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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The present solutions actually continue those refer to only the CSRc test, given by Kourkoulis et al. (2018) and Markides et al. (2018), by expanding them to include also the case of CSRt test. It is of course mentioned that both the previous and the present works are based on the solutions for the curved beam given by Muskhelishvili (1963) and long ago by Golovin (1882) and Timpe (1905). For integrity reasons of the present paper, a brief recapitulation of the theoretical approach adopted is quoted next. In this context, it is recalled that the solution for the CSR is achieved with the aid of the solution of the respective uniform circular ring (CR), i.e., CSR is to be understood as a portion of the CR. According to Muskhelishvili (1963), the complex potentials solving the 1 st fundamental problem for the CR are: related to stresses and displacements on the CR and along its boundaries through the well-known expressions:   i2 4 ( ), i ( ) ( ) e ( ) ( ) r r r z z z z z z                      (4)     2 i ( ) ( ) ( ), ( ) ( )d , ( ) ( )d u v z z z z z z z z z z                  (5) Here μ ≡ G is the shear modulus and κ is Muskhelishvili’s constant. Considering a theoretical cut joining the boundaries of the CR (transforming it into a simply connected region), displacements due to Eqs.(5) are multi-valued, i.e., after one circuit L of the CR they do not revert to their original values on the cut undergoing an increase (Muskhelishvili, 1963):       1 1 i i 1 L u v Az a a             (6) If multi-valued displacements are not allowed, then by zeroing their increment given by Eq.(6) leads to: (7) which are the supplementary (to the second one of Eqs.(4) on the boundaries of CR) conditions required to obtain the coefficients a k , a ΄ k of Φ ( z ) and Ψ ( z ); and it is to be mentioned that in that case if the CR is unstressed, i.e., if it holds on the boundaries ( r = R 1 , r = R 2 ) of the CR that: (8) then all the coefficients a k , a ΄ k of Φ ( z ), Ψ ( z ) are zero so that stresses and displacements are identically zero on the CR. But if multi-valued displacements are allowed, which is the present case, Eq.(6) leads to (Muskhelishvili 1963):   1 1 , i (1 ) i A a a                 (9) α , β , ε , are real infinitesimal, positive quantities, called characteristics of dislocations (Volterra, 1907; Love, 1927), de scribing theoretical mismatches between the displacements in the horizontal, vertical and angular sense respectively, on the two sides (+, – ) of the theoretical cut introduced to the CR; the nature of β and ε used here, is shown in Figs.2(b,c), 3(b,c). Given the conditions of Eqs.(9), instead of those of Eqs.(7), it is found that some of the coefficients a k , a ΄ k of Φ ( z ), Ψ ( z ) are not zero, even if Eq.(8) is true, i.e., when the CR is unloaded, with the complex potentials attaining the non-zero forms (the values of the non-zero coefficients a k , a ΄ k , in terms of α , β , ε , are found in Muskhelishvili (1963)): (10) In turn, stresses and displacements are, through Eqs.(4) and (5), non-zero, too. As it was shown by Golovin (1882) and Timpe (1905) and soon after by Muskhelishvili (1963) with the aid of the complex potentials technique (adopted here), those, at first sight, unnatural non-zero stresses and displacements as obtained in an unloaded, but theoretically deformed cut CR due to the assumption of non-zero α , β and ε , effectively describe a number of problems referring to a curved beam, considered as a portion of the cut CR. The CSR configuration, studied here, is considered as such 1   1 a a     0, 0 A  1 2 , r r r R R       i 0 1 1 2 3 0 1 z A z a a z a z      1 1 2 3 ( ) log , ( ) z a z a z a z              ( ) z A z log  , ( ) z        k a z   k k k a z   (3)