University of Liège
Aerospace & Mechanical Engineering
Fracture mechanics, Damage and Fatigue
Linear Elastic Fracture Mechanics – Crack Growth
Fracture Mechanics – LEFM – Crack Growth
Ludovic Noels
Computational & Multiscale Mechanics of Materials – CM3
http://www.ltas-cm3.ulg.ac.be/
Chemin des Chevreuils 1, B4000 Liège
Elastic fracture
• Definition of elastic fracture
– Strictly speaking:
• During elastic fracture, the only changes to the material are atomic separations
– As it never happens, the pragmatic definition is
• The process zone, which is the region where the inelastic deformations
– Plastic flow,
– Micro-fractures,
– Void growth, …
happen, is a small region compared to the specimen size, and is at the crack tip
• Therefore
– Linear elastic stress analysis describes the fracture process with accuracy
Mode I Mode II Mode III
(opening) (sliding) (shearing)
2013-2014 Fracture Mechanics – LEFM – Crack Growth 2
• SIFs (KI, KII, KIII) define the asymptotic solution in linear elasticity
• Crack closure integral
– Energy required to close the crack by an infinitesimal da
– If an internal potential exists
with
• AND if linear elasticity
– AND if straight ahead growth
• J integral – Energy that flows toward crack tip
– If an internal potential exists
• Is path independent if the contour G embeds a straight crack tip
• BUT no assumption on subsequent growth direction
• If crack grows straight ahead G=J
• If linear elasticity:
• Can be extended to plasticity if no unloading (see later)
Summary
2013-2014 Fracture Mechanics – LEFM – Crack Growth 3
Elastic fracture in mode I
• Crack growth criterion in mode I
– For a crack to grow
• The energy released by the system has to be larger than the fracture energy Gc
• Criterion:
• The energy release rate G depends on
– The geometry
– The loading
– The crack size
• What does the fracture energy Gc depend on ?
– For perfectly brittle materials: this is a material dependant constant
– Practically, it has to account for the inelastic deformations happening in the
process zone
– For (an)isotropic materials, Gc is (not) the same in all the directions
» Example, in wood cracks grow along grains
– We consider isotropic linear elastic materials
– For an initial straight crack under mode I: • The crack will grow straight ahead (see next slides)
• So toughness and fracture energy are related
2013-2014 Fracture Mechanics – LEFM – Crack Growth 4
Elastic fracture in mode I
• Crack growth criterion in mode I (2)
– Toughness is defined for plane e state
• It is conservative as
• Near the free surface
– Deformations along thickness
release the stress state
– The SIF is lower and
– Process zone is not negligible
(see lecture on NLFM)
Gc is larger near free surfaces
• Plane strain & elastic fracture can be assumed
if the working zone is small compared
to the specimen and the crack length
– Thick specimen
(see lecture on NLFM)
– Crack large enough
(see lecture on NLFM)
t
Kc
K rupture
Plane s
Plane e
Brittle fracture
t1
Ductile
fracture t2<<t1
Plane e
t
2013-2014 Fracture Mechanics – LEFM – Crack Growth 5
Elastic fracture in mode I
• Crack growth criterion in mode I (3)
– Toughness & fracture energy of brittle materials
Material KC [MPa · m½] GC [J · m-2]
Borosilicate Glass 0.8 9.
Alumina 99% polycrystalline 4. 39.
Zirconia-Toughened Alumina 6. 90.
Yttria Partially Stabilized Zirconia 13. 730.
Aluminum 7075-T6 25. 7800.
AlSiC Metal Matrix Composite 10. 400.
Epoxy 0.4 200.
2013-2014 Fracture Mechanics – LEFM – Crack Growth 6
Elastic fracture in mode I
• Crack growth criterion in mode I (4) – Behavior depends on environmental conditions
• Example: DBTT – High strength steel alloys – Toughness divided by 3 – Fracture energy divided by 9
brittle
ductile
• We know if the crack will grow
• But – How fast ?
– How far ?
– In which direction ?
