4 fracture mechanics
TRANSCRIPT
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Fracture mechanics
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Loading configuration Obreimoff: stable
equilibrium No failure
Griffith: unstableequilibrium Failure only for
uniform tension
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We need to characterize the drivingforce for fracture:
Stress Intensity Factor, K (units: Pa m 0.5 ) Crack extension force, G (units: J m -2)
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Crack displacement modes:
Mode I Opening mode fractureMode II In plane shear fractureMode III Antiplane shear fracture
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Irwins crack tip solutions Defines the
shape of thestress fieldsurrounding the
crack tip
Polar or cartesiancoordinates
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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 stressintensity factor, K
2 / 1
r
2 / 1= Kr
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Stress intensity factor, K Depends on fracture displacement mode
(I, II or III) and crack geometry
cK cK
cK
yz III
yx II
yy I
=
=
=
y
yy
yx
yz
x
z
2c
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Geometry term,
2c 2c
Straight crack = 1
Penny-shaped crack = 2/
cK
cK
cK
yz III
yx II
yy I
=
=
=
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Irwins crack tip solutions give the shape of the stress field
Stress intensity factor gives the magnitude
of the stress field
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Critical stress intensity factor (or fracture toughness), K c
Where the stress intensity factor reaches the energy equilibrium- unstable propagation of the crack
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Critical stress intensity factor, K c There is a K c for each displacement
mode: K Ic K IIc K IIIc
Units of K c are stress x crack length,MPa m 0.5
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Typical values for K Ic ~0.7 MPa m 0.5 for glass
~1.0 MPa m 0.5 for marble ~1.5 MPa m 0.5 for granite ~2.5 MPa m 0.5 for basic rocks ~3.5 MPa m 0.5 for eclogite ~140 MPa m 0.5 for mild steel
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Crack extension force, G Energy per unit area at the crack tip
G is 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
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Problems with the fracturemechanics approach
Crack tip processes lower the crackextension 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
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Measuring K Ic
Easy to prepare Crack growth initially stable Critical crack length is constant no crack
length measurements needed
Chevron notch method
-recommended by ISRM
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