Chapter 5
Analytical
techniques and solutions for linear elastic solids
5.6 Solutions to dynamic problems for
isotropic linear elastic solids
Dynamic problems are even more difficult to solve than static
problems.Â Â Nevertheless, analytical
solutions have been determined for a wide range of important problems.Â There is not space here to do justice to
the subject, but a few solutions will be listed to give a sense of the
general features of solutions to dynamic problems.
5.6.1 Love potentials for dynamic
solutions for isotropic solids
In this section we outline a general potential
representation for 3D dynamic linear elasticity problems.Â The technique is similar to the 3DÂ PapkovichNeuber representation for static
solutions outlined in Section 5.5.
Assume
that
Â The
solid has Youngâ€™s modulus E, mass
density Â and Poissonâ€™s ratio .
Â Define
longitudinal and shear wave speeds (see Sect 4.4.5)
Â Body
forces are neglected (a rather convoluted procedure exists for problems
involving body force)
Â The
solid is assumed to be at rest for t<0
Â Part
of the boundary Â is subjected to time dependent prescribed
displacements
Â A
second part of the boundary Â is subjected to prescribed tractions
The
procedure can be summarized as follows:
1. Find a vector function Â Â and a
scalar function Â which satisfy
as
well as boundary conditions
and
initial conditions .
2. Calculate displacements from the formula
3. Calculate stresses from the formula
You can easily show
that this solution satisfies the equations of motion for an elastic solid, by
substituting the formula for displacements into the CauchyNavier equation
The details are left as an
exercise.Â Â More importantly, one can
also show that the representation is complete,
i.e. all dynamic solutions can be derived from some appropriate combination
of potentials.
5.6.2 Pressure suddenly applied to the
surface of a spherical cavity in an infinite solid
The figure shows a
spherical cavity with radius a in
an infinite elastic solid with Youngâ€™s modulus E and Poissonâ€™s ratio .Â The solid is at rest for t<0.Â Â A time t=0,
a pressure p is applied to the
surface of the hole, and thereafter held fixed.
The
solution is generated by Love potentials
where
The displacements and
stresses follow as
The radial and hoop
stresses at several time intervals are plotted below:
Â Â Â Â Â Â Â Â Â Â Â Â Â Â
Observe that
 A wave front propagates out from the cavity at
the longitudinal wave speed ;
 Unlike the simple 2D
wave problems discussed in Section 4.4, the stress is not constant
behind the front.Â Instead, each
point in the solid experiences a damped oscillation in displacement and
stress that eventually decays to the static solution;
 Both the radial and
hoop stress reverse sign as the wave passes by.Â For this reason dynamic loading can
cause failures to occur in very unexpected places;
 The maximum stress induced by dynamic loading
substantially exceeds the static solution.Â
5.6.3 Rayleigh waves
A Rayleigh wave is a special type of wave which propagates near the
surface of an elastic solid.Â Assume
that
Â The
solid is an isotropic, linear elastic material with Youngâ€™s modulus ,
Poissonâ€™s ratio Â and mass density
Â The
solid has shear wave speed Â Â and longitudinal wave speed
Â The
surface is free of tractions
Â A
Rayleigh wave with wavelength Â propagates in the Â direction
The displacement and stress due can be derived from Love potentials
where
,
Â is the amplitude of the vertical
displacement at the free surface, Â is the wavenumber; ,
;
and Â is the Rayleigh wave speed, which is the
positive real root of
This
equation can easily be solved for Â with a symbolic manipulation program, which
will most likely return 6 roots.Â The
root of interest lies in the range Â for .
The solution can be approximated by Â with an error of less than 0.6% over the
full range of Poisson ratio.
The
nonzero components of displacement and stress follow as
You
can use either the real or imaginary part of these expressions for the
displacement and stress fields (they are identical, except for a phase
difference). Of course, if you choose to take the real part of one of the
functions, you must take the real part for all the others as well. Note that
substituting Â in the expression for Â and setting Â yields the equation for the Rayleigh wave
speed, so the boundary condition Â is satisfied.Â The variations with depth of stress
amplitude and displacement amplitude are plotted below.
Â Â Â Â Â Â
Important
features of this solution are:
 The wave is confined
to a layer near the surface with thickness about twice the
wavelength.Â
 The horizontal and
vertical components of displacement are 90 degrees out of phase.Â Material particles therefore describe
elliptical orbits as the wave passes by.
 The speed of the
wave is independent of its wavelength Â that is to say, the wave is
nondispersive.
 Rayleigh waves are
exploited in a range of engineering applications, including surface
acoustic wave devices; touch sensors; and miniature linear motors.Â They are also observed in earthquakes,
although these waves are observed to be dispersive, because of density
variations of the earthâ€™s surface.
