EN222: Mechanics of Solids

 

 

 

 

 

  

 Division of Engineering

    Brown University

 

 

6.3 Fracture Mechanics

 

The goal of fracture mechanics is to predict the critical loads that will cause catastrophic failure in a structure or component.  It is a large and venerable field with many sub-disciplines.  Some of these are illustrated schematically in the picture below

 

The ultimate goal of work in the field is to be able to design structures or components that are capable of withstanding cyclic or static service loads.  Most engineering decisions are based on semi-empirical design rules, which rely on phenomenological fracture or fatigue criteria calibrated by means of standard tests.  The design rules are based on current understanding of how materials fail, which is derived from extensive observations of failure mechanisms, together with theoretical models that have been developed to describe, as far as possible, these mechanisms of failure.

 

The mechanisms involved in fracture or fatigue failure are complex, and is influenced by material and structural features that span 12 orders of magnitude in length scale. 

 

Continuum mechanics contributes to understanding of failure mechanisms from the sub-micron to km length scales.  Most engineering applications involve structures of the order of mm-km.  For many such applications, it’s 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.  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 continuum mechanics has provided insight into a range of issues in this area.

 

We do not have time to address all these issues in this course.  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

 

 

Mechanisms of fracture and fatigue

 

In order to understand the various approaches to modeling fracture, fatigue and failure, it is helpful to review briefly the features and mechanisms of failure in solids.

 

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 ¼ 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 have such a complex microstructure (especially composites) that it’s hard to classify them as entirely brittle or entirely ductile.

 

Brittle fracture 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 a large crack 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 normally results in failure.

 

Static fatigue

 

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).

 

Failure under cyclic loading

 

Mechanical engineers generally have to design components to withstand cyclic as opposed to static loading.  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. Usually the loading is uniaxial tension, although other cycles are used too (e.g. contact fatigue applications). The cycle can be stress controlled, or strain controlled. A cycle of uniaxial load is 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.  The plot normally shows different regimes of behavior, depending on stress amplitude.  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.  At lower stress levels life has a power law dependence on stress – this is referred to as `high cycle’ fatigue behavior.  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.

 

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 has 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.

Stress and strain based failure and fatigue criteria

 

Brittle fracture criteria

 

The simplest brittle fracture criterion states that fracture is initiated when the greatest tensile principal stress in the solid reaches a critical magnitude,

(The subscript TS stands for tensile strength). 

 

To apply the criterion, you must first measure (or look up)  for the material.   can be measured by conducting tensile tests on specimens – it is important to test a large number of specimens because the failure stress is likely to show a great deal of statistical scatter.  The tensile strength can also be measured using beam bending tests.  The failure stress measured in a bending test is referred to as the `modulus of rupture’  for the material.  It is nominally equivalent to  but in practice usually turns out to be somewhat higher.

 

Then you must calculate the anticipated stress distribution in your component or structure (e.g. using FEM).  Finally, you plot contours of principal stress, and find the maximum value .  If  the design is safe (but be sure to use an appropriate factor of safety!).

 

 

 

Probabilistic Design Methods for Brittle Fracture  (Weibull Statistics)

 

The fracture criterion  is too crude for many applications.  The tensile strength of a brittle solid usually shows considerable statistical scatter, because the likelihood of failure is determined by the probability of finding a large flaw in a highly stressed region of the material.  This makes it difficult to determine an unambiguous value for tensile strength – should you use the median value of your experimental data? (no way!).  Pick the stress level where 95% of specimens survive? (Better!).   It’s better to deal with this problem using a more rigorous statistical approach.

 

Weibull statistics refers to a technique used to predict the probability of failure of a brittle material.  The approach is to test a large number of samples with identical size and shape under uniform tensile stress, and determine the survival probability as a function of stress (survival probability is approximated by the fraction of specimens that survive a given stress level).  The survival probability  is fit by a Weibull distribution

where  is the volume of the specimen, and , m are material constants.  The index m is typically of the order 5-10 for ceramics, and is independent of specimen volume.  The parameter  is the stress at which the probability of survival is exp(-1), (about 37%). This does vary with specimen volume.

