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>
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Basics elements on linear elastic fracture
mechanics and crack growth modeling
Sylvie Pommier,
LMT
(ENS Paris-Saclay, CNRS, Université Paris-Saclay)
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)
2
Foundations of fracture mechanics : The Liberty Ships
Liberty ships – hiver 1941
• 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
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
Liberty Ships, WWII, 1941, Brittle fracture
4
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)
Georges Rankine Irwin
•
•
•
•
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, 417418.
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 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
8
Development of rules for the EASA certification
Aloha April, 28th 1988,
Los Angeles, June, 2nd 2006,
9
Rotor Integrity Sub-Committee (RISC)
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
 AIA Rotor Integrity Sub-Committee (RISC) : Elaboration of AC 33.14-1
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
Elaboration of AC 33.70-2
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
Damage
tolerance
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 ?
13
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.
14
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
15
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
16
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, nonplanar 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,
17
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
Data : critical energy release rate Gc
Griffith’ theory
Wext : work of external forces
DU elastic : variation of the elastic energy of the structure
DU surface: variation of the surface energy of the structure
DU  DU elastic  DU surface  Wext
 DU surface  2 da  Wext  DU elastic
G  2
Criteria :
where
DU elastic  Wext
G 
da
19
Evolution by Bui, Erlacher & Son
dU  Wext  Q
where
TdS  Q  0
dU  dF  TdS  SdT
in isothermal conditions dT  0
TdS  Q  dF  Wext   0
 dFvolume  dFsurface  Wext   0
 G  Gc  da  0
where
Free energy instead of internal energy
Isothermal conditions instead of adiabatic
Second principle
Gc 
DFsurface
 2
da

DFvolume  Wext 
G
da
20
J Integral (Rice)

DFvolume  Wext 
G
da
Eschelby tensor : energy density
J integral , (Rice’s integral if q is coplanar)
q vector: the crack front motion
21
J contour integral
If the crack faces are free
surfaces (no friction, no
fluid pressure …),
y
If volume forces can be
neglected (inertia, electric
field...)
x
Then the J integral is shown
to be independent of the
choice of the selected
integration contour
𝐺 = 𝐽=
Γ
𝜑𝑓𝑟𝑒𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑑𝑦 − 𝜎𝑛.
𝜕𝑢
𝜕𝑥
22
Applications
C. Stoisser, I. Boutemy and F. Hasnaoui
23
• 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
Limitations
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
Preliminary remarks:
From the discontinuity to the singularity
Stress concentration factor Kt of an elliptical hole,
With a length 2a and a curvature radius r
2a
r
 loc
a
K t    1 2

r
2a
 loc
r  0    2 loc

a
r

Singularity
26
Remarks: existence of a singularity
Geometry locally-self-similar → self-similar solution
→ principle of simulitude
r 0
 r ,   f r g  
2a
r*   r
r

f (r * )  q  f (r )
r : distance to the discontinuity
Warning: implicit choice of scale
27
Order of this singularity
For a crack : =-0.5
Linear elasticity:
 r   Br 
r 0
 r   Cr 
r 0
Eelast  A
r 0
  r rmax
 

r rd dr
0
   r  0
Eelast  2A
r 0
Eelast  2A
r 0
r
r rmax
2  1
r
r 0
2 2
rmax
2  2
dr
Eelast
2  2
rmax
 2 A
r 0
2  2
2  2  0    1
  1   280
2  2  0    1
Non linear material behaviour ?
 
   o  
 o 
n
n  1 elastic

n=4
 r   Ar 
r 0
 r   Br n

r 0
Eelast  C
r 0
  r rmax
 n

Eelast
r
rd dr
   r  0
Eelast  2C
r 0
r
r rmax
1n  1
r
dr

r 0
1n   2
rmax
 2C
r 0
1  n   2
1  n   2  0
2
 
1 n
29
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
30
Fracture modes
Planar symmetric
Planar anti-symmetric
Anti-planar
31
Fracture modes
Tubes (pipe line)
32
Fracture modes
Various fractures in compression
33
Fracture modes
Various fractures in torsion
34
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
35
Case of mode I
Analysis of Irwin based on Westergaard’s analysis
and Williams expansions
Planar Symmetric
36
Balance equation
Div  f v  r a
2D problem, quasi-static, no volume force
 xx  xy  xz


