Task 6 - Safety Review and LicensingOn the Job Training on Stress Analysis
Pisa (Italy)June 15 – July 14, 2015
Fracture Mechanics: Linear Elastic Fracture Mechanics 1/2
Davide Mazzini – Ciro Santus
2
Content
• Stress singularity
- Notch degenerating into a crack
- Multi-axial stress at notch root/ crack tip
- The Williams problem
• Linear Elastic Fracture Mechanics (LEFM)
- The Westergaard stress function
- Definition and calculation of the Stress Intensity Factors (SIFs)
- LEFM Validity limitations
Table of content – Class VI.b.1
Pisa, June 15 – July 14, 2015
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Books on Fracture Mechanics
T.L. Anderson, Fracture Mechanics: Fundamentals and Applications,
third edition. CRC Press 2005.
D. Broek. The Practical Use of Fracture Mechanics. Kluwer 1989.
… and many many others
Books
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• Experities and similitude (up to 1700)
• Elastic evaluations (nominal solutions) (Eulero, Cauchy, De SaintVenant, 1800)
• Stress concentrations (Kirsch, Inglis, 1900)
• Theory of plasticity (Prandtl, 1920)
• Sharp tip defects (Griffith, 1922)
History of “Strength of Materials”
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• Griffith’s energy approach for brittle materials (1930)• Practical relevance (1940-1950)• Definition of K, extension to metallic materials, complete develpment
of the Linear Elastic Fracture Mechanics (LEFM) (Williams, Irwin,1950)
• Application of the LEFM to Fatigue (Paris, 1960)• Extension to ductile materials (Elatic Plastic Fracture Mechanics
EPFM) (Irwin, Dugdale, Baremblatt, Wells, Landes, Rice, 1960)• Dynamics and crack arrest (DFM), viscous and (NLFM) (AA.VV. 1980)• Engineering applications, standards for design and testing, NDT,
corrosion, anisotropic materials, Damage Tolerant approaches, ..(ASTM, ASME, ESIS, BS)
History of “Fracture Mechanics”
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History of “Fracture Mechanics”
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2500 liberty ships, hull assembled by the innovative process of welding
Liberty ships – World War II
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700 experienced heavy structural damage, 145 completely destroyed,many lost (complete breakage of the hull)
Liberty ships – World War II
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Liberty ships – World War II
Post-failure analysis
• Failure at low stress (sometimes with the ship in the arbor)
• Quite “brittle” fractures
• Failure more frequent in winter time (ductile to brittle transition
temperature)
• Effect of the technological process (metallurgical, geometrical: weld
crack-like defects)
Fracture mechanics was born to understand these failure!
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Circular hole in a flat plate
Complete analytical solution
Plane stress solution if a>>B
Plane strain if a<<B
Extension to other problems
Kirsch 1898
a
B
0
0
Far boundaries
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Circular hole in a flat plate
Kirsch 1898
Far boundariesx
y
r
2 20
2 2
2 40
2 4
2 20
2 2
1 1 1 3 cos 22
1 1 3 cos 22
1 1 3 sin 22
rr
r
a ar r
a ar r
a ar r
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Circular hole in a flat plate
Kirsch 1898
x/a
0
y/a
-1+3
rr 0
+1
x/a
y/a
Why rr at these points?
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Circular hole in a flat plate, bi-axial loading
Kirsch 1898
Uniaxial Kt = 3
Equibiaxial Kt = 2
Pure shear Kt = 4~
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Elliptical hole in a flat plate
Inglis 1913
Problem definition:Geometry
,
Load, nominalstress(far field stress)
a b
Far boundaries
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Elliptical hole in a flat plate
Inglis 1913
A
At
t
Stress concentration:21
21
Kirsch solution for central hole
3
ab
aKb
b aK
Far boundaries
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Elliptical hole in a flat plate
Inglis 1913
Far boundaries
2
Moresignificant, local radius:ba
a
a
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Elliptical hole in a flat plate
Inglis 1913
t
2
t
t
21
being:
then:
1 2
, are more properly definingthe localgeometrywhen:
2
aKb
ba
aK
a
a
aK
Far boundaries
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Elliptical hole in a flat plate
Inglis 1913
t0 0lim lim 2
and the power of singularity ist
Limit:
square rhe of the locao lo radiut s
aK
Far boundaries0
lim( , ) 0b
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Stress components in-plane, plane stress
Multi-axial stress at notch root
x/a
y/a
a
B<<a
0
0
x
z
y
tK
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Plane stress
Multi-axial stress at notch root
Transversal stressfree surfaces
Almost zero stressat interior points
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Stress components in-plane, plane strain (approx.)
