Micromechanics in Practice

This book concentrates on the application of micromechanics to the analysis of practical engineering problems. Both classical composites represented by carbon/carbon textile laminates and applications in Civil Engineering including asphalts and masonry structures will be considered. A common denominator of these considerably distinct material systems will be the randomness of their internal structure. Also, owing to their complexity, all material systems will be studied on multiple scales. Since real engineering, rather than academic, problems are the main interest, these scales will be treated independently from each other on the grounds of fully uncoupled multi-scale analysis. Attention will be limited to elastic and viscoelastic behaviour and to the linear heat transfer analysis. To achieve this, two different approaches to the homogenization of systems with random microstructures are addressed.In particular, classical averaging schemes based on the Eshelby solution of a solitary inclusion in an infinite medium represented by the Hashin-Shtrikman variational principles, or by the considerably simpler and more popular Mori-Tanaka method, will be compared to detailed finite element simulations of a certain representative volume element (RVE) representing accommodated geometrical details of respective microstructures. These are derived by matching material statistics such as the one- and two-point probability functions of real and artificial microstructures. The latter one is termed the statistically equivalent periodic unit cell owing to the assumed periodic arrangement of reinforcements (carbon fibres, carbon fibre tows, stones or masonry bricks) in a certain matrix (carbon matrix, asphalt mastic, mortar). Other types of materials will be introduced in the form of exercises with emphases to the application of the Mori-Tanaka method in the framework of the previously mentioned uncoupled multi-scale analysis.

Table of Contents:
Contents 1 Random Composites in Practical Applications Concrete, asphalts, cement pastes; Natural wood, ceramic and metallic foams; Plain weave textile composites; Natural fiber-reinforced biodegradable composites 2 Introduction to Micromechanics Electrical conduction; Terminology and notation; Equation of continuity; Constitutive equation; Evaluation of effective electric conductivities; First-order homogenization; Dirichlet boundary conditions; Neumann boundary conditions; Review of basic micromechanical models; Minimum energy principles; Averages of local fields and Hill's lemma; Minimum potential energy principle; Minimum complementary energy principle; Summary of governing equations; Governing equations; Hill lemma; Constitutive equations; Constitutive law for elastic isotropic and transversely isotropic systems; Self-consistent and Mori - Tanaka estimates for N-phase composite; Example problems; Prediction of effective electric properties; Prediction of effective thermal conductivities; Prediction of effective elastic properties 3 Random microstructure Quantitative description of microstructure; Basic concepts and hypotheses; Motivation; Concept of an ensemble; Ergodic hypothesis; Statistical homogeneity; Statistical isotropy; Microstructural descriptors; n-Point probability functions; Lineal path function; Numerical evaluation of microstructural statistics; n-point probability functions; Lineal path function; Examples; Fully penetrable discs; Results; Unidirectional fiber composite; Stochastic Hashin - Shtrikman variational principles; Body with prescribed surface displacements and eigenstresses; Variational principle; Extension to random composites; Approximate solution; Body with prescribed surface tractions and eigenstrains; Variational principle; Extension to random composites; Approximate solution; Examples; Statistically equivalent periodic unit cell; Definition of a periodic unit cell; Carbon - carbon textiles; Geometry description; Verification; Application to real-world material system; Fiber - matrix composites; Masonry structures; Asphalt mixtures; Methods to evaluate random microstructures; Finite Element-based homogenization scheme; Formulation based on strain approach; Formulation based on stress approach; Implementation issues; Examples; FFT-based homogenization scheme; Example; Beyond periodicity 4 Applications of multiscale modeling Textile composites; Laboratory measurements; Phase elastic moduli from nanoindentation; Two-dimensional image analysis; Three-dimensional X-ray microtomography; Large scale laboratory measurements; Effective properties from hierarchical modeling; Mesoscale analysis based on first-order homogenization; Mesoscale analysis based on the Mori - Tanaka method; Macroscopic properties; Verification and validation of hierarchical modeling; Summary; Asphalt mixtures; Laboratory measurements; Generalized Leonov model; Small-scale laboratory measurements; Large-scale macroscopic laboratory measurements; Effective properties of macroscopic constitutive model; Definition of scales; Effective properties from binary images; Generalized Leonov model from two-step homogenization; Application of the Mori - Tanaka method; The Mori - Tanaka method including eigenstrains; Mori - Tanaka predictions based on correspondence principle; Augmented Mori - Tanaka method; Why the Mori - Tanaka method fails in representing creep; Model verification from macroscale analysis; Wheel tracking - numerical simulation; Summary 5 Examples Alkali-activated fly ash; Identification of scales; Identification of intrinsic material properties; Hierarchical homogenization; Theoretical predictions supported by experiments; Natural wood - birch; Identification of scales; Identification of intrinsic material properties; Hierarchical homogenization; Theoretical predictions supported by experiments; Closed cell metallic foam; Identification of scales; Identification of intrinsic material properties; Hierarchical homogenization; Theoretical predictions supported by experiments 6 Conclusions and future Prospects Evaluation of Eshel by tensor; Conductivity; Elasticity; The Fourier transform, fundamental solutions, and microstructural matrices; The Fourier transform; The discrete Fourier transform; The convolution and correlation theorems; Fundamental solutions; The Fourier transform of tensors and; Evaluation of matrices Ars and Brs; Transformation of compliance function to relaxation function