Probabilistic homogenization of solid foams

The present study is concerned with a numerical procedure for prediction of uncertainties in the effective properties of solid foams. The approach is based on the multiple homogenization analysis of small-scale testing volume elements. Their microstructure is defined in terms of random variables with known probability distribution. Using a discretization of the space of the random variables, the probability distribution for the effective properties can easily be determined from the homogenization results and the probability for occurrence of the underlying microstructures of the testing volume elements. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mean and full field homogenization of artificial long fiber reinforced thermoset polymers

Based on two artificial microstructures representing a long fiber reinforced thermoset material, the effective linear elastic material properties are calculated by both a mean and a full field homogenization method. Concerning the mean field method, the effective elastic material properties are approximated using the homogenization scheme by Mori and Tanaka, formulated explicitly in terms of orientation averages. This allows to use orienation tensors of 2nd and 4th order describing the orientation information on the micro level. The full field method is based on the fast Fourier transformation (FFT), for which the effective material properties are determined by volume averaging. The comparison between both methods show good agreements, the deviations are in the range between 2% and 12%. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical assessment of disorder effects in metal foam core sandwich beams based on a local homogenization procedure

The present study is concerned with a numerical procedure for prediction of uncertainty effects in sandwich structures with disordered cores. The approach is based on probability distributions of different material properties and their spatial correlation which are the results of the multiple homogenization analysis of testing volume elements. In order to illustrate the essential difference in the results of material uncertainties between computations using random fields and a deterministic approach both methods are applied to a single edge clamped sandwich beam with a metal foam core which is loaded by a force at the free end. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ALE-based 3D FE simulation of electromagnetic forming

Electromagnetic forming is a contact-free high-speed forming process. The deformation of the work piece is driven by the Lorentz force which results from the interaction of a pulsed magnetic field with eddy currents induced in the work piece by the field itself. The purpose of this work is to present a fully-coupled three-dimensional simulation of this process. For the mechanical structure, a thermoelastic, viscoplastic, electromagnetic material model is relevant, which is incorporated in a large-deformation dynamic formulation. The electromagnetic fields are governed by Maxwell's equations under quasistatic conditions. To consider their reduced regularity at material interfaces Nédélec elements are applied. Coupling takes the form of the Lorentz force, the electromotive intensity and the current geometry of the work piece. A staggered solution scheme based on a Lagrangian mesh for the work piece and an ALE formulation for the electromagnetic field is employed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Numerical analysis of uncertainties in the effective material behaviour of disordered structural foams

The present contribution is concerned with a numerical analysis of the uncertainties in the structural response of threedimensional structural foams with partially open cells. The effective thermo mechanical material response is determined by means of an energy based homogenization procedure. Stochastic effects in the geometry and topology of the microstructure are treated by means of a repeated analysis of small-scale representative volume elements with prescribed relative density and prescribed cell size distribution. The results are evaluated by stochastic methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effective heat conduction of short fibre reinforced materials considering the fibre distribution

This contribution focuses on the effective heat conductivity of short fibre reinforced materials. For this purpose, a representative volume element (RVE), which is able to represent all possible fibre orientation distributions, is introduced and modelled in ABAQUS. Subsequently, the effective heat conductivity of the RVE is derived, employing a numerical homogenisation scheme, and a phenomenological material model is fitted to those results. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

热荷载作用下薄板的优化设计

板壳结构是一大类广泛使用的结构元件.在热荷载作用下,当热膨胀受到约束时,板壳结构产生内力及挠度,严重时影响结构的正常服役.由于热荷载的特殊性,简单地均匀加大板壳结构的厚度并不能有效地减少热变形和热应力,热结构设计因此特别困难.该文研究在给定材料体积的条件下,通过优化板壳结构的厚度分布来减少弹性薄板结构在热载荷下的变形.以结构的变形能为优化目标,在给定材料体积的条件下,建立了设计板壳结构厚度分布的优化问题列式,并采用变分法,推导出优化准则,给出了修改厚度的迭代公式.应用商用有限元软件的热结构分析功能,对程序进行二次开发,从而实现该优化算法.算例结果表明,采用该方法优化弹性薄板的厚度分布,可以大幅度地减小结构热变形,是一种有效的热结构设计方法.     

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)     

Modeling of microscopic failure in heterogeneous materials

The proper modeling of state-of-the-art engineering materials requires a profound understanding of the nonlinear macroscopic material behavior. Especially for heterogeneous materials the effective macroscopic response is amongst others driven by damage effects and the inelastic material behavior of the individual constituents [1]. Since the macroscopic length scale of such materials is significantly larger than the fine-scale structure, a direct modeling of the local structure in a component model is not convenient. Multiscale techniques can be used to predict the effective material behavior. To this end, the authors developed a modeling technique based on representative volume elements (RVE) to predict the effective material behavior on different length scales. The extended finite element method (XFEM) is used to model discontinuities within the material structure independent of the underlying FE mesh. A dual enrichment strategy allows for the combined modeling of kinks (material interfaces) and jumps (cracks) within the displacement field [2]. The gradual degradation of the interface is thereby controlled by a cohesive zone model. In addition to interface failure, a non-local strain driven continuum damage model has been formulated to efficiently detect localization zones within the material phases. An integral formulation introduces a characteristic length scale and assures the convergence of the approach upon mesh refinement [3]. The proposed method allows for an efficient modeling of substantial failure mechanisms within a heterogeneous structure without the need of remeshing or element substitution. Due to the generality of the approach it can be used on different length scales. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental Validation of RVE Based Failure Simulations of Macro-Porous Materials

Connections between inhomogeneities and the failure behavior of brittle material may be investigated by finite element simulations of representative volume elements. Representative volume elements are typically subjected to periodic boundary conditions. Moreover, representative volume elements are often chosen as planar, i. e., two dimensional in order to reach reasonable statistics with regard to random distributions of inhomogeneities. The significance of such strongly simplified simulations needs to be validated, especially if the matrix failure is potentially dominated by defects, as is the case, e. g., in macro-porous ceramics. We propose a quasi-periodic concept to design specimens with cylindrical pores, which reproduce the stress state in a two dimensional representative volume element. This is achieved by a partial periodic replication of the region of interest. We suggest that material models used in simulations can be assessed by comparison between simulated and experimentally observed failure. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A phase field model for fracture

The variational formulation of brittle fracture as formulated for example by Francfort and Marigo in [1], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of Γ-convergence has been presented by Bourdin in [2]. In the regularized Francfort–Marigo model cracks are represented by an additional field variable (secondary variable) s∈[0,1] which is 0 if the material is cracked and 1 if it is undamaged. In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg–Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of electrochemical machining using the boundary element method with no saturation

The simulation of electrochemical machining (ECM) is based on determining the surface shape at each point in time. The change in the shape of the surface depends on the rate of the electrochemical dissolution of the metal (conducting material), which is assumed to be proportional to the electric field strength on the boundary of the workpiece. The potential of the electric field is a harmonic function outside the two domains—the tool electrode and the workpiece. Constant potentials are specified on the boundaries of the tool electrode and the workpiece. A scheme with no saturation in which the strength of the electric field created by the potential difference on the boundary of the workpiece is proposed. The scheme converges exponentially in the number of grid elements on the workpiece boundary. Given the rate of electrochemical dissolution, the workpiece boundary, which depends on time, is found. The numerical solutions are compared with exact solutions, examples of the ECM simulation are discussed, and the results are compared with those obtained by other numerical methods and the ones obtained using ECM machines.     

