PSI - Issue 25

Ch.F. Markides et al. / Procedia Structural Integrity 25 (2020) 214–225 Ch. F. Markides et al., Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 6. The variation of the K = P f (CSRc) ratio against ρ . resembling Eq.(2), where the factor k of Hobbs (1964; 1965) regarding the CR becomes k (CSRc) or k (CSRt) for the CSRc and CSRt respectively, as expressed by the quantities in the curly brackets on the right-hand sides of Eqs.(22, 23). It follows from these expressions that k (CSRc) , k (CSRt) apart from ρ , R 2 , c , are also functions of the P f (CSRc,t) / P f (BD) ratio. Given Eq.(20), k (CSRc) and k (CSRt) are related to each other through an expression similar to that of Eq.(21). Moreover, for the CSRt, the tensile stress σ θ (CSRt) at point A is always larger in absolute value than the maximum compressive stress σ θ at point B , sufficing always tensile fracture (Fig.5). Similarly, for the CSRc the tensile stress σ θ (CSRc) at point B , is comparable in absolute value to the maximum compressive stress σ θ at point A , ensuring again tensile fracture at point B (recall that in brittle materials the compressive strength is much higher than the tensile one). Finally, in the specific case with ρ =2, proposed in applications, Eqs.(18, 19) reduce to the simple formulae: 2.3. The displacement fields in CSRc and CSRt for problems I and II Eqs.(5,10-13) ( R 2 fixed) provide the displacements in the CSRc/CSRt (upper/lower sign) for the overall problem I+II:               2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 log log 1 1 log cos 2 1 1 4 log cos 1 1 sin cos 2 log 1 4 1 1 log c R u P r r hR R r r r                                                                     (26) (CSRt) / P f      2 2 R h CSRc CSRc 19.66 12.88  f t P c R        ,      2 2 R h CSRt CSRt 31.02 25.76 2 f t P c R         (24, 25)



     

    

  

  

2 2

2 2

2 2

R

2

 

r

r







2 1 log

1 cos 2

R r















2





2

2

2

2

1  

1







2

2

1

r



 









   

   

  

   

2 2  

2

4

1

1 log log 1 R  

c







P  



 1 log 

2

sin

v

r

r



















2

2

2

2

4

1

hR











2

2

2 1 4 log

 



 

2

2

2

R

sin

1







 





1 cos r         sin 2 r

log



2

(27)

 





2

1

r





2

2    1 

1 log





     

    

  

   

4

2 2

R

r







2

2

1

sin 2

R

R















2





2

2

2

1





2   

2

1

r