Chapter 9
Modeling
Material Failure
One
of the most important applications of solid mechanics is to design
structures, components or materials that are capable of withstanding cyclic
or static service loads. To do this,
you need to be able to predict the conditions necessary to cause
failure. Materials and structures can
fail in many different ways, including by buckling, excessive plastic flow,
fatigue and fracture, wear, or corrosion.
Calculating the stresses in a structure or component can help to
design against these failures, but is usually not enough it is also necessary to understand and to be
able to predict the effects of
stress.
Fracture mechanics is a sub-discipline of solid mechanics. The goal of the field is to predict the
critical loads that will cause catastrophic failure in a material or
component. Much of fracture mechanics is based on phenomenological fracture
or fatigue criteria, which are calibrated by means of standard tests. The failure criteria are based on current
understanding of how materials fail, which is derived from extensive
observations of failure mechanisms, together with theoretical models that
describe, as far as possible, these mechanisms of failure.
The
mechanisms involved in fracture or fatigue failure are complex, and are
influenced by material and structural features that span 12 orders of
magnitude in length scale, as illustrated in the picture below
Most
engineering applications involve structures of the order of mm-km. For many such applications, it is
sufficient to measure the maximum cyclic or static stress (or perhaps strain)
that the material can withstand, and then design the structure to ensure that
the stress (or strain) remains below acceptable limits. This involves fairly routine constitutive
modeling and numerical or analytical solution of appropriate boundary value
problems.
More
critical applications require some kind of defect tolerance analysis perhaps the material or structure is known
to contain flaws, and the engineer must decide whether to replace the part;
or perhaps it is necessary to specify material quality standards. This kind of decision is usually made using
either linear elastic, or plastic, fracture mechanics, which model flaws
themselves in detail at length scales of cm or below.
Finally,
there is great interest in designing failure resistant materials. In this case the basic question is: how
does the material fail, and what can be done to the material’s microstructure
to avoid failure? This is a more
exploratory field, but solid mechanics has provided insight into a range of
issues in this area.
A
complete discussion of these issues is beyond the scope of this book. Instead, we will summarize some results in
the continuum mechanics of solids that are central to analysis of fracture
and fatigue, and outline briefly their main applications. Specifically, we will give
- A brief review of the mechanisms of failure and
fatigue;
- An overview of phenomenological stress or
strain based failure criteria, primarily used in design applications;
- A brief discussion of the mechanics of cracks
in solids.
9.1 Summary of
mechanisms of fracture and fatigue under static and cyclic loading
Before
discussing the various approaches to modeling fracture, fatigue and failure,
it is helpful to review briefly the features and mechanisms of failure in
solids.
9.1.1 Failure under monotonic loading
If you test a sample of any
material under uniaxial tension it will eventually fail. The features of the failure depend on
several factors, including
The materials involved and their
microsctructure;
The applied stress state (particularly the
hydrostatic stress)
Loading rate
Temperature
Ambient environment (water vapor; or
presence of corrosive environments).
Materials
are normally classified loosely as either `brittle’ or `ductile’ depending on the
characteristic features of the failure.
Examples
of `brittle’ materials include refractory oxides (ceramics) and
intermetallics, as well as BCC metals at low temperature (below about of the melting point). Features of a brittle material are
- Very little plastic
flow occurs in the specimen prior to failure;
- The two sides of the
fracture surface fit together very well after failure.
- The fracture surface
appears faceted you can make out individual grains and
atomic planes.
- In many materials, fracture occurs
along certain crystallographic planes.
In other materials, fracture occurs along grain boundaries
Examples
of `ductile’ materials include FCC metals at all temperatures; BCC metals at
high temperatures; polymers at high temperature. Features of a `ductile’ fracture are
- Extensive plastic flow occurs in the material
prior to fracture
- There is usually evidence of considerable
necking in the specimen
- Fracture surfaces don’t fit together.
- The fracture surface
has a dimpled appearance you can see little holes, often with
second phase particles inside them.
Of
course, some materials (especially composites) have such a complex
microstructure that it’s hard to classify them as entirely brittle or
entirely ductile.
Brittle
fracture often occurs as a result of a single crack propagating through the
specimen. Some materials contain
pre-existing cracks, in which case fracture is initiated when one of these
cracks in a region of high tensile stress starts to grow. In other materials, the origin of the
fracture is less clear various mechanisms for nucleating crack have
been suggested, including dislocation pile-up at grain boundaries; or
intersections of dislocations.
Ductile
fracture occurs as a result of the nucleation, growth and coalescence of
voids in the material. Failure is
controlled by the rate of nucleation of the voids; their rate of growth, and
the mechanism of coalescence. High
tensile hydrostatic stress promotes rapid void nucleation and growth, but
void growth generally also requires significant bulk plastic strain.
A
ductile material may also fail as a result of plastic instability such as necking, or the formation of a shear
band. This is analogous to buckling at a critical strain, the component no
longer deforms uniformly, and the deformation localizes to a small
region of the solid. This is normally
accompanied by a loss of load bearing capacity and a large increase in
plastic strain rate in the localized region, which eventually causes failure.