2013-2014 Fracture Mechanics – LEFM – Crack Growth 7
Crack grow stability
• Considering a cracked body with G ≥ GC: the crack will grow
– If GC is constant: two possible behaviors as the crack grows
• G decreases the crack stops growing (unless the loading increases)
• G increases instability and the specimen fractures
– If GC is not constant: the question becomes
• Which one of G or GC will grow at the highest rate
– The stability of a crack growth depends on
• The geometry
• The loading
• The material behavior
2013-2014 Fracture Mechanics – LEFM – Crack Growth 8
Crack grow stability
• Example: Delamination of composite (DCB specimen)
– Compliance (Linear elasticity)
– Prescribed loading
• As the crack is growing: G increases
• For a perfectly brittle material GC is constant
unstable
– Prescribed displacement
• As the crack is growing: G decreases
• For a perfectly brittle material GC is constant
stable
– Is it a general rule? What happens for a general loading?
a
h
h
Q
Thickness t
a
h
h
u
Thickness t
2013-2014 Fracture Mechanics – LEFM – Crack Growth 9
Crack grow stability
• General loading conditions
– Practically, when a crack propagates
• Part of the structure becomes more compliant
• Part of the load is transferred to other parts
neither fixed load nor fixed displacement
– This corresponds to a compliant machine loading
• Spring of compliance CM
• Generalized loading Q(a) (spring and cracked body)
• Generalized displacement at the cracked body u(a)
• Prescribed displacement uT at one extremity of the spring Q(a)
u(a)
C(a)
uT
Q(a)
CM
2013-2014 Fracture Mechanics – LEFM – Crack Growth 10
Crack grow stability
• General loading conditions (2)
– Variation of the energy release rate
• Dead load
– Spring of infinite compliance (CM → ∞)
– As first term is <0, the value CM → ∞
is the one for which ∂AG is maximum
• As the spring stiffens (toward a fixed grip)
– CM decreases
– ∂AG also decreases, which stabilizes the crack growth
– If ∂AG becomes <0, for a perfectly brittle material
(GC=constant) the crack becomes stable
– A fixed grip is always more stable than a dead load
• Fixed grip
• Dead load
Q(a)
u(a)
C(a)
uT
Q(a)
CM
2013-2014 Fracture Mechanics – LEFM – Crack Growth 11
Crack grow stability
• Non-perfectly brittle material: Resistance curve
– A crack can be stable even if ∂AG > 0
– For non-perfectly brittle materials GC depends on the crack surface
• Therefore GC will be renamed the resistance Rc (A)
• In 2D: Rc (DA)= Rc (t Da)
• Elastoplastic behavior
– In front of the crack tip
there is an active plastic
zone
– Resistance increases when
the crack propagates as
the required plastic work
increases
– This effect is more
important for thin
specimens as the
elasto-plastic behavior
is more pronounced in plane s
– A steady state can be reached (active zone in a plastic wake) or not
– Composites: • As the crack propagates fibers in the wake tend to close crack tip
more and more energy is required for the crack to grow
Plastic
zone
Blunting
Initial crack
growth
Steady state
DA=tDa
Rc
Rc fragile
Rc ducile t1
0
Rc ducile t2<t1
Rc ducile t3<< (Plane s)
Active
plastic zone Plastic wake
2013-2014 Fracture Mechanics – LEFM – Crack Growth 12
Crack grow stability
• Example: Delamination of composite with initial crack a0
Dead load vs Fixed grip
– Dead load Q1
• Perfectly brittle materials: unstable
• Ductile materials: stable, but if a is larger than a** it turns unstable
– Dead load Q2 > Q1
• This is the limit of stability for ductile materials (always unstable for perfectly brittle material)
– For Q3: unstable
a
Rc, G
Rc ducile
u increasing
a0
GC brittle
u1
u2
u3
a
Rc, G
Rc ducile
Q increasing
a0 a* a**
GC brittle
Q1
Q2 Q3
– Fixed grip u1 ,u2 or u3
• The crack is stable for any material
2013-2014 Fracture Mechanics – LEFM – Crack Growth 13
Crack grow stability
• Crack growth stability criterion
– Crack growth criterion is G ≥ GC
– Stability of the crack is reformulated (in 2D)
• Stable crack growth if
• Unstable crack growth if
2013-2014 Fracture Mechanics – LEFM – Crack Growth 14
Mixed mode fracture
• In which direction will the crack grow?
– For anisotropic & isotropic material
– Under mixed mode loading
– In composite
• Cracks follow the path of least resistance
– Important to predict this path during the design
– “Fail safe design”
• In the following we assume homogeneous & isotropic materials under
mixed mode loading
2013-2014 Fracture Mechanics – LEFM – Crack Growth 15
Mixed mode fracture
• Combination of mode I & mode II loadings
– The crack will propagate with a kink angle q
– q=0 only if it corresponds to a weak plane of the material (bond line in
composites e.g.)
– Loadings in mode I and mode II
• Rotation
• Mode I loading (infinite plate):
• Mode II loading (infinite plate):
– How can we determine the kink angle?
