Basics elements on linear elastic fracture mechanics and crack growth modeling

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Basics elements on linear elastic fracture mechanics and crack growth modeling

Sylvie Pommier

To cite this version:

Sylvie Pommier. Basics elements on linear elastic fracture mechanics and crack growth modeling. Doctoral. France. 2017. �cel-01636731�

Basics elements on linear elastic fracture

mechanics and crack growth modeling

Sylvie Pommier

,

LMT

Fail Safe

Damage Tolerant Design

• Consider the eventuality of damage or of the presence of defects,

• predict if these defects or damage may lead to fracture, • and, in the event of failure,

predicts the consequences (size, velocity and trajectory

• 2700 Liberty Ships were built between 1942 and the end of WWII

• The production rate was of 70 ships / day

• duration of construction: 5 days

• 30% of ships built in 1941 have suffered catastrophic failures • 362 lost ships Li berty ship s – hi ve r 1 94

1 The fracture mechanics concepts were still

unknown

Causes of fracture:

• Welded Structure rather than bolted, offering a substantial assembly time gain but with a continuous path offered for cracks to propagate through the structure.

• Low quality of the welds (presence of cracks and internal stresses)

• Low quality steel, ductile/brittle

Foundations of fracture mechanics : The Liberty Ships

LEFM - Linear elastic fracture mechanics

Georges Rankine Irwin “the godfather of fracture mechanics »

• Stress intensity factor

K

• Introduction of the concept of fracture toughness

K

_IC

• Irwin’s plastic zone (monotonic and cyclic)

• Energy release rate

G

and

Gc

(G in reference to Griffith) Geo rge s Ra nki ne Irw in

Historical context

Previous authors

Griffith A. A. - 1920

–

"

The phenomenon of rupture and flow in

solids", 1920, Philosophical Transactions of the Royal Society, Vol.

A221 pp.163-98

Westergaard H. M. – 1939 -

Bearing Pressures and Cracks, Journal

of Applied Mechanics 6: 49-53.

Muskhelishvili N. – 1954

- Ali Kheiralla, A. Muskhelishvili, N.I. Some

Basic problems of the mathematical theory of elasticity. Third revis.

and augmented. Moscow, 1949, J.Appl. Mech.,21

(1954)

, No 4,

417-418.

Fatigue crack growth: De Havilland Comet

3 accidents

26/10/1952, departing from Rome

Ciampino

March 1953, departing from Karachi

Pakistan

10/01/1954, Crash on the Rome-London

flight (with passengers)

Paris & Erdogan 1961

They correlated the cyclic fatigue

crack growth rate

da / dN

with the

stress intensity factor amplitude

DK

Introduction of the

Paris’ law

for

modeling fatigue crack growth

Fatigue remains a topical issue

8 Mai 1842 - Meudon (France) Fracture of an axle by fatigue

3 Juin 1998 - Eschede (Allemagne) Fracture of a wheel by Fatigue

Development of rules for the EASA certification

Los Angeles, June, 2nd 2006,

Aloha April, 28th 1988,

Rotor Integrity Sub-Committee (RISC)



AIA Rotor Integrity Sub-Committee (RISC) : Elaboration of AC 33.14-1

UAL 232, July 19, 1989 Sioux City, Iowa

•

DC10-10 crashed on landing

• In-Flight separation of Stage 1 Fan Disk

• Failed from cracks out of material anomaly

-Hard Alpha produced during melting

• Life Limit: 18,000 cycles. Failure: 15,503 cycles. • 111 fatalities

• FAA Review Team Report (1991)recommended: - Changes in Ti melt practices, quality controls

- Improved mfg and in-service inspections - Lifing Practices based on damage tolerance

Elaboration of AC 33.70-2

DL 1288, July 6, 1996 , Pensacola, Florida

• MD-88 engine failure on take-off roll • Pilot aborted take-off

• Stage 1 Fan Disk separated; impacted cabin • Failure from abusively machined bolthole

• Life Limit: 20,000 cycles. Failure: 13,835 cycles. • 2 fatalities

• NTSB Report recommended ...

