Theory of Elasticity

Transformation properties of vectors and tensors. Analysis of strain at a point in a deformable body. Strain-displacement relations. The Green-Lagrange strain tensor, Cauchy`s small (linear) strain tensor. Small strain tensor invariants. Principal strains. Principal axes of strain. Spherical tensor, strain deviator tensor. Octahedral normal and shear strains. Compatibility of strain conditions. The state of stress at a point in a body. Stress tensor. Invariants of the stress tensor. Principal stresses, principal planes, principal directions of the stress tensor at a point. Spherical tensor and stress deviator. Normal and shear stresses on the octahedral plane. The method of Mohr`s circles. Cauchy`s differential equations of equilibrium. Physical equations for anisotropic, orthotropic, transversely isotropic and isotropic, linearly elastic homogeneous solid. Boundary conditions. Solution of the elastic problem, formulation in terms of displacements - Lamé (Navier) equations, formulation in terms of stresses - Beltrami-Michell equations. Planar problems of the theory of elasticity, plane stress and plane strain. Airy`s stress function, biharmonic differential equation in orthogonal Cartesian coordinates. The planar problem in polar coordinates. Planar axial-symmetric problem. The stress concentration due to a circular hole in an infinite plate of constant thickness. Pure bending of the circular curved bar. Bending of the circular curved bar with the force at the free end. The stress field around an edge dislocation. Line uniform continuous traction on the boundary of the elastic half-space – Flamant`s problem.