Tartu Ülikool

Doctoral defence: Shahid Mubasshar “Natural vibrations of curved nanobeams“

On 29. September at 2.15 pm Shahid Mubasshar defended his thesis “Natural vibrations of curved nanobeams  to obtain the degree of Doctor of Philosophy in Mathematics. 


Prof emer Jaan Lellep, University of Tartu. 


Prof R. Kacianauskas, (Vilnius Gediminas Technical University, Lithuania); 

Prof J. Logo, (Budapest University of Technology and Economics, Hungary). 


The natural vibrations of curved nanobeams or nanoarches are studied in the present work. To achieve authentic results, the study employs the nonlocal theory of elasticity, which accounts for small-length scale effects that are not considered by the classical theory. The impact of crack-like defects on the natural frequency is thoroughly examined, and the obtained results are compared with available data. Nanoarches with constant thickness, where a stable crack is located within the element are studied first of all. Additionally, nanoarches with piecewise constant thickness are analysed, where stable cracks are positioned at the re-entrant corners of steps. A novel method based on linear elastic fracture mechanics is developed to determine the natural frequencies of nanoarches. Different nanoarches such as simply supported, clamped at both edges and cantilevers are considered. The method developed in this study is also applied to investigate the vibrational behaviour of the CNTs of the shapes armchair, zigzag and chiral. To solve the governing equations, a numerical approach utilising the separation of variables is employed. The obtained results are validated by comparing them with existing literature. Furthermore, the study emphasizes the significant role of cracks in altering the frequency behaviour of curved beams. Moreover, the research extensively discusses the influence of various parameters, including thickness, radius and central angle on the eigenfrequency of the nanoarches. The dissertation is organised as follows: Chapter 1 provides a historical review of the papers on the natural vibrations of curved nanobeams, focusing on the application of Eringen's nonlocal theory of elasticity. In Chapter 2, the physical model of the nanoarch is presented. The model includes both defects and steps. The governing equations required for analysing the natural frequency of the nanoarch are also presented in Chapter 2. The natural frequency of the nanoarches of constant thickness is defined in Chapter 2. The natural frequency of the simply supported nanoarches of piecewise constant thickness with a defect is presented in Chapter 4. Chapter 5 focuses on the natural frequency of the nanoarch with both a flaw and a step, considering clamped nanoarches. In Chapter 6, the natural frequency of the cantilever nanoarches is determined. In Chapter 7, the method developed in this study is applied to analyse the natural vibrations of CNTs. Finally, the concluding remarks of the dissertation summarising the key findings and contributions of the research work are provided. 

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