3. Beam as a basis of supporting

Kelvin-Voigt model are given by ε = ε<sup>e</sup> + ε<sup>v</sup>

Figure 1. Viscoelastic model of three parameters.

352 Numerical Simulations in Engineering and Science

the total deformation is obtained as

concurrently with increasing strain:

, where ε<sup>e</sup> is the deformation of the elastic model,

<sup>v</sup> (1)

<sup>v</sup> (3)

ε ¼ σ<sup>0</sup> (4)

(6)

ε (2)

(5)

and ε<sup>v</sup> is the deformation of the Kelvin-Voigt model. When differentiated with respect to time,

which is the constitutive equation of the elastic and Kelvin-Voigt models, respectively. Consid-

E<sup>0</sup> þ E<sup>0</sup> η1

<sup>σ</sup> <sup>¼</sup> <sup>E</sup>0ε<sup>v</sup> <sup>þ</sup> <sup>η</sup>1ε\_

where σ = 0 for t < 0 and σ = σ<sup>0</sup> for t > 0, with t representing the time and t = 0 the instant of loading application. As the stress remains constant, the stress derivate with respect to time is zero. Applying the previous stress condition, the following ordinary differential equation is found:

> E0E<sup>0</sup> η1

Obviously, if the stress level remains constant, the modulus of elasticity should decrease

1 � e �E0 η1 t

<sup>E</sup><sup>0</sup> 1 � e

�E0 η1 t

are found. From the previous equations, one derives the following differential equation:

E0ε\_ þ

for which the general solution for t > 0, taking the initial condition ε(0) = σ0/E0, is

1 E0 þ 1 E0

E tðÞ¼ <sup>1</sup> 1 <sup>E</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup>

εðÞ¼ t σ<sup>0</sup>

σ ¼ E0ε\_ þ

E0E<sup>0</sup> η1

<sup>e</sup> <sup>þ</sup> <sup>ε</sup>\_

ε ¼ ε\_

ering E1 = E0 as the modulus of elasticity for both parts of the rheological model,

<sup>σ</sup> <sup>¼</sup> <sup>E</sup>0ε<sup>e</sup> andσ\_ <sup>þ</sup>
