## Fachbereich Mathematik

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- Dissertation (1) (entfernen)

- Effective mechanical properties of technical textile materials via asymptotic homogenization (2012)
- The goal of this work is to develop a simulation-based algorithm, allowing the prediction of the effective mechanical properties of textiles on the basis of their microstructure and corresponding properties of fibers. This method can be used for optimization of the microstructure, in order to obtain a better stiffness or strength of the corresponding fiber material later on. An additional aspect of the thesis is that we want to take into account the microcontacts between fibers of the textile. One more aspect of the thesis is the accounting for the thickness of thin fibers in the textile. An introduction of an additional asymptotics with respect to a small parameter, the relation between the thickness and the representative length of the fibers, allows a reduction of local contact problems between fibers to 1-dimensional problems, which reduces numerical computations significantly. A fiber composite material with periodic microstructure and multiple frictional microcontacts between fibers is studied. The textile is modeled by introducing small geometrical parameters: the periodicity of the microstructure and the characteristic diameter of fibers. The contact linear elasticity problem is considered. A two-scale approach is used for obtaining the effective mechanical properties. The algorithm using asymptotic two-scale homogenization for computation of the effective mechanical properties of textiles with periodic rod or fiber microstructure is proposed. The algorithm is based on the consequent passing to the asymptotics with respect to the in-plane period and the characteristic diameter of fibers. This allows to come to the equivalent homogenized problem and to reduce the dimension of the auxiliary problems. Further numerical simulations of the cell problems give the effective material properties of the textile. The homogenization of the boundary conditions on the vanishing out-of-plane interface of a textile or fiber structured layer has been studied. Introducing additional auxiliary functions into the formal asymptotic expansion for a heterogeneous plate, the corresponding auxiliary and homogenized problems for a nonhomogeneous Neumann boundary condition were deduced. It is incorporated into the right hand side of the homogenized problem via effective out-of-plane moduli. FiberFEM, a C++ finite element code for solving contact elasticity problems, is developed. The code is based on the implementation of the algorithm for the contact between fibers, proposed in the thesis. Numerical examples of homogenization of geotexiles and wovens are obtained in the work by implementation of the developed algorithm. The effective material moduli are computed numerically using the finite element solutions of the auxiliary contact problems obtained by FiberFEM.