Modeling Material Failure
9.3 Modeling failure by crack growth linear elastic fracture mechanics
damage models are useful in design applications, but they have many
extensive experimental testing to calibrate the model for each application;
They provide no
insight into the relationship between a materials microstructure and its
more sophisticated approach is to model the mechanisms of failure
directly. Crack propagation through the
solid, either as a result of fatigue, or by brittle or ductile fracture, is by
far the most common cause of failure.
Consequently much effort has been devoted to developing techniques to
predict the behavior of cracks in solids.
Below, we outline some of the most important results.
Crack tip fields in an isotropic, linear elastic solid.
of the techniques of fracture mechanics rely on the assumption that, if one
gets sufficiently close to the tip of the crack, the stress, displacement and
strain fields always have the same distribution, regardless of the geometry of
the solid and how it is loaded. The
fields near a crack tip are a fundamental result in fracture mechanics.
The picture shows an infinitely large linear elastic solid, with
Young’s modulus E and Poisson’s ratio
which contains a crack. The solid is
loaded at infinity. Note that
tip fields are most conveniently expressed in terms of cylindrical-polar
coordinates with origin at the crack tip;
The displacement and stress near the crack tip
can be characterized by three numbers ,
known as stress intensity factors. By
with the limit taken along .
The stress intensity factors depend on the
detailed shape of the solid, and the way that it is loaded. To calculate stress
intensity factors, you need to find the full stress field in the solid, and
then compute the limiting values in the definition. These calculations can be difficult you can try to find the solution in standard
tables of stress intensity factors, or if this fails use a numerical method
(such as FEM). A short table of stress intensity factors for various crack geometries
can be found in Section 9.3.3, and FEM techniques are discussed in 9.3.4.
Stress intensity factors have the bizarre
units of .
significance of the three stress intensity factors is illustrated in the
picture below. The `Mode I’ stress
intensity factor quantifies the crack opening displacements and
stresses; the `Mode II’ stress intensity factor characterizes in-plane shear
displacements and stress; and the `Mode III’ stress intensity factor quantifies
out-of-plane shear displacement of the crack faces and anti-plane shear
stresses at the crack tip.
The stress field near the
crack tip is
Equivalent expressions in rectangular coordinates are
while the displacements can be calculated by integrating the strains,
with the result
that the formulas for in-plane displacement components are valid for plane strain deformation
9.3.2 The assumptions and application of
phenomenological linear elastic fracture mechanics
The objective of linear elastic fracture
mechanics is to predict the critical loads that will cause a crack in a solid
to grow. For applications involving
fatigue or dynamic fracture, the rate and direction of crack growth are also of
The phenomenological theory is based on the
following qualitative argument. Consider
a crack in a reasonably brittle, isotropic solid. If the solid is ideally elastic, we expect
the asymptotic solution listed in the preceding section to become progressively
more accurate as we approach the crack tip.
Away from the crack tip, the fields are influenced by the geometry of
the solid and boundary conditions, and the asymptotic crack tip field is not
accurate. In practice, the asymptotic
field will also not give an accurate representation of the stress fields very
close to the crack tip. The crack may not be perfectly sharp at its tip, and if
it were, no solid could withstand the infinite stress predicted by the
asymptotic linear elastic solution. We
therefore anticipate that in practice the linear elastic solution will not be
accurate very close to the crack tip itself, where material nonlinearity and
other effects play an important role. So
the true stress and strain distributions will have 3 general regions
1. Close to the crack tip,
there will be a process zone, where the material suffers irreversible
2. A bit further from the crack
tip, there will be a region where the linear elastic asymptotic crack tip field
might be expected to be accurate. This
is known as the `region of K dominance’
3. Far from the crack tip the
stress field depends on the geometry of the solid and boundary conditions.
Material failure (crack growth or fatigue) is
a consequence of the ugly stuff that goes on in the process zone. Linear elastic fracture mechanics postulates
that one doesn’t need to understand this ugly stuff in detail, since the fields
in the process zone are likely to be controlled mainly by the fields in the
region of K dominance. The fields in
this region depend only on the three stress intensity factors . Therefore, the state in the process zone can
be characterized by the three stress intensity factors.
