2. Energy domain integral method:
- Formulated by Shih et al. (1986):
- Generalized definition of the J- integral (nonlinear materials, thermal strain, dynamic effects).
- Relatively simple to implement numerically, very efficient.
Elements of Theory
Finite element (FE) code ABAQUS version 6.5
CF Shih, B. Moran and T. Nakamura, “Energy release rate along a three-dimensional crack front in a
thermally stressed body”, International Journal of Fracture 30 (1986), pp. 79-102
ABAQUS: - suite of powerful engineering simulation programs
- based on the finite element method
- for simple linear analyses and most challenging nonlinear simulations
3. Abaqus 6.5 :
creates input files (.inp) that will be processed by Abaqus standard.
products associated with Abaqus:
For details see the Getting Started Manual of Abaqus 6.5
can be used for producing/ importing the geometry to be analyzed.
is useful to monitor/control the analysis jobs and display the results (Viewer).
Abaqus Standard : general-purpose analysis product that can treat a wide range of problems.
CAE : interactive, graphical environment allowing models to be created quickly.
Optional capabilities (offshore structures,
design sensitivity calculations)
Abaqus Explicit : intended for modeling brief, transient dynamic events (impact)
uses an explicit dynamic finite element formulation.
4. In 2D, under quasistatic conditions, J may be expressed by
and,
The contour surrounds the crack tip.
The limit indicates that shrinks onto the crack tip.
For details see the Theory Manual of Abaqus 6.5, section 2.16
n : unit outward normal to
q : unit vector in the virtual crack extension direction.
w : strain energy density
displacement gradient tensor
x1 , x2 Cartesian system
0
n H q
T
J lim ds
H I u σ
T
w
Energy Domain Integral :
H : Eshelby’s elastic energy-momentum tensor (for a non-linear elastic solid)
: Cauchy stress tensor
5. With q along x1 and the field quantities expressed in Cartesian components, i.e.
1
0 1
j
ij i
u
J lim wn n ds
x
The expression of J (see eq. 6.45) is recovered
1
0
q
In indexed form, we obtain
1 1
1 2
11 12
21 22 2 2
1 2
0
0
HT
u u
x x
w
w u u
x x
1
2
n
n
n
1 1
1 2
11 12
1 2 1 2
21 22 2 2
1 2
0 1 1
0 0 0
u u
x x
w
n n n n
w u u
x x
n H q
T
Thus,
1 2
n ds dx dy
with
The previous equation is not suitable for a numerical analysis of J.
Transformation into a domain integral
6. Following Shih et al. (1986),
is a sufficiently smooth weighting
function in the domain A.
on
on C
q
q
0
m = -n on
A includes the crack-tip region as 0
m : outward normal on the closed contour C C C
q q
i
q x
Note that,
m H q t u q
T
C C C C C
ds ds
A
: the surface traction on the crack faces.
t σ m
0
n H q
T
J lim ds
1
q
1
0
i
on
q x on C
otherwise arbitrary
with
(*)
7.
m H q m I u σ q
T
T T
C C C C
ds w ds
m I σ u q
C C
w ds
σ σ
T
since
m q m σ u q
C C C C
w ds ds
m σ u q
C C
ds
σ m u q
T
C C
ds
Derivation of the integral expression
0
t u q
C C
ds
t σ m
since
(*)
m H q m H q m H q m H q m H q
T T T T T
C C C C C
J ds ds ds ds ds
Noting that,
=
0
=
q
→ Line integral along the closed contour enclosing the region A.
C C C
8. Using the divergence theorem, the contour integral is converted into the domain integral
H q t u q
T
A C C
J div dA ds
A C C
J dA ds
H : q h q t u q
T
div div
A a A a + A: a
Under certain circumstances, H is divergence free, i.e. , 0
ki i
H
indicates the path independence of the J-integral.
In the general case of thermo-mechanical loading and with body forces and crack face
tractions: , 0
ki i
H
the J-integral is only defined by the limiting contour 0
,
k ki i
h H
or
Introducing then the vector,
div
h = H in A
Using next the relationship,
Contributions due to crack face tractions.
9. - This integral is evaluated using ring elements surrounding the crack tip.
- Different contours are created:
In Abaqus:
First contour (1) = elements directly connected to crack-tip nodes.
The second contour (2) are elements sharing nodes with the first, … etc
8-node quadratic plane
strain element (CPE8)
1
2
Refined mesh Contour (i)
Crack
1
q
0
q nodes outside
nodes inside
q
0 1
q
Exception: on midside nodes (if they exist) in the outer ring of elements
10. J-integral in three dimensions
Local orthogonal Cartesian coordinates at the point s on the crack front:
0
n H q
T
J s lim d
Point-wise value
J defined in the x1- x2 plane crack front at s
For a virtual crack advance (s) in the plane of a 3D crack,
L : length of the crack front under consideration.
: surface element on a vanishingly small tubular surface enclosing
the crack front along the length L.
dA ds d
L
T
11. Numerical application (bi-material interface):
x
y
Material 1
Material 2
b
a
2h
and h/b = 1
a = 40 mm
b = 100 mm
h = 100 mm
• SEN specimen geometry (see annex III.1):
Material 1:
a/b = 0.4
MP
a.
Remote loading:
Materials properties (Young’s modulus, Poisson’s ratio):
Plane strain conditions.
E1 = 3 GPa
1 = 0.35
Material 2: E2 = 70 GPa
2 = 0.2
12. • Typical mesh:
Material 1= Material 2
Refined mesh around the crack tip
Number of elements used: 1376
Type: CPE8 (plane strain)
Material 1
Material 2
15. Isotropic Bi-material
KI KII KI KII
Annex III 0.746 0. / /
Abaqus 0.748 0. 0.752 0.072
Results:
Material 1 Material 2 Bi-material
J (N/mm) J (N/mm) J (N/mm)
Abaqus 0.1641 0.0077 0.0837
MPa m
SIF given in
(*) same values on the contours 2-8
for the isotropic case (i =1,2).
2
2
1
I
i i
K
J
E ( )
• Ones checks that:
(*)
16. - For an interfacial crack between two dissimilar isotropic materials (plane strain),
where
and 3 4
i i
2 1
i
i
i
E
G
2
1
i
i
i
E
E
• Relationship between J and the SIF’s for the bi-material configuration:
plane strain, i = 1,2
H. Gao, M.Abbudi and D.M. Barnett, “Interfacial Crack-tip fields in anisotropic elastic solids
thermally stressed body”, Journal of the Mechanics and Physics of Solids 40 (1992), pp. 393-416
- Extracted from the Theory Manual of Abaqus 6.5, section 2.16.2.
Disagreement with the results of Smelser et al.
KI and KII are defined here from a complex intensity factor, such that
with
