Issue 68

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 68 (2024) 1-18; DOI: 10.3221/IGF-ESIS.68.01

the variable ζ = ξ +i η (via the mapping of Eqns.(1)), is expressed in terms of Φ 2 ( ζ ), Ψ 2 ( ζ ). Taking into account Eqns.(1) for η =0, and Eqns.(13), the boundary condition for the stresses on the parabolic notch L reads, in terms of the two functions Φ 2 and Ψ 2 , as follows:

2

 ( ξ i α )

(14)



 ( ξ i α ) Φ ( ξ ) ( ξ i α ) Φ ( ξ )  



  Φ ( ξ ) ( ξ i α ) Ψ ( ξ )



ο σ ξ

2

2

2

2

2

Following Muskhelishvili [13], the solution to the functional equation of Eqn.(14) is:

ξ

ξ

 ( ζ i α ) 2( ζ i α )  2



 σ ξ d ξ ζ i α ξ ζ ζ i α    o

1

o σ ξ d ξ

1

 o

 o

(15)

 Φ ( ζ )







Φ ( ζ )

, Ψ ( ζ )

Φ ( ζ )

2

2

2

2

 2 π i( ζ i α )

 ξ ζ

 2 π i( ζ i α )





ξ

ξ

o

o

After some relatively lengthy algebra it is obtained that:

  

  

  ζ ξ ζ ξ

σ

(16)





Φ ( ζ )

ζ log

2 ξ

o

ο

2

ο

 2 π i( ζ i α )

ο

   

   

  

  

2

  2 6 α ζ i α ζ ξ ζ 3

2

2

2

 

  ( ζ i α ) 

 

σ

3i αζ

2i αζ 5 α

ζ ( ζ i α )

1

1

(17)









Ψ ( ζ )

log

ξ

o

ο

2

ο

3

3

  2( ζ i α ) ζ ξ ζ ξ 2

 2( ζ i α )

2 π i

ζ ξ

ο

ο

ο

Stretching of the strip with the stress free edge notch L Substituting from Eqns.(4), (16) and (17) in Eqn.(3), the solution of the problem in question (i.e., that of stretching a finite strip 2bx2h with the stress free edge notch L, (Fig.3d or Fig.1)), is obtained as:

  

  

  ζ ξ ζ ξ

σ

σ

(18)

  o



Φ ( ζ )

ζ log

2 ξ

o

ο

ο

 4 2 π i( ζ i α )

ο

   

   

  

  

2

  2 6 α ζ i α ζ ξ ζ 3

2

2

2

 

  ( ζ i α ) 

 

σ σ

3i αζ

2i αζ 5 α

ζ ( ζ i α )

1

1

(19)

  o







Ψ ( ζ )

log

ξ

o

ο

ο

3

3

  2( ζ i α ) ζ ξ ζ ξ 2

 2( ζ i α )

2 2 π i

ζ ξ

ο

ο

ο

Inversing the transformation of Eqns.(1) yields:

   ζ i α iz

(20)

Substituting for ζ from Eqn.(20) in Eqns.(18) and (19), the solution in terms of the variable z=x+iy=re i θ is obtained as:

  

  

     

σ σ

i

i α i α

iz ξ iz ξ





(21)

  o

 



Φ (z)

i α

iz log

2 ξ

o

ο

ο

4 2 π i z

ο

  σ σ 3 α z 4i α

3

2

     

 iz 4 α

i α i α

iz ξ iz ξ

  o



Ψ (z)

i log

2 i ξ

o

ο

 

ο

3 2

3 2

2 4 π

z

z

ο

(22)

     2

3 iz 4i α 2 ξ z

5 α z iz iz 8 α

  



ο





2

2 ο

   iz

i α

ξ

The stress field in the stretched notched strip At one’s convenience, the stress components in the stretched, notched strip (Fig.1 or Fig.3d), may be expressed either in the Cartesian (x, y) or the curvilinear ( ξ , η ) reference sytem, via the well-known formulae [13]:

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