The course is optional but highly recommended
The course is taught in English 

 

Team   

Coord. Véronique Aubin (veronique.aubin@centralesupelec.fr, CentraleSupelec)

Objectives

This basic course has the aim of providing insight in the fundamental concepts of kinematics and the different stress measures in order to formulate the equilibrium of a deformable body undergoing a finite motion. The necessary complements on tensor algebra and analysis are reviewed at the beginning of the course. Starting from the finite deformations framework, the more familiar small transformations are deduced by linearization. In particular, a discussion on the objectivity (frame indifference) of the physical quantities introduced is proposed. A proper understanding of the mechanical concepts used throughout the course requires mathematics of tensor algebra and tensor calculus. The symbolic (modern) notation is typically used, completed by the index notation at the appropriate point. The last part is devoted to linear elasticity with a brief discussion of finite elasticity. Some examples of application are finally given. This course should provide the student of the master MAGIS with a reasonable background in mechanics to learn efficiently more specific subjects during the semester.

Content

 

Chapter 0. Tensor algebra

• Body, configuration and motion – Material and spatial description
• Material derivative, velocity, acceleration
• Deformation gradient, deformation tensors and invariants
• Polar decomposition, Volume and area change– Conservation of mass – Distortion
• Homogeneous deformations and motion of rigid body
• Linearized kinematics – Small displacements and tensor of infinitesimal strains
• Deformation rate
• Objectivity of kinematics quantities

Chapter 2. Kinematics of continuous media

• Body, configuration and motion – Material and spatial description
• Material derivative, velocity, acceleration
• Deformation gradient, deformation tensors and invariants
• Polar decomposition, Volume and area change– Conservation of mass – Distortion
• Homogeneous deformations and motion of rigid body
• Linearized kinematics – Small displacements and tensor of infinitesimal strains
• Deformation rate
• Objectivity of kinematics quantities

Chapter 3. Stress and equilibrium

• Body and contact forces – Postulate of Cauchy
• Translational and rotational equilibrium of a continuum
• Properties of the Cauchy stress tensor – Deviatoric and pressure components
• Examples of stress states
• Piola-Kirchhoff stress tensors
• Objective stresses – Principal of virtual work

Chapter 4 : Elasticity and applications

• Recalls on infinitesimal and linear elasticity
• Elasticity tensor in material and spatial description
• Examples of applications

References

• Bonet J. and Wood R. D., Nonlinear continuum mechanics for finite element analysis, Cambridge University Press, 2000
• Botsis J. et Deville M., Mécanique des milieux continus: une introduction, Presses Universitaires Romandes, 2005
• Curnier A., Mécanique des solides déformables (Tome 1), Presses Universitaires Romandes, 2004†
• Fung Y. C. and Pin Tong, Classical and computational solid mechanics, World Scientific Publishing Co. Pte. Ltd. 2001
• Gurtin M., An introduction to Continuum Mechanics, Academic Press, 1981
• Haupt P., Continuum Mechanics and Theory of Materials, Springer, 2000 †
• Holzapfel G.A., Nonlinear Solid Mechanics, John Wiley & Sons, LTD, 2000
• Lemaitre J. et Chaboche J. L., Mécanique des matériaux solides, Dunod, 1988
• Simmonds J.G., A Brief on Tensor Analysis, 2nd edn, Springer-Verlag, New York, 1994

† : advanced books