Dynamics of Discrete Media

The course describes the processes in materials for which the discreteness of their internal structure is important. The examples include behavior of nano-sized objects, chemical reactions, fracture, phase transitions, dynamics of granular materials, crowd dynamics, evolution of astrophysical systems, etc. The course includes theoretical and practical parts. Students will study the fundamental aspects of mechanics of discrete media. At practice students will learn to solve the applied problems using modern simulation techniques, such as molecular dynamics, particle dynamics, discrete element method, smoothed particles hydrodynamics, etc.

The course includes the following items:

• Discrete models.
  • Discrete models. Ab initio Molecular Dynamics (D. Marx An Introduction to Ab Initio Molecular Dynamics Simulations). Classical molecular dynamics. Molecular dynamics (Ehrenfest). Molecular dynamics (Born-Oppenheimer). Molecular dynamics (Carr-Parinello). Methods of determining the electronic structure. Hart-Fock method (a method of self-consistent field). Density functional method (Hohenberg-Kohn-Sham)

• Classical molecular dynamics.

• • The system of material points. The basic laws. The interactions in the system of material points. Paired interaction potentials (Lennard-Jones, Morse, MI). Many-particle potentials (EAM, Stillinger-Weber Terzoffa, Brenner). Calculation of forces (two approaches). System Phone-points. The basic laws. The interactions in the body-points. Linear theory. Stiffness tensors. Nonlinear theory (tensors rotation, rotation vector, the vectors are rigidly connected to the body). Mixed system (material points + freewheeling body-point)

• Communication with the continuum model. The main hypotheses.
  • The main hypotheses. Long-wave approximation. Traffic Separation (averaging over time, in space, the frequency spectrum). Chain models.

• The transition to a three-dimensional membrane continuous medium.
  • Virial theorem (Virial theorem, Heat theorem, article Zhou). The method based on the use of long-wave approximation (kinematics, dynamics, energy balance equation). Paired interaction. Generalization to many-particle interactions. Generalization to complex lattice method. Problems of Hardy method (smoothed particle method). Key ideas and hypotheses. Mass balance equation. The momentum balance equation, th equation of energy balance. Problems of method for calculating the temperature (Gibbs distribution)

• The transition to the theory of plates.
  • The system of material points with many particle interactions. The system of body-points with moment interactions.

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