The Fracture Mechanics Method (da/dN- û - eFatigue · A crack becomes unstable (fracture) when the stress intensity factor, K I, exceeds the critical, for given material, stress

$Page 1: The Fracture Mechanics Method (da/dN- û - eFatigue · A crack becomes unstable (fracture) when the stress intensity factor, K I, exceeds the critical, for given material, stress$
© 2010 Grzegorz Glinka. All rights reserved. 1

The Fracture Mechanics Method (da/dN- K)

June 2nd – 6th, 2014,Aalto University,

Espoo, Finland

Prof. Grzegorz GlinkaUniversity of Waterloo, Canada

Information path for fatigue life estimation based onthe da/dN- K method

LOADING

F

t

GEOMETRY, Kt

PSO

MATERIAL

0

E

Stress-StrainAnalysis

Damage Analysis

Fatigue Life

MATERIAL

n

DKDKth

dadN


Stress intensity factor, K(indirect method)

Weight function, m(x,y)

, ,A

n

K x y m x y dxdy

KYa

Stress intensity factor, K(direct method)

2I yFE FE

n

K xor

dUK E EGda

KYa

(x, y)

f)

a

g)

01

mi i i

N

f ii

i

a C K N

a a a

N N

Integration of Paris’ equationh)

af

ai

Number of cycles , N

Cra

ckde

pth,

a

Fatigue Life

i)

Steps in Fatigue Life Prediction Procedure Based on theda/dN- K Approach (cont’d)


The Similitude Concept states that if the stressintensity K for a crack in the actual componentand in the test specimen are the same, then thefatigue crack growth response in the componentand in the specimen will also be the same andcan be described by the material fatigue crackgrowth curve da/dN - K.

a) Structure

Q

H

F

ab) Weld detail

c) Specimena

P

P

K 10-12

10-11

10-10

10-9

10-8

10-7

10-6

1 10 100

Cra

ckG

row

thR

ate,

m/c

ycle

K MPa m,

The Similitude Concept in the da/dN – K Method



Crack tip stress dependence on the stress intensity factor K

yS

x

xy

yy

xx

uy

ra

S

SS

3

2yyK

x2

2yyK

x

K1 < K2 < K3

Stress components, ij, at thecrack tip depend on the stressintensity factors KI which isinfluenced by:- the load, S-crack dimensions, a-geometry, Y

The stress field , ij, aroundthe crack tip can be describedby one universal functionvalid for all cracks of Mode I,i.e. for =0

yyK

x2

II

II

S aY KG plane stressE E

S aY KG plane strain

E E

22

22

2 21 1

G. Irwin’s fundamental Fracture Mechanics principles:1. The near crack tip stress field expressions above are universal, i.e. the stressdistributions in the vicinity of the crack tip have the same general mathematicalform regardless of the crack geometry, loading and geometrical shape of the body.

2. The strain energy release rate GI is related to the stress intensity factor KI andtherefore it is justified (and easier) to calculate the strain energy release rate (andthe critical stress) from the purely elastic local (near the crack tip) stressdistribution (i.e. from the Stress Intensity Factor).


A crack becomes unstable (fracture) when the stress intensity factor,KI , exceeds the critical, for given material, stress intensity factor KIc!

KI > KIc

Strengthparameterstraditionallyused for thesttrengthanalysis ofengineeringcomponentsand structures:

Y Y

cr ut

I

c

Ics

P KKK

PP

Force [MN] Stress [MPa]Stress IntensityFactor [MPa m]

Leonardo da Vinci17th-century

Euler, Cauchy19th -century

Irwin20th-century

FractureMechanicsparametersused for thestrengthanalysis ofengineeringcomponentsand structures:


General Stress Intensity Factor Expressions for Cracks in Mode I

The stress intensity factor is defined as:

in which S is the stress (usually the nominal) away from the crack. The geometryfactor, Y , accounts for the effect of geometry of the crack and the body, theboundary conditions and the type of loading.Determining stress intensity factor means in essence the derivation of the functiondescribing the geometrical factor Y. One of the confusing issues while determiningstress intensity factors is that the remotely applied stress S and the geometry factorY are inter-related. The value of parameter Y depends on the definition of the remote(termed often as nominal) stress S. In cases where the nominal or hot spot stress iswell defined there is no problem in the definition of the remote stress S. However, ifthe stress distribution is non-uniform it may not be clear which stress should beused in the expression for the stress intensity factor. Theoretically, any referencestress S can be chosen for the determination of the geometrical factor Y, as far asthe stress varies proportionally with the applied load. However, the user of givenexpression for K has to use the same definition of the reference stress whilecarrying out fatigue and fracture analyses. Nominal or the maximum stress in thecase of non-uniform stress distributions is most often used in stress intensity factorexpressions. Therefore, it is a good professional practice to define the referencestress S when quoting the geometry factor Y.