0
x
y
z
 xy  yy  yz


0
x
y
z
 xz  yz  zz


0
x
y
z
37
Linear isotropic elasticity : E, n
1 n
n

  Tr   1
E
E
E
 xx n yy 
 xx 
2
1 n
E
 yy n xx 
 yy 
2
1 n
E
 xy 
 xy
1 n
38
Compatibility equations
 2 xx  3u x

2
2
 y x y
u x
 xx 
x
u y
 yy 
y
 2 yy
 x
2
1  u x u y 

 xy  

2  y x 
  xy

 2 xy
 3u y
y x
2
 3u y
 3u x
2


2
2
xy y x x y
  yy
 2 xx
2
 2  2
xy
 x
 y
2
2
39
Combination
Compatibility
 2 xy
 2 yy
 2 xy
 2 yy
 2 xx
2
 2  2
xy  x
 y
 2 xx
2
 2  2
xy
 x
 y
Balance equations
+
Linear elasticity
E
 xx n yy 
2
1 n
E
 yy n xx 
 yy 
2
1 n
E
 xy 
 xy
1 n
 xx 
+
 xx  xy

0
x
y
 xy  yy

0
x
y
= 3 Equations, 3 unknowns
40
Airy function F(x,y)
Balance equation
 xx  xy  xy  yy



0
x
y
x
y
-1862Compatibility
 2 xy
 2 yy
 2 xx
2
 2  2
xy
 x
 y
Assuming
2F
 xx  2
y
2F
 yy  2
x
2F
 xy  
xy
4F
4F
4F
2 2 2  4 0
4
 x
 x y  y
1 equation, 1 unknow
F(x,y)
41
Z(z) , z complex,
4F
4F
4F
2 2 2  4 0
4
 x
 x y  y
F=F(x,y)
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)
4Z 4Z
 4
4
 x  z
4Z
4Z
 4
2
2
 x y
 z
4Z 4Z
 4
4
 y  z
Z (z) always fulfill all the
equations of the problem
Z(z) must verify the symmetry
and the boundary conditions
42
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
43
Irwin’s or Westergaard’s analyses
S
y
S
2a
x
S
S
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
2D problem, plane (x,y) : Szz=n(Sxx+Syy)
Symmetric with respect to y=0 & x=0
Away from the crack (x & y >> a) : sxx= S syy= S & sxy= 0
Singularities in y=0 x=+a & y=0 x=-a
44
Boundary conditions & Symmetries
 xx   yy  S ,  xy  0
&
2F
 xx  2
y
2F
 yy  2
x
2F
 xy  
xy


S 2 2
F  y  x  a2 x  a3 y  a4
2

symmetries


S 2 2
F  y  x  a4
2

45
Construction of Z(z)
F 


S 2 2
y  x  a4
2
S 2
Z  z  a4
2

Relation
 Z 
 Z 
F  Re Z   yRe    Re Z   yI m  
 z 
 y 
2F
 xx  2
y
 2Z 
 3Z 
 xx  Re  2   yI m  3 
 z 
 z 
2F
 yy  2
x
2F
 xy  
xy
 2Z 
 3Z 
 yy  Re  2   yI m  3 
 z 
 z 
 3Z 
 xy   yRe  3 
 z 
46
Solution
At infinity
At infinity
    S,   0

xx
Solution:

yy

xy
S 2
Z  z  a4
2

 2Z 
 3Z 
 xx  Re  2   yI m  3 
 z 
 z 
 2Z 
 3Z 
 yy  Re  2   yI m  3 
 z 
 z 
 3Z 
 xy   yRe  3 
 z 
Valid for any 2D problem, with symmetries along the
planes y=0 & x=0, and biaxial BCs
47
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
48
Exact solution for a crack
Singularities
in y=0 x=+a
& y=0 x=-a
 2Z 
 3Z 
 xx  Re  2   yI m  3 
 z 
 z 
 2Z 
 3Z 
 yy  Re  2   yI m  3 
 z 
 z 
S 2
Z  z  a4
2
Z 
 Sz
z

