• Mechanics, Semester 1, position 2

To introduce students continuum mechanics as applied form of classical mechanics. Aim of this subject is to students overcome and understand terms of continuum mechanics, i.e., to familiarize basic principles Euler’s and Lagrange’s approach to continuum, as well as basic of tensor calculus.

Upon successful completion of this course, students should be able to: • form Green (Lagrangian) strain tensor; • form Eulerian strain tensor; • form velocity strain tensor; • determine the stress tensor components; • compose general equation of motion (Navia) of any deformable medium; • form continuity equation (conservation of mass); • apply the theorem of the change in total energy of a continuous medium in integral form.

Continuum hypothesis. Lagrange’s and Euler’s approach to continuum. Material derivative. Surface and volume forces. Stress tensor. Symmetry of stress tensor. Cauchy’s principle. Major stress and directions of major stress. Extreme values of main stresses. Mohr’s circle. Deformation gradient. Deformation tensor. Displacement vector. Infinitesimal deformation and rotation. Deformation energy. Hooke’s law. Characteristic of fluids. Divergence and rotor of velocity vector. First Helmholtz’s theorem. Velocity of deformation. Acceleration – Kelvin’s theorem. Vortex and nonvortex circulations. Law of conservation of mass – continuity equation. Sources and abysses. Euler’s equation. Laws of change of momentum and angular momentum. Inner forces. Stress assumptions. Navier-Stokes equations. Constitutive equations.

Application of tensor algebra and analysis. Determination of stress components. Deformations in Lagrange and Euler sense. Calculation of major deformations. Stress and displacement tensor. Continuity equation. Navier-Stokes equations. Constitutive equations.

Defined by curriculum.

Handouts

Total assigned hours: 75

New material: 20
Elaboration and examples (recapitulation): 10

Auditory exercises: 30
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 0
Review and grading of the project: 0
Test: 10
Test: 0
Final exam: 5

Activity during lectures: 0
Test/test: 60
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Final exam: 40
Requirement for taking the exam (required number of points): 30

Zorić N., Tomović A., Mehanika kontinuuma za inženjere, Univerzitet u Beogradu, Mašinski fakultet, Beograd 2022. ISBN 978-86-6060-144-7; Reddy, J.N., An Introduction to Continuum Mechanics, Second Edition, Cambridge University Press, 2013. ISBN 978-1-107-02543-1