Basics elements on linear elastic fracture mechanics and ...

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Basics elements on linear elastic fracture mechanics andcrack 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�

https://hal.archives-ouvertes.fr/cel-01636731

https://hal.archives-ouvertes.fr

Basics elements on linear elastic fracture mechanics and crack growth modeling

Sylvie Pommier, LMT

(ENS Paris-Saclay, CNRS, Université Paris-Saclay)

2

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

of the fragments)

• 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

Lib

ert

y s

hip

s–

hiv

er

19

41

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

transition around 0°C

Foundations of fracture mechanics : The Liberty Ships

4

Liberty Ships, WWII, 1941, Brittle fracture

LEFM - Linear elastic fracture mechanics

Georges Rankine Irwin “the godfather of fracture mechanics »

• Stress intensity factor K

• Introduction of the concept of fracture toughness KIC

• Irwin’s plastic zone (monotonic and cyclic)

• Energy release rate G and Gc

(G in reference to Griffith)

Georg

es R

ankin

e I

rwin

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.

n.b. Joseph Staline died in 1953

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 1961They 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

8

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

9

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

approach) and are introduced on purpose

for lab. tests and certification

Fracture mechanics

13

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 ?

14

Classes of material behaviour : relevant variables

Linear elastic behaviour: linear elastic fracture mechanics (K)

Nonlinear behavior: non-linear fracture mechanics

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. Pineau, G. Rousselier, A. Needleman, Tvergaard , S. Pommier etc.

15

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

16

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 (materialhardening or softening, continuum damage, ductile damage...)

• Scale of the structure (wave propagation …)

Atomic cohesion

energy

10 J/m2Brittle fracture

energy

10 000 J/m2

17

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’ theoryThreshold 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

Data : critical energy release rate Gc

da

WUG

where

G

extelastic D

2

19

Criteria :

Griffith’ theory

elasticextsurface

extsurfaceelastic

UWdaU

WUUU

DD

DDD

2

: work of external forces

: variation of the elastic energy of the structure

: variation of the surface energy of the structure

extW

elasticUD

surfaceUD

0

0

0

0

extsurfacevolume

ext

ext

WdFdF

WdFQTdS

dTconditionsisothermalin

SdTTdSdFdU

QTdSwhereQWdU

20

Evolution by Bui, Erlacher & Son

da

WFG

da

FG

where

daGG

extvolume

surface

c

c

D

D

2

0

Free energy instead of internal energy

Isothermal conditions instead of adiabatic

Second principle

21

Eschelby tensor : energy density

J Integral (Rice)

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

q vector: the crack front motion

da

WFG extvolume D

22

J contour integralIf 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 𝜑𝑓𝑟𝑒𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦𝑑𝑦 − 𝜎𝑛.𝜕𝑢

𝜕𝑥

x

y

23

C. Stoisser, I. Boutemy and F. Hasnaoui

Applications

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

aK loc

t 21

26

2a

r

Preliminary remarks:

From the discontinuity to the singularity

2a

r

r

aloc

loc 20

Singularity

)()(

,

0

*

*

rfqrf

rr

grfr

r

27

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

Crrr 0

Brrr 0

222

2

22max

0

0

12

0

00

max

max

rAE

drrAE

drrdrrAE

relast

rr

rr

elast

rr

rr

elast

28

Order of this singularity

Linear elasticity:

011022

1022

222

0

22

max

0

rAE

relast

For a crack : =-0.5

elasticn

n

o

o 1

n

rBrr0

212

2

21max

0

0

11

0

00

max

max

n

rCE

drrCE

drrdrrCE

n

relast

rr

r

n

relast

rr

r

n

relast

Arrr 0

29

n

n

1

2

021

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 asymptoticdevelopment

E. Stress intensity factor

F. Williams analysis

G. Fracture Toughness

H. Irwin’s plastic zones

30

31

Fracture modes

Planar symmetric Anti-planarPlanar anti-symmetric

32

Tubes (pipe line)

Fracture modes

33

Various fractures in compression

Fracture modes

34

Various fractures in torsion

Fracture modes

LEFM

KI, KII, KIII

T, Tz, G

A. Modes








35

36

Case of mode I Analysis of Irwin based on Westergaard’s analysisand Williams expansions

Planar Symmetric

afDivv

r

37

2D problem, quasi-static, no volume force

Balance equation

0

0

0

zyx

zyx

zyx

zzyzxz

yzyyxy

xzxyxx

38

Linear isotropic elasticity : E, n

xyxy

xxyyyy

yyxxxx

E

E

E

n

nn

nn

1

1

1

2

2

11

n

n

TrEE

39

Compatibility equations

x

u

y

u

y

u

x

u

yxxy

y

yy

xxx

2

1

yx

u

y

xxx

2

3

2

2

xy

u

x

yyy

2

3

2

2

yx

u

xy

u

yx

xyxy

2

3

2

32

2

yxyx

xxyyxy

2

2

2

22

2

40

Combination

= 3 Equations, 3 unknowns

Balance equations

Compatibility Linear elasticity

yxyx

xxyyxy

2

2

2

22

2

xyxy

xxyyyy

yyxxxx

E

E

E

n

nn

nn

1

1

1

2

2

yxyx

xxyyxy

2

2

2

22

2

+

0

0

yx

yx

yyxy

xyxx

+

41

Airy function F(x,y) -1862-

1 equation, 1 unknow

F(x,y)

Balance equation Compatibility

Assuming

0

yxyx

yyxyxyxx

yxyx

xxyyxy

2

2

2

22

2

024

4

22

4

4

4

y

F

yx

F

x

F

yx

F

x

F

y

F

xy

yy

xx

2

2

2

2

2

42

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

024

4

22

4

4

4

y

F

yx

F

x

F

z

Z

y

Z

z

Z

yx

Z

z

Z

x

Z

4

4

4

4

4

4

22

4

4

4

4

4

LEFM

KI, KII, KIII

T, Tz, G

A. Modes








43

44

Irwin’s or Westergaard’s analyses

Away from the crack (x & y >> a) : sxx= S syy= S & sxy= 0

2aS

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) : Szz=n(Sxx+Syy)

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

45

Boundary conditions & Symmetries

symmetries

0,

xyyyxx S

432

22

2ayaxaxy

SF

4

22

2axy

SF

&

yx

F

x

F

y

F

xy

yy

xx

2

2

2

2

2

46

Construction of Z(z)

Relation

4

22

2axy

SF

4

2

2az

SZ

z

ZyIZR

y

ZyRZRF meee

yx

F

x

F

y

F

xy

yy

xx

2

2

2

2

2

3

3

3

3

2

2

3

3

2

2

z

ZyR

z

ZyI

z

ZR

z

ZyI

z

ZR

exy

meyy

mexx

47

Solution

Solution:

4

2

2az

SZ 0,

xyyyxx 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

ZyR

z

ZyI

z

ZR

z

ZyI

z

ZR

exy

meyy

mexx

At infinity

LEFM

KI, KII, KIII

T, Tz, G

A. Modes








48

49

Exact solution for a crackSingularities

in y=0 x=+a

& y=0 x=-a

+ +

Exact solution

az

1

az

1

Szz

Z

azS

Z

4

2

2

21

22 azSz

Z

3

3

3

3

2

2

3

3

2

2

z

ZyR

z

ZyI

z

ZR

z

ZyI

z

ZR

exy

meyy

mexx

50

Asymptotic solution - Irwin-

x

r

yLocal coordinates (r,), r → 0

21

222

2

az

Sz

z

Z

23

22

2

3

3

az

Sa

z

Z

ireaz

2

212

2

22

i

i

er

aS

are

Sa

z

Z

2

3

23

2

3

3

2

1

2

i

i

er

aS

rare

Sa

z

Z

Exact Solution

21

22 azSz

Z

51

Westergaard’s stress function :

