A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction
A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction
A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction
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H. T. Banks, S. H. Hu <strong>and</strong> Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 45<br />
Forcing Function<br />
f(t) (units:N/m 2 )<br />
3<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
−3<br />
−4<br />
0 0.002 0.004 0.006 0.008 0.01<br />
4 x t (units: s)<br />
Figure 21: Sinusoidal input function.<br />
displacement return to the baseline in all <strong>of</strong> the following figures due to the restoring<br />
<strong>for</strong>ce present in the input. Since the displacement returns to the baseline, we can also<br />
infer that our model is linearly dependent on the input.<br />
We remark that the input term <strong>and</strong> the equations we use <strong>for</strong> demonstration purposes<br />
are only a convenient approximation to physical reality in a small displacements<br />
case. One could implement the actual physical situation by allowing <strong>for</strong> a moving upper<br />
boundary at z p0 . In this case, the input would only be the first, positive part <strong>of</strong> the<br />
sine wave, as the differential equation dynamics coupled with the moving boundary<br />
would return the soil to near its original position. Since a moving boundary is more<br />
difficult to implement computationally, <strong>for</strong> this demonstration we chose to implement<br />
the simpler stationary boundary at the impact site z p0 . In reality, the ground boundary<br />
will not remain stationary under impact, but it will instead move first in the positive<br />
(downward) direction <strong>and</strong> then rebound in the negative (upward) direction. The rebound<br />
is due in part to the viscoelastic properties <strong>of</strong> the soil column <strong>and</strong> also to the<br />
physical soil column interacting with the surrounding soil (e.g., shear which is not<br />
represented explicitly in the one dimensional dynamics). We model this restoring motion<br />
as the second half <strong>of</strong> the input signal as depicted in Fig. 21. That is, the positive<br />
first half <strong>of</strong> the sinusoid represents the <strong>for</strong>ce imparted by the thumper <strong>and</strong> the negative<br />
latter half is modeling the rebounding <strong>of</strong> the soil boundary that one would (<strong>and</strong><br />
we did in field experiments) see in reality. This permits use <strong>of</strong> the stationary boundary<br />
at z p0 while still approximating (with reasonable accuracy) the true dynamics at the<br />
surface.<br />
4.2.1 Results <strong>for</strong> Case 1: only soil present in the column<br />
The first situations we examine are when holding one parameter (<strong>of</strong> (κ, ρ)) constant<br />
<strong>and</strong> increase the other parameter. The results are depicted in the panels <strong>of</strong> Fig. 22.<br />
In the left panel, we hold density ρ constant <strong>and</strong> change the elastic modulus κ. As