2a
x
y
s∞
s∞
b q
2013-2014 Fracture Mechanics – LEFM – Crack Growth 16
Mixed mode fracture
• Method of the maximum circumferential stress
– Erdogan & Sih, 1963
– Assumptions:
• The crack will grow in the direction where
the SIF related to mode I in the new frame
is maximal
• It will grow only if this SIF is larger than
the toughness of pure mode I loading
– So the new criterion is
with &
2a
x
y
s∞
s∞
b q* err eqq
2013-2014 Fracture Mechanics – LEFM – Crack Growth 17
Mixed mode fracture
• Method of the maximum circumferential stress (2)
– Mode I
– Mode II
– Mixed mode I & mode II loadings
2013-2014 Fracture Mechanics – LEFM – Crack Growth 18
Mixed mode fracture
• Method of the maximum circumferential stress (3)
– Mixed mode I & mode II loadings in new frame
2013-2014 Fracture Mechanics – LEFM – Crack Growth 19
Mixed mode fracture
• Method of the maximum circumferential stress (4)
– Maximum hoop stress
• Direction of loading defined by
• In case of the infinite plate b* corresponds to b
• The maximum hoop stress becomes
2a
x
y
s∞
s∞
b*
q* err eqq
– For b*=90°:q*=0, which
corresponds to a straight
ahead growth under pure
mode I
– For b*>0 the maximum hoop
stress is reached for q*<0
-100 0 100-4
-2
0
2
4
q [°]
(2
r)1
/2 s
qq /
KI
b*=15°
b*=30°
b*=45°
b*=60°
b*=90°
q*
Maximum is for q <0
q [deg.]
2013-2014 Fracture Mechanics – LEFM – Crack Growth 20
Mixed mode fracture
• Method of the maximum circumferential stress (5)
– Kink angle obtained when hoop stress is maximum
• So the kink angle q* is reached for
• This prediction is in good agreement
with experimental results obtained
by Erdogan and Sih (1963)
0 20 40 60 800
20
40
60
80
b* [°]
- q*
AnalyticExperimental
b* [deg.]
2013-2014 Fracture Mechanics – LEFM – Crack Growth 21
Mixed mode fracture
• Method of the maximum circumferential stress (6)
– Failure envelope
• Criterion
becomes
with
• Resolution of this linear system of
2 equations with 2 unknowns
leads to the failure envelope
• Toughness tests for different
loading directions gives
comparisons points
• Maximum circumferential stress
theory is conservative 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
KI / K
C
KII
/ K
C
AnalyticExperimental
KI / KC
2013-2014 Fracture Mechanics – LEFM – Crack Growth 22
Mixed mode fracture
• Method of the maximum circumferential stress (7)
– Pure mode II theory (b * = 0, KI = 0)
• Kink angle q*|b*=0 ~ -70.53°
• Failure if KII = 0. 87KC
• N.B.: For pure mode II the case of a plate in tension is meaningless
– One should consider shearing
2a
x
y
s∞
s∞
b*
q* err eqq
2a
x
y
t∞
t∞
q* err eqq
2013-2014 Fracture Mechanics – LEFM – Crack Growth 23
Mixed mode fracture
• Method of the maximum circumferential stress (8)
– Pure mode II: plate under shearing
• Circumferential stress:
– Maximum for q =-45°
• Theory predicts a kink angle q*|b*=0 ~ -70.53°
• Why is there a difference?
– Due to the stress field at crack tip
the maximum circumferential stress
is at -70.53° crack will grow
with this angle
– As it grows, the crack will not remain
under pure mode II loading
– Crack will turn until growing
with a -45-degree angle
– Second order theory
• For a plate in shearing it has been shown
that the kink |angle| will be reduced as the
crack propagates
• What is the general rule?
2a x
y t∞
t∞
2a x
y t∞
t∞
2013-2014 Fracture Mechanics – LEFM – Crack Growth 24
Mixed mode fracture
• Method of the maximum circumferential stress (9)
– Second order theory (2)
• Order 0 has to be added to the asymptotic
solution (in 1/r1/2) when r ~ rc
• Asymptotic solution characterized by SIFs
• Order 0 characterized by T along Ox e.g.
