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)     

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)     

Numerical Prediction of Macroscopic Material Failure

The aim of this contribution is the numerical determination of macroscopic material properties based on constitutive relationships characterising the microscale. A macroscopic failure criterion is computed using a three dimensional finite element formulation. The proposed finite element model implements the Strong Discontinuity Approach (SDA) in order to include the localised, fully nonlinear kinematics associated with the failure on the microscale. This numerical application exploits further the Enhanced–Assumed–Strain (EAS) concept to decompose additively the deformation gradient into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. This finite element formulation is then used for the modelling of the microscale and for the discretisation of a representative volume element (RVE). The macroscopic material behaviour results from numerical computations of the RVE. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of the effective nonlinear material behaviour of composites using X-FEM - Analysis of the matrix material behaviour

For a consequent 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, the modelling approach using numerical homogenization techniques based on the simulation of representative volume elements which are modelled by the extended finite element method (X–FEM) is currently extended to nonlinear material behaviour. While the glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law accounts for strain–rate dependence and inelastic deformation of the matrix material. This paper describes the process of finding suitable constitutive relations for the polymeric matrix material Polypropylene in the small–strain regime. (© 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)     

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)     

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)     

Micromechanical strength analysis of fiber reinforced plastics

Strength of an unidirectional lamina is computed with a representative volume element. An approximative “voxel” meshing method is used in conjunction with continuum damage mechanics to simulate crack growth in the RVE. Two material models for the nonlinear material behaviour of the epoxy resin are compared. (© 2006 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)     

Micro-macro modelling of fiber reinforced composites exhibiting elastoplastic deformation

Fiber reinforced plastics such as carbon fiber-reinforced composites are typically characterized by their high siffness to weight ratio making them particularly attractive in lightweight construction. In addition, the architecture of these materials means that the correct modelling of their orthotropy is very important. In this work, volume averaged stress-strain responses are generated from a micro representative volume element (RVE). A nonlinear macro constitutive material model accounting for anisotropic plasticity is proposed. The model is fitted and compared to the micro stress-strain response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational modeling of shape memory fibers in conforming and non-conforming compound structures

In this work a material model for shape memory alloy (SMA) fibers is presented. A constitutive model is provided which aims for computational use. The presented model incorporates all relevant material nonlinear phenomena. It takes pseudoplasticity into account as well as pseudoelasticity and further the shape memory effect (SME). The constrained SME (CSME) and the two-way SME are covered by the presented material model. The constitutive model is implemented in a one-dimensional truss formulation and in a 3D-rebar element. Both formulations are used to model fiber composite structures. Those are described by the use of a non-conforming and a conforming mesh on the mesoscale. The numerical examples show the capability of the formulation. Different meshing strategies for the fiber–matrix compound are discussed. (© 2013 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)     

Topological sensitivity analysis of a multi‐scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi‐scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non‐penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth‐order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi‐scale framework changes when a small circular inclusion is introduced at the micro‐scale level. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical study in stochastic homogenization for elliptic partial differential equations: Convergence rate in the size of representative volume elements

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.     

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)     

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)     

Estimation of material properties of cancellous bone using multiscale FEM

Cancellous bone is a spongy type of bone with voids filled by blood marrow. Without much loss of generality it can be modeled as a material with periodic microstructure where overall parameters can be calculated using homogenization methods. Here the multiscale finite element method is applied and the assumed representative volume element (RVE) is a cube with solid frame and fluid core. From the point of view of the finite element method the RVE is a combination of solid and shell elements. As the acoustic excitation is considered, a complex stiffness matrix and complex displacements appear in the solution of the problem. Calculation of overall properties is repeated for different geometries of the solid frame. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Microstructural Model of a Closed-Cell Foam on the Basis of Image Analysis

The focus of this work is the identification of a unit cell that is able to represent the microstructure of a closed-cell solid foam to predict the effective behaviour of the foam numerically. For the investigation, a finite element model consisting of a repeating unit cell with periodical boundary conditions is implemented. A tetrakaidecahedral foam microstructure is considered as simplified cell geometry, and a strain-energy based homogenisation concept is utilized. On the basis of image analysis imperfections are applied to the cell. The obtained model is used as a representative volume element (RVE) for further investigations of the postbuckling behaviour of the foams. Different analyses are performed and the results are compared to literature and experimental data. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)