Open cell metal foams can be represented by a network of beams. Due to the heterogeneity of the geometry, the length scale of the representative volume element is often nearly of the same order as the length scale of structures made of metal foam. Therefore, classical homogenization techniques for the computation of effective properties can not be applied. Statistical volume elements lead to apparent material properties that depend on the boundary conditions. Here, we introduce a model for structures made of metal foam that consists of two domains, an interior region and a boundary region. For both regions, unique random fields are identified by simulations of the microstructure. The model is validated by comparison with Finite Element simulations and experiments.

We propose a procedure for the computation of natural frequencies for structures made of metal foam. Because the heterogeneity of the foam geometry has an influence on macroscopic properties, the irregular geometry of the foam has to be taken into account. This is done by adapting a Laguerre tessellation to statistical descriptors of the geometry obtained from CT image analysis. As the length scale of the representative volume element is nearly of the same order as the length scale of the structures under consideration, classical homogenization techniques for the computation of effective properties cannot be applied. Therefore, we introduce a stochastic homogenization method based on empirical marginal distribution functions and correlation functions of apparent properties. This information allows us to define random fields for elastic properties and the mass density on the macroscopic level. Statistical properties of the natural frequencies can then be inferred.

Due to their useful properties, metal foams became an interesting, often utilized and investigated material. Recent applications are in areas, where dynamic processes play a significant role. In the huge amount of literature about metal foam, mainly the material properties like strength and stability are investigated but the dynamic behavior is rarely the object of research. Therefore this work investigates the principal vibration behavior of heterogeneous metal foam by examining the eigenfrequencies of bending beams in dependence on the irregular microstructure of the foam.First of all the linear elastic properties of metal foam and their variations have to be investigated. Therefore, mesoscopic three dimensional stochastic volume elements are sampled including the effects of inhomogeneities like varying thickness along a ligament, pre-deformed ligaments, imperfections, partially closed cell faces or non-planar cell faces. In order to perform the step from the meso- to the macroscale, the mechanical properties are expressed as normally distributed random fields with a determined autocorrelation function or the power spectral density.These random fields are used to predict the eigenfrequencies of Timoshenko beams made of metal foam. Therefore the Karhunen-Loeve expansion and the Spectral Representation are derived analytically for the determined data of metal foam and used as two realization generators in Monte-Carlo-Simulations. The results are compared with experiments.

A metal foam may consist of a very heterogeneous structure, such that the size of the representative volume element is rather large. Therefore, macroscopic properties of components made of metal foams might show a large scatter.To predict the scatter of eigenfrequencies for bending beam structures, a consistent formulation from image analysis to the distribution of macroscopic properties is developed. With the help of computed tomography, statistical characteristics of the cell geometry of open cell foams are estimated. This information allows to fit a random tessellation model to the material, which reproduces the statistical properties of the cell geometry. To compute the linear elastic properties as well as the mass density of metal foams, three dimensional volume elements from random model realizations are analyzed and distributions of apparent properties are computed. The covariance function is estimated by considering volume elements at different locations of the macrostructure. Having a description of random fields for the apparent properties at hand, Monte Carlo simulations are applied to predict the eigenfrequencies, their scatter and the associated eigenforms of beams made of metal foams. The procedure is validated by experiments. (c) 2011 Elsevier Ltd. All rights reserved.

AbstractIn this contribution, a simulation chain for the prediction of the structural dynamics behavior of metal foam is proposed: Starting with investigations of the mesoscopic structure and topology, the linear elastic material properties and their scatter are computed with the help of mesoscopic non‐representative volume elements. Also their probability functions are estimated. In order to be able to describe the mechanical properties as random fields, the autocorrelation function and the power spectral density are determined.The next step in the proposed simulation chain is to generate realisations of the random fields with the help of the Karhunen‐Loeve expansion or the Spectral Representation. Afterwards, these realisations are used in Monte‐Carlo‐Simulations on the macroscale in order to predict the eigenfrequencies of beams and their scatter.With this multiscale approach, the mechanical properties of an open‐celled Copper foam are determined by simulations and the eigenfrequencies are compared to experimental results. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

AbstractIn this contribution, a way of simulating the influence of the mesoscopic irregular structure of metal foams on the macroscale is shown. To this end, mesoscopic periodic volume elements of a foam are derived in order to compute the mechanical properties including the effects of inhomogenities like imperfections, irregular structure and varying cell wall thicknesses. With the help of these volume elements, which are analysed via the finite element method, and their varying mechanical properties, a local varying stiffness can be computed and inserted into the macromechanical model. In this way the propagation of uncertainities from the mesoscale to the macroscale can be assessed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

