Problems for Chapter 5

Analytical Techniques and Solutions for Linear Elastic Solids

5.6.  Solutions to Dynamic Problems

5.6.1.      Consider the Love potentials ${\Psi }_{i}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =A\mathrm{sin}\left({p}_{i}{x}_{i}-{c}_{L}t\right)$, where ${p}_{i}$ is a constant unit vector and A is a constant.

5.6.1.1.            Verify that the potentials satisfy the appropriate governing equations

5.6.1.2.            Calculate the stresses and displacements generated from these potentials.

5.6.1.3.            Briefly, interpret the wave motion represented by this solution.

5.6.2.      Consider the Love potentials ${\Psi }_{i}={U}_{i}\mathrm{sin}\left({p}_{k}{x}_{k}-{c}_{s}t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =0$, where ${p}_{i}$ is a constant unit vector and ${U}_{i}$ is a constant unit vector.

5.6.2.1.            Find a condition relating ${U}_{i}$ and ${p}_{i}$ that must be satisfied for this to be a solution to the governing equations

5.6.2.2.            Calculate the stresses and displacements generated from these potentials.

5.6.2.3.            Briefly, interpret the wave motion represented by this solution.

5.6.3.      Show that ${\Psi }_{i}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{1}{R}f\left(t-R/{c}_{L}\right)$ satisfy the governing equations for Love potentials.  Find expressions for the corresponding displacement and stress fields.

5.6.4.      Calculate the radial distribution of Von-Mises effective stress surrounding a spherical cavity of radius a, which has pressure ${p}_{0}$ suddenly applied to its surface at time t=0.  Hence, find the location in the solid that is subjected to the largest Von-Mises stress, and the time at which the maximum occurs.

5.6.5.      Calculate the displacement and stress fields generated by the Love potentials

${\Psi }_{i}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{A}{R}\mathrm{sin}\left\{\omega \left(t-\left(R-a\right)/{c}_{L}\right)+\xi \right\}$

Calculate the traction acting on the surface at r=a.  Hence, find the Love potential that generates the fields around a spherical cavity with radius a, which is subjected to a harmonic pressure $p\left(t\right)={p}_{0}\mathrm{sin}\omega t$.  Plot the amplitude of the surface displacement at r=a (normalized by a) as a function of $\omega a/{c}_{L}$.

5.6.6.      Calculate the distribution of kinetic and potential energy near the surface of a half-space that contains a Rayleigh wave with displacement amplitude ${U}_{0}$ and wave number k.  Take Poisson’s ratio $\nu =0.3$.  Calculate the total energy per unit area of the wave (find the total energy in one wavelength, then divide by the wavelength).  Estimate the energy per unit area in Rayleigh waves associated with earthquakes.

5.6.7.      The figure shows a surface-acoustic-wave device that is intended to act as a narrow band-pass filter.  A piezoelectric substrate has two transducers attached to its surface $–$ one acts as an “input” transducer and the other as “output.”  The transducers are electrodes: a charge can be applied to the input transducer; or detected on the output.  Applying a charge to the input transducer induces a strain on the surface of the substrate: at an appropriate frequency, this will excite a Rayleigh wave in the solid.  The wave propagates to the “output” electrodes, and the resulting deformation of the substrate induces a charge that can be detected.   If the electrodes have spacing d, calculate the frequency at which the surface will be excited.   Estimate the spacing required for a 1GHz filter made from AlN with Young’s modulus 345 GPa and Poisson’s ratio 0.3

5.6.8.      The figure shows a thin elastic strip, which is bonded to rigid solids on both its surfaces. The strip has shear modulus $\mu$ and wave speed ${c}_{s}$, and acts as a wave-guide.  The goal of this problem is to calculate the displacement field associated with transverse wave propagation down the strip.

