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Today, Fracture Mechanics is a well known topic within the scientific community. Applications of Fracture Mechanics can be found in various fields ranging from solid mechanics and structures to materials sciences and computational mechanics. However, most of these results apply only to linear fracture mechanics of two-dimensional and homogeneous isotropic solids. Therefore there are still incompletely solved problems; such as non-linearity, frictional contact cracks, residual stresses in fracture mechanics, three-dimensional crack geometry, coupled cracked solid/fluid, etc. Recently, new topics related to crack detection based on different physical phenomena have appeared. This book is an attempt to present, in a unified manner, different topics of Continuum and Fracture Mechanics: energy methods, conservation laws, mathematical methods to solve two-dimensional and three-dimensional crack problems. Moreover, a series of new subjects is presented in a straightforward manner, accessible to under-graduate students. These new topics take into consideration the thermodynamics of continuous media, including thermal and dynamical aspects. In addition, the book introduces the notion of duality or symmetry in Solids Mechanics. The loss of symmetry is exploited to provide a unique and powerful tool, called the reciprocity gap functional introduced by the author's groups, to solve explicitly some important inverse problems arising in crack determination as well as in the earthquake inverse problem.With its emphasis, initially on physical or experimental back-grounds, and then on analysis and theoretical results, rather than on numerical computations, this monograph is intended to be used by students and researchers in solids mechanics, mechanical engineering and applied mathematics.

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Title:Fracture Mechanics: Inverse Problems and SolutionsFormat:PaperbackDimensions:398 pagesPublished:October 10, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048172071