+
1
za
 3Z 
 xy   yRe  3 
 z 
Z

z
1
za

+
 S z a
2
2
Exact solution

1
2
49
Asymptotic solution
y
- Irwin-
Local coordinates (r,), r → 0
z  a  re
r
Exact Solution
i
Z 

x
2Z
Sz

2
z
z2  a2


1
2
3Z
 Sa 2

3
3
2
2 2
z
z a


z
2Z
Sa

2
z
2arei


 S z a
2
2

1
2


1
2
S a i 2

e
2r
3
3Z
 Sa 2
1 S a i 2


e
3
3
z
r 2r
2arei 2


50
Asymptotic solution
- Irwin
2Z
S a i 2

e
2
z
2r
 2Z 
 3Z 
 xx  Re  2   yI m  3 
 z 
 z 
 2Z 
 3Z 
 yy  Re  2   yI m  3 
 z 
 z 
3
3Z
1 S a i 2

e
3
z
r 2r
 3Z 
 xy   yRe  3 
 z 
Westergaard’s stress function :
y
r

x
 xx 
S a


3 
cos 1  sin sin 
2
2
2
2 r
 yy 
S a


3 
cos 1  sin sin 
2
2
2
2 r
 xy 
S a  

3 
 cos sin cos 
2
2
2
2 r 
51
Error associated to this Taylor development along =0
𝜎𝑦𝑦 𝑟, 𝜃 = 0 =
Exact solution
𝑆𝑦𝑦 𝑎 + 𝑟
=
𝑟 2𝑎 + 𝑟
Asymptotic solution
𝜎𝑦𝑦
Error
𝐾𝐼
3 𝑟
5 𝑟
𝑟, 𝜃 = 0 =
1+
+
4 𝑎
32 𝑎
2𝜋𝑟
2
+𝑂
5
𝑟2
𝐾𝐼 𝑎 + 𝑟
𝜋𝑎𝑟 2𝑎 + 𝑟
3𝑟
𝑒𝑟𝑟𝑜𝑟~
4𝑎
1 term
0.1
0.01
Erreur = 1%
2 terms
0.001
1 term
𝑟
= 𝟎. 𝟎𝟏𝟑
𝑎
2 terms
𝑟
= 𝟎. 𝟐𝟗
𝑎
3 terms
𝑟
= 𝟎. 𝟔𝟗
𝑎
10 4
0.1
0.2
0.3
r/a
0.4
0.5
52
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
53
Mode I, non equi-biaxial conditions
Equibiaxial
Biaxial (Superposition)

  

K I  S yy a
T  S xx  S yy

 


54
Stress intensity factors
Similitude principle
 xx 
(geometry locally planar, with a
straigth crack front, self-similar,
singularity)
KI


