The teaching unit is mandatory.
The teaching unit is taught in French.
The teaching unit is taught in English.

Outline

3 ECTS / 10 sessions of 3 hours

 Courses: 10 sessions of 1h

Exercises : 4 sessions of 2h

Practice : 6 sessions of 2h

Team

Coord. Andrea Barbarulo (andrea.barbarulo@centralesupelec.fr, CentraleSupelec)

Guillaume Puel (guillaume.puel@centralesupelec.fr, CentraleSupelec)

Objectives

This course presents numerical methods in structural computations and mechanics of materials. The discussed topic is structured in two large parts: (i) Nonlinear-Structural Computations and (ii) formulation and implementation of complex material laws in programs for structural computations. The objectives are therefore to provide the students with (i) basic knowledge to perform numerical computations (essentially using the finite element method) and (ii) advanced knowledge of implementing a material behavior in standard finite element programs.

Targets

The cours is dedicated to engineering jobs in design research and development for various industries, such as aeronautics, transportations, civil, mechanical or electrical engineering. Theses industries employ and produce currently complex materials involving microscopic and macroscopic scales.

There is a frequent need to employ structural computations in order to evaluate the stress, strain or damage fields. In the past, most of the structures have been computed in the elastic domains. However, present applications demand more and more that the behavior of structures should be estimated in complex nonlinear situations. The present lectures are devoted to give the proper tools to the students in order to respond efficiently for these type of questions.

Bibliography

• Besson, J., Cailletaud, G, Chaboche, J.-L., Forest, S.,Mécanique non-linéaire des matériaux, Hermes (2001)
• Besson J., Billon N., Cantournet S. - Matériaux pour l'ingénieur, Editions Mines, 2006
• Bonnet M, Frangi A. - Analyse des solides déformables par la méthode des éléments finis, Editions de l’Ecole Polytechnique, 2006
• Constantinescu A, Korsunsky AMK – Elasticity with Mathematica, Cambridge University Press, 2007
• Dhondt, G., The Finite Element Method for Three-Dimensional Thermomechanical Applications, 2004
• Hughes, T. J. R. Finite Element Method - Linear Static and Dynamic Finite Element Analysis
• Prentice-Hall, Englewood Cliffs, 1987
• Georges Cailletaud - Modélisation mécanique d'instabilités micro-structurales en viscoplasticité cyclique à température variable (Note technique ONERA) (Broché - 1979)
• Simo, J. C. and Hughes, T. J. R. Computational Inelasticity, Springer 1999
• Suquet P. - Rupture et plasticité, Editions de l’Ecole Polytechnique, 2006

List of finite element books is available here.

Content

Session 1 : Review of the finite element method (Course + Exercises)

Balance equations for elastic bodies. Principles of virtual strains. Minimum Principles and weak formulation of the equlibrium.

Session 2 : The Finite Element Method in linear elasticity (Course + Practice)

Construction of the approximate elasticity, Discret system of equations and its numerical resolution. Static and Dynamic balance. Boundary conditions and computations of the reactions. Numerical computations of stress intensity factors. Computation of the energy release rate.

Session 3 : Solids with nonlinear material behaviour (Course + Exercises):

Review of large strains. Nonlinear equations and Newton type iterative algorithms.

Session 4 : Solids with nonlinear material behaviour (Course + Exercises):

Review of elastoplastic material behaviour. Nonlinear equations and Newton type iterative algorithms.

Session 5 : Solids with elastoplastic material behavior: local aspects (Course + Practice)

Computation of an elastoplastic structure: problem definition. Local integration of the constitutive law. Examples.

Session 6 : Solids with elastoplastic material behavior: integration algorithm (Course + Practice)

Programming of an example.

Session 7 : Solids with elastoplastic material behavior: identification of the material behaviour (Course + Practice)

Testing possibilities. Problem formulations and identification techniques. Sensitivity computations.

Session 8 and 9 : Programing Exercices (Practice)

Session 10 : Exam

Evaluation :

Written examination and Continuous control – homework, numerical projects

Computer Programs :

Castem, Zebulon, Calculix, Abaqus / Scilab, Matlab, Mathematica