IK YS a


(Y. Murakami et. al)

Center Crack Plate under Uniform Tension

2a Wt

Reference: C.F. Federsen (1), H. Tada (2)

Method: Empirical formula based on Isida’s results

Accuracy: +0.3% for 2a/W 0.7 and 1.0% for 2a/W=0.8

I I I

I

aK a F FW

Y

F

2( ), , ( )

( ) sec2

Accuracy: Better than 0.2% for any value ofIK a F 2 4( ) 1 0.025 0.06

or

(2)

(1)


2a/W 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0100.200 1.025 1.026 1.026 1.026 1.026 1.027 1.027 1.027 1.028 1.028 1.0580.210 1.028 1.028 1.029 1.029 1.029 1.029 1.030 1.030 1.030 1.031 1.0310.220 1.031 1.031 1.032 1.032 1.032 1.032 1.033 1.033 1.033 1.034 1.0340.230 1.034 1.034 1.035 1.035 1.035 1.035 1.036 1.036 1.036 1.037 1.0370.240 1.037 1.037 1.038 1.038 1.038 1.039 1.039 1.039 1.040 1.040 1.0400.250 1.040 1.041 1.041 1.041 1.042 1.042 1.042 1.043 1.043 1.043 1.0440.260 1.044 1.044 1.045 1.045 1.045 1.046 1.046 1.046 1.047 1.047 1.0470.270 1.047 1.048 1.048 1.049 1.049 1.049 1.050 1.050 1.051 1.051 1.0510.280 1.051 1.052 1.052 1.052 1.053 1.053 1.054 1.054 1.054 1.055 1.0550.290 1.055 1.056 1.056 1.056 1.057 1.057 1.058 1.058 1.059 1.059 1.0590.300 1.059 1.060 1.060 1.061 1.062 1.062 1.062 1.062 1.063 1.063 1.0640.310 1.064 1.064 1.064 1.067 1.066 1.066 1.066 1.067 1.067 1.068 1.0680.320 1.068 1.069 1.069 1.070 1.070 1.071 1.071 1.072 1.072 1.072 1.0730.330 1.073 1.073 1.073 1.074 1.075 1.075 1.076 1.076 1.077 1.077 1.0780.340 1.078 1.078 1.078 1.079 1.080 1.080 1.081 1.081 1.082 1.082 1.0830.350 1.083 1.083 1.083 1.085 1.085 1.086 1.086 1.087 1.087 1.088 1.0880.360 1.088 1.089 1.089 1.090 1.090 1.091 1.092 1.092 1.093 1.093 1.0940.370 1.094 1.094 1.094 1.096 1.096 1.097 1.097 1.098 1.098 1.099 1.1000.380 1.100 1.100 1.100 1.101 1.102 1.103 1.103 1.104 1.104 1.105 1.1060.390 1.106 1.106 1.107 1.107 1.108 1.109 1.109 1.110 1.111 1.111 1.1120.400 1.112 1.112 1.113 1.114 1.114 1.115 1.116 1.116 1.117 1.118 1.1180.410 1.118 1.119 1.120 1.120 1.121 1.122 1.122 1.123 1.124 1.124 1.1250.420 1.125 1.126 1.126 1.127 1.128 1.128 1.129 1.130 1.131 1.131 1.1320.430 1.132 1.133 1.133 1.134 1.135 1.136 1.136 1.137 1.138 1.138 1.1390.440 1.139 1.140 1.141 1.141 1.142 1.143 1.144 1.144 1.145 1.146 1.1470.450 1.147 1.148 1.148 1.149 1.150 1.151 1.151 1.152 1.153 1.154 1.1550.460 1.155 1.155 1.156 1.157 1.158 1.159 1.159 1.160 1.161 1.162 1.1630.470 1.163 1.164 1.164 1.165 1.166 1.167 1.168 1.169 1.170 1.170 1.1710.480 1.171 1.172 1.173 1.174 1.175 1.176 1.176 1.177 1.178 1.179 1.1800.490 1.180 1.181 1.182 1.183 1.184 1.185 1.186 1.186 1.187 1.188 1.1890.580 1.277 1.279 1.280 1.281 1.283 1.284 1.285 1.287 1.288 1.289 1.2910.590 1.291 1.292 1.293 1.295 1.296 1.297 1.299 1.300 1.302 1.303 1.3040.600 1.304 1.306 1.307 1.309 1.310 1.311 1.313 1.314 1.316 1.317 1.3190.610 1.319 1.200 1.322 1.323 1.325 1.326 1.328 1.329 1.331 1.332 1.3340.620 1.334 1.335 1.337 1.338 1.340 1.342 1.343 1.345 1.346 1.348 1.3500.630 1.350 1.351 1.353 1.354 1.356 1.358 1.359 1.361 1.363 1.364 1.3660.640 1.366 1.368 1.370 1.371 1.373 1.375 1.376 1.378 1.380 1.382 1.3830.650 1.383 1.385 1.387 1.389 1.391 1.392 1.394 1.396 1.398 1.400 1.402

