8) Recent developments in the fatigue crack growth analysisutmis.org.loopiadns.com/media/2016/08/Weldments6_Fatigue... · 2018-11-02 · fatigue cracks Use of appropriate SIF using

© 2008 Grzegorz Glinka. All rights reserved. 536

8) Recent developments in the fatigue 8) Recent developments in the fatigue crack growth analysiscrack growth analysis• the UniGrow fatigue crack growth model for spectrum loading


FATIGUE CRACK GROWTH ANALYSES BASED ON FATIGUE CRACK GROWTH ANALYSES BASED ON THE THE ‘‘UniGrowUniGrow’’ TWO PARAMETER FATIGUE CRACK TWO PARAMETER FATIGUE CRACK

GROWTH MODELGROWTH MODEL

G. GlinkaUniversity of Waterloo,

Mechanical and Mechatronics Engineering

Waterloo, Ontario,

Canada


•The crack is modeled as a sharp notch with finite tip radius ρ*.

•Fatigue crack growth is regarded as successive crack increments (re-initiation) over distance ρ*.

•The number of cycles N*necessary to break the material over the distance ρ* can be determined from the cyclic (Ramberg-Osgood) and fatigue material curve (Manson-Coffin) obtained experimentally from smooth specimens.

•The instantaneous fatigue crack growth rate can be determined as:

dadN N

ρ∗

∗

=

Fatigue Crack Growth ModelFatigue Crack Growth Model

σr(x)

ρ* ρ*

2a

S

ρ*

aS

S


ρ*

ρ*

Idealized discrete structure of the material and Idealized discrete structure of the material and the continuum mechanics based stress fieldthe continuum mechanics based stress field

σyy

xρ*

ρ* ρ*ρ* ρ*

a)

crack

σyy (x)


The principal idea of the local stressThe principal idea of the local stress--strain approach to strain approach to fatigue life prediction methodfatigue life prediction method

0

Stress -Strain Curve

ε

σ

E

x

σpeak

Stre

ss

y

S

S

ρ∗

log

(Δε/

2)

log (2Nf)

σf /E

εf

0 2Ne

σe/E

Strain - Life Curve

c

b

2N*

( )( )

peak

peak

f , geometry,mtl . properties

g , geometry,mtl . properties

S

S

σ

ε

=

=

Need to be determined !!

NdadN

ρ ∗

∗=

σ, ε

x

y

z

σ, ε


Phenomena to be accounted for in the case of Phenomena to be accounted for in the case of elasticelastic--plastic stressplastic stress--strain analysis of growing strain analysis of growing

fatigue cracksfatigue cracksUse of appropriate SIF using Handbook or any other solution, Kappl.

Determination of the linear elastic crack tip stresses σelast. from applied stress intensity factor Kappl is relatively easy (Creager-Paris solution).

However, FOUR important phenomena need to be additionally accounted for while determining the elastic-plastic strains ad stresses around the crack tip, i.e.

•the non-linear elastic-plastic material behaviour (Ramberg-Osgood cyclic stress-strain curve),

•material memory while applying cyclic loading (the effect of the load path or the sequence of applied loading cycles),

•the interaction between the crack tip plastic zone and the surrounding linear elastic stress field,

•the structural memory associated with the propagation of the crack tip (assessment of the length of the past stress history affecting the crack tip at the current position).


Approximate crack tip geometry, the cyclic plastic Approximate crack tip geometry, the cyclic plastic zone and the crack tip stresszone and the crack tip stress--strain responsestrain response Global (remote) stress history, S

Local crack tip stress and strains

Rem

ote/

Glo

bal S

tres

s ‘S

’

0

time

A

D DD

C’B’

A’A’ A’

D

B’’

C’’BC

A

B’’

C→S=0, σr ≠ 0D

C’

B’

0

σtot

εtot

A’

B

C’’

SA > SB >0B)

Cyclic plastic zone

Sc =0

C)

Cyclic plastic zone

SD < 0D)

Cyclic plastic zone

SA > 0

crack

σtot, εtot

A)