AbstractDue to their useful properties in lightweight construction and due to their excellent behavior in energy absorption for example in crash mechanics, metal foams became an interesting, often utilized and investigated material. For the determination of the mechanical properties of foams without the help of expensive experiments, a way for computing these properties is searched. The problem in doing so is that foams can be composed out of randomly distributed edges and faces with varying thickness and of other inhomogeneities on the mesoscale like imperfections. The goal in this paper is, to investigate the influence of these irregularities on the mechanical, linear elastic properties of a metal foam on the macroscale and to determine the size of a representative volume element, for which the irregularities on the mesoscale do not have a great influence on the linear elastic properties. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

This article presents an analytical investigation on stability and bifurcation behaviour due to an exponential and a generalized friction characteristics in the sliding domain of a simple friction oscillator, which is commonly referred to as 'mass-on-a-belt' oscillator. The friction is described by a friction coefficient which depends on the relative velocity between the two tribological partners.The standard way of examining the steady-state only gives very rough insight in the behaviour and is not able to provide further informations about the steady-state's basin of attraction or about limit-cycles. It is found that the system may undergo bifurcations of Hopf type. Hereby, the character of the bifurcations strongly depends on the parameters of the friction characteristic.

This article presents an analytical investigation on stability and local bifurcation behavior due to exponentially decaying friction characteristics in the sliding domain of a simple friction oscillator, which is commonly referred to as "mass-on-abelt"-oscillator. Friction is described by a friction coefficient which in the sense of Stribeck depends on the relative velocity between the two tribological partners.For such a characteristic the stability and bifurcation behavior are discussed. It is shown, that the system can undergo a subcritical Hopf-bifurcation from an unstable steady-state fixed-point to an unstable limit cycle, which separates the basins of the stable steady-state fixed-point and the self-sustained stick-slip limit cycle.Therefore, only a local examination of the eigenvalues at the steady-state, as is the classical approach when investigating conditions for the onset of friction-induced vibrations, may not give the whole picture, since the stable region around the steady-state fixed-point may be rather small.Furthermore, the results of above considerations are applied to a brake-noise problem. It is found that, in contrast to squeal, a decaying friction characteristic may be a satisfying explanation for the onset low-frequency groan. The analytical results are compared with experimental measurements. (C) 2006 Elsevier B. V. All rights reserved.

AbstractThis article deals with different types of friction models and their influence on the behavior of a simple 1 degree‐of‐freedom (DOF) sliding friction oscillator which is in literature commonly referred to as “mass‐on‐a‐belt”‐oscillator. The examined friction characteristics are assumed to be proportional to the applied normal force and only dependend on the relative velocity between the mass and the belt. For an exponential and a generalized cubic friction characteristic, the linear stability of the steady‐state and the bifurcation behavior in the sliding domain are examined. It is shown that the resulting phase plots of the observed system are strongly dependent on the chosen friction characteristic. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Human romantic relationships are studied via system dynamics methodology. Starting point is a time-invariant linear model of two individuals without interaction with environment. Specifically, time-dependent fluctuations both in the source terms and the system parameters are introduced and examined in their consequences where also more realistic nonlinear modeling is proposed and analyzed.

This article deals with analytical investigations on stability and bifurcations due to declining dry friction characteristics in the sliding domain of a simple disc-brake model, which is commonly referred to as “mass-on-a-belt”-oscillator. Sliding friction is described in the sense of Coulomb as proportional to the normal force, but with a friction coefficient μS which depends on the relative velocity. For many common friction models this latter dependence on the relative velocity can be described by exponential functions. For such a characteristic the stability and bifurcation behavior is discussed. It is shown, that the system can undergo a subcritical Hopf-bifurcation from an unstable steady-state fixed point to an unstable limit cycle, which separates the basins of the stable steady-state fixed point and the self sustained stick-slip limit cycle. Therefore, only a local examination of the eigenvalues at the steady-state, as is the classical ansatz when investigating conditions for the onset of friction-induced vibrations, may not give the whole picture, since the stable region around the steady state fixed point may be rather small. The analytical results are verified by numerical simulations. Parameter values are chosen for a model which corresponds to a conventional disc-brake.