Prof. V. Ungureanu. First Semester
Description: This course intends to provide students a comprehensive knowledge on the theory of elasticity and plasticity. The course focuses on the following topics: continuous medium, Cartesian tensors, deformation, displacement and strain tensors, compatibility conditions, external and internal forces, equilibrium, stress tensors, principal stresses, invariants and stress deviators, constitutive law, plasticity theory, yield and failure criteria, laws of mechanics, plane stress and plane strain problems. Students will obtain theoretical knowledge on stress in spatial and planar structural elements, walls, plates and shells on elastic and plastic properties of materials. By the end of the course, the students should have an idea about for what kinds of mechanics problem the analytical solutions exist and for which ones we must depend on numerical solutions. They have to be able to solve analytically a number of classical problems of elasticity.
Content: Torsion with restraint warping in thin-walled bars with non-circular open cross section; Basic equations of elasticity theory. Three-dimensional problem of stress and strain state in the surrounding of a point in elastic body. Introduction, The State of Stress at a Point, The State of Strain at a Point, Basic Equations of Elasticity, Methods of Solution of Elasticity Problems, Spherical Co-ordinates, Principal Stresses and Principal Planes; Two dimensional problems in the theory of elasticity in Cartesian co-ordinates; Introduction, Formulation of all governing equations; Navier’s equations; 2D stress states in Cartesian coordinates: plane stress, plane strain, Airy stress function: Bending of a cantilever loaded at the end; Two-dimensional problems in polar co-ordinates. Basic equations, 2D stress states in cylindrical coordinates: axisymmetry, strain-displacement relations, Airy stress function: Stress-concentration due to a Circular Hole in a Stressed Plate (Kirsch Problem), finite-difference methods (FDM); Analysis of rectangular plates –theory and methods, application of numerical methods. Introduction, Cylindrical Bending of Rectangular Plates, Type of stresses on rectangular flat plates, Determination of bending and twisting moments on any plane, finite-difference methods (FDM), Navier’s solution for simply supported rectangular plates; Analysis of rotationally symmetric circular plates – methods of solution. Introduction, symmetrical bending of a circular plate, governing Equations for symmetric bending of circular plates, some typical solutions; Theory of rotationally symmetric thin shells – membrane and bending theory. Introduction, statics of shells - the stress resultants and stress couples in shells, the equilibrium equations, membrane theory of shells, geometry of shells of revolution; Basic theory of plasticity. Theory of rupture: (i) Plasticity in one dimensional stress states: material models for uniaxial tension/compression; Bauschinger effect; Strain hardening; (ii) Yield criteria in two and three dimensional stress states: general expression of yield criterion; yield criteria for ductile material: Tresca, von Mises; Fracture criteria for brittle materials: Rankine, Mohr-Coulomb, Drucker-Prager; (iii) Deformations in plastic regime: yield surface; Incremental deformations; Convexity of the yield surface; Normality of plastic deformation increment to yield surface; Plastic potential and flow rule.