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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 84
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 184

Singular Trefftz Functions for Modelling Material Reinforced by Hard Particles

V. Kompiš1, M. Kompiš2, M. Kaukic3 and D. Hui4

1Academy of Armed Forces of General M.R. Štefánik, Liptovský Mikuláš, Slovakia
2Building Testing and Research Institute, Zilina, Slovakia
3Faculty of Management Science and Informatics, University of Zilina, Slovakia
4University of New Orleans, United States of America

Full Bibliographic Reference for this paper
V. KompiĀš, M. KompiĀš, M. Kaukic, D. Hui, "Singular Trefftz Functions for Modelling Material Reinforced by Hard Particles", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 184, 2006. doi:10.4203/ccp.84.184
Keywords: computational simulation, composite material, small particles, short fibres, boundary point method, continuous dipole method.

Modern composite materials will offer many advanced features over composites used in the past. Materials reinforced by hard nanoparticles or fibres (nanotubes) will be important materials for structural, electromagnetic and many other applications with very attractive properties. These composites can even create products that we have not considered yet. Nanocomposites offer the chance to combine organic polymers with inorganic ceramics or to add nanoparticles to polymers and ceramics for improved properties. A material could even be made with properties that would vary with a changing electrical field, magnetic field, heat flux, or mechanical stress. These materials could change their colour, or conductive properties.

The reinforcing particles with a small percentage of additional weight improves the matrix properties by tens of per cents. By orienting the carbon nanotubes in a nanocomposite properly, the material can be made to be a thermal conductor or insulator. The scientists discovered that if all the nanotubes were lined up parallel to each other and a heat flux was applied parallel to the direction of the nanotubes, the material is an excellent thermal conductor. However, for the same material, if the heat is applied perpendicular to the direction of the nanotubes, then the material reflects the heat, and is an insulator. Similarly, the material can be very stiff parallel to the nanotubes and weak in perpendicular direction. The material can be designed so that its properties will change in the volume as required and so light structures with optimal parameters can be obtained.

Numerical simulation is a tool which enables the understanding of the behaviour of these materials in different situations. Because of very different material properties of the matrix and reinforcing particles, very large gradients are present in all field variables. Numerical simulation of such problems requires special techniques in order to obtain efficient models. This paper is concerned with the simulation of these materials under static loading.

The correct analysis of the interaction of the particles with the matrix, particles among each other, particles with the domain boundary, or with the other matrix phase (e.g. composite material reinforced with nano-particles) is important for the optimal design and for damage assessment of such structures. Classical methods such as the finite element method (FEM) and the boundary element method (BEM) will need models with many millions, or billions of equations for the simulation of such problems.

Methods which satisfy the homogeneous governing equations inside the domain (known as Trefftz, or T-methods) and decay to zero in same power in the infinity as the real field are much more efficient for modelling problems such as a material with inclusions or holes than the other methods. A similar accuracy can be achieved by models which lead to system of equations smaller by several orders than the other methods. The simplest representative of these methods is the method of fundamental solutions (MFS), which is a boundary collocation method. It does not need any elements and any integration. Weak points of the method are 1) that the global equilibrium is not satisfied and 2) the efficiency decreases also for domains with a complicated form, or domains with a large aspect ratio. A method using dipoles, i.e. two forces applied in the same point in opposite directions and acting at a point inside the inclusion, i.e. outside the domain of the matrix is very effective for modelling materials reinforced with particles which have not too different dimensions in different directions and the technique is called the boundary point method (BPM) here.

The fast multi-pole method (FMM), which is a form of the BEM solution, reduces considerably the model of composite material reinforced with nano-particles by expansion of the integral equation terms into a Taylor series. The BPM unlike the FMM does not solve the problem by integral equations but it interpolates (similar to the MFS) the boundary conditions in collocation points. If the aspect ratio of the particle is too large like in composite materials reinforced by fibres, however, then the BPM will not be very effective and too many collocation points in the fibre direction would be necessary in order to obtain good accuracy with the model. A technique using distributed (continuous) source functions (multipoles, or 3D dipoles) and called here the method of continuous dipoles (MCD) was found to be an efficient tool for such problems. Application of both the BPM and the MCD to simulation of the interaction of particles with a matrix in composite materials is introduced in the paper.

As usually very many particles have to be included in the model, the direct solution of such problems is inefficient and iterative schemes contribute to increase the efficiency. Moreover, the strain and stress fields in micro- and nano-scale can be introduced as superposed by constant components of the far field effects and the local effects. The intensities of the dipoles in the first step of solution need not include any interaction effects, but can be defined for the form of corresponding particles and the far field strain. All the other effects can be included in the following iteration steps of the solution.

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