The J-integral is used to characterize cracks in materials. It is defined as a contour integral involving the strain energy density and traction terms. For a double cantilever beam under a distributed load P, the J-integral can be found by evaluating the integral along different segments of the contour path surrounding the crack tip. Each segment along the bending regions or crack surfaces has a J-integral value of zero, so the total J-integral is simply the value along the constant moment region. The J-integral is shown to be path independent by applying divergence theorem to an arbitrary closed contour.

1. J INTEGRAL FOR DOUBLE CANTILEVER BEAM
Presented by
A. Sabiha Khathun
M.TECH (Machine Design)
16481D1501
Subject: Fracture Mechanics
GUDLAVALLERU ENGINEERING COLLEGE
Sheshadri Rao Knowledge Village, Gudlavalleru,PIN:521356

2. What is j integral ?
• the J-Integral is a parameter to characterize a crack.
• It is applicable to linear and non-linear elastic materials.
• It is very useful to characterize materials, exhibiting
elastic-plastic behavior near the crack tip.

3. Continued…
• It is defined as
and Tij = σij ni
Г contour
W strain energy density
Ti  traction vector at a point on the path Г
u i displacement vector point on the path Г
ds length increment along path
where

4. J integral for double cantilever beam, if each cantilever is
pulled by a distributed load P, as shown
The chosen path Γ is BCDEFH and it coincides with the body contour.

5. Contour of the crack
Therefore,
J=JBC+JCD+JDE+JEF+JFH

6. J=JBC+JCD+JDE+JEF+JFH
To find JBC
As bending moment is zero,
bending stress is zero.
So, W=0
Assuming,
• A small element of length dy on the path then ds = dy
• Along y-direction ui =V
• Length is h so limits are 0 to h.

7. J=JBC+JCD+JDE+JEF+JFH
• To find JCD
dy is negligible and Ti = 0
•To find JEF
 JCD = 0
 JEF = 0

8. J=JBC+JCD+JDE+JEF+JFH
• To find JDE
stresses are very small, which in turn, make Wand Ti negligible.
 JDE = 0
• To find JFH
On segments BC and FH, W is negligible,

9.  Hence, J = JBC+JCD+JDE+JEF+JFH
Now we can find using the bending moment equation
Bending moment BM = P*x
1

10. On applying boundary conditions at x=a and x=0(at crack tip)
At x = a, = 0

11. At x=0
And using the relation I = Bh3/12,
we have
Substitute it in equation 1
But on face FH

12. To prove that j integral is path independent
• The J-Integral is carried out along an arbitrary path Γ, which
starts from one crack face and ends up on the other crack
face, while going around the crack tip. It would be proved,
it is path independent.

13. First evaluate j along closed path S
By divergence theorem,
On using the relation, we get
and
2

14. Substitute in the equation 2
For elastic materials
 =0

15. Continued…
• Consider an area A, shown in which is closed by four
segments, BCD, DE, EFG and GB
• segments DE and GB coincide with the crack surfaces.
Segments BCD and EFG can be chosen arbitrarily
• We now split the integral of equation into four parts, as

16. Hence for a closed contour BCDEFGB, J integral =0
Integrations along DE and GB do not
contribute because dy = 0 and Ti = 0
and hence,
Then,


17. if then
If BCD contour Г1 and GFE contour Г2
Therefore, j integral is path independent.

18. conclusion
• The integration path Γ can be chosen very close to the
crack tip so that it passes through the plastic zone. In any
case, the J-Integral remains the same, provided there is no
unloading.
• This suggests that the j integral is fully capable of
characterizing a crack.