MEC430 - Mécanique des milieux déformables (2023-2024)

This course presents the fundamental concepts of the Mechanics of deformable continuums within the simplified framework of slender structures. Introducing these concepts in a limited geometric framework in order to get to applications quickly and deal with a wide range of phenomena with a lighter mathematical formalism.

The approach is similar to the one followed in more specialized courses, in particular to the MEC431 one in the case of three-dimensional structures: we will cover the notions of internal forces and external forces, equilibrium equations, boundary conditions, strain, constitutive laws and boundary problems.

We will next get interested in the resolution of mathematical problems obtained and in the highlighting of the related physical phenomena, in statics and dynamics, in both small and large displacements.

We will study the problems of statics of wires, rods, beams or elastic arches which will enable us to deal with the classical problems on the strength of materials, but also more advanced problems such as buckling instabilities or boundary layers.

Finally, we will introduce the variational approach which offers both a different viewpoint of the physical laws governing the mechanics of structures and that provides mathematical and numerical tools for solving equations. This will enable us in particular to get fundamental energy proprieties, define stability concepts and provide an introduction to the finite element method.

Course language: French

MEC430T - Mécanique des milieux déformables - Tutorat (2023-2024)

MEC431 - Mécanique des solides (2023-2024)

This course is about solid mechanics within the general framework of deformable continuum systems in three spatial dimensions. It relies on notions of tensor calculus, kinematics in large deformations, conservation laws, behavior laws, particularly in the elastic domain, boundary problems and energy methods. As will be demonstrated throughout the course via numerous examples, these tools allow the analysis and design of complex structures made up of a broad range of materials and of industrial, medical, and environmental interests.

 

No prerequisites are required, except for a basic mathematical basis in differential calculus and linear algebra. The course Mécanique des Milieux Déformables MEC430 is not a pre-requisite but eases the understanding of the first lectures.

Textbooks:

  • Patrick Le Tallec (2023) Mécanique des Solides https://moodle.polytechnique.fr/course/view.php?id=14794

Other useful references:

  • W. OGDEN (1997). Non-Linear Elastic Deformations. Dover.
  • E. GURTIN (2003). An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, Vol 158.
  • LOPEZ-PAMIES (2021). The Mechanics of Solids. Editions de l'Ecole Polytechnique.
  • TRUESDELL, W. NOLL (2004). The Non-Linear Field Theories of Mechanics. Third Edition, Springer.
  • LE TALLEC (2009). Modélisation et Calcul des Milieux Continus. Editions de l'Ecole Polytechnique.

Synopsis:

The course will be organized in 10 sessions. The lectures will be taught in French and the PC will be in French or in English depending on the choice of the student.

MEC432 - Mécanique des fluides (2023-2024)

Fluid mechanics is at the center of many engineering and research applications.

The course aims to establish and consolidate the fundamentals of this discipline. It is designed to appeal both to students who wish to limit themselves to obtaining a general and rigorous education on the subject, and to students who plan to pursue one or more in-depth studies in this specialty next year.

The MEC432 fluid mechanics course lays the basics of this discipline. The course begins by putting the notion of fluid into perspective with regard to physics and thermodynamics:

what distinguishes a fluid from a solid? What distinguishes different fluids? What is a Newtonian fluid?

We will then explain the fundamental principles of fluid flow kinematics, those of dynamics and energetics, to arrive at the fundamental equations of Newtonian fluid mechanics: the Navier-Stokes equations. These equations, although reputed to be among the most difficult in physics, are analyzed and calculated by engineers and researchers on a daily basis. We will present the approximations that ease this, notably the one of the "incompressible fluid" and of the "perfect fluid", insisting on their physical conditions of validity.

We'll then study how can the complexity of these problems be further reduced, sometimes to the point of no mathematical resolution at all, thanks to dimensional analysis and the principles of similarity on which fluid mechanics experimentation is based.

We'll then take a closer look at the perfect-fluid approximation that forms the major part of many historical applications of fluid mechanics. The analysis of perfect-fluid flows highlights the special role of vorticity, and we will explore "vorticity dynamics", which can be considered a discipline.

We will then show how the perfect-fluid approximation can be connected to another approximation that concentrates most of the effects of viscosity, the boundary-layer approximation, in order to obtain a uniformly valid description of any flow.

Finally, the course will end with an introduction to turbulence, one of the main frontiers in our understanding of fluid mechanics today.