 

Given m,  and the corresponding specimen volume , the  survival probability of a volume  of material subjected to uniform uniaxial stress  follows as

To see this, note that the volume V can be thought of as containing  specimens.  The probability that they all survive is .

 

The survival probability of a solid subjected to an arbitrary stress distribution with principal values  can be computed as

where

 

This approach is quite successful in some applications, for example, it explains why brittle materials appear to be stronger in bending than in uniaxial tension.  Like many statistical approaches it has some limitations as a design tool.  The problem is that we can predict quite nicely the stress that gives 30% probability of failure.  But who the hell buys a product that has a 30% probability of failure?  (Yeah, I know – Microsoft users).  For design applications we need to predict the probability of 1 failure in a million or so.  It is very difficult to measure the tail of a statistical distribution accurately, and a distribution that was fit to predict 63% failure probability may be wildly inaccurate in the region of interest.

 

Static Fatigue of Brittle Materials

 

A fracture mechanics approach (discussed in more detail below, where we address the mechanics of cracks) can be used to develop a suitable phenomenological static fatigue law.  We assume that at time t=0  the material contains a crack of initial length 2 , and is subjected to a stress .  The stress will cause the crack to increase in length, until it becomes long enough to trigger brittle fracture.  A corrosion crack grows at a rate determined by the crack tip stress intensity factor  - we’ll define this shortly, but for present purposes it is sufficient to know that for a crack of length 2a subjected to stress  the stress intensity factor is .   Experiments suggest that the crack growth rate can be approximated by

where m is of order 10-20.  We therefore obtain the following expression for crack length as a function of time

where 2  is the crack length at time t=0.   The solid will fracture when the crack tip stress intensity factor reaches a critical value , so that the tensile strength at time t=0   and at time t satisfy

Eliminating the crack length and simplifying gives

Assuming that the operating stress is well below the fracture stress, we can approximate this by

where  is a constitutive parameter, to be determined by experiment. For a component that is subjected to a constant operating stress

This expression can be used in design calculations to estimate the degradation in tensile strength (or the Weibull stress , if you prefer) with time.  The constants  and m must be determined from experiment. 

 

Note that the structure fails when , giving

for the time to failure.  Thus, m and  can be conveniently determined by measuring the time to failure of a material as a function of stress under constant loading.

 

 

 

 

Constitutive laws for crushing failure of brittle materials

 

Brittle materials are generally used in applications where they are subjected primarily to compressive stress.  Brittle materials are very strong in compression, but they will fail if subjected to combined hydrostatic compression and shear (e.g. by loading in uniaxial compression).  Failure in compression is a consequence of distributed microcracking in the solid – large numbers of small cracks form, propagate for a short while and then arrest.  Failure occurs as a result of coalescence of these cracks.   Failure in compression is less catastrophic than tension, and in some respects qualitatively resembles metal plasticity.

 

 

 

 

 

This type of crushing is usually modeled using an extended classical plasticity theory.  Its most common application is to model concrete.  A simple constitutive law of this type has the form

 

1.      Decomposition of strain into elastic and irreversible (damage) parts

 

2.      A pressure dependent failure surface of the form

 

where , , c is a material constant controlling the variation of strength with hydrostatic pressure,  is the accumulated irreversible strain, and  is a functional fit to the stress strain curve in uniaxial compression;

 

3.      An associated flow rule

 

 

These constitutive equations are used only in regions where the hydrostatic stress is compressive . 

In regions of hydrostatic tension, a tensile brittle fracture criterion should be used – for example, the material could be assumed to lose all load bearing capacity if the principal tensile stress exceeds a critical magnitude.

 

 

 

 

Ductile Fracture

 

Ductile fracture in tension occurs by the nucleation, growth and coalescence of voids in the material.  A crude criterion for ductile failure could be based on the accumulated plastic strain, for example

at failure, where  is the plastic strain to failure in a uniaxial tensile test.

 

This failure criterion does not account for the substantial reduction in strength caused by the presence of tensile hydrostatic stress.  A more sophisticated approach uses a state variable plasticity law, in which the void volume fraction is an explicit state variable. 

