Mean field homogenization and experimental investigation of short and long fiber reinforced polymers

The effective elastic material properties of short fiber reinforced polypropylen are determined by means of the self-consistent (SC) method and the interaction direct derivative (IDD) method. In order to account for thermoelastic effective material properties, a Hashin-Shtrikman (HS) based two-step homogenization method with variable reference stiffness is used. The influence of the reference stiffness, dependent on a scalar parameter is investigated. Information on the microstructure are derived by computed tomography scans (µCT) and considered within the homogenization schemes. Thermomechanical properties of a long fiber reinforced polymer (LFRP) and a short fiber reinforced polymer (SFRP) are obtained by means of dynamic mechanical analysis (DMA). Simulation results for SFRP are compared to experimental results. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

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)     

Experimental investigation and approximation of the temperature-dependent stiffness of short-fiber reinforced polymers

In order to predict the effective material properties of a short-fiber reinforced polymer (SFRP), homogenization of elastic properties with the self-consistent (SC) scheme and the interaction direct derivative (IDD) method is performed by means of µCT data describing the microstructure of the composite material. Using dynamic mechanical analysis (DMA), the material properties of both, polypropylene and fiber reinforced polypropylene are investigated by tensile tests under thermal load. The measured storage modulus of the matrix material is used as input parameter for the homogenization scheme. The effective properties of SFRP are compared to experimental results from DMA. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Multiscale modeling of continuous fiber and fabric reinforced rubber components

Fabric or continuous fiber reinforced rubber components (e.g. tires, air springs, industrial hoses, conveyor belts or membranes) are underlying high deformations in application and show a complex, nonlinear material behavior. A particular challenge depicts the simulation of these composites. In this contribution we show the identification of the stress and strain distributions by using an uncoupled multiscale modeling method, see [1]. Within this method, two representation levels are described: One, the meso level, where all constituents of the composite are shown in a discrete manner by a representative volume element (RVE) and secondly, the macro level, where the structural behavior of the component is defined by a smeared anisotropic hyperelastic constitutive law. Uncoupled means that the RVE does not drive the macroscopic material behavior directly as in a coupled approach, where a RVE boundary value problem has to be solved at every integration point of the macro level. Thus an uncoupled approach leads to a tremendous reduction in numerical effort because the boundary value problem of a RVE just has to be solved at a point of interest, see [1]. However, the uncoupled scale transition has to fulfill the HILL–MANDEL condition of energetic equivalence of both scales. We show the calibration of material parameters for a given constitutive model for fiber reinforced rubber by fitting experimental data on the macro level. Additionally, we demonstrate the determination of effective properties of the yarns. Finally, we compare the energies of both scales in terms of compliance with the HILL–MANDEL condition by using the example of a biaxial loaded sample and discuss the consequences for the mesoscopic level. (© 2014 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)     

Microscale modelling and homogenization of fiber structured materials

Various phenomena occurring on the macrosscale result from physical and mechanical behaviour on the microscale [1]. For the mechanical modeling and simulation of the heterogeneous composition of fiber structured material, in addition to the material properties the contact between the fibers has to be taken into account. The material behaviour is strongly influenced by the material properties of the fiber, but also by the geometrical structure. Periodically arranged fibers like woven, knitted or plaited fabrics and randomly oriented ones like fleece can be distinguished in their arrangement. In consideration of different lengthscales the problem involves, it is necessary to introduce a multiscale approach based on the concept of a representative volume element (RVE). The macro-micro scale transition requires a method to impose the deformation gradient on the RVE by suited boundary conditions. The reversing scale transition, based on the HILL-MANDEL condition, requires the equality of the macroscopic average of the variation of work on the RVE and the local variation of the work on the macroscale [2]. For the micro-macro transition the averaged stresses have to be extracted by a homogenization scheme. From these results an effective material law can be derived. Beside the theoretical aspects, we present the stress-strain relation for RVE-models and different boundary conditions. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some basic ideas for the reconstruction of statistically similar microstructures for multiscale simulations