Cyclic plastic zone


Geometrical/Mathematical Geometrical/Mathematical model for linear elastic stressmodel for linear elastic stress--strain analysis of fatigue crackstrain analysis of fatigue crack

2 2min,

11,min,

22,mi

2 2

2 4mi

4nn,

2,

3 * *12

* 3 *22

applnet

neappl

t

Kx xa

Kx xa

σ

σ

ρ ρπ

ρ ρπ

⎛ ⎞= ⋅ −⎜ ⎟

⎝ ⎠⎛ ⎞

= ⋅ + +⎜ ⎟⎝ ⎠

S-min

2ρ* 2ρ*

min,netσ −min,netσ −

S-min

time0

Kap

plor

S

K or S+ +Δ Δ

K or S− −Δ Δ

Kmax,appl or Smax

Kmin,appl or Smin

( )

( )

max,

m

11,m

ax,

ax,

22,max,

1 ....2 0.52

1 ....2 0.52

apnet

net

pl

appl

K

xx

K

xx

σ

σ

ρρπ

ρρπ

∗

∗

∗

∗

⎛ ⎞⎜ ⎟= − + +⎜ ⎟−⎝ ⎠⎛ ⎞⎜ ⎟= + +⎜ ⎟−⎝ ⎠

Compression

Creager & Paris, valid for x>½ρ* !

S+max

2a

ρ* ρ*

max,netσ +max,netσ +

S+max

Tension

2

1

½ ρ*


Hypothetical linearHypothetical linear--elastic stress field near a sharp notch elastic stress field near a sharp notch ((CreagerCreager--Paris solution)Paris solution)

( )

( )

( )

max,max

max,max

,max

1

122

122

0

xx

yy

xy

st reversal

Kx

xx

Kx

xxx

ρσπ

ρσπ

τ

∗

∗

⎛ ⎞= − +⎜ ⎟

⎝ ⎠⎛ ⎞

= +⎜ ⎟⎝ ⎠

=

( )

( )

( )

2

122

122

0

xx

yy

xy

nd reversal

Kxxx

Kxxx

x

ρσπ

ρσπ

τ

∗

∗

⎛ ⎞ΔΔ = − +⎜ ⎟

⎝ ⎠⎛ ⎞Δ

Δ = +⎜ ⎟⎝ ⎠

Δ =

xxi0

σyy(ρ/2)

a

xi+1

σxx,i

σyy,i

σyy(x)

σxx(x)

ρ*


⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= +'

1

'n

E Kσσε

The INPUT: The cyclic stressThe INPUT: The cyclic stress--strain and fatigue strainstrain and fatigue strain--life curves life curves obtained from smooth cylindrical specimens tested under cyclic obtained from smooth cylindrical specimens tested under cyclic

strain controlled strain controlled uniuni--axial stress stateaxial stress stateε

σσ

0


ε

σ

Elo

g (Δ

ε/2)

log (2Nf)

σf /E

εf

0 2Ne

σe/E

Strain - Life Curve

c

b

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Δ = +'

'2 22

b cf N Nf f fE

σε ε


Fi

σija,εij

a

PlasticZone

Calculation of the crack tip stresses and strain Calculation of the crack tip stresses and strain ––the the elastoelasto--plastic analysis (the Neuber rule)plastic analysis (the Neuber rule)

( ), ,, ,

, ,

a aij tot ij t

e eij ne ot

a aij tot ij to

t ij net

tf

σ ε

ε σ

σ ε⎧ =

=⎪⎨⎪⎩

Multiaxial stress state

'

, ,

,

,

,, '

, 22 22

1

22 2222

22 22a a

tot tot

a a ntot tota

e ene

to

t net

t E K

σ ε

σ

σ

σε

ε

⎛ ⎞= + ⎜ ⎟⎜ ⎟

⎧ =⎪

⎝

⎨

⎠

⎪

⎪⎪⎩

Uniaxial stress state

εnet

netnet E

σε =

σnet

E

0

Fi

σij,tot, εij,tot

Elastic body

εtotσ

tot

E

( )tottot totf

Eσ

ε σ= +

0


NeuberNeuber’’s rules rule ESED methodESED method

( )ε

σ ε

σ

ε σ

σ

⎧ =⎪⎨

= +⎪⎩

22 22

2222 22

22 22a a

aa a

e e

fE ( )

