4 Fracture mechanics

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 I Mode II Mode III

Opening mode fracture In plane shear fracture Antiplane shear 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 σ  ∝

−1 / 2

r 

• The constant of proportionality is the stress intensity factor, K  σ  =

−1 / 2

Kr 

Stress intensity factor, K  • Depends on fracture displacement mode (I, II or III) and crack geometry σyy

σyz

σyx

K  I 

= ψσ  yy

π c

K  II 

= ψσ  yx

π c 2c

K  III 

= ψσ  yz

π c

y

z

x

Geometry term,

2c

Straight crack ψ  = 1

ψ 

K  I 

= ψσ  yy

π c

K  II 

= ψσ  yx

π c

K  III 

= ψσ  yz

π c

2c

Penny-shaped crack ψ  = 2/π

• Irwin’s crack tip solutions give the shape of  the stress field • Stress intensity factor gives the magnitude of the stress field

Critical stress intensity factor (or  fracture toughness), K c  Where the stress intensity factor  reaches the energy equilibrium - unstable propagation of the crack

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 m0.5

Typical values for K Ic  • ~0.7 MPa m0.5 for glass • ~1.0 MPa m0.5 for marble • ~1.5 MPa m0.5 for granite • ~2.5 MPa m0.5 for basic rocks • ~3.5 MPa m0.5 for eclogite • ~140 MPa m0.5 for mild steel

Crack extension force, G  • Energy per unit area at the crack tip • G  is related to the stress intensity factor, K  by: 2 K I  G I  =  E  (for plane stress and mode I fractures only)

G=

dU m dC 

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 K Ic  Chevron notch method -recommended by ISRM

• Easy to prepare • Crack growth initially stable • Critical crack length is constant – no crack length measurements needed

Measuring K Ic 

Double torsion test

Hertzian fracture test