the edge of a deep-drawn cup. (g) The different
proportional strain paths shown in Figure 3.2 plotted in an engineering strain diagram.
which β = 1. As stated above, the principal tension and principal stress in the direction,
1, will always be tensile or positive, i.e. σ1 ≥ 0. For the extreme case in which σ1 = 0
we find from Equations 2.6 and 2.14, that α = −∞ and β = −2. Therefore all possible
straining paths in sheet forming processes will lie between 0A and 0E in Figure 3.3(a)
and the strain ratio will be in the range −2 ≤ β ≤ 1.

3.4.1 Equal biaxial stretching, β = 1
The path 0A indicates equal biaxial stretching. Sheet stretched over a hemispherical punch
will deform in this way at the centre of the process shown in Figure 3.3(b). The membrane
strains are equal in all directions and a grid circle expands, but remains circular. As β = 1,
the thickness strain is ε3 = −2ε1, so that the thickness decreases more rapidly with respect
to ε1 than in any other process. Also from Equation 2.19(c), the effective strain is ε = 2ε1
and the sheet work-hardens rapidly with respect to ε1.
3.4.2 Plane strain, β = 0
In this process illustrated by path, 0B, in Figure 3.3(a), the sheet extends only in one
direction and a circle becomes an ellipse in which the minor axis is unchanged. In long,
trough-like parts, plane strain is observed in the sides as shown in Figure 3.3(c). It will
be shown later that in plane strain, sheet is particularly liable to failure by splitting.
3.4.3 Uniaxial tension, β = −1/2
The point C in Figure 3.3(a) is the process in a tensile test and occurs in sheet when the
minor stress is zero, i.e. when σ2 = 0. The sheet stretches in one direction and contracts
in the other. This process will occur whenever a free edge is stretched as in the case of
hole extrusion in Figure 3.3(d).

3.4.4 Constant thickness or drawing, β = −1
In this process, point D, membrane stresses and strains are equal and opposite and the sheet
deforms without change in thickness. It is called drawing as it is observed when sheet is
drawn into a converging region. The process is also called pure shear and occurs in the
flange of a deep-drawn cup as shown in Figure 3.3(e). From Equation 3.1(b), the thickness
strain is zero and from Equation 2.19(c) the effective strain is ε = 2/√3 ε1 = 1.155ε1
and work-hardening is gradual. Splitting is unlikely and in practical forming operations,
large strains are often encountered in this mode.
3.4.5 Uniaxial compression, β = −2
This process, indicated by the point E, is an extreme case and occurs when the major
stress σ1 is zero, as in the edge of a deep-drawn cup, Figure 3.3(f). The minor stress is
compressive, i.e. σ2 = −σf and the effective strain and stress are ε = −ε2 and σ = −σ2
respectively. In this process, the sheet thickens and wrinkling is likely.
3.4.6 Thinning and thickening
Plotting strains in this kind of diagram, Figure 3.3(a), is very useful in assessing sheet
forming processes. Failure limits can be drawn also in such a space and this is described
in a subsequent chapter. The position of a point in this diagram will also indicate how
thickness is changing; if the point is to the right of the drawing line, i.e. if β > −1, the
sheet will thin. For a point below the drawing line, i.e. β < −1, the sheet becomes thicker.
3.4.7 The engineering strain diagram
In the sheet metal industry, the information in Figure 3.3(a) is often plotted in terms of
the engineering strain. In Figure 3.3(g), the strain paths for constant true strain ratio paths
have been plotted in terms of engineering strain. It is seen that many of these proportional
processes do not plot as straight lines. This is a consequence of the unsuitable nature of
engineering strain as a measure of deformation and in this work, true strains will be used
in most instances. Engineering strain diagrams are still widely used and it is advisable
to be familiar with both forms. In this work, true strain diagrams will be used unless
specifically stated.
3.5 Effective stress–strain laws
In the study of a process, the first step is usually to obtain some indication of the strain
distribution, as in Figure 3.2(c). As mentioned, this may be done by measuring grids or
from some geometric analysis. The next step is to determine the stress state associated with
strain at each point. To do this, one must have stress–strain properties for the material
and Chapter 2 indicates how the tensile test data can be generalized to apply to any
simple process using the effective stress–strain relations. In numerical models, the actual
stress–strain curve can be used as input, but in a mechanics model it is preferable to use
a simple empirical law that approximates the data. Here we consider some of these laws.

The effective strain ε for any deformation process such as the one illustrated in
Figure 3.1 can be calculated from the principal strains and the strain ratio using
Equation 2.19(c). As shown in Section 2.8, if the material is isotropic, the effective
stress–strain curve is coincident with the uniaxial true stress–strain curve and a variety
of mathematical relations may be fitted to the true stress–strain data. Some of the more
common empirical relations are shown in Figure 3.4 and in these diagrams elastic strains
are neglected. In the diagrams shown, the experimental curve is represented by a light
line, and the fitted curve by a bold line.

3.5.1 Power law
A simple power law

will fit data well for some annealed sheet, except near the initial yield; this is shown in
Figure 3.4(a). The exponent, n, is the strain-hardening index as described in Section 1.1.3.
The constants, K and n, are obtained by linear regression as explained in the section
referred to. The only disadvantage of this law is that at zero strain, it predicts zero stress
and an infinite slope to the curve. It does not indicate the actual initial yield stress.
3.5.2 Use of a pre-strain constant
Although it requires the determination of three constants, a law of the type

is useful and will fit a material with a definite yield stress as shown in Figure 3.4(b).
The constant ε0 has been termed a pre-strain or offset strain constant. If the material

has been hardened in some prior process, this constant indicates a shift in the strain axis
corresponding to this amount of strain as shown in Figure 3.4(b). In materials which are
very nearly fully annealed and for which ε0 is small, this relation can be obtained by first
fitting Equation 3.6 and then, using the same values of Kand n, to determine the value
of ε0 by fitting the curve to the experimentally determined initial yield stress using the
equation