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 .Â Â  Vertical displacement of the membrane is prevented at .Â  The membrane is subjected to a uniform out-of-plane pressure with magnitude p on its surface.Â Â  Calculate the displacement field in the membrane, assuming small deflections.Â  This problem can be solved quite easily using Cartesian coordinates.

10.7.2.    The figure shows a thin circular plate with thickness h, Youngâ€™s modulus E and Poissonâ€™s ratio .Â  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 Youngâ€™s modulus E, Poissions ratio .Â  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 Â are modified Bessel functions of the first and second kinds, with order n, while Â and Â 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 Â at r=R, and hence show that the wave numbers Â are roots of the equation

10.7.3.4.           Show that the corresponding mode shapes are given by

10.7.3.5.           Calculate Â for 0<n<3, 1<m<3 and tabulate your results.Â Â  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.Â Â  You should find that the wave numbers Â are the roots of the equation

Calculate the natural frequencies for .

10.7.5.    Use Rayleighâ€™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 Rayleighâ€™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 Â is deposited on the surface of a circular wafer with radius R and thickness h.Â  Both film and substrate have Youngâ€™s modulus E, and Poissonâ€™s ratio . An inelastic strain Â 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.Â  These expressions are only valid if the substrate curvature is small.Â  For mismatch strains, the wafer buckles, as shown in the figure.Â  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.Â  It is best to do this calculation using a symbolic manipulation program.

10.7.7.2.           Calculate the values of Â 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 Â as a function of a suitably normalized measure of mismatch strain, for a Poissonâ€™s ratio .

10.7.8.    The figure shows an elastic plate with Youngâ€™s modulus E, Poissonâ€™s ratio Â and thermal expansion coefficient .Â  The plate is circular, with radius R and its edge is clamped.Â  The plate is initially stress free and is then heated to raise its temperature by , inducing a uniform internal forceÂ  Â in the plate.Â  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 Â with .Â  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 Â is the wave number for the nth buckling mode. Find an expression for Â in terms of Â and relevant geometric and material properties.

10.7.8.3.           Show that the wave number Â 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.Â Â  You should find that the wave numbers Â 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.Â  The film is initially stress free, then heated to raise its temperature by , inducing a uniform biaxial stress fieldÂ  Â in the film.Â  At a critical temperature, the film buckles as shown in the figure.Â Â  For R>>h the buckled film can be modeled as a plate with clamped edge.Â  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 Â and the curvature change tensor Â for the film in terms of Â and , and hence find a formula for the internal force Â and moment Â Â in terms of , , Â and material properties.

10.7.10.2.       Hence, show that the equations of equilibrium in terms of Â and Â reduce to

10.7.10.3.       Assume that Â at .Â  Hence, calculate the smallest value of Â 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 Â exceeds the critical value.Â Â  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, Youngâ€™s modulus E, Poissonâ€™s ratio Â and thermal expansion coefficient . Let Â be a cylindrical-polar coordinate system and let Â 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 .Â Â  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 Â

10.7.12. The Von-Karman equations for a circular plate cannot be solved analytically, even for the case of axially symmetric deformations.Â  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 spreadsheetâ€™s solver function to minimize the potential energy. To this end, consider a circular plate with radius R, thickness h, Youngâ€™s modulus E, Poissonâ€™s ratio Â and thermal expansion coefficient .Â  Assume that the plate is initially stress free, and is then heated to increase its temperature uniformly by .Â  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 Â and Â are related to the displacements by the equations listed in the preceding problem.

10.7.12.2.       To obtain a numerical solution, let Â , with i=0,1,2â€¦N and , denote the position and displacements of a set ofÂ  N+1 equally spaced points along a radius of the disk.Â Â  The derivatives of u and w can be approximated using simple difference formulas.Â  For example, define ,Â  ,Â  i=1,2â€¦N to be the derivatives of u and w at ; similarly, approximate the second derivative of w as Â at each i=1,2â€¦N-1.Â  The value of Â at r=R can be estimated by extrapolation as Â Â Â 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.Â  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 Â and Â that minimize the potential energy.Â Â  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 .Â  You will need to enforce the clamped boundary condition using a constraint.Â Â  You can set Â but you will find that this makes the plate slightly too stiff.Â Â  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 .Â Â Â  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 Â  This is because the in-plane strains begin to significantly stiffen the plate.Â Â  Plot a graph showing the deflection of the center of the plate (normalized by plate thickness) as a function of normalized pressure .

10.7.13. Â The interface between a thin film and its substrate contains a circular crack with radius R.Â  The film has Youngâ€™s modulus E, Poissonâ€™s ratio Â and thermal expansion coefficient .Â  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  Â in the film.Â  At a critical temperature, the film buckles as shown in the figure.Â Â  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.Â  When the film buckles, some of the strain energy in the film is relaxed.Â  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 Â Â Â Â Â Â Â Â Â

so that

where Â and Â 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 Poissonâ€™s ratio Â and the dimensionless parameter .Â  Use the spreadsheet developed in problem 10.7.12 to plot a graph showing Â as a function of Â for a film with .Â  Verify that the critical value of Â corresponding to Â is consistent with the solution to problem 10.7.8.

10.7.13.3.       Find an expression for the crack tip energy release rate Â and the dimensionless function .Â  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 .Â  The dome is open at its top, so that the shell is bounded by spherical polar angles .Â  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 .Â  Calculate the internal forces induced by gravitational loading of the structure, using the membrane theory of shells in Section 10.7.7.Â Â  Use the cylindrical-polar coordinates Â as the coordinate system.

(c) A.F. Bower, 2008
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