- Changes in inspection methods, shop practices - Fracture mechanics based damage tolerance

Damage

tolerance

Why ?

• To prevent fatalities and disaster

Where ?

• Public transportation (trains, aircraft, ships…)

• Energy production (nuclear power plant, oil extraction and transportation …)

• Any areas of risk to public health and environment

How ?

• Critical components are designed to be damage tolerant / fail safe

• Rare events (defects and cracks) are assumed to be certain (deterministic

Fracture mechanics

One basic assumption :

The structure contains a singularity (ususally a

geometric discontinuity, for example: a crack)

Two main questions :

What are the

relevant variables

to characterize the

risk of fracture and to be used in fracture criteria ?

What are the

suitable criteria

to determine if the

crack may propagate or remain arrested, the crack

growth rate and the crack path ?

Classes of material behaviour : relevant variables

Linear elastic behaviour

: linear elastic fracture mechanics (K)

Nonlinear behavior: non-linear fracture mechanic

s

Hypoelasticity : Hutchinson Rice & Rosengren, (J)

Ideally plastic material : Irwin, Dugdale, Barrenblatt etc.

Time dependent material behaviours

: viscoelasticity,

viscoplasticity (C*)

Complex non linear material behaviours :

Various local and non local approaches of failure, J. Besson, A.

Classes of fracture mechanisms : criteria

•

Brittle fracture

•

Ductile fracture

•

Dynamic fracture

•

Fatigue crack growth

•

Creep crack growth

•

Crack growth by corrosion, oxydation, ageing

•

Coupling between damage mechanisms

Mechanisms acting at very different scales of time and

space, an assumption of scales separation

•

Atomic scale (surface oxydation, ageing, …)

•

Microstructural scale (grain boundary corrosion, creep,

oxydation, persistent slip band in fatigue etc… )

•

Plastic zone scale or damaged zone (material

hardening or softening, continuum damage, ductile

damage...)

•

Scale of the structure (wave

propagation …)

Atomic cohesion energy 10 J/m2 Brittle fracture energy 10 000 J/m2

Classes of relevant assumptions : application of

criteria

Long cracks

(2D problem, planar crack with a straight crack)

Curved cracks, branched cracks, merging cracks

(3D problem,

non-planar cracks, curved crack fronts)

Short cracks

(3D problem, influence of free surfaces, scale and gradients

effects)

Other discontinuities and singularities

:

•

Interfaces / free surfaces,

•

Contact front in partial slip conditions,

•

acute angle ending on a edge,

Griffith’ theory

Threshold for unsteady crack growth

(brittle or ductile)

Relevant variable :

energy release rate

G

Criteria

:

An unsteady crack growth occurs if the cohesion

energy released by the structure because of the creation of new

cracked surfaces reaches the energy required to create these

new cracked surfaces

G = Gc

da

W

U

G

where

G

ext elastic



D







2



Criteria :

Griffith’ theory

elastic ext surface ext surface elastic

U

W

da

U

W

U

U

U

D







D





D



D



D



2

:

work of external forces

:

variation of the elastic energy of the structure

:

variation of the surface energy of the structure

ext

W

elastic

U

D

surface

U

D









0

0

0

0



































ext surface volume ext ext

W

dF

dF

W

dF

Q

TdS

dT

conditions

isothermal

in

SdT

TdS

dF

dU

Q

TdS

where

Q

W

dU

Evolution by Bui, Erlacher & Son









da

W

F

G

da

F

G

where

da

G

G

ext volume surface c c



D







D











2

0

Free energy instead of internal energy

Isothermal conditions instead of adiabatic

Eschelby tensor : energy density

J Integral (Rice)

J integral , (Rice’s integral if q is coplanar)

q vector: the crack front motion





da

W

F

J contour integral

If the crack faces are free

surfaces (no friction, no

fluid pressure …),

If volume forces can be

neglected (inertia, electric

field...)

Then the J integral is shown

to be independent of the

choice of the selected

integration contour

𝐺 = 𝐽=

_Γ

𝜑

_{𝑓𝑟𝑒𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦}

𝑑𝑦 − 𝜎𝑛.

