Stochastic Mechanics of Discrete Media

ISBN-13: 9783642514876 / Angielski / Miękka / 2012 / 335 str.

David R. Axelrad

Opis

For the past three decades the mechanics of structured media, frequently called micromechanics, has been recognized as an important new approach in the analysis of material behaviour. This book discusses the modern use of mathematical analysis to the stochastic mechanics of discrete media. The theoretical study is therefore based on set and measure theory and the application of point processes.

Spis treści

1 Introduction to the Probabilistic Analysis.- 1.1 Historical Remarks on Probability and its Application in Mechanics.- (i) The notion of probability.- (ii) Continuity and discontinuity.- (iii) Determinism and Probabilism.- 1.2 Topological Spaces, Sets and Operators.- (i) Topological Spaces and Sets.- (ii) Vector spaces and convexity.- (iii) Linear operators and bilinear forms.- 1.3 Probability and Random Variables.- (i) Probability.- (ii) Random variables.- 1.4 Probability Measures.- (i) General remarks on measures.- (ii) Probability measures.- 1.5 Dependence of Random variables.- (i) Independent random variables.- (ii) Dependent random variable.- 1.6 Stochastic Processes.- (i) Characteristics of stochastic processes.- (ii) Regularity and Continuity.- (iii) Some basic stochastic processes.- (iv) Random fields.- 2 Phenomenology of Discrete Media.- 2.1 Classification of Materials.- (i) Introduction.- (ii) Classification of microstructures.- (iii) Idealized microstructures and fundamental concepts.- 2.2 Statistical Models of Discrete Media.- (i) Disorder effects.- (ii) The local approach.- (iii) Molecular dynamics and correlation functions.- (iv) Lattice models.- (v) Percolation models.- 2.3 Probabilistic Models of Discrete Media.- (i) Introduction.- (ii) Observables and States.- (a) Observables.- (b) State-space representation.- (iii) The state-space and constitutive maps.- 2.4 Markov Processes and Stochastic Differential Equations.- (i) Markov processes.- (ii) Stochastic differential equations.- 3 Random Evolution and Geometric Probabilities.- 3.1 Wide-sense Markov Processes.- (i) Wide-sense Markov processes.- (ii) Partially observed Markov processes.- (iii) Random evolution of discrete media.- (a) Transient behaviour of a structured solids.- (b) Evolution relations for simple fluids.- 3.2 Interaction effects in Discrete Media.- (i) Interaction potentials.- (ii) Stochastic models of interfacial behaviour in solids.- (iii) Markov models of bond failure and fracture in solids.- (iv) Interfaces in fluids.- 3.3 Introduction to Geometric Probabilities.- (i) Introduction.- (ii) Random sets.- (iii) Random point models.- (a) The Boolean model.- (b) Other point models.- 3.4 Some Fundamental Concepts of Stereology.- 4 Applications of the Stochastic Analysis.- 4.1 The Response Behaviour of Discrete Solids.- (i) A general stochastic deformation theory.- (ii) Deformational stability of structured solids.- (iii) The inelastic behaviour multi-component solids.- (iv) General remarks on material operators.- 4.2 The response of Polycrystalline Solids.- (i) The elastic response including interactions.- (ii) Inelastic behaviour of MC-systems (application of Point processes).- (iii) Dynamics of structured solids.- (a) Stochastic models of wave propagation.- (b) Application of the Monte-Carlo simulation.- 4.3 The Stochastic Analysis of Fibrous and Polymeric Networks.- (i) Stochastic mechanics of fibrous structures.- (ii) Stochastic analysis of polymer melts.- (a) Poissonian behaviour of entanglement of the polymers.- (b) Local balance relations and flow dynamics.- 4.4 Simple Fluids and the Flow in Fully Saturated Porous Media.- (i) The dynamics of discrete fluids.- (ii) Markov theory in the mechanics of discrete fluids.- (iii) Flow through a fully saturated porous medium.- References.