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Imagine a large, soft, 3D linear-elastic medium containing a small, hard object in the middle (e.g., a marble). If the marble is displaced from equilibrium and then released, it will oscillate back and forth since the medium provides a Hooke's law-type restoring force.

My questions are on definitions/what they mean:

  1. If the medium is indeed perfectly linear-elastic and homogenous, will the marble simply oscillate back and forth forever (with no damping--i.e., an SHO), or is damping still possible?
  2. If the answer to #1 is that the marble will behave like an SHO, what would be the correct name to give for a velocity-dependent damping force in such a material? (In other words, what is the minimal change I would have to make of my "linear-elastic" assumption in order for the marble to behave like a damped harmonic oscillator with a linear damping coefficient?)
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Imagine a large, soft, 3D linear-elastic medium [emph. added]

(I think the intended meaning here is instead compliant: easily allowing elastic deformation. In contrast, "soft" means easily allowing permanent deformation.)

Elasticity, of course, is an idealization. A perfectly elastic medium and embedded object, all other physical phenomena aside, would allow the object to oscillate forever as you describe, as mediated by the stiffness and density of the medium. This is identical to ideal-spring behavior.

We don't see this behavior because all solids exhibit so-called mechanical hysteresis or internal friction, a type of anelasticity. These are some searchable terms for a variety of damping effects associated with otherwise elastic deformation.

Some references are Zener's series of articles, starting in 1937; Nowick discusses the topic in metals here and Zdaniewski et al. in glasses here. There is also a text by Puškár called Internal Friction of Materials, which starts by giving a few of the phenomena of interest :

"The frequency-dependent components include the thermoelastic phenomenon, the movement of valency electrons, the viscosity of grain boundaries, the movement of interstitial atoms in interstitial solid solutions, the movement of interstitial atoms in substitutional solid solutions, the change of the orientation of paired defects, and dislocation relaxation."

One option for you is therefore to select a single one of these effects, or perhaps a generic representation with a certain deformation rate and temperature dependence, for example, as you desire.

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