Crack Tip Fields for Power Law Hardening Materials—HRR Fields
Refs. Rice and Rosengren (1968) & Hutchinson (1968); fracture notes pg. 46
σ Y εY ⎛ ε
W→
⎜
1 + N ⎜⎝ ε Y
P
eff
1+ N
⎞
⎟⎟
⎠
,
ε P / ε Y = (σ / σ Y ) n − 1
ε =σ / E
ε effP = 2ε ijPε ijP / 3 , N = 1/ n
J=
εY = σ Y / E
σ
σY
Deformation theory fit to stress-strain
curve with power law plasticity
εY
∫ (Wnx − σ ij n j ui, x ) ds
ε
Tensile stress-strain curve
ΓC
=∫
π
−π
(W cos θ − σ
ij
n j ui , x )rdθ
If J non-zero this requires that (W cos θ − σ ij n j ui , x ) ∝ ( J / r ) since J is independent
of the path and ΓC can be shrunk down to the tip, i.e. r → 0.
This implies the following as r → 0:
W ∝ (J / r)
⇒ ε ijP ∝ ( J / r ) n /( n +1) ⇒ σ ij ∝ ( J / r )1/( n +1)
The crack tip fields are of the form:
1/( n +1)
⎛
⎞
J
=
,
σ
σ
σ
( ij e ) Y ⎜ σ ε I r ⎟
⎝ Y Y n ⎠
(σ
ij
(θ , n), σ e (θ , n) ) , ε P ij
π ⎡ 1
In = ∫ ⎢
εeffP
(
−π 1 + N
⎣
⎛
⎞
J
= εY ⎜
⎟
I
r
σ
ε
⎝ Y Y n ⎠
)
1+ N
⎤
cos θ + (....) ⎥ dθ
⎦
n /( n +1)
ε ijP (θ , n)
More next slide
Crack Tip Fields for Power Law Hardening--continued
Plane strain, mode I fields:
The θ -variations, σ (θ ,n), ε (θ ,n) and I n , depend on:
1) the mode (I, II or III); 2) whether plane strain or plane stress holds; 3) the hardening exponent n.
See the references for computing the θ -variations.
Crack Tip Fields for Power Law Hardening--continued
Plane stress, mode I fields:
Isochromatic contours
Plastic Zones in Plane Strain Small Scale Yielding under Combined Modes I and II
Mode I+II
Mode I
Mode I+II
Mode II
Ref. Shih, 1976
Plastic Zones in Plane Stress Small Scale Yielding under Combined Modes I and II
Mode I
Mode I+II
Mode I+II
Mode II
Ref. Shih, 1976
J Characterization of Initiation of Crack Growth and JIC
The crack tip fields:
1/( n +1)
n /( n +1)
⎛
⎞
⎛
⎞
J
J
P
=
=
,
(
,
),
(
,
)
,
σ
σ
σ
σ
θ
σ
θ
ε
ε
n
n
( ij e ) Y ⎜ σ ε I r ⎟ ( ij
) ij Y ⎜ σ ε I r ⎟ εijP (θ , n)
e
⎝ Y Y n ⎠
⎝ Y Y n ⎠
Given plane strain (or plane stress) and mode I, J is a unique measure of the amplitude, or intensity,
of the crack tip stress and strain distribution.
For plane strain, mode I, a criterion for crack growth initiation is
J = J IC
with J IC as a material property (toughness).
In small scale yielding, J = (1 −ν 2 ) K 2 / E , and thus,
J IC = (1 −ν 2 ) K IC 2 / E ≡ GIC
Since J is not limited to small scale yielding, it can be used in
testing to determine toughness or in applications under large scale yielding.
This requires knowledge of how J depends on load or displacement.
How does one determine J ?
Determination of J—Analytical Methods and Experimental Methods
Various methods have been proposed to estimate J as a function of applied load or load point displacement.
Some of these are surveyed in the fracture notes. It now common to use a finite element code to solve
large scale yielding crack problems and to evaluate J directly using the path-independent integral. The
situation is not unlike that for evaluating K or G for linear elastic problems, except that the nonlinearity
due to plasticity makes the problems more difficult. We will not pursue this further in this course, however,
we will derive one basic result for deeply-cracked bend specimens which allows direct experimental
evaluation of J when such a specimen is used for determining JIC.
J for a deeply cracked bend specimen (pg. 55 of notes)
M ,θ / 2
M ,θ / 2
c
a
b
L
For a deeply cracked beam (b >> c), θ cr depends on c
M
but not b. By dimensional analysis:
Recall, (with P → M & ∆ → θ )
, a)
M ∂θ ( M
⎛ ∂PE ⎞
J = −⎜
dM
⎟ = ∫0
⎛ M σY ⎞
∂a
⎝ ∂a ⎠ M
θ cr = f ⎜
, , n ⎟ In linear elasticity,
2
θ = θ no crack + θ cr
⎝ σY c
E
⎠
θ cr = 16 M /( Ec )
θ
2
continued
Determination of J—Experimental Method for Bend Specimens (continued)
, c)
, a)
M ∂θ ( M
M ∂θ ( M
⎛ ∂PE ⎞
cr
because ∂ ( ) = − ∂ ( )
J = −⎜
dM
dM
=
=
−
⎟
∫0 ∂a
∫0
a
∂
∂c
∂a
∂c
⎝
⎠M
∂f M
M
∂f 1
⎛ ∂θ cr ⎞
⎛ ∂θ cr ⎞
Now note: ⎜
, with µ =
, and ⎜
⎟ = −2
⎟ =
3
2
2
c
µ
σ
c
σ
c
M
µ
σ
c
∂
∂
∂
∂
⎝
⎠M
⎝
⎠
c
Y
Y
Y
Thus,
⎛ ∂θ cr ⎞
⎛ ∂θ cr ⎞ 2 M
=
−
, and
⎜
⎟
⎜
⎟
⎝ ∂c ⎠ M
⎝ ∂M ⎠c c
2 M
J = ∫ Mdθ cr
c 0
Conclusion: For the deeply cracked bend specimen
J can be determined directly as the area under the
moment-rotation (cr) curve.
1
cJ
2