Geometry correction factor, FI( ) = Y

The SIF geometry correction factor Y; KI = S Y; (central crack)


Example

A thick center-cracked plate of a high strength aluminum alloy is 200 mm wide and contains a crackof length 80 mm. If it fails at an applied stresses of 100 MPa, what is the fracture toughness of thealloy? What value of applied stress would produce fracture for the same length of crack in:

a) an infinite plate

b) a 120 mm wide plate?

2a

2W


) 1202 40 0.6666

60

2 0.6666 1.413

; 39.42 0.04 1.41339.42 78.70.04 1.413

handbook example

c

c Plate mm widea a

W W

aYW

K K

MPa

)2 200 , 2 80 , 100

; ;

:2 2

2

2 40 0.4100

2 0.4 1.1

handbook example example

handbook example

a Finite width plateW mm a mm MPa

aK a Y Y fW

See notation in the Handbooka a a

W W W

a aW W

aYW

12

100 0.04 1.112 39.42

39.42

)

; 1

; 39.42 0.0439.42 111.20

0.04

c

c

K a Y MPa m

K K MPa m

b Ininitely wide plate

K a Y and Y

K K

MPa



Geometry Effects on the Stress Intensity FactorStress Intensity factors for cracks in a butt weldment and flat plate of the same thickness

S

t

S

a

S

a

t t

S

x

y

0

(x)

t

S

(x)=SIK aaS

0

(a)

a/t 0.5

Y

IK aaS

0

(b)

(a)

a/t 0.5

Y

1

2 2 2

1 1

1 312 2

1 11 1

11 2 3

1 312

2 22 2

2

2

1 2 3

2 1 1 1 12

2, 1 1 1

,

2,

,

1P

A

A

P

P x x xK

a M M Ma a aa

a M M Ma

P x x xK m x Px

m x Px a aa

2121 21 ,, ,,A

PP am xK m xP Pa

0

,A

ax m x a dxK x

y

AP1

x1

x

a

t

x2

P2

(x)

S

S

The Weight Function method for calculating Stress Intensity Factors


A

Px

x

a

W

y

1

1 312 2

1 2 32 1 1 1 1

2, ,P

A a M Mx x Ma a

xK m xa x

Pa

A

P xx

2a

2W

y

x

Geometrical parameters and notation for weight functions


2 3 4 5 6 7

1

2 3

2

0.06987 0.40117 5.5407 50.0886 200.699 395.552 377.939 140.218

0.09049 2.14886 22.5325 89.6553 210.599

a a a a a a aMw w w w w w w

a a a aMw w w w

4 5 6

2 3 4 5 6 7

3

239.445 111.128

0.427216 2.56001 29.6349 138.40 347.255 457.128 295.882 68.1575

a aw w


Central through crack in a finite width plate subjected to symmetric loading

2 3 4 5 6 7

1

8 9 10

2

0.0719768 1.51346 61.1001 1554.95 14583.8 71590.7 205384 356469

368270 208233 49544

0.246984 6.47543


a a aw w w

aMw

2 3 4 5 6 7

8 9 10

2

3

176.457 4058.76 37303.8 181755 520551 904370

936863 531940 127291

0.529659 22.3235 532.074 5479

a a a a a aw w w w w w

a a aw w w

a aMw w

3 4 5 6 7

8

.53 28592.2 81388.6 128746 106246

35780.7

a a a a aw w w w w

aw

Edge crack in a finite width plate


(x)

t

S

a

y

x

(x)

S

at

y

x

S

S

at

y

x

a) b) c)

Ka = Kb

The Weight Function method for calculatingStress Intensity Factors

The Stress Intensity Factor for any loading case is equal to the stress intensityfactor obtained by applying to the crack faces the stresses that used to be therewhen there was no crack.


