Fatigue Crack Growth

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Hasan G PashaME06M045

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DETERMINATION OF FATIGUE CRACKGROWTH RATE

OBJECTIVE

Determine the fatigue crack growth rate for the given aluminum alloy under fluctuating

tensile load

Plot the fatigue crack growth rate against the stress intensity factor range

APPARATUS

1. Universal material testing machine with control unit

2. Traveling microscope

THEORY ON FATIGUE AND FRACTURE MECHANICS

[The contents of this section were composed using excerpts from Wikipedia, eFunda and a hostof other resources available on the World Wide Web]

Stress concentration

A stress concentration is a location in an object where stress is concentrated. An object is

strongest when force is evenly distributed among it, but an area of stress concentration resultsin a localized increase in stress at that location.

Geometric discontinuities cause an object to experience a local increase in the intensity of astress field. The examples of shapes that cause these concentrations are: cracks, sharp corners,holes and, changes in the cross-sectional area of the object.

High local stresses can cause the object to fail more quickly than if it was not there.

Fatigue

Fatigue is the progressive, localized, and permanent structural damage that occurs when a

material is subjected to cyclic or fluctuating strains at nominal stresses that have maximumvalues less than (often much less than) the static yield strength of the material.

The resulting stress may be below the ultimate tensile stress, or even the yield stress of thematerial, yet cause catastrophic failure.

Characteristics of Fatigue failures

The following characteristics are common to fatigue in all materials:

The process starts with a microscopic crack, termed the initiation site, which then widenswith each subsequent movement, a phenomenon analyzed in the topic of fracturemechanics

Failure is essentially probabilistic. The number of cycles required for failure varies betweenhomogeneous material samples. Analysis demands the techniques of survival analysis.

The greater the applied stress, the shorter the life

Damage is cumulative. Materials do not recover when rested.

Fatigue life is influenced by a variety of factors, such as temperature and surface finish, in

complicated ways

Some materials (e.g., some steel and titanium alloys) exhibit an endurance limit or fatigue

limit, a limit below which repeated stress does not induce failure, theoretically, for aninfinite number of cycles of load. Most other non-ferrous metals/alloys (e.g., aluminumand copper alloys) exhibit no such limit and even small stresses will eventually cause

failure.


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As a means to gauge fatigue characteristics of non-ferrous and other alloys that do notexhibit an endurance limit, a fatigue strength is frequently determined, and this is typically

the stress level at which a component will survive 107 loading cycles.

High-cycle fatigue

Most attention is focused on situations that require more than 10

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cycles to failure where stressis low and deformation primarily elastic.

S-N curve

In high-cycle fatigue situations, materials performance is commonly characterized by an S-Ncurve, also known as a Whler curve. This is a graph of the magnitude of a cyclical stress (S)against the cycles to failure (N).

S-N curves are derived from tests on samples of the material to be characterized (often termedcoupons) where a regular sinusoidal stress is applied by a testing machine which also counts the

number of cycles to failure. This process is sometimes known as coupon testing.

Each coupon test generates a point on the plot though in some cases there is a run-out where

the time to failure exceeds that available for the test. Analysis of fatigue data requirestechniques from statistics, especially survival analysis and linear regression.

Fatigue and Fracture Mechanics

The account above is purely phenomenological. Though it allows life prediction and design

assurance, it does not enable life improvement or design optimization. For the latter purposes,an exposition of the causes and processes of fatigue is necessary. Such an explanation is givenby fracture mechanics in stages.

Fracture Mechanics

Fracture mechanics is a method for predicting failure of a structure containing a crack. It usesmethods of analytical solid mechanics to calculate the driving force on a crack and those of

experimental solid mechanics to characterize the material's resistance to fracture.

Fracture mechanics is an important tool in improving the mechanical performance of materialsand components. It applies the physics of stress and strain, in particular the theories of

elasticity and plasticity, to the microscopic crystallographic defects found in real materials inorder to predict the macroscopic mechanical failure of bodies.

Crack Initiation, Propagation and Failure

Failure of a material due to fatigue may be viewed on a microscopic level in three steps:

i. Crack Initiation: The initial crack occurs in this stage. The crack may be caused by surfacescratches caused by handling, or tooling of the material; thread fasteners; slip bands ordislocations intersecting the surface as a result of previous cyclic loading or work hardening.

ii. Crack Propagation: The crack continues to grow during this stage as a result of continuouslyapplied stresses

iii. Ultimate ductile failure: Failure occurs when the material that has not been affected by thecrack cannot withstand the applied stress. This stage happens very quickly.

Linear Elastic Fracture Mechanics

Linear Elastic Fracture Mechanics (LEFM) assumes that the material is isotropic and linearelastic. Based on the assumption, the stress field near the crack tip is calculated using the


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theory of elasticity. When the stresses near the crack tip exceed the material fracturetoughness, the crack will grow.

