PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 402–415 Author name / Structural Integrity Procedia 00 (2019) 000–000

406

5

1 h z h   

.

(8)

2

2

The mechanical behaviour of the functionally graded material in the beam i -th layer is treated by using a linear viscoelastic model which consists of two springs and a dashpot. The model is shown in Fig. 3. The modulii of elasticity of the springs are denoted by i E 1 and i E 2 as shown in Fig. 3.

Fig. 3. Linear viscoelastic model consisting of two springs and a dashpot.

The coefficient of viscosity is i  (Fig. 3). Since each layer of the beam is functionally graded along its width, the modulii of elasticity and the coefficient of viscosity are distributed along the width of the i -th layer according to following exponential laws:



i 1 1 1 1 1   i i y y

q y y

1

Li i E E e 1 1 

,

(9)

i



i 1 1 1 1 1   i i y y

q y y

2

Li i E E e 2 2 

,

(10)

i



i 1 1 1 1 1   i i y y

q y y

3

Li e   

,

(11)

i

i

where

1 1  i i y y y . 1 1  

(12)

1 y is the horizontal centroidal axis of the dross-section of the right-hand delamination i y 1 and 1 1  i y are, respectively, the coordinates of the left-hand and right-hand lateral surfaces

In formulae (10) – (12),

crack arm (Fig. 2), of the i -th layer,

i E 1 and

i E 2 , and

Li E 1 ,

Li E 2 and Li  are, respectively, the values of modulii of elasticity,

the coefficient of viscosity at the left-hand lateral surface of the layer, i q 1 , control the material gradient along the width of the layer. The strain,  , that is involved in (7) increases with the time at constant speed,

i q 2 and

i q 3 are parameters which

 v , i.e.