Mathematical modelling of interacting fibre structures and non-woven materials

Non–woven materials consist of many thousands of fibres laid down on a conveyor belt under the influence of a turbulent air stream. To improve industrial processes for the production of non–woven materials, we develop and explore novel mathematical fibre and material models. In Part I of this thesis we improve existing mathematical models describing the fibres on the belt in the meltspinning process. In contrast to existing models, we include the fibre–fibre interaction caused by the fibres’ thickness which prevents the intersection of the fibres and, hence, results in a more accurate mathematical description. We start from a microscopic characterisation, where each fibre is described by a stochastic functional differential equation and include the interaction along the whole fibre path, which is described by a delay term. As many fibres are required for the production of a non–woven material, we consider the corresponding mean–field equation, which describes the evolution of the fibre distribution with respect to fibre position and orientation. To analyse the particular case of large turbulences in the air stream, we develop the diffusion approximation which yields a distribution describing the fibre position. Considering the convergence to equilibrium on an analytical level, as well as performing numerical experiments, gives an insight into the influence of the novel interaction term in the equations. In Part II of this thesis we model the industrial airlay process, which is a production method whereby many short fibres build a three–dimensional non–woven material. We focus on the development of a material model based on original fibre properties, machine data and micro computer tomography. A possible linking of these models to other simulation tools, for example virtual tensile tests, is discussed. The models and methods presented in this thesis promise to further the field in mathematical modelling and computational simulation of non–woven materials