Issue 59

S.K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 59 (2022) 405-422; DOI: 10.3221/IGF-ESIS.59.27

A NALYTICAL SOLUTION

F

rom the theoretical point of view, the problem of the finite circular disc with a rectangular notch (of finite length and width) with rounded corners, squeezed between the curved jaws suggested by the respective ISRM standard (Fig.1a), is here confronted as a first fundamental problem of the plane theory of linear elasticity. In this context, and assuming, in addition, that the material of the disc is homogeneous and behaves isotropically, Muskhelishvili’s com plex potentials technique is adopted [9]. The general solution of the problem was obtained, relatively recently, by Markides and Kourkoulis [4,10] by taking advantage of the following mapping function, provided by Savin [11] (Fig.1b):



1

 

 

 

  

  

  

c

b

      R   

  1



1   



z

R

c

,



2 1  



2





1

1





 

  4













2

2

2

4   4

2       2

     

5

2

  

2 

 









c

c

c

c

;

;

;

;

1

2

3

4

24

80

896

(1)

  5

  2

  14

  5









5        5 3   3  

6  

6  

4   4

2       2

7

21

4





c

c

;

5

6

2304

11264















2 i k

ik

ik

ik

ik



0 k for w    0,

 



1 

2 

3 

 

4 , 



e

e

e

e

e

(

,

,

,

)

The above function maps the region outside the area A 1 A 2 A 3 A 4 (see Fig.1b) to the region outside γ in the mathematical plane. In other words, one introduces Eq.(1) into the already available solution [4, 10] of the circular ring of outer radius R O . Then, demanding that the rectangular notch A 1 A 2 A 3 A 4 must be free from stresses, and assuming, also, that the notch has too little affection on the boundary of the disc, the solution of the problem is sought in the form:             ; * * o o z z z z z z           (2) where   o z  and   o z  are given by the following expressions :

B

B

      

  

 

P

B

B

B

 2 2 1 n  

 2 2 1 n 4 1 n   











  z

 

 

4 1 n

4 1 n



4 1 n 

4 3 n 

3 z B z  2

1



2     0 b z c

n

n

4

4









o 

z

z

z

z

 



4 1 n 

4 1 n 

4 3 n 

3



n

1

  z

o 



(3)







B

B

B

      

  

 

 

4 



P

B

B

3        1 2 z B z b z 





 2 2 1 n 



  n







 4 1 n

 

2 2 1 n   4 1 n 

4 1

 

 

4 3 n



4 1 n 

4 3 n 

3

c

n

4









z

z

z

z

z

 



0

2



4 1 n 

4 3 n 

4 3 n 

3

3



n

1

providing the solution of the circular ring, as it is analytically described by Kourkoulis et al. [12]. In the above expressions φ * (z) and ψ * (z) are analytic functions perturbing the ring’s solution due to the introduction of the notch. Alternatively, instead of the ring the intact disc could have been used as the solution basis. In this direction, it is assumed that the inner radius of the ring tends to zero. Combining Eqs.(1)-(3) and introducing them in the boundary condition for zero stresses on the rectangular notch:

    s s



  s

  s   

  0 s

(4)





 

 

(where s is the point ζ on γ ) one obtains (after some relatively lengthy algebra) the quantities φ * (z) and ψ * (z) solving the problem [10]. The configuration of this alternative problem is shown in Fig.2, where, in accordance with the solution of the circular ring, the disc with the notch is to be subjected to a parabolic pressure along two symmetric arcs of its peri phery. In the same figure the introduction of the curvilinear coordinate system ρ =const., θ =const., at any point of the disc with the notch is shown, as it is dictated by the mapping function described in Eq.(1).

407