PSI - Issue 58

Victor Rizov et al. / Procedia Structural Integrity 58 (2024) 137–143 V. Rizov / Structural Integrity Procedia 00 (2019) 000–000

139 3

t , is represented by formula (1), i.e.   t    sin 0  ,

(1)

where 0  and  are parameters.

Fig. 2. Schema of viscoelastic model.

The non-linear viscoelastic mechanical behaviour of the beam under consideration is taken to be represented by the theoretical viscoelastic model shown in Fig. 2. The model combines two linear springs with moduli of elasticity, 1 E an 2 E , one linear dashpot with coefficient of viscosity,  , as well as non-linear spring and dashpot marked with ( ) nls and ( ) nld , respectively (Fig. 2). The stress-strain relation of the non-linear spring is given by







,

(2)

nls

B D 



where nls  is the stress,  is the strain, B and D are material properties. The constitutive law of the non-linear dashpot is taken to be represented by the following expression:

  P Q 





,

(3)

nld





where nld  is the stress, P and Q are material properties. The beam under consideration is functionally graded along its thickness, i.e. in  z direction. Thus, the continuous distribution of the properties of the viscoelastic model along the beam thickness is governed by exponential laws

h 2

z



1 

g E E e  1 1

h

,

(4)

h 2

z



2 

g E E e  2 2

h

,

(5)

h 2

z



g e 3   



h

,

(6)