Course detail

Continuum Mechanics

FSI-S1KAcad. year: 2026/2027

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.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. Ing. Tomáš Profant, Ph.D.

Department

Institute of Solid Mechanics, Mechatronics and Biomechanics (ÚMTMB)

Entry knowledge

In the field of mechanics: Knowledge of basic concepts of the theory of elasticity (stress, principal stress, deformation, strain, Hooke law). Principle of virtual displacements, principle of virtual work. In the field of mathematics: Partial differential equations of 2nd order. Elements of variational calculus. Elements of functional analysis (functional spaces, Hilbert space L2).

Rules for evaluation and completion of the course

Written test examining the knowledge of basic concepts - examination paper containing 3 examples to be solved - oral discussion over examination papers with an optional additional question.
Attendance is required. One absence can be compensated by attending a seminar with another group in the same week, or by elaboration of substitute tasks. Longer absence is compensated by special tasks according to instructions of the tutor. Course-unit credit is awarded on the following conditions: - active participation in the seminars, - good results in seminar tests of basic knowledge, - solution of additional tasks in case of longer excusable absence. Seminar tutor will specify the form of these conditions in the first week of semester.

Aims

The goal of the course is to familiarise students with basic concepts and relations of the continuum mechanics of solid state and the ways of formulation and solution of boundary-value problems for elastic and elastic/plastic bodies. Apart from classical formulation, the emphasis is placed on the variational formulation of problems with consideration of consecutive numerical solution. Another goal is for students to learn the basics of finite deformation theory and constitutive equations, which enables better understanding of the application of advanced FEM systems simulating complex processes under consideration of large strains and nonlinear properties of materials.
Students will be made familiar with basic methods applied for the determination of stress-strain fields in general bodies stemming from differential and variational approach. Knowledge of physical origin of variational formulation of problems of continuum mechanics together with knowledge gained in the course “Numerical Methods III” allows to choose a suitable approach to the preparation of numerical computation. Mastering the basics of the constitutive equation theory offers a good orientation among various material models. Students are also provided with knowledge of a negative influence of cracks upon the durability of cracked bodies.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

http://www.old.umt.fme.vutbr.cz/~tprofant/vyuka.html#continuum-mechanics (EN)

Recommended reading

Oden, J.T., Mechanics of elastic structures, McGraw-Hill, 1967. (EN)
Hwu, C., Anisotropic elastic plates, Springer, 2010. (EN)
Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, Springer Dordrecht, 1977. (EN)
Ting, T.C.T., Anisotropic elasticity, Oxford University Press, 1996. (EN)
Reddy, J.N., Energy principles and variational methods in applied mechanics, John Wiley & Sons, 2017. (EN)

Classification of course in study plans

  • Programme N-MAI-P Master's 2 year of study, winter semester, elective

Type of course unit

 

Lecture

39 hours, optionally

Teacher / Lecturer

doc. Ing. Tomáš Profant, Ph.D.

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

Teacher / Lecturer

doc. Ing. Tomáš Profant, Ph.D.

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.