5.6.4 Love waves
Love
waves are a second form of surface wave, somewhat similar to Rayleigh waves,
which propagate through a thin elastic layer bonded to the surface of an
elastic half space.Â Love waves involve
motion perpendicular to the plane of the figure. Assume that
Â The
layer has thickness H, shear
modulus Â and shear wave speed
Â The
substrate has shear modulus Â and shear wave speed
Â The
wave speeds satisfy Â
The
displacement and stress associated with a harmonic Love wave with wavelength Â which propagates in the Â direction can be derived from Love
potentials
where
Â is the amplitude of the vertical
displacement at the free surface, Â is the wavenumber; ,
;
and Â is the wave speed, which is given by the
positive real roots of
This
relationship is very unlike the equations for wave speeds in unbounded or
semiinfinite solids, and leads to a number of counterintuitive
results.Â Â Note that
 The wave speed
depends on its wavelength.Â A wave
with these properties is said to be dispersive,
because a pulse consisting of a spectrum of harmonic waves tends to
spread out;
 The wave speed is
always faster than the shear wave speed of the layer, but less than the
wave speed in the substrate;
 If a wave with
wavenumber Â propagates at speed c, then waves with wavenumber Â where n is any integer, also propagate at the same speed.Â Â These waves are associated with
different propagation modes for
the wave.Â Each propagation mode
has a characteristic displacement distribution through the thickness of
the layer, as discussed below.
 A wave with a
particular wave number can propagate at several different speeds,
depending on the mode.Â The number
of modes that can exist at a particular wave number increases with the
wave number.Â You can see this in
the plot of wave speed v
wave number below.
 Dispersive wave
motion is often characterized by relating the frequency of the wave to its wave number, rather than by
relating wave speed to wave number.Â Â
The (angular) frequency is related to wave number and wave speed
by the usual formula .Â Substituting this result into the
equation for wave speed yields the Dispersion
Relation for the wave
The nonzero displacement
component is
The
nonzero stresses in the layer can be determined from ,
but the calculation is so trivial the result will not be written out
here.Â The wave speed is plotted as a
function of wave number below, for the particular case ,
.Â Â The displacement amplitude a function of
depth is also shown for several modes.Â
Â Â Â Â Â Â Â Â Â Â Â
5.6.5 Elastic waves in waveguides
The
surface layer discussed in the preceding section is an example of a
waveguide: it is a structure which causes waves to propagate in a particular
direction, as a result of the confining effect of its geometry.
The picture shows a much
simpler example of a waveguide: it is a thin sheet of material, with
thickness H and infinite length in
the Â and Â directions.Â
The strip can guide three types of wave:
 Transverse waves, which propagate in the Â direction with particle motion in the Â direction;
 Flexural waves, which propagate in the Â direction with particle motion in the Â direction;
 Longitudinal waves, which propagate in the Â direction with particle motion in the Â direction.
The
solutions for cases 2 and 3 are lengthy, but the solution for case 1 is
simple, and can be used to illustrate the general features of waves in
waveguides.Â For transverse waves:
 The wave can be any member of the following
family of possible displacement distributions
where n=0,1,2â€¦,
and you can use either the real or imaginary part as the solution. This
displacement represents a harmonic wave that has wavenumber Â where Â is the wavelength in the Â direction, which propagates in the Â direction with speed c.Â The variation of
displacement with Â at any fixed value of Â is a standing wave with wavelength Â and angular frequency .
Each value of n corresponds to a
different propagation mode.
 The speed of wave propagation (usually referred
to as the phase velocity of
the wave)Â satisfies
 The wave speeds for modes with n>0 depend on the wave number:
i.e. the waves are dispersive;
 There are an
infinite number of possible wave speeds for each wave number.Â Each wave speed is associated with a
particular propagation mode n.
 The formula for wave
speed can be rewritten as an equation relating the angular frequency Â to the wave number k.Â
This
is called the Dispersion relation
for the wave.
 Dispersive waves
have a second wave speed associated with them called the Group Velocity.Â
This wave speed is defined as the slope of the dispersion
relation Â (in contrast, the phase velocity is Â ).Â
For the waveguide considered here
The group velocity, like the phase velocity, depends
on the propagation mode and the wave number.Â
The group velocity has two physical interpretations (i) it is the
speed at which the energy in a harmonic wave propagates along the waveguide;
and (ii) it is the propagation speed of an amplitude modulated wave of the
form
where Â and Â are the wave number and frequency of the
modulation, and Â are the wavenumber and frequency of the
carrier wave.Â Â The carrier wave
propagates with speed ,
while the modulation (which can be regarded as a `groupâ€™ of wavelets)
propagates at speed .Â Note that for a nondispersive wave, the
group and phase velocities are the same.
