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The work consists of two parts.
In the first part an optimization problem of structures of linear elastic material with contact modeled by Robin-type boundary conditions is considered. The structures model textile-like materials and possess certain quasiperiodicity properties. The homogenization method is used to represent the structures by homogeneous elastic bodies and is essential for formulations of the effective stress and Poisson's ratio optimization problems. At the micro-level, the classical one-dimensional Euler-Bernoulli beam model extended with jump conditions at contact interfaces is used. The stress optimization problem is of a PDE-constrained optimization type, and the adjoint approach is exploited. Several numerical results are provided.
In the second part a non-linear model for simulation of textiles is proposed. The yarns are modeled by hyperelastic law and have no bending stiffness. The friction is modeled by the Capstan equation. The model is formulated as a problem with the rate-independent dissipation, and the basic continuity and convexity properties are investigated. The part ends with numerical experiments and a comparison of the results to a real measurement.
The manuscript divides in 7 chapters. Chapter 2 briefly introduces the reader to the elementary measures of classical continuum mechanics and thus allows to familiarize with the employed notation. Furthermore, deeper insight of the proposed first-order computational homogenization strategy is presented. Based on the need for a discrete representative volume element (rve), Chapter 3 focuses on a proper rve generation algorithm. Therein, the algorithm itself is described in detail. Additionally, we introduce the concept of periodicity. This chapter finalizes by granting multiple representative examples. A potential based soft particle contact method, used for the computations on the microscale level, is defined in Chapter 4. Included are a description of the used discrete element method (dem) as well as the applied macroscopically driven Dirichlet boundary conditions. The chapter closes with the proposition of a proper solution algorithm as well as illustrative representative examples. Homogenization of the discrete microscopic quantities is discussed in Chapter 5. Therein, the focus is on the upscaling of the aggregate energy as well as on the derivation of related macroscopic stress measures. Necessary quantities for coupling between a standard finite element method and the proposed discrete microscale are presented in Chapter 6. Therein, we tend to the derivation of the macroscopic tangent, necessary for the inclusion into the standard finite element programs. Chapter 7 focuses on the incorporation of inter-particle friction. We select to derive a variational based formulation of inter-particle friction forces, founded on a proposed reduced incremental potential. This contribution is closed by providing a discussion as well as an outlook.