Complex Variable Approach for Thermoelastic Boundary Value Problem Using Rational Mapping Techniques

Mai Taha, Mohamed A. Abdou, Amnah E. Shammaky, Abeer A. Al-Dohiman and Eslam M. Youssef ()
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Mai Taha: Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21256, Egypt
Mohamed A. Abdou: Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21256, Egypt
Amnah E. Shammaky: Department of Mathematics, Faculty of Science, Jazan University, Jazan 21944, Saudi Arabia
Abeer A. Al-Dohiman: Department of Mathematics, Faculty of Science, Jouf University, Sakaka 2014, Saudi Arabia
Eslam M. Youssef: Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21256, Egypt

Mathematics, 2025, vol. 13, issue 19, 1-17

Abstract: This article presents a novel approach to looking at steady-state thermoelastic boundary value problems in isotropic elastic plates with curvilinear holes using a complex variable approach and rational conformal mappings. The physical domain with a non-circular opening is mapped conformally to the unit disk. A thermoelastic potential combines the temperature distribution, which is determined by the Laplace equation with Neumann boundary conditions. Gaursat functions, which are shown as truncated power series, show the complicated stress and displacement fields. They are found by putting boundary constraints at certain collocation points. This procedure presents us with a linear system that can be solved using the least squares method. The method is applied in an annular shape that is exposed to a radial temperature gradient. This experiment shows how changes at the boundary affect the distribution of stress. According to numerical simulations, stress distributions are more uniform when boundaries are smoother, but stress concentrations increase with the size of geometric disturbances. The suggested approach remarkably captures the way geometry and thermal effects interact in two-dimensional thermoelasticity. It is a reliable tool for researching intricate, heated elastic domains.

Keywords: conformal mapping; complex variables; Gaursat functions; boundary value problems; curvilinear hole; stress analysis; collocation method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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