This volume contains the Proceedings of the IUTAM Symposium held in Liverpool in 2002. It includes the articles presenting the results of recent work in mathematical modelling that covers the following areas of continuum mechanics and theoretical physics: perturbation problems for partial differential equations and their applications in mechanics; analysis of singular fields; homogenisation theory in models of composite structures; mathematical models of cracks in solids; wave propagation, scattering; models of photonic and phononic band gap composite structures; and advanced numerical techniques. This book is intended to contribute towards wider dissemination of the recent developments in asymptotic analysis, mathematical modelling in continuum mechanics and engineering applications.

Table of Contents

PREFACE CHAPTER 1: Wave Propagation and ScatteringJ. Kaplunov, V. Kovalev and M. Wilde, Asymptotic Analysis of Higher Order Peripheral Modes in Acoustic Wave Scattering by an Elastic Cylinder or Sphere C.M. Linton, Embedding Formulas and Singularities in Acoustic Scattering A. Nicolet, S. Guenneau, F. Zolla, C. Geuzaine, B. Kulhmey and G. Renversez, Numerical Investigation of Photonic Crystal Fibers by Spectral and Multipole methods Y.A. Antipov and V.V. Silvestrov, Method of Hyperelliptic Surfaces for Vector Functional-Difference Equations S.V. Vladimirov, M.P. Hertzberg and N.F. Cramer, Dynamics of Charge Rotators and Lattice Waves in a Plasma Environment O. Avila-Pozos, A.B. Movchan and S.V. Sorokin, Propagation of Elastic Waves Along Interfaces in Layered Beams S.B. Platts and N.V. Movchan, Low Frequency Band Gaps and Localised Modes for Arrays of Coated Inclusions J.S. Jensen and O. Sigmund, Phononic Band Gap Structures as Optimal Designs I.M. Mohamed-Guled, J.B. Lawrie, A Parametric Investigation of the Acoustic Power in a Two-dimensional Waveguide with Membrane Bounded Cavity J. Servant, S. Guenneau, A.B. Movchan and C. Poulton, Vibrations of a Circular Cylinder in Oblique Incidence RevisitedI.F. Bercial-Velez, Asymptotic Analysis of a "Crack" in a Layer of Finite ThicknessV. Zalipaev, Summation of Gaussian Beams in 3D Problems of Radiation and Scattering of Elastic Waves CHAPTER 2: Asymptotics for Eigenvalue Problems A.G. Aslanyan, A.B. Movchan and O. Selsil, Asymptotics for the First Six Eigenfrequencies of a 1D-3D Multi-structure E. Babenkova and J. Kaplunov, The Two-term Interior Asymptotic Expansion in the Case of Low-frequency Longitudinal Vibrations of an Elongated Elastic Rectangle C.G. Poulton, S. Guenneau, A.B. Movchan and A. Nicolet, Transverse Propagating Waves in Perturbed Periodic Structures D.J. Hasanyan, G.T. Piliposian, A.H. Kamalyan and M.I. Karakhanyan, Anti-Plane Harmonic Problems for a Class of Elastic Materials with Functional Inhomogeneity J.B. Lawrie and R. Kirby, On Analysing the Performance of a Dissipative Silencer: a Mode-matching Approach CHAPTER 3: Green's Functions and Defect Modes C.G. Poulton, R.C. McPhedran, N.A. Nicorovici and L.C. Botten, Localized Green's Functions for a Two-Dimensional Periodic Material P.G. Martinsson, G.J. Rodin, Boundary Algebraic Equations for Lattice Problems CHAPTER 4: Mathematical Models of Cracks J.R. Willis, Dynamic Perturbation of a Propagating Crack: Implications for Crack Stability O. Obrezanova, A.B. Movchan and J.R. Willis, Dynamic Crack Stability O. Selsil, I.S. Jones and A.B. Movchan, A Thermoelasticity Problem in a Domain with an Edge Crack: Asymptotic Analysis I.S. Jones, Methods of Assessment of Thermal Striping Fatigue Damage M. Gei, K. Radi, Near-tip Fields of Mode III Steady-state Crack Propagation in Elastic-plastic Strain Gradient Solids G.S. Mishuris, Mode III Interface Crack Lying