334 / 352
330
In that case significant energy
γ
p
is spent for development of plastic deformation
before final fracture, even greater than surface energy
γ
. Orowan /13/ proposed the
extension of basic formula of Griffith in the form:
2
p
F
E(
)
a
γ
γ
σ
π
+
=
(12)
The expression
γ
+
γ
p
, called the fracture energy, can include plastic, viscoelastic or
viscoplastic effects, depending on the material type.
The efficient parameter in describing crack in plane stress condition with expressed
plastic deformation is path independent, contour
J
integral /12/. It is also possible to
apply it as a measure of fracture toughness,
J
Ic
, according to ASTM E813 and similar
later introduced standards, enabling to cover a wide range of stress conditions by only
one parameter.
By definition
J
integral (Fig. 9) is an energy criterion, given in the form:
i
i
u
Wdy T ds
x
J
Γ
∂
⎛
⎞
− ⎜
⎟ ∂
⎝
⎠
= ∫
v
(13)
with
W =
∫σ
ij
d
ε
ij
- strain energy density (
σ
ij
is stress tensor,
ε
ij
strain tensor);
Γ
- integ-
ration path;
ds
- element of segment length;
T
i
=
σ
ij
n
j
- traction vector on the contour;
u
i
–
displacement vector,
n
i
- normal unit on contour
Γ
. Rice has shown that
J
integral is path
independent /12/ for two-dimensional plane problems with no volume and inertial forces,
and for non-linear elastic material, homogeneous at least in crack growth direction. In
that case
J
integral can be presented as the energy, released on crack tip for unit area
crack growth,
Bda
, e.g. following expression is valid
i
i
u
Wdyda B T dsda
x
JBda B
Γ
Γ
∂
−
∂
= ∫
∫
v
v
(14)
where
B
is specimen thickness. The member
B
∫
Wdyda
denotes the energy obtained (and
released) along the contour
Γ
for crack increase,
da
. Second member in Eq. (14) repre-
sents the work of traction forces on contour displacement for crack growth
da
. As an
energy criterion (the value
JBda
is the total energy at crack tip available for crack growth
Δ
a
)
,
it can be related to the Griffith's energy
G
, and stress intensity factor,
K
, so:
2
K J G
E
= =
′
;
E'
=
E
(plane stress)
E'
=
E
/ (1 -
ν
2
) (plane strain condition) (15)
The behaviour of elastic-plastic material during stable crack growth can be described
by diagram
J
-
Δ
a
, (J-R curve), where
Δ
a
stands for crack extension, Fig. 10.
Generally,
J
integral is seen to be the potential energy difference between two bodies
having identical boundary tractions, and incrementally different crack lengths. This is
expected, since in
J
integral the energy terms are considered, similar as in Griffith’s
approach, in which the total energy of cracked body is calculated. It has been proved
experimentally that J integral is path independent also in the plastic range. Therefore,
J
integral can be used as a general FM parameter, in both, elastic and plastic ranges, and
the procedure for J - R curve determination is accepted in ASTM E1152 standard.
It is to mention that crack opening displacement (COD) is another popular FM
parameter in elastic and in plastic range, applied for welded structures.