Fracture, Macroscopic&Microscopic Aspects

8/2/2019 Fracture, Macroscopic&Microscopic Aspects

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M A C R O S C O P I C & M I C R O S C O P I CA S P E C T S

FRACTURE

Karla Ixchel Garca Garca Enero 2012

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C E N T R O D E I N V E S T I G A C I N E N M A T E R I A L E SA V A N Z A D O S


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Fracture of a material by cracking can occur in many ways:

1. Slow application of external loads.

2. Rapid application of external loads (impact).

3. Cyclic or repeated loading (fatigue).

4. Environmental effects (stress corrosion cracking, hydrogenembrittlement, liquid metal embrittlement, etc.)

The process of fracture:

1. Damage accumulation.

2. Nucleation of one or more cracks or voids.3. Growth of cracks or voids. (This may involve a coalescence of thecracks or voids.)

When the local strength or ductility is exceeded, a crack (two freesurfaces) is formed. On continued loading, the crack propagatesthrough the section until complete rupture occurs.

MACROSCOPIC ASPECTS

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To describe the resistance to fracture of a material is the plane strainfracture toughness, defined as the critical stress intensity factor underplane strain conditions and mode I loading. This is the stress intensityfactor at which a crack of a given size starts to grow in an unstablemanner. The fracture toughness is related to the applied stress by :

KIc is the fracture toughness in mode I loading, a is the characteristicdimension (semilength) of the crack and Y is a factor that depends onthe geometry of the specimen, the location of the crack, and theloading configuration.

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Orowan developed a simple method for obtaining the theoretical tensile strengthof a crystal. With his method, no stress concentrations at the tip of the crack areassumed; instead, it is assumed that all atoms separate simultaneously once theirseparation reaches a critical value.

Distance, a

MAX

Stress

a0 d

The area under the curve is the work required to cleavethe crystal.

This work of deformation cannot be lower than theenergy of the two new surfaces created by thecleavage.

THEORETICAL TENSILE STRENGTH

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Assuming that the elastic deformation is restricted to thetwo planes and that the material is isotropic, thefractional change in the distance between the planes,

da/a0, is defined as the incremental strain d.


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However, d is not known; to determine d, the area under the curve has tobe equated to the energy of the two surfaces created

The maximum value of is equal to the theoretical cleavage stress.

Sustituyendo d en la ec.

anterior

Sustituyendo K en la siguienteecua

cin

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STRESS CONCENTRATION

It is almost impossible to measure the stress at the tip of the crack. Anequivalent criterion, called the Griffith criterion, is more useful andpredicts the force that must be applied to a body containing a crackfor the propagation of the crack

Stress Concentration Factor

His analytical model was based on the elastic solution ofa cavity elongated in the form of an ellipse. The maximumstress occurs at the ends of the major axis of the cavityand is given by Ingliss formula

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In the case of an extremely flat ellipse or a very narrow crack of length 2aand having a radius of curvature = b2/a

Kt=

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GRIFFITH CRITERION

Elastic strain energy is released in a volume of material, and two newcrack surfaces are created, which represent a surface-energy term

Strain energy Surface energy

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fracture toughness

A necessary condition for crackpropagation is:

NOTE: if the local stress at the crack tip is not sufficiently large to breakthe atomic bonds, the energy criterion of Griffith will be inadequate

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CRACK PROPAGATION WITH PLASTICITY

The mechanics of fracture will, then, depend on the magnitude of

p,the plastic work done, which in its turn depends on the crack speed,temperature, and the nature of the material.

In the case of plastic deformation, the work done in the propagation ofa crack per unit area of the fracture surface is increased from sto (s+p)

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Localized plastic deformation at the crack tip therefore improves thefracture toughness of the material. This is the conventional treatment ofthe plastic work contribution to the fracture process, where in p isconsidered to be a constant.

Irwin made a fundamental contribution to the mechanics of fracture whenhe proposed that fracture occurs at a stress that corresponds to a criticalvalue of the crack extension force

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LINEAR ELASTIC FRACTURE MECHANICS

Linear elastic fracture mechanics (LEFM) was create enabling us to obtain aquantitative measure of the resistance of a brittle material to unstable orcatastrophic crack propagation. Extension of these efforts into nonlinearelastic and plastic regimens has led to the development of elasto-plasticfracture mechanics (EPFM), also called post-yield fracture mechanics

1. The far-field stress, .2. The characteristic crack length, a.3. The inherent material resistance tocracking,KR.

K (stress intensity) is to stress. Stress and stress intensity factor vary with theexternal loading conditionsKR (fracture toughness) is to strength. strength and toughness are materialparameters, independent of loading and specimen size considerations.

Fracture Toughness

13http://mae.ucdavis.edu/vlasaponara/equations_fracture_fatigue.pdf


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Crack-Tip Separation Modes

Stress Field in an Isotropic Material in the Vicinity of a Crack Tip

KI, KII, and KIII represent stress intensity factors in modes I, II, and III,respectively.

Mode I Mode II Mode III

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Details of the Crack-Tip Stress Field in Mode I

Thus, K will characterize the external conditions (i.e., the nominal appliedstress and half the crack length a) that correspond to fracture whenstresses and strains at the crack tip reach a critical value.

Kc depends on the dimensions of the specimen. In the case of a thinsample (plane-stress conditions), Kc depends on the thickness of thesample, whereas in the case of a sufficiently thick sample (plane-strainconditions), K is independent of the thickness of the specimen and is

designated as KIc. 15


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for a central through-the-thickness crack of length 2a, in a plate of widthW

for an infinite solid, a/W= 0, and we have ,as expected. For

an edge crack in a semi-infinite plate, we have

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Plastic-Zone Size Correction

When the whole of the reduced section yields, the plastic zone spreads tothe edges of the sample, and K does not have any validity as a parameterdefining the stress field.

