Probabilistic homogenization of hexagonal honeycombs with perturbed microstructure

The present study is concerned with a numerical determination of the effective mechanical properties of hexagonal honeycombs with irregular random cell geometry. Based on a regular hexagonal model, a perturbation technique is employed for generation of randomized microstructures by repositioning of the cell wall intersections within prescribed areas. Using a large number of numerical experiments, the entire set of testing volume elements is statistically representative for the random microstructure of the honeycomb material. Subsequently, the effective mechanical properties of the microstructural model are determined by means of a strain energy based homogenization procedure. Both the effective stiffness and the effective strength are examined. The stochastic information about the scatter in the effective properties is gained from repeated numerical experiments on small scale testing volume elements for the microstructure. Compared to a single analysis of a large scale, statistically representative volume element, the repeated analysis of small scale testing volume elements proves to be rather efficient. Furthermore, the statistical distributions of the effective properties can be determined in addition to their mean values.     

Numerical prediction of disorder effects in solid foams using a probabilistic homogenization scheme

The present study is concerned with a numerical prediction of uncertainties in the macroscopic mechanical properties of microheterogeneous materials with uncertain microstructure. As a model material, solid foams are employed. The stochastic information on the uncertainty is gained in multiple numerical homogenization analyses of small-scale testing volume elements. The local relative density, the cell size distribution, the cell geometry and the spatial orientation of the testing volume elements are assumed to form the set of the relevant stochastic variables. Selected microstructural cases are analyzed for their macroscopic material response. Based on the probability distributions for the stochastic variables defining the microstructures of the testing volume elements, the probability distributions for the mesoscopic material properties are obtained. For the numerical homogenization of the testing volume elements, an enhanced finite element technique is employed, where the components of the macroscopic deformation gradient are introduced as generalized degrees of freedom. Assuming periodic boundary conditions, the global degrees of freedom interact with the conventional displacement degrees of freedom of the discretized microstructure via special boundary coupling elements. The mesoscopic stresses are obtained in a rather efficient manner as the generalized reaction forces for the global degrees of freedom.     

Strain-energy based homogenisation of two- and three-dimensional hyperelastic solid foams

A possible classification of cellular solids can be made based on the dimension into honeycombs and foams. In numerical simulations 2D models that are employed primarily to study honeycombs can also be used to model open-cell foams. Thereby, a loss of information regarding the 3D connectivity of the microstructure is involved. To answer the question how the missing third dimension in 2D models affects the overall properties, spatially periodic 2D and 3D model foams are adopted. From the point of homogenisation, a strain-energy based scheme is used for adequately determining the effective mechanical properties at large strains. The key idea behind this method is to use directly the equivalence condition between the meso-strain energy and the macro-strain energy. In a first step a representative volume element with the given microstructure and a corresponding volume element containing the effective medium are subjected to equivalent states of deformation. Subsequently, the macroscopic stress-strain relationships are determined by volume-averaging of the stored strain energy. The results of some fundamental loading cases indicate that both model foams represent the deformation characteristics of hyperelastic solid foams like localized bending and elastic buckling. In addition, the development of anisotropy due to microstructural changes at large strains can be traced with both model foams. Nevertheless, the different cell morphology affects the stress-strain curves in a quantitative manner.     

Laguerre tessellations for elastic stiffness simulations of closed foams with strongly varying cell sizes

The dependency of the elastic stiffness, i.e., Young’s modulus, of isotropic closed-cell foams on the cell size variation is studied by microstructural simulation. For this purpose, we use random Laguerre tessellations which, unlike classical Voronoi models, allow to generate model foams with strongly varying cell sizes. The elastic stiffness of the model realizations is computed by micro finite element analysis using shell elements. The main result is a moderate decrease of the effective elastic stiffness for increasing cell size variations if the solid volume fraction is assumed to be constant.     