22

2

2

i

er

aS

z

Z

2

3

3

3

2

1

i

er

aS

rz

Z

2

3cos

2sin

2cos

2

2

3sin

2sin1

2cos

2

2

3sin

2sin1

2cos

2

r

aS

r

aS

r

aS

xy

yy

xx

3

3

3

3

2

2

3

3

2

2

z

ZyR

z

ZyI

z

ZR

z

ZyI

z

ZR

exy

meyy

mexx

x

r

y

Asymptotic solution - Irwin-

52

Error associated to this Taylor development along =0

Exact solution 𝜎𝑦𝑦 𝑟, 𝜃 = 0 =𝑆𝑦𝑦 𝑎 + 𝑟

𝑟 2𝑎 + 𝑟=

𝐾𝐼 𝑎 + 𝑟

𝜋𝑎𝑟 2𝑎 + 𝑟

Asymptotic solution

𝜎𝑦𝑦 𝑟, 𝜃 = 0 =𝐾𝐼

2𝜋𝑟1 +

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

r/a

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








53

54

Mode I, non equi-biaxial conditions

Equibiaxial Biaxial (Superposition)

aSK yyI

yyxx SST

55

Stress intensity factors

KI &T

Crack geometry and

boundary conditions

Spatial distribution, given

once for all, in the crack

front region

gij() f(r)=r

Similitude principle(geometry locally planar, with a

straigth crack front, self-similar,

singularity)

Same KI & T → Same local field 2

3cos

2sin

2cos

2

2

3sin

2sin1

2cos

2

2

3sin

2sin1

2cos

2

r

K

r

K

Tr

K

Ixy

Iyy

Ixx

56

von Mises stress field

Plane stress, Mode I, T=0 Plane strain, Mode I, T=0

13

,,,

rTrrr

D ,:,

2

3, rrr

DD

eq

57

Plane strain, Mode I

T / K = 0 m-1/2 T / K = 10 m-1/2T / K = -10 m-1/2 T / K = 5 m-1/2T / K = -5 m-1/2

Mechanisms controlled by shear

Plasticity,

Visco-plasticity

Fatigue


yyxx SST

58

Hydrostatic pressure

Plane stress, Mode I, T=0 Plane strain, Mode I, T=0

,rTr

Fluid diffusion (Navier Stokes),

Diffusion creep (Nabarro-Herring)

Chemical diffusion

59Plane strain, Mode I

T / K = 0 m-1/2 T / K = 10 m-1/2T / K = -10 m-1/2 T / K = 5 m-1/2T / K = -5 m-1/2

Hydrostatic pressure

,rTrFluid diffusion (Navier Stokes),

Diffusion creep (Nabarro-Herring)

Chemical diffusion

yyxx SST

60

Other T components, in Mode I

General triaxial

loadingEquibiaxial

plane strain

Superposition

non equibiaxial

conditions

Superposition

non plane strain

conditions

61

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

Mode I

cos2

sin22

cos2

cos22

2

3cos

2sin

2cos

2

2

3sin

2sin1

2cos

2

2

3sin

2sin1

2cos

2

rKu

rKu

r

K

r

K

Tr

K

Iy

Ix

Ixy

Iyy

Ixx

cos22

cos22

cos22

sin22

2

3sin

2sin1

2cos

2

2

3cos

2cos

2sin

2

2

3cos

2cos2

2sin

2

rKu

rKu

r

K

r

K

r

K

IIy

IIx

IIxy

IIyy

IIxx

Mode II

2sin

22

4

2cos

2

2sin

2

rKu

r

K

r

K

IIIz

IIIyz

IIIxz

G

Mode III

)43( n

n

zyyxxzz T

Déformation plane

nn

1

)3(

Contrainte plane

62

Mode I Mode II


13

,,,

rTrrr

D

,:,2

3, rrr

DD

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








64

65

Williams expansion

A self-similar solution in

the form is sought directly

as follows :

x

r

y

02 4

4

4

22

4

4

4

F

y

F

yx

F

x

F

grrF 2,

4

4

2

2222224

4

42

2

222

2

2222224

2

22

2

2

2

2

22

22

211

gggrF

gr

gr

grgrF

grgr

F

rr

Fr

rrF

66

x

r

y

022

4

4

2

2222224

gg

grF

02222

2

222

4

4

g

d

gd

d

gd

Dans ce cas g() doit vérifier

Williams expansion

A self-similar solution in

the form is sought directly

as follows :

grrF 2,

67

The solution is sought as follows :

x

r

y

ipAeg

2

20222222222224

p

p

pppp

02222

2

222

4

4

g

d

gd

d

gd

Williams expansion

68

x

r

y

222Re, iiii DeCeBeAerrF

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

rr

F

rr

rr

Fr

r

Williams expansion

69

x

r

y

222Re, iiii DeCeBeAerrF

022Re

022Re0,

0Re

0Re0,

22

22

22

22

iiii

iiii

r

iiii

iiii

DeCeBeAe

DeCeBeAer

DeCeBeAe

DeCeBeAer

Williams expansion

70

A sery of eligible solutions is

obtained :

x

r

y

12

sin12

sin

12

cos12

cos

12

nC

nAg

nD

nBg

n

n even

n odd

grrFn

12,

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 expansion

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








72

73

J contour integralThe 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 − 𝜈2

𝐸𝐾𝐼2 + 𝐾𝐼𝐼

2 +1 + 𝜈

𝐸𝐾𝐼𝐼𝐼2

𝐺𝑐 =1 − 𝜈2

𝐸𝐾𝐼𝑐2

Energy release rate

Fracture toughness

LEFM

KI, KII, KIII

T, Tz, G

A. Modes








74

75

Mode I, LEFM, T=0

Syy

Syy

Syy

Syy

76

LEFM stress field (Mode I)

Von Mises equivalent deviatoric stress

77

Irwin’s plastic zones size, step 1: rY

Along the crack plane, =0

00,,2

20,,2

0,0,

n

rr

Kr

r

Krr xy

Izz

Iyyxx

n

13

2

20,

r

Krp I

H

r

Kr I

eq

n

2

210,

Yield criterion : YYeq r 0,

2

22

2

21

Y

IY

Kr

n

78

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

rY rp

79

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

80

Irwin’s plastic zones size, step 2: balance

yy(r,=0)

r

Elastic field

Y

rY rp

r

rr Y

I

rr

r

Y

r

r

I

pm

pm

drrr

Kdrdr

r

K

n

2212

max

00

max

2

2221

2Y

IYpm

Krr

n

81

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

82




83




84

Mode I, Monotonic and cyclic plastic zones

Plastic strain (%)

Str

es

s (

MP

a)

85

Mode I, Monotonic and cyclic plastic zones

Monotonic plastic zone

Cyclic plastic zone

2

2

max2

21

Y

I

mpz

Kr

n

2

22

4

21

Y

I

cpz

Kr

n D

86

T-Stress effect

aSK yyI

yyxx SST

87

T-Stress effect

88

T-Stress effect

Irwin’s plastic zone, Y=400 MPa, KI=15MPa.m1/2

aSK yyI

yyST

89

Ductile fracture

Measurement

of the crack tip

opening angle

at the onset of

fracture

90

Example of the effect of a T-Stress for long cracks

910.48 % Carbon Steel [Hamam,2007]

Example of the effect of a T-Stress for long cracks

Fatigue, and crack growth modeling

93

Measurements

J.Petit

COD

Potential drop

Direct optical measurements

Digital image correllation

F

COD

Crack length increasing

94

a

a

NLoad cycle N

Fmax

Fmin

Fop

R=Fmin/Fmax

DF

DFeff

da/dN = f(a)

Paris’ lawICI KK max

95

A - threshold regime

B – Paris’ regime

C - unstable fracture

theff KK DD

Subcritical crack

growth if DK is over

the non propagation

threshold mMPaKeffD

[Neumann,1969]

