Microstructural Randomness and Scaling in Mechanics of Materials. Martin Ostoja-Starzewski. University of Illinois at Urbana-Champaign

Transcription

1 Microstructural Randomness and Scaling in Mechanics of Materials Martin Ostoja-Starzewski University of Illinois at Urbana-Champaign

2 Contents Preface ix 1. Randomness versus determinism ix 2. Randomness and scales in mechanics of materials x 3. Outline of contents xiii Chapter 1. Basic Random Media Models 1 1. Probability measure of geometric objects Definitions of probability Probabilities on countable and Euclidean sample spaces Random points, lines and planes Basic point fields Bernoulli trials Example model of a fiber structure of paper Generalization to many types of outcomes Binomial and multinomial point fields Bernoulli lattice process Poisson point field Directional data Basic concepts Circular models Random fibers, random line fields, tessellations Poisson random lines in plane Finite fiber field in plane Random tessellations Basic concepts and definitions of random microstructures General Towards mathematical morphology 34 Chapter 2. Random Processes and Fields Elements of 1D random fields Scalar random fields Vector random processes Mechanics problems on 1D random fields Propagation of surface waves along random boundaries Fracture of brittle micro-beams Elements of 2D and 3D random fields Scalar and vector fields 53 iii

3 iv CONTENTS 3.2. Random tensor fields Mechanics problems on 2D and 3D random fields Mean field equations of random materials Mean field equations of turbulent media Ergodicity Basic considerations Computation of (5.1) Conditions for (5.1) to hold Existence of the limit in (5.1) The maximum entropy method Cracks in plates with holes Disorder and information entropy 70 Chapter 3. Planar Lattice Models Periodic Topologies and Elastostatics One-dimensional lattices Simple lattice and elastic strings Micropolar lattice and elastic beams Planar lattices: classical continua Basic idea of a spring network representation Anti-plane elasticity on square lattice In-plane elasticity: triangular lattice with central interactions In-plane elasticity: triangular lattice with central and angular interactions Triple honeycomb lattice Applications in mechanics of composites Representation by a fine mesh Solutions of linear algebraic problems Example: simulation of a polycrystal Planar lattices: non-classical continua Triangular lattice of Bernoulli-Euler beams Triangular lattice of Timoshenko beams From stubby beams to a perforated plate model Hexagonal lattice of Bernoulli-Euler beams Square lattice of Bernoulli-Euler beams Non-local and gradient elasticity on a lattice with central interactions Plate-bending response Extension-twist coupling in a helix Constitutive properties Harmonic waves Viscoelastic helices 116 Chapter 4. Lattice Models: Rigidity, Randomness, Dynamics, and Optimality 119

4 CONTENTS v 1. Rigidity of networks Structural topology and rigidity percolation Application to cellulose fiber networks Spring network models for disordered topologies Granular-type media Solutions of truss models Mesoscale elasticity of paper Damage patterns and maps of disordered composites Particle models Governing equations Examples Michell trusses: optimal use of material Study via hyperbolic system Study via elliptic system 151 Chapter 5. Two- versus Three-Dimensional Classical Elasticity Basic relations Isotropic relations Plane elasticity models Special planar orthotropies The CLM result and stress-invariance Isotropic materials Anisotropic materials and the null-lagrangian Mulitply connected materials Applications to composites Extension of stress invariance to presence of eigenstrains Poroelasticity 170 Chapter 6. Two- versus Three-Dimensional Micropolar Elasticity Micropolar elastic continua Force transfer and degrees of freedom Equations of motion and constitutive equations Isotropic micropolar materials Virtual work principle Hamilton s principle Reciprocity relation Elements of micropolar elastodynamics Non-centrosymmetric micropolar elasticity Classical vis-à-vis non-classical (elasticity) models A brief history The ensemble average of a random local medium is non-local Planar Cosserat elasticity First planar problem Characteristic lengths in isotropic and orthotropic media Chapter written jointly with I. Jasiuk

