Problems for Chapter 5
Analytical Techniques and Solutions for
Linear Elastic Solids
5.3. Complex
Variable Solutions to Static Linear Elasticity Problems
5.3.1.
A long cylinder
is made from an isotropic, linear elastic solid with shear modulus . The solid is loaded so
(i) the resultant forces and moments acting on
the ends of the cylinder are zero;
(ii) the body force in the interior of the solid acts parallel to
the axis of the cylinder; and (iii) Any tractions or displacements imposed on the sides of the cylinder are
parallel to the axis of the cylinder.
Under
these conditions, the displacement field at a point far from the ends of the
cylinder has the form ,
and the solid is said to deform in a state of anti-plane shear.
5.3.1.1.
Calculate the
strain field in the solid in terms of u.
5.3.1.2.
Find an
expression for the nonzero stress components in the solid, in terms of u and material properties.
5.3.1.3.
Find the
equations of equilibrium for the nonzero stress components.
5.3.1.4.
Write down
boundary conditions for stress and displacement on the side of the cylinder
5.3.1.5.
Hence, show that
the governing equations for u reduce
to
5.3.2.
Let be an analytic function of a complex number . Let ,
denote the real and imaginary parts of .
5.3.2.1.
Since is analytic, the real and imaginary parts must
satisfy the Cauchy Riemann conditions
Show
that
5.3.2.2.
Deduce that the
displacement and stress in a solid that is free of body force, and loaded on
its boundary so as to induce a state anti-plane shear (see problem 1) can be
derived an analytic function ,
using the representation
5.3.3.
Calculate the
displacements and stresses generated when the complex potential
is substituted into the representation described in
Problem 2. Show that the solution represents the displacement and stress in an
infinite solid due to a line force acting in the direction at the origin.
5.3.4.
Calculate the
displacements and stresses generated when the complex potential
is substituted into the representation described in
Problem 2. Show that the solution represents the displacement and stress due to
a screw dislocation in an infinite solid, with burgers vector and line
direction parallel to .
(To do this, you need to show that (a) the displacement field has the correct
character; and (b) the resultant force acting on a circular arc surrounding the
dislocation is zero)
5.3.5.
Calculate the
displacements and stresses generated when the complex potential
is substituted into the representation described in
Problem 2. Show that the solution represents the displacement and stress in an
infinite solid, which contains a hole with radius a at the origin, and is subjected to anti-plane shear at infinity.
5.3.6.
Calculate the
displacements and stresses generated when the complex potential
is substituted into the representation described in
Problem 2. Show that the solution represents the displacement and stress in an
infinite solid, which contains a crack with length a at the origin, and is subjected to a prescribed anti-plane shear
stress at infinity. Use the procedure
given in Section 5.3.6 to calculate
5.3.7. Consider complex potentials ,
where a, b, c, d are complex numbers.
Let
be
a displacement and stress field derived from these potentials.
5.3.7.1.
Find values of a,b,c,d that represent a rigid displacement where are (real) constants representing a
translation, and is a real constant representing an
infinitesimal rotation.
5.3.7.2.
Find values of a,b,c,d that correspond to a state of
uniform stress
Note
that the solutions to 7.1 and 7.2 are not unique.
5.3.8.
Show that the
complex potentials
give the stress and displacement field in a
pressurized circular cylinder which deforms in plane strain (it is best to
solve this problem using polar coordinates)
5.3.9.
The complex
potentials
generate the plane strain solution to an edge
dislocation at the origin of an infinite solid.
Work through the algebra necessary to determine the stresses (you can
check your answer using the solution given in Section 5.3.4). Verify that the resultant force exerted by
the internal tractions on a circular surface surrounding the dislocation is
zero.
5.3.10. When a stress field acts on a dislocation, the
dislocation tends to move through the solid.
Formulas for these forces are derived in Section 5.9.5. For the particular case of a straight edge
dislocation, with burgers vector ,
the force can be calculated as follows:
Let denote the stress field in an infinite solid
containing the dislocation (calculated using the formulas in Section 5.3.4
Let denote the actual stress field in the solid
(including the effects of the dislocation itself, as well as corrections due to
boundaries in the solid, or externally applied fields)
Define denote the difference between these
quantities.
The force can then be calculated as .
Consider two edge dislocations in an infinite solid,
each with burgers vector . One dislocation is located at the origin, the
other is at position . Plot contours of the horizontal component of
force acting on the second dislocation due to the stress field of the
dislocation at the origin. (normalize the force as ).
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5.3.11. The figure shows an edge dislocation below the surface
of an elastic solid. Use the solution
given in Section 5.3.12, together with the formula in Problem 5.3.9 to
calculate an expression for the force acting on the dislocation.
