Direct solution of plane elasticity problems by using the Muskhelishvili functional equation and computer algebra software

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2018
• Θέματα: Plane isotropic elasticity; First fundamental problem; Second fundamental problem; Boundary value problems; Complex potentials of Kolosov–Muskhelishvili; Muskhelishvili functional equation; Complex variables; Analytic functions; Cauchy integral formula; Numerical methods; Collocation method; Mathematica; Επίπεδη ισότροπη ελαστικότητα; Πρώτο θεμελιώδες πρόβλημα; Δεύτερο θεμελιώδες πρόβλημα; Προβλήματα συνοριακών τιμών; Μιγαδικά δυναμικά των Kolosov–Muskhelishvili; Συναρτησιακή εξίσωση του Muskhelishvili; Μιγαδικές μεταβλητές; Αναλυτικές συναρτήσεις; Ολοκληρωτικός τύπος του Cauchy; Αριθμητικές μέθοδοι; Μέθοδος του συντοπισμού

The classical Muskhelishvili functional equation for the solution of plane isotropic elasticity problems is revisited. The first complex potential φ(z) of Kolosov–Muskhelishvili is assumed to have the approximate form of a polynomial with unknown coefficients inside the elastic medium. Then the boundary conditions permit the expression of the boundary values for the second complex potential ψ(z) of Kolosov–Muskhelishvili in terms of the same coefficients. This potential, ψ(z), is then obtained by using the classical Cauchy integral formula in complex analysis. Finally, the unknown coefficients are determined by using an appropriate number of collocation points for ψ(z) outside the elastic medium. Both cases of the first and the second fundamental problems are considered. The present method was implemented and applied to the problem of an elliptical elastic medium by using the computer algebra system Mathematica and its modern and powerful language. Generalizations of the present results are also possible and suggested in brief. It is hoped that the present approach will be considered as an alternative to the classical methods of solution of plane isotropic elasticity problems like the finite and boundary element methods.