PSI - Issue 79

Victor Rizov et al. / Procedia Structural Integrity 79 (2026) 109–116

112

q a m Dn Dn =− , q a m D D   =− , q a m Hn Hn =− , q a m H H   =− ,

(8)

(9)

(10)

(11) where m is the mass per unit length of the structure. The quantity, m , changes continuously along the structure length due to the continuous material inhomogeneity. Equation (12) is used for describing the change of m along the length.

l l m m L  ) ( 1 2 4 + − L



m m

x

= +

1

,

(12)

m

1

L

m

where

1 2 0 x l l   + .

(13)

Here, 1 L m and 4 L m are the mass per unit length in points, 1 L and 4 L , of the structure, x is the longitudinal axis, m  is a parameter.

Fig. 2. Viscoelastic model.

The structure under the inertia loads which intensities are given in Eqs. (8), (9), (10) and (11) has non-linear viscoelastic behavior. The latter is treated by the model depicted in Fig. 2. The dashpot of the model has linear viscoelastic behavior that obeys the law given in Eq. (14).      = , (14) where   is the stress in the dashpot,  is the coefficient of viscosity,    is the time derivative of the strain in the dashpot. The behavior of the spring of the model is non-linear elastic. The non-linear stress-strain relation in Eq. (15) is applied for treating the spring behavior. r R R R   = , (15) where R  and R  are the stress and strain in the spring, R and r are material properties. The coefficient of viscosity and the material properties, R and r , change along the length according to the laws given in Eqs. (16), (17) and (18), respectively.

    l l L L ) 1 2 4 + − l l R R L  ) ( 1 2 4 + − 1 L



x

 

= +

,

(16)



1

L

(



R R

x

= +

1

,

(17)

R

1

L

R

l l r r L 1 2 4 + −

r 

r r

x

= +

1

L

,

(18)

1

L

r  )

(

where