PSI - Issue 2_B

V. Crupi et al. / Procedia Structural Integrity 2 (2016) 1221–1228 Author name / Structural Integrity Procedia 00 (2016) 000–000

1225

5



 0



1









p

,

(6)

 F : ~p ~ : ~ ~p ~p  

T q T

 



 



p e cT q r Q Q       ,

(7)

       1

  

1

~ : ~ 

~ F : ~p

T e Q TF : ~ e    



e

p

p  

where

- heat production due to thermoelastic effect;

-

Q



p



represents the inelastic part to the heat production; c - the specific heat capacity. To assume the linear links between thermodynamic forces and the thermodynamics fluxes, we obtain the constitutive equations   ~p p ~ ~ p l F F ~ l F e p e p          , (8)   e p e p ~ ~p p ~ l F ~p l F F        , (9) where the function p F follows from the presentation of free energy given by the statistical model of solid with mesodefects. To consider the influence of diffusion processes on defect nucleation and evolution and to study the localization effects of the damage accumulation, we have introduced in the expression of the total free energy F the term describing spatially-nonuniform distribution of microcrack density tensor ik p   2 2 2 p F p K ik      F = . (10)

      r

2 1

3 1

2 1    

   

e ll

     e ik

e ll

ik

ik

p F , we have to consider the equation (11) as a functional determined for a

In order to evaluate tensor

representative material volume. For one dimension problem we can write

  

   

   F F  

ik p / x     F



F

.

(11)



p

p p x   

ik

ik

l

l

The system (8)–(9) in the case of uniaxial cyclic loading (

   zz , e e zz  , p p zz  ) takes the form

  

   ,

 p x F  



  x D p

p



(12)

  l p 

l







p

p







  

   

p x F   



x D p  

    p l

,

(13)

p l



p

 p



where D is the coefficient of self-diffusion. 4. Numerical simulation of damage to fracture transition and temperature distribution (surface and subsurface fatigue crack initiation) To describe the bulk defect (pores, microcracks) evolution specimen we will consider a one component of defect induced strain. Under high cyclic and gigacyclic fatigue we can propose a weak interaction of defect accumulation and microplastisity processes and write the equations (12),(13) as follows ( 0   p p l  ) (14) To describe the defect evolution in critical cross-section of the sample tested in ultrasonic testing machine let write the equation (14) in cylindrical coordinates  r, :    .     x D p         i zz i zz zz p zz x p F l p  

   

   

2

2

   

    

   

   

2

2

p

p

p

  

   

F













  

   

  

   

zz

zz

zz

zz

zz

zz

zz

p 

l

 r D p

D p

Dp

r D p

r D p

.









zz

p

zz

zz

2

2

2

p

r









r







zz

We can used the following boundary conditions