Chapter 3
Constitutive Models: Relations between
Stress and Strain
3.2. Linear
Elastic Constitutive Equations
3.2.1. Using the table of values given in Section 3.2.4, find
values of bulk modulus, Lame modulus, and shear modulus for steel, aluminum and
rubber.
3.2.2. A specimen of an isotropic, linear elastic material
with Young’s modulus E and is placed
inside a rigid box that prevents the material from stretching in any
direction. This means that the strains
in the specimen are zero. The specimen
is then heated to increase its temperature by . Find a formula for the stress in the
specimen. Find a formula for the strain
energy density. How much strain energy
would be stored in a sample of steel if its temperature were
increased by 100C? Compare the strain
energy with the heat required to change the temperature by 100C the specific heat capacity of steel is about
470 J/(kg-C)
3.2.3. A specimen of an isotropic, linear elastic solid is
free of stress, and is heated to increase its temperature by . Find expressions for the strain and
displacement fields in the solid.

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3.2.4. A thin isotropic, linear elastic thin film with
Young’s modulus E, Poisson’s ratio and thermal expansion coefficient is bonded to a stiff substrate. The film is stress free at some initial
temperature, and then heated to increase its temperature by . The substrate prevents the film from
stretching in its own plane, so that ,
while the surface is traction free, so that the film deforms in a state of
plane stress. Calculate the stresses in the film in terms of material
properties and temperature, and deduce an expression for the strain energy
density in the film.
3.2.5.
A cubic material
may be characterized either by its moduli as
or
by the engineering constants
Calculate
formulas relating to and ,
and deduce an expression for the anisotropy factor in terms of .
3.2.6. Suppose that the
stress-strain relation for a linear elastic solid is expressed in matrix form
as ,
where ,
and represent the stress and strain vectors and
the matrix of elastic constants defined in Section 3.2.8. Show that the material has a positive
definite strain energy density ( ) if and only if the eigenvalues of are all positive.
3.2.7. Write down an expression for
the increment in stress resulting from an increment in strain applied to a
linear elastic material, in terms of the matrix of elastic constants . Hence show that, for a linear elastic
material to be stable in the sense of Drucker, the eigenvalues of the matrix of
elastic constants must all be positive or zero.
3.2.8. Let ,
and represent the stress and strain vectors and
the matrix of elastic constants in the isotropic linear elastic constitutive
equation
3.2.8.1.
Calculate the eigenvalues of the stiffness matrix for an isotropic solid in terms of Young’s
modulus and Poisson’s ratio. Hence, show
that the eigenvalues are positive (a necessary requirement for the material to
be stable see problem ??) if and only if and .
3.2.8.2.
Find the eigenvectors of and sketch the deformations associated with
these eigenvectors.
3.2.9. Let ,
and represent the stress and strain vectors and
the matrix of elastic constants in the isotropic linear elastic constitutive
equation for a cubic crystal
Calculate the eigenvalues of the stiffness matrix and hence find expressions for the admissible
ranges of for the eigenvalues to be positive.
3.2.10. Let denote the components of the elasticity tensor
in a basis .
Let be a second basis, and define . Recall that the components of the stress and
strain tensor in and are related by .
Use this result, together with the elastic constitutive equation, to show that
the components of the elasticity tensor in can be calculated from
3.2.11. Consider a cube-shaped specimen of an anisotropic, linear elastic material.
The tensor of elastic moduli and the thermal expansion coefficient for the
solid (expressed as components in an arbitrary basis) are ,
. The solid is placed inside a rigid box that
prevents the material from stretching in any direction. This means that the strains in the specimen
are zero. The specimen is then heated to
increase its temperature by . Find a formula for the strain energy density,
and show that the result is independent of the orientation of the material with
respect to the box.

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3.2.12. The figure shows a cubic crystal. Basis vectors are aligned perpendicular to the faces of the
cubic unit cell. A tensile specimen is cut from the cube the axis of the specimen lies in the plane and is oriented at an angle to the direction. The specimen is then loaded in
uniaxial tension parallel to its axis. This means that the stress components in the
basis shown in the picture are
3.2.12.1.
Use the basis
change formulas for tensors to calculate the components of stress in the basis in terms of .
Use the stress-strain equations in Section 3.1.16 to
find the strain components in the basis, in terms of the engineering constants for the cubic crystal. You need only calculate
.
3.2.12.2.
Use the basis
change formulas again to calculate the strain components in the basis oriented with the specimen. Again, you need only calculate . Check your answer by setting - this makes the crystal isotropic, and you
should recover the isotropic solution.
3.2.12.3.
Define the
effective axial Young’s modulus of the tensile specimen as ,
where is the strain component parallel to the direction.
Find a formula for in terms of .
3.2.12.4.
Using data for
copper, plot a graph of against . For copper, what is the orientation that
maximizes the longitudinal stiffness of the specimen? Which orientation minimizes the stiffness?