Mathematics and physics on continuous media

Language

French

Prerequisites

This course assumes acquired the basic concepts of analysis and algebra as taught in the LEPL1101, LEPL1102, and LEPL1105 courses.

Main themes

This is an introductory and discovery course, which focuses on the description of continuous media, associated fields (deformations and strain, strain rate, stresses, scalar and vector fields), as well as some fundamental physical phenomena that depend on space, and also on time in some cases; and  are therefore expressed by partial differential equations (PDEs): transport PDE, Laplace PDE, Poisson PDE, diffusion PDE, wave PDE.
The link between the physical phenomena that take place in continuous media and their mathematical description by PDEs is paramount and is presented in an interwoven and structured way. The mathematical methods for solving fundamental physical problems are also introduced progressively, and are applied in simple cases in 1-D (unbounded medium and bounded medium) and in 2-D (unbounded medium and bounded medium of simple geometry: rectangle, circle, annulus, segment of circle or annulus). Induced fields (solution of Poisson's PDE) in unbounded 2-D and 3-D media are also covered, as they are applications in electrostatics, electromagnetism, and gravitation.
Note: the notion of "constitutive relation" to, for example, link stresses and deformations in elasticity, or link stress and deformation rates for fluids, is also introduced, but in a limited/simplified way; because otherwise the problem posed is not complete, and therefore not soluble. This also allows students to be able to grasp a variety of advanced problems in a downstream course(s).

Learning outcomes

At the end of this learning unit, the student is able to :

At the end of this course, the student will be able to: ​ Understand the concepts of continuous medium and fields; Use tensor "tools" to generalize concepts and simplify notations; Use the properties of tensors, obtain the invariants, directions and principal values, know how to perform a change of reference; Use tensor analysis for fields (scalar and vector) and kinematic and stress tensors; Describe the kinematics of continuous media (reference and current configurations, descriptions of motion, material derivative), strain measurement tensors (uniaxial and multiaxial), velocity gradient tensors (strain rate and rotation rate); Use the stress tensor, obtain the invariants, directions and principal values, their decomposition (spherical and deviatoric components), the boundary conditions (normal and shear stress), use Mohr's circles for simple cases; Understand the laws of conservation (global and local forms), ​the concepts of material volume and control volume and the use of the Reynolds transport theorem and its variants, for the conservation of mass, momentum, and angular momentum, total internal energy (including thermodynamic considerations, first law of thermodynamics); Understand problem closure by introducing constitutive laws and forming a PDE system, use these concepts for heat diffusion laws and linear elasticity; Understand and differentiate the fundamental physical phenomena governed by order 1 (transport equation) and order 2 (Laplace equation, Poisson equation, diffusion equation, wave equation) PDEs. ​ Define and apply appropriate initial and/or boundary conditions (Dirichlet, Neumann, Robin); Apply the method of characteristics to solve the 1-D transport equation and the 1-D wave equation; Understand the concept of self-adjoint operators, eigenvalues and eigenfunctions, orthogonality of eigenfunctions, development of a function in series of eigenfunctions; Apply the separation of variables method to solve Laplace's equation in a rectangle, a (sector of) circle, a (sector of) annulus, and also Poisson's equation; Apply the separation of variables method to solve the diffusion equation in a segment, a rectangle, a circle; Apply the separation of variables method to solve the wave equation in a segment, a rectangle, a circle.