3 
cos 1  sin sin   T
2
2
2
2 r
 yy 
KI


3 
cos 1  sin sin 
2
2
2
2 r
Same KI & T → Same local field
 xy 
KI
 
3
cos sin cos
2
2
2
2 r
KI &T
Crack geometry and
boundary conditions
Spatial distribution, given
once for all, in the crack
front region
gij()
f(r)=r
55
von Mises stress field
Tr  r ,  
 r ,    r ,  
1
3
D
Plane stress, Mode I, T=0
3 D
 eq r ,  
 r ,  :  D r , 
2
Plane strain, Mode I, T=0
56
von Mises stress field
Mechanisms controlled by shear
 Plasticity,
 Visco-plasticity
 Fatigue
T  S xx  S yy
T / K = -10 m-1/2 T / K = -5 m-1/2 T / K = 0 m-1/2 T / K = 5 m-1/2 T / K = 10 m-1/2
Plane strain, Mode I
57
Hydrostatic pressure
Tr  r ,  
Fluid diffusion (Navier Stokes),
Diffusion creep (Nabarro-Herring)
Chemical diffusion
Plane stress, Mode I, T=0
Plane strain, Mode I, T=0
58
Hydrostatic pressure
Tr  r ,  
Fluid diffusion (Navier Stokes),
Diffusion creep (Nabarro-Herring)
Chemical diffusion
T  S xx  S yy
T / K = -10 m-1/2 T / K = -5 m-1/2 T / K = 0 m-1/2 T / K = 5 m-1/2 T / K = 10 m-1/2
Plane strain, Mode I
59
Other T components, in Mode I
General triaxial
loading
Equibiaxial
plane strain
Superposition
non equibiaxial
conditions
Superposition
non plane strain
conditions
60
Full solutions KI, KII, KIII, T, Tz & G
Mode I
 xx 
KI


3 
cos 1  sin sin   T
2
2
2
2 r
KI


3 
 yy 
cos 1  sin sin 
2
2
2
2 r
 xy 
KI
 
3
cos sin cos
2
2
2
2 r
Mode II
 K II


3 
 xx 
sin  2  cos cos 
2
2
2
2 r
 xz 
K


3
 yy  II sin cos cos
2
2
2
2 r
 yz 
K


3 
 xy  II cos 1  sin sin 
2
2
2
2 r
uz 
ux 
KI r

cos   cos  
2 2
2
ux 
K II
2
r

sin 2    cos  
2
2
uy 
KI r

sin   cos  
2 2
2
uy 
K II
2
r

cos 2    cos  
2
2
Déformation plane
 zz  n  xx   yy   Tz
  (3  4n )
Mode III
 K III

sin  G
2
2 r
K III

cos
2
2 r
4 K III
2
r

sin
2
2
Contrainte plane

(3 n )
1 n 
61
von Mises stress field
Tr  r ,  
 r ,    r ,  
1
3
D
Mode I
3 D
 eq r ,  
 r ,  :  D r , 
2
Mode II
62
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
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
64
Williams expansion
4F
4F
4F
4

2



F 0
4
2
2
4
 x
 x y  y
A self-similar solution in
the form is sought directly
as follows :
F r ,   r   2 g  
y
r

x
2
1   F  1  2 F
2 
  g  
 F
r
  2 2    2 r g    r
r r  r  r 
 2
2
2
2
4





g


g


g  
2
 2
 2


 4 F    2 2 r   2 g    2 r   2


2
r

r
 2
 2
 4
2
4




g


g   
2
2
4
 2
2
2

 F  r    2  g      2  

2
4 

 

2


65
Williams expansion
y
r
A self-similar solution in
the form is sought directly
as follows :

F r ,   r   2 g  
x
 F r
4
 2
2

 4 g   
2 2
2
2  g  
   2   g      2   
0

2
4 

 



Dans ce cas g() doit vérifier
2
d 4 g  
d
g   2
2
2
2









2




2
g    0
4
2
d
d


66
Williams expansion
y
r

x
2
d 4 g   2
d
g   2
2
2









2




2
g    0
4
2
d
d


The solution is sought as follows :


g    Aeip


 p 4  2    2  p 2  2   2   0  p 2  2 p 2    2 
p  
p    2 
2
2
2
67

Williams expansion
y
r

x


F r ,    Re r   2 Aei  Be i  Ce i   2   De  i   2 

Boundary conditions are defined along the crack faces which are defined as
free surface (fluid pressure & friction between faces are excluded)
2F
  r ,     2 r ,     0
r
  F 
 r r ,      
r ,     0
r  r 
68
Williams expansion
y
r