• New kink angle q * (b*, T, rc, KI) depends on stress field at rc
– Useful for FE computation
– rc determined by experiment (Aluminum alloy: rc ~1.5 mm)
2a
x
y
T
T
b*
q* err eqq
2013-2014 Fracture Mechanics – LEFM – Crack Growth 25
Mixed mode fracture
• Method of the maximum circumferential stress (10)
– Second order theory (3)
• Results obtained numerically can be summarized as: after initial kinking
– For compression (T < 0) or low traction: crack will tend to be aligned with T
– For traction of the order of KI /rc1/2 or larger: there is bifurcation
2a
2a 2a
2013-2014 Fracture Mechanics – LEFM – Crack Growth 26
Mixed mode fracture
• Method of the maximum energy release rate
– In thermodynamics: equilibrium corresponds to the lowest potential energy
• So the crack will propagate to minimize the potential energy
• As the energy release rate is defined by
the crack will grow in the direction maximizing G
– Methodology
• ki: SIFs at the crack tip before kink propagation
• Ki: SIFs at extremity of a kink of infinitesimal length and angle a
• Ki(a) can be deduced from ki: (Cotterell & Rice, 1980)
– First order accurate in a
• As the kink grows straight ahead:
• Fracture criterion and growth direction are defined such that
, &
kI & kII a
KI & KII
2013-2014 Fracture Mechanics – LEFM – Crack Growth 27
Mixed mode fracture
• Method of the maximum energy release rate (2)
– Loading direction defined by cot b* = kII/kI
– It gives results similar to the method of maximum hoop stress
2a
x
y
s∞
s∞
b
a* err eqq
-100 0 1000
5
10
15
20
25
a [°]
G( a
) /
G(0
)
b*=15°
b*=30°
b*=45°
b*=60°
b*=90°
a*
a [deg.]
2013-2014 Fracture Mechanics – LEFM – Crack Growth 28
Mixed mode fracture
• Method of the maximum energy release rate (3)
– Comparison against the method of maximum hoop stress
• Recall we used a formula first order accurate in a
– < 5% error for a < 40°
– ~ 5% error on KI for a > 40° (Cotterell & Rice, 1980)
– For the method of maximum energy release rate:
• Before the crack propagates: kI & kII can be ≠ 0
• After kinking: KII = 0 the crack propagates under pure mode I
0 20 40 60 800
20
40
60
80
b* [°]
- q*
Max. hoopMax. G
b* [deg.]
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
KI / K
CK
II /
KC
Max. hoopK
i for Max. G
ki for Max. G
KI / KC
2013-2014 Fracture Mechanics – LEFM – Crack Growth 29
Fatigue failure
• Under cyclic loadings a crack can
– Be initiated for loadings with s < sp0
– Propagate for Ki < KC
• Total life approaches
– Unable to account for inherent defects
• Example: design of the De Havilland Comet
using total life approach. Defects resulting
from punched riveting of square windows
caused failure of aircrafts
– Unable to predict crack propagation
P
P
a
2013-2014 Fracture Mechanics – LEFM – Crack Growth 30
Fatigue failure
• Microscopic observations for cycling loading
– I. Crack initiated at stress concentrations (nucleation)
– II. Crack growth resulting into surface striations
– III. Failure of the structure when the crack reaches a critical size
– Example of a crank axis
• Development of damage tolerant design
– Assume cracks are present from the beginning of service
– Predict crack growth and end of life
2013-2014 Fracture Mechanics – LEFM – Crack Growth 31
Fatigue failure
• I. Crack initiation
– Nucleation: cracks initiated for K << KC
• Surface: deformations result from
dislocations motion along slip planes
• Can also happen around a
bulk defect
P
P
a
N cycles
Pmin Pmax
2013-2014 Fracture Mechanics – LEFM – Crack Growth 32
Fatigue failure
• II. Crack growth
– Stage I fatigue crack growth:
• Along a slip plane
– Stage II fatigue crack growth:
• Across several grains
– Along a slip plane in each grain
– Straight ahead macroscopically
• Striation of the failure surface:
corresponds to the cycles
• III. Structure failure – As the crack growth K tends toward KC
– For a critical size of the crack there is a static failure
Perfect crystal
Cra
ck g
row
th
2013-2014 Fracture Mechanics – LEFM – Crack Growth 33
Fatigue failure
• Crack growth rate
– Tests: conditioning parameters
• DP &
• Pmin / Pmax
– SSY assumption
• As fatigue happens for
Ki < KC this assumption is
usually satisfied, at least
during crack growth stage
– Therefore, as SSY assumption
holds, fatigue failure can be
described uniquely by
