Course detail

Continuum Mechanics

FSI-S1K-AAcad. year: 2026/2027

The course deals with the following topics: Introduction, basic terminology, bodies, motions, configurations. Foundation of the theory of finite strains. General equation of balance. Cauchy's I. and II. law of continuum mechanics. Geometrical equations, compatibility conditions, boundary conditions. Thermodynamic background of the theory of constitutive relations. Models of elastic behaviour. Hyperelastic materials. Isotropic elasticity and thermoelasticity. Anisotropic elasticity. Classical formulation of an elastic problem using differential approach. Deformation theory and incremental theory of plasticity. Variational principles in the infinitesimal strain theory. Weak solution. Axisymmetric problems. Plane strain/plane stress. Solution of two-dimensional elasticity problems. Airy's stress function. Foundation of the theory of plates and shells. Fundamentals of linear fracture mechanics. Remarks on Ritz method and FEM in continuum mechanics problems.

Language of instruction

English

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)

Offered to foreign students

Of all faculties

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

Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, Springer Dordrecht, 1977. (EN)
Oden, J.T., Mechanics of elastic structures, McGraw-Hill, 1967. (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-AIM-A Master's 2 year of study, winter semester, elective
  • Programme N-MAI-A Master's 2 year of study, winter semester, elective

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

doc. Ing. Tomáš Profant, Ph.D.

Syllabus

. 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 hod., 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.