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where
U
is the deformation energy for thickness unit, and
Δ
deformation in the direction
of loading. Under LEFM conditions
J
become identical with energy release rate (
J
=
G
).
Based on this,
J
becomes parameter describing stress intensity similar to
K
in LEFM.
Hutchinson and Rice and Rosengren independently derived a crack tip stress and strain
solution for nonlinear elastic materials, which became known as the HRR solution. The
HRR solution was derived for materials with power law stress-strain relationship
n
o
o
ε
σ
α
ε
σ
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
(15)
where
σ
0
is a reference value of stress (usually),
ε
ο
=
σ
ο
/E
, and
α
and
n
are material
constants. The HRR stress and strain fields near the tip of the crack are given by
1/( 1)
2
( , )
n
ij
o
ij
o n
EJ
g n
I r
σ
σ
θ
ασ
+
⎛
⎞
= ⎜
⎟
⎜
⎟
⎝
⎠
(16)
/( 1)
2
( , )
n n
o
ij
ij
o n
EJ
h n
E
I r
σ
ε
α
θ
ασ
+
⎛
⎞
= ⎜
⎟
⎜
⎟
⎝
⎠
(17)
where
I
n
is a dimensionless constant that depends on
n
,
and
g
ij
and
h
ij
are dimensionless
functions of
n
and
θ
. These parameters also depend on stress state (plane stress or plane
strain). For a linear elastic material,
n = 1
and stress varies as
J/r (
=
G/r
or
/
K r
)
which is consistent with the elastic solution for
K
.
As with the stress intensity factor
K
for linear elastic materials,
J
completely
characterizes the crack tip stresses and strains in nonlinear elastic materials. Thus
J
is an
adequate fracture mechanics parameter for elastic-plastic materials, as long as the
nonlinear elastic assumption gives a reasonable approximation of real material behaviour.
Real relationships are, as shown in Fig. 10, somewhat more complex. Part (a)
illustrates the state in case of small scale yielding that corresponds to LEFM regime. The
region moderately close to the crack tip, where the crack tip stress and strain fields are in
conformance with the calculation with
K
, is the region controlled by
K
and with slope that
corresponds to the solution based on
1 /
r
dependence. Somewhat closer to the crack tip,
in the plastic zone, is the
J
controlled region where the slope of the change corresponds to
1
/(n+
1
).
The small area very close to the crack tip is the large strain region where the
HRR solution is no longer valid. A
log-log plot has been selected to show these
relationships more clear. It should not forget, that in the solution for
K
(7) there are other
terms of high order neglected that outside the
K
controlled region gain in the importance.
However, in case of small scale yielding crack can be characterised with
K
or
J
, because
of analytical relationship between this two parameters in form
2
K J
E
=
(18)
With the loading increase the
K
controlled region gradually disappears (b in Fig. 10),
but there is still remain the region in which the HRR solution is valid. This also means
that the application of
K
is no more adequate and the situation at the crack tip can be only
characterised by
J
. However, the evaluation of
K
values based on
'
eff
K JE
=
(19)