x


F r ,    Re r   2 Aei  Be i  Ce i   2   De  i   2 
 
 r

r ,     0 
Re Ae
Re Ae
r ,     0 
Re Ae


 0

Re Aei  Be i  Ce i   2   De i   2   0
i
 Bei  Ce i   2   Dei   2 
i
i


 0
 Bei    2Ce i   2     2Dei   2   0
 Be i    2Ce i   2     2De i   2 
69
Williams expansion
y
r

x
2  1  n
A sery of eligible solutions is
obtained :
F r ,   r
n
1
2
n even
 n  
 n  
g    B cos   1   D cos   1 
 2  
 2  
n odd
 n  
 n  
g    A sin    1   C sin    1 
 2  
 2  
g  
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
70
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
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
72
J contour integral
The J integral is shown to be
independent of the choice of the
selected integration contour
𝐺 = 𝐽=
Γ
𝜑𝑑𝑦 − 𝜎𝑛.
y
𝜕𝑢
𝜕𝑥
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
Energy release rate
1 − 𝜈2
𝐺=
𝐸
x
𝐾𝐼2
1 − 𝜈2
2
𝐺𝑐 =
𝐾𝐼𝑐
𝐸
+ 𝐾𝐼𝐼2
1+𝜈
2
+
𝐾𝐼𝐼𝐼
𝐸
Fracture toughness
73
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
LEFM
KI, KII, KIII
T, Tz, G
H. Irwin’s plastic zones
74
Mode I, LEFM, T=0
Syy
Syy
Syy
Syy
75
LEFM stress field (Mode I)
Von Mises equivalent deviatoric stress
76
Irwin’s plastic zones size, step 1: rY
Along the crack plane, =0
KI
KI
 xx r ,  0   yy r ,  0 
,  zz r ,  0  2n
,  xy r ,  0  0
2 r
2 r
KI 2
1 n 
pH r ,   0  
2 r 3
Yield criterion :
K I 1  2n 
 eq r ,  0 
2 r
 eq rY ,  0   Y
rY
2

1  2n 

2
K I2
 Y2
77
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)
Y
Elastic field
rY
rp
r
78
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
79
Irwin’s plastic zones size, step 2: balance
yy(r,=0)
rpm  2rY
2

1  2n 


K I2
 Y2
Y
Elastic field
rY
r 

r 0
max
I
K
dr 
2 r
r
rp
r  rpm
Y
r 
 1  2n  dr  
r 0
r  rpm
K Imax
dr
2 r  rY 
80
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
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
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
83
Stress (MPa)
Mode I, Monotonic and cyclic plastic zones
Plastic strain (%)
84
Mode I, Monotonic and cyclic plastic zones
Monotonic plastic zone

1  2n 

2
rmpz

K I2 max
 Y2
Cyclic plastic zone

1  2n 

2
rcpz

DK
4
2
I
2
Y
85
T-Stress effect
T  S xx  S yy

 



  

K I  S yy a
86
T-Stress effect
87
T-Stress effect
K I  S yy a
T   S yy
Irwin’s plastic zone, Y=400 MPa, KI=15MPa.m1/2
88
Ductile fracture
Measurement
of the crack tip
opening angle
at the onset of
fracture
89
Example of the effect of a T-Stress for long cracks
90
Example of the effect of a T-Stress for long cracks
0.48 % Carbon Steel [Hamam,2007]
91
Fatigue, and crack growth modeling
Measurements
F
Crack length increasing
COD
J.Petit
Potential drop
Direct optical measurements
Digital image correllation
COD

93
a
da/dN = f(a)
a
N
Load cycle N
Fmax
Fop
Fmin
R=Fmin/Fmax
DF
DFeff
94
Paris’ law
K Imax  K IC
A - threshold regime
B – Paris’ regime
C - unstable fracture
DK eff  DK th

DK eff MPa m

Subcritical crack
growth if DK is over
the non propagation
threshold
95
Fatigue – Threshold regime
[Neumann,1969]
96
Fatigue – Threshold regime
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
97
Fatigue – Threshold regime – fracture surface
“pseudo-cleavage” facets at the initiation site
98
Fatigue – Threshold regime – fracture surface
INCO 718
99
Paris’ law
K Imax  K IC
A - threshold regime
B – Paris’ regime
C - unstable fracture
DK eff  DK th