On Variationally-Consistent Homogenization Approaches in Multi-Phase Magnetic Solids

It is well known that classical homogenization schemes, such as the Taylor/Voigt and Reuss/Sachs assumptions, can also be interpreted as energetic bounds. Furthermore, energy relaxation concepts have been established that determine stable effective material responses based on appropriate (convex, quasi-convex, rank-one) energy hulls for non-convex energy landscapes associated with multi-phase materials, see [1–3] and references therein. Our goal is to propose analogous relaxation based homogenization schemes for magnetizable solids. More specifically, we propose a magnetic potential perturbation scheme which yields relaxed effective free energy densities that simultaneously satisfy magnetic induction and magnetic field strength compatibility requirements—i.e. the magnetostatic Maxwell equations—at the phase boundary. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of reinforced concrete structures subjected to dynamic loads with a viscoplastic Drucker–Prager model

The behavior of reinforced concrete structures subjected to dynamic loads is analyzed. The concrete material is modelled by an elasto-viscoplastic law, whose inviscid counterpart is the Drucker–Prager model. A viscous regularization is introduced in order to avoid the mesh dependency effects that usually appear when strain softening occurs. The model is implemented in a general finite element computer code for fast transient analysis of fluid-structure systems, based on an explicit central difference scheme. The model is activated to both continuum elements and layered shell elements. So, realistic numerical analyses of complex 3-D engineering problems are simple and efficient. Three examples, two of which are modelled with layered shell elements, are presented below.     

About the Microstructural Effects of Polycrystalline Materials and their Macroscopic Representation at Finite Deformation

In the sheet bulk metal forming field, the strict geometrical requirements of the workpieces result in a need of a precise prediction of the material behaviour. The simulation of such forming processes requires a valid material model, performing well for a huge variety of different geometrical characteristics and finite deformation. Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically observable plastic deformation is traced back to dislocations within considered slip systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level. A finite crystal plasticity model is used to model polycrystalline materials in representative volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the elastic contribution. In order to circumvent singularities stemming from the linear dependency of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the plastic slips and slip resistances. The model is validated with experimental microstructural data under deformation. The validation on the macroscopic scale is performed through the reproduction of the experimentally calculated initial yield surface. Additionally, homogenised stress-strain curves from the microstructure build the outcome for a suitable effective material model. Through optimisation techniques, effective material parameters can be determined and compared to results from real forming processes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Martensitic transformations and damage: A combined phase field approach

A combined continuum phase field model for martensitic transformations and damage is introduced. The present approach considers the eigenstrain within the martensitic phase which leads in the surrounding material to both tensile and compressive stresses. The damage model needs to account for an appropriate differentiation thereof, since compressive stresses should not promote fracture. Interactions between micro crack propagation and the formation of the martensitic phases are studied in two dimensions. In agreement with experimental observations, martensite forms at the crack tip and influences the crack formation. For the numerical implementation finite elements are used while for the transient terms an implicit time integration scheme is employed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional mixed finite element-finite volume approach for the solution of density-dependent flow in porous media

The density-dependent flow and transport problem in groundwater on three-dimensional triangulations is solved numerically by means of a mixed hybrid finite element scheme for the flow equation combined with a mixed hybrid finite element-finite volume (MHFE-FV) time-splitting-based technique for the transport equation. This procedure is analyzed and shown to be an effective tool in particular when the process is advection dominated or when density variations induce the formation of instabilities in the flow field. From a computational point of view, the most effective strategy turns out to be a combination of the MHFE and a spatially variable time-splitting technique in which the FV scheme is given by a second-order linear reconstruction based on the least-squares minimization and the Barth–Jespersen limiter. The recent saltpool problem introduced as a benchmark test for density-dependent solvers is used to verify the accuracy and reliability of this approach.     

An alternative perspective on the concept of stress in classical continuum mechanics

Within a variational formulation of continuum mechanics, as proposed for instance by Germain [1], the internal virtual work contribution of a continuum is postulated as a smooth density integrated over the deformed configuration of the body. In this smooth density the stress field appears as dual quantity to the gradient of the virtual displacement field. Since the mathematical definition of the volume integral naturally provides a macro-micro relation between infinitesimal volume elements and the continuous body, we propose in this paper an alternative definition of stress on the micro level of the infinitesimal volume elements. In particular, the stress is defined as the internal forces of the body that model the mutual force interaction between neighboring volume elements. The existence of the stress tensor on the macro level is then obtained from the summation of all virtual work contributions within the body, followed by a limit process in which the volume elements are sent to zero. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)