ppt on fracture mechanics

8/11/2019 ppt on fracture mechanics

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TOPIC : FATIGUE CRACK PROPAGATION

CONTENTS:

Similitude in fatigue

Empirical Fatigue Crack Growth Equations

Crack Closure Effect and Fatigue Threshold

Variable Amplitude Loading and Retardation

Fatigue Crack Growth of short cracks

Experimental Measurement of Fatigue Crack Growth

Damage Tolerance Design Methodology

SOURCE:T. L. Anderson, Fracture Mechanics : Fundamentals and Applications, 2/e,

CRC Press, 1995.

One of the most successful applications of the theory of fracture mechanics is the

characterization of fatigue crack propagation - Suresh.


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Fatigue Crack Propagation


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FATIGUE CRACK PROPAGATION : FRACTURE

MECHANICS APPROACH

So far, we have dealt with static or monotonic loading of cracked bodies.

Here, we consider crack growth in the presence of cyclic stresses. The

focus is on fatigue analysis of metals, but the concepts apply to other

materials too.

In the early 1960s, Paris demonstrated that Fracture Mechanics is a

useful tool for characterizing Fatigue Crack Growth (FCG). Since then,

the application of Fracture Mechanics to fatigue analysis has become

almost routine. There are, however, a number of controversial issues

and unanswered questions in this field.


4/81

The methods and procedures for the analysis of constant amplitude

fatigue under small scale yielding conditions at the crack tips are

fairly well established, although a number of uncertainties remain.

Variable amplitude loading, large scale plasticity and short cracksintroduce added complexities and are not fully understood.

We now present Fracture Mechanics Approach to Fatigue Crack

Propagation.


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FATIGUE CRACK PROPAGATION:

SIMILITUDE

Similitude implies that the crack tip conditions are uniquely

defined by a single parameter such as the SIF (K). In the case of

a stationary crack, two configurations will fail at the same

critical K value, provided an elastic singularity zone exists at

the crack tip. Under certain conditions, fatigue crack growth can

also be characterized by the SIF.


6/81

Consider crack growth in the presence of a constant amplitude

cyclic stress(Fig 10.1). A cyclic plastic zone develops at the

crack tip, and the growing crack leaves behind a plastic wake.

If the plastic zone size is sufficiently small that it is embedded

within an elastic singularity zone, the conditions at the crack

tip are uniquely defined by the current K values, and the crack

growth rate is characterized by Kmin and Kmax. It is convenient

to express the functional relationship for crack growth in the

form

--------------------- (1) RKfdN

da ,1


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where K = (KmaxKmin)

R = Kmin/ Kmax

and is the crack growth per cycle.dNda

Fig 10.1 Constant amplitude fatigue crack growth under small

scale yielding conditions


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A number of expressions for f1have been proposed, most of which are

empirical!

Eqn. (1) can be integrated to estimate fatigue life. The number of

stress cycles N required to propagate a crack from an initial length, a0

to a final length, af is given by

------------------- (2) fa

aRKf

da

0

,1


9/81

If Kmax and Kminvary during cyclic loading (variable amplitude loading),

the crack growth in a cycle may depend upon the loading

history as well as the current value of Kmaxand Kmin

da/dN = f2 (K, R, H) -------------- (3)

where H indicates the history dependence, which results from

prior plastic deformation. Eqn. (3) violates the similitude assumption,

two configurations cyclically loaded at the same K and R will not

exhibit the same crack growth rate unless both configurations are

subject to the same prior history.


10/81

Figure 10.2 illustrates several loading cases where the similitude

assumption is invalid. In each case, prior loading history influences

the current conditions at the crack tip. We present a Fatigue Crack

Propagation model that accounts for loading history later.

Fatigue Crack Growth Analysis became considerably more complicated

when prior loading history is taken into account. Consequently, FCG

models of the form of eqn. (1) are applied whenever possible. However,

such Fatigue analyses are only approximate in case of Variable

Amplitude Loading.


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Fig. 10.2 Examples of cyclic loading that violate similitude

K - increasing

Random loading

K - decreasing


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Empirical FCG (Fatigue Crack Growth) Equations:

Figure 10.3 illustrates typical fatigue crack growth behavior in

metals. The sigmoidal curve contains three distinct regions I, II, III.

