Issue 49

V. Matveenko et alii, Frattura ed Integrità Strutturale, 49 (2019) 225-242; DOI: 10.3221/IGF-ESIS.49.23

where

1 2 3 4 5 6 , , , , , C C C C C C are the constants defined by the prescribed combination of boundary conditions (3) –( 5).

E XAMPLE OF CALCULATION OF CHARACTERISTIC INDICES FOR CONICAL BODIES

T

0 k  and

1 k  under the prescribed combination of the boundary conditions

he general solutions obtained for

were used to construct a homogeneous system of algebraic equations with respect to constants i C for the examined conical body. The coefficients of this system of equations depend on the vertex angles of conical bodies, elastic characteristics of the materials and characteristic index  . The indices  , defining the character of stress singularity at the vertices of conical bodies are calculated from the condition of the existence of non-zero solution to the system of linear algebraic equations. Solid cone Let us consider a solid cone ( 0 0 , 0 , 0 r           ). For this case, in general solutions (32), (37), (45) it is necessary to retain terms with particular solutions (15) at 0 x  (or 0   ). Subject to this condition the solutions take the following form: axisymmetric rotation ( 0 k  )     (1) (1) 0 0 0 w x C w x   (46)

0 k  )

axisymmetric deformation (

    0 , u x C u x C u x v x C v x C v x                 (1) (1) (2) (2) 0 0 0 0 0 (1) (1) (2) (2) 0 0 0 0

(47)

0 k  )

non-axisymmetric deformation (

    u x C u x C u x C u x v x C v x C v x C v x w x C w x C w x                                   (1) (1) (2) (2) (5) (5) (1) (1) (2) (2) (5) (5) (1) (1) (2) (2) k k k k k k k k k k k k k k

(48)

k

k k

k

k

The substitution of these solutions in one of the boundary conditions (3)–(5) at 0 homogeneous algebraic equations for the unknown quantities, (1) (2) (1) (2) (5) 0 0 , , , , k k k

q q = yields a system of linear

C C C C C , which have the following

representation: axisymmetric rotation (

0 k  )

(1) C   (1,1)

0

(49)

0

0

0 k  )

axisymmetric deformation (

(1)

(1,1) C             (2) (1,2) 0 0 0 (2,1) (2) (2,2) C

C C

0,

0

(50)

(1)

0.

0

0

0

0

0 k  )

non-axisymmetric deformation (

236