One of the most important engineering tasks over the years has been the design and manufacture of increasingly sophisticated structural materials as a result of the requirements related to the technological progress. In the last decades, the growing needs for improved properties of products have been partially solved through the development of composite materials. A key to the success of many modern structural components is the tailored behavior of the material to given applications. Therefore, research efforts in material science engineering have been focused in the design of new materials either through the creation of new structures at the scale of single atoms and molecules or through the development of structural materials by changing the composition, size, arrangement and topology of the constituents at larger scales: the microscopic/mesoscopic level.

The development of new materials has been linked to the development of a new theoretical field within the mechanics of solids. This branch of the mechanical, known as Continuum Micromechanics, introduces a series of new concepts that are key to the definition of the macroscopic properties of composite materials on the basis of the definition of the characteristics of its components. Starting from the premise of separation of scales and the concept of Representative Volume Element, defined the so-called homogenization methods, whose number has been increasing as the Micromechanics is gone extend over the years. Such methods are many and varied, although especially there have been two that have been used and developed by the majority of authors: the so-called Mean-Homogenization techniques and the multi-scale based on Finite Element Approaches.

Mean-field homogenization schemes are an efficient way to predict the behavior of heterogeneous materials. They range from the simplest hypotheses of the stress or strain sharing among the phases which do not require analytical solution on the associated boundary-value problem to more involved geometric models based on the solution of a boundary-value problem involving a single or composite inclusion embedded in an equivalent homogenized medium whose elastic module become part of the solution procedure. In general, they are based on analytical solutions of the boundary value problem defined in the microstructure level of the inhomogeneous material and provide good predictions for the mean values over the RVE. Although originally designed for elastic materials, some approaches to deal with elastoplastic materials and even with viscoplastic materials have been developed over the years and compared with the results obtained using Finite Element Approaches. The comparison between different methods of homogenization allows the definition of a range of validity between the different methods, which helps to discover the limitations of the various methods and aspects to take into account for future developments and research.

The main goal of this work is, firstly, to present a general overview of the different techniques that have been developed in the last years in order to obtain a prediction of the behavior of elastoplastic composites by taking into account geometrical and mechanical aspects. Secondly, a comparison between the different approaches is carried out through a numerical implementation of such techniques. Both objectives will be carried out through eight different chapters. The first chapter serves as an introduction and historical review of the advances that have been made in the field of micromechanics. On the other hand, the second chapter deals with some important theoretical background that is important in the field of Continuum Micromechanics, as well as a short introduction of the different approaches that traditionally have been considered to solve the problem. One group of methods, based on analytical solutions - the so-called Mean Field Analysis - will be commented in chapter 3. Chapter 4 is devoted to the implementation and validation of a numerical tool that solves the mean-field homogenization using analytical schemes for elastoplastic materials. Subsequent chapters are devoted to the comparison of the results with the results given by the Finite Element Method. The general formulation of such method - applied to multi-scale problems - is presented in chapter 5 from a theoretical point of view, as well as the corresponding numerical examples. Finally, last chapter will be dedicated to enumerate some conclusions extracted from the present work, including some aspects that can be object of future works or improvements.

The current work presents some important aspects about the theoretical concepts and the numerical implementation of some key approaches for solving the mechanical problem regarding composite materials. There exist a large number of possibilities to approximate the response of such complex materials, based in different assumptions. This document shows the general efficiency of the so-called mean-field homogenization schemes to capture correctly the macroscopic behavior of composites. Although these techniques show some limitations, like the incapability to provide results for the distribution of the different variables over the microgeometry or the low accuracy in the case of complex microgeometries (like porous materials), they represent an efficient way to predict the main general behavior of a composite material spending low computational effort. They are specially indicated to be used in the previous steps of an analysis or as a tool to validate the results with more involved approaches.

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