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Problems
for Chapter 5
Analytical
Techniques and Solutions for Linear Elastic Solids
5.4. Solutions
to 3D Static Problems
5.4.1.
Consider the
Papkovich-Neuber potentials
5.4.1.1.
Verify that the
potentials satisfy the equilibrium equations
5.4.1.2.
Show that the
fields generated from the potentials correspond to a state of uniaxial
stress, with magnitude  acting parallel to the  direction of an infinite solid
5.4.2.
Consider the
fields derived from the Papkovich-Neuber potentials
5.4.2.1.
Verify that the
potentials satisfy the equilibrium equations
5.4.2.2.
Show that the
fields generated from the potentials correspond to a state of hydrostatic tension
5.4.3.
Consider the
Papkovich-Neuber potentials
5.4.3.1.
Verify that the
potentials satisfy the governing equations
5.4.3.2.
Show that the
potentials generate a spherically symmetric displacement field
5.4.3.3.
Calculate
values of  and  that generate the solution to an internally
pressurized spherical shell, with pressure p acting at R=a and
with surface at R=b traction free.
5.4.4.
Verify that the
Papkovich-Neuber potential
generates the fields for a point force  acting at the origin of a large (infinite)
elastic solid with Young’s modulus E
and Poisson’s ratio . To this end:
5.4.4.1.
Verify that the
potentials satisfy the governing equation
5.4.4.2.
Calculate the
stresses
5.4.4.3.
Consider a
spherical region with radius R surrounding
the origin. Calculate the resultant
force exerted by the stress on the outer surface of this sphere, and show
that they are in equilibrium with a force P.
5.4.5.
Consider an
infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio . Suppose that the solid contains a rigid
spherical particle (an inclusion) with radius a and center at the origin.Â
The particle is perfectly bonded to the elastic matrix, so that  at the particle/matrix interface. The solid is subjected to a uniaxial
tensile stress  at infinity.Â
Calculate the stress field in the elastic solid. To proceed, note that the potentials
generate
a uniform, uniaxial stress  (see problem 1). The potentials
are
a special case of the Eshelby problem described in Section 5.4.6, and
generate the stresses outside a spherical inclusion, which is subjected to a
uniform transformation strain.  Let ,
where A and B are constants to be determined.Â
The two pairs of potentials can be superposed to generate the required
solution.Â
5.4.6.
Consider an
infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio . Suppose that the solid contains a spherical
particle (an inclusion) with radius a
and center at the origin. The particle
has Young’s modulus  and Poisson’s ratio ,
and is perfectly bonded to the matrix, so that the displacement and radial
stress are equal in both particle and matrix at the particle/matrix
interface. Â The solid is subjected to a
uniaxial tensile stress  at infinity.Â
The objective of this problem is to calculate the stress field in the
elastic inclusion.
5.4.6.1.
Assume that the
stress field inside the inclusion is given by . Calculate the displacement field in the
inclusion (assume that the displacement and rotation of the solid vanish at
the origin).
5.4.6.2.
The stress
field outside the inclusion can be generated from Papkovich-Neuber potentials
where
,
and C and D are constants to be determined.
5.4.6.3.
Use the
conditions at r=a to find expressions for A,B,C,D in terms of geometric and material properties.
5.4.6.4.
Hence, find the
stress field inside the inclusion.
5.4.7.
Consider the
Eshelby inclusion problem described in Section 5.4.6. An infinite
homogeneous, stress free, linear elastic solid has Young’s modulus E and Poisson’s ratio .
The solid is initially stress free. An inelastic strain distribution   is introduced into an ellipsoidal region of
the solid B (e.g. due to thermal
expansion, or a phase transformation).Â
Let  denote the displacement field,   denote the total strain distribution, and
let  denote the stress field in the solid.
5.4.7.1.
Write down an
expression for the total strain energy  within the ellipsoidal region, in terms of ,
 and .
5.4.7.2.
Write down an
expression for the total strain energy outside the ellipsoidal region,
expressing your answer as a volume integral in terms of  and . Using the divergence theorem, show that the
result can also be expressed as
where
S denotes the surface of the
ellipsoid, and  are the components of an outward unit vector
normal to B. Note that, when applying the divergence
theorem, you need to show that the integral taken over the (arbitrary)
boundary of the solid at infinity does not contribute to the energy  you can do this by using the asymptotic
formula given in Section 5.4.6 for the displacements far from an Eshelby
inclusion.
