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

祹

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

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