ε ε

σ ε

σε σ

σ ε⎧

=⎪⎪⎨⎪

= +⎪⎩

∫ ∫22 22

22 22

22

22 22

22 22

0 0

e a

a a

aa

e e

a

d d

fE

ε22e ε22

a

σ22e

σ22a

0

σ

ε

B

A

0

σ

ε

B

A A

ε22e ε22

a

σ22e

σ22a


{ }

{ }

11 11 22 33 11 22 33

22 22 33 11 22 33 11

33 33

( )1 1( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2

( )1 1( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2

1( )

aeqa a a a a a a

aeq

aeqa a a a a a a

aeq

a a

f xx x x x x x x

E x

f xx x x x x x x

E x

xE

σε σ ν σ σ σ σ σ

σ

σε σ ν σ σ σ σ σ

σ

ε σ

=

=

⎡ ⎤ ⎧ ⎫⎣ ⎦⎡ ⎤ ⎡ ⎤− + + − +⎨ ⎬⎣ ⎦ ⎣ ⎦⎩ ⎭

⎡ ⎤ ⎧ ⎫⎣ ⎦⎡ ⎤ ⎡ ⎤− + + − +⎨⎭

=

⎬⎣ ⎦ ⎣ ⎦⎩

{ }11 22 33 11 22

11 11

22 22

33 33

11 11

22 22

33 33

( ) 1( ) ( ) ( ) ( ) ( )

( ) ( )

( ) (

( )( ) 2

( ) ( )

( ) ( )

( ) () ) )

)

( (

aeqa a a a a

a

e e

e e

e

e

e

q

a a

a a

a a

f xx x x x x x

x

x x

x x

x

x

x x

x x x

x

σν σ σ σ

σ ε

σ σσ

σ ε

σ ε

σσ ε

σ ε

ε

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪ =

⎡ ⎤ ⎧ ⎫⎣ ⎦⎡ ⎤ ⎡ ⎤− + + − +⎨ ⎬⎣ ⎦ ⎣

⎪⎪ =⎪⎪ =⎩

⎦⎩ ⎭

⋅

⋅

⋅

⋅

⋅

⋅

Equations of the multiaxial NeuberEquations of the multiaxial Neuber’’s rule and s rule and HenckyHencky’’ss equations of plasticityequations of plasticity

1'

2 2 2

11 22 22 33 33 11

( )( )

'

1( ) ( ) ( ) ( ) ( ) ( ) ( )

:

2

naeqa

eq

a a a a a a aeq

xf x

K

x x

wher

x

e

an x x x xd

σσ

σ σ σ σ σ σ σ

⎡ ⎤⎡ ⎤ = ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − + − + −⎣ ⎦ ⎣ ⎦ ⎣ ⎦


The hypothetical linear elastic and the actual The hypothetical linear elastic and the actual elasticelastic-- plastic strains and stresses in the notch plastic strains and stresses in the notch

tip regiontip region

0 < R < 0.5Kmax appl

Kmin applApp

lied

load

his

tory

F

0 Time

xxi0

σyy(ρ/2)

a xi+1

σxx,i

σyy,i

σyy(x)

σxx(x)

ρ*

F

F

The hypothetical linear elastic stress field

The elastic and actual elastic-plastic distribution of the stress component σyyin the crack/notch tip region.

σmax,tot

Stre

ss n

ear t

he c

rack

tip

σyy

σmin,tot

0

σr(x)Distance from the notch tip, x

σmax, elast.

σmin, elast.


The procedure for calculating σmax,tot and Δεtot is the same as in the case of calculating the residual stress but it is carried out using the total stress intensity factors.

