Issue 47

S. K. Kourkoulis et alii, Frattura ed Integrità Strutturale, 47 (2019) 247-265; DOI: 10.3221/IGF-ESIS.47.19

From Eqns. (11-13), it is immediately seen that by considering any isolated part of the cut CR, one has a curved beam, the only non-zero stress-component on the straight edges of which is σ θ ( ε ) . The distribution of that σ θ ( ε) , attaining its extrema at the beam’s peripheries, is statically equivalent to counterbalancing couples at the beam’s straight edges of magnitude [20]:

2

  

  

R R





2

2 2 2 1 2 R R R R 2 1 4 2  

2

log

R

(     )

h

2

( ) 



1







h

d r r

2

(14)



2 R R 

2

  

2 ) 

2 (

2

1

R

1

Taking into account that according to the assumptions adopted this magnitude should be equal to Pc , then equating Eqn. (14) to Pc and solving for ε , one obtains:

2 R R 

2

(   

2 ) 

Pc h

2

2

1







(15)

0

2

(     )

  

  

R R





2

2 2 2 1 2 R R R R 2 1 4 2  

2

log

1

It is seen that for the above positive value of ε , and by considering the part of the cut CR between θ =±π/2, one obtains the second problem in question for the NCSR under bending by couples – Pc , Pc applied to its straight edges ED and E΄D΄. (c) The overall problem of the NCSR for bending by transverse forces and couples applied to its straight edges The overall stress field in the NCSR under transverse forces – P , P and couples Pc , –P c at its straight edges ED and E΄D΄, approaching that of the CSR-specimen, is given by superposing the respective stress components: σ r = σ r (β) + σ r (ε) (Eqns. (6, 11)), σ θ = σ θ (β) + σ θ (ε) (Eqns. (7, 12)) and τ rθ = τ rθ (β) + τ rθ (ε) (Eqns. (8, 13)), with β and ε given by Eqs.(10, 15), respectively. It is mentioned that these stresses depend on the magnitude P of the transverse force, the eccentricity c , the ratio ρ = R 2 / R 1 , the thickness 2 h of the CSR, the loading conditions adopted (namely plane stress or plain strain), and, also, on the material (appearing in the expressions for the dislocations β , ε ). It must be, also, mentioned that assigning to the parameters of the analysis arithmetic data corresponding to very brittle materials, the values calculated for β and ε (depending on the values of P and c considered) are extremely small, satisfying the principal assumptions governing the present study, i.e., the assumption of linear elasticity. Effectiveness of the analytic solution The effectiveness of the analytic solution was assessed in terms of the data for the displacement field, as it was obtained from a recent experimental protocol [25]. CSR-specimens made of PMMA ( E =3.2 GPa, ν =0.36) with 2 h =10 mm, R 1 =25 mm, R 2 =50 mm, and c =6.25 mm were tested according to the scheme shown in Fig. 1a. A 10 kN electromechanical frame was used to load the specimens and the displacement field was measured with the aid of the Digital Image Correlation (DIC) technique. The experimental set-up is shown in Fig. 4a. All tests were quasi-static, implemented under displacement-control conditions, at a constant rate of 0.1 mm/min. A typical fractured specimen is shown in Fig. 4b. During loading, series of successive photos of the specimens were taken permitting the determination of the displacement field by means of the software of the DIC system. The experimental data for the displacements were then compared with the ones provided by

the analytic solution (as it is analytically described by Markides et al. [25]), which read as: (a) Displacements due to bending of the NCSR by transverse forces applied to its straight edges

  

2

    

(    )

3 1 







r

3

( ) 





1  

u

r

log

 

2 R R 

2

  

2 ) 

  

  

4 (

1

2

(16)

   

  

2 2 1 2

2

R R

r

1

1

2





cos 2





2

2 R R 

2

  

r

2 R R 

2

2

1

2

1

2

252