Micromechanics in Practice

Contents 1Random Composites in Practical Applications Concrete, asphalts, cement pastes; Natural wood, ceramic and metallic foams; Plain weave textile composites; Natural fiber-reinforced biodegradable composites 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