Aluminium foam–polymer composites: processing and characteristics

Dissemination of closed cell metal foam unique properties (low density, efficient energy absorption, high vibration/sound attenuation) in real life products has often been difficult to realise. With advanced pore morphology (APM) aluminium foam–polymer hybrids a new and simplified process route targeted at application in foam-filled structures (e.g. automotive A-pillar) has been introduced. APM foams are made from spherical, small volume foam elements joined to each other in a separate process step. Joining the aluminium foam elements by adhesive bonding delivers composite foam with approximately 80–95 wt.% aluminium foam and 5–20 wt.% adhesive (polymer). Setting up cellular structures from spherical foam elements allows for automatic part production, good pore morphology control and cost effective aluminium foam application. An automated production line is displayed and discussed. Mechanical properties of APM aluminium foam–polymer hybrids are similar to other closed cell aluminium foams. Integration of APM foams in profiles resulted in significantly improved properties as observed for conventional closed cell aluminium foam fillings. The unique properties of APM composite foams make them an attractive alternative as a cost effective and easily applicable material of construction with targeted uses such as energy absorbing reinforcement of composite structures.     

Aspects of the reproducibility of mechanical properties in Al based foams

Al foams have been manufactured via a PM route and compression tested. Testing has shown that density-properties relationships can be constructed which are then valid for the prediction of mechanical response for a sample of given density. The scatter in the density can also be used to predict, with reasonable confidence, the scatter in properties. Testing has shown that little or no difference in processing time can give rise to foams with significantly different densities and hence an undesirable, but nevertheless quantifiable and predictable, scatter in mechanical properties. This demonstrates the sensitivity of the very rapid foaming process and highlights the requirement for improving foam stability. The mechanical response of foams with similar densities is, however, reproducible suggesting that this is a more suitable way in which to control the process rather than by fixing the foaming time.     

Variability response functions for apparent material properties

The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions.     

Computation of permeability of a non-crimp carbon textile reinforcement based on X-ray computed tomography images

The paper demonstrates the possibility of a correct (within the experimental scatter) calculation of a textile reinforcement permeability based on X-ray micro-computed tomography registration of the textile internal architecture, introduces the image segmentation procedures to achieve the necessary precision of reconstruction of the geometry and studies variability of the geometry and local permeability. The homogenized permeability of a non-crimp textile reinforcement is computed using computational fluid dynamics with voxel geometrical models. The models are constructed from X-ray computed tomography images using a statistical image segmentation method based on a Gaussian mixture model. The computed permeability shows a significant variability across different unit cells, in the range of (0.5…3.5) × 10⁻⁴ mm², which is strongly correlated with the solid volume fraction in the unit cell.     

多孔材料辐射-传热耦合性能的统计二阶双尺度计算

对多孔材料辐射-传热耦合计算的数学模型, 即Rosseland方程, 给出了一种统计的二阶双尺度分析方法, 并针对典型问题进行了数值模拟。建立了考虑辐射项的统计二阶双尺度计算公式, 给出了统计意义下热流密度极值的预测算法, 并通过与理论解的比较对算法进行了验证, 利用本文中方法研究了孔洞体分比和空间分布状态对陶瓷多孔材料热传导系数、 辐射传导系数和热流密度极值的影响。结果表明: 孔洞体积分数的增加将导致有效热传导系数下降; 热流密度极值随孔洞体积分数的增加而变大, 并且在高温时辐射的作用明显增大; 数值试验表明, 使用统计二阶双尺度方法及其有限元算法预测孔洞随机分布复合材料结构的热性能是有效的。     

Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis

This paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a nonconcurrent multiscale approach. The geometry of the microstructure is random. The nonconcurrent multiscale approach for micro‐macro nonlinear mechanics is extended to the stochastic case. Because the nonconcurrent multiscale approach is based on the use of a tensorial decomposition, which is then submitted to the curse of dimensionality, we perform an analysis with respect to the stochastic dimension. The technique uses a database describing the strain energy density function (potential) in both the macroscopic Cauchy green strain space and the geometrical random parameters domain. Each value of the potential is numerically computed by means of the FEM on an elementary cell whose geometry is given by the random parameters and the corresponding macroscopic strains being prescribed as boundary conditions. An interpolation scheme is finally introduced to obtain a continuous explicit form of the potential, which, by derivation, allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations. Two numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Effect of scatter on reconstructed image quality in cone beam computed tomography: evaluation of a scatter-reduction optimisation function

The effect of scatter on reconstructed image quality in cone beam computed tomography was investigated and a function which can be used in scatter-reduction optimisation tasks was tested. Projections were calculated using the Monte Carlo method in an axially symmetric cone beam geometry consisting of a point source, water phantom and a single row of detector elements. Image reconstruction was performed using the filtered backprojection method. Image quality was assessed by the L2-norm-based difference relative to a reference image derived from (1) weighted linear attenuation coefficients and (2) projections by primary photons. It was found that the former function was strongly affected by the beam hardening artefact and did not properly reflect the amount of scatter but the latter function increased with increasing beam width, was higher for the larger phantom and exhibited properties which made it a good candidate for scatter-reduction optimisation tasks using polyenergetic beams.     

Computational homogenization of nonlinear elastic materials using neural networks

In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three‐dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two‐scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd.     

Strain rate dependent compression response of Ni-foam investigated by experimental and FEM simulation methods

Metal foams can be used as structural materials for impact energy absorption applications, due to the extended plateau stress they exert under compressive loads. The compressive behaviour of Ni-foams was studied by experimental and computational methods at various strain rates. The geometry of the porous material was reconstructed based on X-ray computed tomography measurements and used in a FEM simulation software package, facilitating large unconstrained plastic deformation, to determine its response under variable strain rates. SEM in situ compression tests were employed to measure the load-displacement response of the foam, while allowing the acquisition of images illustrating the deformed metal foam struts. The results of the study indicated that the introduced FEM model provides reliable insight with regard to the response of metal foams under various compressive strain rates. Additionally, the FEM model facilitates a holistic overview of the deformation phenomena occurring within the porous structure on both macro- and micro-scales.     

Indentation Plastometry of Particulate Metal Matrix Composites,Highlighting Effects of Microstructural Scale

Herein, it is concerned with the use of profilometry-based indentation plastometry (PIP) to obtain mechanical property information for particulate metal matrix composites (MMCs). This type of test, together with conventional uniaxial testing, has been applied to four different MMCs (produced with various particulate contents and processing conditions). It is shown that reliable stress–strain curves can be obtained using PIP, although the possibility of premature (prenecking) fracture should be noted. Close attention is paid to scale effects. As a consequence of variations in local spatial distributions of particulate, the “representative volume” of these materials can be relatively large. This can lead to a certain amount of scatter in PIP profiles and it is advisable to carry out a number of repeat PIP tests in order to obtain macroscopic properties. Nevertheless, it is shown that PIP testing can reliably detect the relatively minor (macroscopic) anisotropy exhibited by forged materials of this type.     

Effects of particle clustering on the tensile properties and failure mechanisms of hollow spheres filled syntactic foams: A numerical investigation by microstructure based modeling

Particle clustering originated from manufacturing process is thought to be one of the critical factors to the mechanical performance of hollow spheres filled syntactic foams. Although experimental evidence provides a qualitative understanding of the effects of particle clustering on the mechanical properties of syntactic foams, a quantitative assessment cannot be made in the absence of an appropriate micromechanical modeling strategy. In this study, three-dimensional microstructures of syntactic foams with different degrees of particle clustering were reconstructed based on random sequential adsorption (RSA) method. Three-phase finite element models considering the progressive damage behavior of the microsphere–matrix interface were accordingly developed by means of representative volume element (RVE) to quantitatively investigate the effects of particle clustering on the tensile properties and failure mechanisms of syntactic foams. The simulation results indicate that the elastic behavior of syntactic foams is insensitive to the degree of particle clustering, but the strength properties as well as the failure mechanisms are significantly influenced by the degree of particle clustering. From the micromechanical viewpoint, the clustered regions containing higher concentration of microspheres than the average volume fraction would serve as crack initiation sites due to stress concentration, and consequently lead to a negative effect on tensile strength, fracture strain, and interfacial damage of syntactic foams.     

Microscopic examination of the microstructure and deformation of conventional and auxetic foams

Auxetic materials have a negative Poisson’s ratio, that is, they expand laterally when stretched longitudinally. One way of obtaining a negative Poisson’s ratio is by using a re-entrant cell structure. Auxetic foam was fabricated from a conventional polymeric foam. Assuming similar mechanical properties for the solid material comprising the foams, the principle variable affecting the properties of the foam is the geometry of the cells. This means that the unusual mechanical properties of auxetic foams are attributed to the deformation characteristics of re-entrant microstructures. In this paper, the results of optical- and scanning electron-microscopic studies of the geometrical parameters for the different foams examined are presented. Examples of the microstructural deformation mechanisms observed are also presented. Comparison between the conventional foams and their auxetic conversions are also made. This revised version was published online in November 2006 with corrections to the Cover Date.