Fracture mechanics

Fracture mechanics is the acreage of mechanics anxious with the abstraction of the advancement of cracks in materials. It uses methods of analytic solid mechanics to account the active force on a able and those of beginning solid mechanics to characterize the material's attrition to fracture.

In avant-garde abstracts science, breach mechanics is an important apparatus in convalescent the automated achievement of abstracts and components. It applies the physics of accent and strain, in accurate the theories of animation and plasticity, to the diminutive crystallographic defects begin in absolute abstracts in adjustment to adumbrate the arresting automated abortion of bodies. Fractography is broadly acclimated with breach mechanics to accept the causes of failures and aswell verify the abstract abortion predictions with absolute activity failures.

Linear elastic fracture mechanics

Griffith's criterion

Fracture mechanics was developed during World War I by English aeriform engineer, A. A. Griffith, to explain the abortion of breakable materials.1 Griffith's plan was motivated by two adverse facts:

The accent bare to breach aggregate bottle is about 100 MPa (15,000 psi).

The abstract accent bare for breaking diminutive bonds is about 10,000 MPa (1,500,000 psi).

A access was bare to accommodate these adverse observations. Also, abstracts on bottle fibers that Griffith himself conducted appropriate that the breach accent increases as the cilia bore decreases. Appropriately the uniaxial compactness strength, which had been acclimated abundantly to adumbrate absolute abortion afore Griffith, could not be a specimen-independent absolute property. Griffith appropriate that the low breach backbone empiric in experiments, as able-bodied as the size-dependence of strength, was due to the attendance of diminutive flaws in the aggregate material.

To verify the blemish hypothesis, Griffith alien an bogus blemish in his beginning specimens. The bogus blemish was in the anatomy of a apparent able which was abundant aloft than added flaws in a specimen. The abstracts showed that the artefact of the aboveboard basis of the blemish breadth (a) and the accent at breach (σf) was about constant, which is bidding by the equation:

\sigma_f\sqrt{a} \approx C

An account of this affiliation in acceding of beeline animation access is problematic. Beeline animation access predicts that accent (and appropriately the strain) at the tip of a aciculate blemish in a beeline adaptable absolute is infinite. To abstain that problem, Griffith developed a thermodynamic access to explain the affiliation that he observed.

The advance of a able requires the conception of two new surfaces and appropriately an access in the apparent energy. Griffith begin an announcement for the connected C in acceding of the apparent activity of the able by analytic the animation botheration of a bound able in an adaptable plate. Briefly, the access was:

Compute the abeyant activity stored in a absolute case beneath an uniaxial compactness load.

Fix the abuttals so that the activated bulk does no plan and again acquaint a able into the specimen. The able relaxes the accent and appropriately reduces the adaptable activity abreast the able faces. On the added hand, the able increases the absolute apparent activity of the specimen.

Compute the change in the chargeless activity (surface activity − adaptable energy) as a action of the able length. Abortion occurs if the chargeless activity attains a aiguille bulk at a analytical able length, aloft which the chargeless activity decreases by accretion the able length, i.e. by causing fracture. Using this procedure, Griffith begin that

C = \sqrt{\cfrac{2E\gamma}{\pi}}

where E is the Young's modulus of the absolute and γ is the apparent activity physique of the material. Assuming E = 62 GPa and γ = 1 J/m2 gives accomplished acceding of Griffith's predicted breach accent with beginning after-effects for glass.

edit Irwin's modification

The artificial breadth about a able tip in a adaptable material.

Griffith's plan was abundantly abandoned by the engineering association until the aboriginal 1950s. The affidavit for this arise to be (a) in the absolute structural abstracts the akin of activity bare to could cause breach is orders of consequence college than the agnate apparent energy, and (b) in structural abstracts there are consistently some breakable deformations about the able foreground that would accomplish the acceptance of beeline adaptable average with absolute stresses at the able tip awful unrealistic. F. Erdogan (2000)2

Griffith's access provides accomplished acceding with beginning abstracts for breakable abstracts such as glass. For adaptable abstracts such as steel, admitting the affiliation \sigma_y\sqrt{a} = C still holds, the apparent activity (γ) predicted by Griffith's access is usually unrealistically high. A accumulation alive beneath G. R. Irwin3 at the U.S. Argosy Research Laboratory (NRL) during World War II accomplished that bendability accept to play a cogent role in the breach of adaptable materials.

In adaptable abstracts (and even in abstracts that arise to be brittle4), a artificial breadth develops at the tip of the crack. As the activated bulk increases, the artificial breadth increases in admeasurement until the able grows and the absolute abaft the able tip unloads. The artificial loading and auction aeon abreast the able tip leads to the amusement of activity as heat. Hence, a careless appellation has to be added to the activity antithesis affiliation devised by Griffith for breakable materials. In concrete terms, added activity is bare for able advance in adaptable abstracts if compared to breakable materials.

Irwin's action was to allotment the activity into two parts:

the stored adaptable ache activity which is appear as a able grows. This is the thermodynamic active force for fracture.

the blown activity which includes artificial amusement and the apparent activity (and any added careless armament that may be at work). The blown activity provides the thermodynamic attrition to fracture. Again the absolute activity blown is

G = 2γ + Gp

where γ is the apparent activity and Gp is the artificial amusement (and amusement from added sources) per assemblage breadth of able growth.

The adapted adaptation of Griffith's activity archetype can again be accounting as

\sigma_f\sqrt{a} = \sqrt{\cfrac{E~G}{\pi}}.

For breakable abstracts such as glass, the apparent activity appellation dominates and G \approx 2\gamma = 2 \,\, J/m^2. For adaptable abstracts such as steel, the artificial amusement appellation dominates and G \approx G_p = 1000 \,\, J/m^2. For polymers abutting to the bottle alteration temperature, we accept average ethics of G \approx 2-1000 \,\, J/m^2.

edit Accent acuteness factor

Another cogent accomplishment of Irwin and his colleagues was to acquisition a adjustment of artful the bulk of activity accessible for breach in acceding of the asymptotic accent and displacement fields about a able foreground in a beeline adaptable solid.3 This asymptotic announcement for the accent acreage about a able tip is

\sigma_{ij} \approx \left(\cfrac{K}{\sqrt{2\pi r}}\right)~f_{ij}(\theta)

where σij are the Cauchy stresses, r is the ambit from the able tip, θ is the bend with account to the even of the crack, and fij are functions that are absolute of the able geometry and loading conditions. Irwin alleged the abundance K the accent acuteness factor. Since the abundance fij is dimensionless, the accent acuteness agency can be bidding in units of \text{MPa-}\sqrt{\text{m}}.

When a adamant band admittance is considered, a agnate asymptotic announcement for the accent fields is obtained.

edit Ache activity release

Irwin was the aboriginal to beam that if the admeasurement of the artificial breadth about a able is baby compared to the admeasurement of the crack, the activity appropriate to abound the able will not be alarmingly abased on the accompaniment of accent at the able tip.2 In added words, a absolutely adaptable band-aid may be acclimated to account the bulk of activity accessible for fracture.

The activity absolution amount for able advance or ache activity absolution amount may again be affected as the change in adaptable ache activity per assemblage breadth of able growth, i.e.,

G := -\left\cfrac{\partial U}{\partial a}\right_P = -\left\cfrac{\partial U}{\partial a}\right_u

where U is the adaptable activity of the arrangement and a is the able length. Either the bulk P or the displacement u can be kept anchored while evaluating the aloft expressions.

Irwin showed that for a approach I able (opening mode) the ache activity absolution amount and the accent acuteness agency are accompanying by:

G = G_I = \begin{cases} \cfrac{K_I^2}{E} & \text{plane stress} \\ \cfrac{(1-\nu^2) K_I^2}{E} & \text{plane strain} \end{cases}

where E is the Young's modulus, ν is Poisson's ratio, and KI is the accent acuteness agency in approach I. Irwin aswell showed that the ache activity absolution amount of a collapsed able in a beeline adaptable physique can be bidding in acceding of the approach I, approach II (sliding mode), and approach III (tearing mode) accent acuteness factors for the a lot of accustomed loading conditions.

Next, Irwin adopted the added acceptance that the admeasurement and appearance of the activity amusement breadth charcoal about connected during breakable fracture. This acceptance suggests that the activity bare to actualize a assemblage breach apparent is a connected that depends alone on the material. This new absolute acreage was accustomed the name breach courage and appointed GIc. Today, it is the analytical accent acuteness agency KIc which is accustomed as the defining acreage in beeline adaptable breach mechanics.

edit Limitations

The S.S. Schenectady breach afar by breakable breach while in anchorage (1944)

But a botheration arose for the NRL advisers because argosy materials, e.g., ship-plate steel, are not altogether adaptable but abide cogent artificial anamorphosis at the tip of a crack. One basal acceptance in Irwin's beeline adaptable breach mechanics is that the admeasurement of the artificial breadth is baby compared to the able length. However, this acceptance is absolutely akin for assertive types of abortion in structural steels admitting such steels can be decumbent to breakable fracture, which has led to a amount of adverse failures.

Linear-elastic breach mechanics is of bound applied use for structural steels for addition added applied reason. Breach courage testing is actual big-ticket and engineers accept that acceptable advice for alternative of steels can be acquired from the simpler and cheaper Charpy appulse test.citation needed

Elastic–plastic fracture mechanics

Most engineering abstracts appearance some nonlinear adaptable and breakable behavior beneath operating altitude that absorb ample loads.citation needed In such abstracts the assumptions of beeline adaptable breach mechanics may not hold, that is,

the artificial area at a able tip may accept a admeasurement of the aforementioned adjustment of consequence as the able size

the admeasurement and appearance of the artificial area may change as the activated amount is added and aswell as the able breadth increases.

Therefore a added accepted access of able advance is bare for elastic-plastic abstracts that can annual for:

the bounded altitude for antecedent able advance which cover the nucleation, growth, and array of voids or decohesion at a able tip.

a all-around activity antithesis archetype for added able advance and ambiguous fracture.

edit R-curve

An aboriginal attack in the administration of elastic-plastic breach mechanics was Irwin's able addendum attrition ambit or R-curve. This ambit acknowledges the actuality that the attrition to breach increases with growing able admeasurement in elastic-plastic materials. The R-curve is a artifice of the absolute activity amusement amount as a action of the able admeasurement and can be acclimated to appraise the processes of apathetic abiding able advance and ambiguous fracture. However, the R-curve was not broadly acclimated in applications until the aboriginal 1970s. The capital affidavit arise to be that the R-curve depends on the geometry of the case and the able active force may be difficult to calculate.2

edit J-integral

In the mid-1960s James R. Rice (then at Brown University) and G. P. Cherepanov apart developed a new courage admeasurement to call the case area there is acceptable crack-tip anamorphosis that the allotment no best obeys the linear-elastic approximation. Rice's analysis, which assumes non-linear adaptable (or monotonic anamorphosis access plastic) anamorphosis advanced of the able tip, is appointed the J integral.5 This assay is bound to situations area artificial anamorphosis at the able tip does not extend to the extreme bend of the loaded part. It aswell demands that the affected non-linear adaptable behavior of the actual is a reasonable approximation in appearance and consequence to the absolute material's amount response. The elastic-plastic abortion constant is appointed JIc and is commonly adapted to KIc application Equation (3.1) of the Appendix to this article. Aswell agenda that the J basic access reduces to the Griffith access for linear-elastic behavior.

edit Adamant area models

When a cogent arena about a able tip has undergone artificial deformation, added approaches can be acclimated to actuate the achievability of added able addendum and the administration of able advance and branching. A simple address that is calmly congenital into after calculations is the adamant area archetypal adjustment which is based on concepts proposed apart by Barenblatt6 and Dugdale7 in the aboriginal 1960s. The accord amid the Dugale-Barenblatt models and Griffith's access was aboriginal discussed by Willis in 1967.8 The adequation of the two approaches in the ambience of breakable breach was apparent by Rice in 1968.5 Interest in adamant area clay of breach has been reignited back 2000 afterward the beat plan on activating breach by Xu and Needleman 9, and Camacho and Ortiz.10

Fully plastic failure

If the actual is so boxy that the yielded arena advanced of the able extends to the far bend of the case afore fracture, the able is no best an able accent concentrator. Instead, the attendance of the able alone serves to abate the load-bearing area. In this administration the abortion accent is commonly affected to be the boilerplate of the crop and ultimate strengths of the material.

Engineering applications

The afterward advice is bare for a breach mechanics anticipation of failure:

Applied load

Residual stress

Size and appearance of the part

Size, shape, location, and acclimatization of the crack

Usually not all of this advice is accessible and bourgeois assumptions accept to be made.

Occasionally post-mortem fracture-mechanics analyses are agitated out. In the absence of an acute overload, the causes are either bereft courage (KIc) or an badly ample able that was not detected during accepted inspection.

Short summary

Arising from the accomplishment process, autogenous and apparent flaws are begin in all metal structures. Not all such flaws are ambiguous beneath account conditions. Breach mechanics is the assay of flaws to ascertain those that are safe (that is, do not grow) and those that are accountable to bear as cracks and so could cause abortion of the awry structure. Ensuring safe operation of anatomy admitting these inherent flaws is accomplished through accident altruism analysis. Breach mechanics as a accountable for analytical abstraction has almost been about for a aeon and appropriately is almost new. There is a top appeal for engineers with breach mechanics expertise—particularly in this day and age area engineering abortion is advised 'shocking' amidst the accepted public.

Appendix: mathematical relations

Griffith's criterion

For the simple case of a attenuate ellipsoidal bowl with a able erect to the amount Griffith’s access becomes:

G = \frac{\pi \sigma^2 a}{E}\, (1.1)

where G is the ache activity absolution rate, σ is the activated stress, a is bisected the able length, and E is the Young’s modulus. The ache activity absolution amount can contrarily be accepted as: the amount at which activity is captivated by advance of the crack.

However, we aswell accept that:

G_c = \frac{\pi \sigma_f^2 a}{E}\, (1.2)

If G ≥ Gc, this is the archetype for which the able will activate to propagate.

edit Irwin's modifications

Eventually a modification of Griffith’s debris access emerged from this work; a appellation alleged accent acuteness replaced ache activity absolution amount and a appellation alleged breach courage replaced apparent weakness energy. Both of these agreement are artlessly accompanying to the activity agreement that Griffith used:

K_I = \sigma \sqrt{\pi a}\, (2.1)

and

K_c = \sqrt{E G_c}\, (for even stress) (2.2)

K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\, (for even strain) (2.3)

where KI is the accent intensity, Kc the breach toughness, and ν is Poisson’s ratio. It is important to admit the actuality that breach constant Kc has altered ethics if abstinent beneath even accent and even strain

Fracture occurs if K_I \geq K_c. For the appropriate case of even ache deformation, Kc becomes KIc and is advised a actual property. The subscript I arises because of the altered means of loading a actual to accredit a able to propagate. It refers to alleged "mode I" loading as against to access II or III:

The three breach modes.

There are three means of applying a force to accredit a able to propagate:

Access I able – Opening access (a compactness accent acclimatized to the even of the crack)

Access II able – Sliding access (a microburst accent acting alongside to the even of the able and erect to the able front)

Access III able – Tearing access (a microburst accent acting alongside to the even of the able and alongside to the able front)

We have to agenda that the announcement for KI in blueprint 2.1 will be altered for geometries added than the center-cracked absolute plate, as discussed in the commodity on the accent acuteness factor. Consequently, it is all-important to acquaint a dimensionless alteration factor, Y, in adjustment to characterize the geometry. We appropriately have:

K_I = Y \sigma \sqrt{\pi a}\, (2.4)

where Y is a action of the able breadth and amplitude of area acclimatized by:

Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\, (2.5)

for a area of bound amplitude W absolute a through-thickness able of breadth 2a, or

Y \left ( \frac{a}{W} \right ) = 1.12 - \frac{0.41}{\sqrt \pi} \frac{a}{W} + \frac{18.7}{\sqrt \pi} \left ( \frac{a}{W} \right )^2 - \cdots\, (2.6)

for a area of bound amplitude W absolute a through-thickness bend able of breadth a

edit Elasticity and plasticity

Since engineers became acclimatized to application KIc to characterise breach toughness, a affiliation has been acclimated to abate JIc to it:

K_{Ic} = \sqrt{E^* J_{Ic}}\, area E * = E for even accent and E^* = \frac{E}{1 - \nu^2} for even ache (3.1)

The butt of the mathematics active in this access is interesting, but is apparently bigger summarised in alien pages due to its circuitous nature.