Vladimír Kompiš

Method of Continuous Source Functions for Modelling of Matrix Reinforced by Finite Fibres

Fibres are the most effective reinforcing material. Simulation of the interaction of matrix with fibres and fibre with other fibres is a most important problem for understanding the behaviour of fibre-reinforced composites (FRC). Large gradients in all displacement, stress and strain fields and their correct simulation for near and far field action are essential for effective computational modelling. Because of the large aspect ratios in fibre type reinforcing particles, methods using volume discretization are not efficient. Source functions (forces, dipoles, dislocations) describe correctly both near and far field activities and thus help to simulate all interactions very precisely. The method of continuous source functions allows us to satisfy the continuity of fields between very stiff fibres and much more flexible matrix by 1D continuous functions along the fibre axis and local 2D functions in the end parts of a fibre with only few parameters. Two types of examples with rows of non-overlapping sheets of fibre and with overlapping fibres show that the interaction of the end parts of fibres is crucial for evaluation of the mutual interaction of fibres in the composite. Correct simulation of all parts is important for evaluation of stiffness and strength of the FRC.

Efficient Solution for Composites Reinforced by Particles

A very effective method for some kind of problems is the Method of Fundamental Solutions (MFS). It is a boundary meshless method which does not need any mesh and in linear problems only nodes (collocation points) on the domain boundaries and a set of source functions (fundamental solutions, i.e. Kelvin functions) in points outside the domain are necessary to satisfy the boundary conditions. Another kind of source functions can be obtained from derivatives of the Kelvin source functions. Dipoles are the derivatives of Kelvin function in direction of acting force.

Thermal Properties of Short Fibre Composites Modeled by Meshless Method

Computational model using continuous source functions along the fibre axis is presented for simulation of temperature/heat flux in composites reinforced by short fibres with large aspect ratio. The aspect ratio of short fibres reinforcing composite material is often as large as 103 : 1–106 : 1, or even larger. 1D continuous source functions enable simulating the interaction of each fibre with the matrix and also with other fibres. The developed method of continuous source functions is a boundary meshless method reducing the problem considerably comparing to other methods like FEM, BEM, meshless methods, or fast multipole BEM formulation.

Computational Modal and Solution Procedure for Inhomogeneous Materials with Eigen-Strain Formulation of Boundary Integral Equations

In the present study a novel computational modal and solution procedure are proposed for inhomogeneous materials with the eigenstrain formulation of the boundary integral equations. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix with various shapes and material properties via the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. As unknowns appeared in the final equation system are on the boundary of the solution domain only, the solution scale of the inhomogeneity problem with the present model is greatly reduced. This feature is considered to be significant because such a traditionally time-consuming problem with inhomogeneities can be solved most cost-effectively with the present procedure in comparison with the existing numerical models such as finite element method (FEM) and boundary element method (BEM). Besides, to illustrate computational efficiency of the proposed model, results of overall elastic properties are presented by means of the present eigenstrain model and the newly developed boundary point method for particle reinforced inhomogeneous materials over a representative volume element. The influences of scatted inhomogeneities with a variety of properties and shapes and orientations on the overall properties of composites are computed and the results are compared with those from other methods, showing the validity and the effectiveness of the proposed computational modal and the solution procedure.

Computational Simulation Methods for Composites Reinforced by Fibres

Trefftz-FEM (T-FEM), Adaptive Cross Approximation BEM (ACA BEM) and Method of Continuous Source Functions (MCSF) are presented for simulation of Composites Reinforced by Short Fibres (CRSF) with the aim to show possibilities of reduction the problem of complicated and important interac- tions in such composite materials. Fibres are the most effective reinforcing material. Outstanding mechanical, ther- mal and electro-mechanical properties of Carbon Nano-Tubes (CNT), carbon fi- bres and some other fibres are well known. Composites Reinforced by Short Fi- bres/tubes (CRSF) are often defined to be materials of future with excellent elec- tro-thermo-mechanical (ETM) properties. Understanding the behaviour of such composite materials is essential for structural design. Computational simulations play an important role in this process. Usually, strength, stiffness, thermal and electrical conduction of fibres are much larger than those of the matrix material. Very large is also the aspect ratio of the short fibres. Because of these properties very large gradients are localized in all ETM fields along the fibres and in the ma- trix. The fields define the interaction of the fibres with the matrix, with the other fibres, with the boundaries of the domain/structure. Accurate computational simu- lation of the fields is important for correct assessment of the material behaviour. ∗

MFS with RBF for Thin Plate Bending Problems on Elastic Foundation

In this chapter a meshless method, based on the method of fundamental solutions (MFS) and radial basis functions (RBF), is developed to solve thin plate bending on an elastic foundation. In the presented algorithm, the analog equation method (AEM) is firstly used to convert the original governing equation to an equivalent thin plate bending equation without elastic foundations, which can be solved by the MFS and RBF interpolation, and then the satisfaction of the original governing equation and boundary conditions can determine all unknown coefficients. In order to fully reflect the practical boundary conditions of plate problems, the fundamental solution of biharmonic operator with augmented fundamental solution of Laplace operator are employed in the computation. Finally, several numerical examples are considered to investigate the accuracy and convergence of the proposed method.

Finite Displacements in Reciprocity Based Multi-Domain BE/FE Formulations

In this chapter, Trefftz (T-) functions are used for the development of Multi-Domain (MD) BEM/FEM based on the reciprocity relations. This reciprocity principles are well known from the Boundary Element formulations, however, using the Trefftz functions (polynomials, fundamental solutions with the source point defined outside the sub-domain, or other type of non-singular T-function) in the reciprocity relations instead of the fundamental solutions yields the non-singular integral equations for the evaluation of corresponding sub-domain relations. A weak form satisfaction of the equilibrium is used for the inter-domain connectivity relations. For linear problems, the element stiffness matrices are defined in the boundary integral equation form. In non-linear problems the total Lagrangian formulation leads to the evaluation of the boundary integrals over the original (related) sub-domain evaluated only once during the solution and to the domain integrals containing the non-linear terms. Considering the examples of simple tension, pure bending and tension of fully clamped rectangular 2D domain (2D stress/strain problems) for large strain-large rotation problems, the use of the initial stiffness, the Newton-Raphson procedure, and the incremental Newton- Raphson procedure is considered.

Rigid inclusions solved using non-singular reciprocity based BEM

Publisher Summary The local field solution shows large gradients in deformations and in stresses because of the rigid inclusions. These gradients are difficult to solve using FEM techniques — a very fine mesh is required. More convenient for handling such large gradients are reciprocity based boundary element techniques based on nonsingular integral equation formulations. If material of inclusion is much stiffer than the matrix, the chapter considers its deformation to be negligible. It can be modeled as rigid. A special technique using nonsingular boundary elements for solution of the inclusions with smooth surfaces and boundary integral collocation enables to solve such problems efficiently. The formulation uses the boundary traction nodal points to be the integration nodes. For computation of stress fields low, second order Trefftz polynomial is used to obtain the stresses on, or near the surfaces, and inside the domain.