Czech

4

doc. Ing. Tomáš Profant, Ph.D.

Institute of Solid Mechanics, Mechatronics and Biomechanics

In the field of mechanics: Knowledge of fundamental concepts from elasticity and strength of materials I or II (stress, strain, Hooke’s law).

In the field of mathematics: Ordinary differential equations, partial differential equations, integral calculus, and the fundamentals of the theory of functions of a complex variable.

Examination Requirements:

• maximum/minimum score: 100/50 points,
• written overview test covering fundamental knowledge and concepts (40 points),
• written solution to 2 problems (30 points each),
• oral discussion regarding the written work, potentially including a supplementary question.

Requirements for Course Credit:

• active participation in seminars/exercises,
• satisfactory performance on ongoing assessments of fundamental knowledge,
• completion of make-up assignments in the event of an excused absence.

The specific format for fulfilling these requirements will be determined by the seminar instructor during the first week of the semester. Attendance at seminars is mandatory. Instructors will monitor student attendance, participation, and grasp of fundamental knowledge on an ongoing basis. An unexcused absence is grounds for denial of course credit. A single absence may be compensated for by completing make-up assignments; longer periods of absence require the completion of make-up assignments as directed by the instructor.

The objective of the course Continuum Mechanics is to introduce students to fundamental concepts, relationships, formulations, and the solution of boundary-value problems in elasticity. In addition to the classical formulation of problems, emphasis is placed on variational formulations, serving as a theoretical foundation for the future application of advanced numerical tools in engineering practice. General elasticity equations and problems are first reduced to two-dimensional plane cases for both isotropic and anisotropic materials, a context historically closely linked to the use of complex potentials, which play a pivotal role in fracture mechanics. By further simplifying the general elasticity equations into one-dimensional models, students build upon the theory covered in classical courses on the elasticity of structural members (beams/bars), while extending the scope to focus on energy-based and variational principles, as well as methods for calculating stress and deformation. These methods are rooted in the principle of virtual displacements and virtual forces, as well as the variation of strain energy and complementary energy. The course culminates in the study of direct variational methods, which form the basis of the Finite Element Method, and their application to one-dimensional elasticity problems.

http://www.old.umt.fme.vutbr.cz/~tprofant/vyuka_cm.html

Programme N-MAI-P: Mathematical Engineering, Master's, elective

39 hours, compulsory

• Kinetics – stress at a point (Exercises 1–2).


• Kinematics – strain at a point (Exercise 3).


• Boundary-value problems in continuum mechanics; existence and uniqueness of solutions; equations for structural members under tension, bending, and torsion; plane elasticity (Exercises 4–5).


• Anisotropic plane elasticity – Lekhnitskii-Eshelby-Stroh (LES) formalism and its application in fracture mechanics (Exercises 6–7).


• Isotropic plane elasticity – Muskhelishvili’s complex potentials and their application in fracture mechanics (Exercise 8).


• Work and energy; strain energy and complementary strain energy; Hamilton’s principle (Exercise 9).


• Method of virtual displacements, method of virtual loads, Castigliano’s first and second theorems, Betti’s and Maxwell’s reciprocal theorems (Exercises 10–11).


• Direct variational methods – Ritz and Galerkin methods (Exercises 12–13).