• DK = Kmax- Kmin &
• R = Kmin/Kmax
– There is DKth such that if DK ~ DKth:
• The crack has a growth rate lower than one atomic spacing per cycle (statistical value)
• Dormant crack
Interval of striations
P
P
a
a (
cm)
Nf (105) 1 2 3 4 5 6 7
5
4
3
2
DP1
DP2 > DP1
Structure inspection
possible (for DP1) Rapid crack
growth
Rupture
2013-2014 Fracture Mechanics – LEFM – Crack Growth 34
Fatigue failure
• Crack growth rate (2)
– Zone I
• Stage I fatigue crack growth
• DKth depends on R
– Zone II
• Stage II fatigue crack growth: striation
• 1963, Paris-Erdogan
– Depends on the geometry, the loading, the frequency
– Steel: DKth ~ 2-5 MPa m1/2 , C ~ 0.07-0.11 10-11 [m (MPa m1/2)-m], m ~ 4
– Steel in sea water: DKth ~ 1-1.5 MPa m1/2, C ~ 1.6 10-11 [idem], m ~ 3.3
• Be careful: K depends on a integration needed to get a(Nf)
– Infinite plate, mode I :
– Zone III
• Rapid crack growth until failure
• Static behavior (cleavage) due to the effect of Kmax(a)
• There is failure once af is reached, with af such that Kmax(af) = Kc
da
/ d
Nf
logDK DKth
Zone I Zone II Zone III
R
1
m
2013-2014 Fracture Mechanics – LEFM – Crack Growth 35
Fatigue failure
• Experimental results
– Parameters in Paris law
Material DKth [MPa · m½] m [-] C [m (MPa . m1/2)-m]
Mild steel 3.2-6.6 3.3 0.24 . 10-11
Structural steel 2.0-5.0 3.85-4.2 0.07-0.11 . 10-11
Structural steel is sea water 1.0-1.5 3.3 1.6 . 10-11
Aluminum 1.0-2.0 2.9 4.56 . 10-11
Aluminum alloy 1.0-2.0 2.6-2.9 3-19 . 10-11
Copper 1.8-2.8 3.9 0.34 . 10-11
Titanium alloy (6Al-4V, R=0.1) 2.0-3.0 3.22 1 . 10-11
2013-2014 Fracture Mechanics – LEFM – Crack Growth 36
Fatigue failure
• Effect of R=Kmin/Kmax: crack closure – During loading phases at maximum stress
• There is an active plastic zone (phase transformation can happen)
• The crack opening allows fluid or products to enter
– If R <0.7 or negative, at low stress crack lips can enter into contact due to
• Plasticity
– Residual plastic strain resulting
from plastic wake will close
the crack
• Roughness
– Non flat lips prevent sliding (mode II)
• Corrosion
– Corrosion products fill the opening
• Viscous fluid
– Lubricant fluids fill the opening
• Phase transformation
– Phase transformation at crack tip can put material
in compression
Plastic wake Active plastic zone
2013-2014 Fracture Mechanics – LEFM – Crack Growth 37
Fatigue failure
• Effect of crack closure on fatigue
– When the loading decreases local compressive effects parts of the cracks are kept opened
– The stress intensity factor at the minimum of the cycle is more important than predicted the effective DK is actually reduced
– The crack closure effect is therefore beneficial to structure life
t
smax
smin
Ds = smax-smin
t
Kmax
Kmin
DK DKeff
2013-2014 Fracture Mechanics – LEFM – Crack Growth 38
Fatigue failure
• Effect of R=Kmin/Kmax on threshold (Zone I)
– Due to Plasticity Induced Crack Closure (PICC)
– DKth decreases when R increases
2013-2014 Fracture Mechanics – LEFM – Crack Growth 39
Fatigue failure
• Effect of R=Kmin/Kmax on crack growth rate (Zone II)
– Due to crack closure life of structure is improved for low R
• Example of 2024-T3 aluminum alloy*
– DKeff depends on many parameters (loading, environment, …)
• Example: model of Elber & Schijve for Al. 2024-T3
– DKeff = (0.55 + 0.33 R + 0.12 R2) DK for -1<R<0.54
• Models can be inaccurate in non-adequate circumstances
*J.C. Newman Jr, E.P. Phillips, M.H. Swain, Fatigue-life prediction methodology using small-crack theory, International Journal of Fatigue 21 (1999)
2013-2014 Fracture Mechanics – LEFM – Crack Growth 40
Fatigue failure
• Overload effect
– What happens if there is a few (or a moderate) number of overloads ?
– Plastic wake is temporarily increased, until the active plastic zone at crack tip
passed the plastic zone created by the overload
– So which effect ?
Active plastic zone
t
Kmax
Kmin
DK
Plastic wake
2013-2014 Fracture Mechanics – LEFM – Crack Growth 41
Fatigue failure
• Overload effect (2)
– As the plastic wake is temporarily increased, DKeff is reduced due to PICC
there is a retard effect in the crack propagation
• Infrequent overloads help
• Frequent overloads may help
• Too frequent overloads are
damaging, as it actually
corresponds to increasing Kmax
– There exist overload models
• Wheeler e.g.
