PSI - Issue 68

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

142 4

! ! " " # µ

"

!

!

" "

= +

,

(10)

"

!

! " ! ! ! !

,

(11)

and , refer to the centre and surface of the shaft, , , , , , for the crack is extracted from the complementary strain energy, ! ! ! ! ! ! µ

!

where subscripts,

and

are parameters.

! !

!

The SERR,

, in the shaft by (12).

"

!$

%

=

,

(12)

! !

# !" !

where

" !!! = " ! ! "

#

# ! $ #" $

%

.

(13)

! " ! "

The complementary strain energy density, and (4) with respect to the corresponding strain. The shear strains, , and

, in a shaft portion is extracted by integrating the physical laws (3)

! !

! !

! !

, on the surface of the portions of the vertical member of the shaft, the angle of

! " ! "

!

! !

twist of section

and the curvature,

, of the horizontal member of the shaft which are used to calculate

are

derived from Eqs. (14) – (18) by the MatLab.

"

"

"

(

)

# = + #

# ! # $

! + + !

! !

#

"

,

(14)

"

!

"

"

"

"

#

#

! " # = + !

! !

,

(15)

!

"

!

#

"

#

# " ! $ !! !! # " % "

# " ! # !!

!"#

!"#

$ "

=

"

,

(16)

#

+ !!

!

A%"

A%" #

"

"

+ =

,

(17)

%

$

!

# " % "

# " ! $ !!

# " ! $ !!

A"#

!"#

#

=

"

,

(18)

#

!

! !

! !

! !

where

,

and

are the shear stresses in the portions of the vertical member,

is the horizontal axis of the

" ! ! !

! !

! ! ! !

shaft cross-section in portion

, the lengths ,

and

of the shaft portions are defined in Fig. 1.

The SERR obtained by (12) is confirmed by investigating the energy balance in the shaft. 3. Results Results for the normalized SERR are exposed in the next three figures. The results are derived for m, m, m, m, m , , , "#!"" !#!"! !#!!" #"! = ! #"! = ! #"! = ! "#!""

"#!""

$ = !

$ = !

#"! = !

" = !

$ = !

$ = !

,