PSI - Issue 80
Xinpeng Tian et al. / Procedia Structural Integrity 80 (2026) 451–461 Author name / Structural Integrity Procedia 00 (2019) 000 – 000
457
7
(
)
(
)
,
(22)
0
U
U
, jk k P dA
, P dA P dA − 3,3 j j
=
−
=
−
,
,
j
j
( )
( )
( )
A
A
A
1,2 = . Making use the Gauss divergence theorem for
where the Greek subscripts are used for plane coordinates
planar integrations, one can easily show
U
U
U
U
dA
dA
1 n d
1 nd
j
=
=
=
1
,
,1
j
( )
( )
( )
A
A
A
−
(
)
(
)
, i i j t u R n D d u + + , il i lj , j
, i jl j u D d +
, j P dA n P d = j
n
u
=
−
+
il
=
, (23)
, i l l
, i j
i
( )
( )
( )
A
A
A
−
where t n =
(
) , ,
R n =
,
(24)
−
i
i
i
il
il
cr + and
cr − in (23
, and the integrations over
Furthermore, we have utilized that 1 0 n = on cr
2 ) are cancelled out.
1 j = , we may write
Now, in view of (22) and (23) for
(
)
(
)
:
J
u
u
=
U
U
n t u R n D d − + +
n t u R n D d − + +
3,3 P dA j
,1
=
,1
−
.
(25)
1
,1
, 1
1
,1
, 1
i i
il i l
i i
il i l
( )
A
Note that the J-integral is not path independent, since its far-field contour contribution is supplemented with a domain integral depending on the choice of the far-contour . The J-integral can be expressed in terms expansion coefficients occurring in the asymptotic solution near the crack front, while its numerical value can be calculated by employing the far-field expression (25) combined with the numerical solution of concrete boundary value problem with a crack. 4. Numerical examples Using the present CMFEM, the flexoelectric effect in a circular cylinder containing a penny-shaped crack is studied here as shown in Fig. 3. The height of the cylinder is H = 60 nm and the radius is R = 50 nm, a central circular penny shaped crack is located at the middle of the height with radius r = 20 nm. Transversely isotropy ZnO is considered with its isotropic plane perpendicular to the x 3 axis and the material coefficients are consistent with those in Sladek et al. (2022 ) . Due to symmetries, only one eighth of the circular cylinder is analysed numerically as shown in Fig. 3 (b) and the normal components of displacements are constrained to be zero at each plane of symmetry (e.g. 1 1 0 at 0 u x = = ). Moreover, the FEM mesh is also shown in Fig. 3(b) with 6480 elements, yielding 29165 nodes along with 116660 DOFs. To accurately capture the strain gradients and the flexoelectric field, refined meshes are employed at the crack tip vicinity. The other boundary conditions are set as follows: axial tensile loadings are applied uniformly at the top and bottom surfaces of the cylinder with the electric potential fixed to be zero there.
(a) (c) Fig. 3. The schematic of a flexoelectric circular cylinder containing a penny-shaped crack. (b)
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