PSI - Issue 82

Victor Rizov et al. / Procedia Structural Integrity 82 (2026) 246–252 V. Rizov/ Structural Integrity Procedia 00 (2026) 000–000

250

5

! = !

$

#!

" " $ #

,

(19)

!

"

"

"

!

"

"

!

where

and

are found by Eq. (6) and Eq. (10), respectively. The dissipated energy,

, is derived by

! !

! !

integrating of the unit dissipated energy in the volume of the beam structure

#

! " = = + " ! " # " !

!

& # #

'

% #$#

=

.

(20)

"

"

"

Equation (20) can be applied to determine the dissipated energy in the multilayered functionally graded viscoelastic beam under torsion at various values of time. The integration in Eq. (16) is carried-out by the MatLab. 3. Numerical results

"#!"" = !

!#!!" = ! "

The numerical results presented here are obtained by using the following data:

m,

m,

! = !

! #%$ "# ! " =

#"! = ! "

#"! = ! "

# !

, 1/s. The main aim of the analysis is to clarify the influence of the time-dependent coefficient of viscosity and the material gradient on the dissipated energy in the viscoelastic beam configuration depicted in Fig. 1. In order to achieve this aim, calculations of the dissipated energy are performed at different values of and various and ratios (such calculations are needed since the dependency of the coefficient of viscosity on time is controlled by (refer to Eq. (1)) while the material gradient is characterized by and ratios). The results derived are presented in graphic form. ! " ! ! # " $ $ ! ! ! # " A A ! ! " , and

! # " $ $ !

! ! # " A A !

!

! ! " " ! # #

#"!

#"!

! = !

$ = !

Fig. 3. The non-dimensional dissipated energy presented as a function of

ratio (curve 1 - at

, curve 2 - at

#"!

$ = !

and curve 3 - at

).