PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 134–144 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

136

3

3 vt   ,

(1)

where v is a parameter that controls the change of  . The lower crack arm is free of stresses. The mechanical behaviour of the beam is treated by the viscoelastic model shown in Fig. 2. The model is structured by two linear springs with modules of elasticity, 1 E and 2 E , and a linear dashpot with coefficient of viscosity, 1  . Besides, a non-linear spring, nf , and a non-linear dashpot, ng , are placed in the model as shown in Fig. 2. The viscoelastic model is under strain,  .

Fig. 2. Non-linear viscoelastic mechanical model.

The change of  with time is described by the following law: 3 v t    ,  v is a parameter that governs the change of strain. The constitutive law of the non-linear spring is written as m nf E P      3 , where

(2)

(3)

where nf  is the stress, 3 E , P and m are material properties. The behaviour of the non-linear dashpot obeys the following constitutive law: s ng D         2 ,

(4)

where ng  is the stress, 2  , D and s are material properties,   is the first derivative of the strain with respect to time. The stresses in the linear springs and linear dashpot are derived in the following way. First, the equation of equilibrium is written as

1    1 E  and

   ,

(5)

E

E

1

2

2 E , respectively. The

1 E and

2 E  are the stresses in the springs with modules of elasticity,

where

stress in the dashpot is 1   . The strains are related as



1  E

1  

,

(6) (7)

    2 1 E E



,