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5.3.12. The figure shows an edge dislocation with burgers
vector that lies in a strained elastic film with
thickness h. The film and substrate have the same elastic
moduli. The stress in the film consists of the stress due to the dislocation,
together with a tensile stress . Calculate the force acting on the
dislocation, and hence find the film thickness for which the dislocation will
be attracted to the free surface and escape from the film. You will need to use the formula given in
problem 5.3.10 to calculate the force on the dislocation.
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5.3.13. The figure shows a dislocation in an elastic solid
with Young’s modulus E and Poisson’s
ratio ,
which is bonded to a rigid solid. The
solution can be generated from complex potentials
where
is the solution for a
dislocation at position in an infinite solid, and
corrects the solution to satisfy the zero displacement
boundary condition at the interface.
5.3.13.1.
Show that the
solution satisfies the zero displacement boundary condition
5.3.13.2.
Calculate the
force acting on the dislocation, in terms of h and relevant material properties. You will need to use the formula from 5.3.10
to calculate the force on the dislocation.
5.3.13.3.
Calculate the
distribution of stress along the interface between the elastic and rigid
solids, in terms of h and
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5.3.14. The figure shows a rigid cylindrical inclusion with
radius a embedded in an isotropic
elastic matrix. The solid is subjected
to a uniform uniaxial stress at infinity.
The goal of this problem is to calculate the stress fields in the
matrix.
5.3.14.1.
Write down the
boundary conditions on the displacement field at r=a
5.3.14.2.
Show that the
boundary conditions can be satisfied by complex potentials of the form
where
are three real valued coefficients whose
values you will need to determine. The
algebra in this problem can be simplified by noting that on the boundary of the inclusion.
5.3.14.3.
Find an
expression for the stresses acting at the inclusion/matrix boundary
5.3.14.4.
The interface
between inclusion and matrix fails when the normal stress acting on the
interface reaches a critical stress . Find an expression for the maximum tensile
stress that can be applied to the material without causing failure.
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5.3.15. The figure shows a cylindrical inclusion with radius a and Young’s modulus and Poisson’s
ratio embedded in an isotropic elastic matrix
with elastic constants . The solid is subjected to a uniform uniaxial
stress at infinity.
The goal of this problem is to calculate the stress fields in both the
particle and the matrix. The analylsis
can be simplified greatly by assuming a
priori that the stress in the particle is uniform (this is not obvious, but
can be checked after the full solution has been obtained). Assume, therefore, that the stress in the
inclusion is ,
where are to be determined. The solution inside the particle can
therefore be derived from complex potentials
In addition, assume that the solution in the matrix
can be derived from complex potentials of the form
where
are three real valued coefficients whose
values you will need to determine.
5.3.15.1.
Write down the
boundary conditions on the displacement field at r=a. Express this boundary
condition as an equation relating to in terms of material properties.
5.3.15.2.
Write down
boundary conditions on the stress components at r=a.
Express this boundary condition as an equation relating to .
5.3.15.3.
Calculate expressions
for ,
in terms of and material properties.
5.3.15.4.
The inclusion
fractures when the maximum principal stress acting in the particle reaches a
critical stress . Find an expression for the maximum tensile
stress that can be applied to the material without causing failure.
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5.3.16. The figure shows a slit crack in an infinite
solid. Using the solution given in
Section 5.3.6, calculate the stress field very near the right hand crack tip
(i.e. find the stresses in the limit as ). Show
that the results are consistent with the asymptotic crack tip field given in
Section 5.2.9, and deduce an expression for the crack tip stress intensity
factors in terms of and a.
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5.3.17. Two identical cylindrical roller bearings with radius
1cm are pressed into contact by a force P
per unit out of plane length as indicated in the figure. The bearings are made from 52100 steel with
a uniaxial tensile yield stress of 2.8GPa.
Calculate the force (per unit length) that will just initiate yield in
the bearings, and calculate the width of the contact strip between the bearings
at this load.
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5.3.18. The figure shows a pair of identical involute spur
gears. The contact between the two gears
can be idealized as a line contact between two cylindrical surfaces. The goal of this problem is to find an
expression for the maximum torque that can be transmitted through the
gears. The gears can be idealized as
isotropic, linear elastic solids with Young’s modulus E and Poisson’s ratio .
As a representative configuration, consider the
instant when a single pair of gear teeth make contact exactly at the pitch
point. At this time, the geometry can be
idealized as contact between two cylinders, with radius ,
where is the pitch circle radius of the gears and is the pressure angle. The cylinders are pressed into contact by a
force .
5.3.18.1.
Find a formula
for the area of contact between the two gear teeth, in terms of ,
,
,
b and representative material
properties.
5.3.18.2.
Find a formula
for the maximum contact pressure acting on the contact area, in terms of ,
,
,
b and representative material
properties.
5.3.18.3.
Suppose that the
gears have uniaxial tensile yield stress Y. Find a formula for the critical value of required to initiate yield in the gears.