 Problems for Chapter 4

Solution to Simple Problems

4.4.  Simple Dynamic Problems for Elastic Solids

4.4.1.      Calculate longitudinal and shear wave speeds in (a) Aluminum nitride; (b) Steel; (d) Aluminum and (e) Rubber.

4.4.2.      A linear elastic half-space with Young’s modulus E and Poisson’s ratio $\nu$ is stress free and stationary at time t=0¸ is then subjected to a constant pressure ${p}_{0}$ on its surface for t>0.

4.4.2.1.            Calculate the stress, displacement and velocity in the solid as a function of time

4.4.2.2.            Calculate the total kinetic energy of the half-space as a function of time

4.4.2.3.            Calculate the total potential energy of the half-space as a function of time

4.4.2.4.            Verify that the sum of the potential and kinetic energy is equal to the work done by the tractions acting on the surface of the half-space.

4.4.3.      The surface of an infinite linear elastic half-space with Young’s modulus E and Poisson’s ratio $\nu$ is subjected to a harmonic pressure on its surface, given by $p\left(t\right)={p}_{0}\mathrm{sin}\omega t$ t>0, with p=0 for t<0

4.4.3.1.            Calculate the distribution of stress, velocity and displacement in the solid.

4.4.3.2.            What is the phase difference between the displacement and pressure at the surface?

4.4.3.3.            Calculate the total work done by the applied pressure in one cycle of loading.

4.4.4.      A linear elastic solid with Young’s modulus E Poisson’s ratio $\nu$ and density $\rho$ is bonded to a rigid solid at ${x}_{1}=a$.  Suppose that a plane wave with displacement and stress field

${u}_{2}\left({x}_{1},t\right)=\left\{\begin{array}{c}\frac{\left(1-2\nu \right)\left(1+\nu \right)}{\left(1-\nu \right)}\frac{{\sigma }_{0}}{E}\left({c}_{L}t-{x}_{1}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}<{c}_{L}t\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}>{c}_{L}t\end{array}$              ${\sigma }_{11}=\left\{\begin{array}{c}-{\sigma }_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}<{c}_{L}t\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}>{c}_{L}t\end{array}$

is induced in the solid, and at time $t={x}_{1}/a$ is reflected off the interface.   Find the reflected wave, and sketch the variation of stress and velocity in the elastic solid just before and just after the reflection occurs.

4.4.5.      Consider the plate impact experiment described in Section 4.4.8

4.4.5.1.            Draw graphs showing the stress and velocity at the impact face of the flyer plate as a function of time.

4.4.5.2.            Draw graphs showing the stress and velocity at the rear face of the flyer plate as a function of time

4.4.5.3.            Draw graphs showing the stress and velocity at the mid-plane of the flyer plate as a function of time

4.4.5.4.            Draw a graph showing the total strain energy and kinetic energy of the system as a function of time. Verify that total energy is conserved.

4.4.5.5.            Draw a graph showing the total momentum of the flyer plate and the target plate as a function of time.  Verify that momentum is conserved.  4.4.6.      In a plate impact experiment, two identical elastic plates with thickness h, Young’s modulus E, Poisson’s ratio $\nu$, density $\rho$ and longitudinal wave speed ${c}_{L}$ are caused to collide, as shown in the picture.   Just prior to impact, the projectile has a uniform velocity ${v}_{0}$.   Draw the (x,t) diagram for the two solids after impact.  Show that the collision is perfectly elastic, in the terminology of rigid body collisions, in the sense that all the energy in the flyer is transferred to the target.

