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
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
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