# TESSELLATED CONTINUUM MECHANICS: VIBRATION OF PERFORATED PLATES

• Zainab Sedqi

Student thesis: Phd

### Abstract

An important method for lightening the weight of structures is with the incorporation of perforated materials but a good understanding of their dynamic behaviour is required. The more interesting types of perforated structure in the field of engineering are those representable by fractals. Fractals permit the representation of intricate perforated geometries, but their application is recognised to be beset with difficulties, which stem from an inability to define traditionally derived physical quantities such as stress. The presented research provides a novel methodology based on transport theory for pre-fractals, which facilitates the modelling of complex perforated structures. This approach is called tessellated continuum mechanics, which has recently been developed at the University of Manchester. Tessellated continuum mechanics is an approach that enables known analytical and numerical continuum solutions to be immediately applied to the fractal space. A feature of the approach is the representation of fractal structures in equivalent continuum spaces, which can be readily analysed by available numerical techniques. The tessellated approach links pre-fractal elements to tiles in a tessellated continuum by means of a hole-fill map; so called because when applied to a perforated structure (i.e. a pre-fractal) it closes all holes to form a tessellation. It is shown in the thesis how pre-fractals and tessellations can be created independently and very efficiently using iterated function schemes. Such schemes when used in tessellated continuum mechanics involve the recursive application of contraction maps and the exact same number of maps is required for forming tessellations and pre-fractals. To accommodate any discontinuous physics that arises on a tessellation it is necessary to imbue it with a discontinuity network. Jumps in displacement, velocity and derivatives are permitted on a discontinuity network. The research presented in the thesis tests the hypothesis that the dynamic behavior of perforated plates and beams can be analysed to high accuracy on a tessellated continuum. The work also examines the role of similitude, which is an integral feature of the tessellated approach. Similitude enables the physics of a tile on a tessellated continuum to be related to an element of a pre-fractal. In this way it can be demonstrated that a collection of tiles which forms a tessellation has the same behavior as a pre-fractal structure. The work confirms the validity of tessellated continuum mechanics by means of extensive numerical trials using commercial FE software on pre-fractal beams and plates along with corresponding tessellations.
Date of Award 31 Dec 2019 English The University of Manchester Keith Davey (Supervisor) & Rooholamin Darvizeh (Supervisor)

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