1. Introduction to tensorial calculus.

1.1 Definition of a tensor.

1.2 Indicial notation and summation convention.

1.3 Change of base formula.

1.4 Algebraic tensors operations.

1.5 Tensor analysis : differential operators and integral formula.

2. Continuum mechanics.

2.1 Mathematical background.

2.2 Kinematics for finite transformations.

2.3 Stress, Cauchy theorem

2.4 Anisotropic elasticity and constitutive laws.

Langue de l'enseignement

FRANÇAIS / FRENCH

1 Vibrations.

1.1 Equations and fundamental problems of dynamics.

1.2 Free responses and fundamental parameters.

1.3 Deterministic forced responses and linear filters.

1.4 Time-evolution problem with initial conditions.

1.5 Stationary random vibration.

1.6 Vibration transmissibility and vibration isolation.

1.7 Structural modes of vibration and modal analysis.

2. Waves in elastic bodies.

2.1 Wave propagation in one-dimensional medium.

2.2 Propagation of elastic waves in an infinite homogeneous linear medium.

2.3 Waves in semi-infinite media.

2.4 Wave transmission and reflection with the inclusion of a boundary.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Elliptic, parabolic or hyperbolic boundary value problems.

2. Discretization by the finite differences method.

3. Weak formulation and admissible space.

4. Interpolation on a finite element basis; convergence.

5. Numerical integration and reference element.

6. Discretization by the finite elements method.

7. Resolution scheme in time (Euler, Newmark).

8. Nonlinear elasticity (Newton-Raphson, Riks-Crisfield, MAN).

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Classes of behavior.

2. Mechanics of brittle fracture.

3. Fatigue of materials.

4. Time-dependent behavior : viscoelasticity and viscoplasticity.

5. Elastoplasticity and limit analysis.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Classification elastic symmetries.

2. Identification of elastic properties.

3. Formulation of an anisotopic and heterogeneous linear thermoelastic problem.

4. Basic concepts of homogenization theory.

5. Basic homogenization methods for composites.

6. Multilayered composite beams and plates.

7. Spherical and cylindrical multilayered composite shells.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Beams and plates 1.1. Kinematic assumptions of beams

1.2. Cases of elementary solicitations

1.3. Internal efforts: calculations and diagrams

1.4. Energy theorems

1.5. Force method and displacement method

1.6. Kinematics of thin and thick plates

1.7. Strain tensor and generalized stress tensor

1.8. Rectangular and circular plates bending

1.9. Approximate resolution method

1.10. Numerical resolution method

2. Instabilities of elastic beams

2.1. Euler method.

2.2. Instability of discrete systems: energy method.

2.3. Numerical method for solving instability problems related to elastic structures.

Langue de l'enseignement

FRANÇAIS / FRENCH

The internship, lasting at least 2 months, must be done in a private company or in a research laboratory. The objective of the internship is to understand the functioning of business or the research activities and to put in practice the knowledge learned during the courses. Students have to prepare a dissertation and an oral defense.

1. Affine transformation of a solid.

2. Analysis of large homogeneous strain.

3. General transformation of a solid. Lagrangian and Eulerian objectivity.

4. Torsors, forces, stress and equilibrium equations.

5. Elastic and hyperelastic constitutive laws.

6. Formulation of boundary value problems.

7. problems with and without internal linkage.

8. Theory of beams with large displacement and small strain.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Continuity and Navier-Stokes equations.

2 . Kinetic energy equation and Bernoulli equations.

3 . Isentropic compressible flows of an ideal gas and shock waves.

4. Non Newtonian fluid flows.

5. Laminar external flows and dynamic boundary layers.

6. Introduction to turbulence.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Harmonic functions.

2. Complex analysis.

3. Fourier analysis.

4. Methods of Green functions.

5. Asymptotic series expansion methods.

6. Introduction to symbolic computation on MAPLE software.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Direct static characterization of real samples.

2. Prediction of transport properties from three-dimensional periodic unit-cells.

3. Frequency-dependent visco-inertial and thermal responses.

4. Comparison with standing wave tube measurements.

5. Effective elastic properties of the porous medium.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Generality and Principles of reinforced concrete.

2. Properties of the materials.

3. Actions and loadings.

4. Basic hypotheses (ELU et ELS).

5. Design of elements under uniaxial compression.

6. Design of elements under simple bending.

7. Design of elements under shear loading.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Soil Mechanics.

1.1 Soil characterization and classification.

1.2 Soil permeability.

1.3 Effective stress and pore water pressure.

1.4 Stress distribution in soils due to surface loads.

1.5 Compressibility and consolidation.

1.6 Shear strength of soils.

2. Foundation engineering

2.1 Earth retaining structures.

2.2 Slope stability.

2.3 Geotechnical investigations.

2.4 Shallow foundations.

2.5 Deep foundations.

Langue de l'enseignement

FRANÇAIS / FRENCH

1. Introducttion to software Autodesk Robot Structural Analysis.

2. Modelling and simulation of structures made up of beam, plates, shell and 3D elements.

3. Design of structures according to Eurocode 2 et Eurocode 3.

Langue de l'enseignement

FRANÇAIS / FRENCH

Finite difference/finite elements methods for transient/stationary diffusion/thermal problems; finite elements method for linear/non-linear problems; finite elements method for coupled problems: poroelasticity, elasticity/diffusion coupling.

