Advances in Mechanics and Mathematics: Volume II

ISBN-13: 9781461379591 / Angielski / Miękka / 2011 / 314 str.

David Yang Gao; Raymond W. Ogden

Opis

As any human activity needs goals, mathematical research needs problems -David Hilbert Mechanics is the paradise of mathematical sciences -Leonardo da Vinci Mechanics and mathematics have been complementary partners since Newton's time and the history of science shows much evidence of the ben eficial influence of these disciplines on each other. Driven by increasingly elaborate modern technological applications the symbiotic relationship between mathematics and mechanics is continually growing. However, the increasingly large number of specialist journals has generated a du ality gap between the two partners, and this gap is growing wider. Advances in Mechanics and Mathematics (AMMA) is intended to bridge the gap by providing multi-disciplinary publications which fall into the two following complementary categories: 1. An annual book dedicated to the latest developments in mechanics and mathematics; 2. Monographs, advanced textbooks, handbooks, edited vol umes and selected conference proceedings. The AMMA annual book publishes invited and contributed compre hensive reviews, research and survey articles within the broad area of modern mechanics and applied mathematics. Mechanics is understood here in the most general sense of the word, and is taken to embrace relevant physical and biological phenomena involving electromagnetic, thermal and quantum effects and biomechanics, as well as general dy namical systems. Especially encouraged are articles on mathematical and computational models and methods based on mechanics and their interactions with other fields. All contributions will be reviewed so as to guarantee the highest possible scientific standards."

Spis treści

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-à-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.