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Course, academic year 2024/2025
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Mathematical modeling in geomechanics - MG452P70
Title: Matematické modelování v geomechanice
Czech title: Matematické modelování v geomechanice
Guaranteed by: Institute of Hydrogeology, Engineering Geology and Applied Geophysics (31-450)
Faculty: Faculty of Science
Actual: from 2009
Semester: summer
E-Credits: 4
Examination process: summer s.:combined
Hours per week, examination: summer s.:2/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Level: specialized
Note: enabled for web enrollment
Guarantor: prof. RNDr. David Mašín, Ph.D.
Annotation -
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.
Last update: Datel Josef, RNDr., Ph.D. (01.06.2009)
Literature - Czech

Herle, I. (2003) Základy matematického modelování v geomechanice. UK Praha, Karolinum.

Muir Wood, D., 2004, Geotechnical modelling. Ed. Applied Geotechnics, Spon Press, London

Crisfield, M.A. (1997) Non-linear finite element analysis of solids and structures. Vol. I: Essentials. Wiley, Chichester.

Last update: Datel Josef, RNDr., Ph.D. (01.06.2009)
Requirements to the exam - Czech

Písemná zkouška z teoretických znalostí, praktická zkouška řešení zadané geotechnické úlohy pomocí metody konečnách prvků.

Last update: Mašín David, prof. RNDr., Ph.D. (01.11.2011)
Syllabus -

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.

Last update: Datel Josef, RNDr., Ph.D. (01.06.2009)
 
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