4 Fracture mechanics

wafflecanadianΜηχανική

18 Ιουλ 2012 (πριν από 5 χρόνια και 3 μήνες)

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Fracture mechanics
Loading configuration
•Obreimoff: stable
equilibrium
–No failure
•Griffith: unstable
equilibrium
–Failure only for
uniform tension
Irwin’s generalization of the Griffith
concept: Fracture mechanics
•Approach whereby the crack is idealized
as a mathematically flat and narrow slit
contained within a linear elastic medium
•Analyse the stress field around a crack
•Macroscopic strength is determined from:
–intrinsic strength of the material
–applied stresses
–crack tip stresses
We need to characterize the driving
force for fracture:
•Stress Intensity Factor, K(units: Pa m0.5)
•Crack extension force, G(units: J m-2)
Crack displacement modes:
Mode IOpening mode fracture
Mode IIIn plane shear fracture
Mode IIIAntiplaneshear fracture
Irwin’s crack tip solutions
•Defines the
shape of the
stress field
surrounding the
crack tip
•Polar or
cartesian
coordinates
Stress intensity factor, K
•The stress surrounding a crack is proportional
to one over the square root of the distance, r
from the crack, hence
•The constant of proportionality is the stress
intensity factor, K
2/1−
∝r
σ
2/1−
=Kr
σ
Stress intensity factor, K
•Depends on fracture displacement mode
(I, II or III) and crack geometry
cK
cK
cK
yzIII
yxII
yyI
πψσ
πψσ
πψσ
=
=
=
y
σ

σyx
σyz
x
z
2c
Geometry term,
ψ
2c
2c
Straight crack
ψ
= 1
Penny-shaped crack
ψ
= 2/π
cK
cK
cK
yzIII
yxII
yyI
πψσ
πψσ
πψσ
=
=
=
•Irwin’s crack tip solutions give the shapeof
the stress field
•Stress intensity factor gives the magnitude
of the stress field
Critical stress intensity factor (or
fracture toughness), Kc
Where the stress intensity factor
reaches the energy equilibrium
-unstable propagation of the crack
Critical stress intensity factor, Kc
•There is a Kc
for each displacement
mode:
–KIc
–KIIc
–KIIIc
•Units of Kc
are stress x √crack length,
MPam
0.5
Typical values for KIc
•~0.7 MPam
0.5
for glass
•~1.0 MPam
0.5
for marble
•~1.5 MPam
0.5
for granite
•~2.5 MPam
0.5
for basic rocks
•~3.5 MPam
0.5
for eclogite
•~140 MPam
0.5
formild steel
Crack extension force, G
•Energy per unit area at the crack tip
•Gis related to the stress intensity factor, K
by:
E
K
G
I
I
2
=
(for plane stress and mode I fractures only)
dC
dU
G
m
=
G can be related to specific surface energy γ
Problems with the fracture
mechanics approach
•Crack tip processes lower the crack
extension force:
–distributed cracking
–plastic flow
•The crack behind the tip is assumed to be
cohesionless
–ok for mode I fractures
–problematic for mode II and III
Measuring KIc
•Easy to prepare
•Crack growth initially stable
•Critical crack length is constant –no crack
length measurements needed
Chevron notch method
-recommended by ISRM
Measuring KIc
Double torsion test
Hertzianfracture test