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
220
7
2
2
R
R
r
r
2
2
1 log
1 log
1 log 1
1 log
2 2
2 2
R
R
r
r
2
2
4
4
c
II
II
2
2
,
,
Pc
P
r
(17)
h
h
2
2
2
2
2
2
2 1 4 log
2
2
2 1 4 log
R
R
2
2
II
0
r
In the frame of linear elasticity, the overall solution for the stresses is obtained by adding the respective components of Eqs.(14, 17). In case of zero eccentricity c =0, which sometimes reflects the experimental set-up, the stress fields for the CSRc and CSRt are given solely by Eqs.(14) (with the appropriate sign for each case as mentioned previously). As an illustration, the variation of the stress components for the overall problem I+II in the CSRc, are next plotted as functions of r in Fig.4 along an arbitrary radius ( θ =30 o ) and in Fig.5 along the line AB ( θ =0 o ), in juxtaposition to the respective ones of the CSRt. The dimensions of the CSR were R 2 =0.05 m, R 1 =0.025 m ( ρ =2), and h =0.01 m. For comparison reasons, eccentricity c =0.012 m was the same for both the CSRc and CSRt. The material considered was Dionysos marble used in the restoration of Parthenon Temple of the Acropolis of Athens, with modulus of elasticity E =62.5 GPa and Poisson’s ration ν =0.25. Moreover, it was arranged that the value of P provided for both tests the average tensile strength of the specific material, equal to about 6 MPa (Kourkoulis et al., 1999). In other words, it was considered that P f (CSRc) =170 N for the CSRc and P f (CSRt) =90 N for the CSRt. As it was expected, the values of σ θ at the outer and inner peripheries of the CSRc and the CSRt along the radius at 30 o remain lower than the respective ones along AB (equal to 6 MPa, namely, the tensile strength of Dionysos marble). In addition, in Fig.4 the shear component of the stress τ r θ differs from zero, attaining values well comparable to those of σ r . 2.2. The critical stresses in CSRc and CSRt Of great importance is the stress field at the critical points of the CSR where the onset of failure is expected to take place. These points are, obviously, different for the CSRc and the CSRt configurations. Namely, for the CSRc the critical point is B ( R 2 , 0) while for the CSRt the critical point is A ( R 1 , 0). Indeed, setting in Eqs.(14, 17) r = R 2 and θ =0 o , it is seen that the only non-zero stress component at point B is σ θ for both problems I and II, attaining positive values in the CSRc and negative ones in the CSRt. What is more, as it is seen from previous Fig.5, σ θ attains at B its maximum value when considering the CSRc. Thus, σ θ at B could stand as the tensile strength of the material only when the CSRc configuration is considered, reading, after superimposition of problems I and II, and for P = P f (CSRc) , as:
CSRc
2 2
2
2log 1
c
2
P
2
1
t
I+II
CSRc
f
, 0
R
(18)
2
R h R
2
2
2
1 log
1
2
2
2 1 4 log
2
2
On the other hand, setting in Eqs.(14, 17) r = R 1 , θ =0 o , it is seen that the only non-zero stress component at point A is σ θ again for both problems I and II, attaining negative values in the CSRc and positive ones in the CSRt. Moreover, as it is seen from Fig.5, σ θ attains at A its maximum value when considering the CSRt. Thus, σ θ at A stands as the tensile strength of the material considering the CSRt, reading, after superimposing problems I and II, and for P = P f (CSRt) , as:
CSRt
2 2 2 log 1 c
2
2
P
2
1
t
I+II
CSRt
f
, 0
R
(19)
1
R h R
2
2
2
1 log
1
2
2
2 1 4 log
2
2
Assuming now that both the CSRc and the CSRt provided the tensile strength of the material, it must hold that:
t
CSRc
CSRt
(20)
t
Then equating the left-hand sides of Eqs.(18, 19) yields:
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