PSI - Issue 25

Ch.F. Markides et al. / Procedia Structural Integrity 25 (2020) 214–225

219

6

Ch. F. Markides et al., Structural Integrity Procedia 00 (2019) 000 – 000

a curved beam. In this context, introducing Eqs.(10) (as particularized for problems I and II to be solved) in Eqs.(4, 5) provides the stresses and displacement in the CSRc and CSRt configurations. 2.1. The stress fields in CSRc and CSRt for problems I and II Consider first problem I, i.e., bending of the CSR by transverse forces P , applied to its ends θ =± π /2, simultaneously for closing- (i.e., for – β single dislocation ( α = ε =0) corresponding to the problem I for the CSRc (Fig.2b)), and opening mode (i.e., for + β dislocation corresponding to the problem I for the CSRt (Fig.3b)). According to Muskhelishvili (1963), the non-zero coefficients in the expressions of Φ ( z ), Ψ ( z ) will be:                 2 2 1 2 1 1 1 3 2 2 2 2 1 2 1 2 2 2 , , 1 1 1 R R R R R R                                (11) where dislocation β equals (recalling that h here denotes the whole CSR thickness) to:         2 2 1 2 2 2 2 2 2 1 2 1 2 1 2 log P R R R h R R R R R                     (12) Combining Eqs. (4, 10-12) and introducing the ratio (considering R 2 is fixed at all times):

R R

 

2

(13)

1

the stress field in the CSRc (upper sign) and in the CSRt (lower sign) for problem I, reads as:

2

2

2   2 r

2   2 r

R

R

1

3

1













2

2

2

3

2

3

r

r

R

r

R

r

P

P





  I

  I

cos ,

cos , 





 



2

2





r







h

h

 

 

2

2  

1 log

1





 

2

2  

1 log

1





 

(14)

2

2   2 r

R

1







2 3

2

r

R

r

P



  I

sin







2





r



h

2

2  

1 log

1





 

Next, consider problem II of bending of the CSR by couples M = Pc , applied to its ends θ = ± π /2, simultaneously again for both closing- (i.e., for + ε single dislocation ( α = β =0) corresponding to the problem I Ι for the CSRc (Fig. 2c)), and opening-mode (i.e., for – ε dislocation corresponding to the problem II for the CSRt (Fig.3c)). Then, the non-zero coefficients in the expressions of Φ ( z ), Ψ ( z ) will be:           2 2 2 2 1 2 2 2 1 1 2 0 2 2 2 2 2 1 2 1 2 1 log log 1 2 , log 1 1 R R R R R R R R R R R R                              (15)

where dislocation ± ε equals to:  4 2     

 



2 Pc R R 

2

2

1

  

(16)



   

2

  

  

R R

  



2



2 R R R R   2 2 2 1 2 2 1 4

log

h

   

2

 

1

Combination of Eqs.(4, 10, 13, 15, 16), provides the stress field in the CSRc (upper sign) and in the CSRt (lower sign) for problem II, as: