Multiscale Modelling of the Effective Viscoplastic Material Behaviour of Textile-Reinforced Polymers Using XFEM

In this contribution a modelling approach using numerical homogenisation techniques is applied to predict the effective nonlinear material behaviour of composites from simulations of a representative volume element (RVE). Numerical models of the heterogeneous material structure in the RVE are generated using the eXtended Finite Element Method (XFEM) which allows for a regular mesh. Suitable constitutive relations account for the material behaviour of the constituents. The influence of the nonlinear matrix material behaviour on the composite is studied in a physically nonlinear FE simulation of the local material behaviour in the RVE ­ effective stress-strain curves are computed and compared to experimental observations. The approach is currently augmented by a damage model for the fibre bundle. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of micro-structure on fibre push-out tests

In fibre reinforced materials, the interface toughness has a significant influence on strength and damage resistance. Especially in discontinuous fibre reinforced composites, where high densities of fibre ends are apparent, it has been shown that additives which improve the interfacial toughness can increase the effective strength of the materials [1]. Due to the fact that interface properties are strongly dependent on the manufacturing process, only experimental techniques providing the possibility to take these influences into account, are promising for an integrated material characterization. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective Dynamic Material Properties of Foam-like Microstructures

The effective material parameters of a microstructured material can be found using homogenization procedures based on calculations of a Representative Volume Element (RVE) of the material. In our approach the RVE is calculated in frequency domain and inertia is taken into account, leading to a frequency dependent behavior of the RVE.With the frequency response of the RVE, effective dynamic properties of the material are calculated using an optimization procedure. Due to the frequency dependent material behavior on the microscale a viscoelastic constitutive equation is applied on the macroscale. An example calculation is presented for an auxetic 2-D foam-like microstructure which is modelled as a plane frame structure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the single fibre pullout and pushout problem: Effect of fibre anisotropy

Based on our recent theoretical model of the interfacial debonding and fibre pullout/pushout stresses, an improved analysis including the effect of fibre anisotropy has been performed for single fibre pullout and pushout. During fibre-matrix debonding the debonded region is considered as an interfacial crack and its extension is dependent on a fracture energy criterion being satisfied. The elastic properties of the fibre are taken to be transversely isotropic about the fibre axis and the matrix is assumed to be isotropic. The results show that the maximum debond stress for fibre pullout and the initial frictional pullout stress increase as the ratio of the Young's moduli of the fibre between the axial and radial directionsE _f /E _f ^r increases, and this is particularly so at large embedded fibre lengths. However, in the case of fibre pushout these debond and pushout stresses decrease asE _f /E _f ^r increases.     

Homogenization of fibre composites using X-FEM

A successful material design process for novel textile reinforced composites requires an integrated simulation of the material behaviour and the estimation of the effective properties used in a macroscopic structural analysis. In this context the Extended Finite Element Method (X-FEM) is used to model the behavior of materials that show a complex structure on the mesoscale efficiently. A homogenization technique is applied to compute effective macroscopic stiffness parameters. This contribution gives an outline of the implementation of the X-FEM for complex multi-material structures. A modelling procedure is presented that allows for the automated generation of an extended finite element model for a specific representative volume element. Furthermore, the problem of branching material interfaces arising from complex textile reinforcement architectures in combination with high fibre volume fractions will be addressed and an appropriate solution is proposed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective elastic properties of particulate-reinforced compossite materials

A numerical approach for determination of the effective properties of particulate composite materials has been developed. A representative volume element (RVE) of the composite material is analyzed with help of the finite-element method. Uniform boundary displacements or tractions are applied on the boundaries of the RVE for introducing the known average strain in the RVE. Local stress and strain distributions in the RVE are calculated using the finite-element method. Different effective elastic constants can be calculated by averaging the local fields corresponding to different sets of boundary conditions. The present approach allows us to determine the effective properties of particle-reinforced composites with acceptable accuracy. The calculated effective properties of the composite are between the upper and lower Hashin—Shtrikman bounds. The results based on the present approach lead to higher stiffness of composites in comparison with analytical approaches.Institute fur Werkstoffwissenschaften, Fachberech Werkseoffwissenschaften, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 450–459, July–August, 1997.     

Aspects of computational homogenisation of microheterogeneous materials including decohesion at finite strains

During the last years, the development and application of new composite materials gained more and more importance. For engineering applications it is necessary to get effective material properties of such materials. In this contribution we present some aspects of computational homogenisation procedures of microheterogeneous materials which can show decohesion in a cohesive zone around the particles. Due to the decohesion we get finite deformations and .nite strains within the RVE. The geometrical and material nonlinearities cause the main dif.culties. The homogenization procedure leads to an effective stress strain curve for the RVE, and for the nonlinear elastic case one can also obtain effective material parameters. It is necessary to do statistical tests in order to get a representative result. Here we set a special focus on the adaptive numerical model, the statistical testing procedure and the different boundary conditions (pure tractions and pure displacements) applied on the RVE. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite presentability of normal fibre products

We use Bieri–Neumann–Strebel–Renz invariants to determine when a normal fibre product in the product of two finitely presented groups is finitely presented. We give conditions that imply and in some cases characterize the existence of such finitely presented fibre products.     