If this is true, the conditions for crack
growth, or the rate of crack growth, will be only a function of stress
intensity factor and nothing else. We
can measure the critical value of required to cause the crack to grow in a
standard laboratory test, and use this as a measure of the resistance of the
solid to crack propagation. For fatigue
tests, we can measure crack growth rate as a function of or their history, and characterize the
relationship using appropriate phenomenological laws.
Having characterized the material, we can
then estimate the safety of a structure or component that containing a
crack. To do so, calculate the stress
intensity factors for the crack in the structure, and then use our
phenomenological fracture or fatigue laws to decide whether or not the crack
For example, the fracture criterion under mode I loading is written
for crack growth, where is the critical stress intensity factor for
the onset of fracture. The critical
stress intensity factor is referred to as the fracture toughness of the
Experimentally, it is found that this
approach works quite well, provided that the assumptions inherent in linear
elastic fracture mechanics are satisfied.
Careful tests have established the following conditions
for the applicability of linear elastic fracture mechanics.
1. All characteristic specimen
dimensions must exceed 25 times the expected plastic zone size at the crack
2. For plane strain conditions
at the crack tip the specimen thickness must exceed at least the plastic zone
For a material with yield stress Y loaded in Mode I with stress intensity factor
the plastic zone size can be estimated as
application of linear elastic fracture mechanics to in design
To apply LEFM in a design application, you need to be able to:
1. Design a laboratory specimen
that can induce a prescribed stress intensity factor at a crack tip
2. Measure the critical stress
intensity factors that cause fracture in the laboratory specimen, or measure
fatigue crack growth rates as a function of static or cyclic stress intensity
3. Estimate the anticipated
size and location of cracks in your structure or component
4. Calculate the stress
intensity factors for the cracks in your structure or component under
anticipated loading conditions
5. Combine the results of steps
2 and 4 to predict the behavior of the cracks in the structure of interest, and
make appropriate design recommendations.
These steps are outlined in more detail below.
9.3.3 Calculating stress intensity factors
Calculating stress intensity factors is a critical step in fracture
mechanics. Various techniques can be
used to do this, including
1. Solve the full linear
elastic boundary value problem for the specimen or component containing a crack,
and deduce stress intensities from the asymptotic behavior of the stress field
near the crack tips;
2. Attempt to deduce stress
intensity factors directly using energy methods or path independent integrals,
to be discussed in Section 9.4;
3. Look up the solution you
need in tables;
4. Use a numerical method boundary integral equation methods are
particularly effective for crack problems, but FEM can be used too.
Analytical solutions to some crack problems
Calculating stress intensity factors for a
crack in a structure or component involves the solution of a standard linear
elastic boundary value problem. Once the
stresses have been computed, the stress intensity factor is deduced from the
definitions given in Section 9.3.1 Exact solutions are known for a few simple
geometries. A couple of examples are
2D Slit crack in an infinite solid The figure shows a 2D crack
with length 2a in an infinite solid,
which is subjected to a uniform state of stress at infinity.
The complex variable solution to this problem can be found in Section
5.3. The solution is most conveniently
expressed in terms of the polar coordinates centered at the origin, together with the
auxiliary angles and distances and shown in the figure. When evaluating the formulas, the angles and must lie in the ranges ,
respectively. The complete displacement
and stress fields in the solid are
The stress intensity factors are easily
computed to be
Penny shaped crack in an infinite solid The figure shows a circular
crack with radius a in an infinite
solid, subjected to uniaxial tension at infinity. The displacement field, in cylindrical-polar
The displacement of the upper crack face can
be found by setting in these expressions, which gives
The stress intensity factor can be found directly from the displacement
of the crack faces. The asymptotic
formulas in 9.3.1 show that
which shows that
It is not always necessary to solve the full
linear elastic boundary value problem in order to compute stress intensity
factors. Energy methods, or the
application of path independent integrals, can sometimes be used to obtain
stress intensity factors directly. These
techniques will be discussed in more detail in Section 9.4.
Vast numbers of crack problems have been
solved to catalog stress intensity factors in various geometries of
interest. Two excellent (but expensive)
sources of such solutions are Tada’s Handbook of Stress Intensity Factors, and
Murakami “Stress intensity factors handbook,” Pergamon Press,
New York (1987). A few important (and relatively simple)
results are listed below.