Kmax, tot and ΔKtot → Neuber rule → σmax,tot and Δεtot

Smith-Watson-Topper (SWT) parameter

SWT = σmax,tot · Δεtot/2

The number of cycles to failure Nf:

( ) ( ) ( )2

ma

'2 ' '

x, 2 22

b b cff f

tott f fo ftSWT N N N

Eε

εσσ

σ+

=Δ

= + ⇒

Fatigue crack growth rate da/dN

*

f

dadN N

ρ=


∗∗

⎧ ⎛ ⎞⋅⎧ ⎪⎛ ⎞ =⎜ ⎟⋅⎪ ⎜ ⎟⎪=⎜ ⎟ πρ⎝ ⎠⎜ ⎟⎪ ⎪πρ⎪ ⎪⎝ ⎠⎨ ⎨⎪ ⎪⎛ ⎞⎪ ⎪ ⎛ ⎞= + ⎜ ⎟ = +⎪ ⎪ ⎜ ⎟⎝ ⎠⎩ ⎝ ⎠

ΔΔσ ⋅ Δε

σ ⋅ ε

σ σΔε Δσ Δσε

⎪⎩

22 1

1

11

11

22

2 2 2

''

tottot totmax,tot

max,tot max,tot

max,tot max,tottot tot totmax,t t

'o

nn

'

xx

EE

K

E K

K

E K

Complete set of equations for calculating fatigue crack growth rComplete set of equations for calculating fatigue crack growth rateate(constant (constant amplidueamplidue loading)loading)

Maximum stress and strain at the crack tip

Fatigue strain – life curve

( ) ( ) ( ) +Δεσ = + ε

σσ

2'2to bt

max,b c' '

f ftot ff

f2N2 E

2N

Stress and strain range at the crack tip

ρ ∗=

f

dadN N

Fatigue crack growth rate


'

'

max,max, max,

max,

2

1

1

'

2

1

1

max,max

'

,

12

12

2 2 2

tottot tot

tot tottot

tottot tot

tot tot to

n

t n

xE

E K

E

E K

K

K x

σ ε

σ σε

σ ε

ε σ σ

πρ

πρ

∗

∗

⎧ ⎛ ⎞⋅⎪ ⎜ ⎟ =⎜ ⎟⎪⎪ ⎝ ⎠

⎨⎪ ⎛ ⎞⎪ = + ⎜ ⎟⎪ ⎝ ⎠⎩

⎧ ⎛ ⎞⋅⎪ ⎜ ⎟ =⎜ ⎟⎪ ⎝ ⎠⎪⎪

⎨⎪⎪ ⎛ ⎞= +⎪ ⎜ ⎟

⎝ ⎠⎪⎩

⋅

ΔΔ ⋅ Δ

Δ Δ Δ

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

'

'''

'

'

'

'

'''

'

' '

'

1 12'1 1

12 11

1 1'

1 12'1 1

1`2 11

1 1

2max, max,

2max,

'

max,

2

2

2

2

24

24

nn

nnn

nn

tot tot

t

nn

nnn

n n

ot tot

tot tot

t t tn

ot o

K xE

xEK

K xE

K

K

Kx

E

K

K

π ρ

π ρ

π ρ

π

σ

ε

σ

ερ

+

+∗

++

∗

+

+∗

++

∗

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟

⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟

⎜⎠

Δ

⎟⎝

Δ

Δ Δ

Calculation of the actual (total) Calculation of the actual (total) elastoelasto--plastic maximum stress, plastic maximum stress, σσmax,totmax,tot, and strain, , and strain, εεmax,totmax,tot,, over the first element, over the first element, ρρ**,, near the crack tip near the crack tip

(constant (constant amplidueamplidue loading, analysis based on the plastic strain evolution)loading, analysis based on the plastic strain evolution)

Maximum stress and strain at the crack tip

Fatigue strain – life curve

( ) ( ) ( )'

2'

m x,

2

a'2

22

b ctottot f f

bff fN

ENσ

εσ

σε

+= +

Δ

Stress and strain range at the crack tip

fNf

dadN N

ρ ∗=


( ) ( ) [ ]m

1

ax,tot t

p

op

tKd C CdN

Ka γγκ

−⎡ ⎤= =⎢ ⎥⎣Δ

⎦Δ

( )'

'

'

1

21*

3* ' '1

1 ;1

2 ;

2

2

b c

nn

f f

pn

b c

C

E

γ

ψρ

π ρ σ ε

−+

+

+

=+

= −+

⎛ ⎞⎜ ⎟

= ⎜ ⎟⎜ ⎟⎝ ⎠

( )( )

122

1*2* '

0.5;1 ;

24

b

f

p

b

C

γ

ψρ

πρ σ

−

=

= −

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

Near threshold Region I Plastic strain dominated region II

Fatigue crack growth expression based on the local crack tip Fatigue crack growth expression based on the local crack tip stressstress--strain history and the SWT fatigue damage parameterstrain history and the SWT fatigue damage parameter


The approximate analytical analysis resulted in two different crack growth expressions depending on the nature of the dominating strain at the crack tip (i.e. elastic or plastic).

