Constitutive Models Relations between Stress and Strain
3.7 Small Strain, Rate Independent Plasticity: Metals
loaded beyond yield
many design calculations, the elastic constitutive equations outlined in
Section 3.1 are sufficient, since large plastic strains are by and large
undesirable and will lead to failure.
There are some applications, however, where it is of interest to predict
the behavior of solids subjected to large loads, sufficient to cause permanent
plastic strains. Examples include:
forming, machining or other manufacturing processes
theory was developed to predict the behavior of metals under loads exceeding
the plastic range, but the general framework of plasticity theory has since
been adapted to other materials, including polymers and some types of soil
(clay). Some concepts from metal
plasticity are also used in modeling concrete and other brittle materials such
as polycrystalline ceramics.
3.7.1 Features of the inelastic
response of metals.
begin by reviewing the results of a typical tension/compression test on an
annealed, ductile, polycrystalline metal specimen (e.g. copper or Al). Assume that the test is conducted at moderate
temperature (less than the melting point of the solid e.g. room temperature) and at modest strains
(less than 10%), at modest strain rates ( ).
The results of such a test
stresses (and strains) the solid responds elastically. This means the stress is proportional to the
strain, and the deformation is reversible.
If the stress
exceeds a critical magnitude, the stressstrain curve
ceases to be linear. It is often
difficult to identify the critical stress accurately, because the stress strain
curve starts to curve rather gradually.
If the critical
stress is exceeded, the specimen is permanently changed in length on unloading.
If the stress
is removed from the specimen during a test, the stressstrain curve
during unloading has a slope equal to that of the elastic part of the stressstrain
curve. If the specimen is re-loaded, it
will initially follow the same curve, until the stress approaches its maximum
value during prior loading. At this
point, the stressstrain curve once
again ceases to be linear, and the specimen is permanently deformed further.
If the test is
interrupted and the specimen is held at constant strain for a period of time,
the stress will relax slowly. If the
straining is resumed, the specimen will behave as though the solid were
unloaded elastically. Similarly, if the
specimen is subjected to a constant stress, it will generally continue to
deform plastically, although the plastic strain increases very slowly. This phenomenon is known as `creep.’
If the specimen
is deformed in compression, the stressstrain curve is a
mirror image of the tensile stressstrain curve (of
course, this is only true for modest strains.
For large strains, geometry changes will cause differences between the
tension and compression tests).
If the specimen
is first deformed in compression, then loaded in tension, it will generally
start to deform plastically at a lower tensile stress than an annealed specimen. This phenomenon is known as the `Bauschinger
response to cyclic loading can be extremely complex. One example is shown in the picture above in this case, the material hardens
cyclically. Other materials may soften.
shape of the plastic stressstrain curve
depends on the rate of loading, and also on temperature.
also need to characterize the multi-axial response of an inelastic solid. This is a much more difficult experiment to
do. Some of the nicest experiments were
done by G.I. Taylor and collaborators in the early part of the last
century. Their approach was to measure
the response of thin-walled tubes under combined torsion, axial loading and
The main conclusions of
these tests were
The shape of
the uniaxial stress-strain curve is insensitive to hydrostatic pressure. However, the ductility (strain to failure)
can be increased by adding hydrostatic pressure, particularly under torsional
are volume preserving, i.e. the plastic strain rate must satisfy
loading, the principal components of the plastic strain rate tensor are
parallel to the components of stress acting on the solid. This sounds obvious until you think about
it… To understand what this means,
imagine that you take a cylindrical shaft and pull it until it starts to deform
plastically. Then, holding the axial
stress fixed, apply a torque to the shaft.
Experiments show that the shaft will initially stretch, rather
than rotate. The plastic strain
increment is proportional to the stress acting on the shaft, not the stress
increment. This is totally unlike
multi-axial loading, most annealed polycrystalline solids obey the Levy-Mises
flow rule, which relates the principal components of strain rate during
plastic loading to the principal stresses as follows
In this section, we will outline the simplest
plastic constitutive equations that capture the most important features of
metal plasticity. There are many
different plastic constitutive laws, which are intended to be used in different
applications. There are two broad
1. Rate independent plasticity which is used to model metals deformed at low
temperatures (less than half the material’s melting point) and modest strain
rates (of order 0.01-10/s). This is the
focus of this section.
