Subjects(version: 901)
Course, academic year 2022/2023

Mathematical modeling in geomechanics I - MG451P65E
Title: Mathematical modeling in geomechanics I Matematické modelování v geomechanice I Institute of Hydrogeology, Engineering Geology and Applied Geophysics (31-450) Faculty of Science from 2014 summer 3 summer s.: summer s.:2/1 C [hours/week] unlimited unlimited no taught English enabled for web enrollment
Guarantor: prof. RNDr. David Mašín, Ph.D.
 Annotation
Last update: Mgr. Zdeňka Sedláčková (13.05.2014)
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.
 Syllabus
Last update: Mgr. Zdeňka Sedláčková (13.05.2014)

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.

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