 

The `Gurson’ constitutive law is an example.  This is a conventional metal plasticity model, except that the yield stress is taken to be a function of the volume fraction  of voids in the material.  In the original model, the yield criterion is given by

where  is a functional fit to the uniaxial stress-strain curve of the fully dense material (  ),  and . More recent variants introduce a few more adjustable parameters in the yield function to provide a better fit to numerical simulations of voided materials. Note that the yield stress is pressure dependent (decreasing with hydrostatic tension), and the yield stress decreases as the volume fraction of voids increases, dropping to zero at .  The uniaxial stress strain curve  for a porous plastic metal would have to be determined from a compression test on the fully dense solid, because in a tension test you’d nucleate voids and consequently underestimate the yield stress.

 

The model uses an associated flow law

where  is the magnitude of the plastic strain increment and  is a constant, which must be determined from a consistency condition on the plastic dissipation

which yields a scalar equation that can be solved for C.

 

Finally, the model is completed by specifying the void volume fraction as a function of strain.  The void volume fraction can increase due to nucleation of new voids, or due to growth of existing voids.  The void volume fraction can also decrease if the voids are closed up by compressive straining.  To account for both effects, one can set

where the first term accounts for void growth, and the second accounts for strain controlled void nucleation.  Any sensible function can be used for A (it’s very difficult to determine experimentally).  You could assume the voids nucleate at a uniform rate.  A more sophisticated approach might be to assume that the void nucleation rate initially increases with plastic strain, reaches a maximum at some critical strain, and then drops off again as the void nucleation sites are exhausted.

 

 

 

Localization

 

If you test a cylindrical specimen of a very ductile material in uniaxial tension, it will initially deform uniformly. At a critical load the specimen will start to neck, as shown in the picture.  Necking, once it starts, is usually unstable – there is a concentration in stress near the necked region, increasing the rate of plastic flow near the neck compared with the rest of the specimen, and so increasing the rate of neck formation.   The strains in the necked region rapidly become very large, which will quickly lead to failure.

Neck formation is a consequence of geometric softening.  A very simple model explains the concept of geometric softening.  Consider a cylindrical specimen with cross sectional area A. Assume that the material is perfectly plastic and has a true stress-strain curve (Cauchy stress –v- logarithmic strain) that can be approximated by a power-law

with n<1.  Suppose that at some time t the specimen is subjected to a load P, and has length L, strain  and cross sectional area A.   We now increase the length of the specimen by an infinitesimal displacement dL.  This causes an increment in Logarithmic strain , increasing the Cauchy stress to .  At the same time, the cross sectional area of the bar decreases to .  (To see this note that AL=constant to preserve volume).  Consequently, the load applied to the specimen after stretching is

The first term is the result of strain hardening, and tends to increase the load.  The second term is a consequence of the lateral contraction of the bar, and tends to decrease the load.  This second term is referred to as geometric softening – the effect of the specimen’s geometry is to reduce the load required to stretch the specimen.

 

Notice that there is a critical critical strain  such that

Consequently, the load reaches a peak value at strain , and the maximum load the specimen can withstand follows as

where  is the initial cross sectional area of the bar.

 

It turns out that the point of maximum load coincides with the condition for unstable neck formation in the bar.  This is plausible – a falling load displacement curve is always a sign that there might be a possibility of non-unique solutions – but a rather sophisticated calculation is required to show this rigorously.

 

There are two important points to take away from this discussion.

   Plastic localization, as opposed to material failure, may limit load bearing capacity

   If you measure the strain to failure of a material in uniaxial tension, it is possible that you have learned absolutely nothing about the inherent strength of the material – your specimen may have failed due to a geometric effect;

 

 

Plastic localization can occur for many reasons.  There are two general classes of localization – it may occur as a consequence of changes in specimen geometry (i.e. geometric softening); or it may occur due to a natural tendency of the material itself to soften at large strains.

 

Examples of geometry induced localization are

(i)                  Neck formation in a bar under uniaxial tension;

(ii)                Shear band formation in torsional or shear loading at high strain rate due to thermal softening as a result of plastic heat generation

 

Examples of material induced localization are

(i)                  Localization in a Gurson solid due to the softening effect of voids at large strains;

(ii)                Localization in a single crystal due to the softening effect of lattice rotations;

(iii)               Localization in a brittle microcracking material due to the reduction in elastic compliance caused by the cracks.

 

From a modeling standpoint, localization is the easiest form of failure to deal with, because it does not rely on any empirical failure criteria.  A straightforward FEM computation, with an appropriate constitutive law and proper consideration of finite strains, will predict localization if it is going to occur – the only thing you need to worry about is to be sure you understand what triggered the localization.  Localization can start at a geometric imperfection in the model, in which case your prediction is meaningful (but may be sensitive to the nature of the imperfection).  It may also be triggered by numerical errors, in which case the predicted failure load is meaningless.  It is usually exceedingly difficult to compute what happens after localization.  Fortunately it’s rather rare to need to do this for design purposes.

 

 

 

High Cycle Fatigue

 

 

Empirical stress or strain based life prediction methods are extensively used in design applications.  The approach is straightforward – subject a sample of the material to a cycle of stress (or strain) that resembles service loading, in an environment representative of service conditions, and measure its life as a function of stress (or strain) amplitude.

 

Here we will review criteria that are used to predict fatigue life under proportional cyclic loading. A typical stress cycle is parameterized by its amplitude  and the mean stress

 

For tests run in the high cycle fatigue regime with any fixed value of mean stress, the relationship between stress amplitude  and the number of cycles to failure N is fit well by Basquin’s Law

where the exponent b is typically between 0.05 and 0.15.  The constant C is a function of mean stress.

 

There are two ways to account for the effects of mean stress.  Both are based on the same idea: we know that if the mean stress is equal to the tensile strength of the material, it will fail in 0 cycles of loading.  We also know that for zero mean stress, the fatigue life obeys Basquin’s law.  We can interpolate between these two points. 

 

Goodman’s rule uses a linear interpolation, giving

where  is the constant in Basquin’s law determined by testing at zero mean stress.

 

Gerber’s rule uses a parabolic fit

In practice, experimental data seem to lie between these two limits.  Goodman’s rule gives a safe estimate.

 

 

 

Low Cycle Fatigue

 

If a fatigue test is run with a high stress level (sufficient to cause plastic flow) the specimen fails very quickly (less than 10 000 cycles).  This regime of behavior is known as `low cycle fatigue’.  The fatigue life correlates best with the plastic strain amplitude rather than stress amplitude, and it is found that the Coffin Manson Law

gives a good fit to empirical data (the constant C and b do not have the same values as for Basquin’s law, of course)

 

 

Cumulative Damage

 

Fatigue tests are usually done at constant stress (or strain) amplitude.  Service loading usually involves cycles with variable (and often random) amplitude.  Fortunately, there’s a remarkably simple way to estimate fatigue life under variable loading using constant stress data.

Suppose the load history is comprised of a set of  load cycles at a stress amplitude , followed by a set of  cycles at load amplitude  and so on.  For the ith set of cycles at load amplitude , we could compute the number of cycles that would cause the specimen to fail using Basquin’s law

 

The Miner-Palmgren failure criterion assumes a linear summation of damage due to each set of load cycles, so that

at failure. In terms of stress amplitude

The same approach works under low cycle fatigue conditions, in which case

 

 

The criterion is often used under random loading. To do so, we need to find a way to estimate the number of cycles of load at a given stress level.  There are various ways to do this – one approach is to count the peaks in the load history, and compute the probability  of finding a peak at stress level . (Of course, this only works if the signal’s autocorrelation function is differentiable at the origin, which is not the case for white noise, for example).

 

Miner’s rule then predicts that the number of cycles to failure satisfies

 

 

 

Mechanics of Cracks

 

Phenomenological damage models are useful in design applications, but they have many limitations, including

 They require extensive experimental testing to calibrate the model for each application;

 They provide no insight into the relationship between a materials microstructure and its strength;

 

A 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 homogeneous Elastic Solids.

 

Many 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 look the same (differing only in magnitude).  The fields near a crack tip are a fundamental result in fracture mechanics.