𝜕𝑢

_𝜕𝑥

x y

C. Stoisser, I. Boutemy and F. Hasnaoui

Limitations

• The crack faces must be free surfaces

(no friction, no fluid pressure)

• Gc is a material constant (single

mechanism, surfacic mechanism only)

• What if non isothermal conditions are

considered ?

• Unsteady crack growth criteria, non

applicable to steady crack

propagation,

• The surfacic energy 2 may be

negligible compared with the energy

dissipated in plastic work or continuum

damage / localization process

Linear Elastic Fracture

Mechanics (LEFM)

Characterize the state of the structure

where useful (near the crack front where

damage occurs) for a linear elastic

behavior of the material

Stress concentration factor

Kt

of an elliptical hole,

With a length

2a

and a curvature radius

r

r





a

K

_t



loc_



1

2

2a

r

Preliminary remarks:

From the discontinuity to the singularity

2a











_

r







r

loc _loc

a

2

0

Singularity

 

   

 

(

)

)

(

,

0

* *

r

f

q

r

f

r

r

g

r

f

r











r











2a

Remarks: existence of a singularity

r : distance to the discontinuity

Warning: implicit choice of scale

Geometry locally-self-similar → self-similar solution

→ principle of simulitude

 





r

Cr

r 0





 





r

Br

r 0





 

2

2

2

2

2 2 max 0 0 1 2 0 0 0 max max









           



 









       

r

A

E

dr

r

A

E

dr

rd

r

r

A

E

r elast r r r r elast r r r r elast

Order of this singularity

Linear elasticity:

0

1

1

0

2

2

2

2

2

0

2 2 max 0































 















r

A

E

r elast

For a crack : =-0.5



n

elastic



n o o





1





















 





n r

Br

r

0 



 

   



1



2

2

2

2 1 max 0 0 1 1 0 0 0 max max











             



 









       

n

r

C

E

dr

r

C

E

dr

rd

r

r

C

E

n r elast r r r n r elast r r r n r elast

 





r

Ar

r 0









n

n















1

2

0

2

1





n=4

Non linear material behaviour ?



LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

Fracture modes

Tubes (pipe line)

Various fractures in compression

Various fractures in torsion

LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

Case of mode I

Analysis of Irwin based on Westergaard’s analysis

and Williams expansions

a

f

Div

v

r







2D problem, quasi-static, no volume force

Balance equation

0

0

0























































z

y

x

z

y

x

z

y

x

zz

yz

xz

yz

yy

xy

xz

xy

xx



















Linear isotropic elasticity : E, n









xy

xy

xx

yy

yy

yy

xx

xx

E

E

E



n



n



n



n



n



















1

1

1

2

2

 

1

1

n

_

n

_



Tr

E

E







Compatibility equations





































x

u

y

u

y

u

x

u

y x xy y yy x xx

2

1







y

x

u

y

x xx 2 3 2 2















x

y

u

x

y yy 2 3 2 2















y

x

u

x

y

u

y

x

x y xy 2 3 2 3 2

2

























y

x

y

x

xx yy xy 2 2 2 2 2

2























_

Combination

=

3

Equations,

3

unknowns

Balance equations

Compatibility

Linear elasticity

y

x

y

x

xx yy xy 2 2 2 2 2

2























_

_

_





xy xy xx yy yy yy xx xx

E

E

E



n



n



n



n



n



















1

1

1

2 2

y

x

y

x

xx yy xy 2 2 2 2 2

2























_

+

0

0

























y

x

yy xy xy xx









+

Airy function F(x,y)

-1862-1

equation,

1

unknow

F(x,y)

Balance equation Compatibility

Assuming

0

























y

x

y

x

yy xy xy xx









y

x

y

x

xx yy xy 2 2 2 2 2

2























_

0

2

₄ 4 2 2 4 4 4





















y

F

y

x

F

x

F

y

x

F

x

F

y

F

xy yy xx























2 2 2 2 2







Z(z) , z complex,

A point in the plane is defined by a complex number

z = x + i y

Z

a function of

z

:

Z(z)=F(x,y)

F=F(x,y)