A short table of stress intensity factors
stress intensity factors for cracks in nonuniform stress fields
The solutions for cracks loaded by point forces acting on their faces
are particularly useful, because they allow you to calculate stress intensity
factors for a crack in an arbitrary stress field using a simple superposition
argument. The procedure works like
1. We start by computing the stress field in a solid
without a crack in it. This solution
satisfies all boundary conditions except that the crack faces are subject to
2. We could correct solution by applying pressure (and
shear) to the crack faces that are just sufficient to remove the unwanted
3. If we know the stress intensity factors induced by
point forces acting on the crack faces, we can superpose an appropriate
distribution of point forces on the crack faces calculate stress intensity
factors induced by the corrective pressure distribution.
As an example, suppose that we want to calculate stress intensity
factors for a crack in a linearly varying stress field (such as would be
induced by bending a beam, for example), as illustrated in the figure to the
1. In the uncracked solid, the stress field is
2. The traction acting along the line of the crack is The sign convention for p is that a positive p
acts downwards on the upper crack face, and upwards on the lower crack face
3. To remove the traction from the crack faces, we must
superpose an equal and opposite distribution of point forces on the crack faces. The stress intensity factor induced at the
left (L) and right (R) crack tips are
Evaluating the integrals gives
Actually, this solution is not quite right - note that the stress
intensity factor at the left crack tip is predicted to be negative. This cannot be correct from the asymptotic stress field we know that
if the stress intensity factor is negative, the crack faces must overlap behind
the crack tip (the displacement jump is negative).
With a bit of cunning, we can fix this problem. The cause of the error in the quick estimate
is that we removed tractions from the entire crack this was a mistake; we should only have
removed tractions from parts of the crack faces that open up. So let’s suppose that the crack closes at ,
and put the left hand crack tip there.
The stress intensity factors are then
for the stress intensity factor at the left hand crack tip. The stress must be bounded at where the crack faces touch, so that . This gives . The stress intensity factor at the right hand
crack tip then follows as
This is not very different to our earlier estimate. This illustrates a general feature of the
field of fracture mechanics. There are
many opportunities to do clever things, but often the results of all the
cleverness are irrelevant.
9.3.4 Calculating stress intensity
factors using finite element analysis
solids with a complicated geometry, finite element methods (or boundary element
methods) are the only way to calculate stress intensity factors. It is conceptually very straightforward to
calculate stress intensities using finite elements you just need to solve a routine linear
elastic boundary value problem to determine the stress field in the solid, and
then deduce the stress intensity factors by taking the limits given in Section
this is easier said than done. The
problem is that the stress and strain fields at a crack tip are infinite, and so standard finite element
methods have problems calculating the stresses accurately. Two special procedures have been developed to
help deal with this:
- Special crack tip
elements are available to approximate the singular strains at a crack tip;
- Special techniques are
available to calculate stress intensity factors from stresses far from the crack tip (where they
should be accurate) instead of using the formal definition.
methods can both give very accurate values for stress intensity factors and can be used together to obtain the best
Crack tip elements
A very simple procedure can
be used to approximate the strain singularity at a crack tip.
1. The solid near the crack tip must be meshed with
quadratic elements (8 noded quadrilaterals or 6 noded triangles in 2D, or 20
noded bricks/10 noded tetrahedral in 3D).
2. The elements connected to the crack tip must be
quadrilateral or brick elements
3. One side of each element connected to the crack tip is
collapsed to make the three nodes on the side coincident, as shown in the
figure to the right
4. The mid-side nodes on the elements connected to the
crack tip are shifted to point positions, as shown in the figure
5. If the coincident nodes a,b,c on each crack-tip element are constrained to move together,
this procedure generates a singularity in strain at the crack tip (good
for linear elastic problems). If the
nodes are permitted to move independently, a singularity in strain is produced (good for
problems involving crack tip plasticity)
Calculating stress intensity factors
using path independent integrals
Energy methods in fracture
mechanics are discussed in detail in Section 9.4 Two crucial results emerge from this analysis
1. The ‘energy release rate’ for a mode I crack in a
linear elastic solid with Young’s modulus E
and Poisson’s ratio is related to the mode I stress intensity
2. The energy release rate for a crack can be calculated
by evaluating the following line integral for any contour that starts on one crack face and ends on the other
where is the strain energy density, is the stress field, is the displacement field, is a unit vector normal to ,
and the basis vector is parallel to the direction of
results are ideally suited for FEM calculations. The path independent integral can be
calculated for a contour far from the crack tip, where the stresses are
accurate, and then the relationship between G
and can be used to deduce the stress intensity
factors. Analogous, but rather more
complex, procedures exist to extract all three components of stress intensity
factor, as well as to compute stress intensity factors for 3D cracks, where the
stress intensity factor is a function of position on the crack front.