Those expressions can be presented as straight lines in log-log co-ordinates!

Fatigue crack growth curves: da/dN=f(Δκ)

Note that exponent p varies!

( ) ( )1

max,

p ptot totK Kκ

−Δ = Δ

Log(

da/d

N)

Log(Δκ)Δκth

γ

Elastic strain ignored

Plastic strain ignored

( ) ( )0.5 0.5max, tot totK KκΔ = Δ

Both strain terms included


Evolution of the crack tip plastic zone ahead of a stationary anEvolution of the crack tip plastic zone ahead of a stationary and fatigue crackd fatigue crack

Pmax

Time, t

Load

, P

0 Time, t

Load

, P

0Crack length ‘a’ Crack length ‘a’

Pmin=0

Stationary crack of length ‘a’ under load ‘P’

Crack tip plastic zone at P=Pmax

P=Pmin

Pmax

Pmin=0

Crack tip plastic zone at P=Pmax

Envelope of plastically deformed material in the wake

of a growing fatigue crack

Fatigue crack of length ‘a’ under load ‘P’

Pmin=0


uy (COD) vs. x (for various P), R2 - unloading

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

5.8 5.85 5.9 5.95 6

Distance from the crack tip, x [mm]

CO

D, u

y [m

m]

P=3800P=3420P=3040P=2660P=2280P=1900P=1520P=1140P=760P=380P=0

P

P

x0

Unloading S2Unloading S2--S3S3

Pmax=3.8 kN

0Pmin=0

2Pmax=7.6kN

P

S1

1

2

3

S

S2

S3

S5

S7

S6 S8

S9

S10

Crack tip position


Hypothetical evolution of the crack tip geometry; Hypothetical evolution of the crack tip geometry; 22ndnd reversal reversal –– unloading, 3unloading, 3rdrd reversal reloadingreversal reloading

4

6 7

Δδlo

adin

g

8

Δδun

-load

ing

σres

5

σys

1

2

3

4

5

7

8

σys

σys6

ε

σ

09

10

1112

1

2

4

3 5

0

7

8Strs

. Int

. Fac

tor K

Time

6

Kr

9

10

Km

in, a

ppl

Km

ax, t

ot

ΔKto

t

ΔKap

pl

11

12

Km

in, t

ot

Km

ax, a

ppl

Kappl

Ktot


StressStress--strain evolution near a growing crack tipstrain evolution near a growing crack tip(constant amplitude loading(constant amplitude loading -- according to the according to the UniGrowUniGrow model)model)

0

ayσ

ayε

Kr

Kmin,appl ≈ K min,tot

ΔK

tot

ΔK

appl

Kmax. tot

Δε fatigue

Δεstationary

Stationary crack

Fatigue crack


( ) ( )γ−⎡ ⎤= Δ − −⎢ ⎥⎣ ⎦

p 1 pappl maxr r,appl

da C K KKdN

K

( )'

'

'

1

21*

3* ' '1

1 ;1

2 ;

22

b c

nn

f f

pn

b c

xC

E

γ

ρ

π ρ σ ε

−+

++

=+

= −+

⎛ ⎞⎜ ⎟

= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

( )( )

122

1*2* '

0.5;1 ;

24

b

f

p

b

xC

γ

ρπρ σ

−

=

= −

⎛ ⎞⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠

Near threshold Region I Plastic strain evolution dominated region II

Combination of applied and residual stress intensity factorsCombination of applied and residual stress intensity factors


Crack tip displacement field and corresponding residual stress fCrack tip displacement field and corresponding residual stress field ield