2. Rate dependent plasticity, or viscoplasticity used to model high temperature creep (steady
accumulation of plastic strain at contstant stress) and also to model metals
deformed at high strain rates (100/s or greater), where flow strength is
sensitive to deformation rate.
Viscoplasticity will be discussed in Section 3.3.
There are also various different models within these
two broad categories. The models
generally differ in two respects (i) the yield criterion; (ii) the strain
hardening law. There is no completely
general model that describes all the features that were just listed, so in any
application, you will need to decide which aspect of material behavior is most
important, and then choose a model that accurately characterizes this behavior.
Key ideas in
modeling metal plasticity
key concepts form the basis of almost all classical theories of
plasticity. They are
1. The decomposition of strain
into elastic and plastic parts;
2. Yield criteria, which
predict whether the solid responds elastically or plastically;
3. Strain hardening rules,
which control the way in which resistance to plastic flow increases with
4. The plastic flow rule, which
determines the relationship between stress and plastic strain under multi-axial
5. The elastic unloading
criterion, which models the irreversible behavior of the solid.
concepts will be described in more detail in the sections below.
simplicity, we will at this stage restrict attention to infinitesimal deformations.
we adopt the infinitesimal strain tensor
as our deformation measure. We have no need to distinguish between the
various stress measures and will use to denote stress.
It is also important to note that the plastic strains in a solid depend on the
load history. This means that
the stress-strain laws are not just simple equations relating stress to
strain. Instead, plastic strain laws
must either relate the strain rate in the solid to the stress and stress rate, or else
specify the relationship between a small increment of plastic strain in terms of strain, stress and stress
increment . In addition, plasticity problems are almost
always solved using the finite element method.
Consequently, numerical methods are used to integrate the plastic stress-strain
3.7.2. Decomposition of strain into
elastic and plastic parts
Experiments show that under uniaxial loading, the
strain at a given stress has two parts: a small recoverable elastic strain, and
a large, irreversible plastic strain, as shown in the picture. In uniaxial tension, we would write
Experiments suggest that the reversible part is
related to the stress through the usual linear elastic equations. Plasticity theory is concerned with
characterizing the irreversible part.
For multiaxial loading, we generalize this by
decomposing a general strain increment into elastic and plastic parts, as
elastic part of the strain is related to stress using the linear elastic
equations (discussed in detail in 3.1)
yield criterion is used to determine the critical stress required to cause
permanent deformation in a material. There
are many different yield criteria here we will just list the simplest ones. Let be the stress acting on a solid, and let denote the principal values of stress. In addition, let denote the yield stress of the material in
uniaxial tension. Then, define
Tresca yield criterion
both cases, the criteria are defined so that the material deforms elastically
and plastically for . The yield stress may increase during plastic straining, so we
have shown that Y is a function of a
measure of total plastic strain ,
to be defined in Section 3.2.5
An alternative form of VonMises criterion.
For a general stress state, it is a nuisance having to compute the principal
stresses in order to apply von Mises yield criterion. Fortunately, the criterion can be expressed
directly in terms of the stress tensor
are the components of the
`von Mises effective stress’ and
`deviatoric stress tensor’ respectively.
These yield criteria are
based largely on the following experimental observations:
hydrostatic stress (all principal stresses equal) will never cause yield, no
matter how large the stress;
(2) Most polycrystalline metals are isotropic. Since the yield criterion depends only on the
magnitudes of the principal stresses, and not their directions, the yield
criteria predict isotropic behavior.
suggest that von Mises yield criterion provides a slightly better fit to
experiment than Tresca, but the difference between them is very small. Sometimes it simplifies calculations to use
Tresca’s criterion instead of von Mises.
3.7.4 Graphical representation of the
arbitrary stress state can be plotted in
`principal stress space,’ with the three principal stresses as axes.
criterion is plotted in this way in the picture to the right. The yield criterion is a cylinder, radius ,
with its axis parallel to the line
the state of stress falls within the cylinder, the material is below yield and
responds elastically. If the state of
stress lies on the surface of the cylinder, the material yields and deforms
plastically. If the plastic deformation
causes the material to strain harden, the radius of the cylinder
increases. The stress state cannot lie
outside the cylinder this would lead to an infinite plastic strain.
the yield criterion defines a surface in stress space, it is often
referred to as a yield surface. The
yield surface is often drawn as it would appear when viewed down the axis of
the cylinder, as shown below. The Tresca yield criterion can also be plotted in
this way. It looks like a cylinder with
a hexagonal cross section, as shown.