4.4.7.      In a plate impact experiment, an elastic plates with thickness h, Young’s modulus E, Poisson’s ratio $\nu$, density $\rho$ and longitudinal wave speed ${c}_{L}$ impacts a second plate with identical elastic properties, but thickness 2h, as shown in the picture.   Just prior to impact, the projectile has a uniform velocity ${v}_{0}$.   Draw the (x,t) diagram for the two solids after impact. 4.4.8.      A “ Split-Hopkinson bar” or “Kolsky bar” is an apparatus that is used to measure plastic flow in materials at high rates of strain (of order 1000/s).  The apparatus is sketched in the figure.  A small specimen of the material of interest, with length a<<d, is placed between two long slender bars with length d.  Strain gages are attached near the mid-point of each bar.  At time t=0 the system is stress free and at rest.  Then, for t>0 a constant pressure p is applied to the end of the incident bar, sending a plane wave  down the bar. This wave eventually reaches the specimen.  At this point part of the wave is reflected back up the incident bar, and part of it travels through the specimen and into the second bar (known as the ‘transmission bar’). The history of stress and strain in the specimen can be deduced from the history of strain measured by the two strain gages.  For example, if the specimen behaves as an elastic-perfectly plastic solid, the incindent and reflected gages would record the data shown in the figure.  The goal of this problem is to calculate a relationship between the measured strains and the stress and strain rate in the specimen.  Assume that the bars are linear elastic with Young’s modulus E and density $\rho$, and wave speed ${c}_{B}\approx \sqrt{E/\rho }$, and that the bars deform in uniaxial compression.

4.4.8.1.            Write down the stress, strain and velocity field in the incident bar as a function of time and distance down the bar in terms of the applied pressure p and relevant material and geometric parameters, for $t.

4.4.8.2.            Assume that the waves reflected from, and transmitted through, the specimen are both plane waves.  Let ${\epsilon }_{R}$ and ${\epsilon }_{T}$ denote the compressive strains in the regions behind the reflected and transmitted wave fronts, respectively.  Write down expressions for the stress and velocity behind the wave fronts in both incident and transmitted bars in terms of ${\epsilon }_{R}$ and ${\epsilon }_{T}$, for $2d/{c}_{B}>t>d/{c}_{B}$

4.4.8.3.            The stress behind the reflected and transmitted waves must equal the stress in the specimen.  In addition, the strain rate in the specimen can be calculated from the relative velocity of the incident and transmitted bars where they touch the specimen.   Show that the strain rate in the specimen can be calculated from the measured strains as $\left({\epsilon }_{I}-{\epsilon }_{R}-{\epsilon }_{T}\right){c}_{L}/a$, while the stress in the specimen can be calculated from $\sigma =E{\epsilon }_{T}$.  4.4.9.      In a plate impact experiment, two plates with identical thickness h, Young’s modulus E, Poisson’s ratio $\nu$ are caused to collide, as shown in the picture.  The target plate has twice the mass density of the flyer plate. Find the stress and velocity behind the waves generated by the impact in both target and flyer plate.  Hence, draw the (x,t) diagram for the two solids after impact.

4.4.10.  In a plate impact experiment, two plates with identical thickness h, Young’s modulus E, Poisson’s ratio $\nu$, and density $\rho$ are caused to collide, as shown in the picture.  The flyer plate has twice the mass density of the target plate. Find the stress and velocity behind the waves generated by the impact in both target and flyer plate.  Hence, draw the (x,t) diagram for the two solids after impact. 4.4.11.  The figure shows a pressure-shear plate impact experiment. A flyer plate with speed ${v}_{0}$ impacts a stationary target. Both solids have identical thickness h, Young’s modulus E, Poisson’s ratio $\nu$, density $\rho$ and longitudinal and shear wave speeds ${c}_{L}$ and ${c}_{S}$.    The faces of the plates are inclined at an angle $\theta$ to the initial velocity, as shown in the figure.  Both pressure and shear waves are generated by the impact. Let $\left\{{e}_{1},{e}_{2}\right\}$ denote unit vectors  Let ${\sigma }_{11}={\sigma }_{0}$ denote the (uniform) stress behind the propagating pressure wave in both solids just after impact, and ${\sigma }_{12}={\tau }_{0}$ denote the shear stress behind the shear wave-front.  Similarly, let $\Delta {v}_{1}^{f},\Delta {v}_{2}^{f}$ denote the change in longitudinal and transverse velocity in the flier across the pressure and shear wave fronts, and let $\Delta {v}_{1}^{t},\Delta {v}_{2}^{t}$ denote the corresponding velocity changes in the target plate.  Assume that the interface does not slip after impact, so that both velocity and stress must be equal in both flier and target plate at the interface just after impact.  Find expressions for ${\sigma }_{0}$, ${\tau }_{0}$, $\Delta {v}_{1}^{f},\Delta {v}_{2}^{f}$, $\Delta {v}_{1}^{t},\Delta {v}_{2}^{t}$ in terms of ${v}_{0},\theta$ and relevant material properties.

