T. Daxner

Mesoscopic simulation of inhomogeneous metallic foams with respect to energy absorption

Abstract The subject of this work is metallic foams under crush and crash loading, the focus being on the influence of inhomogeneities of the apparent density on the maximum stresses and the energy absorption behavior during compressive deformation. Based on an analytical description of the uniaxial stress–strain relationship of cellular materials, which is fitted to experimental results, a relation between the effective density and the static compression behavior of a certain brand of metallic foams can be obtained. This relation is implemented into a mathematical model, which represents the material as an array of point masses connected by longitudinal, nonlinear springs and transversal, rigid cross-bridges, which can be opened or closed as required. Several distributions of mesoscale inhomogeneities are studied and assessed with respect to their influence on the energy absorption and impact damage protection potential. It is shown that only a mesoscopically homogeneous foam fully exploits the energy absorption potential of a foam of a given apparent density. The effects of inertia are shown by simulating impact events. The stress waves propagated and reflected in the homogenized foam material and their influence on the impact response are described.

Adaptation of density distributions for optimising aluminium foam structures

Abstract Metallic foams show some potential for being produced with controlled spatial variations in their density. This suggests employing them as graded materials in space filling lightweight structures designed in analogy to cortical bone, a natural cellular material, that displays increased density in regions of high loading. In the present study the influence of the mechanical properties of aluminium foams on the results of an optimisation of the foam density distribution with regard to structural strength and stiffness was examined. Regression formulae for the relationships between stiffness and strength of metallic foams on one hand and effective density on the other hand can be fitted to the results of uniaxial compression tests of a certain brand of metallic foam. These results and additional assumptions such as overall isotropy and a yield surface suitable for cellular materials can be implemented into a finite element program adapted for performing stiffness or strength optimisation on the basis of a density adaptation similar to the remodelling of bone. Some applications are presented that show how foams with gradients in the apparent density may be employed to obtain optimal structural behaviour for classical design problems.

Finite Element Modeling of Cellular Materials

Cellular materials are characterized by a low apparent, density and a discrete micro-structure which is on a distinctly lower length-scale than the one of components made from them. Consequently, the effective behavior of cellular materials is rooted in the mechanical behavior of the struts and/or cell walls on the length scale of individual cells. These lecture notes describe methods of modeling cellular materials by the finite element method. Topics include the setup of micro-mechanical models for open and closed-cell foams as well as sintered hollow sphere foam, the transition between the different mechanical length scales, and the optimization of the density distribution in components ma.de from functionally graded foam.

Micromechanical models of metallic sponges with hollow struts

Coating of a precursor structure, which is subsequently removed by chemical or thermal treatment, is a technology for producing cellular materials with interesting properties, for example in the form of metallic sponges with hollow struts. In this paper idealized models for determining the effective elastic properties of such materials are presented. The chosen models for the structures are space-filling, periodically repeating unit cell models based on idealized models of wet foams, which were generated with the program ‘Surface Evolver’. The underlying topology is that of a Weaire-Phelan structure. The geometry of the micro-structures can be described by two principal parameters, viz. the volume fraction of solid material in the precursor structures, which determines the shape of the final structures, and the thickness of the metallic coating, which defines their apparent density. The influence of these two parameters on the macro-mechanical behavior is investigated. The elastic properties of the micro-structures are described by three independent elastic constants owing to overall cubic material symmetry. The dependence of the effective Young’s modulus on the direction of uniaxial loading is investigated, and the elastic anisotropy of the structures is evaluated.

Numerical Simulations of the Creep Deformation of MMCs in 4-Point Bending Mode

The 4-point bending test is a widely used method to determine material parameters. No commonly accepted evaluation methodology is available for materials showing non-linear deformation mechanisms. In the present study micro- and macro-mechanical simulation models of continuously reinforced metal matrix composites are employed to investigate thermo-elasto-plasticity and creep in such experiments. The overall deflection behavior and the underlying mechanisms are identified revealing the interaction of various micromechanical phenomena. Comparisons to a set of experimental results are presented.

Optimization of Corrugated Paperboard under Local and Global Buckling Constraints

An important design criterion for containers (boxes) made from corrugated paperboard is their resistance against buckling under compressive loads such as those arising from gravity as boxes are piled up. Since such boxes are typically means of packaging goods for transport, their weight should be as low as possible. These demands are taken into account in the presented optimization procedure for reducing the area-specific weight of corrugated paperboard under global, i.e., box wall buckling constraints and local buckling constraints pertaining to the buckling of flute and liner. The critical load with respect to global buckling is correlated to the effective bending stiffness of the paperboard (obtained by homogenization). Local buckling is predicted by a unit cell approach in combination with the finite element method. The stiffness homogenization procedure as well as the unit cell approach for computing the buckling loads are embedded into an optimization process. The geometrical parameters describing the meso-scale geometry of the corrugated paperboard act as optimization parameters. The presented approach is applied to a specific configuration of corrugated paper board, as it is used in packaging. Substantial weight saving could be achieved by the proposed optimization scheme. A further consideration concerns the post-buckling behavior. Once the side walls of corrugated paperboard containers have buckled, they typically show the formation of folds. As demonstrated in non-linear finite element analyses, these folds are the result of the localization of the initially periodic local buckling pattern.

Analytical and Numerical Methods for Modeling the Thermomechanical and Thermophysical Behavior of Microstructured Materials

Basic application-related aspects of two important groups of approaches to continuum micromechanics of inhomogeneous materials are presented, viz., mean field schemes and methods based on discrete microstructures. Emphasis is put on handling both thermomechanical and thermal conduction problems.

On Micro/Meso/Macro Modeling of Composite Lightweight Materials and Structures

The effective, i.e. overall mechanical response of multi-phase or other microstructured materials is influenced by the topology and the geometry of the individual phases as well as by the mechanical behavior of the material and interface at lower length scales, i.e. meso-, micro- and submicroscales. The present contribution shows examples for the application of finite element methods to study problems related to the mechanical behavior of inhomogeneous materials by unit cell models and hierarchical micro-meso-macro concepts. After basic considerations of the formulation of proper unit cells and corresponding homogenization procedures, some typical applications to highly porous metals (metal foams) and composites are discussed. The main body of the paper deals with the calculation of the stiffness and the failure behavior of perforated composite laminates, used as so-called acoustic skins for sound absorbing components in airplane structures. Based on the above mentioned unit cell concepts effective stiffness properties as well as a macroscopic failure-initiation surface in the space of macro-membrane forces is determined for the use in structural analyses. Both inter- and intra-laminar failure as well as free edge effects are taken into account on the meso-level. In this way the deformation and strength evaluation of structures containing such perforated laminates can be performed simply on the structural, i.e. macroscopic level. This procedure is very efficient: the structural analysis does not need to account for of the perforation, neither in the finite element discretization nor in the evaluation of local stress fields.

Stability of rod-shaped nanoparticles embedded in an elastic matrix

In many biological tissues as well as in some technical materials we find nano-sized rod-shaped particles embedded in a relatively soft matrix. Loss of stability of equilibrium, i.e. buckling, is one of the possible failure modes of such materials. In the present paper different kinds of load transfer between matrix and reinforcing particles, which are typical for rod-shaped nanostructures in biological tissues, are considered with respect to stability of equilibrium. Two regimes of matrix stiffnesses leading to different modes of buckling, and a transition regime in between, have been found: soft matrix materials leading to the so-called ‘flip mode’ (also called ‘tilt mode’) and hard matrix materials resulting in ‘bending mode’ buckling. The transition regime is of particular interest for biological tissues. Numerical and semi-analytical as well as asymptotic concepts are employed leading to results for estimating the critical load intensities both in the form of closed form solutions and diagrams. The analytical solutions are compared with results of finite element analyses. From these comparisons indications are gained for deciding which of the different analytical approaches should be chosen for a particular nanostructure configuration in terms of the associated buckling modes.

Mechanical Properties of Semi-expanded Hollow Sphere Structures

Recently, technologies for the production of cellular materials have been proposed that allow for a seamless change of the geometries of such materials from being similar to sintered hollow sphere structures to resembling comparatively regular polyhedra. In order to investigate, how the linear elastic properties of such materials are effected by this change in geometry, analyses of the expansion process are carried out by the finite element method for obtaining cell wall thickness distributions. These material distributions are then transferred to finite element unit cell models that are suitable for predicting the effective, macroscopic elastic properties. It is found that Young’s modulus, as well as the shear modulus increase monotonically with increasing degree of expansion. Furthermore, negative Poisson’s ratios are predicted for configurations that are comparable to sintered hollow sphere structures, while positive Poisson’s ratios are observed for highly expanded configurations.