PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 105–114

108

m t i  6  ,

q q

(8) (9)

Dxi

6 

m t i 

.

Dzi

The moving beam has non-linear viscoelastic behavior under the inertia load. The non-linear viscoelastic model shown in Fig. 2 is applied for treating the beam. The dashpot and the spring that is parallel to it have linear behavior.

Fig. 2. Schematic of the viscoelastic model.

The second spring (that is ahead of dashpot and spring with linear behavior) has non-linear elastic behavior. The model is loaded by stress, i  . The latter is t i i    , (10)

i  is a parameter. The strain in the viscoelastic model is

where

nli Ei i      .

(11)

The notations used in Eq. (11) are: Ei  - strain in the linear-elastic spring;

nli  - strain in the non-linear elastic

spring. The stresses in the linear dashpot and spring,

i   and Ei  , are

i       , i i

(12) (13)

E 

Ei  

.

i Ei

In Eqs. (12) and (13) the notations used are as follows: i  - coefficient of viscosity of the dashpot; i E - modulus of elasicity of the spring; i    is the time derivative of the strain in the dashpot. Since the strains in the linear dashpot and spring are equal, Eq. (12) takes the form

i Ei i      .

(14)