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

1. A brief review of the mechanisms of failure and fatigue;
2. An overview of phenomenological stress or strain based failure criteria, primarily used in design applications;
3. 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.

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)

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

1. Very little plastic flow occurs in the specimen prior to failure;
2. The two sides of the fracture surface fit together very well after failure.
3. The fracture surface appears faceted  you can make out individual grains and atomic planes.
4.  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

1. Extensive plastic flow occurs in the material prior to fracture
2. There is usually evidence of considerable necking in the specimen
3. Fracture surfaces don’t fit together.
4. 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).

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.

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.

(c) A.F. Bower, 2008
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