If a spherical object of certain radius and mass (let's say 10 ft, 3,000kg) is placed in a granular medium such as sand. How can you calculate the max depth to which the object would sink?
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As stated, the question is not answerable. There are two missing pieces of information.
1) What is the bulk density of the material, and
2) Is the material static or is it being perturbed (by vibration, for instance)?
If the material is static, and it is at maximum density (that is, it has been firmly packed) the weight will not necessarily sink at all.
At the other extreme, if the material is not compacted and is being constantly vibrated, it will behave pretty much like a liquid, and you can calculate the sink depth by applying Archimedes's principle.
The intermediate stages are extremely difficult to quantify. In a liquid, which can be modeled as a granular medium with molecules as the particles, but zero friction between them, the molecules are easily displaced by the weight, and the weight sinks into the liquid. For real particles, the friction is not zero, although by vibrating the assembly you can get a dynamic condition which approximates it. For low vibration levels there is little effect, but at some point increasing vibration will begin to allow displacement of particles, and the medium will begin to act as a high-viscosity liquid. The greater the vibration levels, the less the viscosity. This will be complicated by the effects of the weight, which will tend to hold the medium directly below it in place due to the effects of pressure on particle irregularities and increased static friction.
answered Jul 11, 2015 at 13:01
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$\begingroup$ So is it safe to say that, if conditions are uncertain regarding vibration or bulk density, the maximum amount of sinking would be similar to that of an object placed in a liquid? The goal of this question is to determine the amount of sinking of a spherical "house" (with the typical static and dynamic loads), built without any foundations, in different soil types. And to try to distill that amount of sinking into a simple equation with all the significant parameters (vibration, density, mass of building, etc.) as variables in the equation. $\endgroup$– BigPicCommented Jul 11, 2015 at 16:35
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1$\begingroup$ Yes. Treat the soil as a liquid with a known density, and assume the house is floating in it. Buildings built on wet soil are notorious for subsiding at a very slow rate. See Mexico City for examples. Likewise, consider the Leaning Tower of Pisa. $\endgroup$ Commented Jul 11, 2015 at 16:48
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$\begingroup$ Final question: would effect of rains on soil over time have any additional effect on the building sinking that is not already accounted for by the "vibrations" described above? $\endgroup$– BigPicCommented Jul 13, 2015 at 12:34
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1$\begingroup$ By reducing inter-grain friction and acting as a lubricant, water from rain will tend to speed up any subsidence process, but will not make a significant difference in how far the building finally sinks. $\endgroup$ Commented Jul 13, 2015 at 14:24
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$\begingroup$ Ohhh.. I was amazed when you said "material being vibrated". I wish to know more. Do you have any references indicating the physics of granular material/fluids? And also about when they are being perturbed? $\endgroup$ Commented Oct 16, 2015 at 3:31