In Linear Elastic Fracture Mechanics, most formulas are derived for either plane stresses orplane strains, associated with the three basic modes of loadings on a cracked body:

i. Opening a tensile stress normal to the plane of the crack

ii. Sliding a shear stress acting parallel to the plane of the crack and perpendicular to the

crack front

iii. Tearing a shear stress acting parallel to the plane of the crack and parallel to the crackfront

LEFM is valid only when the inelastic deformation is small compared to the size of the crack,which is termed small-scale yielding. If large zones of plastic deformation develop before thecrack grows, Elastic Plastic Fracture Mechanics (EPFM) must be used.

Crack Tip Stresses

Crack tips produce a 1/r singularity. The stress fields near a crack tip of an isotropic linear

elastic material can be expressed as a product of 1/r and a function of with a scaling factor

K.

Crack tip stresses

In actual cases, the maximum stress near the crack tip is of primary interest and whether itsvalue exceeds the fracture toughness. Thus, the stress intensity factor K is commonly expressed

in terms of the applied stresses at r 0 and = 0.


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Crack infinite plate under uniform tension

For a through crack in an infinite plate under uniform tension, the stress intensity factor is

KI Stress intensity factor under tensile loading

Stress

a Half the width of the crack

Stress-intensity factor

Stress-intensity factor, K, is used to more accurately predict the stress state (stress intensity)near the tip of a crack caused by a remote load or residual stresses. When this stress state

becomes critical a small crack grows (extends) and the material fails. The load at which thisfailure occurs is referred to as the fracture strength.

The experimental fracture strength of solid materials is 10 to 1000 times below the theoreticalstrength values, where tiny internal and external surface cracks create higher stresses nearthese cracks, hence lowering the theoretical value of strength.

Critical stress intensity factor and Fracture toughness

Based on the linear theory, the stresses at the crack tip are infinity but in reality there is alwaysa plastic zone at the crack tip that limits the stresses to finite values. It is very difficult to model

and calculate the actual stresses in the plastic zone and compare them to the maximumallowable stresses of the material to determine whether a crack is going to grow or not.

An engineering approach is to perform a series of experiments and reach at a critical stressintensity factor Kc for each material, called the fracture toughness of the material. The crack

stability can then be determined directly by comparing K and Kc.


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Mode I

Mode I: Opening


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Mode II

Mode II: Sliding

Mode III

Mode III: Tearing


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For linear elastic materials, the principle of superposition applies. A mixed-mode problem can betreated as the summation of each mode.

TEST DETAILS AND TERMINOLOGY

[Excerpted from ASTM Standard reference# E 647]

Crack size, a: a linear measure of a principal planar dimension of a crack. This measure iscommonly used in the calculation of quantities descriptive of the stress and displacement fieldsand is often also termed crack length or depth.

Cycle/Stress Cycle: under constant amplitude loading, the force variation from the minimum to

the maximum and then to the minimum force.

Fatigue-crack-growth rate, da/dN crack extension per cycle of loading

Force range P, the algebraic difference between the maximum and minimum forces in a cycle

is expressed as:

P = Pmax Pmin

Force Ratio, R the algebraic ration of the minimum to maximum forces in a cycle.

R = Pmin/Pmax

Maximum force, Pmax the highest algebraic value of the applied force in a cycle. Tensile forcesare considered positive and compressive forces are considered negative.

Maximum stress-intensity factor, Kmax the maximum value of the stress-intensity factor in acycle. This corresponds to Pmax.

Minimum force, Pmin the lowest algebraic value of the applied force in a cycle. Tensile forces

are considered positive and compressive forces are considered negative.

Minimum stress-intensity factor, Kmin the minimum value of the stress-intensity factor in a

cycle. This corresponds to Pmin when R > 0 and is taken to be zero when R 0.

Stress intensity factor range, K the variation of the stress intensity factor in a cycle

K = Kmax KminK = (1-R) Kmax for R 0 andK = Kmax for R 0


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Applied-K curve a curve obtained from a fracture mechanics analysis for a specific specimenconfiguration. The curve relates the stress-intensity factor to crack size and either applied force

or displacement.

The resulting analytical value is termed a K calibration and frequently available in data books forstress intensity factors.

Fatigue crack growth threshold, Kth that asymptotic value of K for which da/dN approacheszero.

Fatigue crack growth, K, da/dN the rate of crack extension cased by fatigue loading andexpressed in terms of average crack extension cycle per cycle.

Normalized K gradient, C = (1/K). dK/da the fractional rate of change of K with increasingcrack size

K-Decreasing test a test in which the value of C is nominally negative. In this test method K-Decreasing tests are conducted by shedding force, either continuously or by a series of

decremental steps, at the crack grows.

K-Increasing test a test in which the value of C is nominally positive. For the standard

specimens in this method the constant-force-amplitude test will result in a K-Increasing testwhere the C value increased but is always positive.

Summary of test method

This test method involves cyclic loading of notched specimen which have been acceptably pre-

cracked in fatigue.

Crack size is measured, either visually or by an equivalent method, as a function of the elapsed

fatigue cycles and these data are subjected to numerical analysis to establish the rate of crackgrowth.