Stepwise Procedure for the Stress IntensityCalculation using the Weight Function Method

1. Calculate stress distribution (x) in the prospective crack plane in theabsence of the crack (un-cracked body, linear elastic analysis).

x f x0 ,

2. Apply the stress distribution (x) to the crack surface as tractions.

x x xm x a M M Ma a aa x

1/2 1 3/2

1 2 32( , ) 1 1 1 1

2

4. Integrate the product of the stress distribution (x) and the weight function m(x,a).

a

a

K x m x a dx( , )

3. Choose appropriate weight function, i.e. parameters M1, M2 and M3.


The superposition principle for calculation of stressintensity factors using the weight function approach;

r

t

T2

o

SProspective crack plane (y)

a)

x

y

t

T2

(y)

crack

b)

y

x

a

IaK m y y dyt

0

,

a) stress distribution in the prospective crack plane in the un-cracked body;b) the “un-cracked stress field” applied to the crack surfaces of identical

body with a crack;


Through the plate thickness stress distributions in a T-butt weldmentobtained for r/t = 1/25, = 45o (in the weld toe cross section)

(y)/

n

y / T

FEM


Geometrical Stress Intensity Correction Factor “Y” foran Edge Crack Emanating from the Weld Toe

(Comparison of WF and FEM data)

T-butt welded joint; Tension loading T-butt welded joint; Bending load


Calculation of SIF for cracksat notches using the weightfunctions for edge andthrough cracks

(x)

(x)

(x)

(x)

t 2t

d b

b

2aa

b b

(x)

b b


Fracture Mechanics Approach to Fatigue CrackGrowth Analysis

•Fatigue crack growth equations

•Integration of fatigue crack growth expressions

•The effect of the initial crack size

•The effect of the weld geometry

•Residual stress effect

•Example


A sharp crack in a tension stress field causes a high stressconcentration at the its tip resulting in slip and plastic deformation inthe crack tip vicinity. The material above and below the crack tip mayslip along a favorable slip plane in the direction of maximum shearstress.

(a)

(b)

(c)

(d)

(e)Time

Stre

ss

0

(C. Laird, ASTM,STP 415, 1967)

(R.Pelloux, ASTM,STP 415, 1967)

a

b

c

d

ea

Fatigue Crack Growth Micro-Mechanism


The ‘a vs. N’ data is obtained inpractice by periodic measurement ofthe crack length, a, together with thenumber of cycles, N. The raw data isusually given in the form of series ofpoints as shown in the figure.

Stre

ss,S

Time, t0

SNumber of applied cycles, N0

Cra

ckle

ngth

,a

a0

S1 > S2 > S3

S3

S2S1

N

a

af , Nf

da/dN= a/ N

Applied nominal stress history

Experimental data for the determination of the fatigue crack growth curve

Number of applied cycles, N

ai, Ni

da/dN

- Experimental test dataS

S

a

Cra

ckle

ngth

,a

0

a0


The Fracture Mechanics approach to fatigue or the da/dN - K method is a technique based on theanalysis of fatigue crack growth. The combination of load/stress and geometry parameters, necessaryfor the quantification of damage due to crack growth, is represented by the stress intensity factor, K,in the case of monotonic load and by the range of the stress intensity factor, K, in the case of cyclicloading.The fatigue material properties are characterized by the threshold stress intensity range, Kth, thefatigue crack growth rate relationship, da/dN vs. K, and the critical stress intensity factor, Kc, to beoften the same as the fracture toughness, KIc. The crack growth rate is then described by anexpression being function of the stress intensity range:

da f KdN ,

The stress intensity range associated with a stress cycle is calculated as:

max min max minK K K S a Y S a Ywhere – a is the crack size, Smax and Smin is the maximum and minimum nominal (or reference) stressrespectively, characterizing a stress cycle, and Y is the geometry correction factor. The aim of thefinal analysis of the da/dN- K data is to determine necessary constants and parameters appearing inexpression f( K).It should be noted that the ‘da/dN - K’ curve in fracture mechanics represents the material fatigueresistance similarly to the S-N curve in the nominal stress approach or the ‘ - N’ relationship in thelocal strain-life methodology.As soon as the crack growth curve for the material of interest is known the fatigue life of the structuralcomponent can be determined as shown in the figure below.


