On the applicability of linear elastic fracture mechanics ...
INTRODUCTION TO LINEAR ELASTIC FRACTURE …INTRODUCTION TO LINEAR ELASTIC FRACTURE MECHANICS Master...
Transcript of INTRODUCTION TO LINEAR ELASTIC FRACTURE …INTRODUCTION TO LINEAR ELASTIC FRACTURE MECHANICS Master...
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
23
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