University of Liège
Aerospace & Mechanical Engineering
Fracture Mechanics, Damage and Fatigue
Linear Elastic Fracture Mechanics - Asymptotic Solution
Fracture Mechanics - LEFM - Asymptotic Solution
Ludovic Noels
Computational & Multiscale Mechanics of Materials – CM3
http://www.ltas-cm3.ulg.ac.be/
Chemin des Chevreuils 1, B4000 Liège
LEFM assumptions
• Balance of body B – Momenta balance
• Linear
• Angular
– Boundary conditions • Neumann
• Dirichlet
• Small deformations with linear elastic, homogeneous & isotropic material
– (Small) Strain tensor , or
– Hooke’s law , or
with
– Inverse law
with
b
T
n
2m l = K - 2m/3
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 2
Resolution method
• Examples of problems to be solved
– Static
– 2D &
• Plane s
• Or plane e
2a 2b x
y
s∞
s∞
syy syy
y
2a x
s∞
s∞
syy syy
2a x
y
s∞
s∞
syy syy
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Resolution method
• Static & 2D assumptions – Plane s ( ) or plane e ( & )
– Hooke’s law
• Plane e :
&
• Plane s :
&
– Greek subscripts substitute for x or y (Roman for x, y or z)
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Resolution method
• Static & 2D assumptions (2) – Plane s ( ) or plane e ( & )
• Linear momentum
– If b = 0, there exists an Airy function F:
• Compatibility
– Plane e :
• In terms of Airy function
• Compatibility
– Plane s :
• In terms of Airy function
• Compatibility
– For both plane states:
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Resolution method
• Application: infinite plate with a hole
– Cylindrical coordinates
• Boundary conditions
– At r = a
– At r = b→∞: only syy= s∞ is non zero
y
2a x
s∞
s∞
syy syy
y
s∞
s∞
r=a x
r q
r =
b→∞
Load case 1
independent
from q
Load case 2
dependent on cos 2q (symmetry with
respect to x)
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Resolution method
• Application: infinite plate with a hole (2)
– For both loading cases: find potential F such that
• Bi-harmonic equation is verified
• Stress field satisfies the boundary conditions
– Superposition of cases 1 and 2 (see appendix 1)
• Stresses:
• For q = 0: syy= sqq
• Stress intensity factor: Kcircle = 3 0
0.5
1
1.5
2
2.5
3
0 2 4 6
x/a
syy
/s∞
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Resolution method
• Application: infinite plate with an elliptical hole
– Elliptical coordinates
with, on the boundary,
– Resolution using complex functions of z (see next slides)
with the complex variable
• 1913, Inglis
– Stress intensity factor
with r the curvature radius at the tip
– Singularity when r → 0
2a 2b x
y
s∞
s∞
syy syy
x
y
a b
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Asymptotic solution
• Fracture modes
– 1957, Irwin, 3 fracture modes
• Boundary conditions
Mode I Mode II Mode III
(opening) (sliding) (shearing)
y
x
r q
Mode I Mode II Mode III
or or
y x
z
y x
z
y x
z
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Asymptotic solution
• Method for linear and elastic materials – Use of
• Complex variables
• Complex functions
with a & b real functions
– If z is differentiable (analytic)
• This yields the Cauchy-Riemann relations
&
• And the functions satisfy the Laplace equation
&
2a
x
y
syy syy
y
x
r
q
z
z
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Asymptotic solution
• Method for linear and elastic materials (2) – What becomes the bi-harmonic equation?
• Change of variables: x, y → z, z
• The Airy function F(x, y) = F(z, z)
– Let us define Y such that:
– The bi-harmonic equation is rewritten
– Since Y satisfies the Laplace equation, it is the real part of a function z(z):
with z = z(z) a function of z only
» But
– Let 4W’(z) = z(z) and w(z) be the unknown functions
which satisfies
N.B. W = W(z) and w = w(z) are functions of z only
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Asymptotic solution
• Method for linear and elastic materials (3)
– What become the stresses?
• Since , &
• The stresses are rewritten
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Asymptotic solution
• Method for linear and elastic materials (4)
– Displacements can be deduced from the stresses
• Complex form:
• Using Hooke’s law (see appendix 2)
–
– Plane s :
– Plane e:
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Asymptotic solution
• Method for linear and elastic materials (5)
– Choice of complex functions
• Owing to the discontinuity for q = p we choose
Indeed, for l=-1/2 the displacement is discontinuous
• The exact solution is actually a series in u = Sl ….
• u is finite l >-1
y
x
r q
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Asymptotic solution
• Application to mode I (opening)
– Mode I:
– Symmetry: &
&
or y
x
r q
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 15
Asymptotic solution
• Application to mode I (opening) (2)
– Stresses
– Boundary conditions
y
x
r q
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Asymptotic solution
• Application to mode I (opening) (3)
– Solution
• System of equations
• Non trivial solution
l = n/2, with n = -1, 0, 1, …. since l >-1
• Displacements in rl+1 and stresses in rl
the dominant term near the crack tip is obtained for l=-1/2
only this term is considered in the asymptotic solution
• Determination of the constants:
&
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Asymptotic solution
• Application to mode I (opening) (4)
– Asymptotic solution (terms of higher order, l = 0, 1/2, …, neglected)
• Displacement field
• Stress field
y
x
r q
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Asymptotic solution
• Application to mode I (opening) (5)
– Asymptotic stress field
• For mode I, the stress concentration
is obtained for syy(q = 0)
• As the stress tends to infinity,
the Stress Intensity Factor (SIF) is defined as
• C1 & so KI depend on both the geometry and loading
• Unit of the SIFs in Pa m1/2
y
x
r q
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Asymptotic solution
• Application to mode I (opening) (6)
– Asymptotic displacements
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 20
-0.1
0
0.1
-0.1
0
0.1-0.5
0
0.5
xy
ux E
/(K
I(1+
))
uxE
/(K
I(1+
))
-0.1
0
0.1
-0.1
0
0.1-0.5
0
0.5
xy
uy E
/(K
I(1+
))
opening uyE
/(K
I(1+
))
Asymptotic solution
• Application to mode I (opening) (7)
– Asymptotic stress field
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
sxx
/KI
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
sxy
/KI
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
syy
/KI
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Asymptotic solution
• Application to mode I (opening) (8)
– Validity of the asymptotic solution
• This solution is only valid in
a restricted region
syy
x
Asymptotic syy
True syy
Zone of asymptotic
solution (in terms of
1/r1/2) dominance
Structural
response Plasticity
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Asymptotic solution
• Application to mode II (sliding)
– Mode II:
• Proceeding as for mode I, with new BCs (C1 = C3 = 0)
&
or
y
x
r q
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Asymptotic solution
• Application to mode II (sliding) (2)
– Asymptotic displacements
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 24
-0.1
0
0.1
-0.1
0
0.1-0.5
0
0.5
xy
ux E
/(K
II(1
+
))
sliding uxE
/(K
II(1
+
))
-0.1
0
0.1
-0.1
0
0.1-0.5
0
0.5
xy
uy E
/(K
II(1
+
))u
yE/(
KII(1
+
))
Asymptotic solution
• Application to mode II (sliding) (3)
– Asymptotic stress field
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
sxx
/KII
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
syy
/KII
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
sxy
/KII
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Asymptotic solution
• Application to mode III (shearing)
– Mode III:
– Equations
• Hooke’s law
• Equilibrium
– Laplace equation satisfied uz is the imaginary part of a function z(z)
• Choice of function
is anti-symmetric
• Cylindrical CV
y
x
r q
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Asymptotic solution
• Application to mode III (shearing) (2)
– Strain field
– Stress field
• Hooke’s law
y
x
r q
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Asymptotic solution
• Application to mode III (shearing) (3)
– Displacement field
•
• Should be finite l > 0
– Boundary conditions
•
• Crack is stress free , n = 1, …
– Asymptotic solution for l = ½
&
y
x
r q
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Asymptotic solution
• Application to mode III (shearing) (4)
– Asymptotic fields
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
sxz
/KII
I
-0.1
0
0.1
-0.1
0
0.1-5
0
5
xy
syz
/KII
I
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-0.1
0
0.1
-0.1
0
0.1-0.5
0
0.5
xy
uz E
/(K
III(1
+
))
shearing uzE
/(K
III(
1+
))
Asymptotic solution
• Summary in 2D
– Principle of superposition holds as linear responses have been assumed
• Ki depends on
– The geometry and
– The loading
• u, s can be added for
– ≠ modes: u = u mode I + u mode II, s = s mode I + s mode II
– ≠ loadings u = u loading 1 + u loading 1, s = s loading 1 + s loading 2
• Ki can be added
– For ≠ loadings of the same mode Ki = Kiloading 1 + Ki
loading 2
• But since f and g depend on the mode K ≠ Kmode 1 + Kmode 2
Mode I Mode II Mode III
(opening) (sliding) (shearing)
y x
z
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• New failure criterion
– 1957, Irwin, crack propagation
• smax → ∞ s is irrelevant
• If Ki = KiC crack growth
– Toughness (ténacité) KIc
• Steel, Al, … : see figures
• Concrete : 0.2 - 1.4 MPa m1/2
(brittle failure)
Fracture toughness
brittle
ductile
brittle
ductile
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• Measuring KIc
– Done by strictly following the ASTM E399 procedure
• A possible specimen is the Single Edge
