2. Principles of Fracture Mechanics
• Quantitative study of failure is called fracture mechanics
• Fracture mechanics is the relationship between materials
properties, stress, crack-producing flaws and mechanisms
of crack propagation
• Possible to design in order to prevent structural failure
Quantify relationships between:
material properties
stress level
presence of cracks/defects in material
types of crack-propagating mechanisms
Objective is to achieve better design
3. Modes of Crack Displacement
• Three modes that a load can act on a crack:
– mode I is tensile (most common)
– mode II and III are sliding or tearing actions
Stress distribution at crack tips depends on how crack is being
extended (Mode of cracking).
FIGURE 8.9 The three modes of crack surface displacement. (a) Mode I, opening or
tensile mode; (b) mode II, sliding mode; and (c) mode III, tearing mode.
4. c)
b)
a)
τ
τ
τ
τ
σ
σ
(III)
(II)
(I)
τ
τ
τ τ
z
z
z
x
x
x y
y
y
σ
σ
Mode I is opening or tensile mode where
the crack surfaces move directly apart.
Modes of Crack Displacement
Mode II is sliding or in-plane shear mode where the
crack surfaces slide over one another in a direction
perpendicular to the leading edge of the crack.
Mode III is tearing and antiplane shear mode
where the crack surfaces move relative to one
another and parallel to the leading edge of the
crack.
5. Modes of Crack Displacement
In practice, crack propagation is not just limited to the three basic modes
Cracks often propagate under so called mixed modes, which are the
combinations of the above mentioned modes, such as I-II, I-III, II-III and
so on.
In practice crack propagation under ………… is the most dangerous.
Under Mode I, it is easier for crack propagation to trigger a brittle
fracture, so it has been studied extensively.
FIGURE 8.9 The three modes of crack surface displacement. (a) Mode I, opening or
tensile mode; (b) mode II, sliding mode; and (c) mode III, tearing mode.
Mode I
6. Stress Analysis of Cracks
• in mode I, the stresses (σx, σy……)
acting on the crack as a function of
radial distance r and angle θ from
crack tip are:
)
(
)
(
)
(
2
2
2
θ
θ
θ
π
τ
π
σ
π
σ
xy
xy
y
y
x
x
f
r
K
f
r
K
f
r
K
=
=
=
K is the stress intensity factor and
defines the stress around a crack or
flaw (similar to but NOT the same as
the stress concentration factor Kt )
−
=
2
3
sin
2
sin
1
2
cos
)
(
θ
θ
θ
θ
x
f
+
=
2
3
sin
2
sin
1
2
cos
)
(
θ
θ
θ
θ
y
f
2
3
cos
2
cos
2
sin
)
(
θ
θ
θ
θ =
xy
f
Where:
7. Fracture Toughness
• When the plate is thin, plane stress exists because σz=0
• When the plate is thick, σz=ν(σx+ σy) and a state of plane strain exists, εz= 0
in the z direction
• stress intensity factor (K) is related to the applied stress and crack length
by:
a
Y
K π
σ
=
Where,
Y is a function of the crack and
specimen size and geometry
(dimensionless parameter)
units of K are MPa m½
Thicker, more rigid pieces of a given material
have a ……… fracture toughness than thin ones.
lower
What is the difference between plane stress or plane strain conditions?
8. Plane Stress and Plane Strain Conditions
Plane stress condition:
In a thin body, the stress through
the thickness (σz) cannot vary
appreciably due to the thin
section.
Because there can be no stresses
normal to a free surface, σz = 0
throughout the section and a
biaxial state of stress results.
Plane strain condition:
In a thick body, however, the
material is constrained in the z
direction due to the thickness of
the cross section and εz = 0.
So due to the Poisson effect, a
stress, σz, is developed in the z
direction.
σz = ν(σx + σy)
9. Fracture Toughness
• When stress level reaches some
critical level, σc, have crack
propagation and fracture
• critical stress intensity factor, Kc,
at the crack tip exists:
Y is a function of a (crack
length) and W (component
width)
( ) a
W
a
Y
K c
c π
σ
/
=
W
Geometric Factor
a) Y(a/W) = 1 (infinite plate)
b) Y(a/W) ≈ 1.1 (semi-infinite plate)
Fracture Toughness, Kc is a critical value of stress intensity
factor, K, upon which fracture will occur.
