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The goal of the course is to teach students theoretical (tectonophysical) fundamentals of structural analysis, fundamentals of the deformation and tensions theory, reologic models of real geological objects and massifs, criteria of solidness and mechanism of origin of fractural damages, mechanical mathematical setting of tasks and methods of solving them using modern computer technology.


  1. Introduction

Subject and tasks of the course, role of structural analysis in geology. Overview of literature. Geological objects and apparatus of mathematical physics, application of mechanics of unified environment, fundamental mechanical notions. Main hypotheses, physical laws and principles of theoretic research. Stages in solving problems of tectonophysics.

2-3. Strain at point.

Concentrated and distributed forces. Mass (volume) and surface forces, reactions. Internal and external forces. Strain at point: vector of tensions, normal and dotted strains. Projection of a tension vector to coordinates axes. Stress tensor. Coordinates systems and formulas of transformations. Main axes of the tension tensor and main normal strains. Tensor invariants. Spherical tensor and deviator of strains. Diagram of Morr. Main dotted strains. Octahedron strains and strain intensiveness. Units of measurements.

^ 4. Partial cases of strain state.

One-weight and flat strain state. Strain of stretching-compression and displacement. Plain displacement, twisting, longitudinal and transverse folding. Force factors for layers. Three-dimensional strain condition. Equation of balance and borderline conditions in strain-stress. Dynamic problems, inertial and impact loads.

^ 5. Displacements and deformations.

Vector of displacements. Deformation in 1-dimensional cases. Tensor of deformation. Invariant of deformation tensor. Deformations of change of form and volume. Representation of deformation components through displacement, geometric interpretation. Small and finite deformation. Deformations velocity. Deformation intensiveness. Determining deformation as per set displacements and reverse problem. Compatibility of deformations. Flat deformation.

^ 6. Connection between strain and deformations.

Diagram of deformations – strain in 1-dimension case. Law of strain – generalized Gook’s law. Strain constants: modules of stretching-compression and displacement, Poisson’s coefficient. Types of anisotropy of environment: isotropic, transverse-isotropic and orthotropic environment. Determination of strain constants. Non-linear strain.

^ 7. Plastic deformation.

Plastics deformations. Theory of small plastic deformations and theory of flow. Criteria of plasticity (plastic flow), strengthening of rocks. Theory of sliding. Theory of plasticity for cases of flat deformation, lines of sliding and their properties.

^ 8. Reologic models. Creep and relaxation.

Models of Kelvin, Foygte, Maxwell. Schematic representation and determinative interrelationships of physical law. Dependence of deformation on time. Curves of creep. Equations of creep.

9. Viscosity of real rocks

Modules of instant strain, strain residual action, and module of balanced deformation. Coefficient of effective viscosity. Time of relaxation. Dependence on strain and temperature.

^ 10. Strength of rocks

Mechanism of origination of fractures, fractures of tearing-off and splitting-off. Theories of strength, theory of Mor, physical theory of origin of tectonic fractures. Modern vision of mechanism of destruction. Cyclic processes and long lasting strength.

^ 11. Deformations of simple shift.

Conditions of origination of deformations of simple shift. Ellipsoid of deformation and its use for characteristics of strain and plastics deformations. State of strain and fractures of simple shift.

12. Introduction to physical modeling.

Principles of physical modeling. Conditions of similarity. Scale coefficients. Equivalent materials. Familiarization with the results of some physical experiments: fold formation, fractures formation, forced structures. Importance and limitations of physical modeling.

^ 13. Mathematical modeling of tectonic processes and geological structures.

Stages of mathematical modeling. Setting a task of linear theory of strain. Borderline conditions in displacements and strains. Static, quasistatic and dynamic task. Physical and geometrical non-linearity. Mathematical aspects of existence and unity of solution. Notion of direct and reverse task of determining state of strain. Fitting non-linear problems into sequence of linear ones. Incremental load, method of sequential approximations. Dependence of solution on trajectory of load.

^ 14. Validation of mathematical models of real processes.

Taking into account anisotropy and heterogeneity of mountainous layers. Taking into account high pressures and temperatures rising with increase of distance from day surface. Introduction of viscous-strain-plastic model. Combined and multiple component mathematical models (considering thermal mass displacement, magnetic strain, hydromechanical models). Non-classic borderline conditions. Contact tasks of mining mechanics.

^ 15-16. Numerical methods of structural analysis.

Familiarization with main ideas of universal numeric methods for solving initial regional tasks for differential equations in partial derivatives. Schemes of differences: approximation of derivatives, compiling system of equations, problems of precision, strength, convergence. Familiarization with fundamentals of method of finite elements: variation and energetic principles, triangulation of region, local approximation based on the element of unknown functions by means of key value, compiling global system of equations and solving it. Familiarization with method of borderline elements: decrease of area, borderline elements, fundamental solution, compiling system of equations and solving it.

17. Familiarization with software for solving structural analysis problems.

Interface of program for calculation of mechanical deformations by method of finite elements. Design of network of finite elements, setting borderline conditions and loads, setting material properties. Compiling system of equations and solving it. Receiving solution in displacement and strain. Visualization of results: design of surfaces of displacements and isolines of strains, design of main strains.

18. Mathematical and computer modeling in tectonophysics.

Solving modeling structural problems within the flat theory of strain. Modeling of folding. Modeling of fractural damages and heterogeneity. Considering gravitational forces and temperatures. Analysis of results and specification of models.


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