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Problems
for Chapter 10
Approximate
Theories for Solids with Special Shapes:
Rods,
Beams, Membranes, Plates and Shells
10.7. Solutions
to Problems Involving Membranes, Plates and Shells
10.7.1. A thin circular membrane with radius R is subjected to in-plane boundary
loading that induces a uniform biaxial membrane tension with magnitude  .  Vertical displacement of the membrane is
prevented at . The membrane is subjected to a uniform
out-of-plane pressure with magnitude p
on its surface.  Calculate the
displacement field in the membrane, assuming small deflections. This problem can be solved quite easily
using Cartesian coordinates.
10.7.2. The figure shows a thin circular plate with
thickness h, Young’s modulus E
and Poisson’s ratio . The edge of the plate is clamped, and its
surface is subjected to a uniform out-of-plane pressure with magnitude p.Â
Calculate the displacement field and internal moment and shear force
in the plate, assuming small deflections.Â
This problem can easily be solved using Cartesian coordinates.
10.7.3. The figure shows a circular elastic plate with
Young’s modulus E, Poissions ratio . The plate has thickness h and radius R, and is clamped at its edge.Â
The goal of this problem is to calculate the mode shapes and natural
frequencies of vibration of the plate.Â
The solution to 10.6.2 may be used.
10.7.3.1.
Show that the
general solution to the equation of motion for the freely vibrating plate is
given by
where
A,B,C,D, Â Â are arbitrary constants, Â are Bessel functions of the first and second
kinds, and  are modified Bessel functions of the first
and second kinds, with order n,
while  and  are a wave number and vibration frequency
that are related by
10.7.3.2.
Show that most
general solution with bounded displacements at r=0 has the form
10.7.3.3.
Write down the
boundary conditions for  at r=R,
and hence show that the wave numbers  are roots of the equation
10.7.3.4.
Show that the
corresponding mode shapes are given by
10.7.3.5.
Calculate  for 0<n<3, 1<m<3 and tabulate
your results.  Compare the solution to
the corresponding membrane problem solved in Section 10.7.2.
10.7.3.6.
Plot contours
showing the mode shapes for the modes with the lowest four frequencies.
10.7.4. Repeat the preceding problem for a plate with a
simply supported edge.  You should
find that the wave numbers  are the roots of the equation
Calculate
the natural frequencies for .
10.7.5. Use Rayleigh’s method, with a suitable guess for the
mode shape, to estimate the fundamental frequency of vibration of the
circular plate described in Problem 10.7.3. Compare the approximate solution
with the exact solution.
10.7.6. Use Rayleigh’s method, with a suitable guess for the
mode shape, to estimate the fundamental frequency of vibration of the
circular plate described in Problem 10.7.4. Compare the approximate solution
with the exact solution.
10.7.7. A thin film with thickness  is deposited on the surface of a
circular wafer with radius R
and thickness h. Both film and substrate
have Young’s modulus E, and
Poisson’s ratio .
An inelastic strain  is introduced into the film by some external
process (e.g. thermal expansion), which generates stresses in the film, and
also causes the substrate to bend.
In Section 10.7.3, expressions were derived relating
the substrate curvature to the mismatch strain in the film. These expressions are only valid if the
substrate curvature is small. For
mismatch strains, the wafer buckles, as shown in the figure. The goal of this problem is to estimate the
critical value of mismatch strain that will cause the wafer to buckle.
10.7.7.1.
Assume that the
displacement of the mid-plane of the wafer-film system is
Calculate the distribution of strain in the film and
substrate, and hence deduce the total strain energy density of the
system. It is best to do this calculation
using a symbolic manipulation program.
10.7.7.2.
Calculate the
values of  that minimize the potential energy of the
plate, and hence show that the two curvatures satisfy
and
find a formula for the constant C.
10.7.7.3.
Hence, plot a
graph showing the equilibrium values of the normalized curvature  as a function of a suitably normalized
measure of mismatch strain, for a Poisson’s ratio .
