PSI - Issue 52
T. Profant et al. / Procedia Structural Integrity 52 (2024) 455–471 T. Profant et al / Structural Integrity Procedia 00 (2023) 000 – 000
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Fig. 3. Matching the outer solution ,⋅ and the inner solution ,⋅ , where a dot in the upper index means or . The traction vector ,⋅ of the outer solution is still valid at the distance 0 from the fictitious tip of the outer crack. On the other hand, the Cauchy type stresses in the total stress vector ,⋅ of the inner solution prevails with respect to the remaining higher-order stresses at the distance = 0 −1 from the true tip of the crack. Consequently, the scaling-up of the inner domain allows both solutions to be matched along these boundaries. The important role of the parameter 0 is obvious from expressions (25) and (26). The parameter 0 is known as the boundary layer in the matched asymptotic expansion analysis. The path independency of the -integral allows the parameter 0 to be fixed by comparing the classical elasticity J -integral and the gradient elasticity J -integral. The expression of the -integral for loading modes I and II of the flexoelectric material with the crack obeying the conducting and insulating boundary conditions, respectively, can be found in (Mao & Purohit, 2015) and written as
(
) 2
5 4 7 6
l
− − −
2
(
)
2 8 , 12
ID
11 3 2 C C C − + 12
J
=
( 16 1
)
1
2
−
1
−
(27)
(
)
(
) 2
) ( 2
(
)
4 1 5 4 7 6 − − − −
2
1 1 2 − − − −
2
(
)
2
2
IIE
1 − l
,
J
C
C
=
+
(
)( 1 2 − )
1
12
22
(
)
2
1
−
2
2
2
3 1
− +
where
3/2
1/2
l R
l R
( ) , , u R C , in ID
(
) , ,
, in ID
lim
lim
C
R
=−
=−
11
12
0
0
R
R
→
→
(28)
1/2
1/2
l R
l R
( )
( ) ,0 R
, in IIE
, in IIE
lim
,0 , R C
lim
C
=
=−
r
21
22
0
0
R
R
→
→
and
1
1
(
)
, in u u = − − r r u , , 2 (29) The dot in the upper script of quantities in (29) again means the symbols or of mode I or mode II loading, respectively. Substituting (25) and (26) into (28) leads to R in in in ,
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