Issue 59

E.S.M.M. Soliman, Frattura ed Integrità Strutturale, 59 (2022) 471-485; DOI: 10.3221/IGF-ESIS.59.31

K

K









r

r

3

3

  

  

  

  

I

II











 7 8 sin sin  



 3 8 cos cos  

u y

 

 

4 2

2

2 4 2

2

2

(13)

    3 2 O r

    r T 

   1





 





B

sin 4 1

cos

n

E

where: B n is the coefficient relevant to Williams’ expansion in anti-symmetric mode. µ is the shear modulus; E is Young’s modulus; ν is Poisson’s ratio; T is a constant that acts parallel to the crack surfaces; u x and u y are displacement fields, for the plane strain condition; (r, θ ) is the polar coordinate and K I and K II are the mode I and mode II SIF. The determination of SIF using the crack flank displacements may be obtained as [2]:

 

   ,

 u r y   ,



 

 





K

u r y

(14)

I

 1 2 r 

 

   ,

 u r x   ,







K

u r x

(15)

 

 

II

 1 2 r 

The first order stress field equations for mode I can be expressed as [13]:

  3 1 sin sin 2 2 3 cos 1 sin sin 2 2 2       

        

     

     

 

      

 xx K I yy

(16)



r

2



xy

 

3 sin cos 2 2

The stress in the z-direction can be written as [13]:                2 2 0 (for plane stress) K I xx yy r zz

cos (for plane strain) 2

(17)

 

For a double edge cracked plate with two symmetric edge cracks having two symmetrical axes, the general stress functions can be written as [23]:

   

   





  m

  m



z b

cos

(2 )

 2 1 b

 2 1 k

 2 1 k

2 2

  z

(18)



 

  E i z c



E z k

F z k

sin

0

2







k

k

1

1



 1 sin

a b

( 2 )

   

   





  m

  m



z b

cos

(2 )

 2 1 b

 2 1 k

 2 1 k

2 2

  z

(19)

 

  E i z c





F z k

F z k

sin

0

2







k

k

1

1

 1 sin



a b

(2 )

475