96

Fatigue – Threshold regime

97

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

Fatigue – Threshold regime

98

“pseudo-cleavage” facets at the initiation site

Fatigue – Threshold regime – fracture surface

99

INCO 718

Fatigue – Threshold regime – fracture surface

Paris’ lawICI KK max

100

A - threshold regime

B – Paris’ regime

C - unstable fracture

theff KK DD

Subcritical crack

growth if DK is over

the non propagation

threshold mMPaKeffD

Paris’ regime : crack growth by the striation process

101

31

6L

TA

6V

OF

HC

[Laird,1967], [Pelloux, 1965]

INC

O 7

18

102

103

104

105

106

107

108

109

110

111

112

Crack growth is governed by crack tip plasticity

• the quantities of LEFM (KI, KII, KIII) control the behavior of the K-dominance area

• which controls the behavior of the plastic zone

• which controls crack growth by pure fatigue

113

Consequences

Outline

• Introduction

• History effects in mode I• Observations

• Long distance effects

• Short distance effects

• Modelling

• History effects in mixed mode

• Observations• Crack growth rate

• Crack path

• Simulation

• Modelling114

115

Long distance effect (overload)

CCT, 0.48% carbon steel, [Hamam et al. 2005]

Cra

ck le

ng

th(m

m)

Number of cycles

Constant amplitude fatigue

idem + 1 OL (factor 1.5)

idem + 1 OL (factor 1.8)

116

Long distance effect (residual stresses)

openingK

Outline

• Introduction




• Modelling



• Crack path

• Simulation

• Modelling117

118CT, 316L austenitic stainless steel, [Pommier et al]

Cra

ck le

ng

th–

aO

L(m

m)

Number of cycles

Constant amplitude fatigue

idem after 1 OL (factor 2)

idem after 10 OL (factor 2)

Short distance effect (repeated overloads)

119

991

100 9900

Short distance effect (block loadings)

• If the plastic zone is well

constrained inside the K-

dominance area

• It is subjected to strain controlled

conditions by the elastic bulk,

• Mean stress relaxation

• Material cyclic hardening

Outline

• Introduction




• Modelling



• Crack path

• Simulation

• Modelling121

122

• Issues

• A very small plastic zone produces very large effects on

the fatigue crack growth rate and direction

• Finite element method : elastic plastic material, very fine

mesh required, 3D cracks, huge number of cycles to be

modelled, tricky post-treatment

• Fastidious and time consuming

Linear elasticFE analyses

for 3D cracks

elasticplastic

FE + POD

A simplified approach is needed: the elastic-plastic behaviour

of the plastic zone is condensed a non-local elastic-plastic

model tailored for cracks

Method

Scale transition Generation of evolutions

of r (CTOD) versus KI

Expérimental input n°1

Constitutive model

LOCAL

Tensile Push

pull test

,...

fdt

d

..., II KdKgdt

d

r

Fatigue crack growth

experiment

Expérimental input n°2

dt

d

dt

da r

Crack growth model,

including history effects,

dt

d

dt

da rCTOD

dN

daD

125

Adjust the coefficient

a using one constant

amplitude fatigue

crack growth

experiment

da/dt : rate of production of cracked area per unit length

of the crack front

126

Single overload : long range retardation

127

Block loading : short range retardation

128

Stress ratio (mean stress) effect (R>0)

129

Stress ratio (mean stress) effect (R<0)

X2

130

number of blocks

Random loading simulations

Outline

• Introduction




• Modelling



• Crack path

• Simulation

• Modelling131

132

Growth criteria in mixed mode conditions ?

𝑑𝑎

𝑑𝑁= 𝐶∆𝐾𝑒𝑞

𝑚

𝛥𝐾𝑒𝑞 = ∆𝐾𝐼𝑛 + 𝛽∆𝐾𝐼𝐼

𝑛 + 𝛾∆𝐾𝐼𝐼𝐼𝑛 1 𝑛

Same values of Kmax, Kmin, DK for each mode

Fatigue crack growth experiments

Crack growth rate

Crack path

133

Load paths in mixed mode I+II

134

Load paths in mixed mode I+II+III

135

𝐾𝐼∞

𝐾𝐼𝐼∞

𝐾𝐼𝐼𝐼∞

=

𝑓𝐼(2𝑎) 𝑓𝐼(2𝑎) 0𝑓𝐼𝐼(2𝑎) −𝑓𝐼𝐼(2𝑎) 0

0 0 𝑓𝐼𝐼𝐼(2𝑎)

𝐹𝑋𝐹𝑌𝐹𝑍

136

Experimental protocol

6 actuators hydraulic testing machine - ASTREE

137

Fatigue crack growth in mixed mode I+II+III

138

Crack path – mode I+II+III

139

Mode III contribution

140


141


142

FE model and boundary conditions

Periodic BC along the two faces normal to the crack front

Prescribed displacements based on LEFM stress intensity

factors

𝑲𝑰∞𝒖𝒃𝒄_𝒏𝒐𝒎

𝑰 , 𝑲𝑰𝑰∞𝒖𝒃𝒄_𝒏𝒐𝒎

𝑰𝑰 ,𝑲𝑰𝑰𝑰∞ 𝒖𝒃𝒄_𝒏𝒐𝒎

𝑰𝑰𝑰

Elastic plastic material constitutive behaviour (kinematic and

isotropic hardening identified experiments)

143

Crack : locally self similar geometry → locally self similar

solution 𝒇 𝜶𝒓 =𝒌 𝜶 𝒇 𝒓

Small scale yielding 𝒇 𝒓𝒓→∞

𝟎

V𝐞𝐥𝐨𝐜𝐢𝐭𝐲 𝐟𝐢𝐞𝐥𝐝 ∶ 𝒇 𝒓 = 𝟎 𝐟𝐢𝐧𝐢𝐭𝐞

𝒇𝒊 𝒓 = 𝒇𝒊 𝟎 𝒆−𝒓𝒑

144

Cumulated equivalent plastic strain

145

radial distribution𝑷𝑶𝑫𝟐 → 𝒖𝒊

𝒄(𝑷) ≈ 𝐟 𝒓 𝒈𝒊𝒄(𝜽)

𝒇𝒊 𝒓 = 𝒇𝒊 𝟎 𝒆−𝒓𝒑

146

𝑲𝒊 𝒕 = 𝑷𝝐𝑫𝒗

𝑬𝑭_𝒊 𝑷, 𝒕 . 𝒖𝒊𝒆(𝑷)

𝑷𝝐𝑫𝒖𝒊𝒆(𝑷). 𝒖𝒊

𝒆(𝑷)𝝂𝒊𝒆 𝑷, 𝒕 = 𝑲𝒊 𝒕 𝒖𝒊

𝒆(𝑷)

𝒗𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 = 𝒗𝑬𝑭_𝒊 𝑷, 𝒕 − 𝝂𝒊𝒆 𝑷, 𝒕

POD based post treatment

𝑢𝑖𝑒(𝑃)

Solution of an elastic FE analyses with𝑲𝒊∞=1MPa.m1/2 for each mode

147

𝒗𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 = 𝒗𝑬𝑭_𝒊 𝑷, 𝒕 − 𝝂𝒊𝒆 𝑷, 𝒕

𝑷𝑶𝑫𝟏 → 𝒗𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 ≈ 𝝆𝒊 𝒕 . 𝒖𝒊𝒄(𝑷)

𝑃𝑂𝐷2 → 𝑢𝑖𝑐(𝑃) ≈ f 𝑟 𝑔𝑖

𝑐(𝜃)

𝑔𝐼𝑦𝑐 𝜃 = 𝜋 = −𝑔𝐼𝑦

𝑐 𝜃 = −𝜋 =1

2

lim𝑟→0

𝑓 𝑟 =1


148


𝒗 𝑷, 𝒕 =

𝒊=𝟏

𝟑

𝑲𝒊 𝒕 . 𝒖𝒊𝒆(𝑷)

𝝂𝒊𝒆 𝑷,𝒕

+ 𝝆𝒊 𝒕 . 𝒖𝒊𝒄(𝑷)

𝝂𝒊𝒄 𝑷,𝒕

𝒖𝒊𝒆(𝑷)

Intensity factors, non-local variables

𝒖𝒊𝒄(𝑷)

𝑲𝒊 𝒕

𝝆𝒊 𝒕

Field basis / weigthing functions tailored for cracks in elastic plastic materials

149

FE Simulations and results

150

Crack propagation law

151

𝒂𝒏∗ = 𝜶 𝒕 ⋀ 𝝆

In mode I, this lawderives from the CTOD equation

In mode I+II+III, itderives from the Li’smodel

152

FE Simulations and results

153

154

Intensity factor evolutions

155

Mode III contribution ?

