Multiscale Modelling and Optimisation of Materials and Structures. Tadeusz Burczynski

      1 Chapter 1Figure 1.1 Multiscale concept diagram which illustrates ‘macro’, ‘meso’, ‘mi...Figure 1.2 Illustration of computational homogenization concept.Figure 1.3 (a) Internal variable method as a precursor of upscaling and (b) t...

      2 Chapter 2Figure 2.1 The Lennard‐Jones atomic potential and force.Figure 2.2 The Morse potential.Figure 2.3 Typical material responses to plastic deformation.Figure 2.4 Schematic illustration of FE model application for a prediction o...Figure 2.5 Schematic illustrations of selected plastometric tests.Figure 2.6 Selected examples of comparison of loads recorded for the two DP ...Figure 2.7 Microstructures at various stages of recrystallization.Figure 2.8 Flow chart of calculations of microstructure evolution in microal...Figure 2.9 Classification of selected phase transformation models: computing...Figure 2.10 Comparison of the CCT diagrams obtained from measurements (fille...Figure 2.11 Comparison of the CCT diagrams (a) and volume fractions of phase...Figure 2.12 Comparison of the CCT diagrams obtained from measurements (fille...Figure 2.13 Thermal profile (dotted line) and changes of volume fractions of...Figure 2.14 Thermal profile (dotted line) and changes of volume fractions of...Figure 2.15 Transcrystalline fracture (a) and intercrystalline fracture (b)....Figure 2.16 Ductile and brittle material behaviour.Figure 2.17 Interpretation of the J integral.Figure 2.18 Schematic illustration of the RVE approach based on homogenizati...Figure 2.19 Geometric measure of damages proposed by Kachanov [46] for mater...Figure 2.20 Macroscopic features of damages proposed by Chaboche for materia...Figure 2.21 Mechanical models of ideal elastic (a) and ideal viscous (b) mat...Figure 2.22 Kelvin (a) and Maxwell (b) material rheological models.Figure 2.23 Typical creep curve with marked first (I), second (II), and thir...Figure 2.24 Schematic relation between loading stress and the time to damage...

      3 Chapter 3Figure 3.1 Representation of a 2D domain by a collection of triangles and qu...Figure 3.2 Discontinuity path (a) and domains (b) created in [86].Figure 3.3 Illustration of the displacement field decomposition [86].Figure 3.4 Illustration of the finite element approximation [86].Figure 3.5 Illustration of the unit jump function finite element approximati...Figure 3.6 The elastic body bounded by the boundary.Figure 3.7 An area near the boundary of the body.Figure 3.8 Discretization with boundary elements.Figure 3.9 A body discretized using finite and boundary elements.Figure 3.10 The infinite body discretized using finite and boundary elements...Figure 3.11 Homogenization of the non‐homogeneous structure.Figure 3.12 Models of a structure with locally periodical microstructures: (...Figure 3.13 Computational homogenization scheme: the average strain and stre...Figure 3.14 Rupture of the SLMoS₂ lattice during the tensile test along the ...Figure 3.15 Stress–strain relation of the SLMoS₂ lattice obtained in armchai...Figure 3.16 Illustration of the cutoff radius and the ‘skin’ parameter.Figure 3.17 Illustration of the periodic boundary conditions.Figure 3.18 The reflecting wall.Figure 3.19 Rectangular plate with a crack under tensile load. A constant li...Figure 3.20 Results (a), (b), and (c) after, respectively, 6, 7, 8 ps of sim...Figure 3.21 Interaction forces between the pair of atoms.Figure 3.22 Homogenous deformation of the atomic lattice.Figure 3.23 Initial (X _i , X _j ) and final (x _i , x _j ) positions of the...Figure 3.24 Assembly of the global system of equations: M and N denote bondi...Figure 3.25 The algorithm of solving the molecular statics problem.Figure 3.26 Shearing of the HCP lattice: (a) is the initial state; (b), (c),...Figure 3.27 Examples of cellular automata space division into (a) squares, (...Figure 3.28 Examples of (a) n × n and (b) n × m CA space composed of square ...Figure 3.29 Examples of CA neighbourhoods in the 2D space: (a) Moore and (b)...Figure 3.30 Examples of other types of CA neighbourhoods, (a) extended Moore...Figure 3.31 Examples of CA neighbourhoods in the 3D space: von Neumann and M...Figure 3.32 A 3D representation of random hexagonal neighbourhood.Figure 3.33 Cellular automata space is a random cellular automata method.Figure 3.34 Source of information for the definition of transition rules.Figure 3.35 Concept of the periodic boundary conditions with Moore neighbour...Figure 3.36 Example of n × n MC computation domain with random assignment of...Figure 3.37 The examples of objective functions.Figure 3.38 The flowchart of descent algorithm.Figure 3.39 (a) Individual genes; (b) the flowchart of evolutionary algorith...Figure 3.40 Mechanism of simple crossover operator.Figure 3.41 Flowchart of the clonal algorithm, which is one of the artificia...Figure 3.42 Flowchart of particle swarm optimization.

      4 Chapter 4Figure 4.1 Unit cells: (a) simple cubic, (b) body‐centred, (c) face‐centred,...Figure 4.2 Controlled cooling of the Al cluster: (a) unstable initial config...Figure 4.3 3D nanocrystalline structures of diameter approximately 20 nm obt...Figure 4.4 3D nanocrystalline structures obtained during fast (a) and slow (...Figure 4.5 Effects of fast and proper cooling: (a) amorphous phase and (b) H...Figure 4.6 The shape of the Morse potential functions for D _e = 1 ev, r ₀ ...Figure 4.7 2D nanocrystalline structures obtained for: (a) short (α = 8...Figure 4.8 3D nanocrystalline structures obtained for various ranges of inte...Figure 4.9 3D nanocrystalline structures obtained for various ranges of inte...Figure 4.10 Squeezing of the set of 2D nanocrystalline structures: (a) initi...Figure 4.11 2D structures obtained during squeezing: (a) proper polycrystal;...Figure 4.12 The set of eight FCC balls: (a) general view; (b) cross‐section ...Figure 4.13 Obtained polycrystalline structure: (a) general view; (b) cross‐...Figure 4.14 Molecular model with (a) vacancies and (b) large voids.Figure 4.15 Molecular models used in tensile and shear tests: (a) ideal FCC ...Figure 4.16 Stress–strain curves – tensile test: (a) ideal FCC monocrystal; ...Figure 4.17 Stress–strain curves – shear test: (a) ideal FCC monocrystal; (b...Figure 4.18 Damaged structures during tensile and shear tests: (a) ideal FCC...Figure 4.19 Structure of the SLMoS₂ 2H lattice.Figure 4.20 Stress–strain relations of the ideal SLMoS₂ 2H lattice – (a) zig...Figure 4.21 SLMoS₂ lattice before final failure: arrows indicate series of s...Figure 4.22 SLMoS₂ lattices with different concentrations of vacancies.Figure 4.23 Illustration of (a) concept of the Delaunay triangulation steps ...Figure 4.24 2D digital material representation of polycrystalline microstruc...Figure 4.25 3D digital material representation of polycrystalline microstruc...Figure 4.26 3D digital material representation of periodic polycrystalline m...Figure 4.27 Examples of CA grain growth within the hexagon‐based discretizat...Figure 4.28 Subsequent stages of CA grain growth algorithm.Figure 4.29 CA grain growth results with different types of neighbourhoods: ...Figure 4.30 Digital microstructure with elongated grains obtained with the m...Figure 4.31 Digital microstructure with gradient grain size obtained with th...Figure 4.32 Digital microstructure with uniform grain distribution obtained ...Figure 4.33 Microstructure with 10 grains generated (a) without

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