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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

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