1.Continuum mechanics

Mathematical background. Tensorial calculus, tensor invariants, trace, devaitor. Continuum mechanics. Cauchy stress, stress invariants, Mohr's circle, octahedral plane. Strain. Small strain, strain invariants. Large strain, stretching tensor, objective stress rate. 2. Constitutive models

Linear isotropic elasticity. Rate formulation, stiffness matrix, calibration of parameters. Linear anisotropic elasticity. Trasversal isotropy. General formulation with five parameters, simplified formulation by Graham-Houlsby with three parameters. Non-linear elasticity, Ohde equation for oedometric compression, hyperbolic elasticity for prediction of shear tests, Duncan-Chang model, small-strain stiffness models.

Ideal plasticity. Elasto-plastic stiffness matrix, yield surface, plastic potential, plastic multiplier. Mohr-Coulomb, Drucker-Prager, Matsuoka-Nakai yield surfaces. Mohr-Coulomb model, calibration of parameters, shortcommings.

Hardening plasticity. Plasticity modulus, calculation of stiffness matrix from consistency condition. Isotropic hardening, cap-type models. Modified Cam clay model. Incoropration of critical state concept, calibration of parameters. Kinematic and mixed hardening. Bounding surface plasticity.

Hypoplasticity. Rate formulation, basic features.

Rheological models. Kelvin's model, Maxwell's model. Viskoplasticity. 3. Numerical methods

Mass-balance equations, momentum conservation. Boundary conditions, initial conditions. Well-possedness.

Finite difference method.

Finite element method. Simple example with springs, formulation of finite elements, Finite element equations, assemblage and solution methods - Newton-Raphson method, initial stiffness method. 4. Numerical methods for discontinuum

Distinct element method. Principles, advantages and shortcommings.

Part 1 of the 2 - term lecture. The course covers foundations of the mathematical modelling needed for solving boundary value problems in geomechanics.

Special attention is paid to the formulation of constitutive models for soils and to the overview of numerical methods used in modern software. Exercises with the FE code Tochnog stimulate individual training of the subject.