Z (z)

always fulfill all the

equations of the problem

Z(z)

must verify the symmetry

and the boundary conditions

0

2

₄ 4 2 2 4 4 4





















y

F

y

x

F

x

F

z

Z

y

Z

z

Z

y

x

Z

z

Z

x

Z

4 4 4 4 4 4 2 2 4 4 4 4 4



































LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

Irwin’s or Westergaard’s analyses

Away from the crack (x & y >> a) : s_xx= S s_yy= S & s_xy= 0

2a S S S S x y

Singularities in y=0 x=+a & y=0 x=-a Symmetric with respect to y=0 & x=0 2D problem, plane (x,y) : S_zz=n(S_xx+S_yy)

6 boundary or symmetry conditions 2 singularities,

0 boundary conditions along the crack faces Exact solution

Taylor’s development with respect to the distance to the crack front

Separated variables Similitude principle

Boundary conditions & Symmetries

symmetries

0

,







   xy yy xx



S









2 3 4 2 2

2

y

x

a

x

a

y

a

S

F

















4 2 2

2

y

x

a

S

F









&

y

x

F

x

F

y

F

xy yy xx























2 2 2 2 2







Construction of Z(z)

Relation





4 2 2

2

y

x

a

S

F









4 2

2

z

a

S

Z







 

 









































z

Z

yI

Z

R

y

Z

yR

Z

R

F

_{e e e m}

F

x

F

y

F

yy xx























2 2 2 2 2





























































































3 3 3 2 2 3 3 2 2

Z

yR

z

Z

yI

z

Z

R

z

Z

yI

z

Z

R

m e yy m e xx







Solution

Solution:

4 2

2

z

a

S

Z







0

,







   xy yy xx



S





At infinity

Valid for any 2D problem, with symmetries along the

planes y=0 & x=0, and biaxial BCs





























































































3 3 3 3 2 2 3 3 2 2

z

Z

yR

z

Z

yI

z

Z

R

z

Z

yI

z

Z

R

e xy m e yy m e xx







At infinity

LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

Exact solution for a crack

Singularities

in y=0 x=+a

& y=0 x=-a

+

+

Exact solution

a

z 

1

a

z 

1

Sz

z

Z

a

z

S

Z











  4 2

2



_z

2

_a

2



1

2

S

z

Z







































































































3 3 3 3 2 2 3 3 2 2

z

Z

yR

z

Z

yI

z

Z

R

z

Z

yI

z

Z

R

e xy m e yy m e xx







Asymptotic solution -

Irwin-x r _ y

Local coordinates (r,), r → 0



2 2



12 2 2

a

z

Sz

z

Z











2 2



32 2 3 3

a

z

Sa

z

Z











 i

re

a

z









12 2 2 2

2

2

  i i

e

r

a

S

are

Sa

z

Z















2 3 2 3 2 3 3

2

1

2

  i i

e

r

a

S

r

are

Sa

z

Z















Exact Solution



_z

2

_a

2



12

S

z

Z











Westergaard’s stress function :

2 2 2

2

 i

e

r

a

S

z

Z









2 3 3 3

2

1

i 

e

r

a

S

r

z

Z



































 













 



2

3

cos

2

sin

2

cos

2

3

sin

2

sin

1

2

cos

2

2

3

sin

2

sin

1

2

cos

2





































a

S

r

a

S

r

a

S

xy yy xx





























































































3 3 3 3 2 2 3 3 2 2

z

Z

yR

z

Z

yI

z

Z

R

z

Z

yI

z

Z

R

e xy m e yy m e xx







x r _ y

Irwin-Error associated to this Taylor development along



=0

Exact solution

𝜎𝑦𝑦 𝑟, 𝜃 = 0 = 𝑆_𝑦𝑦 𝑎 + 𝑟 𝑟 2𝑎 + 𝑟 = 𝐾_𝐼 𝑎 + 𝑟 𝜋𝑎𝑟 2𝑎 + 𝑟

Asymptotic solution

𝜎_𝑦𝑦 𝑟, 𝜃 = 0 = 𝐾𝐼 2𝜋𝑟 1 + 3 4 𝑟 𝑎 + 5 32 𝑟 𝑎 2 + 𝑂 𝑟52 0.1 0.2 0.3 0.4 0.5 10 4 0.001 0.01 0.1 Error _{1 term} 2 terms

