PSI - Issue 32

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

28

3

damage ( δ = 1 – undamaged material, δ = 0 – fully damaged); r is the distance between defects, r 0 is the defect size according to the statistical model proposed by Naimark et al. (2000,2017) and Bayandin et al. (2010). Then the effective elastic modulus can be written as E eff =E δ . The kinetic equation for the introduced damage parameter δ in (1), according to the model developed by Bilalov et al. (2019), can be written as follows:

2

2 p δ = −Γ δ 

(2)

,

where Γ is a kinetic coefficient, which generally depends on the state parameters, p is the intensity of the defects (microcracks) density tensor, which in its sense is the deformation caused by defects. It was shown in Bayandin et al. (2016) that σ ~ p 2 . Then equation (2) can be rewritten as:

2 eff σ δ



(3)

.

δ = −Γ

In the present model, the dependence for Γ is assumed to be:

0 exp  σ  Γ = Γ    σ  , c

(4)

where σ c is the fatigue limit stress, σ is the applied stress, Γ 0 is the initial value of the kinetic coefficient. Substituting (4) and (1) in (3) we obtain:

C δ       σ

C      

C      

E σ δε

E σ ε

σ σ σ δ



exp

exp

exp

(5)

δ = −Γ

= −Γ

= −Γ

.

0

0

0

2

σ δ

During cyclic loading, the stress is defined by the harmonic law:

  

2 π

  

 

  

(

)

(6)

,

sin 2 

1

t R πν − + +

σ = σ

A

where σ A is the stress amplitude, R is the cycle asymmetry coefficient, ν is the loading frequency. Substituting (6) into (5) and taking into account that < σ > = σ A (R+1) and ≤ exp(<(∙)>), where <∙> is the averaging operator over one loading cycle, we obtain:

  

2 π

  

 

 

(

)

sin 2

1

t πν − + +   R

σ

(

)

1  σ +  R

A



A

(7)

exp δ = −Γ 

.

 

0

σ

δ



c

The solution of the differential equation (7) is the dependence for structural-scaling parameter: