Effective parameters of cancellous bone

Cancellous bone is a two–component structure consisting of the bone frame and interstitial blood marrow. In the scope of this presentation, the multiscale finite element method is used for its modeling. This method results from a combination of homogenization theory and the theory of finite elements and is based on the calculation of effective material parameters by investigating representative volume elements (RVEs). For the particular kind of material considered here, a cubic two–phase RVE is assumed where the dry skeleton is modeled in different ways. Apart from the variations of the geometry, the influence of the usage of different types of finite elements is studied in this context. Note that the presence of a liquid phase requires dynamic investigation including the viscous phenomena. To this end, acoustic excitation and an analysis in the complex domain are chosen. The method permits calculation of the effective material parameters such as Young's modulus, bulk modulus and Poisson's ratio and furthermore the simulation of the behaviour of the complete bone or of its parts. (© 2008 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)     

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)     

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)     

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)     

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)     

A multi-mesh finite element method for Lagrange elements of arbitrary degree

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviors of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid–solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier–Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy way to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.     

四段式有限元法分析热耦合的流体-固体相互作用问题

给出了一种流(体)-热-结构综合的分析方法,固体中的热传导耦合了粘性流体中的热对流,因而在固体中产生热应力.应用四段式有限元法和流线逆风Petrov-Galerkin法分析热粘性流动,应用Galerkin法分析固体中的热传导和热应力.应用二阶半隐式Crank-Nicolson格式对时间积分,提高了非线性方程线性化后的计算效率.为了简化所有有限元公式,采用3节点的三角形单元,对所有的变量:流体的速度分量、压力、温度和固体的位移,使用同阶次的插值函数.这样做的主要优点是,使流体-固体介面处的热传导连接成一体.数个测试问题的结果表明,这种有限元法是有效的,且能加深对流(体)-热-结构相互作用现象的理解.     

A finite element method for steady-state conduction-advection phase change problems

A new finite element: technique is developed to solve steady-state conduction-advection problems with a phase change. The energy balance equation at the solid/liquid interface is employed to calculate the velocity of the solid/liquid interface in the Lagrangian frame. The position of the solid/liquid interface in the Eulerian frame is determined based on the composition of the velocity of the solid/liquid interface in the Lagrangian frame and the steady-state velocity of a rigid body. The interface position and the finite element mesh are continuously updated during an incremental process. No artificial diffusion is needed with this new finite element approach. An analytical solution for solidification of a pure material with a radiative boundary condition is also developed in this paper. Numerical experimentation is conducted and the corresponding results are compared with analytical solutions. The numerical results agree well with analytical solutions.     

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)     

Stability analysis of frame structures by bubble-function method

In the stability analysis of frame structures, the results by conventional finite element method (FEM) in which one member is taken as one element are sometimes unavailable. This paper took a new basic function system with bubble functions as the shape function of a bar element to develop a bubble function finite element method (BFEM), in which the bending and the geometric stiffness matrices were derived from the principle of virtual work. Bubble functions are finite element modes that are located entirely within a single element and are zero on boundaries of the element, but are nonzero at the other points. BFEM is as concise as conventional bar FEM but has better accuracy, and is adaptable to the buckling analysis of all kinds of frame structures. The use of bubble functions significantly improves the convergence of finite element analysis, and efficiently reduces the computation cost for the buckling analysis of frame structures. Numerical results show that using bubble functions in finite element for the stability analysis of structures is very efficient, especially for high-rise and large-scale frame structures.     

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)     

Extended multiscale FEM for design of beams and frames with complex topology

A numerical method for design of beams and frames with complex topology is proposed. The method is based on extended multi-scale finite element method where beam finite elements are used on coarse scale and continuum elements on fine scale. A procedure for calculation of multi-scale base functions, up-scaling and downscaling techniques is proposed by using a modified version of window method that is used in computational homogenization. Coarse scale finite element is embedded into a frame of a material that is representing surrounding structure in a sense of mechanical properties. Results show that this method can capture displacements, shear deformations and local stress-strain gradients with significantly reduced computational time and memory comparing to full scale continuum model. Moreover, this method includes a special hybrid finite elements for precise modelling of structural joints. Hence, the proposed method has a potential application in large scale 2D and 3D structural analysis of non-standard beams and frames where spatial interaction between structural elements is important.     

Computational Multi-Scale Stability Analysis of Periodic Electroactive Polymer Composites at Finite Strains

This work is dedicated to multi-scale stability analysis, especially macroscopic and microscopic stability analysis of periodic electroactive polymer (EAP) composites with embedded fibers. Computational homogenization is considered to determine the response of materials at macro-scale depending on the selected representative volume element (RVE) at micro-scale [4, 5]. The quasi-incompressibility condition is considered by implementing a four-field variational formulation on the RVE, see [7]. Based on the works [1–3, 6, 8] the macroscopic instabilities are determined by the loss of strong ellipticity of homogenized moduli. On the other hand, the bifurcation type microscopic instabilities are treated exploiting the Bloch-Floquet wave analysis in context of finite element discretization, which allows to detect the changed critical size of periodicity of the microstructure and critical macroscopic loading points. Finally, representative numerical examples are given which demonstrate the onset of instability surfaces, the stable macroscopic loading ranges, and further a periodic buckling mode at a microscopic instability point is presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An analysis of the finite element method for natural convection problems

We derive stability properties and error estimates for the finite element method when used to approximate heat flow in a fluid enclosed by a solid medium. The coupled Navier Stokes system involves the Boussinesq equations in the fluid-filled cavity linked through an interface with heat conduction in the solid enclosing the fluid. As we assume no extra regularity then can be shown to hold under mild restriction on the data (at least over a small time interval in R ³), we focus primarily on low order finite element spaces.     

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.     

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)