1. Computational Fracture Mechanics
Anderson’s book, third ed. , chap 12

2. Elements of Theory Energy domain integral method:
- Formulated by Shih et al. (1986):
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
- Generalized definition of the J- integral (nonlinear materials, thermal strain, dynamic effects).
- Relatively simple to implement numerically, very efficient.
Finite element (FE) code ABAQUS

3. Energy Domain Integral :
In 2D, under quasistatic conditions, J may be expressed by
The contour G surrounds the crack tip.
The limit indicates that G shrinks onto the crack tip.
n : unit outward normal to G.
x1 , x2 Cartesian system
q : unit vector in the virtual crack extension direction.
w : strain energy density
and,
s : Cauchy stress tensor
displacement gradient tensor
H : Eshelby’s elastic energy-momentum tensor (for a non-linear elastic solid)
For details see Abaqus Theory Manual, section

4. With q along x1 and the field quantities expressed in Cartesian components, i.e.
Thus,
In indexed form, we obtain
The expression of J (see eq. 6.45) is recovered
with
The previous equation is not suitable for a numerical analysis of J.
Transformation into a domain integral

5. Following Shih et al. (1986),
(*)
m : outward normal on the closed contour
: the surface traction on the crack faces. A
is a sufficiently smooth weighting function in the domain A.
m = -n on G
with
Note that,
A includes the crack-tip region as

6. (*)
Derivation of the integral expression = = q
→ Line integral along the closed contour enclosing the region A.
Noting that,
since
since

7. Using the divergence theorem,
the contour integral is converted into the domain integral
Under certain circumstances, H is divergence free, i.e.
indicates the path independence of the J-integral.
In the general case of thermo-mechanical loading and with body forces and crack face tractions:
the J-integral is only defined by the limiting contour
Introducing then the vector,
in A or
Using next the relationship,
Contributions due to crack face tractions.

8. - Different contours are created:
In Abaqus:
- This integral is evaluated using ring elements surrounding the crack tip.
- Different contours are created:
First contour (1) = elements directly connected to crack-tip nodes.
The second contour (2) are elements sharing nodes with the first, … etc
Refined mesh
Contour (i)
nodes outside
nodes inside
Crack 2 1
8-node quadratic plane strain element (CPE8)
Exception:
on midside nodes (if they exist) in the outer ring of elements

9. J-integral in three dimensions
Local orthogonal Cartesian coordinates at the point s on the crack front:
J defined in the x1- x2 plane
crack front at s L
Point-wise value
For a virtual crack advance l(s) in the plane of a 3D crack, T
L : length of the crack front under consideration.
: surface element on a vanishingly small tubular surface enclosing the crack front along the length L.

10. Numerical application (bi-material interface):
SEN specimen geometry (see Appendix III.1): s
a = 40 mm
b = 100 mm
h = 100 mm
a/b = 0.4
Material 1
and h/b = 1
s = 1 MPa.
Remote loading: 2h a b
Materials properties (Young’s modulus, Poisson’s ratio):
Material 1:
E1 = 3 GPa
n1 = 0.35
Material 2 x y
Material 2:
E2 = 70 GPa
n2 = 0.2 s
Plane strain conditions.

11. Typical mesh:
Material 1= Material 2
Material 1
Material 2
Refined mesh around the crack tip
Number of elements used: 1376
Type: CPE8 (plane strain)

12. Results: Remarks: Material 1 Material 2 Bi-material J (N/mm) Abaqus
0.1641
0.0077
0.0837
(*)
(*) same values on the contours 2-8
Isotropic
Bi-material KI
KII
Annex III
0.746 0. /
Abaqus
0.748
0.752
0.072
SIF given in
Remarks:
for the isotropic case (i =1,2).
One checks that:

13. Relationship between J and the SIF’s for the bi-material configuration:
- Extracted from Abaqus Theory Manual, section
- For an interfacial crack between two dissimilar isotropic materials (plane strain),
where
and
plane strain, i = 1,2
KI and KII are defined here from a complex intensity factor, such that
with