Erreur = 1%

1 term 𝑟 𝑎 = 𝟎. 𝟎𝟏𝟑 2 terms 𝑟 𝑎 = 𝟎. 𝟐𝟗 3 terms 𝑟 𝑎 = 𝟎. 𝟔𝟗

𝑒𝑟𝑟𝑜𝑟~

3

4

𝑟

𝑎

LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

Mode I, non equi-biaxial conditions

Equibiaxial Biaxial (Superposition)

              S_xx S_yy T

Stress intensity factors

KI &T

Crack geometry and boundary conditions

Spatial distribution, given once for all, in the crack front region

g_ij() f(r)=r

Similitude principle

(geometry locally planar, with a

straigth crack front, self-similar,

singularity)

Same

KI & T

→ Same local field

₂ cos ₂sin ₂ cos3₂

2 3 sin 2 sin 1 2 cos 2 2 3 sin 2 sin 1 2 cos 2                r K r K T r K I xy I yy I xx                  

von Mises stress field

Plane stress, Mode I, T=0

Plane strain, Mode I, T=0

 

 



 



1

3

,

,

,













D

r



r



Tr

r



 





 

,



:



 

,



2

3

,

r

r

r

D D eq



Plane strain, Mode I

T / K = 0 m

-1/2

_{T / K = 10 m}

-1/2

T / K = -10 m

-1/2

_{T / K = -5 m}

-1/2

_{T / K = 5 m}

-1/2

Mechanisms controlled by shear

 Plasticity,

 Visco-plasticity

 Fatigue

von Mises stress field

yy xx

S

S

T





Hydrostatic pressure

Plane stress, Mode I, T=0

Plane strain, Mode I, T=0

 





r

,





Tr

Fluid diffusion (Navier Stokes),

Diffusion creep (Nabarro-Herring)

Chemical diffusion

Plane strain, Mode I

T / K = 0 m

-1/2

_{T / K = 10 m}

-1/2

T / K = -10 m

-1/2

_{T / K = -5 m}

-1/2

_{T / K = 5 m}

-1/2

Hydrostatic pressure

 





r

,





Tr

Fluid diffusion (Navier Stokes),

Diffusion creep (Nabarro-Herring)

Chemical diffusion

yy xx

S

S

Other T components, in Mode I

General triaxial loading Equibiaxial plane strain Superposition non equibiaxial conditions Superposition non plane strain

Full solutions KI, KII, KIII, T, Tz & G

Mode I







 



                       cos 2 sin 2 2 cos 2 cos 2 2 2 3 cos 2 sin 2 cos 2 2 3 sin 2 sin 1 2 cos 2 2 3 sin 2 sin 1 2 cos 2                       r K u r K u r K r K T r K I y I x I xy I yy I xx







 



                       cos 2 2 cos 2 2 cos 2 2 sin 2 2 2 3 sin 2 sin 1 2 cos 2 2 3 cos 2 cos 2 sin 2 2 3 cos 2 cos 2 2 sin 2                         r K u r K u r K r K r K II y II x II xy II yy II xx Mode II 2 sin 2 2 4 2 cos 2 2 sin 2          r K u r K r K III z III yz III xz   G    Mode III





) 4 3 ( n    n       _{xx yy z} zz T Déformation plane



n



n     1 ) 3 ( Contrainte plane

Mode I

Mode II

von Mises stress field

 

 



 



1

3

,

,

,













D

r



r



Tr

r



 



₂



 

,



:



 

,



3

,

r

r

r

D D eq



Summary

- Exact solutions for the 3 modes, determined for one specific geometry

- Taylor development, 1st order → asymptotic solution generalized to any other cracks

- First order

- Solution expressed with separate variables f (r) g () and f (r) self-similar - Solution : f (r) a power function, r, with  = - 1/2

