On Treating of Surface Cracks in Finite Layers of Fractional Materials
Keywords:
Fracture mechanics, Fourier integral transform, System of Fredholm integral equations, Cauchy kernel, Jacobi polynomial.Abstract
The plane strain problems of the bounded layer medium composed of three different materials contain a crack on one of the interface are considered. Using Fourier integral transform, the boundary value problem leads to a mixed integral equation with Cauchy kernel in position and continuous kernel in time. In addition, using a quadratic numerical method we have a system of Fredholm integral equations with Cauchy kernel in position. Then, the Jacobi polynomials method, according to the index of integral equation, is used to solve the system of Fredholm integral equations. Moreover, the developing program is used to computing the approximate solution and the estimated error.
References
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[19] M. A. Abdou, M. M. EL-Borai, M. M. El-Kojok, Toeplitz matrix method and nonlinear integral equation of Hammerstein type, J. Comp. Appl. Math. Vol. 223, (2009) 765 – 776
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[2] F. Erdogan, A. C. Kaya,” Stress intensity factor and COD in an orthotropic strip” International journal of Fracture, vol. 16(1980)1971-1980
[3] F. Erdogan,”Stress Intensity Factors”, ASME Applied Mechanics, vol . 50(1983)276-288.
[4] M. Matbuly, M.S. Nassr,” Electrostatic analysis of edge cracked orthotropic plates” D.M.,J. of Eng. Appl. Sci. vol . 50(2005)376-388.
[5] S.Y. Shin, H. Lees, S.Y. Han,” Influence of silicon in low density Fe-C-Mn-Al stell” Metal. Mat. Trans. A, vol. 41(2010)138-148.
[6] G. W. Jia, L. Hau,” A note on the discontinuity problem in heston's stochastic volatility model” j. Material Process Technology, vol.187(2007)562-565
[7]A.A. Rizk, M. Hrairi, “Edge- cracked biomaterial systems under thermal heating”, j. Eng.Fra. Mech. Vol.(2009)1648-1653.
[8] H. Parkus, Thermo elasticity, Spring-Verlag, Barlen, 1976
[9] N. Noda, R.B. Hetnarski, Y. Tanigawa, Thermal Stress, Taylor and Francis, 2003.
[10] R.B. Hetnarski,J. Ignaczak, Mathematical Theory of Elasticity, Taylor and Francis, 2004.
[11] M. A. Abdou, Fundamental problems for finite plate with a curvilinear hole having finite poles, Appl. Math. Compute. 125 (2002) 79-91
[12] M. A. Abdou, S. A. Asseri, Gaursat functions for an infinite plate with a generalized curvilinear hole in zeta plane, Appl. Math. Compute. 212 (2009) 23-36
[13] M. A. Abdou, S. A. Asseri, Closed forms of Gaursat functions in presence of heat for curvilinear holes, J. Thermal stress, Vol.32, (2009) 83 - 101
[14] M. A. Abdou, S. J. Monaquel, Integro differential equation and fundamental problems of an infinite plate with a curvilinear hole having a strong pole, Int. J. Contemp. Math. Sciences, Vol. 6, (2011), no. 4 199 - 208
[15] M. A. Abdou, F. S. Bayones, Integro – differential equation and an infinite elastic plate with a curvilinear hole in S-plane, IJRRAS 10 (3) March (2012)1-10
[16] M. A. Abdou, A. R. Jaan, An infinite elastic plate weakened by a generalized curvilinear hole and Gaursat functions, Applied Mathematics, 2014, 5, 428-443
[17] M. A. Abdou, W. G. El Said, E. I. Deebs” A solution of a nonlinear integral equation, Appl. Math. Compute. 160 (2005)1-14
[18] M. A. Abdou, A. A. Badr, M. B. Soliman, Chebyshev Polynomials and Fredholm-Volterra integral equation, Int. J. Appl. and Mech. Vol.4 (2008), 78-92.
[19] M. A. Abdou, M. M. EL-Borai, M. M. El-Kojok, Toeplitz matrix method and nonlinear integral equation of Hammerstein type, J. Comp. Appl. Math. Vol. 223, (2009) 765 – 776
[20] M. A. Abdou, A. A. Badr, M. M. El-Kojok, On the solution of mixed nonlinear integral equation, Appl. Math. Compute. 217(2011) 5466-5475
[21] M. A. Abdou, F. A. Salama, The solution of mixed integral equation of the first kind using Toeplitz matrix method, J. of Advanced in Mathematics, Vol. 9, No. 6 (2014)2723-2732
[22] M.A. Abdou, Osama L. Mustafa, Fredholm–Volterra integral equation in contact problem, J. Appl. Math. Comp.199–215, 138 (2003).
[23] A. Jose, Cuminato, Numerical solution of Cauchy - type integral equations of index - 1 by collocation methods, Adv. Comp. Math. 6 (1996)47-64.
[24] G. Szegö, Orthogonal Polynomials, AMS Colloquium Publications. 23, American Mathematical Society, New York, 1999.
[25] A. Erdelyi, Higher Transcendental Functions, Vol.2 McGraw-Hill. New York 1973.
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Published
2017-08-12
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Khayyat, H. (2017). On Treating of Surface Cracks in Finite Layers of Fractional Materials. American Scientific Research Journal for Engineering, Technology, and Sciences, 34(1), 213–230. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/3270
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