Mathias H. Luxner

A finite element study on the effects of disorder in cellular structures

The susceptibility to deformation localization of simple cubic arrangements of struts, which are a simple approximation of the micro-architecture in cancellous bone, is analyzed. The coherence between structural disorder and the tendency towards deformation localization is investigated and its relevance from a biological point of view is discussed. A systematic study on the spatial deformation distribution of regular and disordered open cell structures is carried out. To this end, finite element models are employed which account for elastic-plastic bulk material and large strain theory, and a methodology for the estimation of the degree of deformation localization is introduced.

An Incremental Mori-Tanaka Homogenization Scheme for Finite Strain Thermoelastoplasticity of MMCs

The present paper aims at computational simulations of particle reinforced Metal Matrix Composites as well as parts and specimens made thereof. An incremental Mori-Tanaka approach with isotropization of the matrix tangent operator is adopted. It is extended to account for large strains by means of co-rotational Cauchy stresses and logarithmic strains and is implemented into Finite Element Method software as constitutive material law. Periodic unit cell predictions in the finite strain regime are used to verify the analytical approach with respect to non-proportional loading scenarios and assumptions concerning finite strain localization. The response of parts made of Metal Matrix Composites is predicted by a multiscale approach based on these two micromechanical methods. Results for the mesoscopic stress and strain fields as well as the microfields are presented to demonstrate to capabilities of the developed methods.

Regular, low density cellular structures - rapid prototyping, numerical simulation, mechanical testing

Cellular solids form the basis of many biological and engineering structures. Most models use the relative density and the mechanical properties of the bulk material as the main parameter for the prediction of the mechanical properties of such structures. In this work the inuence of the architecture of periodic cellular solids on the mechanical properties is investigated numerically and experimentally. Using computer aided design, structures with 8x8x8 base cells are designed and fabricated. The physical prototypes which are tested experimentally are made from thermosetting and thermoplastic polymers by employing Rapid Prototyping (RP) techniques. Various RP techniques are compared regarding their suitability for the fabrication of cellular materials. For numerical simulation of the cellular structures, linear Finite Element analysis is employed. Three-dimensional models are set up using higher order beam elements. In a rst step, the structure is treated as an innite medium and homogenization via a ’periodic micro-eld approach’ is used. The entire elastic tensors for different relative densities are evaluated, from which the directional dependencies of the Young’s moduli are derived. In a second step, simulations of nite structures are performed for direct comparison with experiments. Samples consisting of several basic cells are modeled which leads to a better correspondence to the experimental setup. Finite structures of different numbers of cells are modeled to study the inuence of the sample size. The experimental and numerical results correspond very well and form a consistent picture of the problem. The multi-disciplinary approach leads to a comprehensive view of effects which govern the mechanical behaviour of the investigated cellular structures.

Linear and Nonlinear Numerical Investigations of Regular Open Cell Structures

Linear and nonlinear Finite Element simulations of various regular three-dimensional cellular solids (lattice structures) with relative densities ranging from 10% to 20% are presented. The structures consist of polymeric struts with circular cross sections. Two different Finite Element modeling techniques are employed. Beam element based models and continuum element based models are utilized and their applicability is assessed. Beam element based models compromise about the numerical model size and the detail resolution of the problem. Continuum element based models are used for highly detailed unit cell analyses. For simulations of the overall behavior the structures are treated as infinite media by a periodic microfield approach. The entire overall elasticity tensors are computed for the constitutive characterization of the effective mechanical behavior of the micro-structures. Overall stress‐strain curves are predicted for uniaxial compressive loading, taking into account finite strains and elasto-plastic strut material. The predicted properties are evaluated with respect to direction dependence and density dependence.

Forming Simulations of MMC Components by a Micromechanics Based Hierarchical FEM Approach

The present work deals with computational simulations of an elastoplastic particulate metal matrix composite undergoing finite strains. Two different approaches are utilized for homogenization and localization; an analytical constitutive material law based on a mean field approach, and a periodic unit cell method. Investigations are performed on different length scales. The Finite Element Method is employed to predict the macroscopic response of a component made from a metal matrix composite. Its constitutive material law, based on the incremental Mori Tanaka approach, has been implemented into an Finite Element Method package, and is extended to the finite strain regime. This approach gives access to the mesoscale fields as well as to approximations for the microscale fields in the individual phases of the composite. Selected locations within the macroscopic model are chosen and their entire mesoscopic deformation history is applied to unit cells using the periodic microfield approach. As a result, mesos...

Three-Dimensional Open Cell Structures: Evaluation and Fabrication by Additive Manufacturing

Biological materials (e.g. wood, trabecular bone, marine skeletons) rely heavily on the use of cellular architecture, which provides several advantages: (1) The resulting structures can bear the endurable mechanical loads using a minimum of a given bulk material, thus enabling the use of lightweight design principles. (2) The inside of the structures is accessible to body fluids which deliver the required nutrients. (3) Furthermore cellular architectures can grow organically by adding or removing individual struts or by changing the shape of the constituting elements. All these facts make the use of cellular architectures a reasonable choice for nature. Using Additive Manufacturing Technologies (AMT) it is now possible to fabricate such structures for applications in engineering and biomedicine. In this book chapter we present methods which allow the 3D-analysis of the mechanical properties of cellular structures with open porosity. Various different cellular architectures are studied. In order to quantify the influence of architecture, the apparent density is always kept constant. Various lithography based AMT are described and compared regarding their suitability for the fabrication of cellular structures.

Influence of defects and perturbations on the performance of 3D open cell structures

Regular and irregular highly porous open cell structures with a relative density of 12.5% are investigated by the Finite Element Method. The three-dimensional models are based on beam elements and account for the material distribution and the constrained deformation in the vertices [1].