Crack growth rates are expressed as a function of the stress-intensity factor range, K, which iscalculated from expressions based on linear elastic stress analysis.

Significance and Use

Fatigue crack growth rate expressed as a function of crack-tip stress-intensity factor range,

da/dN versus K, characterizes a materials resistance to stable crack extension under cyclicloading.

This test method can serve to establish:

i. The influence of the fatigue crack growth on the life of components subjected to cyclic

loading, provided data are generated under representative conditions and combined with

appropriate fracture toughness fat, defect characterization data and stress analysisinformation.

ii. Material selection criteria and inspection requirements for damage tolerant applications

iii. Quantitative terms for the individual and combined effects of metallurgical, fabrication,environmental, and loading variables on fatigue crack growth


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Specimen configuration, size and preparation

Standard specimens: While selecting the specimen adequate care should be taken in measuringand factoring the residual stresses and stress relief.

The residual stresses can be minimized by selecting symmetrical specimen configurations.

Specimen size: In order for results to be valid according to this test method it is required thatthe specimen be predominantly elastic at all vales of applied force. The minimum in planespecimen sizes to meet this requirement are based primarily on empirical results and are specificto the specimen configuration.

An alternative size requirement may be employed for high-strain hardening materials. The un-cracked ligament requirement for specific specimen geometry may be relaxed by replacing theyield strength with a higher and effective yield strength which accounts for the material strainhardening capacity.

Notch preparation: The machined notch (crack starter notch) for standard specimens can be

made by EDM, milling, broaching grinding or saw-cutting.

DESCRIPTION

Specimen Details

Type Aluminum alloy

Notch preparation method Saw-cutting

Thickness B 6 mm

Width W 50 mm

General Details

Test machine Universal material testing machine with electro-hydraulic actuators for loading the specimen usingcomputer generated sinusoidal load commandsand wedge grips for supporting the specimen

Equipment used to measure crack size Traveling microscope

Type of loading Tensile constant force-amplitude loading insinusoidal form

Technique used to convert a versus N

to da/dN

Secant and 7-point polynomial methods

Apart from the above, it would be desirable to give an account the following details:

Test material characterization in terms of heat treatment, chemical composition andmechanical properties, product size and form, method of stress relief if applicable

The crack plane orientation If specimen is removed from a large product form, itslocation with respect to the parent product should be given

The terminal values of K, R and crack size from fatigue pre-cracking. If pre-crack forceswere stepped-down, the procedure employed and the amount of crack extension at the

initial force level shall be given.


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Environmental variables: temperature, chemical composition, pH for liquids and pressure

for gases and vacuum. For tests in air, the RH should be reported. For tests in inertreference environments, such as fry argon, estimates of residual levels of water andoxygen in the test environment shall be given. Nominal values for all of the aboveenvironmental variables, as well as max deviations through out the duration testing, shall

be reported.

Specific procedure used to correct for crack curvature and magnitude of crack curvaturecorrection

PROCEDURE

1. A notch was prepared by saw-cutting on the aluminum alloy specimen and it was pre-cracking under fatigue

2. The specimen was held in the testing machine and constant force-amplitude loading insinusoidal form was given using computer at a 10 Hz cycle

3. Using a traveling microscope, the crack growth was observed for a 0.2 mm length

4. After 0.2 mm growth readings for cumulative number of cycles, maximum and minimumload applied were recorded. A loading cycle with 1 Hz frequency was given during this

period.

5. The following information was observed and tabulated for a total crack length of 17.6

mm: a, N, K and da/dN

PICTURES

Material testing machine Wedge grips


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Crack growth in specimen

PLOT

The da/dN (ordinate) values as a function of K (abscissa) were plotted in log-log co-ordinates.

The value of da/dN was obtained by using both secant and 7-point methods.

As per the recommendation by ASTM, for optimum data comparisons the size of the K logcycles should be two or three times larger than da/dN log cycles

FORMULAE

K =[(P)/A] * (* a) * f(a/w) MPa (m)

f(a/W) =0.857 + 0.265 (a/ W)

0.265*[1(a/W)]^4 + [1-(a/W)]^(3/2)

P = Pmax Pmin N

K Stress intensity factor range MPa (m)

F(a/W) Geometric factor

A Cross-sectional area m2

a Crack length M

P Force range N


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SAMPLE CALCULATION

a = 11 mmW = 50 mmPmax = 24571 NPmin = 1958 N

f(a/W) =

=

0.857 + 0.265 (11/50)0.265*[1(11/50)]^4 +

[1-(11/50)]^(3/2)

1.427

K =

=

=

[(P)/A] * (* a) * f(a/w)

24571-1958

* ((* 11 * 10-3) * 1.427)50 * 6 * 10-6

MPa (m)

MPa (m)

RESULT

The fatigue crack growth rate for given aluminum alloy was determined

The values of fatigue crack growth rate were was plotted against the stress intensity

factor range by using the Secant and 7-point polynomial methods

Fatigue Crack Growth

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