Multiscale Stochastic Computation of Natural Frequencies for Beams Made of Metal Foam Based on Stochastic Geometry Models

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Aim of the project is to develop a multiscale stochastic computation method which, starting from CT geometry data, allows for the determination of natural frequencies of beams made of metal foam as far as possible without ad hoc assumptions. To this end, the computation method developed by the applicants is extended and modified.With regard to the stochastic modeling of the microstructure geometry, the goal is to develop a statistical model fitting technique which avoids a prior selection of a tessellation model. For this purpose, a Laguerre tessellation is reconstructed from the cell system of the foam using a suitable distance measure. The set of generators of this tessellation is interpreted as a marked point process. Using classical point process statistics, a model is fit to this point process. The Laguerre tessellation of model realizations then provides three-dimensional samples of the microstructure geometry. The distance measure used in the model fitting steps has to be chosen such that both the geometrical characteristics of the structure as well as the material properties are reproduced correctly.By means of the geometry samples, boundary effect independent properties of the apparent material parameters are determined. For this purpose, by introduction of an additional length scale a difference is made between the volume element on which the boundary conditions are applied, and the volume element, over which stresses and strains are averaged. This allows to obtain boundary effect independent histograms of the one-dimensional marginal distributions and estimates of the autocorrelation function of material parameters. Using these data, a random field is modeled without assuming a distribution class for the marginal distributions. The goal is to develop a novel approach that starts from the truncated Karhunen-Loève expansion of the random field and generates iteratively realizations of the non-Gaussian random variables contained in the expansion. This approach offers the advantage that the random variables enter linearly in the Stochastic-FE-formulation of the eigenvalue problem for the determination of the natural frequencies, which facilitates a non-intrusive solution of the FE problem.Another goal of the project is the statistical description and the modeling of the occurrence of special inhomogeneities such as variable strut thickness distribution and defects in order to better understand the effect of the combination of these effects on the structural behavior.The developed computation method is validated step by step by means of sample images and results of experimental modal analysis.Another major objective of the proposed approach is to gain insights that are also transferable for the modeling of other heterogeneous materials. This applies particularly to the investigations of the generator set and of the boundary effect independent properties of apparent material parameters as well as to the random field modeling.