10.7.8. The figure shows an elastic plate with Young’s
modulus E, Poisson’s ratio  and thermal expansion coefficient . The plate is circular, with radius R and its edge is clamped. The plate is initially stress free and is then heated to raise its temperature by ,
inducing a uniform internal force  in the plate. The goal of this problem is to calculate
the critical temperature that will cause the plate to buckle, using the
governing equations listed in Section 10.6.2.
10.7.8.1.
Assume an
axially symmetric buckling mode, so that  with . Show that w satisfies the governing equation
10.7.8.2.
Show that the
general solution to this equation is
where A,B,C,D, are arbitrary constants, Â are Bessel functions of the first and second
kind of order zero, and  is the wave number for the nth buckling mode. Find an expression
for  in terms of  and relevant geometric and material
properties.
10.7.8.3.
Show that the
wave number  satisfies ,
and deduce an expression for the critical temperature change at which the
plate can first buckle.
10.7.8.4.
Plot the
buckling mode associated with this temperature.
10.7.9. Repeat the preceding problem for a plate with a
simply supported edge.  You should
find that the wave numbers  are the roots of the equation
Calculate
the critical buckling temperature and plot the corresponding buckling mode
for .
10.7.10. The interface between a thin film and its substrate
contains a tunnel crack with length 2a. The film is initially stress free, then
heated to raise its temperature by ,
inducing a uniform biaxial stress fieldÂ
 in the film.Â
At a critical temperature, the film buckles as shown in the
figure.  For R>>h the buckled film can be modeled as a plate with
clamped edge. The goal of this problem
is to calculate the critical temperature required to cause the film to
buckle, and to calculate the crack tip energy release rate as the buckled
film delaminates from the substrate.Â
Assume that the displacement of the mid-plane of the film has the form
,
and use the Von-Karman plate bending theory of Section 10.6.3.
10.7.10.1. Write down expressions for the mid-plane strain  and the curvature change tensor  for the film in terms of  and ,
and hence find a formula for the internal force  and moment   in terms of ,
,
 and material properties.
10.7.10.2. Hence, show that the equations of equilibrium in
terms of  and  reduce to
10.7.10.3. Assume that  at . Hence, calculate the smallest value of  for which a solution with nonzero w exists.
10.7.10.4. Deduce an expression for the critical temperature
required to cause the film to buckle
10.7.10.5. Assume that  exceeds the critical value.  Calculate the resulting displacement field
10.7.10.6. Hence calculate the decrease in energy of the film
during the buckling
10.7.10.7. Use the preceding result to deduce the crack tip
energy release rate during delamination.
10.7.11. Consider a thin circular plate with thickness h, Young’s modulus E, Poisson’s ratio  and thermal expansion coefficient .
Let  be a cylindrical-polar coordinate system and
let  be a triad of unit vectors parallel to the natural (covariant) basis vectors
for the coordinate system. Assume that the deformation of the plate is
axially symmetric, so that the displacement field can be expressed as .  Show that the Von-Karman equations
governing the motion of the plate can be reduced to the following form
10.7.11.1. Mid-plane strain-deflection relation
10.7.11.2. Displacement-curvature relation
10.7.11.3. Membrane stress-strain relation
10.7.11.4. Moment-curvature relations
10.7.11.5.
Equations of
motion
 Â
10.7.12. The Von-Karman equations for a circular plate cannot
be solved analytically, even for the case of axially symmetric
deformations. However, it is very
straightforward to set up a simple spreadsheet to calculate the potential
energy of the plate, and find a numerical solution by using the spreadsheet’s
solver function to minimize the potential energy. To this end, consider a
circular plate with radius R,
thickness h, Young’s modulus E, Poisson’s ratio  and thermal expansion coefficient . Assume that the plate is initially stress
free, and is then heated to increase its temperature uniformly by . At the same time, the surface of the plate
is subjected to a uniform pressure p
acting perpendicular to its surface. Assume that the deformation of the plate
is axially symmetric, so that the displacement field can be expressed as .
10.7.12.1. Show that the potential energy of the plate can be
expressed as
where
 and  are related to the displacements by the
equations listed in the preceding problem.
