An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions

Affiliation(s)

Department of Mathematics, Alexandria University, Alexandria, Egypt.

Department of Mathematics, Umm Al-Qura University, Mecca, KSA.

Department of Mathematics, Alexandria University, Alexandria, Egypt.

Department of Mathematics, Umm Al-Qura University, Mecca, KSA.

ABSTRACT

Complex variables method has been used to
solve the first and second fundamental problems for an infinite plate weakened
by a generalized curvilinear hole *C*. The curvilinear hole is conformally
mapped on the domain outside or inside a unit circle *γ* using a general rational mapping function with
complex constants. Many special and new cases are derived from this work. Some
of the work of the previous authors in this domain will be considered as
special cases of this paper. Also the interesting cases when the shape of the
hole takes different famous shapes are included. The components of stresses for
some examples are obtained.

Cite this paper

Abdou, M. and Jan, A. (2014) An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions.*Applied Mathematics*, **5**, 728-743. doi: 10.4236/am.2014.54070.

Abdou, M. and Jan, A. (2014) An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions.

References

[1] Colton, D. and Kress, R. (1983) Integral Equation Methods in Scattering Theory. John Wiley, New York.

[2] Ya. Popov, G. (1982) Contact Problems for a Linearly Deformable Functions. Kiev, Odessa.

[3] Noda, N., Hentarski, R.B. and Tanigowa, Y. (2003) Thermal Stresses. Taylor and Francis, UK.

[4] Schinzinger, R. and Laura, P.A. (2003) Conformal Mapping Methods and Applications. Dover Publications, New York.

[5] England, A.H. (1971) Complex Variable Method in Elasticity. McGraw, London, New York.

[6] Parkus, H. (1976) Thermo Elasticity. Spring-Verlag, Berlin,

[7] Kalandiya, A.I. (1975) Mathematical Method of Two-Dimensional Elasticity. Mir Publishers, Moscow.

[8] Muskhelishvili, N.I. (1953) Some Basic Problems of Mathematical Theory of Elasticity. Noordroof, Holland.

[9] El-Sirafy, I.H. and Abdou, M.A. (1984) First and Second Fundamental Problems of Infinite Plate with a Curvilinear Hole. Journal of Mathematical and Physical Sciences, 18, 1-12.

[10] Abdou, M.A. and Khar-Eldin, E.A. (1994) An Infinite Plate Weakened by a Hole Having Arbitrary Shape. Journal of Computational and Applied Mathematics, 56, 341-361.

http://dx.doi.org/10.1016/0377-0427(94)90088-4

[11] Abdou, M.A. and Khamis, A.K. (2000) On a Problem of an Infinite Plate with a Curvilinear Hole Having Three Poles and Arbitrary Shape. Bulletin of Calcutta Mathematical Society, 92, 309-322.

[12] Abdou, M.A. (2002) Fundamental Problems for Infinite Plate with a Curvilinear Hole Having Finite Poles. Applied Mathematics and Computation, 125, 177-193. http://dx.doi.org/10.1016/S0096-3003(00)00118-1

[13] Abdou, M.A., Sabbah, A.S. and Ismail, A.S. (2002) An Infinite Plate with a Curvilinear Hole and Flowing Heat. Proceeding Mathematical Physics Society Egypt, 34, 15-27.

[14] Exadaktylos, G.E. and Stavropoulou, M.C. (2002) A Closed form Elastic Solution for Stress and Displacement around Tunnels. International Journal of Rock Mechanics and Mining Sciences, 39, 905-916.

[15] Exadaktylos, G.E., Liolios, P.A. and Stavropoulou, M.C. (2003) A Semi-Analytical Elastic Stress-Displacement Solution for Notched Circular Openings in Rocks. International Journal of Solids and Structures, 40, 1165-1187.

[16] Abdou, M.A. and Asseri, S.A. (2009) Closed Forms of Gaursat Functions in Presence of Heat for Curvilinear Holes. JJournal of Thermal Stresses, 32, 1126-1148.

[17] Abdou, M.A. and Asseri, S.A. (2009) Gaursat Functions for an Infinite Plate with a Generalized Curvilinear Hole in Zeta Plane. Applied Mathematics and Computation, 212, 23-36.

http://dx.doi.org/10.1016/j.amc.2009.01.079

[1] Colton, D. and Kress, R. (1983) Integral Equation Methods in Scattering Theory. John Wiley, New York.

[2] Ya. Popov, G. (1982) Contact Problems for a Linearly Deformable Functions. Kiev, Odessa.

[3] Noda, N., Hentarski, R.B. and Tanigowa, Y. (2003) Thermal Stresses. Taylor and Francis, UK.

[4] Schinzinger, R. and Laura, P.A. (2003) Conformal Mapping Methods and Applications. Dover Publications, New York.

[5] England, A.H. (1971) Complex Variable Method in Elasticity. McGraw, London, New York.

[6] Parkus, H. (1976) Thermo Elasticity. Spring-Verlag, Berlin,

[7] Kalandiya, A.I. (1975) Mathematical Method of Two-Dimensional Elasticity. Mir Publishers, Moscow.

[8] Muskhelishvili, N.I. (1953) Some Basic Problems of Mathematical Theory of Elasticity. Noordroof, Holland.

[9] El-Sirafy, I.H. and Abdou, M.A. (1984) First and Second Fundamental Problems of Infinite Plate with a Curvilinear Hole. Journal of Mathematical and Physical Sciences, 18, 1-12.

[10] Abdou, M.A. and Khar-Eldin, E.A. (1994) An Infinite Plate Weakened by a Hole Having Arbitrary Shape. Journal of Computational and Applied Mathematics, 56, 341-361.

http://dx.doi.org/10.1016/0377-0427(94)90088-4

[11] Abdou, M.A. and Khamis, A.K. (2000) On a Problem of an Infinite Plate with a Curvilinear Hole Having Three Poles and Arbitrary Shape. Bulletin of Calcutta Mathematical Society, 92, 309-322.

[12] Abdou, M.A. (2002) Fundamental Problems for Infinite Plate with a Curvilinear Hole Having Finite Poles. Applied Mathematics and Computation, 125, 177-193. http://dx.doi.org/10.1016/S0096-3003(00)00118-1

[13] Abdou, M.A., Sabbah, A.S. and Ismail, A.S. (2002) An Infinite Plate with a Curvilinear Hole and Flowing Heat. Proceeding Mathematical Physics Society Egypt, 34, 15-27.

[14] Exadaktylos, G.E. and Stavropoulou, M.C. (2002) A Closed form Elastic Solution for Stress and Displacement around Tunnels. International Journal of Rock Mechanics and Mining Sciences, 39, 905-916.

[15] Exadaktylos, G.E., Liolios, P.A. and Stavropoulou, M.C. (2003) A Semi-Analytical Elastic Stress-Displacement Solution for Notched Circular Openings in Rocks. International Journal of Solids and Structures, 40, 1165-1187.

[16] Abdou, M.A. and Asseri, S.A. (2009) Closed Forms of Gaursat Functions in Presence of Heat for Curvilinear Holes. JJournal of Thermal Stresses, 32, 1126-1148.

[17] Abdou, M.A. and Asseri, S.A. (2009) Gaursat Functions for an Infinite Plate with a Generalized Curvilinear Hole in Zeta Plane. Applied Mathematics and Computation, 212, 23-36.

http://dx.doi.org/10.1016/j.amc.2009.01.079