We know that when we apply loads to an object,
like this bracket, if we keep increasing the magnitude of the load, at some point the material
will fail. But how can we predict when static failure
will occur? What level do the stresses in the object need
to reach for it to fail? First we need to define what failure is. For ductile materials failure is usually considered
to occur at the onset of plastic deformation. And for brittle materials it occurs at fracture. These points are easy to define for a uniaxial
stress state, like a tensile test. They occur when the normal stress in the object
reaches the yield strength of the material for ductile materials and the ultimate strength
for brittle materials. But for a more complex case of tri-axial stress,
predicting failure is much less straightforward. In fact it's so difficult to predict that
we don't have one universally applicable method. Instead, we have to predict failure by selecting
the most suitable one of a range of different failure theories, each of which we know works
relatively well under certain circumstances, based on experimentation. Because they fail in fundamentally different
ways, failure theories which apply for ductile materials usually aren't applicable for brittle
materials, and vice-versa. So what does a failure theory do? It's quite simple - it allows us to predict
failure of a material by comparing the stress state in the object we are assessing with
material properties that are easy to determine, like the yield or ultimate strengths of the
material, that we can obtain by performing a uniaxial test. The stress state at a point can be described
using the three principal stresses, so most failure theories are defined as a function
of the principal stresses and the material strength. Probably the simplest failure theory is to
say that failure occurs when the maximum or minimum principal stresses reach the yield
or ultimate strengths of the material. This is called Maximum Principal Stress theory,
or Rankine theory. Although it’s simple to apply, it’s not
actually a great failure theory, particularly for ductile materials, for reasons I'll explain
later. Let’s look at some better failure theories
for ductile materials. Any good failure theory needs to be consistent
with experimental observations we can make about how materials fail. There is one key observation that failure
theories for ductile materials need to capture, which is the fact that hydrostatic stresses
do not cause yielding in ductile materials. Let me explain… A general tri-axial stress state like the
one shown here can be decomposed into stresses which cause a change in volume, and stresses
which cause shape distortion. Stresses that cause a change in volume are
called hydrostatic stresses, because that is the type of stress acting on an object
submerged in liquid. For a hydrostatic stress configuration the
three principal stresses are always equal, and there are no shear stresses. For a tri-axial stress state we can calculate
the hydrostatic component as the average of the three principal stresses. The mechanism that causes yielding of ductile
materials is shear deformation. Since there are no shear stresses for a state
of hydrostatic stress, this component can be very large and still not contribute to
yielding. Yielding is only caused by the stresses which
cause shape distortion. These are called deviatoric stresses. The deviatoric component is calculated by
subtracting the hydrostatic component from each of the principal stresses. The hydrostatic and deviatoric components
can be expressed in matrix form, like this. Here we have described the stress state using
the principal stresses, but we could also describe it for an arbitrary orientation of
the stress element. It can be helpful to use Mohr's circle to
understand the hydrostatic and deviatoric components of a tri-axial stress state. For a hydrostatic stress configuration there
are no shear stresses, and so Mohr’s circle reduces to a single point, equal to the average
of the three principal stresses. Shifting Mohr’s circle horizontally represents
an increase in the hydrostatic component. And increasing the radius of Mohr’s circle
without changing the average stress represents an increase in the deviatoric component. Since failure of ductile materials only depends
on the deviatoric component, a good failure theory for ductile materials should produce
the same result regardless of where Mohr’s circle is located on the horizontal axis. This explains why Maximum Principal Stress
theory is not a good failure theory for ductile materials - it isn’t consistent with the
observation that yielding is independent of hydrostatic stress. Two failure theories which are consistent
with this observation are the Tresca and von Mises failure criteria. These are the two most commonly used failure
theories for ductile materials. Let’s start with the Tresca failure criterion,
which is also called the maximum shear stress theory. It is named after the French engineer Henri
Tresca, and it states that yielding occurs when the maximum shear stress is equal to
the shear stress at yielding in a tensile test. This can be defined mathematically, like this. And graphically using Mohr’s circle, like
this. This theory is consistent with the observation
that hydrostatic stresses don’t affect yield. It doesn't care where Mohr’s circle is
located on the horizontal axis. It’s common to express this theory as a
function of the principal stresses, instead of as a function of the shear stresses. We can see based on Mohr’s circle for a
tri-axial stress state that the maximum shear stress is equal to the radius of the outer
circle, which is the difference between the maximum and minimum principal stresses, divided
by 2. Mohr's circle for a uniaxial tensile test
at yielding looks like this. The intermediate and minimum principal stresses
Sigma-2 and Sigma-3 are equal to zero, and the maximum principal stress Sigma-1 is equal
to the yield strength of the material. The shear stress at yielding is equal to half
of the yield strength of the material, so we can re-write our equation to obtain the
standard formulation for the Tresca theory. Let’s look at the von Mises failure criterion
next, which is also called the maximum distortion energy theory. It was initially developed by the Austrian
scientist Richard von Mises, but a number of others were involved in refining it, so
it is sometimes called the Maxwell–Huber–Hencky–von Mises theory. It states that yielding occurs when the maximum
distortion energy in a material is equal to the distortion energy at yielding in a uniaxial
tensile test. So what is the distortion energy? It is essentially the portion of strain energy
in a stressed element corresponding to the effect of the deviatoric stresses. The distortion energy per unit volume can
be calculated from the principal stresses using this equation. We know that at yielding during a tensile
test the maximum principal stress is equal to the yield strength of the material, and
the two other principal stresses are equal to zero. So we can calculate the distortion energy
at yielding in a tensile test by plugging in the appropriate principal stress values. And by equating the distortion energy at yielding
with the general equation for a tri-axial stress state, and re-arranging a bit, we get
an equation that defines the von Mises failure criterion. Again we can see that this theory considers
the difference between principal stresses, and so is independent of the hydrostatic stress. The term on the left is often called the equivalent
von Mises stress. If it is larger than the yield strength of
the material, yielding is predicted to have occurred. The equivalent von Mises stress is a common
output from stress analysis performed using the finite element method. Contour plots are typically generated to show
the distribution of the von Mises equivalent stress within a component, as these allow
areas at risk of yielding to be identified. When comparing failure theories it can be
useful to plot their yield surfaces. A yield surface is the representation of a
failure theory in the principal stress space, which sounds more complicated than it actually
is. To keep things simple let's start by considering
a plane stress case, where one of the three principal stresses is equal to zero. By convention principal stresses are ordered
from largest to smallest, like this, but since we don't know yet how the stresses will be
ordered, I'll refer to them as Sigma-A, Sigma-B, and Sigma-C instead. We’ll say that Sigma-C is equal to zero,
and so the two axes of our yield surface graph correspond to the two non-zero principal stresses,
Sigma-A and Sigma-B. The yield surface for the Maximum Principal
Stress theory is easy to plot - it says that yielding begins when either of these principal
stresses is equal to the yield strength of the material, so it looks like this. Yielding is considered to have occurred when
the stress state reaches this line. Next let’s plot the yield surface for the
Tresca theory. This theory states that yielding occurs when
the difference between the maximum and minimum principal stresses is equal to the yield strength
of the material. But it’s not as simple as taking the difference
between Sigma-A and Sigma-B. Plane stress is still a three-dimensional stress state. In the top right quadrant of the graph both
Sigma-A and Sigma-B are positive, and so Sigma-C, which is equal to zero, is the minimum principal
stress. The yield surface looks like this. But in the bottom right quadrant Sigma-B is
negative, and so Sigma-B is the minimum principal stress. That gives us a yield surface that looks like
this. Repeating this process for the other two quadrants
completes the Tresca yield surface. von Mises yield theory for plane stress conditions
is defined using this equation when expressed in terms of the principal stresses. We can square both sides of the equation and
see that it is of the same form as an ellipse, which gives us the von Mises yield surface. Let's plot some experimental data for tests
performed on ductile materials and see how each of these theories compares. It’s clear that the Maximum Principal Stress
theory has large areas where its use is potentially unsafe. So let's get rid of it, and compare Tresca
and von Mises. Both agree well with experimental observations,
although von Mises agrees slightly better. The Tresca yield surface lies entirely inside
the von Mises surface, meaning that Tresca is more conservative. If we assume that the principal stresses will
increase in proportion to one another as a load is applied, and so the stress paths are
straight lines, the difference between the Tresca and von Mises theories is largest for
these three configurations. The maximum difference between the two theories
can be calculated to be 15.5%. von Mises is usually preferred to Tresca because
it agrees better with experimental data, but Tresca is sometimes used because it is easier
to apply and it is more conservative. So far we have looked at a plane stress case,
with Sigma-C equal to zero. But for a general three-dimensional stress
state, Sigma-C will be non-zero. We know that the Tresca and von Mises yield
surfaces aren't affected by hydrostatic stresses, and so to obtain the yield surfaces in three
dimensions
we just need to extend the plane stress yield surfaces along the hydrostatic axis. Failure of brittle materials is different
to failure of ductile materials. For brittle materials, failure is considered
to occur by fracture rather than yielding. And unlike ductile materials, brittle materials
tend to have compressive strengths that are much larger than their tensile strengths. This needs to be captured in any failure theory
for brittle materials. It also means that to assess failure of brittle
materials we need to know two separate ultimate strengths, for tension and for compression. Coulomb-Mohr theory is a failure theory often
used for brittle materials. Unlike the theories we have seen for ductile
materials, it considers failure to be sensitive to hydrostatic stresses, and captures the
difference between tensile and compressive ultimate strengths. The easiest way to define this theory is using
Mohr’s circle. We start by drawing the Mohr’s circles corresponding
to failure in the uniaxial tensile and compressive tests. By drawing lines tangent to both circles we
create a failure envelope. Coulomb-Mohr theory states that a material
will fail for a stress-state with a Mohr’s circle that reaches this envelope. The Tresca yield criterion we saw earlier
is actually a special case of the Coulomb-Mohr failure theory, where
we have the same material properties in tension and in compression. The plane stress failure surface for Coulomb-Mohr
theory looks like this. As you can see it doesn’t agree particularly
well with experimental data in the bottom right quadrant. Modified Mohr theory is a slight variation
on the theory, which better fits experimental data. It is one of the preferred general failure
theories for brittle materials. Because failure theories for brittle materials
need to account for the effect of hydrostatic stresses, these failure theories converge
to a point when extended along the hydrostatic axis. And that’s it for this introduction to failure
theories! We've covered the most common theories, but
there are many more out there which may be more appropriate for specific scenarios. If you enjoyed the video and would like me
to cover more topics, please consider supporting the channel on Patreon. I’ll put a link in the description. Thanks for watching!