5.6.8.1.            Assume that the displacement has the form

${u}_{3}=f\left({x}_{2}\right)\mathrm{exp}\left(-ik\left({x}_{1}-ct\right)\right)$

By substituting into the Cauchy-Navier equation, show that

$\frac{{d}^{2}}{d{x}_{2}^{2}}f\left({x}_{2}\right)+{k}^{2}\left(\frac{{c}^{2}}{{c}_{s}^{2}}-1\right)f\left({x}_{2}\right)=0$

Hence, write down the general solution for $f\left({x}_{2}\right)$

5.6.8.2.            Show that the boundary conditions admit solutions of the form

$f\left({x}_{2}\right)=\left\{\begin{array}{c}A\mathrm{sin}\left(n\pi {x}_{2}/H\right)\\ B\mathrm{cos}\left(\left(\pi /2+n\pi \right){x}_{2}/H\right)\end{array}$

where n is an integer, so that

${u}_{3}={U}_{0}\left\{{\mathrm{cos}}^{2}\left(\pi n/2\right)\mathrm{sin}\left(n\pi {x}_{2}/2H\right)+{\mathrm{sin}}^{2}\left(\pi n/2\right)\mathrm{cos}\left(n\pi {x}_{2}/2H\right)\right\}\mathrm{exp}\left(ik\left({x}_{1}-ct\right)\right)$

5.6.8.3.            Find an expression for the phase velocity of the wave, and plot the phase velocity as a function of kH/n

5.6.8.4.            Calculate the dispersion relation for the wave and hence deduce an expression for the group velocity.  Plot the group velocity as a function of kH/n.

5.6.9.      Consider the Love wave described in Section 5.6.4.

5.6.9.1.            Consider first a system with  $\mu ={\mu }_{f}$, ${c}_{s}/{c}_{sf}=2$, as discussed at the end of 5.6.4. Find an expression for the group velocity of the wave, and plot a graph showing the group velocity (normalized by shear wave speed in the layer) as a function of  kH

5.6.9.2.            Consider a system with $\mu ={\mu }_{f}$, ${c}_{s}/{c}_{sf}=1/2$.  Plot graphs showing both the phase velocity and the group velocity in the layer as a function of kH.

5.6.10.  In this problem, you will investigate the energy associated with wave propagation down a simple wave guide.  Consider an isotropic, linear elastic strip, with thickness 2H, shear modulus $\mu$ and wave speed ${c}_{s}$ as indicated in the figure.  The solution for a wave propagating in the ${e}_{1}$ direction, with particle velocity $u={u}_{3}{e}_{3}$ is given in Section 5.6.6 of the text.

5.6.10.1.        The flux of energy associated with wave propagation along a wave-guide can be computed from the work done by the tractions acting on an internal material surface.  The work done per cycle is given by

$〈P〉=\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{-H}{\overset{H}{\int }}{\sigma }_{ij}{n}_{j}\frac{d{u}_{i}}{dt}d{x}_{2}dt$

where T is the period of oscillation and ${n}_{j}=-{\delta }_{j1}$ is a unit vector normal to an internal plane perpendicular to the direction of wave propagation. Calculate $〈P〉$ for the nth wave propagation mode.

5.6.10.2.        The average kinetic energy of a generic cross-section of the wave-guide can be calculated from

$〈K〉=\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{-H}{\overset{H}{\int }}\frac{\rho }{2}\frac{d{u}_{i}}{dt}\frac{d{u}_{i}}{dt}d{x}_{2}dt$

Find $〈K〉$ for nth wave propagation mode.

5.6.10.3.        The average potential energy of a generic cross-section of the wave-guide can be calculated from

$〈\Phi 〉=\frac{1}{T}\underset{0}{\overset{T}{\int }}\underset{-H}{\overset{H}{\int }}\frac{1}{2}{\sigma }_{ij}{\epsilon }_{ij}d{x}_{2}dt$

Find $〈\Phi 〉$ for nth wave propagation mode.  Check that $〈K〉$ = $〈\Phi 〉$

5.6.10.4.        The speed of energy flux down the wave-guide is defined as ${c}_{e}=〈P〉/\left(〈K〉+〈\Phi 〉\right)$.  Find ${c}_{e}$ for the nth propagation mode, and compare the solution with the expression for the group velocity of the wave

${c}_{g}=\frac{d\omega }{dk}=\frac{{c}_{s}kH}{\sqrt{{\left(n\pi /2\right)}^{2}+{k}^{2}{H}^{2}}}$