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.   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 5.5.


Assume that

 The solid has Young’s modulus E, mass density  and Poisson’s ratio .

 Define longitudinal and shear wave speeds (see Sect 4.4.5)


 Body forces are neglected (a rather convoluted procedure exists for problems involving body force)

 The solid is assumed to be at rest for t<0

 Part of the boundary  is subjected to time dependent prescribed displacements  

 A second part of the boundary  is subjected to prescribed tractions  


The procedure can be summarized as follows:

1.       Find a vector function   and a scalar function  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 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 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 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.


The solution is generated by Love potentials




The displacements and stresses follow as


The radial and hoop stresses at several time intervals are plotted below:


Observe that

  1. A wave front propagates out from the cavity at the longitudinal wave speed ;
  2. 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;
  3. 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;
  4. The maximum stress induced by dynamic loading substantially exceeds the static solution. 




5.6.3 Rayleigh waves

A Rayleigh wave is a special type of wave which propagates near the surface of an elastic solid.  Assume that

 The solid is an isotropic, linear elastic material with Young’s modulus , Poisson’s ratio  and mass density  

 The solid has shear wave speed   and longitudinal wave speed  

 The surface is free of tractions

 A Rayleigh wave with wavelength  propagates in the  direction

The displacement and stress due can be derived from Love potentials


where ,  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


This 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.


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  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.



Important features of this solution are:

  1. The wave is confined to a layer near the surface with thickness about twice the wavelength. 
  2. The horizontal and vertical components of displacement are 90 degrees out of phase.  Material particles therefore describe elliptical orbits as the wave passes by.
  3. The speed of the wave is independent of its wavelength  that is to say, the wave is non-dispersive.
  4. 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


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.  Love waves involve motion perpendicular to the plane of the figure. Assume that

 The layer has thickness H, shear modulus  and shear wave speed  

 The substrate has shear modulus  and shear wave speed  

 The wave speeds satisfy  

The displacement and stress associated with a harmonic Love wave with wavelength  which propagates in the  direction can be derived from Love potentials


where  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



This 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

  1. 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 out;
  2. The wave speed is always faster than the shear wave speed of the layer, but less than the wave speed in the substrate;
  3. 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.
  4. 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 number below.
  5. 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 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.  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


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  and  directions.  The strip can guide three types of wave:

  1. Transverse waves, which propagate in the  direction with particle motion in the  direction;
  2. Flexural waves, which propagate in the  direction with particle motion in the  direction;
  3. Longitudinal waves, which propagate in the  direction with particle motion in the  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.  For transverse waves:

  1. The wave can be any member of the following family of possible displacement distributions


where n=0,1,2…, 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.

  1. The speed of wave propagation (usually referred to as the phase velocity of the wave)  satisfies


  1. The wave speeds for modes with n>0 depend on the wave number: i.e. the waves are dispersive;
  2. There are an infinite number of possible wave speeds for each wave number.  Each wave speed is associated with a particular propagation mode n.
  3. The formula for wave speed can be re-written as an equation relating the angular frequency  to the wave number k


This is called the Dispersion relation for the wave.

  1. 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.




(c) A.F. Bower, 2008
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