A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

46 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
the modulus increases, we see that the wave begins its upward slope sooner, meaning
the wave has arrived at the observation point sooner so wave speed has increased.
Also, as the modulus increases, we see less displacement overall. In the case where
density increases, we see that we also get less displacement overall, but the wave
speed decreases. We clearly see the model demonstrates that soil parameters have
multiple realistic effects on overall displacement and wave speed.
In Fig. 23, we examine the situation where we increase both soil parameters. As expected,
the effects of the parameter increases combine to reduce overall displacement.
In the left pane of the figure, we see that the elastic modulus increases more rapidly
than the density and the wave reaches the observation point sooner. In the right pane,
we applied the same percentage increases to the parameters and so the wave reaches
the observation point at the same time for all the parameter combinations. Thus, our
model is able to demonstrate the effects of more complex parameter changes that one
might see in applications. For example one might input multiple impacts at the same
site, which would increase both soil density and stiffness from one impact to the next.
(This in fact was realized in field experiments by scientific colleagues when testing for
repeatability of responses to interrogation impacts.)
Displacement around z=0.3048m (z=1 ft)
Displacement around z=0.3048m (z=1 ft)
u(0.3048,t)) (units: m)
12 x 10−4 t (units: s)
10
8
6
4
2
κ=102000000 ρ=1800
κ=204000000 ρ=1800
κ=408000000 ρ=1800
u(0.3048,t)) (units: m)
10 x 10−4 8
6
4
2
t (units: s)
κ=204000000 ρ=1440
κ=204000000 ρ=1800
κ=204000000 ρ=2250
0
0
−2
0 0.002 0.004 0.006 0.008 0.01
−2
0 0.002 0.004 0.006 0.008 0.01
Figure 22: Wave form at z 10 with varying soil parameters individually (left: variable κ, constant ρ; right:
constant κ, variable ρ).
Displacement around z=0.3048m (z=1 ft)
Displacement around z=0.3048m (z=1 ft)
u(0.3048,t)) (units: m)
14 x 10−4 t (units: s)
12
10
8
6
4
2
κ=102000000 ρ=1440
κ=204000000 ρ=1800
κ=408000000 ρ=2250
u(0.3048,t)) (units: m)
12 x 10−4
10
8
6
4
2
t (units: s)
κ=163200000 ρ=1440
κ=204000000 ρ=1800
κ=255000000 ρ=2250
0
0
−2
0 0.002 0.004 0.006 0.008 0.01
−2
0 0.002 0.004 0.006 0.008 0.01
Figure 23: Wave form at z 10 with both soil parameters κ and ρ varying.