The notation for the cyclic stress history parameters and the steps necessary for the determination ofthe da/dN - K relationship are explained later in the following sections of the notes.

The fatigue life in terms of the number of cycles necessary to propagate the crack from its initial size,a0, to the final or critical crack size, af, is determined by integrating the crack growth equation.

ffaa

a a

da daNf K f S a Y

0 0

The determination of the integral above needs a numerical treatment because the geometrycorrection factor, Y, becomes frequently a complex function of the crack size, a.

Subsequent stages of the fatigue life prediction method based on the crack growth analysis areshown graphically in the Figure.


Constant Amplitude Cyclic Load - Notation

Smin - minimum stress

Smax - maximum stress

S = Smax- Smin - stress range

Sa = S /2 = (Smax- Smin )/2 - mean stress

R = Smin/ Smax - stress ratio

Smax

Smin

Stre

ss,

S

Sm

Time, t

S

S a

1 cycle

0


S

and

K S a Y

K S a Y

Fatigue Crack Growth Rate vs. Stress Intensity Factor


Scatter of fatigue crackgrowth data; Low alloy steel

18G2VA

4 mm thick plate 4 mm thick welded plate


The Fatigue Crack Growth Expression – The Paris equationThe first mathematical relationship relating fatigue crack growth rate and the stress intensity range wasproposed by Paris and Erdogan. This relationship is up to date the most popular mathematical expression usedif various fatigue/fracture mechanics analyses. It was obtained by fitting power law curve into the experimentaldata.

mda C KdN

Where: da/dN - fatigue crack growth rate [in/cycle or m/cycle]C - Paris’ equation parameter (valid for given R)m - Paris’ equation exponent

K - stress intensity range

max min min

max min

max max

min min

00

K K K for KK K for K

K S a Y

K S a Y

Where:a - crack length/depthSmax - maximum stress in a stress cycleSmin - minimum stress in a stress cycleKmax - maximum stress intensity factorKmin - minimum stress intensity factorY - geometry correction factor in the stress intensity factor expression


Complete Fatigue Crack Growth Rate Curve, da/dN - KSoon after the Paris equation gained wide acceptance as a tool for fatigue crack growth analysis, it wasfound that the simple expression proposed by Paris and Erdogan had some limitations. As the Figurebelow illustrates the complete log-log plot of da/dN vs. K is sigmoidal rather then linear and limited by thethreshold stress intensity range, Kth, and the critical stress intensity factor Kc.

At low growth rates, the da/dN vs. K curve becomes steep and appears to approach a vertical asymptotedenoted Kth, which is called the fatigue threshold stress intensity range or fatigue crack growth threshold.This quantity is interpreted as a lower limiting value of the stress intensity factor range K below whichfatigue crack growth does not ordinarily occur. The fatigue crack growth threshold is analogous to thefatigue limit in the S-N approach.

At high growth rates, the da/dN vs. K curve may again become steep. This is due to rapid unstable crackgrowth just prior to final fracture when Kmax Kc. The increase of the fatigue crack rate near the criticalstress intensity factor Kc is due to mixture of static (monotonic -fracture) and fatigue mechanisms drivingthe crack growth.

Also, the fatigue crack growth rate exhibits a dependence on the stress ratio ‘R’. The stress ratio R affectsthe fatigue crack growth rate in a manner analogous to the effects observed in the S-N and -N methods,i.e. for a given K, increasing R-ratio increases the fatigue crack growth rate, and vice-versa.

The effect of the R -ratio (or mean stress) on Fatigue Crack Growth is most often explained using thephenomenon discovered by Elber. By measuring the compliance of specimens with fatigue cracks henoticed that the crack tip got closed during the descending part of the stress cycle in spite of the fact thatthe applied stress/load remained tensile (see Figure). Elber postulated that crack closure decreases thefatigue crack growth rate by reducing the effective stress intensity range.