Langue de l'enseignement

ANGLAIS / ENGLISH

Phenomenological aspects for damage modeling. Damage variables. Effective stresses. Damage measures. Elementary laws and damage criterion; Thermodynamic formulation. Dimensional representation of damage. Isotropic and anisotropic damage theories; Specific models. Plastic ductile damage. Creep damage. Fatigue damage. Interaction effects; Deformation/damage coupling. Elasticity(resp. Plasticity)/Damage coupling; Application to the modeling the behavior of concrete. Behavior in tensile and compression. Relationship between damage and anelastic strains. Strain localization.

Langue de l'enseignement

ANGLAIS / ENGLISH

1) External acoustic

a) Response to prescribed acoustic source

b) Acoustic impedance boundary operator and radiation impedance operator

c) Helmholtz integral representations

d) Boundary integral equations

e) Finite elements discretization

2) External vibroacoustic

3) Internal vibroacoustic

Langue de l'enseignement

ANGLAIS / ENGLISH

Definition of scales in heterogeneous solids; notion of Representative Volume Element(RVE); Kinematically or Statically Uniform Boundary Conditions; RVE stiffness or compliance tensor; Eshelby problem and Green operators. Voigt and Reuss bounds, Hashin-Shtrikman bounds, dilute, Mori-Tanaka and autocoherent models. Homogenization of periodic media, asymptotic methods, Lippmann-Schwinger equation, resolution with iterative méthods based on fast Fourier Transform (FFT)

Langue de l'enseignement

ANGLAIS / ENGLISH

Introduction; probabilistic models for vector-valued random variables​​; construction of probability laws; stochastic fields; methods for solving equations with random parameters; nonparametric probabilistic models for uncertainties in dynamics.

Langue de l'enseignement

ANGLAIS / ENGLISH

English for oral presentation in international scientific events; English for writing scientific articles.

Langue de l'enseignement

ANGLAIS / ENGLISH

Complex fluids vs Newtonian fluids : Modelling and identification of highly viscous behaviors; nonlinear viscosity and Bighman fluids. Hele Shaw approximation and numerical modelling of viscous flows. Thermal aspects (WLF law, self-heating, exchange with tools); numerical simulation of thermo-mechanical coupled problem. Visco-elastic behavior in finite strain.

Langue de l'enseignement

ANGLAIS / ENGLISH

Plane waves in isotropic fluids and solids. Acoustic impedance at normal incidence in fluids, and highly porous materials. Acoustic impedance at oblique incidence in fluids, and highly porous materials. Examples : reflection coefficient and absorption coefficient at normal and oblique incidence for multi-layered materials (Matlab and dedicated software).

Langue de l'enseignement

ANGLAIS / ENGLISH

Perfect thermal interfaces. Hadamard relation for temperature jump. Continuity and discontinuity conditions for temperature/heat flux gradients. Fourier's law for perfect interfaces; Perfect elastic interface. Hadamard compatibility condition for displacements. Continuity/discontinuity conditions for displacements and stresses. Hooke's law for perfect interfaces; Perfect multi physical interface. Hadamard compatibility equation. Generalized Hill interface operators. Solution for perfect interfaces in a composite (multi-physical framework). Homogenization of laminates ; Asymptotic analysis for imperfect interfaces in a multi-physical context. General forms for displacement vectors and generalized constraints jumps. Special cases (Kapitza law, Gurtin-Murdoch relation, ...); Application of imperfect interface models in micromechanics and nanomechanics.

Langue de l'enseignement

ANGLAIS / ENGLISH

Computation of effective properties for linear diffusion/elasticity problems; numerical homogenization of coupled problems: thermoelasticity, poroelasticity and electromechanical coupling; Introduction to advanced techniques for the homogenization of nonlinear heterogeneous structures

Langue de l'enseignement

ANGLAIS / ENGLISH

Basics in probability theory, utility function and limit states, structural reliability, numerical simulations, simplified methods and reliability index, applications.

Langue de l'enseignement

ANGLAIS / ENGLISH

Introduction to topological optimization by homogenization. 1) Reminder of variational principles;2) Formulation of the problem of minimization of compliance and non-existence of a solution to the problem {0.1} ;3) Elements of homogenization and introduction to composite theory; 4) Formulation of the relaxed problem and numerical implementation; 5) Anisotropic elastic materials

Langue de l'enseignement

ANGLAIS / ENGLISH

Le stage doit permettre à l’étudiant de mettre en application l’ensemble des connaissances acquises durant sa formation académique et d'acquérir des compétences additionnelles en matière d’initiation à la recherche (dans le contexte d’un laboratoire ou dans un secteur R&D d’une entreprise).