5.4.7.3.
The Eshelby
solution shows that the strain  inside B is uniform. Write down the displacement field inside
the ellipsoidal region, in terms of  (take the displacement and rotation of the
solid at the origin to be zero).Â
Hence, show that the result of 7.2 can be re-written as
5.4.7.4.
Finally, use
the results of 7.1 and 7.3, together with the divergence theorem, to show
that the total strain energy of the solid can be calculated as
5.4.8.
Using the
solution to Problem 7, calculate the total strain energy of an initially stress-free isotropic, linear
elastic solid with Young’s modulus E and
Poisson’s ratio ,
after an inelastic strain  is introduced into a spherical region with
radius a in the solid.Â
5.4.9.
A steel
ball-bearing with radius 1cm is pushed into a flat steel surface by a force P.Â
Neglect friction between the contacting surfaces. Typical ball-bearing steels have uniaxial
tensile yield stress of order 2.8 GPa. Â
Calculate the maximum load that the ball-bearing can withstand without
causing yield, and calculate the radius of contact and maximum contact
pressure at this load.
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5.4.10. The contact between the wheel of a locomotive and
the head of a rail may be approximated as the (frictionless) contact between
two cylinders, with identical radius R
as illustrated in the figure. The rail and wheel can be idealized as
elastic-perfectly plastic solids with identical Young’s modulus E, Poisson’s ratio  and yield stress Y. Find expressions for the radius of the contact patch, the
contact area, and the contact pressure as a function of the load acting on
the wheel and relevant geometric and material properties. By estimating values for relevant
quantities, calculate the maximum load that can be applied to the wheel
without causing the rail to yield.
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5.4.11. The figure shows a rolling element bearing. The inner raceway has radius R, and the balls have radius r, and both inner and outer raceways
are designed so that the area of contact between the ball and the raceway is
circular. The balls are equally spaced
circumferentially around the ring. The bearing is free of stress when
unloaded. The bearing is then subjected to a force P as shown. This load is
transmitted through the bearings at the contacts between the raceways and the
balls marked A, B, C in the figure
(the remaining balls lose contact with the raceways but are held in place by
a cage, which is not shown). Assume
that the entire assembly is made from an elastic material with Young’s
modulus  and Poisson’s ratio
5.4.11.1.
Assume that the
load causes the center of the inner raceway to move vertically upwards by a
distance ,
while the outer raceway remains fixed.Â
Write down the change in the gap between inner and outer raceway at A,
B, C, in terms of
5.4.11.2.
Hence,
calculate the resultant contact forces between the balls at A, B, C and the
raceways, in terms of  and relevant geometrical and material
properties.
5.4.11.3.
Finally,
calculate the contact forces in terms of P
5.4.11.4.
If the
materials have uniaxial tensile yield stress Y, find an expression for the maximum force P that the bearing can withstand before yielding.
5.4.12. A rigid, conical indenter with apex angle  is pressed into the surface of an isotropic,
linear elastic solid with Young’s modulus  and Poisson’s ratio .Â
5.4.12.1.
Write down the
initial gap between the two surfaces
5.4.12.2.
Find the
relationship between the depth of penetration h of the indenter and the radius of contact a
5.4.12.3.
Find the
relationship between the force applied to the contact and the radius of
contact, and hence deduce the relationship between penetration depth and
force.  Verify that the contact
stiffness is given by
5.4.12.4.
Calculate the
distribution of contact pressure that acts between the contacting surfaces.
5.4.13. A sphere, which has radius R, is dropped from height h
onto the flat surface of a large solid.Â
The sphere has mass density ,
and both the sphere and the surface can be idealized as linear elastic
solids, with Young’s modulus  and Poisson’s ratio . As a rough approximation, the impact can be
idealized as a quasi-static elastic indentation.
5.4.13.1.
Write down the
relationship between the force P acting
on the sphere and the displacement of the center of the sphere below Â
5.4.13.2.
Calculate the
maximum vertical displacement of the sphere below the point of initial
contact.
5.4.13.3.
Deduce the
maximum force and contact pressure acting on the sphere
5.4.13.4.
Suppose that
the two solids have yield stress in uniaxial tension Y. Find an expression for
the critical value of h which will
cause the solids to yield
5.4.13.5.
Calculate a
value of h if the materials are
steel, and the sphere has a 1 cm radius.
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