Problems for Chapter 5
Analytical Techniques and Solutions for
Linear Elastic Solids
5.8.1. A planet that deforms under its own gravitational
force can be idealized as a linear elastic sphere with radius a, Young’s modulus E and Poisson’s ratio that is subjected to a radial gravitational
where g is the acceleration
due to gravity at the surface of the sphere, and R is the radial coordinate. Use the
reciprocal theorem, together with a hydrostatic stress distribution as the reference solution, to calculate the
change in volume of the sphere, and hence deduce the radial displacement of its
5.8.2. Consider an isotropic, linear elastic
solid with Young’s modulus E, mass density ,
and Poisson’s ratio ,
which is subjected to a body force distribution per unit mass, and tractions on its exterior surface. By using the reciprocal theorem, together
with a state of uniform stress as the reference solution, show that the
average strains in the solid can be calculated from
5.8.3. A cylinder with arbitrary
cross-section rests on a flat surface, and is subjected to a vertical
gravitational body force ,
where is a unit vector normal to the surface. The
cylinder is a linear elastic solid with Young’s
modulus E, mass density ,
and Poisson’s ratio .
Define the change in length of the cylinder as
where denotes the displacement of the end of the cylinder. Show that ,
where W is the weight of the
cylinder, and A its cross-sectional
5.8.4. In this problem, you will calculate an expression for
the change in potential energy that occurs when an inelastic strain is introduced into some part B of an elastic solid. The inelastic strain can be visualized as a
generalized version of the Eshelby inclusion problem it could occur as a result of thermal
expansion, a phase transformation in the solid, or plastic flow. Note that B need not be ellipsoidal.
figure illustrates the solid of interest.
solid has elastic constants
body forces act on the solid (for simplicity)
of the surface of the solid is subjected to a prescribed displacement
remainder of the surface of the solid is subjected to a prescribed traction
Let denote the displacement, strain, and stress in
the solid before the inelastic strain is introduced. Let denote the potential energy of the solid in
Next, suppose that some external process introduces an
inelastic strain into part of the solid. Let denote the change in stress in the solid
resulting from the inelastic strain.
Note that these fields satisfy
stress-strain law in B, and outside B
Boundary conditions on ,
and on .
Write down an
expressions for in terms of
Suppose that are all zero (i.e. the solid is initially
stress free). Write down the potential
energy due to . This is called the “self energy” of the
eigenstrain the energy cost of introducing the eigenstrain into a stress-free solid.
Show that the
expression for the self-energy can be simplified to
Now suppose that are all nonzero. Write down the total potential energy of the
in terms of and .
that the total potential energy of the system can be expressed as
Here, the last term is called the “interaction energy”
of the eigenstrain with the applied load. The steps in this derivation are very
similar to the derivation of the reciprocal theorem.
isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio is subjected to a uniaxial tensile stress . As a result of a phase transformation, a
uniform dilatational strain is then induced in a spherical region of the
solid with radius a.
solution to problem 1, and the Eshelby solution, find an expression for the
change in potential energy of the solid, in terms of and relevant geometric and material
Assume that the
interface between the transformed material an the matrix has an energy per unit
area . Find an expression for the critical stress at
which the total energy of the system (elastic potential energy + interface
energy) is decreased as a result of the transformation