5 10 20 50 100

5 10 20 50 100

210

310

410

510

610

710

310

410

510

610

710

810

Cra

ckG

row

thR

ate,

da/d

N[m

m/c

ycle

]

da/d

N.C

rack

Gro

wth

Rat

e,da

/dN

[inch

es/c

ycle

]

K, Stress Intensity Range, MPa m

K, Stress Intensity Range, [ksi in]

A533B-1 steel,u=627 MPa

R=0.1

mda C KdN

thK

(a)

(b)

(c)

Fatigue crack growthrates for a ductile

pressure vessel steel(the Paris equation)

mda C KdN

Source: N. Dowling, ref(2)


The da/dN- K curve is thefatigue material curve

independent of the geometry,i.e. the same curve for allgeometrical crack-body

configurations!

For simplicity reasons thecomplete fatigue crackgrowth rate is usuallyapproximated by three

linear pieces with the twoof them being verticallimiting asymptotes.


3.0126.9 10da KdN

2.25101.4 10da KdN

3.25125.6 10da KdN

Ferritic-Pearlitic Steel:

Martensitic Steel:

Austenitic Stainless Steel:

J. Barsom, “Fatigue Crack Propagation in Steels of Various Yield Strengths”Journal of Engineering for Industry, Trans. ASME, Series B, Vol. 93, No. 4, 1971, 1190-1196

5 10 100

10-7

10-6

10-8

Cra

ckG

row

thR

ate,

m/c

ycle

K, MPa m

ys252273392415

for: da/dN in [m/cycle] and K in [MPa m]

Paris’ equation constants for steel materials at R = 0


Estimation of the Fatigue Crack Propagation LifeBasic Steps:

1. Estimate the initial crack size and shape, ao;

- non-destructive testing - ao

- proof load - ao

2. Estimate the critical crack size ac based on the fracture toughness KIC, i.e. the crack sizethat the component will tolerate when the applied stress reaches its maximum Smax.

3. Using the same expression for the stress intensity factor calculate the stress intensityrange K.

2

maxmax

1 ICIC c c c

c

KK S a Y aS Y

max

0

0 ( 0!!)r

K S a Y for R

K S a Y for R if


4. Substitute K into fatigue crack growth equation (Paris or Forman)

( ) ( )m m mda C K C S a YdN

5. Integrate the equation above from a = ao to a =ac and determine the number ofcycles, N, necessary to grow the crack from the initial crack size of ao to the criticalsize of ac. This is the estimated fatigue crack propagation life of given component!

( )

( ) ( )

cc

o o

m

aa

m

a a

dadNC K

da daNC K C S a Y

Note! In most practical cases the integration requires numerical solution due to thecomplexity of the geometric factor Y.


Integrated Paris’ Equation for a Constant GeometricFactor, Y = const.

/ 2 ( 2) / 2 ( 2) / 2

2 2

( )

2

2 1 1 ;( 2) ( )

21 ln ;

m m m mc

m m

o

c

o

da C K C S aY

for m

Nm C SY a a

for ma

NaC S

dN

Y


0 0 0 0

1 1 1 1 0 0 1 0 1

2 2 2 2 1 1 2 1 2

3 3 3 3 2 2 3 2 3

4 4 4 4 3 3 4 3 4

0 0 ;

1; 1; ; ;

1; 1; ; ;

1; 1; ; ;

1; 1; ; ;

.........................................

m

m

m

m

N a a a

N N a C S a Y a a a

N N a C S a Y a a a

N N a C S a Y a a a

N N a C S a Y a a a

1 1 1

..................................................

1; 1; ; ;m

i i i i i i i i i

i c

N N a C S a Y a a a

until a a

Calculations have to becarried out for each cycle !!

Numerical Integration of the Paris EquationIf the Y factor is not constant a numerical technique has to be applied. The mostoften used is the cycle by cycle technique based on the calculation of crackincrements ai corresponding to each load cycle. In this case, the infinitesimalincrements da and dN are replaced by finite differences a and N= 1.