A superstatistical model for anomalous heat conduction and diffusion

We propose a superstatistical model for anomalous heat conduction and diffusion, which is formulated by the thermal conductivity distribution, overall temperature and heat flux distributions. Our model obeys Fourier's law and the continuity equation at the individual level. The evolution of the thermal conductivity distribution is described by an advection-diffusion equation. We show that the superstatistical model predict anomalous behaviors including the time-dependent effective thermal conductivity and slow long-time asymptotics. The time-dependence of the effective thermal conductivity is determined by the mean square displacement (MSD), which coincides with existing investigations. The superstatistical structure can also be extended into other non-Fourier models including the Cattaneo and fractional-order heat conduction models.     

Consistent macroscopic tangent computation for FFT-based computational homogenization at finite strains

The present work addresses the efficient computation of effective properties of periodic microstructures by the use of Fast Fourier Transforms. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of an RVE, the computation of the associated moduli is not straight-forward. The contribution of the present paper is thus the derivation and implementation of an algorithmically consistent macroscopic tangent operator that comprises the effective properties of the RVE. In contrast to finite-difference based approaches, an exact solution for the macroscopic tangent is derived by means of the classical Lippmann-Schwinger equation. The problem then reduces to the solution of a system of linear equations even for nonlinear material behaviour. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Profinite rigidity for Seifert fibre spaces

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.     

On estimates for the effective shear modulus of cubic crystal aggregates

To avoid time–consuming computations, several analytical approaches can be used to estimate the elastic properties of polycrystalline aggregates. In this paper we compute some well–known bounds and estimates for the effective shear modulus of aggregates of cubic crystals and compare them with results from finite element simulations using a representative volume element (RVE). It is shown that among the evaluated approaches, the singular approximation results in the best agreement with the RVE simulations for the examined materials. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of the nonlinear matrix material behaviour on the effective properties of textile-reinforced thermoplastics

For a consistent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain-rate effects on the mesoscale, is required. Therefore, a modelling approach using numerical homogenization techniques is applied to predict the effective nonlinear material behaviour of the composite based on the finite element simulation of a representative volume element (RVE). In this RVE suitable constitutive relations account for the material behaviour of each constituents. While the reinforcing glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law is applied to represent the strain-rate dependent, inelastic deformation of the matrix material. In order to analyse the influence of the nonlinear matrix material behaviour on the global mechanical response of the composite, effective stress-strain-curves are computed for different load cases and compared to experimental observations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical homogenization of composite structures with rhombic fiber arrangements

A numerical procedure is developed to determine effective material properties of unidirectional fiber reinforced composites with rhombic fiber arrangements. With the assumption of a periodic micro structure a representative volume element (RVE) is considered, where the phases have isotropic or transversely isotropic material characterizations. The interface between the phases is treated as perfect. The procedure handles the primary non-rectangular periodicity with homogenization techniques based on finite element models. Due to appropriate boundary conditions applied to the RVE elastic effective coefficients are derived. Six different boundary condition states are required to get all coefficients of the stiffness tensor. Results are listed and compared with other publications and good agreements are shown. Furthermore new results are presented, which exhibit the orthotropic behavior of such composites caused by the rhombic fiber arrangement. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Curvature homogeneous metrics on principal fibre bundles

Summary We consider a principal fibre bundle P=P(M, , G) with one-dimensional fibre G over a curvature homogeneous Riemannian manifold (M,g) and we give a list of sufficient conditions for some distinguished metric on P to be curvature homogeneous. We give examples to illustrate the result.This work was partially supported by MPI 40% and GNSAGA of the CNR.     

On the fibre product of incidence structures

It is shown that in the category of incidence structures a fibre product (pull back) always exists. Several applications to Klingenberg incidence structures and divisible designs are given.Dedicated to Professor Johannes Böhm on the occasion of his 70^th birthday     

A RVE-based micro mechanical model for short fiber reinforced plastic materials including matrix damage and fiber fracture

The representative volume element (RVE) method is applied to a fiber reinforced polymer material undergoing matrix damage and fiber fracture. Results of RVE computations are compared to uniaxial tensile tests performed with the composite material. It is shown that the macroscopic behavior of the composite material can accurately be predicted by RVE computations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Triangulation of stratified fibre bundles

We prove that for any stratified fibre bundle p:A·M (A being the underlying space of an abstract prestratification and M a smooth manifold) and any triangulation of M there exists a triangulation of A such that p becomes linear with respect to these triangulations. In particular, any abstract prestratification is triangulable. As a corollary we obtain that the orbit space of a smooth action of a compact Lie group is triangulable.This paper was written while the author was a visiting professor at the Institute of Mathematics of the University of Genova.