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2019 | Buch

Computational Homogenization of Heterogeneous Materials with Finite Elements

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This monograph provides a concise overview of the main theoretical and numerical tools to solve homogenization problems in solids with finite elements. Starting from simple cases (linear thermal case) the problems are progressively complexified to finish with nonlinear problems. The book is not an overview of current research in that field, but a course book, and summarizes established knowledge in this area such that students or researchers who would like to start working on this subject will acquire the basics without any preliminary knowledge about homogenization. More specifically, the book is written with the objective of practical implementation of the methodologies in simple programs such as Matlab. The presentation is kept at a level where no deep mathematics are required.​

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The need for materials with higher performances is a strategic issue in engineering. Composite materials, i.e., combining at least two constituents with desired properties like mechanical resistance and lightness, have been developed and applied in many fields of engineering, and are now routinely used in many applications, including automotive industry, aircrafts, drones, biomedicals, wind turbines, sports, and leisure, etc. (see reviews in [16]). On the other hand, heterogeneous materials are found in many other engineering or science fields, such as cementitious materials in civil engineering or biomechanics. More recently, the progress in manufacturing techniques have allowed producing very complex materials like metallic foams (see Fig. 1.1a), or even allowed producing materials with “on demand” microstructures [7, 8] via 3D printing techniques, see Fig. 1.1b. Developing new materials involves synthesis, manufacturing, and testing for certification. This process is long and costly, and usually only involves a “trial and error” procedure, rather than a clear optimization methodology.
Julien Yvonnet
Chapter 2. Review of Classical FEM Formulations and Discretizations
Abstract
The objective of this first chapter is to recall the basics of Finite Elements for simple problems, here, the steady-state and linear elasticity problems, in order to use it as a solver for the localization problems required in computational homogenization in the next chapters. We do not intend to provide a complete framework on FEM here, but present a short introduction and practical aspects which can be used to directly implement a FEM program. For more in-depth about FEM formulations, we refer to classical books on Finite Elements such as [14] for linear problems and [5, 6] for nonlinear problems.
Julien Yvonnet
Chapter 3. Conduction Properties
Abstract
The objective of this chapter is to present the different basic concepts of computational homogenization through the simplest problem: defining the effective conductivity of a heterogeneous medium in steady-state regime. First, the notion of RVE is introduced. Then, the localization problems and the effective quantities are defined, and the numerical procedures using FEM to compute the effective conductivity tensor are presented.
Julien Yvonnet
Chapter 4. Elasticity and Thermoelasticity
Abstract
In this chapter, the definition and computation of effective properties in the context of linear elasticity are presented. First, the localization problem and the different types of boundary conditions are defined. Then, the definition of the effective elastic fourth-order tensor is introduced. The practical calculation of the effective elastic tensor with 2D and 3D FEM is detailed. An extension to thermoelasticity is described. Finally, reference solutions are provided for validation purpose.
Julien Yvonnet
Chapter 5. Piezoelectricity
Abstract
In this section, we present a procedure based on FEM for solving homogenization problems involving two coupled phenomena: electric conductivity and elasticity. In contrast to the case of thermoelasticity where the thermal problem has only an effect on the elastic problem and not the opposite, here both problems depend on the solution of each other. After presenting the localization problem defined over the RVE, the different effective operators are defined and the FEM procedure for their numerical calculation is provided. Finally, a numerical validation example is provided for fibrous piezoelectric composites.
Julien Yvonnet
Chapter 6. Saturated Porous Media
Abstract
In this chapter, two different homogenization problems and their solving strategies with FEM are presented, related to porous media saturated with a Newtonian fluid. The first problem is associated with microstructures where the porous cavities are disconnected (see Fig. 6.1a). In that situation, the fluid cannot flow but induces a pressure on the solid matrix which affects the mechanical response of the solid. The overall behavior is then called “poroelastic”. The second problem is related to the case where the solid contains connected porosities (see Fig. 6.1b), in which a fluid can flow and cross the solid. In this second case, the problem is then to determine the transport properties, or effective permeability of the medium, given the morphology of the porosity network. In this chapter, we present the FEM methodologies to solve these problems and do not go into further theoretical aspects, which can be found in excellent books on this topic, such as [1, 2]. In the presented developments, we restrict to linearized problem, i.e., we assume small strains, small displacements, small variations of porosity of fluid mass density (see [1], p. 113), and constant viscosity (Newtonian fluid).
Julien Yvonnet
Chapter 7. Linear Viscoelastic Materials
Abstract
Viscoelastic materials induce delayed mechanical response when loaded, and are of major interest for designing damping systems or for studying the creep behavior in concrete, among many other engineering applications. Progress in the design of viscoelastic composites require the construction of homogenized models based on microstructural analysis. As compared to the homogenization problems presented in the previous chapters, an additional difficulty arises from the time dependence of the behavior of the individual phases. In this chapter, a method for computing the homogenized behavior of linear viscoelastic materials is presented, based on the work proposed in [1]. The technique operates in the time domain. In this book, we restrict ourselves to this method due to its simplicity, even though other approaches have been proposed based on Laplace–Carson transform. A literature review about the different available methods and their drawbacks/advantages can be found in [1].
Julien Yvonnet
Chapter 8. When Scales Cannot Be Separated: Direct Solving of Heterogeneous Structures with an Advanced Multiscale Method
Abstract
In previous chapters, the assumption of scale separation was adopted. When this assumption does not hold, e.g., when the size of heterogeneities are not much smaller than local dimensions of the structures, classical homogenization methods fail to describe the local fields and up to a certain precision even the global response. More precisely, lack of scale separation occurs when the wavelength associated with the strain and stress fields at the microscale is of the same order of magnitude as the wavelength of the prescribed loads or the characteristic dimensions of the structure [1].
Julien Yvonnet
Chapter 9. Nonlinear Computational Homogenization
Abstract
The homogenization of nonlinear heterogeneous materials is by an order of magnitude tougher than the homogenization of linear ones. The main reason is that in the linear case, the general form of the homogenized (or effective) behavior of heterogeneous materials is a priori known, and it suffices to determine a set of effective moduli by considering a finite number of macroscopic loading modes. In contrast, in the nonlinear case, the general form of the homogenized behavior of heterogeneous materials is unknown and the determination of the homogenized behavior requires solving nonlinear partial differential equations with random or periodic coefficients and entails considering, in principle, an infinite number of macroscopic loading modes. Then, the superposition principle, which was used as a basis in the previous chapters to construct the homogenized behavior no more applies. The central problem is to define the constitutive relationship to be used at the macroscale at each integration point of the structure, given an RVE and a description of the nonlinear behavior of each phase. The development of nonlinear computational homogenization methods has been an active topic of research since the end of the 90’s and many issues still remain at the time this book is written.
Julien Yvonnet
Correction to: Elasticity and Thermoelasticity
Julien Yvonnet
Backmatter
Metadaten
Titel
Computational Homogenization of Heterogeneous Materials with Finite Elements
verfasst von
Prof. Dr. Julien Yvonnet
Copyright-Jahr
2019
Electronic ISBN
978-3-030-18383-7
Print ISBN
978-3-030-18382-0
DOI
https://doi.org/10.1007/978-3-030-18383-7