Issue 49
V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23
If 1 0 , the region under consideration is bounded by only one coordinate surface 0 , and at
0 is assumed to
meet the regularity conditions
/ 0, 0, u u u r
0
(7)
In the framework of the proposed problem formulation we can consider a composite cone occupying a region (1) (2) V V V , where a subregion (1) V (subregion (2) V ) is made of the material with the shear modulus (1) ( (2) ) and Poisson’s ratio (1) ( (2) ) and its geometry is defined by the relations 0 r , 0 2 , 2 0 ( 1 2 ) (Fig. 1b). In particular cases 1 and 0 can be equal to 0 and , respectively. For a composite cone eigensolutions (6) are constructed for each subregion. At the contact line 2 we can prescribe perfect bonding conditions
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
(1)
(2)
(8)
, r u u u u u u , r
r
r
;
,
,
or perfect slip conditions
(1) u u
(2)
(1)
(2)
(1)
(2) (1) (2) r
(9)
r
;
,
0
Upon substituting Eqns. (6) into the equilibrium Eqn. (2) and going to a new independent variable
[20]
(1 cos )/2 x
we obtain the following equations for each harmonic of the Fourier series:
xH x
1
2
k
4
2 d u x
k du x
1
k
1 x x
x
k u x
1 2
x x
2
dx
4
1
dx x x H d x 1 v
(10)
kw x
H
1 v
k
k
2
2
k x x
0
1 x x
dx
2
2
1 x x
xG x
2 k G 1
4
2 d v x
k dv x
2
1
k
1 G x x
1 2 G x
k v x
x x
1
1
2
dx
4
1
dx
(11)
1 2 1
1 k G
G k x
k dw x
d G x x u x 1
1
1
k w x
0
x x
k
3
dx
dx
2
4
1
é ê ë
2 ù - + + 1 1 G k
(
)
( ) (
xG x
2
4
d w x
) ( ) k dw x
ú û
2
1
( )( x
)
( )
k
-
+ -
+
+
x
x
w x
1
1 2
(
)
k
2
dx
-
x x
4
1
dx
(12)
)
( ) (
) ( 1 2 1 )
é ê ê ë
ù ú ú û
1 G k dv x G k x - + -
kG
( ) ( + ê
( ) ú
k
1
1
3
+
+
⋅
=
u x
v x
0
(
)
k
k
(
)
dx
2
-
x x
4
1
x x -
2 1
Here we introduce the following notation: 2 1 1 H
2 / 2 1 ,
3
4 /( 2 1),
H
1
2
2 1 / 1 2 ,
1 ,
2 4 4 / 1 2 .
G
G
G
1
2
3
Using (6) we can reduce boundary conditions (3)-(5) and regularity conditions (7) to condition for
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