Fracture, Toughness and
Strength
by
Gordon Williams
Introduction
• Strength is not a material property
• For ductile materials we have flow and
necking
• For brittle materials we have failure from
flaws
• Surface polishing, a transition from brittle
to ductile
• Griffith ideas
Griffith(1922)
•
•
•
•
•
All bodies contain flaws
Fracture is from these flaws
Used “Energy Release Rate” (see later)
Defined as “G”
G>Gc, energy per unit of created surface
area (J/m^2)
• Gc is a basic material property
Fig 1
s
2a
H
W
b
Griffith
•
G
s pa
2
E
• In general,
 Gc at fracture
K cI  EGc  s 2Y 2 a
• Y2 is a geometric factor, Y2= p for an infinite plate
• To find Gc vary a, measure s, calculate Y2 hence
EGc
• From E find Gc
Griffith
•
•
•
•
From E find Gc
If only stresses needed use Kc
Gc preferred , better physics
The strength problem
s a  const
2
• “a” exists, flaws, hence s is determined
Compliance Method (Composites)
F
F
F
d
F+dF
C(a+da)
a
C
o
d
d+dd
d
da
b
Compliance Method (Composites)
Initial Energy:
Work done on a
Final Energy:
1
U 1 ( a )  Fd
2
a+da,
1
U 2  (2 F + dF )dd
2
1
U 3  ( F + dF )(d + dd )
2
 Change in energy=U1+U2-U3 (Shaded area)
ie
1
dU  ( Fdd  ddF )
2
Compliance Method (Composites)
Compliance:

C (a ) 
d
F
dC
1
dd
dF
 2 (F
d
)
da F
da
da
2
Hence
dU F dC
G

bda 2b da
Energy release rate
Compliance Method (Composites)
1
1
2
U

F
d

CF
• Energy form:
2
2

U 1 dC
U
G (
)
b C da
bW
dC
1
  C[
]
d (a / W )
Used in impact
F
b
h
For DCB
d
h
a
8a 3
C
,
3
Ebh

dC 24a 2

3
da Ebh
12 F 2 a 2
G
2 3
Eb h
Experimental Method
i)
Measure C(a)
dC
da
ii) Measure F at fracture
iii) True for any form
Gc
Compliance Method
From Griffith Solution
G

s pa
2
E
2
F dC
F

, s
2b da
bW
p
a 2
H
C  Co +
( ) , Co 
Eb W
EbW
a
Y ( )
W
2
in General
Plasticity and Size Effects
• Basic method is elastic (LEFM)
• All cracks have a local plastic/damage zone
rr
a
• Let sc be the zone stress
sx
r
Plasticity and Size Effects
• Local stresses,

K
sx 
2pr
(singular)
2
c
K
EGc
rr  » 2 
2 (const., 2p can change)
2psc 2psc
•rr makes response non-linear,
•Must be within limits, e.g F5% , Fmax
Plasticity and Size Effects
• Gc & Kc are dependent on Constraint
• Lowest values are for Plane strain, ez=0
in the plastic zone, i.e. lateral constraint.
• Highest values are for Plane stress, sz=0
Plasticity and Size Effects
Plane stress
z
b
Kc
rb
Plane strain
bc
Plasticity and Size Effects
• For b >> rr, ez0, plane strain
• For b ≈ rr, sz=0, plane stress
Transition:
b<bc
bc  2.5(
Kc
sc
)
2
high value