International Scientific Collaboration on Modelling and Analysis of Nano-Beams in Gradient Elasticity Theory

by Ramazan Özgür İridere | Dec 12, 2024

International Scientific Collaboration Title: Modelling and Analysis of Nano-Beams in Gradient Elasticity Theory

Project Coordinator: Prof. Dr. Teoman Özer (Istanbul Technical University)

Researcher: Prof. Dr. Martin Kröger (ETH Zurich)

This study investigates the search for the exact analytical solutions for the plane stress and displacement fields in linear homogeneous anisotropic nano-beam models of gradient elasticity. It focuses on the solution of the Helmholtz equation, which involves the second-order inhomogeneous linear partial differential equation in the theory of plane gradient elasticity, using polynomial series-type solutions. The analysis is based on the use of gradient Airy stress functions to derive the stress fields in gradient theory. The study aims to obtain closed-form analytical solutions for the Airy stress functions, stress, and strain fields in both classical and gradient theories. The study considers five different types of two-dimensional functionally graded cantilever beams with various boundary conditions. The cases of cantilever anisotropic nano-beam subjected to a concentrated force at the free end, cantilever anisotropic nano-beam under a uniform load, simple anisotropic nano-beam under a uniform load, cantilever anisotropic nano-beam supported under a uniform load and cantilever anisotropic nano-beam with fixed ends under a uniform load are investigated. Exact analytical solutions are provided for the gradient stress and displacement fields of two-dimensional and one-dimensional anisotropic nano-beams under different boundary conditions. The study demonstrates the significant stress gradient size effects at the nanoscale through the analytical solutions derived for anisotropic beams. In addition, we show that the strain gradient theory results for the limiting case of the gradient coefficient c are in good agreement with the results obtained for isotropic and anisotropic materials in elasticity theory and classical theory. Furthermore, we discuss the real-world applications by considering the stress and displacement fields in real anisotropic materials such as TiSi2 single crystals and orthotropic materials, which are special cases of anisotropic materials such as wood and epoxy as documented in the literature.

In addition, the analytical solutions of the stress fields from Airy stress functions for isotropic and anisotropic axisymmetric curved nano-beams in gradient elasticity theory have been obtained by general integration of differential equations without using series extension type solutions. In addition, as a special case, analytical solutions of the stress and displacement fields of a nano-circular ring subjected to internal and external pressure, which is the most general case of the two-dimensional multi-connected body problem in the gradient elasticity theory, have been investigated. In addition to the known initial stress concept in the literature, general expressions of such stresses occurring in both isotropic and anisotropic circular nano-circular rings have been obtained by starting from the gradient initial stress concept, which introduced for the first time in the literature.

Kröger M., Özer T., Applied Mathematical Modelling, 133, 108-147, 2024

Kröger M., Özer T., Applied Mathematical Modelling, 2024 (under review)

İTÜ Faculty of Civil Engineering

It can be accepted that the history of ITU Civil Engineering Faculty was commenced in 1727 by the establishment of Humbarahane during the era of Damat İbrahim Paşa. However, this attempt was to no avail due to the martyrization of the studwents of this school by the Janissaries.

In 1734, during the grand viziership of Topal Osman Paşa, classes related to Civil Engineering was being taught in Hendesehane, which was founded in Üsküdar. Similarly, this school was also closed after three years due to the opposition of the Janissaries. The reasons of the tragedy that occured in Battle of Chesma, 1770 were inadequacy of the ships and the ignorance of the Ottoman Navy.

Although the Atlantic Map (1513) and the Map of Europe and the Mediterranean Sea (1523) by Piri Reis were the leading pieces of the era, Ottoman viziers were so ignorant that they claimed it was impossible for the Russian Baltic Fleet to reach the Mediterranean Sea assuming that there were no sea connection. With the proposal from Gazi Hasan Paşa and Baron de Tott, Mühendishane-i Bahri-i Humayun was established during the time of Sultan Mustafa III, aiming to train naval construction engineers. Since Ottoman Empire did not have any engineering schools at the time, military engineering classes were taught by French engineers beginning from 1784. French lecturers had to return to their country in 1788 with the increasing suppression from Austria and Russia, which were unwilling for the development and thriving of Ottoman Empire. In 1759, during the Grand Viziership of Koca Ragıp Paşa, Humbarahane was reopened in Karaağaç, Haliç (Golden Horn) and continued to teach even though it was incompetent. In 1792, Lağımcı Ocağı and Humbaracı Ocağı (Guild of Sappers) were established.