# 330-0501/01 – 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. Leo Václavek, CSc. Study level undergraduate or graduate Requirement Compulsory Year 1 Semester summer Study language Czech Year of introduction 2015/2016 Year of cancellation 2021/2022 Intended for the faculties FS Intended for study types Follow-up Master
Instruction secured by
LIC098 Ing. Mgr. Dagmar Ličková, Ph.D.
SOF007 doc. Ing. Michal Šofer, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 3+2
Part-time Credit and Examination 8+4

### 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.

Lectures
Tutorials

### Summary

Transformation properties of vectors and tensors. Analysis of strain at a point in a deformable body. Strain-displacement relations. The Green-Lagrange strain tensor, Cauchy`s small (linear) strain tensor. Small strain tensor invariants. Principal strains. Principal axes of strain. Spherical tensor, strain deviator tensor. Octahedral normal and shear strains. Compatibility of strain conditions. 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. Spherical tensor and stress deviator. Normal and shear stresses on the octahedral plane. The method of Mohr`s circles. Cauchy`s differential equations of equilibrium. Physical equations for anisotropic, orthotropic, transversely isotropic and isotropic, linearly elastic homogeneous solid. Boundary conditions. Solution of the elastic problem, formulation in terms of displacements - Lamé (Navier) equations, formulation in terms of stresses - Beltrami-Michell equations. Planar problems of the theory of elasticity, plane stress and plane strain. Airy`s stress function, biharmonic differential equation in orthogonal Cartesian coordinates. The planar problem in polar coordinates. Planar axial-symmetric problem. The stress concentration due to a circular hole in an infinite plate of constant thickness. Pure bending of the circular curved bar. Bending of the circular curved bar with the force at the free end. The stress field around an edge dislocation. Line uniform continuous traction on the boundary of the elastic half-space – Flamant`s problem.

### Compulsory literature:

[1] TIMOSHENKO, S. P.-GOODIER, J. N.: Theory of elasticity. New York-Toronto- London: Mc Graw-Hill, 1951, 3.ed.1970. [2] LEIPHOLZ, H.:Theory of elasticity. Noordhoff International Publishing Leyden, 1974. ISBN 90 286 0193 7

### Recommended literature:

[1] TIMOSHENKO, S. P.-GOODIER, J. N.: Theory of elasticity. New York-Toronto- London: Mc Graw-Hill, 1951, 3.ed.1970. [2] LEIPHOLZ, H.:Theory of elasticity. Noordhoff International Publishing Leyden, 1974. ISBN 90 286 0193 7

### Way of continuous check of knowledge in the course of semester

Test, example solutions

no

### Other requirements

Requirements to the students are solved in exercise

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

Transformation properties of vectors and tensors. Analysis of strain at a point in a deformable body. Strain-displacement relations. The Green-Lagrange strain tensor, Cauchy`s small (linear) strain tensor. Small strain tensor invariants. Principal strains. Principal axes of strain. Spherical tensor, strain deviator tensor. Octahedral normal and shear strains. Compatibility of strain conditions. 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. Spherical tensor and stress deviator. Normal and shear stresses on the octahedral plane. The method of Mohr`s circles. Cauchy`s differential equations of equilibrium. Physical equations for anisotropic, orthotropic, transversely isotropic and isotropic, linearly elastic homogeneous solid. Boundary conditions. Solution of the elastic problem, formulation in terms of displacements - Lamé (Navier) equations, formulation in terms of stresses - Beltrami-Michell equations. Planar problems of the theory of elasticity, plane stress and plane strain. Airy`s stress function, biharmonic differential equation in orthogonal Cartesian coordinates. The planar problem in polar coordinates. Planar axial-symmetric problem. The stress concentration due to a circular hole in an infinite plate of constant thickness. Pure bending of the circular curved bar. Bending of the circular curved bar with the force at the free end. The stress field around an edge dislocation. Line uniform continuous traction on the boundary of the elastic half-space – Flamant`s problem.

### Conditions for subject completion

Part-time form (validity from: 2017/2018 Winter semester, validity until: 2020/2021 Summer semester)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 35  20
Examination Examination 65  16
Mandatory attendence parzicipation:

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### Occurrence in study plans

Academic yearProgrammeField of studySpec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2021/2022 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2021/2022 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2020/2021 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2020/2021 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2019/2020 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2019/2020 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2018/2019 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2018/2019 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2017/2018 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2017/2018 (N2301) Mechanical Engineering (3901T003) Applied Mechanics K Czech Ostrava 1 Compulsory study plan
2016/2017 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2015/2016 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

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