PSI - Issue 82

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

249

4

Combination of Eq. (6) and Eq. (9) yields the following expression for the shear strain in the viscous component of the rheological model: . (10) Equation (10) is applied for describing the viscoelastic behavior of the i -th layer in the multilayered functionally graded beam structure shown in Fig. 2. The radius of the beam cross-section is . The beam is made of an arbitrary number of concentric layers with different thickness and material properties. Each layer is functionally graded in radial direction. Therefore, the material properties vary smoothly in radial direction in each layer. The variations of and in an arbitrary layer obey the following power laws: ( ) ! ! ! + ! = # " ! " " " " " " " A # &B " " " #! " " A B &' B &' A " " # " " ! "

! "

! "

! % % & & ! ! + % % & & ! ! + ! ! $ # ! A # !

#

"

"

!

& &

% % !

= +

,

(11)

!

!

(

)

!

#

!

"

!

!

#

"

"

!

& &

% % !

= +

,

(12)

!

!

(

)

!

!

#

"

!

!

!

where

! ! ! ! " " " ! " $"#"!!!" = !

,

(13)

! +

.

(14)

! + ! "

! "

In Eqs. (11) - (14), is the number of layers,

and

are the radiuses of the internal and external surfaces of

"

! "

!

! " #

! " #

the layer,

is a running radius,

and

are the values of . The parameters,

and at the internal surface. The values at and , in Eq. (11) and Eq. (12) govern the

!

"

! "

! " #

! " #

the external surfaces of the layer are

and

!

! "

! "

variation of , respectively. The length of the beam is (Fig. 2). The beam is clamped in its right-hand end. The beam is under torsion so that the angle of twist, , of the left-hand end increases linearly with time, i.e. , (15) where is the velocity. Since beams of high aspect ratio are under consideration in this paper, the distribution of strains obeys the hypothesis for conservation of plane cross-sections. Therefore, the distribution of strains is written as , (16) ! ! " ! ! ! = ! ! "#$ ! and

&

% ! = !"# !

&

!

!"# !

where

is the shear strain at the beam surface. In order to determine

, the angle of twist of the left-hand end

of the beam is expressed by applying the integrals of Maxwell-Mohr, i.e. . " =

" # $%& !

!

(17)

Combination of Eq. (15) and Eq. (17) yields . ! " =

! $ # " %

(18) In the rheological model (Fig. 1) the energy is dissipated by the viscous component. Therefore, the unit dissipated energy, , in an arbitrary layer of the beam is written as &'(

! " !