WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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good for low to moderate stiffness contrasts (E0 < 50). Interestingly,<br />
as the material contrast grows to infinity the wave splits up in a forward<br />
and backward travelling wave of equal magnitude.<br />
An external force is generally required to change the material<br />
properties, hence the mechanical energy in the wave is not conserved.<br />
Based on the formulas for the forward and backward travelling<br />
waves in [36] the relative change of mechanical energy<br />
occurring at t ¼ t0 is found as<br />
DEmek<br />
¼ 1<br />
2 ðE0 1Þ; ð6Þ<br />
Emek<br />
E = 1 ρ = 1<br />
E = E0 ρ = 1<br />
t < t 0<br />
t > t 0<br />
t = 0.750 s<br />
t = 0.975 s<br />
Fig. 3. Top: one-dimensional medium with an instant change in the stiffness for<br />
t ¼ t0. Bottom: two snapshots of the wave motion with indication of the<br />
propagation direction.<br />
Amplitude<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 1 2 3 4<br />
ln(E0 )<br />
5 6 7 8<br />
Fig. 4. Amplitude of the forward and backward travelling waves versus the stiffness<br />
E0. Comparison between exact (solid lines) and numerical results (discrete<br />
markers).<br />
thus, if the stiffness of the medium is increased (E0 > 1) energy is<br />
supplied to the wave, whereas for E0 < 1 energy is taken out of<br />
the wave. Simulation results show excellent agreement with the<br />
analytical result.<br />
J.S. Jensen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715 707<br />
Fig. 7 shows simulated values for Emek and also Ekin and Epot versus<br />
time for the case of E0 ¼ 3. The instant doubling of the mechanical<br />
energy is noted at the time of instant material property change<br />
– as predicted by (6). Furthermore, oscillations in the kinetic and<br />
potential energy are seen in the short time interval after the material<br />
change for which the forward and backward travelling waves<br />
have a spatial overlap.<br />
Thus, it has been demonstrated that the simple time integration<br />
algorithm produces accurate results in the case of instant material<br />
properties changes, at least in the case of low to moderate stiffness<br />
contrasts.<br />
3.2. Moving material interface<br />
Weekes [35] studied wave propagation in structures with a<br />
moving material interface. Exact results were provided for the<br />
example illustrated in Fig. 5.<br />
Here, the interface between a material with stiffness E ¼ 1 and a<br />
material with stiffness E ¼ E0 moves with constant speed V towards<br />
left. A wave that propagates towards right meets the moving<br />
interface and is partly transmitted and partly reflected at the interface.<br />
It was shown in [35] that the relative amplitudes of the transmitted<br />
wave T and the reflected wave R are independent of the<br />
material interface speed<br />
pffiffiffiffiffi<br />
2<br />
E0 1<br />
T ¼ p ; R ¼ p ; ð7Þ<br />
þ 1 þ 1<br />
ffiffiffiffiffi<br />
E0<br />
ffiffiffiffiffi<br />
E0<br />
which correspond to the known results for transmission and reflection<br />
of waves at an immovable interface.<br />
However, the interface speed V causes a shift in the wavenumber<br />
(or frequency) of the transmitted and reflected waves (Doppler<br />
shift). The relative wavenumber shift of the reflected wave is found<br />
as<br />
Dc<br />
c 0 R<br />
1 þ V<br />
¼ ; ð8Þ<br />
1 V<br />
i.e. the shift depends only on the interface speed. Here, it should be<br />
noted that V is given relative to the wave speed in the material in<br />
which the incoming wave propagates ðc ¼ ffiffiffiffiffiffiffiffi p<br />
E=q ¼ 1Þ. The shift<br />
E = 1 E = E 0<br />
V<br />
t = 0.560 s<br />
t = 0.940 s<br />
Fig. 5. Top: one-dimensional elastic rod with a moving material interface. Bottom:<br />
two snapshots of the wave motion indicating the propagation direction.