Basics elements on linear elastic fracture mechanics … elements on linear elastic fracture...
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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>
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
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
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
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
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
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
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
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
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
von Mises stress field
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
von Mises stress field
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
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
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
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
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
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
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
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
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
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
• 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
• 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
• 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
• 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
• 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
• 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
Mode III contribution
141
Mode III contribution
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
POD based post treatment
148
POD based post treatment
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