A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

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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.
H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 47 4.2.2 Results for Case 2: rigid body present in the soil column In this section we examine the case where we have included a rigid body located at z 10 , modeled as a point mass at the same point as our previous observation point. In the figures, this location is represented by a vertical dashed line. For all the simulation results presented in this section, the values for the soil density, elastic modulus and damping ratio are chosen to be the baseline values specified earlier, with the value for the mass of the target given by M = 34.2671kg, and the contacting surface area taken as S = 0.1580m 2 . In the upper left pane of Fig. 24, we can see that the sinusoidal force has begun imparting displacement in the soil but this displacement has not yet reached the target. In the upper right pane, the displacement has impacted the target. In this simulation, the target imparts much of the energy to the soil below it. In the lower left pane of Fig. 24, we see that most of the energy has passed through the target and is deeper into the soil. However, looking at the domain between z = 0 and the dashed target line, we see the remnant energy that was reflected by the target back toward the surface. In the lower right pane in the figure, we see that this energy has bounced off the soil and impacted the target again, once more passing some energy through the target (as seen Displacement, units: m 12 10 Wave form in z−domain, at time t=0.00067831 14 x 10−5 8 6 4 2 0 −2 0 0.5 1 1.5 2 Distance under ground, units: m Displacement, units: m Wave form in z−domain, at time t=0.0054744 x 10 −4 8 6 4 2 0 0 0.5 1 1.5 2 Distance under ground, units: m Displacement, units: m Displacement, units: m Wave form in z−domain, at time t=0.0020486 x 10 −4 8 6 4 2 0 0 0.5 1 1.5 2 Distance under ground, units: m Wave form in z−domain, at time t=0.0068447 8 x 10−4 7 6 5 4 3 2 1 0 0 0.5 1 1.5 2 Distance under ground, units: m Figure 24: Wave form in the soil column at various times (the buried object location is represented by vertical dashed line).
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A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

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