Notch Bend (SENB)
– Plane strain constraint (thick enough
specimen) conservative (see next slide)
– Specimen machined with a V-notch in
order to start a sharp crack
• Cyclic loading to initiate a fatigue crack
• Toughness test performed with
– A calibrated P, d recording equipment
– The Crack Mouth Opening Displacement
(CMOD=v) is measured with a clipped gauge
– Pc is obtained on P-v curves
» Either the 95% offset value or
» The maximal value reached before
– KIc is deduced from Pc using
» f(a/W) depends on the test (SENB, …)
Fracture toughness
P, d
v/2 v/2
L=4W
W Thickness t
CMOD,
v
v
P
tg 0.95 tg
Pc
P
tg 0.95 tg
Pc
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• Only 2D solutions have been considered but a real crack is clearly 3D
– At any point of the crack line
• A local referential can be defined
• Since the asymptotic solutions hold for
r → 0, at this distance the crack line seems
straight, and the problem is locally 2D
• The crack tip field can be broken into
3 2D problems (3 2D modes)
3D problems
y
x
z
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• 3D effects
– Near the border of a specimen the problem
is plane s, while it is plane e near the center
at the center there is a triaxial state
• The SIF is larger at the center as
no lateral deformations are possible
(see next lecture)
• 2 consequences
– The front will first propagate at the center
– The toughness decreases with the thickness
» Crude approximation
» Later on we will see how to evaluate this effect
– The practical toughness Kc is the plane strain one
3D problems
t K rupture
t
Kc
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• Computation of SIFs (see next lecture) – Computation of the SIFs Ki
• Analytical (crack 2a in an infinite plane)
• For other geometries or loadings
• bi obtained by
– Superposition
– FEM
– Energy approach
» Related to Griffith’s work
» See next lecture
– For 2 loadings a & b: KI = KIa+ KI
b
– BUT for 2 modes K ≠ KI+KII
SIF
s∞ s∞
y
2a
x
s∞
s∞
y
2a
x
t∞
t∞
y
2a
x
t∞
t∞
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• Exercise 1
– Let us consider a mode I state
around a notch of opening p-b
– What are the BCs ?
– What is the asymptotic behavior ?
– When does the SIF concept make sense ?
Exercise
x
y
b
s∞
s∞
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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.
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 37
• One can use the developments of mode I
– Boundary conditions
&
– Change of coordinates
– System of equations
Exercise 1: Solution
x
y
b
s∞
s∞
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 38
• Non trivial solution
– Determinant is equal to zero
– For b=p , it becomes
• Since l>-1 (for the displacements to be finite)
l = -½ , 0, ½, ..
singularity at crack tip
– There is no singularity if the first l >-1 is ≥ 0
• For b=p/2, the equation becomes
l = 0, 1 , ..
no singularity at crack tip
• For b < p/2: no singularity
• For b > p/2: there is a singularity
– Solving the equation with NR
• Considering the minimum l >-1
Exercise 1: Solution
x
y
b
s∞
s∞
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 39
60 80 100 120 140 160 180-0.5
0
0.5
b
lmin
b [deg.]
lm
in
Annex 1: Plate with a hole
• Cylindrical coordinates
– Boundary conditions
• At r = a
• At r = b→∞: only syy= s∞ is non zero
y
2a x
s∞
s∞
syy syy
y
s∞
s∞
r=a x
r q
r =
b→∞
Load case 1
independent
from q
Load case 2
dependent on cos 2q (symmetry with
respect to x)
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 40
• Cylindrical coordinates (2)
– Bi-harmonic equation
with
– Stresses from Airy function
with , &
• Eventually, after substitutions
y
s∞
s∞
r=a x
r q
r =
b→∞
Annex 1: Plate with a hole
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 41
• Load case 1
– Independent from q F(r)
• Bi-harmonic equation
• General solution:
• Stresses:
• Boundary conditions:
&
, &
• Solution: , &
Annex 1: Plate with a hole
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 42
• Load case 2
– Dependent on cos 2q F(r, cos 2q) = g(r) cos 2q
• Bi-harmonic equation
• General solution:
• Stresses:
Annex 1: Plate with a hole
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 43
• Load case 2 (2)
– Stresses
– Boundary conditions:
, , &
Annex 1: Plate with a hole
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 44
• Load case 2 (2)
– Solution:
Annex 1: Plate with a hole
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 45
• Superposition of case 1 and 2
– Stresses:
– For q = 0: syy= sqq
– Stress intensity factor: Kcircle = 3
0
0.5
1
1.5
2
2.5
3
0 2 4 6
x/a
syy
/s∞
Annex 1: Plate with a hole
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Annex 2: Displacement of asymptotic solution
• Displacements for linear and elastic materials
– Complex form:
– In terms of strains:
– Hooke’s law (Plane s or e):
• With m(z) such that the plane s or e condition is satisfied:
• But
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 47
• Displacements for linear and elastic materials (2)
– Out-of-plane state effect
• Plane s :
with
• Plane e:
with
– For both plane states:
Annex 2: Displacement of asymptotic solution
2013-2014 Fracture Mechanics - LEFM - Asymptotic Solution 48