Microstructural Randomness and Scaling in Mechanics of Materials

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Features

  • Presents basic coverage of random geometry and continuum random fields for readers unacquainted with this area
  • Reviews truss-type and beam-type lattices as well as the construction of the corresponding classical and nonclassical continua
  • Explores-for the first time in book form-scaling to the RVE in conductivity, linear or finite elasticity and thermoelasticity, elasto-plasticity, and flow in porous media
  • Provides methods for problems below the RVE via SFEs
  • Studies effects of microscale material randomness on wavefronts in (in)elastic/nonlinear media
  • Includes a number of applications from engineering and science to promote active research in the field
  • Contains numerous problems at the end of the first six chapters
  • Summary

    An area at the intersection of solid mechanics, materials science, and stochastic mathematics, mechanics of materials often necessitates a stochastic approach to grasp the effects of spatial randomness. Using this approach, Microstructural Randomness and Scaling in Mechanics of Materials explores numerous stochastic models and methods used in the mechanics of random media and illustrates these in a variety of applications.

    The book first offers a refresher in several tools used in stochastic mechanics, followed by two chapters that outline periodic and disordered planar lattice (spring) networks. Subsequent chapters discuss stress invariance in classical planar and micropolar elasticity and cover several topics not yet collected in book form, including the passage of a microstructure to an effective micropolar continuum.

    After forming this foundation in various methods of stochastic mechanics, the book focuses on problems of microstructural randomness and scaling. It examines both representative and statistical volume elements (RVEs/SVEs) as well as micromechanically based stochastic finite elements (SFEs). The author also studies nonlinear elastic and inelastic materials, the stochastic formulation of thermomechanics with internal variables, and wave propagation in random media.

    The concepts discussed in this comprehensive book can be applied to many situations, from micro and nanoelectromechanical systems (MEMS/NEMS) to geophysics.

    Table of Contents

    PREFACE

    BASIC RANDOM MEDIA MODELS
    Probability Measure of Geometric Objects
    Basic Point Fields
    Directional Data
    Random Fibers, Random Line Fields, Tessellations
    Basic Concepts and Definitions of Random Microstructures

    RANDOM PROCESSES AND FIELDS
    Elements of One-Dimensional Random Fields
    Mechanics Problems on One-Dimensional Random Fields
    Elements of Two- and Three-Dimensional Random Fields
    Mechanics Problems on Two- and Three-Dimensional Random Fields
    Ergodicity
    The Maximum Entropy Method

    PLANAR LATTICE MODELS: PERIODIC TOPOLOGIES AND ELASTOSTATICS
    One-Dimensional Lattices
    Planar Lattices: Classical Continua
    Applications in Mechanics of Composites
    Planar Lattices: Nonclassical Continua
    Extension-Twist Coupling in a Helix

    LATTICE MODELS: RIGIDITY, RANDOMNESS, DYNAMICS, AND OPTIMALITY
    Rigidity of Networks
    Spring Network Models for Disordered Topologies
    Particle Models
    Michell Trusses: Optimal Use of Material

    TWO- VERSUS THREE-DIMENSIONAL CLASSICAL ELASTICITY
    Basic Relations
    The CLM Result and Stress Invariance
    Poroelasticity

    TWO- VERSUS THREE-DIMENSIONAL MICROPOLAR ELASTICITY
    Micropolar Elastic Continua
    Classical vis-à-vis Nonclassical (Elasticity) Models
    Planar Cosserat Elasticity
    The CLM Result and Stress Invariance
    Effective Micropolar Moduli and Characteristic Lengths of Composites

    MESOSCALE BOUNDS FOR LINEAR ELASTIC MICROSTRUCTURES
    Micro-, Meso-, and Macroscales
    Volume Averaging
    Spatial Randomness
    Hierarchies of Mesoscale Bounds
    Examples of Hierarchies of Mesoscale Bounds
    Moduli of Trabecular Bone

    RANDOM FIELD MODELS AND STOCHASTIC FINITE ELEMENTS
    Mesoscale Random Fields
    Second-Order Properties of Mesoscale Random Fields
    Does There Exist a Locally Isotropic, Smooth Elastic Material?
    Stochastic Finite Elements for Elastic Media
    Method of Slip-Lines for Inhomogeneous Plastic Media
    Michell Trusses in the Presence of Random Microstructure

    HIERARCHIES OF MESOSCALE BOUNDS FOR NONLINEAR AND INELASTIC MICROSTRUCTURES
    Physically Nonlinear Elastic Microstructures
    Finite Elasticity of Random Composites
    Elastic-Plastic Microstructures
    Rigid-Perfectly Plastic Microstructures
    Viscoelastic Microstructures
    Stokes Flow in Porous Media
    Thermoelastic Microstructures
    Scaling and Stochastic Evolution in Damage Phenomena
    Comparison of Scaling Trends

    MESOSCALE RESPONSE IN THERMOMECHANICS OF RANDOM MEDIA
    From Statistical Mechanics to Continuum Thermodynamics
    Extensions of the Hill Condition
    Legendre Transformations in (Thermo)Elasticity
    Thermodynamic Orthogonality on the Mesoscale
    Complex versus Compound Processes: The Scaling Viewpoint
    Toward Continuum Mechanics of Fractal Media

    WAVES AND WAVEFRONTS IN RANDOM MEDIA
    Basic Methods in Stochastic Wave Propagation
    Toward Spectral Finite Elements for Random Media
    Waves in Random 1D Composites
    Transient Waves in Heterogeneous Nonlinear Media
    Acceleration Wavefronts in Nonlinear Media

    BIBLIOGRAPHY
    INDEX