– N.B. 1952, a fuselage of the Comet was tested against fatigue
• Static loading at 1.12 atm, followed by
• 10 000 cycles at 0.7 atm (> cabin pressurization at 0.58 atm)
• Production fuselage without the static loading failed after a few 1000 cycles
(pressurization at 0.58 atm)
a (
cm)
Nf (105) 1 2 3 4 5 6 7
5
4
3
2
Overloads
Retards
2013-2014 Fracture Mechanics – LEFM – Crack Growth 42
Stress corrosion cracking
• A crack can grow due to the combination of stress and chemical attack
– This is not only fatigue but also for static stress with K<KC
– It happens in particular environments
• Salt water
• Hydrogen
• Chlorides
• …
– Mainly for metals
• Steel in salt water, chloride, hydrogen
• Aluminum alloys in salt water
Steel AISA
4335V
2013-2014 Fracture Mechanics – LEFM – Crack Growth 43
KI (MPa m1/2)
da/d
t (m
m/m
in)
10 100
10
1
0.1
0.01
H20
H2
Steel 4043
• Double cantilever beam loaded with a compliant machine
– Assume a perfectly brittle material
– What is the effect of CM on
the crack growth stability?
Exercise 1
a
h
h
Thickness t
x
y
uT
CM
2013-2014 Fracture Mechanics – LEFM – Crack Growth 44
• Edge notch specimen under cyclic loading – Assume titanium alloy 6%Al - 4%V
• See figures below
– Assume a remains << W and << h
– Initial crack size a = 1.5 cm
– Cyclic loading between • Minimum value: 8 MPA
• Maximum value: 80 MPa
– What is the life of the structure?
Exercise 2
y
x
s
s
W h
a
2013-2014 Fracture Mechanics – LEFM – Crack Growth 45
• Improved materials to crack propagation – Periodical deflection of crack
• Depends on structure
– Using the model • Deflected crack with periodic tilts
• Crack opening/closing under cyclic
loading
– Establish
• SIF for growth along the kinks
• Effective SIF
• Apparent crack propagation rate
• Effect of surface mismatch
Exercise 3
Overaged (precipitation from heat treatment),
7475-T7351 aluminum alloy, R=0.1, vacuum
Underaged 7475 aluminum alloy, R=0.1, vacuum
S. Suresh, Metallurgica transactions A, 16A, 1985
S S
S
D D q
Dd
Dd*
uI uII
2013-2014 Fracture Mechanics – LEFM – Crack Growth 46
• Improved materials to crack propagation (2) – Initial nanocrystalline material
• Subjected to cyclic mode I
• No crack deflection observed
• Crack growth data in table
– Grain size is increased to micro-order • Deflections observed
– Represent on a graph the crack growth data • For nanocrystalline materials
• For microcrystalline materials
Exercise 4
S. Suresh, Metallurgica transactions A, 16A, 1985
S S = 0.418 mm
S
D D = 0.255 mm q = /3
DK (MPa m1/2)
da/dN (mm/cycle)
2.573 2.32E-08
2.607 3.68E-08
2.6585 5.84E-08
2.7462 9.27E-08
2.8552 1.47E-07
3.0265 2.24E-07
3.2287 3.42E-07
3.4445 5.22E-07
3.7225 8.27E-07
3.9713 1.21E-06
4.2642 1.92E-06
4.6381 3.05E-06
4.98 4.48E-06
5.3474 7.10E-06
5.7415 1.04E-05
6.125 1.42E-05
6.4088 1.92E-05
6.9707 2.93E-05
7.4366 4.48E-05
2013-2014 Fracture Mechanics – LEFM – Crack Growth 47
References
• Lecture notes
– Lecture Notes on Fracture Mechanics, Alan T. Zehnder, Cornell University,
Ithaca, http://hdl.handle.net/1813/3075
• Other references
– « on-line »
• Fracture Mechanics, Piet Schreurs, TUe, http://www.mate.tue.nl/~piet/edu/frm/sht/bmsht.html
– Book
• Fracture Mechanics: Fundamentals and applications, D. T. Anderson. CRC press, 1991.
• Fatigue of Materials, S. Suresh, Cambridge press, 1998.