In Region II, the curve is linear. In Region III, the crack growth

rate accelerates as Kmax approaches Kcritical, the fracture toughnessof the material. In Region I, da / dN approaches Zero at a threshold

K.


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Fig. 10.3 Typical fatigue crack growth behavior in metals


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Paris Law:Applies to Region II.

--------------------(1)

C and m are material constants that are determined

experimentally. da / dN is insensitive to the R in Region II.

m ranges from 2 to 7 for various metals.


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Formans Law :Valid for Regions II and III; Accounts for R ratio

effects; The crack growth rate approached infinity as Kmaxapproaches

Kcritical.

------------------- (2)

OR

-------------------(3)

KKR

KC

dN

da

crit

m

)1(

).(

1

).(

max

1

K

K

KC

dN

da

crit

m


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Weertmans Semiempirical equation for regions II and III :

-------------------(4)

Both the Forman and weertmans equations do not account for the

threshold.

Klesnil and Lukas equation:Accounts for the threshold

-------------------(5)

The threshold KTh is a fitting parameter to be determined

experimentally.

max

4).(

KK

KC

dN

da

crit

)( mThm KKC

dN

da


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Pridles empirical relationship:An attempt to describe the entire

curve (Region I, II, III) taking account of both the threshold KThand

Kcrit.

-----------------(6)

Mc Evilys equation:can fit the entire curve.

------------------(7)

maxKK

KKC

dN

da

crit

Th

max

21

KK

KKKC

dN

da

crit

Th


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Some Useful Remarks

Each of the equations (1) to (7) can be integrated to estimate

fatigue life N. The most general FCG model contains four

material constants: C, m, Kcirtand KTh(The threshold Stress

Intensity Factor range KThis not a true material constant

since it usually depends on the ratio R).


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None of these equations incorporate a history dependence;which results from prior plastic deformation. Two configurations

cyclically loaded at the same K and R will not exhibit the same

crack growth rate unless both configurations are subject to the

same prior history.

All the equations are strictly valid only for constant amplitude

loading.


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Example: Derive an expression for the number of stress cycles required

to grow a semicircular surface crack from an initial radius a0to a final

size of af

, assuming Paris Law describes the FCG rate. Assume that af

is

small compared to plate dimensions (width, length, thickness) and the

stress amplitude is constant.

The SIF Solution:

which can be integrated to estimate the fatigue life.

aK 663.0

)( minmax

2/.663.0 mm aCdN

da


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fa

a

m

m daa

C

N

0

.

.663.0

12

m

m

f

m

mC

aa

.663.012

21

21

0

Closed form integration is possible in this case because the K expression

is relatively simple. In most instances, numerical integration is required

to estimate fatigue life.

2mfor


22/81

Dowling and Begley applied the J integral to fatigue crack growth

under large scale yielding conditions where the SIF (K) is no longer

valid. They fit the growth rate data to a power low expression in J

(Note )

Here J is the contour integral for cyclic loading, analogous to the

J integral for monotonic loading.

)( minmax JJJ

mJCdN

da .


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FATIGUE CRACK RPOPAGATION:

Crack Closure Effects and The Fatigue Threshold:

Soon after Paris Law gained side acceptance as a predictor of FCG,

many researchers have realized that that this law was not universally

applicable. As figure 10.3, illustrates a loglog plot of (da/dN) Vs (K)

is sigmoidal rather than linear when FCG data are obtained over a

sufficiently wide range of K. Also, FCG exhibits a dependence on

R ratio, especially at both extremes of the curve.


24/81

Elber provide a partial explanation for both the fatigue threshold

and R ratio effects. Elber postulated that crack closure (i.e., contact

between crack faces at loads that were low but greater than zero)

decreased the fatigue crack growth rate by reducing the effective

stress intensity factor range. Figure 10.4(b), illustrates the crack

closure effect. When a cracked specimen is cyclically loaded at

Kmaxand Kmin, the crack faces are in contact below Kop, the SIF at

which the crack opens. Elber postulates that the portion of the stress

cycle that is below Kopdoes not contribute to FCG!.


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Fig 10.4 Crack closure during fatigue crack growth. The crack faces

contact at a positive load (a), resulting in a reduced driving force for

fatigue, keff(b)


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He defines an effective SIF range as

and an effective SIF ratio.

and then proposes a modified ParisErdogan equation:

This equation has been reasonably successful in correlating FCG data at

various R ratios.

minmax

max

KK

KK

K

KU

opeff

meffKCdN

da .

effK opeff KKK max


27/81

Since Elbers original study, numerous researchers have confirmed

that Crack Closure does in fact occur during Fatigue Crack

Propagation. Suresh and Ritchie identified five mechanisms for crack

closure undefatigable.

