PSI - Issue 68

Victor Rizov et al. / Procedia Structural Integrity 68 (2025) 139–145 V. Rizov / Structural Integrity Procedia 00 (2025) 000–000

141 3

! !

! !

The shaft end, , the shaft is supported by a rotational spring located in the horizontal plane, i.e. perpendicular to the shaft longitudinal axis. This spring has non-linear elastic behaviour obeying the physical law (1). , is clamped. In section

! " # $ # ! ! = ! "

,

(1)

! ! !

!

!

where section

is the moment in the spring, is the spring constant, . The shaft has circular cross-section of radius,

is a parameter,

is the angle of twist of

! !

" ! !!

! !

. In portion

the shaft hosts a longitudinal crack

!

! !

representing a circular cylindrical surface of radius

and length . The upper end of the inner crack arm is under

!

!

angle of twist, , whose change with time, , is given by (2).

and ! # " ! = ! !

,

(2)

" ! ! !

where

are parameters. Apparently, the inner crack arm and the shaft portion

are under twist, while

" ! ! !

the horizontal shaft member

is under bending around the vertical axis. The shaft mechanical behaviour under

torsion is non-linear viscoelastic. It obeys the law (3) (Kishkilov and Apostolov, 1994).

!

(

)

"

& B !

#

" "

=

,

(3)

!

!

#$

+

, , ! ! ! ! ,

!

!

!

where

is the shear stress,

is the shear strain,

and

are material parameters. The behaviour of

" ! ! !

the shaft member

under bending obeys the law (4).

!

(

)

"

& % ! # " " ! =

,

(4)

!

!

#$

+

, , ! ! ! ! ,

!

!

where

is the normal stress,

is the strain,

and

are material parameters. The shaft is

, , ! ! ! ! ! , ,

!

functionally graded along its radius. The variation of

and

along the radius is

! ! " " # µ

"

!

!

" "

= +

,

(5)

"

!

! ! " " # µ

"

!

!

" "

= +

,

(6)

"

!

! ! # "# # µ

!

!

# #

= +

,

(7)

"

!

! ! " " # µ

"

!

!

" "

= +

,

(8)

"

!

! ! " " # µ

"

!

!

" "

= +

,

(9)

"

!