330-0531/01 – Theory of Elasticity (TP)
| Gurantor department | Department of Applied Mechanics | Credits | 6 |
| Subject guarantor | doc. Ing. Michal Šofer, Ph.D. | Subject version guarantor | doc. Ing. Michal Šofer, Ph.D. |
| Study level | undergraduate or graduate | Requirement | Compulsory |
| Year | 1 | Semester | winter |
| | Study language | Czech |
| Year of introduction | 2021/2022 | Year of cancellation | |
| Intended for the faculties | FS | Intended for study types | Follow-up Master |
Subject aims expressed by acquired skills and competences
Educate students in basic procedures which are applied for a definition and solving of more exciting engineering technical problems in the sphere of mechanics of solid elastic deformable bodies. Ensure understanding of teaching problems. To learn the students apply gained theoretical peaces of knowledge in praxis.
Teaching methods
Lectures
Individual consultations
Tutorials
Experimental work in labs
Project work
Summary
The aim of the study course is to provide students with a deeper overview in the field of classical theory of elasticity, thanks to which the students will be able to solve complex problems in practice using analytical approach. The course is thematically divided into two parts. In the first part, students become familiar with the problematics of coordinate systems transformation, body deformation, stress and strain, and last but not least with the mathematical description of body compatibility or equilibrium equations in the case of elastostatics and elastodynamics. The second part of the course, on the other hand, deals with the theory of solving 2D problems using the stress function and also with the topic of torsion of arbitrary cross-sections.
Compulsory literature:
Recommended literature:
[1] TIMOSHENKO, S. P.-GOODIER, J. N.: Theory of elasticity. New York-Toronto-London: Mc Graw-Hill, 1951, 3.ed.1970.
Way of continuous check of knowledge in the course of semester
The study course is being completed in form of the examination, which has got oral and written part. In the written part, the student solves problems related to the theory of elasticity, while in the oral part, there´s a discussion on two selected topics from the given set of question related to this study course.
E-learning
Other requirements
The student is required to elaborate a project on a given topic.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
1. Orthogonal transformation. Transformation of coordinate system. Transformation properties of vectors and tensors. Physical components of vectors and tensors.
2. Analysis of strain at a point in a deformable body. Strain-displacement relations. Geometric meaning of individual components of Cauchy strain tensor.
3. The state of stress at a point in a body. Stress tensor. Invariants of the stress tensor. Principal stresses, principal planes, principal directions of the stress tensor at a point.
4. Mohr´s representation of 3D stress state. Extreme shear stresses. Spherical and deviatoric stress tensor. Octahedral normal and shear strains.
5. Strain tensor invariants. Principal strains and directions. Maximum shear strains. Spherical and deviatoric strain tensor. Normal and shear stresses on the octahedral plane.
6. Compatibility and equilibrium equations.
7. Constitutive equations. Hook´s law for anisotropic, orthotropic, transversely isotropic and isotropic material. Pre-heating and initial deformation effect on constitutive equations.
8. Boundary conditions. Solution of the 2D elastic problem, formulation in terms of displacements - Lamé (Navier) equations, formulation in terms of stresses - Beltrami-Michell equations.
9. Two variants of the 2D elastic problem. Plane stress and plane strain problem. Airy`s stress function, biharmonic differential equation in orthogonal Cartesian coordinates.
10. Expression of boundary conditions using Airy´s stress function. Biharmonic equation in polar coordinates.
11. 2D elastic problem with axially symmetric stress distribution. Pure bending of the circular curved bar.
12. Bending of the circular curved bar with the force acting at the free end. The effect of circular hole on the stress field in the plate.
13. The Flamant-Boussinesq problem.
14. Axially symmetric problem in cylindrical coordinates. Force acting in point of infinite isotropic elastic space (Kelvin problem).
15. Free torsion of arbitrary cross-section.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction