Course Details

Prerequisite mathematics: Elements of vector and tensor calculus in cartesian coordinates. Green – Gauss theorem. Basic notions and equations: Traction and stress tensor. Balance laws (mass, momentum, angular momentum). Equation of motion and equilibrium. The symmetry of the stress tensor. The gradient of displacement field. Strain tensor. Strain energy density. Hooke law. Navier – Cauchy equation. Beltrami – Michell equation. The general problem of elasticity: Field equations. Basic boundary value problems. Initial conditions for the dynamical problem. Uniqueness solution in elasticity. The superposition principle. Saint –Venant problem (tension, bending, torsion). Two-dimensional  problems in elastostatics: plane stress, plane strain, anti-plane shear. Airy stress function. Problems in polar coordinates. Exact torsion theory. Prandtl stress function. Stress concentration: Williams eigenfunctions method. Two-dimensional Kelvin solution. Flamant- Boussinesq solution. Contact problems. Generalizations: Energy theorems. Elasticity and thermodynamics. Wave propagation. Viscoelasticity. Thermoelasticity. Representations with potentials: Reminders from the complex function theory. The Kolosoff-Muskhelishvili method. Lame potentials for three dimensional elastostatic problem. Elastoplastic behavior: metals and geomaterials in monotone and cyclic loadings. Elastic and plastic deformations. Interpretation with mechanistic models. Constitutive relations: The notion of flow surface in the stress space. Loading index. The Internal variables. Rate equations. Consistency conditions. Plastic measure. Isotropic plasticity: Generalities. Stress invariants. Special cases, Mises for metals and Coulomb for geomaterials. Isotropic hardening and softening. Anisotropic elasticity: Generalities. kinematic analysis. Orthotropic symmetries. HIL simulation and extensions.