 

 

The state of stress can be calculated by considering a semi-infinite crack in an infinite elastic solid subjected to uniform loading at infinity. 

 

The remote loading can be decomposed into an anti-plane shear state combined with a plane strain state.  For both cases, we seek a solution with bounded stress at infinity;  a displacement jump across the crack faces; and satisfying traction boundary conditions  on .  The solution will have a stress singularity – we saw in Section 4 that singular solutions be unique if the strain energy is unbounded.   To ensure that the solution is unique, we accept only solutions with a bounded strain energy.

 

For anti-plane shear loading, the displacement can be computed from a complex potential (following the procedure outlined in Section 4) as

where C is a constant and .  The stress state follows as

Evidently this solution satisfies all boundary conditions for all values of C.  As expected, the stress distribution is known, but its magnitude is arbitrary.  It is conventional to re-scale the stress and displacement fields by defining the mode III stress intensity factor

giving

for the stress and displacement fields.

 

The equivalent plane problem is most conveniently derived from an Airy function.   The derivation is outlined in detail in the linear elasticity notes.  It is found that the Airy function

(with  arbitrary constants) generates a solution satisfying traction free boundary conditions on the crack faces; with vanishing stress at infinity and bounded strain energy.   The stress state is singular at the crack tip, just like the equivalent anti-plane shear solution.   The unknown stress magnitude is scaled by introducing Mode I and Mode II stress intensity factors

The stresses follow from the Airy function as

Equivalent expressions in rectangular coordinates are

while the displacements can be calculated by integrating the strains, with the result

Note that this displacement field is valid for plane strain deformation only.

 

Observe that the stress intensity factor has the bizarre units of .

 

 

The assumptions 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 fatigue applications or dynamic fracture, the rate and direction of crack growth are also of interest.

 

The phenomenological theory is based on the following loose argument.  Consider a crack in a reasonably brittle, isotropic solid.  If the solid were ideally elastic, we would 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 would not be accurate.  In practice, the asymptotic field will also not give an accurate representation of the stress fields very close to the crack tip either. The crack may not be perfectly sharp at its tip, and if it were, no solid could withstand the infinite stress predicted by our 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 damage. 

2.      A bit further from the crack tip, there will be a region where the linear elastic field 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 for phenomenological purposes 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 its 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 will grow.

 

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 solid.

 

Empirically, 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 standards 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 tip

2.      For plane strain conditions at the crack tip the specimen thickness must exceed at least the plastic zone size.

 

For a material with yield stress Y  loaded in Mode I with stress intensity factor  the plastic zone size can be estimated as

 

 

 

 

Practical application of linear elastic fracture mechanics

 

To apply LEFM in a design application, you need to be able to do three things. 

1.      Measure the critical stress intensity factors that cause fracture in your material, or measure fatigue crack growth rates as a function of static or cyclic stress intensity

2.      Estimate the anticipated size and location of cracks in your structure or component

3.      Calculate the stress intensity factors for the cracks in your structure or component under anticipated loading conditions.

 

A. Fracture toughness measurements

 

For structural applications, standard testing techniques are available to measure material properties for fracture applications.  Two standard test specimen geometries are shown below

 

(a) Compact tension specimen                                            (b) 3 point bend specimen

 

Stress intensity factors for these specimens have been carefully computed as a function of crack length and the results fit by curves

 

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.

 

 For a fracture test, you measure the critical load when the crack starts to grow.  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 below.

 

 

 

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.

 

A few typical values of fracture toughness are tabulated below.

 

 

 

 

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.

 

 

Stable 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 phenomenologically, 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 material.

 

 

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 stress intensity factor 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, consider a large sample of material containing a slit crack of length 2a, subjected to stress .  The stress intensity factor (from the table) is .  Crack growth begins when .   Thereafter, there will be a period of stable crack growth, during which the applied stress increases.  The stress satisfies

The stress will continue to increase as long as the increase in toughness with crack length is sufficient to overcome the increase in stress intensity factor with crack length.  Catastrophic failure (unstable crack growth ) will occur when continued crack growth is possible at constant or decreasing load.  This requires

For the case of a slit crack, this gives