1 1 11

( ) ( ) ; ; ( )N

m m mii i i i i o i i i i i i

ii

a C K C S a Y a a a a C S a Y NN


Subsequent stages of fatigue life prediction method based on the crackgrowth analysis

Analysis of external forces acting on the structure and the component in question (a), Analysis of internal loads in chosen cross section of a component ( b), Selection of individual welded joints in the structure (c), Identification of appropriate nominal or reference stress history (d), Extraction of stress cycles (rainflow counting) or reversals from the stress history

(Fig.e), Determination of the stress intensity factor (i.e. the factor Y) for postulated or existing

crack,- indirect method (Fig.f):

analyze un-cracked weldment and determine the stress field, (x,y), in theprospective crack plane; normalize the calculated stress distribution with respect tothe nominal or any other reference stress, i.e. (x,y)/ n,

choose appropriate weight function,calculate stress intensity factor

determine the stress or displacement field near the crack, or the strain energyrelease rate,

calculate stress intensity factor using. Determination of crack increments for each stress cycle (Fig. h), Determination of the number of cycles, N, necessary to grow the crack from its initial

size, a0, up to the final size, af.

A summary of necessary input data and procedures used in the, da/dN - K, approachto fatigue life estimation is also presented in the Figure.


Example: A very wide SAE 1020 cold-rolled thin plate is subjected to constant amplitude uni-axial cyclicloads that produce nominal stresses varying from Smax=200MPa (29ksi) to Smin=-50 MPa (-7.3ksi). Themonotonic properties for this steel are Y=630 MPa (91 ksi), uts=670 MPa (97 ksi), E=207000 MPa(30000 ksi), Kc =104 MPa m (95 ksi in). What fatigue life would be attained if an initial through-thicknessedge crack existed and was 1 mm (0.04 in) in depth?

a

W >> a

S

S

Smax

Smin

time

Stre

ss

0

The fatigue crack growth data are:Kth(r=0)=6 MPa m, and Paris’ equation

parameters C=6.9 10-12 and m=3.


B. Is Linear Elastic Fracture Mechanics

(LEFM) applicable?

max

max max

2 2max

:0.8 0.8 630 504 !

:

200 0.001 1.12 12.6

1 1 12.6 0.0000635 0.06352 630

0.0635 1 0.125 !1 8

Y

yY

y

Nominal stress levelS MPa YES

Plastic zone size

K S a Y MPa m

Kr m mm

rYES

a

A. What is the stress intensity factor expression?

Semi-infinite plate with an edge crack.

max max max 1.12K S a Y S a


C. The effective stress range

max min min

max min

max min

max

00

200 50

200

S S S for SS S for S

S MPa and S MPathus

S S MPa

D. Is the Paris equation applicable?

0 0

0 0

0

!0.001 .

200 0.001 1.12 12.6

12.6 6 , !

th

th

Paris equation is valid for K KSmallest K K occurs for a a m

K S a Y MPa m

K K MPa m YES Paris equation is applicable


E. What is the critical/final crack size?

max

2 2

max

1 1 104 0.068 68200 1.12

c final c

cc

K K S a Y

Ka m mmS YE. Integration of the Paris equation

Analytical integration is possible because Y= const.

000

2 2

/ 2 ( 2) / 2 ( 2) / 20

3 3 / 2 (3 2) / 2

( )

1( )

2

2 1 1 ;( 2) ( )

2 1 1(3 2) 6.9 10 12 (200 1.12) 0.001 0.06

c c c

m m

a a a

m m m m mm

aaa

m m m mc

da C K C S aYdN

da da daNC S a YC S aYC K

for m and Y const

Nm C S Y a a

N (3 2) / 2

( 2) / 2 ( 2) / 20

8

1 1 1 14631 4631 4631 31.645 3.834 1287920.0316 0.2608m m

c

cyclesa a


© 2007 Grzegorz Glinka. All rights reserved. Page 45

Fatigue Crack Growth under Variable Amplitude Loading:the retardation effect

a

m

max

da C KdNda f K ,KdN

46

Evolution of the crack tip plastic zone ahead ofa fatigue crack & crack tip closure

A

B

CD

Time, t Time, t Time, tTime, t0 0 0 0

Stre

ss,

S

Stre

ss,

S

Stre

ss,

S

Stre

ss,

S

Crack tipopen

Crack tipstarts to close

Extended cracktip closure

Crack tipstarts to open

Crack Crack

Crack tipplastic zone

Envelope of plasticallydeformed material in thewake of a growing crack


Cyclic plastic zonerc

ys

ys

Crack

Stre

ssSt

ress

rp Monotonic plastic zone

x

xCrack

2

max1p

ys

Kr

21

4cys

Kr

at: A

at: B

Time

K

0

A

B

Kmax

Kmin

K

0

A

B

Strain,

Stre

ss,

The stress-strain evolution and the monotonicand plastic zone ahead of a fatigue crack tip


The effect of the crack tip closure

0

Kth

,eff

?

