PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 428–442 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

430

3

n

h z



    2

  

n h E E E E   LW UP

UP

,

(1)

3

m

h z

  





    2

  

UP  

LW

UP

,

(2)

3

m

h

where

3 h z h    .

(3)

2

2

Fig. 1. Functionally graded cantilever beam with lengthwise crack of length, a .

In formulae (1) – (3), UP E and LW E are, respectively, the values of the modulus of elasticity in the upper and lower surfaces of the beam, the values of the coefficient of viscosity in the upper and the lower surfaces of the beam are denoted, respectively, by UP  and LW  , n and m are parameters which control the distributions, respectively, of the modulus of elasticity and the coefficient of viscosity along the beam thickness, 3 z is the vertical centroidal axis of the beam. According to the viscoelastic model under consideration, the change of normal stress,  , with the time, t , is written as (Popov (1998))

Et



E e

 





,

(4)

where  is the strain. In the present paper, a time-dependent solution to the strain energy release rate, G , that accounts for the stress relaxation in the cantilever beam is derived by analyzing the balance of the energy. For this purpose, the balance of the energy is written as



a F u U F   

a Gb a   

,

(5)