4.4.12.  Draw the full (x,t) diagram for the pressure-shear configuration described in problem 11.  Assume that the interface remains perfectly bonded until it separates under the application of a tensile stress.  Note that you will have to show (x,t) diagrams associated with both shear and pressure waves. 4.4.13.  Consider an isotropic, linear elastic solid with Young’s modulus E, Poisson’s ratio $\nu$, density $\rho$ and shear wave speed ${c}_{S}$. Suppose that a plane, constant stress shear wave propagates through the solid, which is initially at rest. The wave propagates in a direction $p=\mathrm{cos}\theta {e}_{1}+\mathrm{sin}\theta {e}_{2}$, and the material has particle velocity $v=V{e}_{3}$ behind the wave-front.

4.4.13.1.        Calculate the components of stress in the solid behind the wave front.

4.4.13.2.        Suppose that the wave front is incident on a flat, stress free surface.  Take the origin for the coodinate system at some arbitrary time t at the point where the propagating wave front just intersects the surface, as shown in the picture.   Write down the velocity of this intersection point (relative to a stationary observer) in terms of V and $\theta$.

4.4.13.3.        The surface must be free of traction both ahead and behind the wave front.  Show that the boundary condition can be satisfied by superposing a second constant stress wave front, which intersects the free surface at the origin of the coodinate system defined in 13.2, and propagates in a direction $p=\mathrm{cos}\theta {e}_{1}-\mathrm{sin}\theta {e}_{2}$.  Hence, write down the stress and particle velocity in each of the three sectors A, B, C shown in the figure.  Draw the displacement of the free surface of the half-space. 4.4.14.  Suppose that a plane, constant stress pressure wave propagates through an isotropic, linear elastic solid that is initially at rest. The wave propagates in a direction $p=\mathrm{cos}\theta {e}_{1}+\mathrm{sin}\theta {e}_{2}$, and the material has particle velocity $v=Vp$ behind the wave-front.

4.4.14.1.        Calculate the components of stress in the solid behind the wave front.

4.4.14.2.        Suppose that the wave front is incident on a flat, stress free surface.  Take the origin for the coodinate system at some arbitrary time t at the point where the propagating wave front just intersects the surface, as shown in the picture.   Write down the velocity of this intersection point (relative to a stationary observer) in terms of V and $\theta$.

4.4.14.3.        The pressure wave is reflected as two waves $–$ a reflected pressure wave, which propagates in direction $p=\mathrm{cos}\theta {e}_{1}-\mathrm{sin}\theta {e}_{2}$ and has particle velocity ${v}_{p}p$ and a reflected shear wave, which propagates in direction $p=\mathrm{cos}{\theta }_{1}{e}_{1}-\mathrm{sin}{\theta }_{1}{e}_{2}$ and has particle velocity ${v}_{s}\left(\mathrm{sin}{\theta }_{1}{}_{\text{ }}{e}_{1}+\mathrm{cos}{\theta }_{1}{e}_{2}\right)$.  Use the condition that the incident wave and the two reflected waves must always intersect at the same point on the surface to write down an equation for ${\theta }_{1}$ in terms of $\theta$ and Poisson’s ratio.

4.4.14.4.        The surface must be free of traction. Find equations for ${v}_{p}$ and ${v}_{s}$ in terms of V, $\theta$, ${\theta }_{1}$ and Poisson’s ratio.

4.4.14.5.        Find the special angles for which the incident wave is reflected only as a shear wave (this is called “mode conversion”)