CONTENTS -- Preface -- Contributors -- Computational Homogenisation for Non-Linear Heterogeneous Solids V. G. Kouznetsova, M. G. D. Geers and W. A. M. Brekelmans -- 1. Introduction -- 2. Basic Hypotheses -- 3. Definition of the Problem on the Microlevel -- 4. Coupling of the Macroscopic and Microscopic Levels -- 4.1. Deformation -- 4.2. Stress -- 4.3. Internal work -- 5. FE Implementation -- 5.1. RVE boundary value problem -- 5.1.1. Fully prescribed boundary displacements -- 5.1.2. Periodic boundary conditions -- 5.2. Calculation of the macroscopic stress -- 5.2.1. Fully prescribed boundary displacements -- 5.2.2. Periodic boundary conditions -- 5.3. Macroscopic tangent stiffness -- 5.3.1. Condensation of the microscopic stiffness: Fully prescribed boundary displacements -- 5.3.2. Condensation of the microscopic stiffness: Periodic boundary conditions -- 5.3.3. Macroscopic tangent -- 6. Nested Solution Scheme -- 7. Computational Example -- 8. Concept of an RVE within Computational Homogenisation -- 9. Extensions of the Classical Computational Homogenisation Scheme -- 9.1. Homogenisation towards second gradient continuum -- 9.2. Computational homogenisation for beams and shells -- 9.3. Computational homogenisation for heat conduction problems -- Acknowledgements -- References -- Two-Scale Asymptotic Homogenisation-Based Finite Element Analysis of Composite Materials Qi-Zhi Xiao and Bhushan Lal Karihaloo -- 1. Introduction -- 2. Mathematical Formulation of First- and Higher-Order Two-Scale Asymptotic Homogenisation -- 2.1. Two-scale expansion -- 2.2. O(ε.2) equilibrium: Solution structure of ui(0) -- 2.3. O(ε.1) equilibrium: First-order homogenisation and solution structure of u(1)m -- 2.4. O(ε0) equilibrium: Second-order homogenisation -- 2.4.1. Solution structure of u(2) -- 2.4.2. Solution of u(0) m -- 2.4.3. Solution of ψmno k (y).

2.4.4. Constraints from higher-order solutions -- 2.5. O(ε1) equilibrium: Third-order homogenisation -- 2.5.1. Solution of u(3) k -- 2.5.2. Constraints from higher-order terms -- 3. Variational Formulation of Problem (29) -- 4. Finite Element Methods -- 4.1. Displacement compatible elements from the potential principle -- 4.2. Element-free Galerkin method from the potential principle -- 4.2.1. MLS interpolant -- 4.2.2. Imposition of the essential boundary conditions -- 4.2.3. Discontinuity in the displacement field -- 4.2.4. Interfaces with discontinuous first-order derivatives -- 4.3. Displacement incompatible element from the potential principle -- 4.3.1. 2D 4-node incompatible element -- 4.3.2. 3D 8-node incompatible element -- 4.4. Hybrid stress elements from the Hellinger-Reissner principle -- 4.4.1. Plane 4-node Pian and Sumihara (PS) 5β element -- 4.4.2. 3D 8-node 18β hybrid stress element -- 4.5. Enhanced-strain element based on the Hu-Washizu principle -- 4.5.1. Plane 4-node enhanced-strain element -- 4.5.2. 3D 8-node enhanced-strain element -- 4.6. Comments on the various methods -- 5. Enforcing the Periodicity Boundary Condition and Constraints from Higher-Order Equilibrium in the Analysis of the RUC -- 6. A Posteriori Recovery of the Gradients -- 6.1. Superconvergent patch recovery (SPR) -- 6.2. Moving Least Squares (MLS) -- 7. Numerical Examples -- 8. Discussion and Conclusions -- References -- Multi-Scale Boundary Element Modelling of Material Degradation and Fracture G. K. Sfantos and M. H. Aliabadi -- 1. Introduction -- 2. Macromechanics -- 2.1. Modelling the continuum -- 3. Artificial Microstructure Generation -- 4. Microstructure Modelling -- 4.1. Grain material modelling -- 4.2. A boundary cohesive element formulation -- 4.3. Grain discretisation -- 5. Grain Boundary Interface -- 6. Microcracking Evolution Algorithm.

6.1. Non-local approach -- 7. Definitions: Averaging Theorems -- 7.1. RVE boundary conditions -- 8. Micro-Macro Interface -- 8.1. Coupling with macro-BEM -- 8.2. Coupling with macro-FEM -- 9. Multiprocessing Algorithm -- 10. Multi-Scale Damage Simulations -- 11. Conclusions -- References -- Appendix -- Non-Uniform Transformation Field Analysis: A Reduced Model for Multiscale Non-Linear Problems in Solid Mechanics Jean-Claude Michel and Pierre Suquet -- 1. Introduction -- 2. Structural Problems with Multiple Scales -- 2.1. Homogenisation and two-scale expansions -- 2.2. Individual constituents -- 2.3. Unit-cell problem: Effective response of heterogeneous materials -- 2.4. An auxiliary elasticity problem -- 3. Non-Uniform Transformation Field Analysis (NTFA) -- 3.1. Motivation: Approximate resolution of the local problem -- 3.2. Non-uniform transformation fields -- 3.3. Reduced variables and influence factors -- 3.4. Constitutive relations for the reduced variables -- 3.5. Choice of the plastic modes -- 3.6. Reduced localisation tensors and influence factors -- 3.7. Time-integration of the NTFA model: Strain control -- 3.8. Time-integration of the NTFA model: Stress control -- 3.9. Example 1: Effective response of a dual-phase inelastic composite -- 3.9.1. Material data -- 3.9.2. Microstructure -- 3.9.3. Plastic modes -- 3.9.4. Discussion of the results -- 4. Application of the NTFA to Structural Problems -- 4.1. Implementation of the NTFA method -- 4.2. Implementation of the NTFA in a Finite Element code (step C) -- 4.3. Localisation rules -- 4.4. Example 2: Fatigue of a metal-matrix composite structure -- 4.4.1. Meshes -- 4.4.2. Loading -- 4.4.3. Plastic modes -- 4.4.4. Accuracy of the NTFA model at the level of a material point -- 4.4.5. Accuracy of the NTFA model at the structure level -- 5. Conclusion -- References.

Multiscale Approach for the Thermomechanical Analysis of Hierarchical Structures Marek J. Lefik, Daniela P. Boso and Bernhard A. Schrefler -- 1. Introduction -- 1.1. Bounds and other estimates -- 2. Asymptotic Theory of Homogenisation -- 2.1. Asymptotic analysis -- 2.2. Statement of the problem and assumptions -- 2.3. Formalism of the homogenisation procedure -- 2.4. Global solution -- 2.5. Local approximation of the stress vector -- 2.6. Finite element analysis applied to the local problem -- 2.7. Asymptotic homogenisation at three levels: Micro, meso, and macro -- 3. Non-Standard Numerical Techniques in Modelling of Hierarchical Composites -- 3.1. Definition of effective behaviour based on numerical or real experiment -- 3.2. Characterisation of the elastic-plastic behaviour of a composite based directly on numerical experiments (virtual testing) -- 3.3. ANN in constitutive modelling -- 3.4. Direct use of an ANN to define the effective material behaviour known from laboratory experiments -- 3.5. Direct use of an ANN to define the effective material behaviour based on numerical experiment -- 3.6. Approximation of dependence of effective material properties on the microstructural parameters by ANN in multiscale homogenisation -- 3.7. ANN as a tool for unsmearing -- 4. Concluding Remarks -- Acknowledgements -- References -- Recent Advances in Masonry Modelling: Micromodelling and Homogenisation Paulo B. Louren¸co -- 1. Introduction -- 2. Masonry Behaviour and Non-Linear Mechanics -- 2.1. Non-linear properties of unit and mortar (tension) -- 2.2. Non-linear properties of the interface (tension and shear) -- 2.3. Non-linear properties of unit, mortar and masonry (compression) -- 3. Modelling Approaches -- 4. Micromodelling Approaches -- 4.1. A combined crack-shear-compression interface model -- 4.1.1. Standard plasticity constitutive model.

4.1.2. Extension for cyclic loading -- 4.2. A combined crack-shear-compression limit analysis model -- 4.3. Applications -- 4.3.1. Modelling masonry under compression -- 4.3.2. Conventional micromodelling -- 4.3.3. The macroblock approach for historical buildings -- 5. Homogenisation Approaches -- 5.1. Homogenised failure surfaces -- 5.2. Applications -- 5.2.1. Masonry shear wall -- 5.2.2. Two-storeyed unreinforced masonry building -- 6. Conclusions -- References -- Mechanics of Materials With Self-Similar Hierarchical Microstructure R. C. Picu and M. A. Soare -- 1. Introduction -- 2. Elements of Fractal Geometry -- 3. Elements of Fractional Calculus -- 3.1. Local fractional differential operators -- 3.2. Fractional integral operators -- 3.2.1. The 1D case -- 3.2.2. Approximate relationship between classical and fractional integrals -- 4. Mechanics BVPs on Materials with Fractal Microstructure -- 4.1. Deterministic fractal microstructures -- 4.1.1. Iterative approaches -- 4.1.2. Approaches based on the reformulation of governing equations -- 4.1.2.1. Non-local fractional operators-based approach -- 4.1.2.2. Local fractional operators-based approach -- 4.2. Stochastic fractal microstructures -- 5. Conclusions -- References -- Index.