 


Course language : French

ECTS credits : 5

MEC433 - Dynamique de l'atmosphère et des océans (2023-2024)

The atmosphere and ocean are at the heart of the planet's environmental issues, on every scale of space or time, as they are the vectors for the transportation of water, energy and chemical compounds. The role of these two fluids contrasts them: the atmosphere, which moves very quickly, determines the spatial organization of temperatures and precipitation on the planet, but has very little memory; the ocean, on the other hand, has considerable inertia (dynamic, thermal) and organizes temporal fluctuations in climate.

They are similar in terms of the physical principles that set them in motion: both the atmosphere and the ocean are fluids that have little vertical extension, are highly stratified in density (because they are heated), and move on a rotating sphere by predominantly horizontal motions. The description of their motion is an important branch of what is known as geophysical fluid mechanics. Coriolis forces and pole-equator contrasts are responsible for the fact that atmospheric and oceanic circulation are primarily organized on a global scale.

This course is a Mechanics course, but also provides an introduction to certain environmental topics: weather forecasting, climate change or, to a lesser extent, contamination and air quality.

 This course is accessible without prerequisites, although it requires a slightly greater effort from students who have not previously taken any Mechanics module.

Course language: French

 

MEC471A - Modal de mécanique (2023-2024)

MODAL in Mechanics is an experimental and project-based course designed to introduce students to a branch of Mechanics (aerodynamics, acoustics, granular materials, etc.), without prerequisites.

During the first sessions, students discover and grasp the basic concepts of a field of Mechanics through concrete experiments.In the following six or seven sessions, students are asked to carry out a specific project or study on a complex problem of their choice.

During this project phase, they apply a scientific approach: posing a problem, comparing their observations with more or less simple models, then carry out new experiments based on these models, etc.

For example, the following themes were offered in previous years:

- AERODYNAMICS
- ACOUSTICS
- BIOMECHANICS
- CIVIL ENGINEERING
- DROPS AND BUBBLES
- GRANULAR
- INTELLIGENT STRUCTURES

CAUTION: the list of themes changes from one year to the next, depending on the number of registrations and the availability of teachers. The choice of themes is made after a presentation a few weeks before the start of MODAL, and we cannot guarantee a student's registration for a specific theme.

 


Evaluation: Written report and poster presentation.

 

MEC472B - Modal de mécanique (2023-2024)

The Mechanics MODAL is an experimental course that aims to make students discover a sector of Mechanics (aerodynamics, acoustics, granular...), without prerequisites.

During the first sessions, students discover and grasp basic concepts of the Mechanics field through concrete experiences. During the following 6 or 7 sessions, students have to realize a project or a specific study on a complex issue of their choice. During this phase of the project, they have to execute a scientific procedure: pose a problem, confront observations to more or less simple models then realize new experiences according to it.

As an example, the following themes were offered in the previous years:

- AERODYNAMICS
- ACOUSTICS
- BIOMECHANICS
- GEOPHYSICAL FLOWS
- GRANULAR
- ROBOTICS
- STRUCTURAL FAILURE
- INTELLIGENT STRUCTURES

CAREFUL: the list of themes changes from one year to another one with the number of people enlisted and the availability of teachers. The choice of the themes is done after the presentation, a few weeks before the start of the MODAL, and we cannot guarantee a specific theme to a student.

Evaluation: Written report and poster presentation

MEC473C - Modal de mécanique (2023-2024)

The Mechanics MODAL is an experimental course that aims to make students discover a sector of Mechanics (aerodynamics, acoustics, granular...), without prerequisites. During the first sessions, students discover and grasp basic concepts of the Mechanics field through concrete experiences. During the following 6 or 7 sessions, students have to realize a project or a specific study on a complex issue of their choice. During this phase of the project, they have to execute a scientific procedure: pose a problem, confront observations to more or less simple models then realize new experiences according to it.

As an example, the following themes were offered the previous years:

- AERODYNAMICS
- ACOUSTICS
- BIOMECHANICS
- GEOPHYSICAL FLOWS
- GRANULAR
- ROBOTICS
- STRUCTURAL FAILURE
- INTELLIGENT STRUCTURES

CAREFUL: the list of themes changes from one year to another one with the number of people enlisted and the availability of teachers. The choice of the themes is done after the presentation, a few weeks before the start of the MODAL, and we cannot guarantee a specific theme to a student.

Evaluation: Written report and poster presentation