Stability analysis of stepped nanobeams with defects

Abstract

Käesolev väitekiri on pühendatud elastsete astmeliste nanovarraste stabiilsuse analüüsile. Eeldatakse, et vaadeldavad nanovardad on ristkülikukujulise ristlõikega ja neile on rakendatud teljesuunaline rõhk ning et astmete nurkades asuvad stabiilsed praod. Nanotala kriitilise koormuse määramiseks kasutatakse Eringeni poolt välja töötatud mittelokaalse elastsusteooria võrrandeid. Prao mõju hindamiseks seotakse nanotala järeleandlikkus pinge intensiivsuse koefitsiendiga prao tipus ning leitakse kriitilised koormused vabalt toetatud ja jäigalt kinnitatud astmeliste varraste, samuti konsooltala jaoks. Numbrilised tulemused on saadud programmi MATLAB abil. Tulemusi võrreldakse teiste autorite töödega homogeensete nanovarraste korral. Väitekiri põhineb autori viiel artiklil (neist neli on trükis ilmunud). Väitekiri koosneb sissejuhatusest, kirjanduse ülevaatest ja kuuest peatükist, kirjanduse nimestikust ja autori elulookirjeldusest. Töö esimeses peatükis esitatakse kirjanduse ülevaade, samuti töö eesmärk ning tema struktuur. Teises peatükis kirjeldatakse nanovarda füüsikalist mudelit ja lokaalse järeleandlikkuse kontseptsiooni detailsemalt. Peatükkides 3, 4 ja 5 rakendatakse seda meetodit vabalt toetatud ja jäigalt kinnitatud nanovarraste ja nanokonsoolide korral. Uuritakse prao pikkuse, samuti teiste parameetrite mõju nanovarraste kriitilisele koormusele. Viimases peatükis rakendatakse kirjeldatud meetodit lisatugedega astmeliste nanovarraste korral.

The nanobeams under consideration are of rectangular cross-section with piecewise constant thicknesses In the current dissertation, the stability analysis of nonlocal elastic nanobeams is carried out. subjected to the axial compressions. It is assumed that the nanobeams are weakened by the defects like steps, cracks and internal supports. The cracks are considered to be stable surface cracks that are located at the re-entrant corners of the steps penetrated throughout the thickness of the nanobeam. To analyse the buckling of nanobeams, an analytical approach has been developed within the framework of Eringen's nonlocal theory of elasticity to embrace the small size effect. The nonlocal theory of elasticity for Euler-Bernoulli nanobeams is combined with the linear fracture mechanics to develop an approximate method for the stability analysis of nanobeams and nanocolumns. The crack effect is considered by coupling the local compliance of the structure with the stress intensity factor which can be calculated by the methods of linear fracture mechanics. The critical buckling loads for axially loaded stepped nanobeams and nanocolumns with cracks and with the internal supports are calculated. The influence of nonlocal parameter, crack parameters, steps, intermediate support location and the other physical parameters on the stability of nanobeams is investigated. The method developed for stepped nanobeams with cracks is applied to the simply supported, clamped and cantilever nanobeams. The case of stepped nanobeam with intermediate rigid support is studied separately. Since conducting experiments at nanolevel is difficult to handle, the accuracy of the presented methods is verified by the comparison of results with the available work in the literature. MATLAB tools are used to provide significant numerical results. The dissertation is based on the five papers of the author (four of these are published during the last three years). The dissertation consists of the review of the obtained results, the copies of the paper, the list of the literature and the CV of the author. The dissertation is organised as follows. Section 1 contains a historic background of the stability analysis of nonlocal beams, the aim and the structure of the dissertation. In section 2, the nonlocal physical model and the local flexibility of stepped nanobeams with cracks are described in detail. In sections 3, 4 and 5, the method is applied to the nanobeams with different support conditions. The cases of simply supported, clamped and cantilever nanobeams are studied in great details. The influence of the crack parameters and the nonlocal parameters on the buckling of nanobeams is investigated through parametric studies. In section 6, the method is applied to the stepped nanobeams with additional internal supports. The influence of the support parameters on the stability of nanobeams has been analyzed. Finally, the concluding remarks of the dissertation are presented in the last section.