Crack Closure reduces FCG rate and introduces a threshold (KTh)

(See fig 10.3)


28/81


Variable Amplitude Loading and Retardation:

Similitude of crack tip conditions, which implies a unique relation

between da / dN, K and R is strictly valid for constant amplitude

loading (i.e., dK / da = 0) only. Real structures / components, however,

seldom conform to this ideal. A real structure experiences a spectrum

of stresses over its life time. In such cases, a FCG rate at any moment

in time depends not only on the current loading conditions, but also

the prior history. i.e., da / dN = f3(K, R, H)


29/81

History effects in FCG analysis are a direct result of the history

dependence of plastic deformation. Figure 10.10, schematically

illustrates the cyclic stress-strain response of an elastic-plastic

material which is loaded beyond yield in both tension and compression.

If we desire to know the stress at a particular strain, *. For the loading

path in fig 10.10, there are three different stresses that correspond to *;

we must specify not only *, but also the deformation history that

preceded this strain.


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Fig. 10.10. Schematic stress-strain response of a material that

is yielded in both tension and compression. The stress at a

given strain, *, depends on prior loading history.


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Figure 10.11, illustrates the crack tip plastic deformation that results

from a single stress cycle. A plastic zone forms at the crack tip when

the structure is loaded to Kmax. Upon unloading, material near the crack

tip exhibits reverse plasticity, which results in a compressive plastic

zone. The compressive stress field at the crack tip influences subsequent

deformation and the crack growth. Retardation of crack growth after an

overload is an example of this effect.


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Fig. 10.11. Formation of a reverse plastic zone during

cyclic loading

(c) Stress field after unloading

(a) Crack loaded to Kmax (b) Superimposed stress field


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We now analyze reverse plasticity by means of the BagdateBarenblatt

strip yield model. The advantage of this model is that it permits

superposition of loading and unloading stress fields. Refer fig 10.11(a),

where the crack is loaded to Kmax; assuming small scale yielding the

size of the plastic zone P is, 2

max

8

ys

KP


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Let us now superimpose a compressive stress intensityK. The

effective yield stress for reverse yielding is2*ysi. Since the

material inside the compressive plastic zone must be stressed toys

from an initial value of + ys. Figure 10.11(b), illustrates the

superposed stress field, and fig.10.11(c) shows the net stress field

after unloading. The estimated size of the compressive plastic zone,

P * , is

Therefore, the compressive plastic zone is ththe size of the

monotonic zone. FEA predicts a much smaller CYCLIC PLASTIC

ZONE SIZE than P *!!

2

28*

ys

KP


35/81

Budiansky and Hutchinson Crack Closure model for FCG analysis

incorporates a plastic wake.

Budiansky, B. and Hutchinson, J.W., Analysis of closure in Fatigue

Crack Growth, ASME Journal of Applied Mechanics, Vol.45, 1978,

Page No. 267276.


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The effect of overloads on FCG

The fatigue loading history illustrated in Fig.10.12, shows a

constant amplitude loading interrupted by a single OVERLOAD.

Prior to the overload, the crack tip plastic zone would have reached

a steady state size, but the overload cycle produces a significantly

larger plastic zone. When the load drops to the original Kmaxand

Kmin, the RESIDUAL STRESSES that result from the overload

plastic zone are likely to influence subsequent FCG behavior.


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Fig. 10.12. A single overload during cyclic loading


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Figure 10.13, shows experimental data, where a single overload is

imposed in an otherwise constant amplitude test. Immediately after

application of the overload, da/dN drops dramatically. The overload

cycle results in compressive residual stress at the crack tip, which

RETARD fatigue crack growth. The FCG rate resumes its original

value once the crack has grown through the overload plastic zone.


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Fig. 10.13. Retardation of crack growth following an overload


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Retardation of FCG rate is a complicated phenomenon. There area number of empirical and semi empirical model for retardation,

which contain one or more fitting parameters that must be

obtained experimentally. Some models assume that crack closureeffects are responsible for retardation while others consider the

plastic zone in front of the crack tip.


