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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48 H. T. Banks, S. H. Hu <strong>and</strong> Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51<br />
in the small amount <strong>of</strong> extra displacement to the right <strong>of</strong> the dashed line) <strong>and</strong> having<br />
some energy reflected. The basic one dimensional model we have used can effectively<br />
model the presence <strong>of</strong> a rigid body in a column <strong>of</strong> soil, in particular the behavior <strong>of</strong><br />
partial reflecting <strong>and</strong> transmitting <strong>of</strong> energy when the object is impacted.<br />
This numerical example has demonstrated the ability <strong>of</strong> a one dimensional model<br />
to capture some <strong>of</strong> the salient features <strong>of</strong> wave propagation in the soil medium, including<br />
sensitivity to soil structure <strong>and</strong> to the presence <strong>of</strong> rigid objects in the soil.<br />
Additional features that would require a two or three dimensional model might be<br />
modeling the presence <strong>of</strong> more than one type <strong>of</strong> body wave (e.g., shear <strong>and</strong> compressional)<br />
as well as modeling surface waves. In such a higher dimensional setting we<br />
could also take into account more complicated buried object geometries. Ultimately,<br />
the one dimensional model still captures much <strong>of</strong> the basic dynamics <strong>of</strong> elasticity in<br />
soil <strong>and</strong> is there<strong>for</strong>e useful in predicting outcomes to physical experiments with some<br />
degree <strong>of</strong> fidelity.<br />
Acknowledgments<br />
This research was supported in part by the Air Force Office <strong>of</strong> Scientific Research<br />
under grant number FA9550-09-1-0226. The ef<strong>for</strong>ts <strong>of</strong> ZRK were supported in part<br />
by the Department <strong>of</strong> Education with a GAANN Fellowship under grant number<br />
P200A070386. The authors are grateful to Dr. Richard Albanese <strong>for</strong> encouragement,<br />
suggestions <strong>and</strong> constructive comments during the course <strong>of</strong> preparation <strong>of</strong> the material<br />
in this manuscript.<br />
References<br />
[1] H. T. BANKS, J. H. BARNES, A. EBERHARDT, H. TRAN AND S. WYNNE, Modeling <strong>and</strong><br />
computation <strong>of</strong> propagating waves from coronary stenoses, Comput. Appl. Math., 21 (2002),<br />
pp. 767–788.<br />
[2] H. T. BANKS, J. B. HOOD, N. G. MEDHIN AND J. R. SAMUELS, A stick-slip/Rouse hybrid<br />
model <strong>for</strong> viscoelasticity in polymers, Technical Report CRSC-TR06-26, NCSU, November,<br />
2006, Nonlinear. Anal. Real., 9 (2008), pp. 2128–2149.<br />
[3] H. T. BANKS AND N. LUKE, Modelling <strong>of</strong> propagating shear waves in biotissue employing an<br />
internal variable approach to dissipation, Commun. Comput. Phys., 3 (2008), pp. 603–640.<br />
[4] H. T. BANKS, N. G. MEDHIN AND G. A. PINTER, Nonlinear reptation in molecular based<br />
hysteresis models <strong>for</strong> polymers, Quart. Appl. Math., 62 (2004), pp. 767–779.<br />
[5] H. T. BANKS, N. G. MEDHIN AND G. A. PINTER, Multiscale considerations in modeling<br />
<strong>of</strong> nonlinear elastomers, Technical Report CRSC-TR03-42, NCSU, October, 2003, J. Comp.<br />
Meth. Engr. Sci. Mech., 8 (2007), pp. 53–62.<br />
[6] H. T. BANKS, N. G. MEDHIN AND G. A. PINTER, Modeling <strong>of</strong> viscoelastic shear: a nonlinear<br />
stick-slip <strong>for</strong>mulation, CRSC-TR06-07, February, 2006, Dyn. Sys. Appl., 17 (2008), pp.<br />
383–406.