Issue 49

V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23 228 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, 0 r u u u         (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) , , ; r r u u u u u u        (1) (2) (1) (2) (1) (2) , , r r                (8) or perfect slip conditions (1) (2) (1) (2) (1) (2) (1) (2) ; , 0 r r u u                     (9) Upon substituting Eqns. (6) into the equilibrium Eqn. (2) and going to a new independent variable (1 cos )/2 x    [20] we obtain the following equations for each harmonic of the Fourier series:                            2 2 1 2 2 2 4 1 1 1 2 4 1 1 v 1 v 0 2 2 1 1 k k k k k k xH x k d u x du x x x x u x dx x x dx x x H d x kw x H x x dx x x x x                                 (10)                               2 2 2 1 1 1 2 1 1 3 4 1 1 1 2 4 1 1 1 2 1 1 0 2 4 1 k k k k k k xG x k G d v x dv x G x x G x v x dx x x dx k G G k x dw x d G x x u x w x dx dx x x                             (11) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 1 1 3 4 1 1 1 1 2 4 1 1 1 2 1 0 2 4 1 2 1 k k k k k k xG x G k d w x dw x x x x w x dx x x dx G k dv x G k x kG u x v x dx x x x x 2 é ù - + + ê ú ë û - + - + + - é ù - + - ê ú + + + ⋅ = ê ú ê ú - - ë û (12) Here we introduce the following notation:         1 2 2 1 1 2 / 2 1 , 3 4 /( 2 1), H H                           1 2 3 2 1 / 1 2 , 1 , 2 4 4 / 1 2 . G G G                 Using (6) we can reduce boundary conditions (3)-(5) and regularity conditions (7) to condition for