Content

Introduction:
Scales (from microscopic to macroscopic), concept of Representative Elementary Volume, fundamental assumptions, and field representation.
Elements of tensor calculus:
Tensors as a “tool” for generalizing concepts (e.g., to 2-D and 3-D) and simplifying notation. Tensors: order, orthonormal bases, component transformation, Einstein's convention. Second-order tensors: motivation, properties, invariants, principal directions and values, change of reference frame. Tensor analysis: scalar field, vector field, coordinates. Examples: kinematic tensors (deformation, strain rate), stress tensors. Solution in displacements for given deformations: compatibility of deformations.
Kinematics of continuous media:
Reference and current configurations. Descriptions of motion: Lagrangian and Eulerian. Material time derivative, material velocity and acceleration. Transformation gradient. Strain measurement tensors, strain measurements (uniaxial and multiaxial). Velocity gradient tensor (decomposition into strain rate and rotation rate).
Stress measurements:
True stress (Cauchy stress) and stress vector. Cauchy stress tensor, invariants, principal directions and values, decomposition (spherical and deviatoric components). Stresses acting on a surface: normal stress and shear stress. Mohr circles, with examples of simple loads.
Conservation laws:
From global to local forms: Material volume and control volumes. Transport: Reynolds transport theorem and variants; link with integral theorems from the Analysis 2 course. Conservation of mass, and examples. Conservation of momentum, and examples. Conservation of angular momentum, symmetry of Cauchy's stress tensor.
Transport PDE: linear 1-D case and solution by the method of characteristics (case with constant velocity c, conservative case with velocity c(x)). Nonlinear case and solution using the method of characteristics; appearance of discontinuity (= jump), jump relation and speed of discontinuity: Burgers equation. LWR model of road traffic and examples (traffic jam, starting at a green light).
Energy and thermodynamic considerations: Kinetic energy theorem, first law of thermodynamics, internal energy, local form.
Diffusion phenomena:
Fourier's law for heat diffusion in a solid or fluid. Obtaining the diffusion PDE for an infinitesimal control volume (1-D and 2-D cases). Writing in index and vector form. Diffusion with the addition of a source term.
Diffusion in unbounded 1-D and 2-D media: obtaining the “Green's function” for point initial conditions (IC). Solution by convolution with Green's function for cases with distributed initial conditions.
Diffusion in a finite 1-D medium (= segment) with general IC and Dirichlet, Neumann, or Robin (= mixed) boundary conditions (BCs); possibly also with source term: steady-state solution (satisfies the BCs); transient solution with homogeneous BCs obtained by the “separation of variables method”; and expressed as a “Fourier series” (with associated “eigenfunctions” and “eigenvalues”, and “orthogonality” of the eigenfunctions).
Diffusion in a finite 2-D medium (= rectangle, circle, annulus): steady-state solution (= Laplace equation, satisfies the BCs) obtained by separation of variables; transient solution with homogeneous BCs obtained by separation of variables.
Induced fields:
Poisson's EDP. 
unbounded media: obtaining the “Green's function” in 2-D and 3-D. Examples: electric field induced by a point electric charge, gravitational field induced by a point mass, magnetic field induced by an electric current. Solution by convolution with Green's function for cases with distributed sources.
Finite 2-D media (rectangle, circle) with homogeneous CLs: solution by separation of variables.
Wave phenomena:
1-D wave PDE. Solution using the method of characteristics. Standing waves and examples (guitar string, electromagnetic wave, acoustic wave). Solution in a finite medium with homogeneous BCs by separation of variables. 
2-D wave PDE. Helmholtz problem with homogeneous BCs and orthogonality of eigenfunctions. Solutions in a rectangle  and in a circle by separation of variables. Vibration of a rectangular, circular, membrane.

Teaching methods

The course is organized into 13 lectures (CM1 to CM13) in a large lecture hall, and 13 exercise-based learning sessions (APE1 to APE13) which are carried out, in part, in tutored groups (with supervision by one assistant tutor per group); and, for the rest, through individual work outside of the tutored groups.
The course slot scheduled for the last week is used partly for a Q&A session and partly to present additional content.

Evaluation methods

The APEs are not corrected. The solutions are posted on the course website as the semester progresses. This allows students to continuously assess their level of understanding and learning.
Students are assessed individually via a written exam.

Online resources

Moodle site of the course

Bibliography

Faculty or entity

> BTCI