2013-2014 Fracture Mechanics – LEFM – Crack Growth 48
• Double cantilever beam loaded with a compliant machine
– Compliant machine
– DCB alone:
• Compliance
• Failure for a crack size a0 with
– Back to compliant machine
• Instability for a perfectly brittle material if if
or again if
• So for a perfectly brittle material it is unstable if CM > 2 C(a), but eventually as C
will grow with the crack, it will become larger than CM / 2 again
Exercise 1: Solution
a
h
h
Thickness t
x
y
uT
CM
2013-2014 Fracture Mechanics – LEFM – Crack Growth 49
• Double cantilever beam loaded with a compliant machine
– Other method
– Normalizing with respect to the case CM=0
• Since it has been normalized
for critical uT of the case CM=0,
for a crack to grow we would
need a larger uT so G reaches GC
• If CM>2C(a0) there will be an
instability as for a/a0 =1 the slope
is positive
• Eventually, as the crack grows the slope
becomes negative again, stabilizing the crack
Exercise 1: Solution
2013-2014 Fracture Mechanics – LEFM – Crack Growth 50
0 1 2 30
0.05
0.1
0.15
0.2
0.25
a/a0
G/G
C
CM
/C=1/2
CM
/C=1
CM
/C=3/2
CM
/C=2
CM
/C=5/2
CM
/C=3
a /a0
G /
GC
• Edge notch specimen under cyclic loading – Material properties at room temperature
• Yield: 830 MPa
• Toughness: 55 MPa · m½
– SIF if a remains < 2% of W
• KI = 1.122 s ( a)1/2
– Plane strain & elastic fracture?
• Yes if specimen thick enough
• Crack is large enough
• Moreover the applied stress << the yield stress
Exercise 2: Solution
y
x
s
s
W h
a
2013-2014 Fracture Mechanics – LEFM – Crack Growth 51
• Edge notch specimen under cyclic loading (2)
– Is the crack critical?
• The crack will not lead to static failure
– Fatigue?
• As we are above the threshold there will be fatigue
• R = 0.1, so crack experiences closing effect
– What is the critical crack length leading to static failure?
Exercise 2: Solution
2013-2014 Fracture Mechanics – LEFM – Crack Growth 52
• Edge notch specimen under cyclic loading (3)
– Parameters in Paris’ law
• There is a phase transition around
DK=17 MPa . m1/2 but we are above
• Paris’ coefficients:
– C=10-11 m (MPa . m1/2)-m
– m=3.22
Exercise 2: Solution
2013-2014 Fracture Mechanics – LEFM – Crack Growth 53
• Edge notch specimen under cyclic loading (4)
– Paris’ law
• Can be integrated explicitly, and, for m≠2, it yields
• So the number of cycles in terms of the crack size is
– Critical size is reached after
1.74 105 cycles, but this value cannot
be used as
• Paris’ law is not valid is zone III
• After 1.5 105 cycles the crack is
growing too quickly for allowing
inspections
• 1.1 105 is a conservative time life
Exercise 2: Solution
2013-2014 Fracture Mechanics – LEFM – Crack Growth 54
0 0.5 1 1.5 2
x 105
0
0.05
0.1
Nf
a [
m]
N x 105
• Edge notch specimen under cyclic loading (5)
– Some remarks
• If there is an error of 10% in the estimation
of the initial crack length size
– Life of the structure is
reduced by 15%
– Using a 1.5 105-cycle life prediction
would actually lead to a crack
size located in zone III
• We have assumed a < 0.02 W, which
corresponds to a six-meter wide
specimen. In practice
– Crack size cannot be considered
infinitely small
– SIF must then be evaluated using
» Either FEM simulations
» Or SIF handbooks
– Paris’ law
» Cannot be integrated in a closed form
» Integration has to be performed numerically
Exercise 2: Solution
2013-2014 Fracture Mechanics – LEFM – Crack Growth 55
0 0.5 1 1.5 2
x 105
0
0.05
0.1
Nf
a [
m]
a0=1.5 cm
a0=1.65 cm
a0=1.8 cm
N x 105
• SIF for growth along the kinks
– Cortell & Rice
– For the problem under consideration
• Global mode I: kI = KIg & kII = 0
• Along kink: KI = KIk & KII = KII
k
Exercise 3: Solution
S. Suresh, Metallurgica transactions A, 16A, 1985
kI & kII a
KI & KII
S S
S
D D q
2013-2014 Fracture Mechanics – LEFM – Crack Growth 56
• SIF for growth along the kinks (2)
– Global mode I: KIg
– Along kink: KIk & KII
k
•
Exercise 3: Solution
S. Suresh, Metallurgica transactions A, 16A, 1985
S S
S
D D q
2013-2014 Fracture Mechanics – LEFM – Crack Growth 57
• Effective SIF
– Global mode corresponds to segment S: KIg
• As crack grows straight ahead on S :
– Along kink D: KIk & KII
k
• As crack grows straight ahead on D:
– Assumption: effective SIF is the weighted average of the SIFs
•
– For cyclic loading
Exercise 3: Solution
S. Suresh, Metallurgica transactions A, 16A, 1985
S S
S
D D q
2013-2014 Fracture Mechanics – LEFM – Crack Growth 58
• Apparent crack propagation rate
– Effective SIF
– Crack growth
• Real: a = D+S
• Apparent: aapp = D cos q + S
– The 2 relations are the equations of LEFM for a periodically deflected crack
– But how to take into account contact during unloading?
