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INTRODUCTION TO LINEAR ELASTIC FRACTURE MECHANICS
Master MAGIS
Theoretical strength of a perfectly ordered crystalline lattice
d0
Energy U
Force F = dU/dd
x
y
A
2b
2a
A
a
b
yy
xx
Stress concentration at the vicinity of an elliptical hole
• Inglis calculation
Uniformly stressed plate
A
A
Infinitely narrow elliptical cavity
Remote unifomr tensile stress field A
Crack length
Cra
ck e
ner
gy U
Us
UM
U
0
Unstable configuration
Crack in uniform tension
2a
F
d
a
h
Crack length
Energ
y U
UM
Us
0
Stable configuration
Cleavage by a wedge
• Obreimoff’s experiment (cleavage of mica)
The Dupré work of adhesion:
w = g1 + g2 – g12
En
erg
yRepulsion
Attraction
Tension
Compression
Str
es
s
d
d
z0
nm
w
w ~ 50 mJ/m2
Equilibrium conditions !No chemistry !
Homogeneous solid w = 2g
Stress-separation function for two atom planes
d
FElectrostatic interfactions
(Van der Walls,…)
Condition for equilibrium fracture : the Griffith criterion
• Energy balance for crack extension
EQUILIBRIUM condition !!
No dissipative processes !!
In most practical situations
- Energy dissipation at the crack tip (plasticity, viscoelasticity) and/or in the bulk- Dependence of actual fracture micro-mechanisms at the crack tip (process zone)……
where Strain energy release rate
Irwin’s approach : Linear elastic crack-tip fields
• The three modes of fracture
• Irwin slit-crack tip in rectangular and polar coordinates
I : opening modeII: sliding mode (in plane shear)III: tearing mode (out of plane shear)
Stress-intensity factors K
K=f (geometry, applied loading)
Mode I crack loading
a da
Equivalence of G and K parameters
• Extension and closure of the crack increment CC’
Fixed grip (u=const) calculation of the strain-energy release:
Cohesive zone models
Dugdale, Barenblatt
• Non linear cohesive stress p(X,u) acting accross the walls of the slip
• Superposition of the K fields arising from the ‘external’ and ’internalcontributions’
l << c << L
Physical origins of the cohesive zone
• Surface forces acting across the crack faces• Platic deformation at the crack tip (Dugdale model)• Crack bridging in crazes (polymer fracture)• ….
Barenblatt model: crack-opening profile
Irwin
Barenblatt
Hyp:Irwin
Path independent integrals about crack tip : The J integral approach
Eshelby, Rice
J integral → holds for any reversible deformation response linear or non linear
→ path independent
Brittle materials : Hertzian fracture / I
Sphere on flat contacts
(ii) Crack initiation at the edge of the contact
(iii) Stable crack propagation
(iv) Instable crack propagation : formation of the Hertzian cone
(v) Stable propagation of the cone
(vi) Crack closing during unloading
r/a
TRACTION
COMPRESSION
int.
ext.
/pm
zz
r
q
r
q
0-1 1
-1
-1.5
Maximum tensile stress:
mmR pp 1.0212
1
pm = mean contact pressure
Hertz contact : surface tractions
Point surface loading: sub-surface stress fields
Half surface
Side view
Stress trajectories Contours
Side view
• Stress-based approach: mR p 212
1
2RPc
Experimentally ! RPc
• Linear elastic fracture mechanics analysis approach
Non homogeneous stress field: the stress intensity factor is evolving as along the crack path
Critical load for the propagation of the Hertzian cone crack
Hertz limiting case
Boussinesq limiting case
K / K
c
Branchs(1) et (3) : instable propagation
Branchs (2) et (4) : stable propagation
*
cc
E
KRP
2
Crack of length cc :Hertzian cone formation
Stability criterion for a Hertzian crack
Critical load
Vickers
Residual stresses during unloading:
→Nucleation and propagation of median cracks
→ Surface propagation of radial cracks
• Plastic deformation
• Radial and median cracks
Brittle materials : cone indentation
Vickers indentation
ccR KaY
23c
PKc
23
21
/cc
P
H
EK
32
0 cot
Toughness measurements using indentation
• Sphère / plan
•Vickers
c >> a P > Pc
: non dimensional parameter dependent on Poisson’s ratio
0 : non dimensional constant function of the strain distribution
Vickers indentation
ccR Ka
23c
PKc
23
21
/cc
P
H
EK
32
0 cot
Determination of fracture toughness from indentation testing
• sphere on flat
• Vickers, cone…
c >> a
Non dimensional parameter depending on Poisson’s ratio
0 Non dimensional parameter depending on indenter’s geometry
Half-penny crack
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Bibliography
• Fracture mechanics
B. Lawn
Fracture of brittle solids, Cambridge Solid State Science Series,Cambridge University Press, 1993
D. Maugis
Contact, adhesion and rupture of elastic solids, Springer, Solid State Sciences, 2000
• Fracture mechanics of polymer materials
J. G. Williams
Fracture mechanics of polymers, Halsted Press, New York, 1984
