Isogeometric Analysis of Small-Scale Plates with Generalised Continua

Nguyen, Hoang (2018) Isogeometric Analysis of Small-Scale Plates with Generalised Continua. Doctoral thesis, Northumbria University.

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Abstract

Small-scale (micro) structures are increasingly used in variety of engineering fields such as micro- and nano-electro-mechanical system, space and bioengineering. Reliable numerical prediction of the behaviours of these structures becomes desirable due to costly experiments in micro scales. In structural modelling, it is known that classical continuum theories fail to capture size effects, instead generalised continuum theories which take into account the material length scale should be used.

In this thesis, the behaviours of small-scale plate structures with size-dependent effects are investigated by means of the numerical method of isogeometric analysis and generalised continuum theories of modified couple stress and strain gradient. While the size effects are efficiently captured by the modified couple stress theory and strain gradient theory, the refined plate theory with four variables is employed to establish the displacement fields of plates. The isogeometric analysis in which non-uniform rational B-splines basis function is used for both geometry representation and field approximation is capable of exactly describing the geometry of the domain and efficiently dealing with the high-order element requirements.

The investigation of small-scale plates focuses on the linear analysis, nonlinear analysis, and crack analysis. While linear analysis includes bending, vibration, and buckling responses with a quasi-3D refined plate theory, nonlinear analysis involves the derivation of von Kámán strains and implementation of iterative Newton-Raphson technique. Meanwhile, the vibration responses of cracked plates are assessed in the platform of the strain gradient theory and extended isogeometric analysis where enrichment functions are included to model the discontinuities and improve the solution accuracy around the crack tip.

The successful convergence and comparison studies as well as benchmark results of the analyses of functionally graded small-scale plates ascertain the validity and reliability of the proposed approaches. In addition, a number of parametric studies have been carried out to investigate the effects of material length scale, material and geometrical parameters on the responses of the plates.

The numerical results show that the inclusion of the material length scale which takes into account the size-dependent effects increases the approximated structural stiffness. Consequently, the displacement of the structure is decreased while the fundamental frequency and critical buckling load are increased.

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