Ingénieur 2A

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 enables us to get to applications quickly, to deal with a wide range of phenomena, and to use a light 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, offering 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

Fluid mechanics is at the center of our experience of the world surrounding us: from oceanic currents to the microscopic flows inside our vascular system, the way fluids move, deform and transport heat or particles profoundly impact our life. Fluid mechanics is central to many engineering and research applications, and plays a major role in several of the most pressing societal challenges (climate, energy, health, defense, food, mobility).

MEC432 aims to provide and reinforce the fundamental bases of this discipline, starting from the students' own experience of fluid motion. The class is geared both for a wide scientifically-curious audience interested in gaining a general and rigourous basis of the discipline, and for providing the technical knowledge and abilities to those who will later wish to continue in this specialty during their training and/or career.

This course is an introduction to the mechanics of deformable solids in three space dimensions. The fundamental concepts that will be covered are: (i) kinematics of deforming bodies, (ii) balance laws (which hold for all bodies), and (iii) constitutive relations (which distinguish between different types of materials). We will emphasize elasticity theory because it is at once central to a majority of engineering applications and the cornerstone on which to build more complex theories of material behavior.

Put together, these ingredients (kinematics, balance laws, and elastic constitutive relations) will allow us to formulate and solve initial-boundary value problems, for small deformations (linear elasticity) as for large deformations (finite elasticity).

In linear elasticity, we will also introduce variational principles, which provides the tools for a qualitative analysis of problems in linearized elastostatics and is the basis for numerical techniques such as the finite element method.

In finite elasticity (which is relevant to such materials as elastomers and biological tissues), we will illustrate how nonlinearity leads to unexpected physical phenomena which cannot be captured by the linearized theory.

Finally, we will conclude the course with an opening towards coupled problems, which include thermoelasticity, piezoelectricity, and chemoelasticity), and are encountered in many applications, such as the electrochemistry of batteries and 3D printing. In these problems, the mechanical response of a solid to thermal, electrical, or chemical loadings is governed by the interplay between elasticity and other physical mechanisms (e.g., heat transfer or the diffusion of chemical species). The modeling of these couplings and the study of their effects will be illustrated through selected examples.