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Fatigue Crack Growth Analysis under Variable Amplitude Loading

Figure 10.2 illustrates several examples of Variable Amplitude

Loading. Similitude is not satisfied in such cases, and history effects

are pronounced.

FCG analysis under Variable Amplitude Loading also is done using

!! If is dK/da is small! Simple FCG laws are

usually conservative when applied to variable amplitude loading.

Retardation effects, which these equations do not consider, tend to

extend the fatigue life of a structure / component.

),(1 RKfdNda


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Variable amplitude loading FCG analysis that account for retardation

can be very complicated and computationally intensive. Some FCG

models require cycle-by-cycle summation of crack growth.

For FCG analysis under Variable Amplitude Loading, the stress input

consists of two components. The spectrum and the sequence. The

spectrum is a statistical distribution of stress amplitudes, which

quantifies the relative frequency of LOW, MEDIUM and HIGH

stress cycles. The sequence, which defines the order of the various

stress amplitudes, can be wither RANDOM or a REGULAR pattern.


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FCG analysis under Variable Amplitude Loading must first be Performed

on the experiment to determine the adjustable parameters that gives the

best prediction of measured crack growth. The model can then be applied

to structure / component life predictions. If a structure / component with

a different stress spectrum is to be analyzed, the FCG analysis model

must be recalibrated with a new experiment.

Newman has developed a FCG analysis model, based on crack closure

effects that can be applied to variable amplitude loading. This model is

capable of a priori predictions, with the empirical approaches are merely

able to correlate crack growth data after the fact. Newmans model is

based on the Dugdale Barenblantt strip yield model and is an expansion

of the superpositionreverse plasticity concept illustrated in Fig 10.11


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Fatigue Crack Closure and Retardation:

Summary:

Under constant amplitude cyclic loading, the normal SIF amplitude

(K = KmaxKmin) and / or the maximum SIF (Kmax) uniquely

govern FCG rates in ductile and brittle solids

There are, however, a variety of circumstances where the effective SIF

range or peak value of crack tip SIF, which is responsible for FCG,

can be markedly different from the nominal value.


45/81

These differences between the apparent and actual driving force for

fatigue fracture are called crack growth retardation mechanisms:

Plasticityinduced crack closure

Oxideinduced crack closure

Roughnessinduced crack closure

Viscous fluidinduced crack closure

Phase transformationinduced crack closure

Additional Retardation Mechanisms

- Retardation following tensile overloads- Transient effects following compressive overloads

- Load sequence effects


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A detailed study of these fatigue crack growth retardation mechanisms

under constant amplitude cyclic loading is essential for

- Developing accurate fatigue life prediction methods

- Improving micro structural design of materials for enhanced

damage tolerance.

Structures / components in practice are subjected to Variable amplitude

spectrum loads. Experimental observations of fatigue crack growth due

to these loads have revealed that the application of periodic overloads

can significantly decelerate the rate of FCG.

Variable amplitude and spectrum loads can have a distinctly different

effect on the fatigue crack growth than on fatigue crack initiation.

Elbers FCG Model:


47/81

Elber s FCG Model:

, if Kmin< Kop

, if Kmin Kop

Fatigue crack closure is a phenomenon whose influence on FCG rates

is strongly dictated by micro structural and environmental factors and

mechanical loading parameters. It is therefore a formidable challenge

to experimental and theoretical attempts aimed at quantifying the effect

of crack closure on fatigue behavior in engineering materials /

components / structures.

meffKCdN

da

opeff KKK max

minmax KK


48/81

The ultimate aim of fatigue research is to develop models for the

prediction of useful life in cyclically loaded structures / components.

It is not surprising that only semi-empirical approaches are

developed.

Many researchers have included the crack closure effects in conjunction

with various cycle counting methods to predict FCG life in variable

amplitude fatigue involving random and block program loads. These

approaches, can predict fatigue lives to within an accuracy of a factor

of two.


49/81

The characteristics approach was first proposed by Paris (1960) for

RANDOM LOADING. The basic hypothesis here is that the random

variation of the crack tip stress fields are described in terms of root-

mean-square value of the SIF range, Krms. The FCG rate under

variable amplitude spectrum loading is given by the relationship

where C and m are material constants.

mrmsKCdNda


50/81

and

Here, kiis the SIF range in the ithcycle in a sequence consisting of

n stress cycles. Krms= K for constant amplitude fatigue loading.Although empirical, this approach is used in a number of fatigue critical

applications.

