8. Rigidity and Elastic Properties: The Discrete Approach 8. 0 Introduction 8. 1 Elastic Networks in Biological Materials 8. 2 Number of Elastic Moduli of a Lattice 8. 3 Numerical Simulation and Finite-Size Scaling 8. 4 Derivation of Elastic Networks from Continuum Elasticity 8. 4. 1 The Born model 8. 4. 2 Shortcomings of the Born model 8. 5 The Central-Force Network 8. 6 Rigidity Percolation 8. 6. 1 Static and dynamic rigidity and ?oppiness of networks 8. 6. 2 The correlation length of rigidity percolation 8. 6. 3 The force distribution 8. 6. 4 Determination of the percolation threshold 8. 6. 4. 1 Moments of the force distribution 8. 6. 4. 2 The pebble game 8. 6. 4. 3 Constraint-counting method 8. 6. 5 Mapping between rigidity percolation and resistor networks 8. 6. 6 Nature of phase transition in rigidity percolation 8. 6. 7 Scaling properties of the elastic moduli 8. 7 Green Function Formulation and Perturbation Expansion 8. 7. 1 E?ective-medium approximation 8. 7. 2 The Born model 8. 7. 3 Rigidity percolation 8. 8 The Critical Path Method 8. 9 Central-Force Networks at Non-zero Temperature and under Stress 8. 10 Shortcomings of the Central-Force Networks 8. 11 Elastic Percolation Networks with Bond-Bending Forces 8. 11. 1 The Kirkwood-Keating model xiv 8. 11. 2 The bond-bending model 8. 11. 3 The percolation thresholds 8. 11. 4 The force distribution 8. 11. 5 Comparison of the central-force and bond-bending networks 8. 11.

This book describes and discusses the properties of heterogeneousmaterials, both linear and nonlinear properties, providingcomprehensive and up to date theoretical and computer simulationanalysis of materials' properties. Volume I covers linear propertieswhile Volume II considers non-linear and fracture and breakdownproperties, as well as atomistic modeling.This multidisciplinary book will appeal to applied physicistsmaterials scientists, chemical and mechanical engineers, chemists, andapplied mathematicians.

Characterization and Modelling of the Morphology.- Characterization of Connectivity and Clustering.- Characterization and Modelling of the Morphology.- Linear Transport and Optical Properties.- Effective Conductivity, Dielectric Constant and Optical Properties: The Continuum Approach.- Effective Conductivity and Dielectric Constant: The Discrete Approach.- Frequency-Dependent Properties: The Discrete Approach.- Rigidity and Elastic Properties: The Continuum Approach.- Rigidity and Elastic Properties: The Discrete Approach.- Rigidity and Elastic Properties of Network Glasses, Prolymers, and Composite Solids: the Discrete Approach.