PSI - Issue 80

Xinpeng Tian et al. / Procedia Structural Integrity 80 (2026) 451–461 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

456

6

where   is a circle with a small radius ε around the crack tip, n is the normal vector, the quantities t i , R i , s i and Q are defined in (16) and (17), and U is the density of the Gibbs free energy functional

1 2

1 2

1 2

U

.

D

=

ij ij     + ijk ijk

+

, k k 

It is possible to show the path independence of the J-integral on an arbitrary domain around the crack tip, where the crack tip is excluded. The outer boundary of this domain  can be chosen as the integration path. Then, due to missing body forces as well as tractions, higher-order tractions and electric charge on crack faces, one can show that the value of the J integral on   and Γ are the same (Sladek et al. 2017)

  

  

u x

s x

 

 





 

U

d

J

n t

i R n D

i − − −

.

(19)

1 

=



j j

i

i

j

j

x



1

1

1

Now, we try to extend the J-integral expression to 3D crack problems. This extension for a pure classical elasticity problem has been done by Huber et al. (1993) and Okada and Ohata (2013). Consider the local Cartesian crack front coordinate system with ( x 1 , x 2 ) - plane being perpendicular to the crack front (Fig.2).

1 3 ( , ) x x ; crack contours

Fig. 2. Illustration of crack with crack front in plane

 ,

  , cr

1 2 ( , ) x x

  in plane

lies in ( x 1 , x 2 ) - plane and ( ) A  is the regular plane domain with the boundary

The contours  ,

  ,

  =

cr x S 

cr

3 0 =

contour + −   =+ − + , i.e. the singularity point at the crack tip is excluded even when 0  → . For such a plane at any point on the crack front, the J-integral expression by (19) were valid, if the field variables would be invariable in the direction tangential to the crack front. Now, bearing in mind the governing equations, we can derive the identity ( ) cr cr A 

, , ikl i klj u u D P    = + + = U , k kj j , ik i kj

,

(20)

, jk k

where

(

) , , u

P

u D +

= −

 

+





.

(21)

, ilk i jl

, k j

jk

ik

ikl l

i j

Applying (20) to a regular domain ( ) A  with arbitrary  , we obtain