3.7.5. Strain hardening laws
Experiments show that if you plastically deform a
solid, then unload it, and then try to re-load it so as to induce further
plastic flow, its resistance to plastic flow will have increased. This is known as strain hardening.
Obviously, we can model strain hardening by relating
the size and shape of the yield surface to plastic strain in some appropriate
way. There are many ways to do
this. Here we describe the two simplest
Rather obviously, the easiest way to model strain
hardening is to make the yield surface increase in size, but remain the same
shape, as a result of plastic straining.
This means we must devise some appropriate
relationship between Y and the plastic strain. To get a suitable scalar measure of plastic
strain we define the accumulated plastic strain magnitude
(the factor of 2/3 is introduced so that in a uniaxial tensile test in which the
specimen is stretched parallel to the direction.
To see this, note that plastic strains do not change volume, so that and substitute into the formula.)
Then we make Y
a function of . People often use power laws or piecewise
linear approximations in practice. A few of the more common forms of hardening
Linear strain hardening
these formulas, ,
h and m are material properties.
These functions are illustrated in the figures below
Perfectly plastic solid
Linear strain hardening solid
Power-law hardening solid
An isotropic hardening law is generally not useful
in situations where components are subjected to cyclic loading. It does not account for the Bauschinger
effect, and so predicts that after a few cycles the solid will just harden
until it responds elastically.
To fix this, an alternative hardening law allows the
yield surface to translate, without changing its shape. The idea is illustrated graphically in the
picture. As you deform the material in
tension, you drag the yield surface in the direction of increasing stress, thus
modeling strain hardening. This softens
the material in compression, however.
So, this constitutive law can model cyclic plastic deformation. The stress-strain curves for isotropic and
kinematic hardening materials are contrasted in the figure below.
To account for the fact that the center of the yield
locus is at a position in stress space, the Von-Mises yield criterion
needs to be modified as follows
Here, Y is
now a constant, and hardening is modeled by the motion of the yield surface. To
do so, we need to relate to the plastic strain history somehow. There are many ways to do this, which can
model subtle features of the plastic response of solids under cyclic and
nonproportional loading. The simplest approach is to set
hardening law predicts that the stress-plastic strain curve is a straight line
with slope c. This is known as linear
more sophisticated approach is to set
where c and are material constants. It’s not so easy to visualize what this does it turns out that that this relation can model
cyclic creep the tendency of a material to accumulate
strain in the direction of mean stress under cyclic loading, as illustrated in
the figure on the right. It is known as
the Armstrong-Frederick hardening law.
There are many other kinematic type hardening laws. New ones are still being developed.
The plastic flow law
To complete the plastic stress-strain relations, we
need a way to predict the plastic strains induced by stressing the material
beyond the yield point. Specifically,
1. The current stress applied to the material
2. The current yield stress
(characterized by for isotropic hardening, or for kinematic hardening)
3. A small increase in stress applied to the solid
we wish to determine the small change in plastic
The formulas are given below, for isotropic and
kinematic hardening. These are just fits
to experiment (specifically, to the Levy-Mises flow rule). The physical significance and reason for the
structure of the equations will be discussed later.
The plastic strains are usually derived from the
yield criterion f defined in 3.6.3,
and so are slightly different for isotropic and kinematic hardening. A material that has its plastic flow law
derived from f is said to have an
`associated’ flow law i.e.the flow law is associated with f.
Isotropic Hardening (Von-Mises yield
denotes the Von-Mises yield criterion, and is determined from the condition that the
yield criterion must be satisfied at all times during plastic straining. This shows that
the slope of the plastic stress-strain curve.
The algebra involved in differentiating f with respect to stress is outlined below.
Linear Kinematic Hardening (Von-Mises
the yield criterion is now
and as before is determined from the condition that the
yield criterion must be satisfied at all times during plastic straining. This shows that
that for linear kinematic hardening the hardening law is
into the Taylor
expansion of the yield criterion and simplifying shows that
of flow law formulas with the Levy-Mises flow rule
Levy-Mises flow law (based on experimental observations) states that principal
values of the plastic strain increment induced by a stress increment are related to
the principal stresses by
is straightforward to show that this observation is consistent with the
predictions of the flow law formulas given in this section. To see this,
suppose that the principal axes of stress are parallel to the directions.
