PSI - Issue 82

Victor Rizov et al. / Procedia Structural Integrity 82 (2026) 253–259 V. Rizov/ Structural Integrity Procedia 00 (2026) 000–000

255 3

! !

lower support (in section, ). The layers of the column have viscoelastic behaviour that is presented by the model in Fig. 2. This model is under a constant strain, . The stress-strain-time relation of the model is given by (Dowling (2007)) !

" " +

$ " !

$

!

$

(

)

$ !

" " " # " " $

$ #

= +

,

(1)

"

#$

$

#$

where

! # # # # " ! + !

#

=

.

(2)

"!

"

!

!

!

!

! " !

! " !

! !

! !

!

Here,

is the stress, is the strain, is time,

and

are the moduli of elasticity,

is the coefficient of

viscosity (the moduli of elasticity and the coefficient of viscosity are shown in Fig. 2).

Fig. 2. Schema of the viscoelastic model.

The layers are continuously inhomogeneous along the length of the column. Formulas (3), (4) and (5) are used for presenting the distribution of , and along the length.

! " !

! " !

! !

( % % & & " ! " # + ! % % & & " ! ! # + ! ) # " ( ) # "

!

& &

$

# " " " = +

" "

# "

,

(3)

"

"

!

!

& &

$

# " ! " = +

! "

# "

,

(4)

!

"

!

# " # " " !

!

$

" "

# " " = +

# " "

,

(5)

(

)

"

!

% % " ! +

! " !

" ! # ! !

" ! # " !

! " !

" ! ! !

! !

where " ! # " !

,

and

are the magnitudes of

,

and

in the lower end of the column, and are parameters, is the column

" ! # " !

, ! !

!

" ! ! !

!

,

and

are the magnitudes in the upper end,

! !

! !

axis, the lengths, and , are shown in Fig. 1. The modulus of relaxation,

! " !

, of the model in Fig. 2 is

" " +

#

$ ! " !

$

!

"

(

)

" " " " # " = +

.

(6)

$

"

$

#$

$

#$

Thus, the stress can be given by . ! " " ! =

(7)

!

!

!

When deriving the SERR,

, for the cracks in the column in Fig. 1, we use (7) for treating the behaviour of the