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.
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.
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.
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:
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:
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
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.
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:
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:
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.
This is called the Dispersion relation for the wave.
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