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Summary
Summary
By the detailed analysis of the modern development of the mechanics of deformable media can be found the deep internal contradiction. From the one hand it is declared that the deformation and fracture are the hierarchical processes which are linked and unite several structural and scale levels. From the other hand the sequential investigation of the hierarchy of the deformation and destruction is not carried out.
The book's aim is filling this mentioned gap and investigates the hot topic of the fracture of non-ideal media. From the microscopic point of view in the book we study the hierarchy of the processes in fractured solid in the whole diapason of practically used scales. According the multilevel hierarchical system ideology under "microscopic" we understand taking into account the processes on the level lower than relative present strata. From hierarchical point of view the conception of "microscopic fracture" can be soundly applied to the traditionally macroscopic area, namely geomechanics or main crack propagation. At the same time microscopic fracture of the nanomaterials can be well-grounded too. This ground demands the investigation on the level of inter-atomic interaction and quantum mechanical description.
The important feature of the book is the application of fibred manifolds and non-Euclidean spaces to the description of the processes of deformation and fracture in inhomogeneous and defected continua. The non-Euclidean spaces for the dislocations' description were introduced by J.F. Nye, B.A. Bilby, E. Kröner, K. Kondo in fiftieth. In last decades this necessity was shown in geomechanics and theory of seismic signal propagation. The applications of non-Euclidean spaces to the plasticity allow us to construct the mathematically satisfying description of the processes. Taking into account this space expansion the media with microstructure are understood as Finsler space media. The bundle space technique is used for the description of the influence of microstructure on the continuum metrics. The crack propagation is studied as a process of movement in Finsler space. Reduction of the general description to the variational principle in engineering case is investigated and a new result for the crack trajectory in inhomogeneous media is obtained. Stability and stochastization of crack trajectory in layered composites is investigated.
The gauge field is introduced on the basis of the structure representation of Lie group generated by defects without any additional assumption. Effective elastic and non-elastic media for nanomaterials and their geometrical description are discussed.The monograph provides the basis for more detailed and exact description of real processes in the material.The monograph will be interesting for the researchers in the field of fracture mechanics, solid state physics and geomechanics. It can be used as well by the last year students wishing to become more familiar with some modern approaches to the physics of fracture and continual theory of dislocations.In Supplement, written by V.V.Barkaline, quantum mechanical concept of physical body wholeness according to H. Primas is discussed with relation to fracture. Role of electronic subsystem in fracture dynamics in adiabatic and non-adiabatic approximations is clarified. Potential energy surface of ion subsystem accounting electron contribution is interpreted as master parameter of fracture dynamics. Its features and relation to non-euclidean metrics of defected solid body is discussed. Quantum mechanical criteria of fracture arising are proposed.
Author Notes
Ihar Miklashevich is Head of Laboratory of System Dynamics and Mechanics of Structure and Materials, National Technical University, Belarus.
Table of Contents
Preface to English Edition | p. ix |
Introduction: selection from preface to the first edition | p. x |
Collective effects in mechanics of deformed bodies | p. x |
Generalized mechanics of the continuum | p. x |
Micromechanics and physics | p. xi |
Acknowledgements | p. xiii |
List of Basic Definitions and Abbreviations | p. xv |
List of Figures | p. xvii |
Chapter 1 Deformation Models of Solids: Description | p. 1 |
1.1 Description of hierarchy systems | p. 1 |
1.1.1 General description of hierarchy structures | p. 1 |
1.1.2 Hierarchical space | p. 6 |
1.2 Peculiarities of parameter space structure associated with fracture | p. 7 |
1.2.1 Continual approximation in damage description | p. 8 |
1.2.2 Deformed continuum fibering | p. 11 |
1.3 Hierarchy in continuum models of a deformed solid | p. 12 |
1.3.1 Mechanical properties of an ideal continuum | p. 13 |
1.4 Hierarchy of systems and structures in fracture mechanics | p. 15 |
1.4.1 Hierarchical pattern of fracture process: applications of general systems theory | p. 19 |
1.4.2 Crack fractality, hierarchy and behavior | p. 21 |
1.4.3 Self-organization processes at plastic deformation and fracture | p. 25 |
Certain Outcomes | p. 25 |
Chapter 2 Space Geometry Fundamentals | p. 27 |
2.1 Construction principles of various space types | p. 28 |