at Thin Nonhomogeneous Anisotropic Interface. Asymptotics near the Crack Tip M.N. Perelmuter, Fracture Criterion for Cracks with Bridged Zone CHAPTER 5: Models of Damage in Solids A. Cherkaev and L. Zhornitskaya, Dynamics of Damage in Two-dimensional Structures with Waiting Links S.E. Mikhailov and I.V. Namestnikova, Local and Non-local Approaches to Fatigue Crack Initiation and PropagationJ. Sivaloganathan and S.J. Spector, A Variational Approach to Modelling Initiation of Fracture in Nonlinear Elasticity H. Gao and B. Ji, Modeling Fracture in Nano-MaterialsM. Brun, D. Bigoni and D. Capuani, Boundary Elements and Shear Bands in Incremental Elasticity CHAPTER 6: Homogenisation Analysis and Models of Composite Media E. Cherkaev, Spectral Coupling of Effective Properties of a Random Mixture M. Hori, T. Ichimura and H. Nakagawa, Homogenization Techniques Applied to Earthquake Problems A. Fadili, P.M.J. Tardy and J.R.A. Pearson, Stochastic Homogenisation of Fluid Flows in Heterogeneous Porous MediaI. Monetto and W.J. Drugan, On Micromechanics-Based Nonlocal Modeling of Elastic Matrices Containing Non-Spherical Heterogeneities F. Zolla and S. Guenneau, Artificial Ferro-magnetic Anisotropy: Homogenization of 3D Finite Photonic Crystals J. Bravo-Castillero, R. Guinovart-Diaz, R. Rodriguez-Ramos, F.J. Sabina and O.C. Valdiviezo-Mijangos, Homogenized Stiffnesses of Periodic Fibre-Reinforced Composites S.E. Mikhailov and J. Orlik, Asymptotic Homogenisation in Strength and Fatigue Durability Analysis of Composites CHAPTER 7: Models Associated with Elastic Plates, Shells, Rods and Thin-walled Composite Structures E. Sanchez-Palencia, On a Kind of Singular Perturbations for Transmission Problems H. Ranarivelo and J. Sanchez-Hubert, Asymptotics of Laminated Shells. Membrane - Bending Coupling and Numerical ImplementationA.J.M. Spencer, Exact Singular Solutions for an Inhomogeneous Thick Elastic Plate D.F. Parker, Stretch, Flexure amid Twist in Finite Elasticity A. Selsil, A.B. Movchan, N.V. Movchan, Asymptotic Analysis of Heat Transfer in a System of Channels Connected by Thin Conducting Walls CHAPTER 8: Asymptotics of Fields near Non-smooth Boundaries D. Esparza and N.V. Movchan, Effect of a Thin Coating on the Stress Singularity at the Vertex of a Thin Conical Inclusion K. Kolk, G. Kuhn and G. Mishuris, 3D Singularities with Applications to Fatigue Crack GrowthCHAPTER 9: Mathematical Models of Gas and Fluid FlowsL.E. Fraenkel and J.B. McLeod, A Diffusing Vortex Circle in a Viscous Fluid D.A. MacDonald, Low Axial Reynolds Number, High Swirl Ratio, Vortex Breakdown in a Slowly Varying Tube C.J. Powles, Supersonic Leading-edge NoiseCHAPTER 10: Non-linear Waves V.N. Biktashev, Envelope Equations for Modulated Non-conservative Waves K.R. Khusnutdinova, Nonlinear Waves in a Bi-Layer and Coupled Klein-Gordon Equations CHAPTER 11: Combustion Models V. Karlin, Nonmodal Instability as the Mechanism of Acceleration of Cellular Flames S.T. Kolaczkowski, Catalytic Combustion: Making the Connection Between the Physical/Chemical Processes and the Mathematical Model of the Reactor CHAPTER 12: Asymptotic Approximations for Orbital Perturbations P.J. Message, The Use of Lie Series Generating Functions to Derive Asymptotic Series Expressions for Orbital Perturbations CHAPTER 13: Advanced Numerical Techniques A. Spence and C.G. Poulton, Inverse Iteration for Nonlinear Eigenvalue Problems in Electromagnetic Scattering M.D. Hughes and K. Chen, Fast Iterative Solution of Coupled 3-Dimensional Fluid--Structure Interaction Problems S.C. Hawkins and K. Chen, Preconditioning for Finger Pattern Matrices Arising from Wavelet Discretisations of Boundary Integral EquationsAPPENDIX: IUTAM Symposium Programme Index