Can bulk modulus ever be negative? Can somebody illustrate with an example! I was particularly confused when my teacher proved that $C_p > C_v$ for substances other than ideal gas for which the difference between the two  is R . However the proof involved assuming that the bulk modulus is always positive. Can there ever be a negative bulk modulus?
 A: The general answer seems to be that everything you know is correct and we cannot have a material with a negative bulk modulus. But we can have materials with a negative incremental bulk modulus. In other words, I can't make a material that will always expand when you try to compress it. But I can make one that will compress for a little while, then expand slightly, then compress some more.
The most common example seems to be polymer foams. In one experiment, they fill a plastic bottle with a foam and begin injecting water into the top of the bottle at controlled rates. There is a pressure sensor inside the foam. Because they know the original volume of the bottle, the original volume of the foam, and the volume of the water they are injecting, they can make a $P$-$V$ diagram.
What they see is a linear decrease in volume as $P$ increases, up to a point. At this point, they decrease the volume of the foam a little bit more (by adding more water) and there is a sudden decrease in pressure despite the decreased volume. At this point, the bulk modulus is negative!
The explanation they give is that the foam cells are buckling. As soon as they attempt to compress the foam ever-so-slightly, the foam cells buckle, release quite a bit of energy and also the entire material settles into a new equilibrium which happens to be at a lower pressure than before. Of course, if the load were taken off the foam at this point it would be permanently deformed with a higher density than it previously had.
The negative property arises during a strong non-linearity in the material (buckling). It also can arise in fluids according to the same paper. Specifically when one gas begins to condense out of another gas, you will see this negative compressibility. Which means any time the temperature outside hits the dew point and it gets foggy, you're actually in a negative compressibility gas!
How do we reconcile this with classical thermodynamics? Again, using the argument in this paper, the negative compressibility in gases occurs due to the interactions of a non-infinite number of non-zero-sized atoms. When we talk about classical thermodynamics, particularly the ideal gas law, we always assume that there is an infinite number of molecules, that they are effectively point-masses and have no size, and that they only interact through collisions. And when we do this, all of those relationships that you learned are true and negative compressibility cannot exist.
However, that's all just a model. There's not an infinite number of molecules, they do have a finite size, and they may interact with each other through forces other than collisions. And this can lead to moments of negative bulk modulus or compressibility, but it is almost always associated with a change in the stability of the system due to non-linearities (buckling, condensation, etc). 
And even in the presence of these non-linearities, if you only look at the start and end (equilibrium) of a system, you will get the properties you expect and you won't have a negative compressibility. But if you did a time-accurate tracking of the properties on the way from state A to state B, you might see some local instances of negative properties.
A: Highly compacted soil is a perfect example of a material that expands upon increased shear loading. The grains are initially between one another, but the only way for the material to strain (change shape) is if the soil grains roll over one another. This causes an expansion in the over all volume (and a decreased pore pressure) which is associated with a negative bulk modulus (at that point in time). 