When the plastic zone is small in relation to the crack length, it can bevisualized as a cylinder of radiusryat the crack tip.

Variation in Fracture Toughness with ThicknessThe material in the plastic zone deforms in such a way that its volume iskept constant. Thus, the large deformations in thex1 andx2 directions tendto induce a contraction in the x3 direction, which is resisted by thesurrounding elastic material.

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FRACTURE TOUGHNESS PARAMETERS

Crack Extension Force G

The change in elastic strain energy with respect to the crack area, in thelimit of the area going to zero, equals the crack extension force

It is convenient to evaluate G in terms of the compliance c of the

sampleCrack Opening DisplacementThe development of a plastic zone at the tip of the crack results in adisplacement of the faces without crack extension. This relativedisplacement of opposite crack edges is called the crack openingdisplacement (COD).

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J Integral

Provides a value of energy required to propagate a crack in an elastic-plastic material.

R Curve

An R curve graphically represents this

resistance to crack propagation of thematerial as a function of crack growth.With increasing load in a crackedstructure, the crack extension force Gat the crack tip also increases.

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POST-YIELD FRACTURE MECHANICS

The crack tip opening displacement (CTOD), , is the parameter thatcontrols crack extension.

The elastic portion is related to K orG. The plastic portion is not strictly amaterial property, in as much as it depends on the specimens

dimensions, constraints

Jel is the elastic portion.Jpl is a function of the geometry of the componentand the crack load corresponding to extensive plastic deformation, andthe material characteristics.

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STATISTICAL ANALYSIS OF FAILURE STRENGTH

The strength distribution of a brittle solid can be explained by astatistical distribution called the Weibull distribution

where is the applied stress and m, 0,and u are material constants for aconstant-flaw population

for an equal probability of survival, the largerthe volume (V2 > V1), thesmaller must be the fracture strength (1 < 2).

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MICROSCOPIC ASPECTS

Ductile failure occurs by (a) the nucleation, growth, and coalescence ofvoids, (b) continuous reduction in the metals cross-sectional area until it is

equal to zero, or (c) shearing along a plane of maximum shear. 23


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High-purity metals, such as copper, nickel, gold, and other very ductilematerials, fail with very high reductions in their areasBrittle fracture is characterized by the propagation of one or more cracksthrough the structure; metals and some polymers undergo irreversibledeformation at the tip of the crack, which affects its propagation.

METALS

The nucleation of a crack in a perfect crystal essentially involves therupture of interatomic bonds.The stress necessary to do this is thetheoretical cohesive stress.

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In ductile materials dislocations can move on a large number of slipsystems and even cross from one plane to another.In crystalline solids, cracks can be nucleated by the grouping ofdislocations piled up against a barrier.

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When cleavage cracks form and propagate at a greater rate thanplastic deformation, the material fails in a brittle manner.

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CERAMICS

Among the approaches developed toenhance the ductility of ceramics are:

1. The addition of fibers to the ceramic toform a composite2. The addition of a second phase thattransforms at the crack tip with a shearand dilational component.3. The production of microcracks ahead ofthe crack.4. Careful processing in such a manner

that all flaws of a size greater than thegrain size are eliminated.

The inability of ceramics to undergo plasticdeformation renders ceramics muchstronger, but their ability to resist thepropagation of cracks is decreaseddrastically 28


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The tensile stress at which a flaw will be activated

The fracture toughness

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In anisotropic materials, the grain boundaries are regions of stressconcentration in which the initiation of a fracture is more likely tooccur than in other regions.

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Thermal stresses induced during cooling can have a profound effect on themechanical strength of the ceramics. The problem is especially crucial incomposites

For constant expansion coefficients, and assuming a constant E

This simple equation expresses the experimentallyobserved fact that thermal anisotropy is much moreeffective in weakening specimens with a large grain sizethan specimens with a small grain size.

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Thermally Induced Fracture in Ceramics

When a ceramic is subjected to external loading, the internal stresses due

to thermal differences interact with the externally applied loads and canconsiderablyreduce the stresses required for fracture.


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POLYMERS

Many polymers fracture in a brittle manner below their glass transitiontemperature.

Frequently, the phenomena of crazing (increase in volume) and shear

yielding (constant volume) precede actual fracture in a polymer.A craze is a region of a polymer in which the normalchain arrangement characteristic of the amorphousstate has been transformed into drawnoutmolecular chains interspersed by voids. The crazedregion is a very small percentage of the total regionof the polymer 33


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Because the polymers show a significant amount of viscoelastic behaviorat their service temperature, the strain rate has a profound effect on theirfracture behavior.

In addition to brittle fracture, polymersundergo a mode of fracture involving crazing,in which the polymer chains ahead of a crackalign themselves along the tensile axis.

Another mode of deformation that is aprecursor to fracture is thephenomenon of shear banding in apolymer. 34


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FRACTURE AND TOUGHNESS OF BIOLOGICALMATERIALS

Fracture toughness (Jc)

Fracture Mechanism Maps

Fracture mechanism maps provide information about mechanicalproperties in a compact form. i.e. One can plot normalized tensilestrength /E against the homologous temperature T/Tm.

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REFERENCIAS

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Mechanical Behavior of MaterialsMarc Meyers and Krishan ChawlaSecond editionChapter 7Chapter 8

http://mae.ucdavis.edu/vlasaponara/equations_fracture_fatigue.pdf

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Fracture, Macroscopic&Microscopic Aspects

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