2arc

σr(x)

rc

σr(x)

ρ* ρ*Plastic zoneFatigue crack and anticipated

displacement fieldPlastic zone

Crack model with equivalent residual stress field

max,

min

max,

in, m ,

;

;applappl

ap appl pl

K aY

K S aY

S π

π

=

=( ) ( ),

c

res

r

r

a

a

K x m x a dxσ+

= ∫


( )( )

1 312 2

1 2 22, 1 1 1 1

2x xm a M M Ma a aa

xxxπ

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥= + − + − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠−

⎣ ⎦

( ) ( )0

,r

a

r m x a dK x xσ⎡ ⎤= ⎣ ⎦∫

Calculation of the residual stress intensity factor Calculation of the residual stress intensity factor KKresres (the weight (the weight function method) and the resultant total stress intensity factorfunction method) and the resultant total stress intensity factors, s,

KKmax,totmax,tot and and ΔΔKKtottot

y

x

a

t

σr(x)

A

x

= −

Δ = Δ −

max,tot max,appl

tot a

r

ppl r

KKand

K K K

K


log

(Δε/

2)

log (2Nf)

σf/E

εf

0 2Ne

Δσth/2E

Strain - Life Curve

c

b

Δεth/2

0


ε

σ

Δεth

Δσ t

h

1

2

112 2

thth

th

thK x K xπ

σσπ

ρρ

∗

∗

ΔΔ =

⎛ ⎞Δ⎜⎝

⇒Δ ⎟

⎠=

Determination of the effective crack tip radiusDetermination of the effective crack tip radius(from the near threshold data (from the near threshold data –– predominantly elastic behavior)predominantly elastic behavior)


Fatigue Crack Growth, St 4340

1E-08

1E-07

1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1E+00

1 10 100 1000

Kmax,appl pΔKappl

(1-p), (MPa√m)

da/d

N(m

m/c

ycle

)R=0.7 DowlingR=0.5 DowlingR=0.1 DowlingR=0 DowlingR=-0.5 DowlingR=-1 DowlingR=0.7 TaylorR=0.05 TaylorR=0.5 NASA(L)R=0.5 NASA(R)R=0 NASA(L)R=0 NASA(R)R=-1 NASA(L)R=-1 NASA(R)R=0.5 WanhillR=0 WanhillR=-1 Wanhill


Fatigue Crack Growth, St 4340

1E-08

1E-07

1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1E+00

1 10 100

Kmaxtot pΔKtot

(1-p), (MPa√m)

da/d

N(m

m/c

ycle

)

R=0.7 DowlingR=0.5 DowlingR=0.1 DowlingR=0 DowlingR=-0.5 DowlingR=-1 DowlingR=0.7 TaylorR=0.05 TaylorR=0.5 NASA(L)R=0.5 NASA(R)R=0 NASA(L)R=0 NASA(R)R=-1 NASA(L)R=-1 NASA(R)R=0.5 WanhillR=0 WanhillR=-1 WanhillApprox Sol'n


da/dN vs. ΔK; Al 7075-T6 (data of Newman and Jiang)

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1 10 100

ΔKappl, [MPa√m].

da/d

N, [

m/c

yc].

R=0.75..0.82R=0.5R=0R=-1R=-2R=0.33


da/dN vs. Δκpl,tot; Al 7075-T6 (data of Newman and Jiang, ρ*=5x10-6 m)

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1 10 100

ΔKtot0.5Kmax,tot

p, [MPa√m].

da/d

N, [

m/c

yc].

R=0.82..0.7R=0.5R=0.33R=0R=-1R=-2


da/dN vs. ΔΚappl Ti-17

1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1 10

ΔΚappl, MPa(m)0.5

da/d

N, m

/cyc

R=0.7R=0.4R=0.1


Ti 17 material

y = 1E-22x57.28

y = 2E-11x7.2072

1E-12

1E-11

1E-10

1E-09

1E-08

1E-07

1 10

ΔΚtot0.5Kmax,tot

p, MPa*m1/2

da/d

N, (

m/c

ycle

)

R=0.7R=0.4R=0.1


Memory rules for propagating fatigue cracksMemory rules for propagating fatigue cracks

1. Only the compressive part of the minimum stress distribution affects the fatigue crack growth rate. The compressive part of the minimum stress distribution induced by the first loading cycle constitutes the minimum initial stress field used for the determination of the residual stress intensity factor.