In this case the only nonzero components of deviatoric stress are
The flow law
we see that
similar expressions for other components.
Some trivial algebra then yields the Levy-Mises flow law.
the yield criterion
the yield criterion requires some sneaky index notation manipulations. Note
Now, recall that
and further that
However, observe that
Elastic unloading condition
There is one final issue to consider. Experiments show that plastic flow is
irreversible, and always dissipates energy.
If the increment in stress is tangent to the yield surface, or brings the
stress below yield, as shown in the picture then there is no plastic strain.
isotropically hardening solid, this unloading condition may be expressed as
both cases, the solid deforms elastically (no plastic strain) if the condition
Complete incremental stress-strain relations for a rate independent
conclude by summarizing the complete elastic-plastic stress strain relations
for an isotropic solid with Von-Mises
Isotropically hardening elastic-plastic solid
The solid is characterized by its elastic constants and by the yield stress as a function of accumulated plastic strain and its slope
this case we have that
may be combined to
is sometimes necessary to invert these expressions. A straightforward but tedious series of index
notation manipulations shows that
This constitutive law is
the most commonly used model of inelastic deformation. It has the following properties:
correctly predict the conditions necessary to initiate yield under multiaxial
correctly predict the plastic strain rate under an arbitrary multiaxial stress
It can model
accurately any uniaxial stressstrain curve
It has the following
It is valid
only for modest plastic strains (<10%)
It will not
predict creep behavior or strain rate sensitivity
It does not
predict behavior under cyclic loading correctly
It will not
predict plastic strains accurately if the principal axes of stress rotate
significantly (more than about 30 degrees) during inelastic deformation
Kinematically hardening solid
The solid is characterized by its elastic constants and by the initial yield stress and the strain hardening rate c. Then,
Finally, the evolution equation for is
constitutive equation is used primarily to model cyclic plastic deformation, or
plastic flow under nonproportional loading (where principal axes of stress
rotate significantly during plastic flow). It has the following limitations:
It is valid only for modest plastic strains
It will not predict creep behavior or strain
It does not predict the shape of the
stress-strain curve accurately
3.7.9 Typical values for yield stress of
elastic constants, the plastic properties of metals are highly variable, and
are also very sensitive to alloying composition and microstructure (which can
be influenced by heat treatment and mechanical working). Consequently, it is impossible to give
accurate values for yield stresses or hardening rates for materials. The table below (again, taken from
`Engineering Materials,’ by M.F. Ashby and D.R.H. Jones, Pergamon Press) lists
rough values for yield stresses of common materials these may provide a useful guide in
preliminary calculations. If you need
accurate data you will have to measure the properties of the materials you plan
to use yourself.
180 - 1320
Aluminum & alloys
52 - 90
49 - 87
60 - 110
Low alloy steels
3.7.10 Perspectives on plastic
constitutive equations - The Principle of Maximum Plastic Resistance
The constitutive law
outlined in the preceding section has an important property, known as the principle of maximum plastic resistance.
Statement of the principle: Let be a stress state which causes plastic
deformation, let be a small change in and let be the resulting strain increment. Now, let be any other stress that can be imposed on the
specimen that either does not reach yield, or else just satisfies the yield
criterion, i.e. with .
Interpretation: The Principle of Maximum Plastic Resistance is a
mathematical statement of the following ideas:
The Mises yield surface is convex
The plastic strain rate is normal to the
It is best to illustrate
these ideas graphically. In principal stress space, the product is represented by the dot product of the
stress and plastic strain rate vectors.
equivalent to the requirement that the angle between the vectors formed by and must be greater than 90 degrees for all
stresses and strain rates. This is only
possible if the yield stress is convex and the strain rate is normal to the
Principle of Maximum Plastic Resistance is important because it is the basis
for a number of very important theorems concerning plastic deformation in
solids. For example, it can be shown
that the stress field in a material that obeys the Principle is always
unique. In addition, the principle leads
to clever techniques to estimate collapse loads for elastic-plastic solids and
Proof of the
principle of maximum
goal is to prove that . The simplest way to do so is to show that ,
where is the plastic strain magnitude, and Y is the yield stress. To this end:
1. Recall the plastic flow rule
Multiply both sides by stress
we have noted that .
