Material which becomes less dense as pressure is applied Is it possible that a material may become less dense as more pressure is applied, without a phase transition?
My initial layman thought is no because it seems to me that "density" is a form of "pressure" within a material (I know this to be true at least for gasses). However some materials do exhibit unintuitive properties, such as becoming less dense when solidifying. In the specific case of water, at some temperatures increasing the pressure on water will turn it into a solid. Other unintuitive material properties include water (yet again) becoming less dense as temperature decreases be below 4°, and fluids which lose viscosity under increased pressure. So maybe there is chemistry for a material that expands (becomes less dense) as pressure is applied.
To be clear, this question can be phrased as: Is there a material which expands as pressure is applied?
 A: Density is mass/volume. The only way to become less dense without losing mass is to increase volume.
Therefore what you are asking for is

*

*"is there a material which becomes larger as pressure is applied".

*"is there a material which loses mass as pressure is applied".

This is not in-principle impossible.

*

*For the first, high explosives (HE) with a pressure-sensitive switch would do it.

*For the second, squeezing a sponge full of water might work.

In either case, these are not "simple" materials with "simple" interactions. The problem really is not that it can't be done, but that there are so many ways to do it, and they are all different.
A: Becoming less dense with pressure is an energy-producing
quality; mass being conserved, volume must increase as pressure
rises for density (mass/volume) to have that response.   So, I'd say
that fulminate of mercury does fit the description;
it explodes when compressed.    It does that once,
and only once, of course, and one might not want
to be present at such an event.
A: It is impossible to have a homogeneous thermodynamically stable substance in which pressure increases along with the volume. If this happens, then the material can lower its energy by shrinking one half and expanding the other half (assuming constant entropy). This instability is exactly the reason, why the Van der Waals gas equation experiences liquid-gas phase separation.
Some materials, though, can exhibit drop in tension if they are stretched beyond a certain limit. This results in such effects as necking and beading (you can check a number of examples from the paper Necking, Beading, and Bulging in Soft Elastic Cylinder). I am not sure about a similar example for positive pressure, but I would not be surprised if such material exists. Some sponges, when one tries to squeeze them from the sides, instead of being deformed uniformly, they first squeeze at the surface, while the bulk remains largely undeformed. Such behavior is an indicator the instability resulting from the strain going down as the deformation increases.
A: 
Is it possible that material may become less dense as more pressure is applied without a phase transition?

The answer to this is YES, it is possible, and this is an active area of research. You can have a material that becomes less dense when you compress it with your bare hands.
You can think of constructing metamaterials  with this property (See this paper). This honeycomb-like material, for instance, gets less dense if you apply pressure in a specific direction. The material is hollow, and the sides of the hexagons are flexible.
Before applying pressure (smaller, denser)


After applying pressure (larger, less dense)


Note that this is just a 2-D slice of the material, and if you can imagine a stack of such slices, you would have a real,3-D material.
