An illustration of the variation of stress intensity factor with crack length
for prescribed load & prescribed displacement
Using results from slide on DCB specimen
4 Pa 3
∆=
Eb3
Pa
K = 2 3 3/ 2 ,
b
3Eb3/ 2
K=
∆
2a 2
(P is the force/thickness)
For fixed P, K increases as a increases.
For fixed ∆, K decreases as a increases.
K versus crack advance
With initial crack length, a0 , load up to P0
4
2 3P0 a0
.
3/ 2
b
If we fix P = P0 and increase the crack length,
3
with ∆ 0 = 4 P0 a03 /( Eb3 ) and K 0 =
K
a
=
K 0 a0
Prescribed P=P
0
2
However, if we fix ∆ = ∆ 0
and increase the crack length,
K ⎛ a0 ⎞
=⎜ ⎟
K0 ⎝ a ⎠
1
Prescribed ∆=∆
2
0
1
See plot for very different trends
0
1.5
2
a/a
2.5
0
3
The concept of small scale yielding for cracks in metallic materials
σ Y ∼ tensile yield stress
L ∼ crack length (a )
or other relevant length
region in which K-field holds
rp
Mode I small scale
yielding plastic
zones
Details of these shapes will
be discussed later
1
rp ≅
3π
⎛K ⎞
⎜
⎟
σ
⎝ Y⎠
2
1⎛ K ⎞
rp ≅ ⎜
⎟
π ⎝ σY ⎠
2
π⎛K ⎞
rp = ⎜
⎟
8 ⎝ σY ⎠
Small scale yielding requires that rp be sufficiently small
For the crack of length 2a in an infinite
compared to L such that there exists a region surrounding
the plastic zone in which the K -field is accurate.
sheetor slab, this condition requires
rp << a / 4.
2
Griffith-Irwin Criterion for Mode I Crack Advance in an Ideally Brittle Material
K
(pg. 14 of notes)
K IC
Quasi-static advance occurs when
K = K IC
K IC is called the fracture toughness (units Pa m )
(1 −ν 2 ) K IC 2
GIC =
is also called the fracture toughness (units J / m 2 )
E
crack advance ∆a
We will discuss experimental methods to measure fracture toughness later. To
use LEFM, the test must satisfy small scale yielding conditions in plane strain. In the table
Below we list representative values of yield strength, fracture toughness and plastic zone
size for four metals ranging from relatively low toughness to high toughness.
material
T (o C ) σ Y (MN / m 2 ) K IC ( MPa m ) GIC ( J / m 2 ) rp (mm)
6061 − T 651( Al ) 20
269
33
14, 000
5
7075 − T 651( Al ) 20
AISI 4340 (Steel) 0
620
1500
36
33
16, 600
4900
0.35
0.05
A533B (Steel)
Typical ceramic
620
??
200
0.3 − 0.6
93
20
180, 000
5 − 20
260
??
From Material Selection in Mechanical Design, M.F. Ashby, Pergamon Press
Crack Advance in Ideally in a Brittle
4340 Steel Compact Tension Specimen
KI =
ASTM compact tensile specimen
P
⎛a⎞
a F1 ⎜ ⎟
bt
⎝b⎠
⎛a⎞
⎛ 0.6 − a / b ⎞
⎛ a / b − 0.4 ⎞
F1 ⎜ ⎟ ≅ 11.7 ⎜
⎟ + 17.6 ⎜
⎟
⎝b⎠
⎝ 0.2 ⎠
⎝ 0.2 ⎠
(see earlier slide)
And from Tada, pg. 62
δ
choose : b = 2in = 0.0508m
h = 0.6b
h1 = 0.275b
c = D = 0.25b
thickness ≡ t = b / 2
K IC = 33MPam −1/ 2 , E = 200GPa, ν = 0.29
P ⎛a⎞
⎛1+ x ⎞
2
3
4
5
δ = V2 ⎜ ⎟ , V2 ( x) = ⎜
⎟ ( 2.163 + 12.219 x − 20.065 x − 0.9925 x + 20.609 x − 9.9314 x )
Et ⎝ b ⎠
⎝1− x ⎠
2
-6
3 10
4
4 10
2.5 10
4
3.5 10
2 10
4
3 10
1.5 10
4
2.5 10
1 10
4
2 10
-6
-6
-6
-6
0.4
0.45
0.5
a/b
0.55
0.6
0.4
0.45
0.5
a/b
0.55
0.6
Thickness and Temperature Dependence of Fracture Toughness
(pg. 16 & 17 of notes)
Schematic of Experimental Thickness dependence
KC
Schematic of Experimental Temperature dependence
for a steel
K IC
Temperature T
thickness t
Lower temperature range governed by cleavage
“Upper shelf” range governed by ductile void
nucleation, growth an coalescence mechanism
ASTM requirement for valid K IC test :
The mechanism transition is referred to as the
brittle to ductile toughness transition.
2
2
⎛K ⎞
⎛K ⎞
t > 2.5 ⎜ IC ⎟ ≅ 25rp & t > 2.5 ⎜ IC ⎟ ≅ 25rp
⎝ σY ⎠
⎝ σY ⎠
Compact tension specimens for valid KIC testing. The smallest specimen is about 2 inches wide. The largest specimen
was used to obtain the toughness of a very tough pressure vessel steel (A533B), as required by LEFM.