2.1.1 Affine space | p. 28 |
2.1.2 Vectors, covectors, and 1-forms and tensors | p. 29 |
2.1.3 Euclidean space | p. 31 |
2.1.4 Generalization: affine connectivity spaces and Riemann space | p. 33 |
2.1.5 Tangent spaces | p. 42 |
2.1.6 Main fibration | p. 44 |
2.1.7 Vertical and horizontal lift | p. 47 |
2.2 Minimum paths | p. 47 |
2.2.1 Covariant differentiation | p. 48 |
2.3 Effect of microscopic defects on continuum | p. 50 |
2.3.1 Basics of continuous approximation for imperfect crystals | p. 53 |
2.4 Finsler geometry and its applications to mechanics of a deformed body | p. 55 |
2.4.1 Finsler space | p. 56 |
2.4.2 h- and v-connectivities | p. 56 |
2.4.3 Fracture geometry of solids | p. 57 |
2.4.4 Metrical Finsler spaces | p. 60 |
2.4.5 Indicatrix and orthogonality condition in Finsler space | p. 62 |
2.5 Description of plastic deformation in generalized space | p. 64 |
2.6 Geometry of nanotube continuum | p. 67 |
Certain Outcomes | p. 69 |
Chapter 3 Microscopic Crack in Defect Medium | p. 71 |
3.1 Fundamentals of quantum fracture theory | p. 73 |
3.1.1 Bond energy and electronic structure | p. 74 |
3.1.2 Thermofluctuation fracture initiation | p. 77 |
3.2 Influence of material defect structure on crack propagation | p. 78 |
3.2.1 Crack-defect interaction in classical elasticity and plasticity theory | p. 78 |
3.2.2 Defect fields and intrinsic metric of continua | p. 79 |
3.2.3 Crack trajectory and characteristics of fracture space | p. 81 |
3.2.4 Crack trajectory in heterogeneous medium with defects in the general case | p. 86 |
3.3 Driving force acting on crack | p. 88 |
3.3.1 Gauge crack theory | p. 91 |
3.4 Connection of defective material structure with crack surface shape | p. 94 |
3.4.1 Equation of crack front as function of metric and field of defects | p. 94 |
3.4.2 Break propagation in a medium | p. 97 |
3.5 Macroscopic group properties of deformation process and gauge fields introduction procedure | p. 99 |
3.5.1 Kinematics | p. 100 |
3.5.2 Dynamics | p. 102 |
3.5.3 Group structure of deformation curve | p. 103 |
3.6 Four-dimensional formalism and conservation laws | p. 108 |
Certain Outcomes | p. 111 |
Chapter 4 Application of General Formalism in Macroscopic Fracture | p. 113 |
4.1 Macroscopic variational approach to fracture | p. 114 |
4.1.1 Thermodynamics of crack growth and influence of weakened bond zone on crack equation | p. 119 |
4.2 Crack trajectory equation as a variational problem | p. 127 |
4.3 Propagation stability and influence of material inhomogeneity on crack trajectory | p. 130 |
4.3.1 Stability of crack trajectory | p. 130 |
4.3.2 Crack propagation in real media | p. 137 |
4.3.3 Trajectory in linear approximation | p. 139 |
4.3.4 Influence of weakened bonds zone on crack trajectory | p. 144 |
4.3.5 Crack propagation across singular border | p. 146 |
4.4 Crack trajectory in media with random structure. Trajectory stochastization | p. 150 |
4.4.1 Correlation function and stochastization length | p. 151 |
4.4.2 Fokker-Planck-Kolmogorov equation and probability description of crack trajectory | p. 153 |
4.4.3 Physical reasons of crack trajectory stochastization | p. 161 |
4.5 Deformation and fracture with account of electromagnetic fields | p. 166 |
4.5.1 Destruction of piezoelectric materials | p. 168 |
4.5.2 Crack propagation in piezoelectric material | p. 170 |
Certain Outcomes | p. 175 |
Chapter 5 Surface, Fractals and Scaling in Mechanics of Fracture | p. 179 |
5.1 Some conceptions of theory of fractals | p. 180 |
5.1.1 Roughness of fracture surface | p. 181 |
5.1.2 Fractal shape of fracture surface | p. 183 |
5.2 Physical way of fractal crack behavior | p. 184 |
5.2.1 Problem statement | p. 184 |
5.2.2 Equation of beam bending | p. 186 |
5.2.3 Solution analysis | p. 187 |
5.3 Crackon | p. 190 |
5.3.1 Movement of dynamical system | p. 190 |
5.3.2 Mass of crackon | p. 191 |
5.4 Crackon in medium | p. 194 |
5.4.1 Acoustic approximation | p. 194 |
5.4.2 Crack-Generated energy flow | p. 196 |
5.5 Fractal dimensionality | p. 198 |
5.6 Internal geometry of the media and fractal properties of the fracture | p. 199 |
5.6.1 Damage mechanics and stable growth of microcrack | p. 200 |
5.6.2 Distribution of microcracks in a sample | p. 201 |
5.6.3 Fractal characteristics of a macroscopic crack | p. 202 |
5.6.4 Exclusion principles and fractal dimension of crack trajectory | p. 207 |
5.6.5 Defect structure and fractal properties of a real crack | p. 211 |
5.7 Microscopic fracture of geo massifs | p. 214 |
5.7.1 Energy flux | p. 214 |
5.7.2 Ray path and energy storage in geomechanics | p. 215 |
5.8 Chaotic hierarchical dynamical systems and application of non-standard analysis for its description | p. 217 |
5.8.1 Iterated function system | p. 218 |
5.8.2 Entropy of IFS | p. 219 |
5.8.3 Entropy of hierarchical space | p. 221 |
Certain Conclusions | p. 222 |
Appendix A Spaces: Some Definitions | p. 225 |
Appendix B Certain Relations of Vector Analysis | p. 227 |
Appendix C Groups: Basic Definitions and Properties | p. 229 |
Appendix D Dimensions | p. 233 |
Bibliography | p. 239 |
Index | p. 255 |