- Higher Orders

- A unique stress intensity factor for all terms

- The exponent of (r/a) increasing with the order of the Taylor’s development - Boundary conditions

- Singularity along the crack front, symmetries, planar crack and straight front - no prescribed BCs along the crack faces,

- Boundary conditions defined at infinity

6 independent components of the stress tensor at infinity → 6 degrees of freedoms in MLER: KI, KII, KIII and T, Tz, and G

LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

Williams expansion

A self-similar solution in the form is sought directly as follows : x r _ y

0

2

₄ 4 4 2 2 4 4 4

























F

y

F

y

x

F

x

F

 





 



g

r

r

F

,



2





 

 





 

  

 

 





  







 

 

_

































































































     4 4 2 2 2 2 2 2 2 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2

2

2

2

2

2

1

1



















































      

g

g

g

r

F

g

r

g

r

g

r

g

r

F

g

r

g

r

F

r

r

F

r

r

r

F

x r _ y



2



  



2





 

 

₄

0

4 2 2 2 2 2 2 2 4



























































g

g

g

r

F

 

_

_

_

_

 

_

_{  }

0

2

2

₂ 2 2 2 2 2 4 4































g

d

g

d

d

g

d

Dans ce cas g() doit vérifier

Williams expansion

A self-similar solution in the form is sought directly as follows :

 





 



g

r

r

F

,



2

The solution is sought as follows : x r _ y

 

_

ip

Ae

g





























2



2

0

2

2

2 2 2 2 2 2 2 2 2 4

















































p

p

p

p

p

p

 

_

_

_

_

 

_

_{  }

0

2

2

₂ 2 2 2 2 2 4 4































g

d

g

d

d

g

d

Williams expansion

x r _ y

 

_



 2



   2  2





Re

,



r



Ae

i



Be

i



Ce

i 



De

i 

r

F

Boundary conditions are defined along the crack faces which are defined as free surface (fluid pressure & friction between faces are excluded)











,





,



0

0

,

,

₂ 2









































































 

r

r

F

r

r

r

r

F

r

r

Williams expansion

x r _ y

 

_



 2



   2  2





Re

,



r



Ae

i



Be

i



Ce

i 



De

i 

r

F







_

_ _{ } _



_





_



_





_

_ _

_



_



_ _



_

0

2

2

Re

0

2

2

Re

0

,

0

Re

0

Re

0

,

2 2 2 2 2 2 2 2

























































                                         





























i i i i i i i i r i i i i i i i i

De

Ce

Be

Ae

De

Ce

Be

Ae

r

De

Ce

Be

Ae

De

Ce

Be

Ae

r

Williams expansion

A sery of eligible solutions is obtained : x r _ y





 

 

_                                                         















1 2 sin 1 2 sin 1 2 cos 1 2 cos 1 2 n C n A g n D n B g n n even n odd

 

r



r

g

 



F

n 1 2

,





La solution en contrainte s’exprime alors à partir des dérivées d’ordre 2 de F, toutes les valeurs de n sont possibles, tous les modes apparaissent

Williams versus Westergaard

- The boundary conditions are free surface conditions along the crack faces (apply on 3 components of the stress tensor), no boundary condition at infinity → absence of T, Tz, and G

- Super Singular terms → missing BCs

- The first singular term of the Williams expansion is identical to the first term of the Taylor expansion of the exact solution of Westergaard

- The stress intensity factors of the higher order terms are not forced to be the same as the one of the first term,

- advantage, leaves some flexibility to ensure the compatibility of the solution with a distant, non-uniform field

- drawbacks, it replaces the absence of boundary conditions at infinity by condition of free surface on the crack, and it lacks 3 BCs, it is obliged to add constraints T, Tz, and G arbitraitement

LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

J contour integral

The J integral is shown to be

independent of the choice of the

selected integration contour

The integration contour G can be

chosen inside the domain of

validity of the Westergaard’s

stress functions to get G in linear

elastic conditions

𝐺 = 𝐽=

_Γ

𝜑𝑑𝑦 − 𝜎𝑛.

