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On the Boundary Value Kirsch's Problem

Published online by Cambridge University Press:  11 December 2015

D. Rezini ,
A. Khaldi  and
Y. Rahmani
D. Rezini*
Affiliation:
Université des Sciences et de la Technologie Mohamed Boudiaf Oran, Algérie
A. Khaldi
Affiliation:
Université des Sciences et de la Technologie Mohamed Boudiaf Oran, Algérie
Y. Rahmani
Affiliation:
Université des Sciences et de la Technologie Mohamed Boudiaf Oran, Algérie
*
*Corresponding author ([email protected])
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Abstract

Analytical closed-form solution to the stress distribution associated with a hole in finite plates subjected to tension has not been obtained yet. Wherefore, a method developed in this paper is based on a Beltrami-Michell methodology analyzing the Kirsch's problem under finite dimensions conditions of both plane stress and plane strain. This aimed ability is achieved by combining the Beltrami-Michell plane equations, isochromatic information on the boundaries only; and the finite difference method into an effectual hybrid method for analyzing rectangular plates of finite width with circular holes. Furthermore, the Beltrami-Michell methodology suggested may be applied on other plate and cut-out forms.

Keywords

Kirsch's problemDirichletPhotoelasticityHybrid method.
Type
Research Article
Information
Journal of Mechanics , Volume 32 , Issue 1 , February 2016 , pp. 1 - 10
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2016 

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References

1.Kirsch, G., “Die Theorie Der Elastizität Und Die Bedürfnisse Der Festigkeitslehre,” Zeitschrift Des Vereins Deutscher Ingenieure, 29, pp. 797–807 (1898).Google Scholar
2.Kolosov, G. V., “On the Application of Complex Functions Theory to a Plane Problem of the Mathematical Theory of Elasticity,” Ph.D. Dissertation, Dorpat (Yuriev) University (1909).Google Scholar
3.Inglis, C. E., “Stresses in a Plate Due to the Presence of Cracks and Sharp Corners,” Transactions of the Institute of Naval Architects, 55, pp. 219–230 (1913).Google Scholar
4.Howland, R., “On the Stresses in the Neighbourhood of a Circular Hole in a Strip Under Tension,” Philosophical Transactions of the Royal Society of London, Series A, 229, pp. 49–86 (1930).Google Scholar
5.Durelli, J. and Murray, W. M., “Stress Distribution Around a Circular Discontinuity in Any Two Dimensional System of Combined Stress,” Proceedings of the 14th Semi-Annual Eastern Photoelasticity Conference, Yale University, New Haven (1941)Google Scholar
6.Durelli, J. and Murray, W. M., “Stress Distribution Around an Elliptical Discontinuity in Any Two-Dimensional, Uniform and Axial System of Combined Stress,” Proceedings of the Society for Experimental Stress Analysis, 1, pp. 19–31 (1943).Google Scholar
7.Heywood, R. B., Designing by Photoelasticity, Chapman & Hall, London (1952).Google Scholar
8.Kawadkar, D. B., Bhope, D. V. and Khamankar, S.D., “Evaluation of Stress Concentration in Plate with Cut-Out and Its Experimental Verification,” International Journal of Engineering Research and Applications, 2, pp. 566–571 (2012).Google Scholar
9.Mohan Kumar, M., Rajesh, S., Yogesh, H. and Yeshaswini, B. R., “Study on the Effect of Stress Concentration on Cut-out Orientation of Plates with Various Cut-outs and Bluntness,” International Journal of Modern Engineering Research, 3, pp. 1295–1303 (2013).Google Scholar
10.Nagpal, S., “Optimization of Rectangular Plate with Central Square Hole Subjected to In-Plane Static Loading for Mitigation of SCF,” International Journal of Engineering Research and Technology, 1, pp. 1–8 (2012).Google Scholar
11.Nilugal, R. and Hebbal, M. S., “A Closed-Form Solution for Stress Concentration Around a Circular hole In a Linearly Varying Stressed Field,” International Journal of Materials Engineering and Technology, 4, pp. 37–48 (2013).Google Scholar
12.Singh, A. K., Rashid, A. and Singh, V. K., “An Infinite Plate Weakened by a Circular Hole,” International Journal of Engineering and Science Invention, 2, pp. 32–39 (2013).Google Scholar
13.Mallikarjun, P., Dinesh, K. I., Parashivamurthy, , “Finite Element Analysis of Elastic Stresses around Holes in Plate Subjected to Uniform Tensile Loading,” Bonfring International, Journal of Industrial Engineering and Management Science, 2, pp.136–142 (2012).Google Scholar
14.Shastry, P. and Venkateswara, Rao G., “Studies on Stress Concentration in Tensile Strips with Large Circular Holes,” Composite Structures, 19, pp.345–349 (1984).Google Scholar
15.Isida, M., “On the Tension of a Strip with a Central Elliptic Hole,” Transactions of the Japan Society of Mechanical Engineers, 21, p. 514 (1955).Google Scholar
16.Kotousov, Wang C. H., “Three-Dimensional Stress Constraint in an Elastic Plate with a Notch,” International Journal of Solids and Structures, 39, pp.4311–4326 (2002).Google Scholar
17.Yang, Z., Kim, C. B., Cho, C. and Beom, H. G., “The Concentration of Stress and Strain in Finite Thickness Elastic Plate Containing a Circular Hole,“ International Journal of Solids and Structures, 45, pp. 713–731 (2008).Google Scholar
18.Dharmin, P., Khushbu, P. and Chetan, J., “A Review on Stress Analysis of an Infinite Plate with Cut-Outs,“ International Journal of Scientific and Research Publications, 2, pp. 1–7 (2012).Google Scholar
19.Pilkey, W. D., Peterson's Stress Concentration Factors, Wiley, New York (1997).Google Scholar
20.Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, New York (1956).Google Scholar
21.Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, 4th Ed., Cambridge University Press, Cambridge (1927).Google Scholar
22.Abdou, M. A. and Monaquel, S. J., “An Integral Method to Determine the Stress Components of Stretched Infinite Plate with a Curvilinear Hole,“ International Journal of Research and Reviews in Applied Sciences, 6, pp. 353–358 (2011).Google Scholar
23.Churchill, R. V., Complex Variables and Applications, McGraw-Hill, New York (1960).Google Scholar
24.England, A. H., Complex Variable Methods in Elasticity, John Wiley, New York (1971).Google Scholar
25.Savin, G. N., Stress Concentration Around Holes, Pergamon Press, New York (1961).Google Scholar
26.Muskhelishvili, N. J., Some Basic Problems of the Mathematical Theory of Elasticity, Wolters-Noordhoff, Groningen (1963).Google Scholar
27.Boresi, P., Chong, K. P. and Lee, J. D., Elasticity in Engineering Mechanics, 3rd Ed., John Wiley and Sons, Inc., New Jersey (2011).Google Scholar
28.Lurie, I., Theory of Elasticity, 4th Ed., translated by Belyaev, , Springer, Berlin (2005).Google Scholar
29.Durelli, J., Phillips, E. A. and Tsao, C. H., Introduction to the Theoretical and Experimental Analysis of Stress and Strain, McGraw-Hill, New York, pp.126–131 (1958).Google Scholar
30.Ramesh, K., Digital Photoelasticity, Springer, NewYork (2000).Google Scholar
31.Rezini, D., “Randisochromaten Als Ausreichende Information Zur Spannungstrennung Und Spannungsermittlung,” “Boundary Isochromatics as Sufficient Information for the Stress Separation and Stress Determination,” PhD. Dissertation, Tech. University of Clausthal, (TUC), Germany (1984).Google Scholar
32.Meleshko, V. V, “Selected Topics in the History of the Two-Dimensional Biharmonic Problem,” Applied Mechanics Reviews, 56, pp. 33–85 (2003).Google Scholar
33.Vihak, and Rychagivskii, , “Solution of a Three-Dimensional Elastic Problem for a Layer,” International Applied Mechanics, 38, pp. 1094–1102 (2002).Google Scholar
34.Tokovyy, V. and Ma, C. C., “Analytical Solutions to the 2D Elasticity and Thermoelasticity Problems for Inhomogeneous Planes and Half-Planes,” Archive of Applied Mechanics, 79, pp. 441–456 (2009).Google Scholar
35.Okumura, I., “The Mathematical Theory of Thick and Thin Elastic Rectangular Disks in the States of Plane Stress and Plane Strain,” Memoirs of the Kitami Institute of Technology, 33, pp. 9–19 (2002).Google Scholar
36.Kuske, A. and Robertson, G., Photoelastic Stress-Analysis, John Wiley & Sons, New York (1974).Google Scholar
37.Shortley, H. and Weller, R., “The Numerical Solution of Laplace's Equation,” Journal of Applied Physics, 9, pp. 334–348 (1938).Google Scholar
38.Buzbee, L., Golub, G. H. and Nielson, C. W., “On Direct Methods for Solving Poisson's Equations,“ SIAM Journal on Numerical Analysis, 7, pp. 627–656 (1970).Google Scholar
39.Saad, Y, Iterative Methods for Sparse Linear Systems, 2nd Ed. Society for Industrial and Applied Mathematics Philadelphia (2003).Google Scholar
40.Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recipes: The Art of Scientific Computing, 3rd Ed., Cambridge University Press, Cambridge (2007).Google Scholar
41.Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, 3rd Ed., McGraw Hill, New York (1970).Google Scholar
42.Frocht, M., Photoelasticity, 1, John Wiley & Sons, New York (1941).Google Scholar
43.Frocht, M., Photoelasticity, 2, John Wiley & Sons, New York (1948).Google Scholar
44.Liebmann, H., “Die Angenäherte Ermittlung Harmonischer Funktionen Und Konformer Abbildungen,” Sitzungsberichte der Bayerischen Akademie der Wissenschaften Mathematisch-Physikalische Klasse, pp. 385–416 (1918).Google Scholar
45.Howland, R., “On the Stresses in the Neighbourhood of a Circular Hole in a Strip Under Tension,” Philosophical Transactions of the Royal Society of London, Series A, 229, pp. 49–86 (1930).Google Scholar
46.Peterson, R. E., Stress Concentration Factors, John Wiley & Sons, Inc., New York (1974).Google Scholar
47.Warren, C. Y. and Richard, G. B., Roark's Formulas for Stress and Strain, 7th Ed., McGraw-Hill, New York (2002).Google Scholar
48.Boresi, P., Schmidt, R. J. and Sidebottom, O. M., Advanced Mechanics of Materials, 5th edition, John Wiley and Sons, Inc., New York (1993).Google Scholar
49.Gao, X-L. and Rowlands, R. E., “Hybrid Method for Stress Analysis of Finite Three-Dimensional Elastic Components,” International Journal of Solids and Structures, 37, pp. 2727–2751 (2000).Google Scholar