DK eff MPa m

Subcritical crack
growth if DK is over
the non propagation
threshold
100
Paris’ regime : crack growth by the striation process
316L
INCO 718
OFHC
TA6V
[Laird,1967], [Pelloux, 1965]
101
102
103
104
105
106
107
108
109
110
111
Crack growth is governed by crack tip plasticity
112
Consequences
• 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
• Introduction
• History effects in mode I
• Observations
• Long distance effects
• Short distance effects
• Modelling
Outline
• History effects in mixed
mode
• Observations
• Crack growth rate
• Crack path
• Simulation
• Modelling
114
Long distance effect (overload)
Crack length (mm)
Constant amplitude fatigue
idem + 1 OL (factor 1.5)
idem + 1 OL (factor 1.8)
Number of cycles
CCT, 0.48% carbon steel, [Hamam et al. 2005]
115
Long distance effect (residual stresses)
K opening
116
• Introduction
• History effects in mode I
• Observations
• Long distance effects
• Short distance effects
• Modelling
Outline
• History effects in mixed
mode
• Observations
• Crack growth rate
• Crack path
• Simulation
• Modelling
117
Crack length – aOL (mm)
Short distance effect (repeated overloads)
idem after 1 OL (factor 2)
idem after 10 OL (factor 2)
Constant amplitude fatigue
Number of cycles
CT, 316L austenitic stainless steel, [Pommier et al]
118
Short distance effect (block loadings)
1
99
100
9900
119
• If the plastic zone is well
constrained inside the Kdominance area
• It is subjected to strain controlled
conditions by the elastic bulk,
• Mean stress relaxation
• Material cyclic hardening
• Introduction
• History effects in mode I
• Observations
• Long distance effects
• Short distance effects
• Modelling
Outline
• History effects in mixed
mode
• Observations
• Crack growth rate
• Crack path
• Simulation
• Modelling
121
• 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
122
A simplified approach is needed: the elastic-plastic behaviour
of the plastic zone is condensed a non-local elastic-plastic
model tailored for cracks
elastic
plastic
FE +
POD
Linear elastic
FE analyses
for 3D cracks
Method
d
dt
 f  ,...
Constitutive model
LOCAL
Scale transition
Generation of evolutions
of r (CTOD) versus KI
Expérimental input n°2
dr
 g dK I , K I ...
dt
da
dr

Tensile Push
pull test
Expérimental input n°1
dt
Fatigue crack growth
experiment
dt
Crack growth model,
including history effects,
da/dt : rate of production of cracked area per unit length
of the crack front
da
 DCTOD
dN

da
dr

dt
dt
Adjust the coefficient
a using one constant
amplitude fatigue
crack growth
experiment
125
Single overload : long range retardation
126
Block loading : short range retardation
127
Stress ratio (mean stress) effect (R>0)
128
Stress ratio (mean stress) effect (R<0)
X2
129
Random loading simulations
number of blocks
130
• Introduction
• History effects in mode I
• Observations
• Long distance effects
• Short distance effects
• Modelling
Outline
• History effects in mixed
mode
• Observations
• Crack growth rate
• Crack path
• Simulation
• Modelling
131
Growth criteria in mixed mode conditions ?
𝑑𝑎
𝑚
= 𝐶∆𝐾𝑒𝑞
𝑑𝑁
𝑛
𝑛
𝛥𝐾𝑒𝑞 = ∆𝐾𝐼 + 𝛽∆𝐾𝐼𝐼 + 𝛾∆𝐾𝐼𝐼𝐼
𝑛 1𝑛
Same values of Kmax, Kmin, DK for each mode
Fatigue crack growth experiments
Crack growth rate
Crack path
132
Load paths in mixed mode I+II
133
Load paths in mixed mode I+II+III
134
𝐾𝐼∞
𝐾𝐼𝐼∞
∞
𝐾𝐼𝐼𝐼
𝑓𝐼 (2𝑎)
= 𝑓𝐼𝐼 (2𝑎)
0
𝑓𝐼 (2𝑎)
−𝑓𝐼𝐼 (2𝑎)
0
0
0
𝑓𝐼𝐼𝐼 (2𝑎)
𝐹𝑋
𝐹𝑌
𝐹𝑍
135
Experimental protocol
6 actuators hydraulic testing machine - ASTREE
136
Fatigue crack growth in mixed mode I+II+III
137
Crack path – mode I+II+III
138
Mode III contribution
139
Mode III contribution
140
Mode III contribution
141
FE model and boundary conditions
Periodic BC along the two faces normal to the crack front
Prescribed displacements based on LEFM stress intensity
factors
𝑰
∞ 𝑰𝑰
∞ 𝑰𝑰𝑰
𝑲∞
𝑰 𝒖𝒃𝒄_𝒏𝒐𝒎 , 𝑲𝑰𝑰 𝒖𝒃𝒄_𝒏𝒐𝒎 ,𝑲𝑰𝑰𝑰 𝒖𝒃𝒄_𝒏𝒐𝒎
Elastic plastic material constitutive behaviour (kinematic and
isotropic hardening identified experiments)
142
Crack : locally self similar geometry → locally self similar
solution 𝒇 𝜶𝒓 =𝒌 𝜶 𝒇 𝒓
Small scale yielding 𝒇 𝒓
𝒓→∞
𝟎
𝒇𝒊 𝒓 = 𝒇𝒊 𝟎 𝒆
−
𝒓
𝒑
V𝐞𝐥𝐨𝐜𝐢𝐭𝐲 𝐟𝐢𝐞𝐥𝐝 ∶ 𝒇 𝒓 = 𝟎 𝐟𝐢𝐧𝐢𝐭𝐞
143
Cumulated equivalent plastic strain
144
radial distribution
𝑷𝑶𝑫𝟐 → 𝒖𝒄𝒊 (𝑷) ≈ 𝐟 𝒓 𝒈𝒄𝒊 (𝜽)
𝒇𝒊 𝒓 = 𝒇𝒊 𝟎 𝒆
𝒓
−𝒑
145
POD based post treatment
𝑢𝑖𝑒 (𝑃)