Finally,
some materials, especially brittle materials such as glasses, and oxide based
ceramics, suffer from a form of time-delayed failure under steady loading,
known as `static fatigue’. Automatic coffee-maker jugs are
particularly susceptible to static fatigue.
You use one for a couple of years, and then one day it shatters if you
tap it against the side of the sink.
This is because the jug’s strength has degraded with time. Static fatigue in brittle materials is a
consequence of corrosion crack growth.
The highly stressed material near a crack tip is particularly susceptible
to chemical attack (the stress increases the rate of chemical reaction). Material near the crack tip may be
dissolved altogether, or it may form a reaction product with very low
strength. In either event, the crack
slowly propagates through the solid, until it becomes long enough to trigger
brittle fracture. Glasses and oxide based ceramics are particularly
susceptible to attack by water-vapor (and perhaps coffee).
9.1.2 Failure under cyclic loading
Mechanical engineers generally have to
design components to withstand cyclic as opposed to static loading. Under cyclic loading, materials fail by
fatigue. Fatigue failure is a familiar
phenomenon, but a detailed understanding of the mechanisms involved and the
ability to model them quantitatively have only emerged in the past 50 years,
driven largely by the demands of the aerospace industry. There are some forms of fatigue failure
(contact fatigue is an example) where the mechanisms involved are still a
mystery.
Fatigue
life is measured by subjecting the material to cyclic loading. The loading is
usually uniaxial tension, but other cycles such as torsion or bending can be
used as well. The cycle can be stress controlled (subjecting the material to
a prescribed stress), or strain controlled. A cyclic uniaxial stress is usually
characterized by
The stress amplitude
The mean
stress
The stress
ratio
A
rotating bending test is a particularly convenient way to subject a material
to a very large number of cycles in a short period of time. The shaft can easily be spun at 2000rpm,
allowing the material to be subjected to cycles in less than 100 hrs. Pulsating tension is more common in service
loading, but a servo-hydraulic tensile testing machine operating at 1Hz takes
nearly 4 months to complete cycles.
The
resistance of a material to cyclic loading is characterized by plotting an
`S-N’ curve showing the number of cycles to failure as a function of stress
amplitude. The characteristic features of an S-N curve are illustrated in the
sketch on the right.
1. The plot normally shows two different regimes of
behavior, depending on stress amplitude.
2. At high stress levels, the material deforms
plastically and fails rapidly. In this
regime the life of the specimen depends primarily on the plastic strain
amplitude, rather than the stress amplitude.
This is referred to as `low cycle fatigue’ behavior.
3. At lower stress levels life has a power law
dependence on stress this is referred to as `high cycle’ fatigue
behavior.
4. In some materials, there is a clear fatigue limit
if the stress amplitude lies below a certain
limit, the specimen remains intact forever.
In other materials there is no clear fatigue threshold. In this case, the stress amplitude at which
the material survives cycles is taken as the endurance limit
of the material. (The term `endurance’ appears to refer to the engineer doing
the testing, rather than the material)
Fatigue
life is sensitive to the mean stress, or R ratio, and tends to fall
rather rapidly as R increases.
It is also influenced by environment, and temperature, and can be very
sensitive to details such as the surface finish of the specimen.
A
tensile specimen that has failed by fatigue looks at first sight as though it
might have failed by brittle fracture.
The fracture surface is flat, and the two sides of the specimen fit
together quite well. In fact, for some
time it was thought that some bizarre metallurgical process was responsible
for turning a ductile material brittle under cyclic loading. (An engineer
named Nevil Norway wrote a successful novel based on this theory. The novel is entitled No Highway, published
under the pseudonym Nevil Shute). A
closer examination reveals several differences, however. You usually don’t see cleavage planes on a
fatigue surface, and instead often observe a set of nearly parallel ridges on
the surface, spaced at distances between a few 100 angstroms to a few tenths
of microns apart. These ridges are
known as `striations’ and are marks left behind by the tip of a fatigue crack
at each cycle of load. In many materials,
there is evidence for local areas of cleavage fracture or void coalescence
interspersed with the striations.
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Fatigue
failures are caused by slow crack growth through the material. The failure process involves four stages
1. Crack initiation;
2. Micro-crack growth (with crack length less than the
materials grain size) (Stage I);
3. Macro crack growth (crack length between 0.1mm and
10mm) (Stage II);
4. Failure by fast fracture.
Cracks
will generally only initiate in the presence of cyclic plasticity. However, bulk plastic flow in the
specimen is not necessary: plastic flow may be caused by local stress
concentrations at notches in the part, due to geometric defects such as dents
or scratches in the surface, or even due to microstructural features such as
large inclusions in the material. In a smooth, clean specimen, the cracks
form where `persistent slip bands’ reach the surface of the specimen. Plastic flow in a material is generally
highly inhomogeneous at the micron scale, with the deformation confined to
narrow localized bands of slip. Where these bands intersect the surface,
intrusions or extrusions form, which serve as nucleation sites for
cracks.
Cracks
initially propagate along the slip bands at around 45 degrees to the
principal stress direction this is known as Stage I fatigue crack growth. When the cracks reach a length comparable
to the materials grain size, they change direction and propagate
perpendicular to the principal stress.
This is known as Stage II fatigue crack growth.
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