A Mode III load step increases the amplitude

of Mode I and of Mode II plastic flow

156

157

ApproachFE model

Material constitutive law,

local and tensorial

𝜀 = 𝑓 𝜎, 𝑒𝑡𝑐.

𝒗 𝑷, 𝒕

𝜌 = 𝜌𝐼 , 𝜌𝐼𝐼

𝐾∞ = 𝐾𝐼∞ , 𝐾𝐼𝐼

∞

Crack tip regionconstitutive law, non-local

and vectorial

𝜌 = 𝑔 𝐾∞, 𝑒𝑡𝑐.

- Elastic domain (internal variables)

- Normal plastic flow rule

- Evolution equations

158

Elastic domain :

generalized Von Mises Criterion

𝑓𝑌 =𝐾𝐼∞ − 𝐾𝐼

𝑋

𝐾𝐼𝑌

2

+𝐾𝐼𝐼∞ − 𝐾𝐼𝐼

𝑋

𝐾𝐼𝐼𝑌

2

− 1

𝑓𝑌 =𝐺𝐼

𝐺𝐼𝑌 +

𝐺𝐼𝐼

𝐺𝐼𝐼𝑌 − 1

𝐺𝑖 =𝑠𝑖𝑔𝑛 𝐾𝑖

∞ − 𝐾𝑖𝑋 𝐾𝑖

∞ − 𝐾𝑖𝑋 2

𝐸∗

Model

159

𝒇 =𝑲𝑰∞ −𝑲𝑰

𝑿 𝟐

𝑲𝑰𝒀 𝟐

+𝑲𝑰𝑰∞ −𝑲𝑰𝑰

𝑿 𝟐

𝑲𝑰𝑰𝒀 𝟐

+𝑲𝑰𝑰𝑰∞ −𝑲𝑰𝑰𝑰

𝑿 𝟐

𝑲𝑰𝑰𝑰𝒀 𝟐

− 𝟏

Yield criterion

𝒇 𝑮𝑰, 𝑮𝑰𝑰, 𝑮𝑰𝑰𝑰 =𝑮𝑰

𝑮𝑰𝒀 +

𝑮𝑰𝑰

𝑮𝑰𝑰𝒀 +

𝑮𝑰𝑰𝑰

𝑮𝑰𝑰𝑰𝒀 − 𝟏

Flow rule

𝝆𝒊 = 𝝀𝒔𝒊𝒈𝒏𝒆 𝑮𝒊

𝑮𝒊𝒀

Evolution equation

𝑲𝑿 = 𝑪 𝝆 −𝚪 𝑲𝑿𝒆𝒒

𝑴−𝟏

𝟏 + 𝚪 𝑲𝑿𝒆𝒒𝑴−𝟏 𝒅 𝝆 𝒅 𝒘𝒉𝒆𝒓𝒆 𝒅 =

𝑲𝑿

𝑲𝒆𝒒𝑿

160

Conclusions

• Fatigue crack growth experiments in Mixed mode I+II+III non

proportionnal loading conditions

• Result : A load path effect is observed on fatigue crack growth

and on the crack path

• Adding a mode III step to mixed mode I+II fatigue cycles

increases the fatigue crack growth rate

• Elastic-plastic FE analyses show that accounting for plasticity

allows predicting the load path effect and the effect of mode III

Plasticity

• A simplified model has been developped to replace non-linear

FE analyses

161

Basics elements on linear elastic fracture mechanics and ...

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