Solution of an elastic FE analyses with
1/2 for each mode
𝑲∞
𝒊 =1MPa.m
𝝂𝒆𝒊 𝑷, 𝒕 = 𝑲𝒊 𝒕 𝒖𝒆𝒊 (𝑷)
𝑲𝒊 𝒕 =
𝑬𝑭_𝒊 𝑷, 𝒕 . 𝒖𝒆 (𝑷)
𝒗
𝑷𝝐𝑫
𝒊
𝒆
𝒆
𝒖
(𝑷).
𝒖
𝑷𝝐𝑫 𝒊
𝒊 (𝑷)
𝒗𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 = 𝒗𝑬𝑭_𝒊 𝑷, 𝒕 − 𝝂𝒆𝒊 𝑷, 𝒕
146
POD based post treatment
𝒗𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 = 𝒗𝑬𝑭_𝒊 𝑷, 𝒕 − 𝝂𝒆𝒊 𝑷, 𝒕
𝑷𝑶𝑫𝟏 → 𝒗𝒓é𝒔𝒊𝒅𝒖_𝒊 𝑷, 𝒕 ≈ 𝝆𝒊 𝒕 . 𝒖𝒄𝒊 (𝑷)
𝑃𝑂𝐷2 → 𝑢𝑖𝑐 (𝑃) ≈ f 𝑟 𝑔𝑖𝑐 (𝜃)
𝑐
𝑔𝐼𝑦
𝜃=𝜋 =
𝑐
−𝑔𝐼𝑦
1
𝜃 = −𝜋 =
2
lim 𝑓 𝑟 =1
𝑟→0
147
POD based post treatment
𝟑
𝑲𝒊 𝒕 . 𝒖𝒆𝒊 (𝑷) + 𝝆𝒊 𝒕 . 𝒖𝒄𝒊 (𝑷)
𝒗 𝑷, 𝒕 =
𝒊=𝟏
𝑲𝒊 𝒕
𝝂𝒆𝒊 𝑷,𝒕
𝝂𝒄𝒊 𝑷,𝒕
Intensity factors, non-local variables
𝝆𝒊 𝒕
𝒖𝒆𝒊 (𝑷)
Field basis / weigthing functions tailored for
𝒖𝒄𝒊 (𝑷) cracks in elastic plastic materials
148
FE Simulations and results
149
150
Crack propagation law
𝒂𝒏∗ = 𝜶 𝒕 ⋀𝝆
In mode I, this law
derives from the
CTOD equation
In mode I+II+III, it
derives from the Li’s
model
151
FE Simulations and results
152
153
Intensity factor evolutions
154
Mode III contribution ?
A Mode III load step increases the amplitude
of Mode I and of Mode II plastic flow
155
156
Approach
FE model 𝒗 𝑷, 𝒕
Material constitutive law,
local and tensorial
𝜀 = 𝑓 𝜎, 𝑒𝑡𝑐.
𝜌 = 𝜌𝐼 , 𝜌𝐼𝐼
Crack tip region
constitutive law, non-local
and vectorial
𝐾 ∞ = 𝐾𝐼∞ , 𝐾𝐼𝐼∞
𝜌 = 𝑔 𝐾 ∞ , 𝑒𝑡𝑐.
-
Elastic domain (internal variables)
Normal plastic flow rule
Evolution equations
157
Elastic domain :
generalized Von Mises Criterion
𝑓𝑌 =
𝐾𝐼∞
−
𝐾𝐼𝑌
𝑋 2
𝐾𝐼
+
𝐾𝐼𝐼∞
−
𝐾𝐼𝐼𝑌
𝑋 2
𝐾𝐼𝐼
−1
𝐺𝐼
𝐺𝐼𝐼
𝑓𝑌 = 𝑌 + 𝑌 − 1
𝐺𝐼
𝐺𝐼𝐼
𝐺𝑖 =
𝑠𝑖𝑔𝑛
𝐾𝑖∞
−
𝐾𝑖𝑋
𝐸∗
𝐾𝑖∞
−
𝑋 2
𝐾𝑖
158
Model
Yield criterion
𝒇=
𝑿
𝑲∞
𝑰 − 𝑲𝑰
𝟐
+
𝟐
𝑲𝒀𝑰
𝒇 𝑮𝑰 , 𝑮𝑰𝑰 , 𝑮𝑰𝑰𝑰 =
𝑿
𝑲∞
𝑰𝑰 − 𝑲𝑰𝑰
𝟐
𝟐
𝑲𝒀𝑰𝑰
+
𝑿
𝑲∞
𝑰𝑰𝑰 − 𝑲𝑰𝑰𝑰
𝟐
𝑲𝒀𝑰𝑰𝑰
𝟐
−𝟏
𝑮𝑰
𝑮𝑰𝑰
𝑮𝑰𝑰𝑰
+
+
𝒀
𝒀
𝒀 −𝟏
𝑮𝑰
𝑮𝑰𝑰
𝑮𝑰𝑰𝑰
Flow rule
𝒔𝒊𝒈𝒏𝒆 𝑮𝒊
𝝆𝒊 = 𝝀
𝑮𝒀𝒊
Evolution equation
𝑲𝑿 = 𝑪 𝝆 −
𝚪 𝑲𝑴−𝟏
𝑿𝒆𝒒
𝟏+𝚪
𝑲𝑴−𝟏
𝑿𝒆𝒒
𝒅𝝆 𝒅
𝑲𝑿
𝒘𝒉𝒆𝒓𝒆 𝒅 = 𝑿
𝑲𝒆𝒒
159
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
160
161