9.3.5 Measuring fracture toughness
For structural applications, standard testing
techniques are available to measure material properties for fracture
applications. Two standard test specimen
geometries are shown below.Stress
intensity factors for these specimens have been carefully computed as a
function of crack length and the results fit by curves, as outlined below
Compact tension specimen:
Three point bend specimen.
Various other test specimens exist.
Conducting a fracture test or fatigue test is
(at least conceptually) straightforward you make a specimen (for fracture tests a
sharp crack is usually created by initiating a fatigue crack at the tip of a
notch); and load it in a tensile testing machine.
principle, the fracture toughness can be determined by measuring the critical
load when the crack starts to grow. In
practice it can be difficult to detect the onset of crack growth. For this reason, the usual approach is to
monitor the crack opening displacement during the test, then plot load as a function
of crack opening displacement. A typical
result is illustrated in the picture on the right.
The load-CTOD curve ceases to be linear when
the crack begins to grow. This point is hard to identify, so instead the convention is to draw a
line with slope 5% lower than the initial curve (the 5% secant line) and use the point
where this line intersects the as the fracture load. The plane strain
fracture toughness of the material, ,
is deduced from the fracture load, using the calibration for the specimen.
After measurement, one must check that is within the limits required for K dominance
in the specimen, following the rules in the preceding section.
9.3.6 Typical values for fracture toughness
A short table of toughness values (From Ashby
& Jones, `Engineering Materials’, Pergammon, 1980) is given below. The values are highly dependent on material
composition and microstructure, however, so if you need accurate data you will
need to measure the toughness of your materials yourself.
Pressure vessel steel (HY100)
High carbon steel
Aluminum and alloys
Co/WC metal matrix composites
Woods, perpendicular to grain
Concrete (steel reinforced)
Ceramics (Alumina, SiC)
Wood, parallel to grain
9.3.7 Stable Tearing Kr curves and Crack Stability
In ideally brittle materials, fracture is a
catastrophic event. Once the load
reaches the level required to trigger crack growth, the crack continues to
propagate dynamically through the specimen.
In more ductile materials, a period of stable crack growth under
steadily increasing load may occur prior to complete failure. This behavior is particularly common in
tearing of thin sheets of metals, but stable crack growth is observed in most
materials even polycrystalline ceramics.
crack growth in metals usually occurs because a zone of plastically deformed
material is left in the wake of the crack.
This deformed material tends to reduce the stresses at the crack
tip. In brittle polycrystalline
ceramics, or in fiber reinforced brittle composites, the stable crack growth is
caused by the formation of a `bridging zone’ behind the crack tip. Some fibers, or grains, remain intact in the
crack wake, and tend to hold the crack faces shut, increasing the apparent
strength of the solid.
In some materials, the increase in load
during stable crack growth is so significant that it’s worth accounting for the
effect in design calculations. The
protective effect of the process zone in the crack wake is modeled by making
the toughness of the material a function of the increase in crack length. The apparent toughness is measured in the
same way as - a pre-cracked specimen is subjected to
progressively increasing load, and the crack length is monitored either
optically or using compliance methods (more on this later). A value of can be computed for the specimen using the
calibrations during crack growth it is assumed that is equal to the fracture toughness of the
The results are plotted in a `resistance
curve’ or `R curve’ for the material.
The fracture toughness is the critical stress intensity factor
required to initiate crack growth. The
variation of toughness with crack growth is denoted .
The resistance curve is then used to predict
the conditions necessary for unstable crack growth through the material. To see how this is done
1. Consider a large sample of
material containing a slit crack of length 2a, subjected to stress . The stress intensity factor for this crack
(from the table in sect 9.3.3) is .
2. Crack growth begins when . Thereafter, there will be a period of stable
crack growth, during which the applied stress increases. During the period of stable growth the stress
intensity factor must equal the apparent toughness
3. The stress will continue to
increase as long as increases more rapidly than with . Catastrophic failure (unstable crack growth)
will occur when continued crack growth is possible at constant or decreasing
load. The crack length at the point of
unstable crack growth follows from the condition that
4. Substituting the crack
length back into the fracture criterion gives the critical stress at unstable fracture
9.3.8 Mixed Mode fracture criteria
Fracture toughness is almost always measured
under mode I loading (except when measuring fracture toughness of a bi-material
interface). If a crack is subjected to
combined mode I and mode II loading, a mixed mode fracture criterion is
required. There are several ways to
construct mixed mode fracture criteria the issue has been the subject of some quite
heated arguments. The criterion of
maximum hoop stress is one example.
