Natural vibrations of curved nanobeams

Abstract

Antud uurimistöö keskendub kõverate nanokaarte omavõnkumistele. Töös kasutatakse elastsuse teooriat, mis võtab arvesse mõjusid, mida klassikaline teooria ei arvesta. Siin uuritakse prao-tüüpi defektide mõju omavõnkesagedusele. Põhivõrrandite lahendamiseks kasutatakse numbrilist meetodit, mis põhineb muutujate eraldamisel. Saadud tulemusi võrreldakse kirjanduses leiduvate tulemustega. Lisaks uuritakse erinevate füüsikaliste ja geomeetriliste parameetrite, sealhulgas paksuse, raadiuse ja kaare kesk-nurga mõju vaba võnkumise sagedusele. Doktoritöö põhineb autori koostatud viiel publikatsioonil, millest üks on avaldatud ning teised on trukki suunatud. Doktoritöö struktuur hõlmab kirjanduse ülevaadet, sisu lühikokkuvõtet, kirjanduse loetelu ja autori elulookirjeldust. Doktoritöö on korraldatud järgmiselt: Esimeses peatükis antakse ajalooline ülevaade kõverate nano-talade omavõnkumiste uurimisest keskendudes Eringeni mitteokaalse elastilisuse teooria rakendamisele. Esitatakse nanokaare füüsikaline mudel, arvestades nii defekte kui ka astmeid, ning esitatakse sageduste leidmiseks vajalikud võrrandid. Teises peatükis kirjeldatakse uuritavat nanokaare mudelit, kus paksus on tükiti konstantne ning millel on praod paksuse hüppekohtades. Kolmandas peatükis leitakse konstantse paksusega kõvera nanotala omavõnkesagedused vabalt toetatud ja jäigalt kinnitatud otste korral. Neljandas ja vüendas peatükis leitakse omavõnkesagedused astmeliste nanokaarte jaoks, millel on defektid. Kuues peatükk on pühendatud niisuguste nanokaarte uurimisele, mille üks ots on jäigalt kinnitatud ja teine täiesti vaba. Viimases osas esitatakse saadud tulemuste põhjal tehtud järeldused.

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.