10.7.12.2. To obtain a numerical solution, let  ,
with i=0,1,2…N and ,
denote the position and displacements of a set of N+1 equally
spaced points along a radius of the disk. Â
The derivatives of u and w can be approximated using simple
difference formulas. For example,
define , , i=1,2…N
to be the derivatives of u and w at ;
similarly, approximate the second derivative of w as  at each i=1,2…N-1. The value of  at r=R
can be estimated by extrapolation as    The integrals in the expression for the
potential energy can be evaluated using a simple piecewise-constant
approximation (or if you prefer piecewise linear) to the integrand. Use this
approach to set up an EXCEL spreadsheet to calculate the total potential
energy of the plate. You should find N=40 sufficient for most practical purposes.
10.7.12.3. Check your spreadsheet by calculating the potential
energy of a heated flat plate with ,
and compare the prediction with the exact solution.
10.7.12.4. The deflection of the plate under loading can be
calculated by using the solver feature of EXCEL to determine the values of  and  that minimize the potential energy.  To test your spreadsheet, use it to
calculate the deflection of a circular plate with thickness h/R=0.02, with a clamped boundary,
which is subjected to a pressure . You will need to enforce the clamped
boundary condition using a constraint. Â
You can set  but you will find that this makes the plate
slightly too stiff.  A better result
is obtained by extrapolating the slope of the plate to r=R based on the slope of the last two segments, and enforcing
the constraint .   Show that, for this pressure, the
numerical solution agrees with the predictions of the exact small-deflection
solution.
10.7.12.5. Show that if the deflection of the center of the
plate is comparable to the plate thickness, the small deflection solution
predicts deflections that are substantially greater than the numerical
solution of the Von-Karman equations. Â
This is because the in-plane strains begin to significantly stiffen
the plate.  Plot a graph showing the
deflection of the center of the plate (normalized by plate thickness) as a
function of normalized pressure .
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10.7.13. Â The interface
between a thin film and its substrate contains a circular crack with radius R.Â
The film has Young’s modulus E,
Poisson’s ratio  and thermal expansion coefficient . The substrate can be idealized as rigid.
The film is initially stress free, then heated to raise its temperature by ,
inducing a uniform biaxial stress fieldÂ
 in the film.Â
At a critical temperature, the film buckles as shown in the
figure.  For R>>h the buckled film can be modeled as a plate with
clamped edge, so that the critical buckling temperature is given by the
solution to problem 10.7.8. When the
film buckles, some of the strain energy in the film is relaxed. This relaxation in energy can cause the
film to delaminate from the substrate.Â
10.7.13.1. Show that the change in strain energy of the system
during the formation of the buckle can be expressed in dimensionless form by
defining dimensionless measures of displacement, position and strain as
         Â
so
that
where
 and  is the total strain energy of the circular
portion of the film before buckling.
10.7.13.2. The implication of 10.5.13.1 is that the change in
strain energy (and hence the normalized displacement field which minimizes
the potential energy) is a function of material and geometric parameters only
through Poisson’s ratio  and the dimensionless parameter . Use the spreadsheet developed in problem
10.7.12 to plot a graph showing  as a function of  for a film with . Verify that the critical value of  corresponding to  is consistent with the solution to problem 10.7.8.
10.7.13.3. Find an expression for the crack tip energy release
rate  and the dimensionless function . Hence, use the results of 10.7.13.2 to plot
a graph showing the crack tip energy release rate (suitably normalized) as a
function of ,
for a film with .
10.7.14. The figure shows a thin-walled, spherical dome with
radius R, thickness h and mass density . The dome is open at its top, so that the
shell is bounded by spherical polar angles . Calculate the internal forces induced by
gravitational loading of the structure, using the membrane theory of shells
in Section 10.7.7.
10.7.15. The figure shows a thin-walled conical shell with
thickness h and mass density . Calculate the internal forces induced by
gravitational loading of the structure, using the membrane theory of shells
in Section 10.7.7.  Use the
cylindrical-polar coordinates  as the coordinate system.
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