ISBN - 13:9789048172078

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Table of Contents

Part I Fracture Mechanics: 1. Deformation and Fracture: 1.1. Deformation: Geometric transforms; Small strain; Compatibility condition; Stress. 1.2. Elasticity : Constitutive law ; Tonti's diagram in elasticity; Plasticity : Experimental yield surfaces; Prandt-Reuss equation; 1.3 Fracture : Introduction to Fracture Mechanics; Stress-intensity Factors; On the physics of separation; Different types of fractures (ductile fracture, fatigue Paris's law, Dangvan's criterion); Brittle fracture criterion. 2. Energetic aspects of fracture 2.1 Griffith's theory of fracture Some expressions of G in quasi-statics (Energy release rate). 2.2 Some expressions of G in quasi-statics (Energy release rate). 2.3 Irwin's formula. 2.4 Barenblatt's cohesive force model 2.5 Berry's interpretation of energies 2.6 Stability analysis of multiple cracks 2.7 An inverse energetic problem 2.8 Path-independent integrals in quasi-statics : The path-independent J-integral ; Associated J-integrals for separating mixed modes; The Tintegral in linear thermoelasticity; Lagrangian derivative of energy and the G0 -integral 2.9 Generalization of Griffth's model in three dimensions : A local model of viscous fracture; A non local model of fracture; A dissipation rate model for non local brittle fracture; Convex analysis of three- dimensional brittle fracture. 3. Solutions of crack problems 3.1 Mathematical problems in plane elasticity : Plane strain and antiplane strain; Plane stress condition revisited ; Complex variables in elasticity; The Hilbert problem. 3.2 The finite crack in an infinite medium : The auxiliary problem ; Dugdale -Barenblatt's model; Remote uniform stress. 3.3 The kinked crack in mixed mode : An integral equation of the kinked crack problem; The asymptotic equation. 3.4 Crack problems in elasto-plasticity: Matching asymptotic solutions; A complete solution plasticity and damage; A review of asymptotic solutions in non-linear materials. 3.5 Inverse geometric problem with Coulomb's friction: Non-uniqueness of solution in friction crack ; Solution to the frictional crack problem without opening ; The energy release rate of a frictional interface crack ; The frictional interface crack problem with an opening zone 4. Thermodynamics of crack propagation 4.1 An elementary example 4.2 Dissipation analysis 4.3 Thermal aspects in crack propagation 4.4 Singularity of the temperature in thermo-elasticity 4.5 Asymptotic solution of the coupled equations 5. Dynamic Fracture Mechanics 5.1 Experimental aspects of crack propagation. 5.2 Fundamental equations 5.3 Steady state solutions 5.4 Transient crack problems : Symmetric extension of a crack ; Semi-infinite crack with arbitrary propagation speed 5.5 The Wiener-Hopf technique ; Diffraction of waves impinging a semi- infinite crack 5.6 . Path-independent integrals for moving crack 5.7 A path-independent integral for crack initiation analysis : Inverse problems in dynamic fracture ; A new experimental method for dynamic toughness. 5.8 Some other applications of dynamic fracture 6. Three-dimensional cracks problems 6.1 Fundamental tensors in elastostatics : The Kelvin-Somigliana's tensor; The Kupradze-Bashelishvili tensor ; Singularity analysis 6.2 Fundamental theorems in elastostatics : Solution of the Neumann boundary value problem ; Solution of the Dirichiet boundary value problem ; Direct methods using Kelvin-Somigliana's tensor 6.3 A planar crack in an infinite elastic medium : The symmetric opening mode I ; The shear modes 6.4 A planar crack in a bounded elastic medium : Singularity analysis; Solutions of some crack problems 6.5 The angular crack in an unbounded elastic medium 6.6 The edge crack in an elastic half-space 6.7 On some mathematical methods for BIE in 31) : The Kupradze elastic potentials theory ; On the regularization of hypersingular integrals; Other regularization methods 6.8 An integral equation in elasto-plasticity 7. Non linear fracture mechanics 7.1 Introduction 7.2 Ductile fracture : Rousselier's model ; The micro-mechanics of plasticity; Gurson's model ; Extension of porous plasticity models to aggregates 7.3 Bifurcation problems in plasticity 7.4 A finite strain theory of cavitation in solids : Abeyratne and Hou's solution in finite elasticity ; Solution for creeping materials 8. The fluid-filled crack 8.1 Introduction 8.2 The Leak Before Break inverse problem : Empirical models of fluid flow in a crack breach ; variable breach area 8.3 Wear mechanics : Wear criterion and wear rate conservation of mass; Rheology of the third body ; The W-equation in the sliding of a punch on an half-plane ; Identification of constants. 8.3 Hydraulic fracturing of rocks : equations in hydraulic fracturing of rocks 8.4 Capillary phenomenon in fracture mechanics : The equilibrium crack partially filled with a fluid ; Capillary stress intensity factor 8.5 Viscous fluid flow solution near the moving crack tip : Equation for the fluid-filled moving crack ; Numerical results Part II Inverse problems and solutions 9. Methods for defect and crack detection by scattering of waves 9.1 Introduction 9.2 Scattering of acoustic waves : Rigid indusion ; Flat cavity ; Finite spectrum and finite number of incident waves 9.3 Diffraction of elastic waves 9.4 Non destructive testing of materials : A case study 9.5 Time-reversal mirror (ThM) : Experimental validation of ThM ; The mathematics of time reversal mirror 10. Tomographic evaluation of materials 10.1 Introduction 10.2 X-rays Tomography : Inverse Radons transform; Example of Crack detection. 10.3 Attenuated Radon transform : Novikovs inversion formula. 10.4 Conical Radon transform in Compton scattering : The Conical Radon transform ; Nguyen & Truong's inversion formula. 11. The Reciprocity Gap Functional (RGF) for crack detection 11.1 Distributed defects and cracks. Calderons solution. 11.2 Planar crack identification in quasi-static * elasticity : Determination of the normal to the crack plane; Determination of the crack plane ; Determination of the crack shape 11.3 The instantaneous RG functional 11.3 Inverse problem for the heat diffusion equation: Solution for the crack plane location ; solution for the crack shape 11.4 Inverse acoustic scattering of a crack in time domain 11.5 Elastodynamic scattering of a crack in time domain : The observation equation in elastodynamics 11.7 The earthquake inverse problem 12. Methods of solution to Inverse Problems 12.1 The ill-posedness of the inverse problem 12.2 General considerations on inverse problems 12.3 Tikhonov's regularization 12.4 Control theory : Control of an evolution equation ; Pontryagin's minimum principle ; Bellman's dynamic programming 12.5 The dynamic formulation of quasi static elasticity. : Smoothing operators ; The transfer matrix operator in elasticity. 12.6 Quasi-reversibility methods : Cauchy problem for elliptic equation. 12.7 Control theory for partial derivative equations : Inverse problem to determine the heat conduction coeficient field; Inverse problem to determine a constitutive law. 12.8 Stochastic inversion methods.Appendix : Problems and solutions Index References

Editorial Reviews

From the reviews:"This monograph is mainly a scope of research results of group of scientists of École Polytechnique-Paris. The results concern crack theory and associated fields as fracture, yielding and material science. . There are many problems for discussions, e.g. the Dugodale-Barenblatt cracks are absolutely different from physical point of view. . Altogether the monograph is an interesting and valuable contribution and can be used by researchers and graduate students." (Jozef Golecki, Zentralblatt MATH, Vol. 1108 (10), 2007)