In this contribution we propose a method for the generation of statistically similar representative volume elements (RVE's). These RVE's are obtained by assuming that given real microstructures may be represented by periodic ones. Then an optimization problem has to be solved where the side condition of equal spectral density of the real microstructure and the statistically similar RVE is taken into account. First numerical results show that the proposed method works in principle and leads to more efficient direct multiscale simulations. (© 2008 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)     

Determination of Effective Properties of MMC by Computational Homogenization

On the macrolevel Metal Matrix Composites (MMCs) resemble a homogeneous material. However, on the microlevel they show an inhomogeneous microstructure. This paper will have how heterogeneities affect the overall properties and the behaviour of a material (i. e. the effective properties). This is done using computational homogenization techniques. Finite element (FE) simulations were conducted in ABAQUS in connection with MATLAB, using material parameters for aluminium alloy AA2124 and SiC to develop a representative volume element (RVE) of the MMC AMC217xe. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

FE-implementation of plane strain gradient elasticity constitutive equations obtained by homogenization

This paper presents a plane strain gradient elasticity model together with a strength criterion based on the maximum hoop stress. It is developed by higher order homogenization based on a cylindrical RVE containing a void and is implemented as mixed–type finite element formulation into the FE–code Abaqus. Numerical simulations are performed in order to study the size effect related to the onset of failure of specimens of porous elastic material. (© 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)     

Spatial Random Material Property Model for Vibration of Composites

Investigation of vibration and buckling of thin walled composite structures is very sensitive to parameters like uncertain material properties and thickness imperfections. Because of the manufacturing process and others, thin walled composite and other structures show uncertainties in material properties, and other parameters which cannot be reduced by refined discretization. These parameters are mostly spatial distributed in nature. Here I introduce a semivariogram type material property model to predict the spatial distributed material property (like young's modulus) over the structure. The computation of semivariogram parameters needs the local material properties over a prespecified gird. The material properties at each grid have been obtained by considering a statistically homogeneous representative volume element (RVE) at each gird. According to random nature of the spatial arrangement of fibers, the statistically homogeneous RVE is obtained using image processing. The effective material properties of the RVE have been obtained numerically with the help of periodic boundary condition. The methodology is applied to a composite panel model and modal analysis has been carried. The results of the modal analysis (eigen values and mode shapes) are compared with experimental modal analysis results which are in good agreement. Using the presented material property model we can better predict the vibration characteristics of the thin walled composite structures with the inherent uncertainties. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromechanical Approach of Martensitic Transformation Problem in Steels

To determine the mechanical behavior of material involving the martensitic phase transformation (for example, steels like 100Cr6), a representative volume element (RVE) model including phase transformation criterion is desireable at micromechanical approach. A framework combining the Eshelby's inclusion theory as well as continuum mechanics with phase-transformation (PT) critical condition at RVE model is presented briefly. And application of this model to estimate the critical aspect ratio of martensitic plate or lath inside homogeneneous stress field is also included, where the RVE can be under uniaxial tension/compression or pure shear loading case. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling additive manufactured materials using a crystal plasticity model

In this contribution the macroscopic behaviour of additively manufactured Inconel 718 will be identified using a crystal plasticity model on the meso-level. The experimentally observed grain structures will be used to generate a representative volume element. Based on this RVE macroscopic mechanical parameters will be identified and compared with experimental results. (© 2016 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)     

Designing Statistically Similar RVEs for 3D Dual Phase Steel Microstructures

This contribution presents a method for the construction of three-dimensional Statistically Similar Representative Volume Elements (SSRVEs) for dual phase steels (DP steels). From such kind of advanced high strength steels, enhanced material properties are observed, which originate in the interaction of the individual constituents of the material on the microscale. Our aim is to directly incorporate the microstructure in the material modeling, which can be accomplished by applying i. e. the FE² method. A RVE representing the real material is used in the microscopic boundary value problem, which is solved at each macroscopic integration point. Since such RVEs usually exhibit a high complexity due to the underlying real microstructure, high computational costs are a drawback of the approach. We replace this RVE with a SSRVE, which has a lower complexity but which is still able to represent the mechanical behavior of the RVE and thus of the real microstructure. Virtual experiments show the performance of the method. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)