PSI - Issue 32

Yuriy Bayandin et al. / Procedia Structural Integrity 32 (2021) 26–31 Author name / Structural Integrity Procedia 00 (2019) 000–000

29

4

1/ 2

(

)

(

)

   

   

1

1

R σ +

R σ +

  

  

  

  

(

) 1 exp

( sin 2 Γ πν

)

A

A

2 πν − σ + R

exp

t Γ πν + σ

t

0

0

A

A

σ

σ

( ) t δ =

C

C

=

(8)

( ) πν

1/ 2

1/ 2

(

)

(

)

(

)

  

   

1

1 sin 2 σ + Γ πν  R t

R σ +

  

  

  

(

) 1 exp

0

A

A

1 2 = − σ +  R

exp

.

t Γ + σ

 

0

A

A

σ

σ

πν

C

C

It is obvious that at high values of t , which at a fixed loading frequency correspond to a large number of loading cycles N, the third term in the subcorrelated expression (8) makes a minor contribution to the damage evolution. Therefore, this term can be neglected, and expression (8) will not depend on the loading frequency ν . Hence, assuming that ν = const, we can pass from the dependence δ ( t ) to δ ( N ), knowing that N = ν t :

1/ 2

(

)

  

   

1  σ +   σ  A C R  

(

)

(

) 1 exp

(9)

, 1 2 δ σ = − σ +  N R

.

N δ

Γ

A

A

In equation (9) Γ δ = Γ 0 /t 0 , where t 0 is the characteristic time equal to t 0 =1/ ν . [ Γ δ ] = Pa -1 . Thus, we have obtained the analytical dependence of the damage parameter on the number of cycles and the amplitude of loading. The solution of the equation ( ) , 0 A N δ σ = will allow to establish the dependence between the number of cycles before failure N f and the amplitude of applied stresses σ A :

(

)

1 −σ +      σ  A C R

exp 2

( ) A

N

(10)

σ =

.

(

)

f

1 σ + Γ R

A

δ

It should be noted that the damage evolution kinetics (5) derived from the principles of thermodynamics, according to Bayandin et al. (2010) and Bilalov et al. (2019), is similar in its structure to the equation for the damage parameter evolution proposed by Kachanov (2013), Kachanov (1992) and Rabotnov (1959, 1969):

m

d C dt

1 ω  σ  =  − ω   

(11)

,

where ω is the damage parameter ( ω = 1 − fully damaged material, ω = 0 − undamaged material), which is essentially equal to ω = 1 −δ . Then at m = 1 we can write: ( ) 1 d σ

, dt d C C dt − ω = − δ σ = − δ C

,

1

− ω

0 exp . C   σ = Γ    σ 

Using the reasoning presented above, it can be shown that well known formula proposed by Baskuin (1910), which is used for describing fatigue curve, follows from the Kachanov −Rabotnov damage model:

N σ

f

(12)

.

const or

A f N β σ =

σ =

A

β

f

It is obvious that equation (12) is a particular case of equation (10). Different authors obtained expressions (11)