Engineering Transactions, 62, 2, pp. 131-169, 2014
10.24423/engtrans.244.2014

Stresses and Displacements in an Elliptically Perforated Circular Disc Under Radial Pressure

Christos F. MARKIDES
National Technical University of Athens, Department of Mechanics, Laboratory for Testing and Mterials
Greece

Stavros K. KOURKOULIS
National Technical University of Athens, Department of Mechanics, Laboratory for Testing and Mterials
Greece

The complex potentials governing the elastic equilibrium of a finite circular disc, elliptically perforated at its center, are obtained using Muskhelishvili’s formulation. The disc is subjected to non-uniform distribution of pressure along two symmetric finite arcs of its periphery. Given the complex potentials, the stress- and displacement-fields can be determined everywhere on the disc by introducing a novel flexible interpretation of the conformal mapping, suitably adjusted to the computational process. The results of this procedure are given for various strategic loci and are critically discussed. The length of the loaded arc is considered similar to that obtained from the solution of the intact disc-circular jaw elastic contact problem assuming that the disc is compressed between the steel jaws of the device suggested by the International Society for Rock Mechanics for the implementation of the Brazilian-disc test. Concerning the distribution of the externally induced pressure along the loaded arcs, it is proven that for the general asymmetric configuration (i.e. the axes of the elliptical hole are neither parallel nor normal to the loading axis) the induced asymmetric displacement field does not permit maintenance of equilibrium of the disc as a whole in case the disc is considered exclusively under a distribution of radial pressure: Additional tractions must be exerted along the loaded arcs, obviously in the form of frictional stresses. Besides, providing full-field, analytic expressions for stresses and displacements, the main advantage of the present solution is that, by properly choosing the ratio of the ellipse’s semi-axes, the solution of three additional configurations, of major importance in engineering praxis, are obtained: These of the intact disc, the circular ring and the cracked disc.
Keywords: circular disc, ring, complex potentials, stress field, displacement field, fracture toughness, cracked Brazilian-disc test.
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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

References

ISRM, Suggested methods for determining tensile strength of rock materials, Inter¬national Journal of Rock Mechanics and Mining Sciences Abstracts, 15(3), 99-103, 1978.

ASTM D3967 - 08, Standard test method for splitting tensile strength of intact rock core specimens, ASTM Volume 04.08 Soil and Rock (I): D420 D5876, 2014.

ISRM, (Coordinator Fowell R.J.), Suggested methods for determining mode-I fracture toughness using CCNBD specimens, International Journal of Rock Mechanics and Mining Sciences, 32(1), 57-64, 1995.

Wang Q.Z., Fan H., Gou X.P., Zhang S., Recalibration and clarification of the formula applied to the ISRM-suggested CCNBD specimens for testing rock fracture toughness, Rock Mechanics and Rock Engineering, 46, 303-313, 2013.

Kourkoulis S.K., Markides Ch.F., Fracture toughness determined by the centrally cracked Brazilian disc test: Some critical issues in the light of an alternative analytic solution, ASTM Materials Performance & Characterization, 3(3), 45-86, 2014.

Markides Ch.F., Kourkoulis S.K., Naturally accepted boundary conditions for the Brazilian disc test and the corresponding stress field, Rock Mechanics and Rock Engineering, 46(5), 959-980, 2013.

Burniston E.E., An example of a partially closed Griffith crack, International Journal of Rock Mechanics and Mining Sciences, 5, 17-24, 1969.

Tweed J., The determination of the stress intensity factor of a partially closed Griffith crack, International Journal of Engineering Science, 793-803, 1970.

Pazis D.N., Theocaris P.S., Konstantellos B.D., Elastic overlapping of the crack flanks under mixed-mode loading, International Journal of Fracture, 37, 303-319, 1988.

Markides Ch.F., Pazis D.N., Kourkoulis S.K., The centrally cracked Brazilian disc: Closed solutions for stresses and displacements for cracks under opening mode, Journal of Engineering Mathematics, 83(1), 143-168, 2013.

Markides Ch.F., Pazis D.N, Kourkoulis S.K., The centrally cracked Brazilian disc: Implications and solutions in case of closing cracks, Journal of the Mechanical Behaviour of Materials, 23(3-4), 59-77, 2014.

Atkinson C., Smelser R.E., J. Sanchez, Combined mode fracture via the cracked Brazilian disk test, International Journal of Fracture, 18, 279-291, 1982.

Markides Ch.F., Pazis D.N., Kourkoulis S.K., Stress intensity factors for the Brazilian disc with a short central crack: Opening versus closing cracks, Applied Mathematical Modelling, 35(12), 5636-5651, 2011.

Theocaris P.S., Sakellariou M., A correction model for the incompatible deformations of the shear internal crack, Engineering Fracture Mechanics, 38, 231-240, 1991.

Kourkoulis S.K., Markides Ch.F., . Chatzistergos P.E, The standardized Brazilian disc test as a contact problem, International Journal of Rock Mechanics and Mining Sciences, 57, 132-141, 2012.

Markides Ch.F., Kourkoulis S.K., The stress field in a standardized Brazilian disc: The influence of the loading type acting on the actual contact length, Rock Mechanics and Rock Engineering, 45(2), 145-158 , 2012.

Kolosov G.V., Application of the Complex Variable to the Theory of Elasticity (in Russian), ONT1, Moscow-Leningrad, 1935.

Muskhelishvili N.I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, The Netherlands, 1963.

Kourkoulis S.K., Markides Ch.F., Pasiou E.D., A combined analytic and experimental study of the displacement field in a circular ring, Meccanica, DOI 10.1007/s11012-013-9846-0, Published on line 06 December 2013.

Markides C.F., Pazis D.N., Kourkoulis S.K., Closed full-field solutions for stresses and displacements in the Brazilian disk under distributed radial load, International Journal of Rock Mechanics and Mining Sciences, 47(2), 227–237, 2010.

Kourkoulis S.K., Markides Ch.F., Chatzistergos P.E., The Brazilian Disc under parabolically varying load: Theoretical and experimental study of the displacement field, International Journal of Solids and Structures, 49 (7-8), 959-972, 2012.

Hondros G., The Evaluation of Poisson’s Ratio and the Modulus of Materials of a low Tensile Resistance by the Brazilian (Indirect Tensile) Test with Particular Reference to Concrete, Australian Journal of Applied Science, 10, 243-268, 1959.

Timoshenko S.P., Goodier J.N., Theory of elasticity (2nd ed.), McGraw-Hill: Engineering societies monographs, New York, 1951.




DOI: 10.24423/engtrans.244.2014