Methods for Solving Elasticity Problems
Theory of Elasticity: Kinematics of solid: The concept of the continuum media and the Lagrange and Euler approaches to the description of its movement; Material derivative of vector and tensor; Strain measures and tensors; Infinitesimal strain tensor and vector of small rotation. Dynamics of solid: Conservation of mass law; Classification of forces acting on the material body; Integral laws of the dynamics; Cauchy’s formula for the stress vector; Stress tensor; Differential equations of the deformable body dynamics. Thermodynamics of solid: First law of thermodynamics in integral and differential forms; Second law of thermodynamics in form of Clausius-Duhem inequality; Universal dissipative inequality; Third law of thermodynamics. Constitutive equations theory elements: Influential parameters, thermomechanical processes and constitutive equations; Material objectivity principle; Isotropic materials; Constitutive equations of isotropic thermoelastic material; Constitutive equations of viscous material; Constitutive equations of ideal plastic material. Constitutive equations in linear mechanics of solid: Classical constitutive equations of form changing processes: elastic material, viscous material, plastic material; Rheological models principle; Kelvin-Voigth material; Maxwell material; Standard linear viscous-elastic material; Ideal elastic-plastic material; Elastic- plastic material with linear hardening; Viscous-elastic-plastic material of Bingham. Basic equations of linear elasticity theory: Closed system of equations and boundary conditions for coupled thermal- elastic problem; Various forms of Hook’s law; Theory of elasticity equations in displacements; Theory of elasticity equations in terms of stresses; Beltrami-Mitchell equations; Theory of elasticity equations and boundary conditions in Pobedria form.