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Material characterization is a very important tool needed to describe and enhance material mechanical properties and to develop optimum material chemistries and microstructures. The usual approach of achieving the above using extensive experimental methods has been shown to be expensive and time consuming. This led to the development of the micro mechanical modeling, which can be used to predict the material behavior without the need for the extensive experimental investigation and is based on microstructural characteristics of the material. / In this work, a micro mechanical model is developed to predict the mechanical properties of dual phase steels consisting of martensite in a matrix of ferrite. This micro model is also used to elucidate the mechanics and mechanisms of deformation, which take place in such materials. DP-steels consisting of several volume fractions of martensite (Vm) representing low, intermediate and high Vm are developed and tested mechanically to obtain their mechanical properties. Metallographical examinations are carried out using image analysis to quantify microstructural material properties of each level of Vm considered. As a validation of the current work, comparison between the model predictions, which include all the significant material behavior investigated in this work and the experimental results, is presented. The comparison demonstrates the ability of the model to capture the behavior of DP-steels up to the instability point. / The Gurson-Tvergaard model, which is the most widely known damage model to describe ductile failure, is coupled with the results of the micro mechanical model, presented in this work to form a complete material model of deformation and fracture of DP-steels. A procedure is developed to determine the parameters in the Gurson-Tvergaard model utilizing the micromechanical model. The results are then implemented to simulate the deformation and failure of tensile bars of DP-steels with different Vm, which shows good agreement with the experimental results at failure.

In this thesis, we propose several algorithms for model reduction of second order dynamical systems. These various projection methods are based on singular value decomposition, Krylov projection, and balanced truncation. In many cases models are given in second order form, and the goal is to produce a reduced order system which is in second order form, gives an accurate approximation of the original system, and maintains some important properties such as stability and passivity. Model reduction on first order linear time invariant dynamical systems has been extensively studied, algorithms and theory are well-developed. People usually study and deal with second order system by transforming to first order form which doubles the dimension. This can be inefficient and generally does not respect the second order form. The reduced model is not realizable as a second order system. So far only a very few algorithms have been proposed for second order model reduction. Most of these are not practical for large scale settings, and no error bounds have been provided. In this thesis, a global error bound is given for some of the algorithms based on SVD and balanced truncation, the error bound is bounded by a constant times the summation of the neglected singular (or Hankel singular) values, that means those second order model reduction algorithms provide accurate approximations to the original systems. The structures of controllability and observability Gramians P and Q are discussed. All algorithms developed in this thesis have been implemented and shown to be numerically efficient, and applicable to large scale settings. All algorithms are implemented in Matlab, some of them are implemented in Fortran and C separately for which we use LAPACK. In this thesis, we apply our algorithms to three real models. The performance of our algorithms is compared with some of the previously existing algorithms. It turns out that most of our algorithms are very competitive with existing methods.

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