SOURCE:S.Suresh, Fatigue of Materials, Second Edition, Cambridge

University Press,2003.

Chapter 14. Retardation and transients in fatigue crack growth.

n

K

K

n

i

i

rms

1

2


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FATIGUE CRACK GROWTH PROPAGATION:

Growth of Short Cracks

Experts consider cracks less than 1 mm long to be short (a 1mm)

The fatigue behavior of short cracks is very different from that of

long cracks.

A number of factors contribute to the anomalous behavior of short

cracks. The fatigue crack growth mechanisms depend on whether the

crack is micro-structurally short or mechanically short.


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Micro-structurally short crack has dimensions on the order of the

grain size less than 100 m. The material behavior cannot be

modeled using continuum mechanics at such length scales. The FCG

is strongly a function of MICROSTRUCTURE. The growth of such

cracks is often very sporadic. The crack may grow rapidly and then

virtually arrest when it encounters barriers (grain boundaries)!


53/81

Mechanically Short Cracks: (100m a 1mm), Continuum

mechanism is applicable

Short cracks grow typically much faster than longer cracks at the

same K particularly near the threshold (Fig. 10.18).

Fig 10.18 Growth of short cracks in a

low carbon steel


54/81

Two factors contributing to the faster growth of short cracks:

crack tip plastic zone and crack closure phenomenon.

When the plastic zone size is comparable to the crack length, an

elastic stress singularity does not exist at the crack tip and hence

the use of SIF, K is invalid. An effective crack driving force can

be inferred by adding an Irwin plastic zone correction.


55/81

An intrinsic crack length has been proposed, which, when added to

the physical crack length, bring short crack FCG data in line with the

corresponding longer crack FCG results. The intrinsic crack length is

merely a fitting parameter, however, and does not correspond to a

physical length scale in the material.

Some researchers have proposed adjusting the data for crack tip

plasticity by characterizing da / dN with J rather than K. Note Jcomputations includes crack tip plasticity effects.


56/81

Short cracks exhibit different crack closure behavior than longer cracks,

and the FCG data [ da/dN (m / cycle) Vs K (Mpa m)] for different

crack length can be rationalized through keff. Figure 10.19 (a), shows

Kopmeasurement for the short and long crack FCG data presented in

Fig 10.18. The closure loads are significantly higher in the longer

cracks, particularly at low K levels. Fig. 10.19 (b) shows the short and

long crack FCG data lie on a common curve when da / dN (m/cycle) is

plotted against keff, thereby lending credibility to the crack closuretheory of short crack behavior.


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Fig. 10.19 Short Crack fatigue crack growth data

(a) Crack closure data for short and long

cracks

(b) Closurecorrected data


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FATIGUE CRACK PROPAGATION: EXPERIMENTAL

MEASUREMENT OF FATIGUE CRACK GROWTH

The ASTM standard E64793 outlines a test method for FCG

measurements. This procedure does not take into account Crack

Closure Effects.

The Standard Test Method for measurement of FCG rates, describes

how to determine da/dN Vs K from an experiment. The crack is

grown by cyclic loading of a test specimen, and Kmin, Kmaxand

crack length a are monitored throughout the test


59/81

The test specimen and fixture are identical to those used for fracture

toughness testing. E647 recommends tests on Compact Tension

specimens and Centre Cracked Tension Panels. The ASTM standard E 647 for FCG measurements requires that the

behavior of the test specimen be predominantly elastic during the

tests. Accordingly specifies the following requirements for the

uncracked ligament of a CT specimen

2

max4

ys

K

aW


60/81

There are no specific requirement of thickness. However, the FCG

rate can depend on thickness much like fracture toughness is

thickness dependent. Consequently, the thickness of the test

specimen should match the section thickness of the structure /

component of interest.

All specimens must be fatigue pre cracked prior to the actual fatigue

test. The Kmaxat the end of fatigue precracking should not exceed

the initial Kmaxin the FCG test, otherwise retardation effects any

influence the measured FCG rate.