Recall that causes yield, and so must satisfy the yield
This shows that
6. Now, recall again that causes yield, while can be at or below yield. The yield criterion therefore requires that .
Substituting these inequalities into (5) shows that . Finally, this shows
proving the principle.
3.7.11 Perspectives on plastic
constitutive equations - Drucker’s Postulate
models of inelastic behavior are based largely on experimental observations of
plastic flow in laboratory specimens.
Similar constitutive laws are used to describe very different materials,
including metals, ceramics, glasses, soils and polymers. The mechanisms of deformation in these
materials are very different, so it is surprising that their response is
perspective on the structure of constitutive laws for inelastic solids was
developed by Drucker in the 1950s.
Drucker introduced the idea of a stable
plastic material, as follows. Consider a deformable solid, subjected to
boundary tractions ,
which induce some displacement field . Suppose that the tractions are increased to ,
resulting in an additional displacement .
The material is said to be stable in the sense of Drucker if the work done by
the tractions through the displacements is positive or zero for all :
can be shown that, for a plastic material to be stable in this sense, it must
satisfy the following conditions:
surface must be convex
strain rate must be normal to the yield surface
The rate of
strain hardening must be positive or zero
a material that is stable in the sense of Drucker must satisfy the principle of
maximum plastic resistance.
does not really explain why the
constitutive law should have this structure, but materials that do not satisfy
the Drucker stability criterion tend to be difficult to work with in
calculations, so there is a strong incentive for choosing a constitutive law
that meets the condition. It is not
surprising, then, that the people developing constitutive laws ended up with a
form that satisfies Drucker stability.
3.7.12 Microscopic Perspectives on
Plastic Flow in Metals
is possible to obtain some insight into the structure of the constitutive laws
for metals by considering the micromechanisms responsible for plastic flow.
flow in metals is caused by dislocation motion.
Dislocations are line defects in crystalline solids you can think of a dislocation as an extra
plane of atoms inserted within a perfect crystal, as shown in the picture on
the crystal is subjected to stress, these defects move through the solid and
rearrange the crystal lattice. For
example, if the model crystal shown above is subjected to a shear stress, the
atoms rearrange so that the top part of the crystal is shifted to the right
relative to the bottom part.
the crystal lattice is distorted near a dislocation, only a modest shear stress
is required to drive the dislocation through the solid, causing permanent
and atomistic simulations suggest that dislocation motion obeys Schmidt’s law:
a dislocation moves through a crystal if the shear stress on its glide plane
exceeds a critical magnitude .
can be shown that a material which deforms by dislocation glide, and which
obeys Schmidt’s law, will satisfy the principle of maximum plastic
resistance. This, in turn, implies that
the yield surface for the solid must be convex and the plastic strain rate must
be normal to the yield surface.
notion of a yield surface and convexity for a material which deforms by
dislocation glide can be illustrated with a simple double-slip model. Consider a single crystal, which contains two
dislocation glide planes oriented at to the principal axes of stress as shown in
the picture on the right. Assume plane
stress conditions, for simplicity.
picture below shows side views of the two slip planes. As an exercise, you
should verify that the shear stresses (tangential component of traction) acting
on the two slip planes are
solid reaches yield if . The resulting yield surface is sketched
below: the planes of the yield surface have been color coded to indicate which
slip plane is active.
Observe that the yield
surface is convex: this is a consequence of Schmidt’s law.
suppose that slip is activated on one of the glide planes. Let denote the tangent to the slip plane, and let denote the normal. To compute the strain
produced by slip on a single slip system, consider the deformation of an
infinitesimal line element dx under
The deformed line element
is given by
or, in index notation
The deformation gradient
and the Lagrange strain
we can approximate the Lagrange strain tensor by the infinitesimal strain
Now, suppose that the
marked on the yield locus shown below.
This activates slip as shown in the picture. The normal and tangent to the appropriate
slip plane are
The strain rate therefore
showing that the plastic strain rate is normal to the yield locus (see the
figure).You could verify as an exercise that that if the stress reaches the
other limiting surfaces of the yield locus, the resulting strain rate will be
normal to the yield locus.