# 339-0501/01 – Theory of Elasticity (TP)

 Gurantor department Department of Mechanics of Materials Credits 6 Subject guarantor doc. Ing. Leo Václavek, CSc. Subject version guarantor doc. Ing. Leo Václavek, CSc. Study level undergraduate or graduate Requirement Compulsory Year 1 Semester winter Study language Czech Year of introduction 2003/2004 Year of cancellation 2014/2015 Intended for the faculties FS Intended for study types Follow-up Master
Instruction secured by
SLA20 Dr. Ing. Ludmila Adámková
VAC10 doc. Ing. Leo Václavek, CSc.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 3+2

### 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

### Další požadavky na studenta

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

Full-time form (validity from: 1960/1961 Summer semester, validity until: 2012/2013 Winter semester)
Task nameType of taskMax. number of points
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (100) 51
Exercises evaluation Credit 35 (35) 0
Project Project 28  0
Other task type Other task type 7  0
Examination Examination 65 (65) 0
Written examination Written examination 30  0
Oral Oral examination 35  0
Mandatory attendence parzicipation:

Show history

### Occurrence in study plans

Academic yearProgrammeField of studySpec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2014/2015 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2013/2014 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2012/2013 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2011/2012 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2010/2011 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2009/2010 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2008/2009 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2007/2008 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2006/2007 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2005/2006 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan
2004/2005 (N2301) Mechanical Engineering (3901T003) Applied Mechanics P Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner