PSI- Issue 9

Ch.F. Markides et al. / Procedia Structural Integrity 9 (2018) 108–115 Author name / Structural Integrity Procedia 00 (2018) 000–000

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2

(see, for example, Fairhurst (1964)), related mainly to the fact that the stress field at the disc’s center is biaxial rather than uniaxial and also, to the stress concentrations at the disc-jaw interfaces, which may cause premature fractures rendering the final outcomes erroneous (Hobbs (1963); Mellor and Hawkes (1971)). As a result, quite a few alternative configurations have been proposed, like for example the flattened Brazilian-disc test (Wang (2014)), the semi-circular bend test (Kuruppu et al. (2014)) and the ring test (Hudson (1969)), each one with its own pro and cons. In the present study a novel configuration is described, curing some of the drawbacks of previously introduced alternative configurations. It consists of a circular semi-ring, of outer radius R 2 and inner radius R 1 and thickness 2h, loaded under diametral compression (Fig.1a.) The main advantages of the specific test, denoted from here on as the Circular Semi-Ring (CSR) test, include the easy preparation of the specimens, the fact that the ratio between R 1 and R 2 may vary within broad limits, the very low fracture load, the existence of a single tensile component at point A (well comparable to the respective compressive one) and, above all, the fact that the ratio between the maximum tensile and compressive stresses developed is controlled by the ratio of R 1 over R 2 . It is here mentioned that the geometry of the specific configuration closely resembles the familiar arc-shaped tension specimen used for the determination of Mode-I fracture toughness according to the E 399 – 90 standard of the ASTM (1997).

(a)

(b)

(c)

y

-P

y

-P

y

c

M

-P

z=re iθ

θ ο

r

R 1 R 2

R 1

A

B A

θ

O

x

O

O

x

R 1

x

R 2

R 2

c

-M

P

P

P

Fig. 1. (a) The concept behind the CSR test; (b) Proposed laboratory implementation of the CSR test; (c) The generalized loading scheme considered in the analytic solution, consisting of both bending moments and transverse diametral forces.

2. Analytical considerations: The displacement field in the CSR under bending and diametral compression For the sake of generality the loading scheme considered here is assumed to consist of both a pair of compressive forces P and, also, a pair of opposite bending moments M=Pc, introduced in order to take into account possible ec centric application of the compressive forces, dictated, perhaps, by practical needs for proper support of the specimens (see Fig.1b,c). Determining stresses and displacements for the CSR test is a first problem of plane linear elasticity, which is here solved by taking advantage of the complex potentials method (Fig.1c). Muskhelishvili (1963), in his milestone book, gave the general solution of the first fundamental problem for a homogeneous, isotropic and linearly elastic circular ring, lying at the z=x+iy=re iθ plane. Namely, assuming that the origin of coordinates is at the center of the ring, the solution was given in series form as:

  

  

k a z 

k (z) A ogz a z ,    

k

(z)  

(1)

k

Apart from the imaginary part of a 0 , that can be arbitrarily fixed, the remaining coefficients a k , a k ΄ may be obtained from the boundary conditions of the problem and the additional conditions of single-valuedness of displacements: