Mechanical design of solid deformable media 2

The aim of the course is to establish the various concepts of the mechanics of deformable solids. In the second part of the course, the equations used in linear elasticity are established (for isotropic and anisotropic materials, plane problems). Elastic limit criteria are presented for isotropic, ductile and brittle materials, and for orthotropic materials.

No

Chapter I - Tensors
Index notation , Curvilinear coordinate systems , Notion of variance , Introduction to tensors , Covariant and contravariant components , Physical components of a tensor , Christoffel symbols , Covariant derivative of a tensor , Components of gradient and divergence operators.
Chapter II - study of second-order tensors
Change of basis , Principal directions and principal values , Invariants of a tensor , Graphical representation of a second-order tensor , Divergent tensor .
Chapter III - kinematics of continuous media
Notion of configuration, gradient tensor of the transformation, deformation of a continuous medium, tensor of infinitesimal deformations, change of basis, principal directions, compatibility equations.
Chapter IV - Local equilibrium equations
local equilibrium equations, stress tensor symmetry, mechanical boundary conditions.
Chapter V- Linear elastic behavior
homogeneity assumption, symmetry conditions, matrix writing, material symmetries, orthotropy, transverse isotropy, isotropy, determination of elastic constants, engineering elastic constants, base change, plane stresses.
Chapter VI - Solving elasticity problems
Students are responsible for writing the course in Chapter 6, based on group work involving the drafting of thematic sheets.
Chapter VII- Dimensioning criteria
Von Mises, Tresca, Tsaï-Hill, Tsaï-Wu, maximum stress criterion.

Physics

"Mécanique des solides déformables", C. Bacon and J. Pouyet
"Introduction à la mécanique des milieux continus", P. Germain and P. Muller
"Ondes élastiques dans les solides 1", D. Royer, E. Masson, 1996.
" Mécanique des solides déformables", P. Lac (course handout)
" Mécanique des milieux continus - élasticité linéaire", E. Ringot
course handout - Université de Toulouse III - Paul Sabatier
"Mécanique des milieux continus et discrets", N. Moës (course handout - ECN)
" Nonlinear continuum mechanics for finite element analysis", J. Bonet and R.D. Wood. Cambridge University Press, 1997.