Continuum mechanics
Objectives
This course allows to assimilate the basic formalism of the mechanics of the continuous environments leading to the writing of the Lamé and Navier-Stokes equations.
At the end of the first part of the course, freshmen will be able to:
- to use the formalism of the linear algebra to follow the demonstrations leading to the equations of the mechanics of the continuous mediums;
- explain the transformations between volumes and surfaces in the balance equations;
- describe behavioral laws for the diffusion of heat or the rheology of elastic solids;
- calculate analytical solutions for simple linear elasticity problems.
At the end of the second part of the course, freshmen will be able:
- to describe the kinematics of the flows using matrices expressing the rotation or the deformation of the particles;
- to formulate the conservation equations of mass, momentum and energy;
- to describe behavioral laws for the Newtonian fluid rheology;
- to calculate analytical solutions for simple fluid mechanics problems.
Description
1) Linear algebra and tensors: Einstein convention, differential operators, the divergence formula
2) The continuum hypothesis: heat flux vector by small tetrahedra, Fourier law and state law leading to the heat equation.
3) Large and small deformations: Jacobian matrice, dilatation tensor and small strains tensor, Jacobian.
4) Stress tensor under small strains: mass conservation in Lagrangian representation, fundamental principle of dynamics, existence and symmetry of the stress tensor.
5) Lamé equations : Hooke's Law, longitudinal and transverse waves in solids.
6) Kinematics: trajectories, streamlines, particle spin.
7) Transport theorems: rotation vector and tensor strain rate, pass on a moving domain.
8) incompressible Navier-Stokes equations: fundamental principle of the dynamics, law of behavior.
9) Compressible Navier-Stokes equations: "theorem" of kinetic energy and power of internal forces, first principle of thermodynamics.
A session of Practical Work (4h): "Hydraulic jump", to illustrate the notion of discontinuity and jump relation.
Bibliography
[1] O. Thual, Mécanique des Milieux Continus, Éd. Ress. Pédago. Ouv. INP 1018 (2012) 48h
[2] Introduction à la Mécanique des milieux continus déformables - Auteur : O. THUAL - Editeur : Cépaduès - Editions , 1997 - ISBN : 2854284550
URL : http://www.cepadues.com/chercher.asp?rapid=thual
[3] Étagère de cours Scholarvox : Mécanique des milieux continus
Session 1 ou session unique - Contrôle des connaissances
Modalité | Nature | Coefficient | Remarques |
---|---|---|---|
CC (contrôle continu) | Oral/Ecrit | 375/10 | Examen Mécanique des Milieux continus |
CC (contrôle continu) | Oral/Ecrit | 375/10 | Examen 2 Mécanique des Milieux continus |
CC (contrôle continu) | Travaux Pratiques | 25% | TP Mécanique des Milieux continus |
Session 2 - Contrôle des connaissances
Modalité | Nature | Coefficient | Remarques |
---|---|---|---|
CC (contrôle continu) | Oral/Ecrit | 375/10 | Examen Mécanique des Milieux continus |
CC (contrôle continu) | Oral/Ecrit | 375/10 | Examen 2 Mécanique des Milieux continus |
CC (contrôle continu) | Travaux Pratiques | 25% | TP Mécanique des Milieux continus |
Contact(s)
THUAL OLIVIERPlaces
- Toulouse