Exercise 1 : Stress-strain relationship in linear elasticity
1. Introduction
Question 1.1 : What is a constitutive equation?
Question 1.2 : What does elasticity mean? When is it
observed?
2. Isotropic elasticity
Question 1.3 : Study the isotropic stress-strain relationship in
linear elasticity inthe following particular loading cases:
(a) uniaxial tensile test
(b) torsion test
(c) hydrostatic compression test
Question 1.4 : Using the principle of superposition, build the
isotropic stress-strainrelationship in linear elasticity in the
three-dimensional case.
Question 1.5 : Establish a relationship between the bulk
modulus, the Young mod-ulus and the Poissons ratio.
Question 1.6 : By making a connection between the stress state
in the eigenbasis ina pure shear test and a biaxial
tensile-compression state, establisha relationship between the
shear modulus, the Young modulus andthe Poissons ratio.
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3. Anisotropic elasticity
Question 1.7 : Linear elasticity is more generally modelled
through a fourth order(rigidity or compliance) tensor. Show first
that only 21 compo-nents are independent, and build a matricial
representation usinga second order tensor basis.
Question 1.8 : We assume now that the elastic material exhibits
one plane of sym-metry (e2, e3) of equation x1 = 0. What is the
name of this mate-rial? How many independent parameters are needed
to describe itselastic behaviour?
Question 1.9 : Adding a second plane of symmetry (e1, e3) of
equation x2 = 0,normal to the first one, determine how many
parameters are neededto describe the behaviour. What is the name of
this material?
Question 1.10 : By introducing successively several elements of
symmetry, showthat isotropy is completely described by two material
parameters.