5 vi CONTENTS 3.3. Restricted continuum vis à-vis micropolar model The CLM result and stress-invariance Isotropic materials Anisotropic materials and the null-lagrangian Multiply connected materials Applications to composites Extensions of stress invariance to presence of eigenstrains and eigencurvatures Effective micropolar moduli and characteristic lengths of composites From a heterogeneous Cauchy to a homogeneous Cosserat continuum Applications 211 Chapter 7. Mesoscale Bounds for Linear Elastic Microstructures Micro, meso and macroscales Separation of scales Basic concepts The RVE postulate Volume averaging A paradigm of boundary conditions effect Hill condition Apparent properties Spatial randomness Stationarity of spatial statistics Ergodicity of spatial statistics Hierarchies of mesoscale bounds Response under displacement boundary condition Response under traction boundary condition Scale-dependent hierarchy Homogenization theory viewpoint Apparent moduli in in-plane elasticity Examples of hierarchies of mesoscale bounds Random chessboards and Bernoulli lattices Disk-matrix composites Functionally graded materials Effective and apparent moduli of multi-cracked solids Moduli of Trabecular Bone 248 Chapter 8. Random Field Models and Stochastic Finite Elements Mesoscale random fields From discrete to continuum random fields Scale dependence via beta distribution Mesoscopic continuum physics due to Muschik Second-order properties of mesoscale random fields 255

6 CONTENTS vii 2.1. Governing equations Universal properties of mesoscale bounds Correlation structure of mesoscale random fields Does there exist a locally isotropic, smooth elastic material? Correlation theory viewpoint Micromechanics viewpoint Closure Stochastic finite elements for elastic media Bounds on global response Example: torsion of a duplex-steel bar An overview of phenomenological SFE studies Method of slip-lines for inhomogeneous plastic media Finite difference spacing vis-à-vis grain size Sensitivity of boundary value problems to randomness Michell trusses in the presence of random microstructure Truss-like continuum vis-à-vis random microstructure Solution via hyperbolic system Solution via elliptic system 282 Chapter 9. Hierarchies of Mesoscale Bounds for Nonlinear and Inelastic Microstructures Physically nonlinear elastic microstructures General Power-law materials Random formation vis-à-vis inelastic response of paper Finite elasticity of random composites Averaging theorems Variational principles and mesoscale bounds Elastic-plastic microstructures Variational principles and mesoscale bounds Matrix-inclusion composites Geodesic properties of shear-band patterns Rigid-perfectly-plastic microstructures Background Bounding on mesoscales via kinematic and traction boundary conditions Random chessboard of Huber-von Mises phases Viscoelastic microstructures Stokes flow in porous media Thermoelastic microstructures Linear case Non-linear case Scaling and stochastic evolution in damage phenomena Comparison of scaling trends 328

7 viii CONTENTS Chapter 10. Mesoscale Response in Thermomechanics of Random Media From statistical mechanics to continuum thermodynamics Dissipation function of the RVE Departure from the second law of thermodynamics Extensions of the Hill condition Hill condition in thermomechanics Homogenization in dynamic response Legendre transformations in (thermo)elasticity Elasticity Thermoelasticity Thermodynamic orthogonality on mesoscale General Homogeneous dissipation functions Quasi-homogeneous dissipation functions Complex versus compound processes the scaling viewpoint General considerations Micropolar plasticity Towards continuum mechanics of fractal media 351 Chapter 11. Waves and Wavefronts in Random Media Basic methods in stochastic wave propagation The long wavelength case The short wavelength case ray method The short wavelength case Rytov method Towards spectral finite elements for random media Spectral finite element for waves in rods Spectral finite element for flexural waves Observations and related work Waves in random 1D composites Motion in an imperfectly periodic composite Waves in randomly segmented elastic bars Transient waves in heterogeneous nonlinear media A class of models of random media Pulse propagation in a linear elastic microstructure Pulse propagation in nonlinear microstructures Acceleration wavefronts in nonlinear media Microscale heterogeneity versus wavefront thickness Wavefront dynamics in random microstructures 395 Appendix. Bibliography 403 Appendix. Index 431