Recall that the crack tip hoop and shear stresses are
The maximum hoop stress criterion postulates
that a crack under mixed mode loading starts to propagate when the greatest
value of hoop stress reaches a critical magnitude, at which point
the crack will branch at the angle for which is greatest (or equivalently the angle for
which ). The
critical angle is plotted as a function of below.
The asymptote for is -70.7 degrees. The resulting failure locus (the critical
combination of and that leads to failure) is also shown
All available criteria predict that, after
branching, a crack will follow a path such that the local mode II stress
intensity factor is zero.
9.3.9 Static fatigue crack growth
For a fatigue test, the crack length is
measured (optically, or using compliance techniques) as a function of time or
number of load cycles. Fatigue laws are deduced by plotting crack growth rate
as a function of applied stress intensity factor.
Typical static fatigue data (e.g. for corrosion
crack growth, or creep crack growth) behavior is shown on the right.
Most materials have a static fatigue threshold
a value of below which crack growth is undetectable. Then there is a range where crack growth rate
shows a power-law dependence on stress intensity factor of the form
where m is typically of order 5-10.
Finally, for values of approaching the fracture toughness, the crack
growth rate increases drastically with .
crack growth law can be used to derive the phenomenological static fatigue
criterion outlined in Section 8.2.3.
Assume that at time t=0 the material contains a crack of initial
length 2 ,
and is subjected to a uniaxial stress . The stress will cause the crack to increase
in length, until it becomes long enough to trigger brittle fracture. The table in Sect 8.3.9 below shows that a
crack of length 2a subjected to stress the stress has a crack tip stress intensity
factor . Substituting into the static fatigue crack
growth law and integrating gives the following expression for crack length as a
function of time
2 is the crack length at time t=0. The solid will fracture when the crack tip
stress intensity factor reaches the fracture toughness ,
so that the tensile strength at time t=0 and at time t must satisfy
Eliminating the crack
length and simplifying gives
that the operating stress is well below the fracture stress, we can approximate
which is the stress
based static fatigue law of Sect 9.2.3.
Cyclic fatigue crack growth
Under cyclic loading, the crack is subjected to a cycle of mode I and
mode II stress intensity factor. Most fatigue tests are performed under a
steady cycle of pure mode I loading, as sketched in the figure.
The results are usually displayed by plotting
the crack growth per cycle as a function of the stress intensity factor
A typical result shows three regions. There
is a fatigue threshold below which crack growth is undetectable. For modest loads, the crack growth rate obeys
where the index n is between 2 and 4. As the maximum stress intensity factor
approaches the fracture toughness of the material, the crack growth rate
In the Paris
law regime, the crack growth rate is only weakly sensitive to the mean value of
stress intensity factor . In the other two regimes, has a noticeable effect - the fatigue
threshold is reduced as increases, and the crack growth rate in regime
III increases with .
9.3.11 Finding cracks in structures
Determining the length of pre-existing cracks
in a component is often the most difficult part of applying fracture mechanics
in practice. For most practical
applications you simply don’t know if your component will have a crack in it,
and it will be expensive if you need to find out. Your options are:
1. Take a wild guess, based on
microscopic examinations of representative samples of material. Alternatively, you can specify the biggest
flaw you are prepared to tolerate and insist that your material suppliers
manufacture appropriately defect free materials.
2. Conduct a proof test
(popular e.g. with pressure vessel applications) wherein the structure or
component is subjected to a load greatly exceeding the anticipated service load
under controlled conditions. If the
fracture toughness of the material is known, you can then deduce the largest crack
size that could be present in the structure without causing failure during
3. Use some kind of
non-destructive test technique to attempt to detect cracks in your
structure. Examples of such techniques
are ultrasound, where you look for echoes off crack surfaces; x-ray techniques;
and inspection with optical microscopy.
If you detect a crack, most of these techniques will allow you to
estimate the crack length. If not, you
have to assume for design purposes that your structure is crammed full of
cracks that are just too short to be detected.