Analytical techniques and solutions for
linear elastic solids
5.6 Solutions to dynamic problems for
isotropic linear elastic solids
problems are even more difficult to solve than static problems. Nevertheless, analytical solutions have been
determined for a wide range of important problems. 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.
The technique is similar to the 3D
Papkovich-Neuber representation for static solutions outlined in Section
solid has Young’s modulus E, mass
density and Poisson’s ratio .
longitudinal and shear wave speeds (see Sect 4.4.5)
forces are neglected (a rather convoluted procedure exists for problems
involving body force)
solid is assumed to be at rest for t<0
the boundary is subjected to time dependent prescribed
part of the boundary is subjected to prescribed tractions
procedure can be summarized as follows:
1. Find a vector function and a scalar
function which satisfy
well as boundary conditions
initial conditions .
displacements from the formula
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 Cauchy-Navier equation
The details are left as an
exercise. More importantly, one can
also show that the representation is complete,
i.e. all dynamic solutions can be derived from some appropriate combination of
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 Young’s modulus E
and Poisson’s ratio . The solid is at rest for t<0. A time t=0, a pressure p is applied to the surface of the hole, and thereafter held fixed.
solution is generated by Love potentials
The displacements and
stresses follow as
The radial and hoop stresses
at several time intervals are plotted below:
- 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. 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. 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.
5.6.3 Rayleigh waves
Rayleigh wave is a special type of wave which propagates near the surface of an
elastic solid. Assume that
solid is an isotropic, linear elastic material with Young’s modulus ,
Poisson’s ratio and mass density
solid has shear wave speed and longitudinal wave speed
surface is free of tractions
Rayleigh wave with wavelength propagates in the direction
displacement and stress due can be derived from Love potentials
is the amplitude of the vertical displacement
at the free surface, is the wavenumber; ,
and is the Rayleigh wave speed, which is the
positive real root of
equation can easily be solved for with a symbolic manipulation program, which
will most likely return 6 roots. The
root of interest lies in the range for .
The solution can be approximated by with an error of less than 0.6% over the full
range of Poisson ratio.
nonzero components of displacement and stress follow as
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 in the expression for and setting yields the equation for the Rayleigh wave
speed, so the boundary condition is satisfied.
The variations with depth of stress amplitude and displacement amplitude
are plotted below.
features of this solution are:
- The wave is confined
to a layer near the surface with thickness about twice the
- The horizontal and
vertical components of displacement are 90 degrees out of phase. Material particles therefore describe
elliptical orbits as the wave passes by.
- The speed of the wave
is independent of its wavelength that is to say, the wave is
- Rayleigh waves are
exploited in a range of engineering applications, including surface
acoustic wave devices; touch sensors; and miniature linear motors. They are also observed in earthquakes,
although these waves are observed to be dispersive, because of density
variations of the earth’s surface.
5.6.4 Love waves
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. Love waves involve
motion perpendicular to the plane of the figure. Assume that
layer has thickness H, shear modulus and shear wave speed
substrate has shear modulus and shear wave speed
displacement and stress associated with a harmonic Love wave with wavelength which propagates in the direction can be derived from Love potentials
is the amplitude of the vertical displacement
at the free surface, is the wavenumber; ,
and is the wave speed, which is given by the
positive real roots of
relationship is very unlike the equations for wave speeds in unbounded or
semi-infinite solids, and leads to a number of counter-intuitive results. Note that
- The wave speed depends
on its wavelength. A wave with
these properties is said to be dispersive,
because a pulse consisting of a spectrum of harmonic waves tends to spread
- 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 propagates at speed c, then waves with wavenumber where n
is any integer, also propagate at the same speed. These waves are associated with
different propagation modes for
the wave. 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. The number
of modes that can exist at a particular wave number increases with the
wave number. You can see this in
the plot of wave speed v- wave
- 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. The (angular)
frequency is related to wave number and wave speed by the usual formula . Substituting this result into the
equation for wave speed yields the Dispersion
Relation for the wave
The nonzero displacement
nonzero stresses in the layer can be determined from ,
but the calculation is so trivial the result will not be written out here. The wave speed is plotted as a function of
wave number below, for the particular case ,
. The displacement amplitude a function of
depth is also shown for several modes.
5.6.5 Elastic waves in waveguides
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 and directions.
The strip can guide three types of wave:
- Transverse waves, which propagate in the direction with particle motion in the direction;
- Flexural waves, which propagate in the direction with particle motion in the direction;
- Longitudinal waves, which propagate in the direction with particle motion in the direction.
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. For transverse waves:
- The wave can be any member of the following
family of possible displacement distributions
and you can use either the real or imaginary part as the solution. This
displacement represents a harmonic wave that has wavenumber where is the wavelength in the direction, which propagates in the direction with speed c. The variation of
displacement with at any fixed value of is a standing wave with wavelength 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
- 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. Each wave speed is associated with a
particular propagation mode n.
- The formula for wave
speed can be re-written as an equation relating the angular frequency to the wave number k.
is called the Dispersion relation for
- Dispersive waves have
a second wave speed associated with them called the Group Velocity. This wave speed is defined as the
slope of the dispersion relation (in contrast, the phase velocity is ).
For the waveguide considered here
The group velocity, like the phase velocity, depends
on the propagation mode and the wave number.
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 and are the wave number and frequency of the
modulation, and are the wavenumber and frequency of the
carrier wave. The carrier wave propagates
with speed ,
while the modulation (which can be regarded as a `group’ of wavelets)
propagates at speed . Note that for a non-dispersive wave, the
group and phase velocities are the same.