Advances in Mechanics and Mathematics

1 Fracture Mechanics of Functionally Graded Materials.- 1 Introduction.- 2 Mechanics Models.- 2.1 Mechanics Modeling.- 2.2 Elasticity Equations of FGMs.- 2.3 Effective Elastic Properties.- 3 Crack Tip Mechanics.- 3.1 Crack Tip Elastic Fields.- 3.2 K - Dominance.- 4 Stress Intensity Factor Solutions.- 4.1 Integral Transform/Integral Equation Method.- 4.2 Numerical Methods.- 5 Fracture Toughness and Crack Growth Resistance Curve.- 5.1 Fracture toughness Based on a Rule of Mixtures.- 5.2 Crack Growth Resistance Curve Based on a Crack Bridging Mechanism.- 5.3 Residual Strength.- 5.4 Crack Kinking under Mixed Mode Conditions.- 6 Thermofracture Mechanics.- 6.1 Heat Conduction Equations of FGMs.- 6.2 Thermoelasticity Equations of FGMs.- 6.3 A Heat Conduction Problem.- 6.3.1 A multi-layered material model.- 6.3.2 Interface temperatures for short times.- 6.3.3 A closed form solution of temperature field for short times.- 6.4 A Thermal Crack Problem.- 7 Stationary Cracks in Viscoelastic FGMs.- 7.1 Correspondence Principle.- 7.2 Relaxation Functions in Separable Form in Space and Time.- 7.3 Viscoelastic Crack Tip Fields.- 7.4 Stress Intensity Factors for FGMs with Variables Separable Relaxation Functions.- 8 Fracture Dynamics.- 8.1 Basic Equations.- 8.2 Stationary Cracks Subjected to Dynamic Loading.- 8.3 Crack Propagation.- 9 Fracture Simulation Using a Cohesive Zone Model.- 9.1 A Cohesive Zone Model.- 9.2 Plasticity of FGMs and Tamura-Tomota-Ozawa Model.- 9.3 Cohesive Elements.- 9.4 Calibration of Cohesive Fracture Parameters.- 9.5 Fracture Simulation.- 9.6 Effect of Peak Cohesive Traction for Ceramic Phase.- 10 Concluding Remarks.- References.- 2 Topics in Mathematical Analysis of Viscoelastic Flow.- 1 Introduction.- 2 High Weissenberg number asymptotics.- 2.1 The Euler equation.- 2.2 High Weissenberg number boundary layers.- 2.3 Flow near a reentrant corner.- 3 Instabilities in viscoelastic flows.- 3.1 Parallel shear flows.- 3.2 Shear flows with curved streamlines.- 3.3 Two-layer flows.- 3.4 Open mathematical questions.- 4 Breakup of viscoelastic jets.- 4.1 One-dimensional theory.- 4.2 The Newtonian case.- 4.3 Suppression of breakup.- 4.4 The Giesekus model.- 4.5 Elastic breakup.- 4.6 The role of inertia.- References.- 3 Selected Topics in Stochastic Wave Propagation.- 1 Basic Methods in Stochastic Wave Propagation.- 1.1 The long wavelength case.- 1.1.1 Elementary considerations.- 1.1.2 Series expansion.- 1.2 The short wavelength case - ray method.- 1.2.1 Fermat's principle.- 1.2.2 Smooth inhomogeneity vis-a-vis local isotropy.- 1.2.3 Eikonal equation.- 1.2.4 Markov character of rays.- 1.3 The short wavelength case - Rytov method.- 2 Towards Spectral Finite Elements for Random Media.- 2.1 Spectral finite element for waves in rods.- 2.1.1 Deterministic case.- 2.1.2 Random case.- 2.2 Spectral finite element for flexural waves.- 2.2.1 Deterministic case.- 2.2.2 Random case.- 2.3 Observations and related work.- 3 Waves in Random 1-D Composites.- 3.1 Motion in an Imperfectly Periodic Composite.- 3.1.1 Random evolutions.- 3.1.2 Effects of imperfections on Floquet waves.- 3.2 Waves in randomly segmented elastic bars.- 4 Transient Waves in Heterogeneous Nonlinear Media.- 4.1 A class of models of random media.- 4.2 Pulse propagation in a linear elastic microstructure.- 4.3 Pulse propagation in nonlinear microstructures.- 4.3.1 Bilinear elastic microstructures.- 4.3.2 Nonlinear elastic microstructures.- 4.3.3 Hysteretic microstructures.- 5 Acceleration Wavefronts in Nonlinear Media.- 5.1 Microscale heterogeneity versus wavefront thickness.- 5.1.1 Basic considerations.- 5.1.2 Mesoscale response.- 5.2 Wavefront dynamics in random microstructures.- 5.2.1 Model with one white-noise.- 5.2.2 Model with two correlated Gaussian noises.- 6 Closure.- References.- 4 Periodic Soliton Resonances.- 1 Introduction.- 2 N-periodic soliton solutions to the KP equation with positive dispersion.- 3 Periodic soliton resonances I: solutions to the KP equation with positive dispersion.- 3.1 Resonant interactions between two y-periodic solitons.- 3.2 Resonant interaction between line soliton and y-periodic soliton.- 3.3 Resonant interaction between algebraic soliton and y-periodic soliton.- 3.4 Resonant interaction between inclined line soliton and periodic soliton.- 4 Periodic soliton solutions to the DS I equation.- 5 Periodic soliton resonances II: solutions to the DSI equation.- 5.1 Resonant interaction between two y-periodic solitons to the DSI equation.- 5.2 Resonsnt interactions between line soliton and y-periodic soliton to the DSI equation.- 5.3 Resonant interaction of modulational instability with a line soliton.- 6 Soliton stability theory due to periodic soliton resonance solution.- 6.0.1 Linear stability.- 6.0.2 Stabilitiy theory due to periodic soliton resonance solution.- 7 Summary.- References.- 5 Nonconvex Semi-Linear Problems and Canonical Duality Solutions.- 1 Nonconvex Problems and New Phenomena.- 1.1 Semi-linear equations and double-well potential.- 1.2 Parameter effects: meta-chaos and trio-chaos.- 1.3 Global optimization and NP-hard problems.- 2 Canonical Duality Theory: A brief Review.- 2.1 Clarke-Ekeland-Lasey duality.- 2.2 Lagrangian duality.- 2.3 Canonical duality theory.- 3 Canonical Dual Theory and Solutions.- 3.1 Canonical dual transformation and perfect dual problem.- 3.2 Complete set of solutions.- 3.3 Global minimizer and local extrema.- 4 Applications to Unconstrained Global Optimization.- 4.1 Quadratic W(?).- 4.2 Concave W(?).- 5 Application to Constrained Quadratic Programming.- 5.1 Canonical dual formulation.- 5.2 KKT points and global minimizers.- 5.3 Examples.- 6 Quadratic Programming Over a Sphere.- 7 Concluding Remarks.- References.