Kc measures the resistance to crack propagation and failure
10. Plane Strain Fracture Toughness, KIc
• Fracture Toughness is a function
of specimen thickness, B
• when Kc in mode I becomes
independent of B, then have:
Plane strain fracture toughness, KIc
a
Y
KIc
π
σ
=
2
5
.
2
≥
y
Ic
K
B
σ
Thick plate is plain strain condition
plane strain condition exists when:
So varies with values of KIC and σys
minimum value → safer for design
12. Design Using Fracture Mechanics
a
Y
KIc
c
π
σ ≤
If design involves a
maximum allowable crack
size then:
2
1
=
Y
K
a
Ic
c
σ
π
• Important parameters are:
KIc, applied stress σ and
crack size a
• If KIc and a are fixed then
the critical stress is:
13. KIc is measured using Mode I crack opening and calculating the Y
parameter.
Since KIc is a fundamental material property and is affected by:
- temperature (KIc …. as T↑)
- strain rate (KIc ….. as SR↑)
- strengthening (usually KIc …… as σy↑)
- microstructure ( KIc ….. as grain size ↓)
Stress intensity factor, K, is the stress intensity in a material
under a specific loading condition
K will vary as a function of the applied stress on a component
When the stress intensity in a material reaches a critical value,
KIc, fracture occurs
- KIc is a material property like yield strength
Summary
↑
↓
↓
↑
14. Design using Fracture Mechanics
a
Y
KIc π
σ
=
Material Property: so
can select material with
appropriate value of KIc
This could be design
stress (including safety
factor) or applied stress
This is the allowable
flaw size or smallest
flaw that can be detected
During design, we have to decide which parameters are
constrained by the application and which can be controlled by
design.
For example, KIc may be fixed because of the need to use a
certain material: low weight, corrosion resistance etc
Or flaw size may be limited by detection equipment available.
BUT ONCE TWO PARAMETERS ARE FIXED, THIRD PARAMETER IS DEFINED!
15. a
Y
KIc
c
π
σ ≤
2
1
=
Y
K
a Ic
c
σ
π
Example: If KIc and “a” are fixed because a particular material
is required, then the design (or critical) stress is limited as:
Or if the stress level and KIc fixed and material has been chosen
then the maximum allowable flaw size in the material is given by:
• So manufacturing techniques must be good enough to produce
flaws less than this size
• Non-destructive testing (NDT) techniques must be able to
measure flaws this size.
Design using Fracture Mechanics
17. Yield-before-failure: plastic deformation can be observed before
formation of crack of critical size and fracture occurs. R
Requires regular inspection to observe yielding etc.
Leak-before-burst: ensure that critical crack size for fracture is
greater than wall thickness. (i.e. set ac = t, allowing for safety).
Crack can grow through thickness of wall without fracture (bursting)
and vessel will leak allowing detection of leak by pressure drop and/or
presence of fluid.
Schematic diagram showing
the cross section of a spherical
tank that is subjected to an
internal pressure p, and that
has a radial crack of length 2a
in its wall.
Use of fracture mechanics to design
pressure vessels.
18. Require circumferential wall (hoop) stress be less than
yield strength.
2
2
2
:
for
solving
then
and
:
then
N,
factor,
safety
a
using
and
2
:
by
given
stress
wall
Assume
=
=
=
y
Ic
c
c
c
y
Ic
K
Y
N
a
a
a
N
Y
K
t
pr
σ
σ
π
π
σ
So look for materials with best
ratio of (KIc/σy)2
Method 1: yield-before-failure
19. Set a = t, then and substitute for t from above:
t
Y
KIc π
σ
=
=
y
Ic
K
r
Y
p
σ
π
2
2
2
Method 2: Leak-before-Burst
a
Y
KIc π
σ
= t
pr
2
=
σ
Example: Aircraft Fuselage: 2024 (T3)
-light weight
So look for materials with best
ratio of KIc
2/σy