Exercise 3: Solution
S. Suresh, Metallurgica transactions A, 16A, 1985
S S
S
D D q
aapp
Dd
Dd*
uI uII
Growth rate of a
straight crack
2013-2014 Fracture Mechanics – LEFM – Crack Growth 59
• Effective SIF with surface mismatch
– Effective SIF without surface mismatch
– Surface mismatch under mode I opening
• During unloading: crack unloads in both mode I and II, leading to uII
– At first time of contact:
• Opening during peak loading is rewritten
• During peak loading, in LEFM:
• During closure:
Exercise 3: Solution
S. Suresh, Metallurgica transactions A, 16A, 1985
Dd
Dd*
uI uII
At peak load During unloading
2013-2014 Fracture Mechanics – LEFM – Crack Growth 60
• Effective SIF with surface mismatch (2)
– Effective SIF without surface mismatch
– Due to surface mismatch under mode I opening
• As
• If
Exercise 3: Solution
S. Suresh, Metallurgica transactions A, 16A, 1985
Dd
Dd*
uI uII
At peak load During unloading
2013-2014 Fracture Mechanics – LEFM – Crack Growth 61
• Microcrystalline materials
– Apparent crack growth rate
– Effective DK
Exercise 4: Solution
S S = 0.418 mm
S
D D = 0.255 mm q = /3
DK (MPa m1/2)
da/dN (mm/cycle)
2.573 2.32E-08
2.607 3.68E-08
2.6585 5.84E-08
2.7462 9.27E-08
2.8552 1.47E-07
3.0265 2.24E-07
3.2287 3.42E-07
3.4445 5.22E-07
3.7225 8.27E-07
3.9713 1.21E-06
4.2642 1.92E-06
4.6381 3.05E-06
4.98 4.48E-06
5.3474 7.10E-06
5.7415 1.04E-05
6.125 1.42E-05
6.4088 1.92E-05
6.9707 2.93E-05
7.4366 4.48E-05
S. Suresh, Metallurgica transactions A, 16A, 1985
2013-2014 Fracture Mechanics – LEFM – Crack Growth 62
• Interpretation
– Data correspond to a straight crack:
• &
– What matters is the apparent propagation
• &
Exercise 4: Solution
DK (MPa m1/2)
da/dN (mm/cycle)
2.573 2.32E-08
2.607 3.68E-08
2.6585 5.84E-08
2.7462 9.27E-08
2.8552 1.47E-07
3.0265 2.24E-07
3.2287 3.42E-07
3.4445 5.22E-07
3.7225 8.27E-07
3.9713 1.21E-06
4.2642 1.92E-06
4.6381 3.05E-06
4.98 4.48E-06
5.3474 7.10E-06
5.7415 1.04E-05
6.125 1.42E-05
6.4088 1.92E-05
6.9707 2.93E-05
7.4366 4.48E-05
S. Suresh, Metallurgica transactions A, 16A, 1985
S S
S
D D q
aapp
2013-2014 Fracture Mechanics – LEFM – Crack Growth 63
100
101
102
10-8
10-7
10-6
10-5
10-4
D K (MPa m1/2
)
da/
dN
(m
m/c
ycl
e)
nano
micro, =0.1
micro, =0.25
micro, =0.75
DK [Mpa m1/2]
• Graph
Exercise 4: Solution
DK (MPa m1/2)
da/dN (mm/cycle)
2.573 2.32E-08
2.607 3.68E-08
2.6585 5.84E-08
2.7462 9.27E-08
2.8552 1.47E-07
3.0265 2.24E-07
3.2287 3.42E-07
3.4445 5.22E-07
3.7225 8.27E-07
3.9713 1.21E-06
4.2642 1.92E-06
4.6381 3.05E-06
4.98 4.48E-06
5.3474 7.10E-06
5.7415 1.04E-05
6.125 1.42E-05
6.4088 1.92E-05
6.9707 2.93E-05
7.4366 4.48E-05
S. Suresh, Metallurgica transactions A, 16A, 1985
2013-2014 Fracture Mechanics – LEFM – Crack Growth 64
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