During the test the crack length must be measured periodically. Crack length


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g g p y g

measurement techniques include optical, unloading compliance and

potential drop. Optical crack length measurements require a traveling

microscope. One obvious disadvantage of the optical method is that it can

only measure growth on the surface, in thick specimens, the crack length

measurements must therefore be corrected for tunneling, which cannot be

done until the test specimen is broken open after the test!. Another

disadvantage of the optical technique is that the crack length measurement

are recorded manually. It may be possible to automate optical crack length

measurements with Image Analysis hardware and software, but most

Material Testing Labs do not have this facility. The unloading compliance

technique requires that the fatigue test be interrupted for each crack length

measurement.


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The ASTM standard E647 outlines two types of Fatigue Tests:

(1) Constant Load Amplitude Tests where K increases(2) K - decreasing test in which the load amplitude decreases

during the test to achieve a negative K gradient.

The K - increasing test is suitable for FCG rates greater than

10-8 m/cycle. The Kdecreasing test is preferable when

nearthreshold data are required


63/81

ASTM Standard E647 requires that the normalized K gradient be

computed and reported.

If the test is computer controlled, the load can be programmed to

decrease continuously to give the desired K gradient.

da

dK

Kda

dK

Kda

Kd

Kda

dK

KG max

max

min

min

1111


64/81

Figure 10.24, illustrates typical crack length versus N curves. These

curves must be differentiated to infer (da/dN). ASTM standard E647

recommends two alternative numerical methods to compute(da/dN)

Linear Differentiation Approach: Computes the slopefrom two neighboring data points (aj, Nj) and (aj+1, Nj+1).

The crack growth rate for is

where

aa

jj

jj

aa NN

aa

dN

da

1

1

21 jj aaa


65/81

Fig 10.24 Schematic fatigue crack growth curves. da

/ dN is inferred from numerical differentiation of

these curves.


66/81

The derivative at a given point on the aVs Ncurve can alsobe found by fitting several neighboring points to a quadratic

polynomial (i.e., a Parabola)

An appendix in ASTM standard E647 lists a FORTRANprogram which performs the curve fitting operation and

solves for da/dN

Closure measurement:


67/81

A number of experimental techniques for measurement of closureloads during Fatigue Crack Growth Test are currently available.

Allison has reviewed the existing procedures.

Allison, J.E., The Measurement of Crack Closure Load during

a Fatigue Crack Growth Tests, ASTM STP 945, 1998, PP 913 933.

Most measurements of crack closure loads are inferred from compliance.Figure 10.25 schematically shows the loaddisplacement behavior of

a FCG test specimen that exhibits crack closure. The precise closure

load is ill-defined; because there a range of loads where the crack is

partially closed.

The closure load can be defined by a deviation in linearity in either the


68/81

The closure load can be defined by a deviation in linearity in either the

fully closed or fully open case (P1and P3, in fig.10.25) or by

extrapolating the fully closed and fully open loaddisplacement

curve to the point of intersection (P2).

Fig 10.25 Alternate definition of the closure load.

Figure 10 26 illustrates the required instrumentation for the three most


69/81

Figure 10.26, illustrates the required instrumentation for the three mostcommon compliance technique for crack closure load measurement. The

closure load can be inferred from CLIP GAUGE displacement at thecrack mouth, back face strain measurements or LASER interferrometry

applied to surface indentations specimen alignment is critical when

inferring crack closure loads from compliance measurements.

Fig. 10.26 Instrumentation for the three most

common closure measurement techniques.

k h i di l i h


70/81

Crack mouth opening displacement measurement with a CLIP GAUGEis relatively simple, but extra care is necessary when attaching the clip

gauge. Displacement measurements away from the crack tip often lack

sensitivity.

A back face strain gauge has a high degree of sensitivity.

Interferrometric techniques provide a local measurement of crackclosure load. Monochromatic light from a Laser is scattered off of two

indentations on either side of the crack plane. The two scattered beams

interfere constructively and destructively, resulting in fringe patterns.

The fringes changes as the indentations more apart.


71/81

Crack closure is a 3D phenomenon. The interior of a FCG test specimenexhibits different closure behavior than the surface. Laser interferrometry

is strictly a surface measurement. The clip gauge and back face strain

gauge methods provide a thicknessaverage measure of crack closure.

More elaborate experimental techniques are also available to studythree-dimensional effects. Optical interferrometry has been applied to

transparent polymers to infer closure behavior through the thickness.

Fleck has also developed a special gauge to measure Crack OpeningDisplacement at interior of FCG test specimens.



72/81

Damage Tolerance Design Methodology

We describe the application of fatigue test data and Fatigue Crack

Growth models to structures / components, as part of a damage tolerance

design methodology.

The term damage tolerance design methodology refers to the application

of fracture mechanics analyses to predict remaining life and fix

inspection intervals. This approach is usually applied to structures /

components that are susceptible to timedependent crack growth

( e.g., fatigue crack growth environmental - assisted cracking, creepcrack growth). As its name suggests, the damage tolerance design

philosophy allows CRACKS to remain in the structure / component,

provided they are well below the critical size ac


73/81

The first task of a damage tolerance analysis is the computation of a

critical crack size ac. Depending on material properties, ultimate

failure may be governed by FRACTURE or PLASTIC COLLAPSE.

Consequently, an elastic-plastic fracture mechanics analysis that also

includes the extremes of brittle fracture and plastic collapse as special

cases is preferable.


74/81

Once the critical crack size achas been estimated, a safety factor is

normally applied to determine the tolerable crack size, at. A rational

definition of the safety factor should be based on uncertainties in the

input parameters (e.g., Stress, fracture toughness) in the fracture

analysis. Another consideration in specifying the tolerable crack size

is the crack growth rate; atshould be chosen such that da/dt at this

crack size is relatively small, and a reasonable length of time is

required for the crack to grow from at to ac.

The NDE provides input to the fracture analysis which in turn helps to


75/81

define inspection intervals. A structure / component is subjected to

NDE at the beginning of its life to determine the size of initial flaws. If

no significant flaws are detected, the initial flaw size is set to an

assumed value, a0, which corresponds to the largest flaw that might be

missed by NDE used.

Fig. 10.28(a), illustrates the procedure for determining the first

inspection interval in the structure / component. The lower curve

defines the true behavior of the worst flaw in the structure, while the

predicted curve assumes the initial crack length is a0. The time ( No. of

cycles) required to grow the crack from a0to at( the tolerable flaw size)

is computed. The first inspection interval, I1, should be less than this time,


76/81

Fig. 10.28. Schematic damage tolerance analysis

(c) Determination of third Inspection level, I3

(c) Determination of second Inspection level, I3(c) Determination of first Inspection level, I3

In order to prelude crack growth beyond at before the next inspection! If


77/81

p g y p

no flaws greater than a0are detected, the second inspection interval, I2, is

equal to I1, as Fig. 10.28(b) illustrates. Suppose that the next inspection

reveals a crack length a1, which is larger than a0. In this instance, a crack

growth analysis must be performed to estimate the time required to

grow the crack from a1to at. The third inspection interval, I3, might be

shorter than I2, as Fig.10.28(c) illustrates. Inspection intervals would

then become progressively shorter as the structure / component

approaches the end of its life. The structure is repaired or taken out of

service when the crack length reaches the maximum tolerable size, or

when required inspections become too frequent to justify continued

operation.

In practice, a variable inspection interval is not recommended,


78/81

p , p ,

inspections must often be carried out at regular times that can be

scheduled well in advance. In such instances a variation of the above

approach is required. The main purpose of any damage tolerance

analysis is to ensure that flaws will not grow to failure between

inspections. The precise methods for achieving this goal depends on

practical circumstances.

Since Retardation effects are not considered, the DTA is simpler and

will tend to overestimate FCG rates!

EXAMPLE: An edge crack of length 3 1 mm is detected in a large plate


79/81

EXAMPLE:An edge crack of length 3.1 mm is detected in a large plate

made out of ferrite - pearlite steel with KIC= 165 Mpa m^ and

subjected to constant amplitude cyclic loading having max= 310 Mpa andmin= 172 Mpa. Determine (i) Propagation life up to failure and

(ii) propagation life of the crack length is not allowed to exceed 25 mm.

SOLUTION:

SIF: aw

afKI

12.1

w

af

MPa138172310

ma 0031.00

22

ICK

a(i) Critical Crack Length,


80/81

20

12

12

0

...12

mmm

m

f

m

P

wa

fCm

aa

N

2max*12.1 ca

fam 0719.0

mKCdN

da

1210*8.6 C 3m

(i) Critical Crack Length,

Life:


81/81

233312

12

31

2

3

*13812.1*10*8.6

1*

123

0719.00031.0PN

= 203.6*103cycles

(ii) Replacing af= 0.0719 by af= 0.25

NP= 166.4*103cycles.

Date post:	03-Jun-2018
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ppt on fracture mechanics

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