Stochastic Mechanics of Discrete Media

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.