Course detail
Continuum Mechanics
FSI-S1K Acad. year: 2026/2027 Winter semester
In the course, students will briefly become familiar with the basic concepts, laws and methods of solving some problems of continuum mechanics, specifically with the theory of elasticity, as its broad subfield. At the beginning of the course, the concepts of stress and deformation at a point of the continuum will be defined, their properties and the laws that must be followed. This will be followed by the formulation of boundary value problems of elasticity, the conditions for the uniqueness of their solution, and their simplified plane and one-dimensional forms will be formulated. In this part of the course, students will also become familiar with some classical methods for solving plane and one-dimensional boundary value problems. In the second half of the course, the concepts of energy and work in continuum mechanics will be introduced, the principle of their virtualization and subsequent use in solving boundary value problems of elasticity using variational methods. A brief description of the theoretical foundations of the theory of elasticity will be supplemented by solved examples from one-dimensional and plane elasticity.
Supervisor
Learning outcomes of the course unit
Prerequisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Language of instruction
Czech
Aims
Specification of controlled education, way of implementation and compensation for absences
The study programmes with the given course
Programme N-MAI-P: Mathematical Engineering, Master's
branch ---: no specialisation, 4 credits, elective
Type of course unit
Lecture
39 hours, optionally
Syllabus
1. Kinetics – stress at point,
2. Kinematics – strain at point.
3. Laws of thermodynamics.
4. Generalized Hook's law, strain energy density, thermoelastic constitutive equations.
5. Boundary-value problems of continuum mechanics, existence and uniqueness of solutions, equations of bars, beams, torsion and plane elasticity.
6. Isotropic plane elasticity – Muskhelishvili's complex potentials, application to fracture mechanics.
7. Anisotropic plane elasticity – LES formalism, application to fracture mechanics.
8. Work and energy, strain energy and complementary strain energy, Hamilton's principle.
9. Unit-dummy-displacement method, unit-dummy-load method, Castigliano's first and second theorem, Betti's and Maxwell's reciprocity theorems.
10. Direct variational methods – Ritz and Galerkin method.
11.-12. Finite element method.
13. Discusion and conlusion of semester.
Exercise
39 hours, compulsory
Syllabus
Kinematical quantities of the continuum mechanics.
Stress tensors. Principal stresses, invariants. Equations of balance.
Constitutive equations in the continuum mechanics. Thermodynamic laws.
Hyperelastic material. Neo-Hooke law, Mooney-Rivlin law. Hooke law for isotropic and anisotropic bodies.
Selected 3D problems of the linear theory of elasticity.
Variational methods in the theory of infinitesimal deformations.
Basic quantities of the continuum mechanics in curvilinear coordinates.
Axial-symmetric problems of the linear elasticity.
Solution of plane problems using Airy stress function.
Circular and circular plate with concentric hole.
Cylindrical shell.
Axisymmetric membrane shell.
Selected simple problem form the theory of plasticity.
Numerical methods in the elasticity problems. Awarding course-unit credits.