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
is stress free and stationary at time t=0¸ is then subjected to a constant
pressure 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 is subjected to a harmonic pressure on its
surface, given by 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 and density is bonded to a rigid solid at . Suppose that a plane wave with
displacement and stress field
is
induced in the solid, and at time 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.

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4.4.6. In a plate impact experiment, two
identical elastic plates with thickness h,
Young’s modulus E, Poisson’s
ratio ,
density and longitudinal wave speed are caused to collide, as shown in the
picture. Just prior to impact, the
projectile has a uniform velocity . 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 ,
density and longitudinal wave speed 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 . Draw the (x,t) diagram for the two solids after impact.

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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’).

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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 ,
and wave speed ,
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 .
4.4.8.2.
Assume
that the waves reflected from, and transmitted through, the specimen are both
plane waves. Let and 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 and ,
for
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 ,
while the stress in the specimen can be calculated from .

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4.4.9. In a plate impact experiment, two plates
with identical thickness h, Young’s
modulus E, Poisson’s ratio 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 ,
and density 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.

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4.4.11. The figure shows a pressure-shear plate impact experiment.
A flyer plate with speed impacts a stationary target. Both solids have
identical thickness h, Young’s
modulus E, Poisson’s ratio ,
density and longitudinal and shear wave speeds and . The faces of the plates are inclined at an
angle to the initial velocity, as shown in the
figure. Both pressure and shear waves
are generated by the impact. Let denote unit vectors Let denote the (uniform) stress behind the
propagating pressure wave in both solids just after impact, and denote the shear stress behind the shear
wave-front. Similarly, let denote the change in longitudinal and
transverse velocity in the flier across the pressure and shear wave fronts, and
let 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 ,
,
,
in terms of 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.

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4.4.13. Consider an isotropic, linear elastic
solid with Young’s modulus E,
Poisson’s ratio ,
density and shear wave speed .
Suppose that a plane, constant stress shear wave propagates through the solid,
which is initially at rest. The wave propagates in a direction ,
and the material has particle velocity 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 .
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 . 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.

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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 ,
and the material has particle velocity 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 .
4.4.14.3.
The
pressure wave is reflected as two
waves a reflected pressure wave, which propagates in
direction and has particle velocity and a reflected shear wave, which propagates
in direction and has particle velocity . 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 in terms of and Poisson’s ratio.
4.4.14.4.
The
surface must be free of traction. Find equations for and in terms of V, ,
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”)