# Thermoelastic Problem of an Infinite Plate Weakened by a Curvilinear Hole

Keywords:
Boundary value problems, thermoelastic plate, Gaursat functionsl, rational mapping, curvilinear hole, AMS (2010), 74B10, 30C20.

### Abstract

Mathematical model is considered, to discuss the analytic solution of the first and second boundary value problems (BVPs), for an infinite plate weakened by a curvilinear hole C having two poles. The elastic plate carries a steady uniformly distributed axial current of density J and is placed in an ambient medium of steady temperature Using a conformal mapping function, the curvilinear hole is conformally mapped on the domain outside (inside) a unit circle Then, the Gaursat functions (GFs) are determined. Moreover, the three components of stresses, in the presence of temperature T distributed around the curvilinear hole are completely determined. Many special and new cases are derived from the work. In addition, many, applications for the first and second BVPs are discussed. Moreover, the three stresses components, in each application, are computed.### References

[1] W. Xie, F.-R. Lin" A Fast Numerical Solution Method for Two Dimensional Fredholm Integral Equations of the Second Kind" Applied Numerical Mathematics , vol. 59, pp 1709-1719, 2009.

[2] A. Tari, M. Rahimi, S. Shahmorad, F. Talati" Solving a Class of Two-Dimensional Linear and Nonlinear Volterra Integral Equations by the Differential Transform Method" Journal Computational Applied Mathematics, vol.228, pp 70-76, 2009.

[3] S. Bazm, E. Babolian" Numerical Solution of Nonlinear Two-Dimensional Fredholm Integral Equations of the Second Kind Using Gauss Product Quadrature Rules" Communication. Nonlinear Science Numerical. Simultaneous, vol. 17, pp 1215-1223, 2012.

[4] K. Malenknejad, Z. JafariBehbahani" Application of Two-Dimensional Triangular Functions for Solving Nonlinear Class of Mixed Volterra-Fredholm Integral Equations" Mathematical . Computational . modeling", vol.55,pp 1833-1844, 2012.

[5] M. A. Abdou, A. A. Badr, M. B. Soliman" On a Method for Solving a Two Dimensional Nonlinear Integral Equation of the Second Kind" Journal Computational Applied Mathematics, vol 235 pp 3589-3598,2011

[6] N. I. Muskhelishvili. Some Basic Problems of Mathematical Theory of Elasticity. Noordroof, Holland, 1953pp250-254

[7] R.B. Hetnarski,J. Ignaczak. Mathematical Theory of Elasticity. Taylor and Francis, 2004 pp176-180.

[8 ] M. A. Abdou, S. A. Asseri " Closed Forms of Gaursat Functions in Presence of Heat for Curvilinear holes" Journal Thermalstress vol. 32:11pp1126-1148,2009.

[9] M. A. Abdou, S. A. Asseri " Gaursat Functions for an Infinite Plate with a Generalized Curvilinear Hole in Zeta Plane" Applied Mathematics Computational" vol . 212 pp23-36, 2009.

[10] M.A. Abdou, A. R. Jaan "An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Gaursat Functions" Journal of . Applied Mathematics. vol.5, 5pp,428-443, 2014

[11] A.I. Kalandiya. Mathematical Method of Two Dimensional Elasticity. Mir- Moscow,1975pp 135-137

[12] G. E. Exadaktylos, M. C. Stavropoulos. " A Closed Form Elastic Solution for Stress and Displacement Around Tunnels". International Journal of Rock Mechanics of Mining science. vol. 39 pp 905-916, 2002

[13] G. E. Exadaktylos, P. A. Liolios, M. C. Stavropoulos. "A Semi-Analytical Elastic Stress-Displacement Solution for Notched Circular Openings in Rocks". International Journal of Solids and Structures. Vol. 40,pp 1165-1187,2003

[14] N. Noda, R. B. Hentarski, Y. Tanigowa. " Thermal Stresses", Taylor and Francis, 2003pp 101-103

[15] R. Schinzinger, P. A. Laura. " Conformal Mapping Methods and Applications". Dover Publications, New York, 2003pp43-50

[16] H. Parkus. "Thermoelasticity". Spring-Verlag, Barlen, 1976pp 20-24.

[2] A. Tari, M. Rahimi, S. Shahmorad, F. Talati" Solving a Class of Two-Dimensional Linear and Nonlinear Volterra Integral Equations by the Differential Transform Method" Journal Computational Applied Mathematics, vol.228, pp 70-76, 2009.

[3] S. Bazm, E. Babolian" Numerical Solution of Nonlinear Two-Dimensional Fredholm Integral Equations of the Second Kind Using Gauss Product Quadrature Rules" Communication. Nonlinear Science Numerical. Simultaneous, vol. 17, pp 1215-1223, 2012.

[4] K. Malenknejad, Z. JafariBehbahani" Application of Two-Dimensional Triangular Functions for Solving Nonlinear Class of Mixed Volterra-Fredholm Integral Equations" Mathematical . Computational . modeling", vol.55,pp 1833-1844, 2012.

[5] M. A. Abdou, A. A. Badr, M. B. Soliman" On a Method for Solving a Two Dimensional Nonlinear Integral Equation of the Second Kind" Journal Computational Applied Mathematics, vol 235 pp 3589-3598,2011

[6] N. I. Muskhelishvili. Some Basic Problems of Mathematical Theory of Elasticity. Noordroof, Holland, 1953pp250-254

[7] R.B. Hetnarski,J. Ignaczak. Mathematical Theory of Elasticity. Taylor and Francis, 2004 pp176-180.

[8 ] M. A. Abdou, S. A. Asseri " Closed Forms of Gaursat Functions in Presence of Heat for Curvilinear holes" Journal Thermalstress vol. 32:11pp1126-1148,2009.

[9] M. A. Abdou, S. A. Asseri " Gaursat Functions for an Infinite Plate with a Generalized Curvilinear Hole in Zeta Plane" Applied Mathematics Computational" vol . 212 pp23-36, 2009.

[10] M.A. Abdou, A. R. Jaan "An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Gaursat Functions" Journal of . Applied Mathematics. vol.5, 5pp,428-443, 2014

[11] A.I. Kalandiya. Mathematical Method of Two Dimensional Elasticity. Mir- Moscow,1975pp 135-137

[12] G. E. Exadaktylos, M. C. Stavropoulos. " A Closed Form Elastic Solution for Stress and Displacement Around Tunnels". International Journal of Rock Mechanics of Mining science. vol. 39 pp 905-916, 2002

[13] G. E. Exadaktylos, P. A. Liolios, M. C. Stavropoulos. "A Semi-Analytical Elastic Stress-Displacement Solution for Notched Circular Openings in Rocks". International Journal of Solids and Structures. Vol. 40,pp 1165-1187,2003

[14] N. Noda, R. B. Hentarski, Y. Tanigowa. " Thermal Stresses", Taylor and Francis, 2003pp 101-103

[15] R. Schinzinger, P. A. Laura. " Conformal Mapping Methods and Applications". Dover Publications, New York, 2003pp43-50

[16] H. Parkus. "Thermoelasticity". Spring-Verlag, Barlen, 1976pp 20-24.

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2017-02-28

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