2. If the compressive part of the minimum stress distribution induced by current loading cycle is completely inside of the previous resultant minimum stress field, the material will not “feel” it and the current minimum stress distribution should be neglected.

3. If the compressive part of the minimum stress distribution of the current loading cycle is fully or partly outside of the previous general minimum stress field they should be combined.

4. All subsequent minimum stress distributions should be considered and included into the resultant field as long as the crack tip is located inside of their compressive stress zones. In other words, when the crack tip propagates across the entire compressive stress zone of given minimum stress field it should be excluded from further analysis.


Material StressMaterial Stress--Strain Response at the Strain Response at the Notch Tip Due to Variable Amplitude Notch Tip Due to Variable Amplitude

Cyclic Loading FCyclic Loading F

Stre

ss, σ

Resultant notch tip stress history

Strain, ε

Resultant notch tip strain history

Resultant notch tip stress-strain path

1

2’

3

4

5

6 7

1’

8

7

2

y

d n

0

T

ρ

σ

StressF F


Memory rules (CA)Memory rules (CA)

Parts of stress distribution induced by both cycles (a and b) need to be accounted for!

Only part of the stress distribution induced by cycle ‘b’ should be accounted for !

x

σr

x

σr

Current cycle a

Previous cycle b

The residual stress distribution created by a loading cycle remains active as long as the propagating crack tip remains in its compressive stress zone!

Kappl

0 Time

a b

a b a b


Calculation of KCalculation of Krr for constant amplitude loadingfor constant amplitude loading

P

N

x

σr

a

Kr

K

N

x

σr

a

Kr

Constant loading Constant SIF

a

a

c

a

cbb


( ) ( )

max,max min, min, ,;

,

appl applap

re

a

pl

rb

pl

s

apK aY K aY

m x a

S

d

S

x xK

π π

σ

= =

= ∫

Fatigue crack model for calculating the applied and the residualFatigue crack model for calculating the applied and the residualstress intensity factorstress intensity factor

Sappl

2a y

x

σr(x)

rc

0

σr(x)

rc

Sappl

Sappl Sappl

2b


Kappl

Time

ab

x

σr

x

σr

Kappl

Time

a

b

ab

b

a

Kappl

Time

a

b

xσr

ba

c

c

Memory rules (Memory rules (CA+OvrldCA+Ovrld))


UnderloadUnderload effecteffect

Under-load may sometime eliminate the beneficial effects of all previous overloads!

σr

x

SS

t t

x


Variable amplitude loading (VA)Variable amplitude loading (VA)

P

Nσr

x

Κr

x

Loading history

Residual stress profile

Residual Kr profile


229

2a=12.7

610

σ

σ

Material: Al 2024 – T3

Source:

1. A. Ray and R. Patankar, Fatigue Crack Growth under Variable-Amplitude Loading, Applied Mathematical Modelling, vol. 25, 2001, pp. 979-994 and pp. 995-1013.

2. H. Xiaoping, T. Moan, C. Weigcheng, An Engineering Model of Fatigue Crack Growth under Variable Amplitude Loading, International Journal of Fatigue, vol. 30, 2008, pp. 2-10.

All dimensions in [mm] !

Thickness t = 4.1 [mm]


Material: Al 2024-T3

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1 10 100

ΔKappl [MPa√m]

da/d

N [

m/c

ycle

]

R=0.5R=0.3R=0.1R=0

·


Material: Al 2024-T3, ρ∗ = 3·10- 6 m

1: da/dN = 2E-11Δκ9.3295

2: da/dN = 5E-10Δκ3.4523

3: da/dN = 5E-12Δκ7.4869

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1 10 100

Δκtot [MPa√m]

da/d

N [m

/cyc

]

R=0.5R=0.3R=0.1R=0


0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120

Stre

ss σ

[MPa

]

No. of cycles

Stress History ‘a’


0

10

20

30

40

50

60

70

80

0 50 100 150 200

No. of Cycles x 103

Cra

ck le

ngth

[mm

]

TestUniGrow

Material: Al 2024 – T3


Stress History 'b'

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60

No. of Cycles x 103

Stre

ss σ

[MPa

]


Material:Al 2024 - T3

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140

No. of Cycles x 103

Cra

ck le

ngth

[mm

]

TestUniGrow


Material 7075Material 7075--T6, P3, ONRT6, P3, ONR


0

P3 aircraft stress historyP3 aircraft stress history


FCA 301, 50th PercentileTension Dominated Spectrum

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000

No. of Cycles

Hal

f Cra

ck L

engt

h [i

n]

Test3-CouponDataTest2-CouponDataUniGrow prediction

Max Stress: 29,755 psiMin Stress: -6,190 psi#EventPoints in Block: 584,263


R8.0

∅60

∅70

R3.0

8

50

62

132

SECTION BBAll dimensions are in mm

Initial Flaw Location2mm radius corner crack

T

S

UK Round Robin Program, Al 7010UK Round Robin Program, Al 7010

© 2008 Grzegorz Glinka. All rights reserved. 5882.

4

138

300

80

10

5

25

25

588

2BA Dia ThreadTyp 4 Places

∅60 ∅70

6 Holes Drill + Ream20.0 +0.10/-0.00 A

A

R 2

2.0

Typ

R 2

2.0

Typ

6

20 +0.00-0.10

50

2

2.4

Section BB on next page

R 8

.0 T

YP

R 8

.0 T

YP

R 3

.0 T

YP

B B

All dimensions are in mm


Pj(x,y)

Γci

ρ3

ρ2

ρ1

Δαiρn

ρPjAi

ρAi Γbi

A1

A2

A3

λ n

λ i

λ 1

λ 2

λ 3

2

2( , ; )j i

iiAc

Ac

b

P A

PPK m x yπρ

×Γ + Γ

=Γ

=


Stress Intensity Factor - the geometric correction factor 'β '

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.005 0.01 0.015 0.02 0.025 0.03

Crack size, [m]

Bet

a β

Round Robin 'Beta' from Irving (FE)

BEASY 'Beta' from Newman (BE)

Resultant 'Beta' from Glinka (WF)


Material: Al 7010 -T73651, UK Round Robin Test

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1 10 100

ΔKappl [MPa√m]

da/d

N

[m/c

ycle

]

R=0.9R=0.7R=0.4R=0.1


Material: Al 7010 -T73651, UK Round Robin Test, ρ* = 2.6x10-6m

y = 6E-11x4.0382

y = 9E-13x7.4114

y = 1E-09x2.9065

y = 2E-13x12.294

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1 10 100

Δκ =ΔKtot1-pKmax,tot

p [MPa√m]

da/d

N

[m]

R=0.9R=0.7R=0.4R=0.1


Astrix spectrum loading, UK Round Robin Test

-20

0

20

40

60

80

100

120

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Number of reversals

Nor

mal

ized

app

l. st

ress


Material: Al 7010-T73651 , UK Round Robin Test

0

5

10

15

20

25

30

0 50 100 150 200 250 300 350 400 450 500

Flight hours [hrs]

Cra

ck s

ize

a [m

m]

Test 1Test 2UniGrow


Summary of the ProcedureSummary of the ProcedureAdvantagesAdvantages DisadvantagesDisadvantages

•Accurate master FCG curve can be generated from limited FCG data,

•Any variable amplitude stress spectra can be analyzed,

•No prior information concerning FCG under variable amplitude loading required,

•Wide variety of real crack/load configurations can be analyzed via applications of the weight function,

•Enables modeling of irregular planar cracks

•Required cyclic stress-strain smooth specimen data

•Required limited CA FCG data

•Requires sometime the use of catalogue SIF solutions or FE SIF data

Documents

8) Recent developments in the fatigue crack growth analysisutmis.org.loopiadns.com/media/2016/08/Weldments6_Fatigue... · 2018-11-02 · fatigue cracks Use of appropriate SIF using