Multi-axial stress at notch root
x/a
y/a
a
B>a
0
( )
x
y
z x y
tK
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Plane strain
Multi-axial stress at notch root
Zero transversaldisplacement:
0z
11 1
1
After imposing 01 ( ) 0
( )
x x
y y
z z
z
x y z
z x y
E
E
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Inglis notch-like, plane stress
ANSYS Wb
Multi-axial stress at notch root
5mm
2 40 mma
t201 2 1 2 55
aK
t 5.5Why a different value here?K
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Inglis notch-like, plane stress
ANSYS Wb
Multi-axial stress at notch root
0 2 4 6 8 10-100
0
100
200
300
400
500
600
x coordinate, mm
Stre
ss c
ompo
nent
s, M
Pa
y
x
zPath on the geometry
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Inglis notch-like, plane strain
ANSYS Wb
Multi-axial stress at notch root
Exercise:
Calculate the Stress components, with ANSYS Workbench, at the notch tip for the large thickness geometry, and then verify the plain strain assumption
Repeat same calculation with imposed (exactly) plain strain constraint
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The Williams problem
Williams 1957
• local geometry :
• governing parameters:
• local polar coordinates:
• useful angular variable: s
s =
=0
, r
r
s
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Williams 1957
2 2 2
2 2Airy function: , xx yy xyx yx y x y
2 2 2 22 2
2 2 2 2
4 4 4
4 2 2 4
Governing equation: 0
2 0
x y x y
x x y y
2 2 2 22 2
2 2 2 2 2 2
4 2 4 3 4 3
4 2 2 4 4 3 2 2 2 3 2
2 2 2
2 2 2
Polar coordinates:
1 1 1 1 0
1 1 1 1 12 2 2 0
1 1 1+ ; ; rr r
r r r r r r r r
r r r r r r r r r r
r r r r r r
2
1r
Stress components
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Williams 1957
• Williams hypothesis for the Airy function:
1. General parameters: c1, c2, c3, c4 and exponent (a dimensionless real number)2. Airy equation fulfilled in the domain for any combination of c1, c2, c3, c4 and
• Corresponding stress field:
• Strain and displacement:
11 2 3 4sin 1 cos 1 sin 1 cos 1r c c c c
1
1
1
1
1
rr
r
F F
F
F
r
r
r
1ij ir u r
r
s
1r F
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r
s
Williams 1957
• In order to keep the displacements bounded:
• Local boundary conditions:
• Boundary conditions in explicit form:
• Homogeneous linear system with unknowns: c1, c2, c3, c4 and the parameter • Typical outcome of several problems: instability, free vibrations, etc.• We are interested in not trivial solutions (eigenvalue problem)• Let’s put the determinant of the system matrix to zero• Characteristic equation with as unknown (infinite solutions)
Tractionfree edges
0 2 0
0 2 0s
r r s
2 4
1 3
1 2 3 4
1 2 3 4
01 1 0
sin 2 1 cos 2 1 sin 2 1 cos 2 1 0
1 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1 0s s s s
s s s s
c cc c
c c c c
c c c c
0iu r
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Williams 1957
where 1, 2,3,......2nn n
12
3 41
2sin 1 sin 1 cos 1 cos 12 2 2 2 2
n
n nn
n n n n nr c cn
11
2 23 4
1
1
2, , , ....n
i ij jij n n ij ijn
r n c c B C rA r
Square root singular term !
s
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Crack as the special case with ψ = 0
• For this case the eigensolutions are
• and the corresponding Airy’s function becomes:
• The infinite couples c3n, c4n are determined by the other boundary conditions (remote geometry of the body, applied loads, constraints)
• Final general expression for the stress components:
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Williams 1957
1, and ij ij iu rr
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General conclusions of the Williams analysis
• Among the (usually) infinite terms of the stress expansion at the notch tip, only the first is unbounded (it goes to infinite as r approaches zero)
• The other terms are bounded or tends to zero approaching the notch tip
• The power of the singular term is a function of the angle 2 of the notch
• The strength of the singularity is the highest when = 0: the crack is the most severe notch
• The power of the leading singular term is universal (the same for any crack), the asymptotic terms of the elastic fields at the tip are:
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Williams 1957
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Exercise – MATLAB:
Implement a parametric calculation for the Willams problem and find the λ solution in the range of angles ψ = 0° - 89°
r
s
2 4
1 3
1 2 3 4
1 2 3 4
01 1 0
sin 2 1 cos 2 1 sin 2 1 cos 2 1 0
1 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1 0
Then thesystem can be put in matrix fo
s s s s
s s s s
c cc c
c c c c
c c c c
1
2
3
4
rm:0 1 0 1 0
1 0 1 0 0sin 2 1 cos 2 1 sin 2 1 cos 2 1 0
01 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1s s s s
s s s s
cccc
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Williams 1957
Exercise – MATLAB:
Write the determinant of the matrix, impose it to zero and solve to find
0 1 0 11 0 1 0
0sin 2 1 cos 2 1 sin 2 1 cos 2 1
1 cos 2 1 1 sin 2 1 1 cos 2 1 1 cos 2 1
...
s s s s
s s s s
r
s
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Williams 1957
Exercise – MATLAB:
r
s
0 0.5 1 1.5 2-4
-2
0
2
42 = 60
= 0.51222
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Williams 1957
Exercise – MATLAB:
r
s
0 20 40 60 800
0.1
0.2
0.3
0.4
0.5
1-
Power-law singularity exponent
11
1ij r
r
Pisa, June 15 – July 14, 2015