𝜕𝑢

_𝜕𝑥

x y

𝐺 =

1 − 𝜈

2

𝐸

𝐾

𝐼2

+ 𝐾

𝐼𝐼2

+

1 + 𝜈

𝐸

𝐾

𝐼𝐼𝐼2

𝐺

_𝑐

=

1 − 𝜈

2

𝐸

𝐾

𝐼𝑐2 Energy release rate

LEFM

KI, KII, KIII

T, Tz, G

A. Modes

B. Airy stress functions

C.

Westergaard’s solution

D.

Irwin’s asymptotic

development

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

Mode I, LEFM, T=0

Syy

Syy

Syy

LEFM stress field (Mode I)

Irwin’s plastic zones size, step 1: r

_Y

Along the crack plane,

=0













,



,

0



0

2

2

0

,

,

2

0

,

0

,























n















r

r

K

r

r

K

r

r

_yy I _zz I _xy xx







n











1



3

2

2

0

,

r

K

r

p

_H I









r

K

r

I eq



n





2

2

1

0

,







Yield criterion :



_eq



r

_Y

,



 0







_Y





2 2 2

2

2

1

Y I Y

K

r





n





Irwin’s plastic zones size, step 2: balance

Hypothesis: when plastic deformation occurs, the stress tensor

remains proportionnal to the LEFM one



yy(r,



=0)

r

Elastic field



_Y

Limitations

Crack tip blunting

modifies the

proportionnality ratio

between the

components of the

stress and strain

tensors

FE results, Mesh size 10 micrometers, Re=350 MPa,

Rm=700 MPa, along the crack plane

Irwin’s plastic zones size, step 2: balance



yy(r,



=0)

r

Elastic field



_Y

r

_Y

r

_p







_

_





      









r r r Y I r r r Y r r I pm pm

dr

r

r

K

dr

dr

r

K



n





1

2

2

2

max 0 0 max





2 2 2

2

1

2

Y I Y pm

K

r

r





n







Irwin’s plastic zone versus FE computations

Ideally elastic-plastic material



_Y

=600 MPa, E=200 GPa, n=0.3

plane strain, along the plane =0

Irwin’s plastic zone versus FE computations

Ideally elastic-plastic material



_Y

=600 MPa, E=200 GPa, n=0.3

plane strain, along the plane =0

Irwin’s plastic zone versus FE computations

Ideally elastic-plastic material



_Y

=600 MPa, E=200 GPa, n=0.3

plane strain, along the plane =0

Mode I, Monotonic and cyclic plastic zones

Stress

(MPa

Mode I, Monotonic and cyclic plastic zones

Monotonic plastic zone

Cyclic plastic zone





2 2 max 2

2

1

Y I mpz

K

r





n









2 2 2

4

2

1

Y I cpz

K

r





n

D





T-Stress effect















S

a

K

_{I yy}



















S

_xx

S

_yy

T

T-Stress effect

Irwin’s plastic zone,



_Y

=400 MPa, K

_I

=15MPa.m

1/2

a

S

K

_I



_yy



yy

S

T





Ductile fracture

Measurement

of the crack tip

opening angle

at the onset of

fracture

Measurements

J.Petit

COD

_

Potential drop

Direct optical measurements

Digital image correllation

F

COD

Crack length increasing

a

a

N

Load cycle N F_max F_min F_op

R=F

_min

/F

_max

DF

DF

_eff

da/dN = f(a)

Paris’ law

IC I

K

K

max



A - threshold regime B – Paris’ regime C - unstable fracture

th

eff

K

K



D

D

Subcritical crack

growth if DK is over

the non propagation

threshold



MPa

m



K

_eff

[Neumann,1969]

Titanium alloy TA6V [Le Biavant, 2000]. The fatigue crack grows along slip planes.

N18 nickel based superalloy at room temperature, [Pommier,1992]. The crack grows at the intersection between slip planes

“pseudo-cleavage” facets at the initiation site

INCO 718

Paris’ law

IC I

K

K

max



A - threshold regime B – Paris’ regime C - unstable fracture

th

eff

K

K



D

D

Subcritical crack

growth if DK is over

the non propagation

threshold



MPa

m



K

_eff