The stresses components in position and time of weakened plate with two holes conformally mapped into a unit circle by a conformal mapping with complex constant coefficients

Sharifah E. Alhazmi; M. A. Abdou; M. Basseem; Sharifah E. Alhazmi; M. A. Abdou; M. Basseem

doi:10.3934/math.2023562

AIMS Mathematics

2023, Volume 8, Issue 5: 11095-11112. doi: 10.3934/math.2023562

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The stresses components in position and time of weakened plate with two holes conformally mapped into a unit circle by a conformal mapping with complex constant coefficients

1.
Department of Mathematics, Al-Qunfudah University College, Umm Al-Qura University, Mecca, Kingdom of Saudi Arabic
2.
Department of Mathematics, Faculty of Education, Alexandria University, Egypt
3.
Department of Mathematics, Faculty of Engineering, Sinai University, Egypt

Received: 07 January 2023 Revised: 10 February 2023 Accepted: 17 February 2023 Published: 09 March 2023
MSC : 30C20, 74B99

In this paper, an infinite elastic plate weakened by two holes are considered and the complex variable method is used to derive a closed form of Gaursat functions for the first and second fundamental problems with variant time. The holes, in all previous works, are conformally mapped outside the unit circle without time. Here, the two holes are conformally mapped into the unit circle $ \varpi $ in the effect of time by the generalized rational mapping function with complex constant coefficients. By using this conformal mapping function, the fundamental problems transfer to an integro-differential equation with Cauchy kernel. Then, after applying the complex variable method, one can obtain a closed form of Gaursat functions. Some applications were discussed and the time effect on the applications was studied. In addition, the different stress components in each application have also been calculated using Maple 2022.1.
- an infinite plate,
- Gaursat functions,
- conformal mapping,
- elastic plane,
- complex variable method,
- curvilinear hole
Citation: Sharifah E. Alhazmi, M. A. Abdou, M. Basseem. The stresses components in position and time of weakened plate with two holes conformally mapped into a unit circle by a conformal mapping with complex constant coefficients[J]. AIMS Mathematics, 2023, 8(5): 11095-11112. doi: 10.3934/math.2023562

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Abstract

In this paper, an infinite elastic plate weakened by two holes are considered and the complex variable method is used to derive a closed form of Gaursat functions for the first and second fundamental problems with variant time. The holes, in all previous works, are conformally mapped outside the unit circle without time. Here, the two holes are conformally mapped into the unit circle $ \varpi $ in the effect of time by the generalized rational mapping function with complex constant coefficients. By using this conformal mapping function, the fundamental problems transfer to an integro-differential equation with Cauchy kernel. Then, after applying the complex variable method, one can obtain a closed form of Gaursat functions. Some applications were discussed and the time effect on the applications was studied. In addition, the different stress components in each application have also been calculated using Maple 2022.1.

References

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