Kef

f

Kth

K

Kmax

Kth

Kopn

Kmin

Stre

ssin

tens

ityfa

ctor

,K

Time, t

min

max

max

11

eff

opn

K

KRKKK

U

or

UR

K


Fatigue Growth of Corner Cracks in a Lug Subjected toa VA Loading History

a

d

2W

(x)

r

F

x

a+r

(x)

rx

Experimental data from: Jong-Ho Kim, Soon-Bok Lee, Seong-Gu Hong, Int. Journal of Fatigue, vol. 40, 2003


The Lug Loading Histories, Pmax=21 kN

0

2

4

6

8

10

12

14

16

18

20

0 500 1000 1500 2000

No. of Reversals

Load

/For

ce[k

N]

90% Clipping

100% Clipping

80% Clipping

The VA Load/Stress HistoryThe VA Load/Stress History


0

2

4

6

8

10

12

14

16

18

20

0 500 1000 1500 2000

No. of Reversals

Load

/For

ce[k

N]

90% Clipping

100% Clipping

80% Clipping


0

2

4

6

8

10

12

14

16

18

20

0 500 1000 1500 2000

No. of Reversals

Load

/For

ce[k

N]

90% Clipping

100% Clipping

80% Clipping

The VA Load/Stress HistoryThe VA Load/Stress History


Fatigue life for 80% and 100% Clipped Loading History+Load Shedding: Al7050 T7

0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Number of blocks

Cra

ckle

ngth

[mm

]

80% Clipped Experimental

80% Corner+EdgeRestrained+Shedding_SP=A045B023q065



Calculations vs. ExperimentCalculations vs. Experiment


Fatigue life for 80% and 100% Clipped Loading History+Load Shedding: Al7050 T7

0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Number of blocks

Cra

ckle

ngth

[mm

]





Calculations vs. ExperimentCalculations vs. Experiment



Crack Shape Evolution;Crack Shape Evolution; quarter circular initial crackquarter circular initial crack

Predicted vs. Measured Crack Shape Evolution in the Lug,100% Clipping

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Surface, a [mm]

Dep

th,b

[mm

]

Predicted vs. Mesured Crack Shape Evolution in the Lug,80% Clipping;

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Surface, a, [mm]

Dep

th,b

,[m

m]


Crack Shape Evolution;Crack Shape Evolution; quarter circular initial crackquarter circular initial crack


0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Surface, a [mm]

Dep

th,b

[mm

]


0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Surface, a, [mm]

Dep

th,b

,[m

m]


0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Surface, a [mm]

Dep

th,b

[mm

]


0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Surface, a, [mm]

Dep

th,b

,[m

m]



2

( , ; )

2i

j i

i

P A

A A

c

c

b

K m x y P

P

Pj(x,y)

ci

3

2

1in

PjAiAi

bi

A1

A2

A3

n

i

1

2

3

Pj(x,y)

Ai

c - inverted contour ofthe crack front;

b - inverted contour ofthe external boundary;

PjAi - distance betweenthe point load P and pointA on the crack front

Weight Function for Arbitrary Planar Cracks


Location of maximumshear stress

Fig. 10. Geometry and dimensions of the spring

D=18.5 mm, d=4 mm


Relative dimensions of the inclusion (d=20-30 m) and the finalcrack size (2af= 700 m)


3 3 3 2

1616 8 82 4.625ref torque

DFT FD Fd d d d

2-D Stress Field in the Spring Critical Cross Section and the Locationof the Initial Crack (non-metallic inclusion)

1.0B ref

0.85A ref

1.55C ref

0 0.14 ref1.0D ref

C

D

0

A

B

ref

0.4 ref

time, t0


-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

BoundaryStart10000002000000300000040000005000000510000052000005250000527000052800005290000529500052960005296900

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-3 -2 -1 0 1 2 3

Fatigue crack growth; d=0.03x0.02 mm, depth 0.25 mm, C, max= 1030 MPa, C, min= 390 MPa

Nf = 5,296,900 cycles

Edge of thecross section


Main steps in fatigue design – flow chart

Data analysis

Collecting data SimulationSelecting criteria

Designing Stress relive!

31 2

3

4

5

Start

TheEnd !


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Documents

The Fracture Mechanics Method (da/dN- û - eFatigue · A crack becomes unstable (fracture) when the stress intensity factor, K I, exceeds the critical, for given material, stress