Issue 48

V. Shlyannikov et alii, Frattura ed Integrità Strutturale, 48 (2019) 77-86; DOI: 10.3221/IGF-ESIS.48.10

1

 

  

*

1

C

1

n



(13)

cr

K

 

cr

cr BI L 

0

n

  1 cr n 





(14)

cr BI L

C K  

0 

cr

n

where C* is the C -integral. The value of this line integral corresponding secondary creep deformation used in [10] as the relevant loading parameter to characterize the local stress-strain rate fields at any instant around the crack front in a cracked body subjected to extensive creep conditions

  

   x u n dxW C i j ij 1 2      ds



(15)

 

*



where the first term in Eq.(15) is the strain energy rate density (SERD) for power-law hardening creep [11,12]

0   kl 

n



(16)

cr W d  

cr

cr cr e e 



ij 



 

cr

cr

ij

1

n



cr

and  is an arbitrary counterclockwise path around the crack tip and the Cartesian coordinate system x i crack tip. Substitution of Eq. (16) into Eq. (15) leads to the expression for C* -integral

is centered at the

  

  









 

  

 

  

cr

cr

cr

cr

n

u 

u 

u 

u 









  1 cr n cr e  





(17)

* C Br 

cr

cr

cr

cr

cos

sin

cos

d  

r

d

  

r 

  

r 











cr

r

r





rr

rr





1

n

r

r















cr









Similarly to the plastic problem, Eqn. (13) for the creep stress intensity factor include a governing parameter for power- law nonlinear viscous materials in the form of cr n I -integral which can be obtained using the numerical method elaborated by the authors [8]. This method was extended by Shlyannikov and Tumanov [13] to analyze the fracture resistance characteristics of creep-damaged material. According to this method, the cr n I -integral value can be determined directly from the FEA distributions of the displacement rate  cr i u and dimensionless angular stress   cr ij functions













(18)

    

cr

FEM

, , , cr t n

, , , cr t n d 

I





 

n

cr

du  

du  

  

  

  

  

  

  

cr

cr

n

   cr e   

1

n



cr   rr u  

cr   r  u  





, , , cr t n

cos

sin

FEM

cr

cr







r  













cr

cr



cr

r



1

n

d

d







(19)

cr

1



cr cr rr r u 

r    u 

cos

cr cr

 







 

1

n



cr

n

n

n

cr        e   cr e

cr        e   cr e

cr        e   cr e

n

cr

cr

cr

cr

1

u 

u 

u 

du

cr

cr



cr

cr



(20)

1

n



cr

u

r









0

i

i

i

i

cr

i

  cr

n







L

L

dt

L

0 

0 

0 

BL K

cr

where t is creep time,   e . In Eqs.(19,20) a dot over a displacement quantity denotes a time differentiations. The  -variation angular functions of the suitably normalized functions   ij and  i u and correspondingly governing parameter cr n I -integral depend on the damage function  and creep exponent n . In the traditional models for creep or creep-fatigue crack growth rate prediction the I n -integral is a function only the creep is the von Mises equivalent stress and    cr  0 cr e e

83