Negative pressure, tension, and energy conditions We have lots of common everyday experience with positive pressure, the canonical example is a gas.  
But other examples of positive pressure are easy to imagine: for instance, a solid that gets compressed to be more compacted than its equilibrium density.
To me it is straightforward that if a solid is instead pulled apart slightly so that it is still connected but at a lower density than its equilibrium density, that it can have a tension that is a negative pressure.
But sometimes people object to negative pressure, so I think we could benefit for a comprehensive answer, that includes good definitions, justifications about why the definitions are good, and even includes comparisons to energy conditions (weak energy condition, strong energy condition, dominant energy condition, etcetera).
An answer does not need to address cosmological constants or dark energy specifically, but I would like the answer to be comprehensive enough that people with questions about those issues can satisfy all their questions about negative pressure itself.
What is negative pressure in general?  How do we know that is the proper and fully general definition?  Is it reasonable in light of known and acceptable physics? How/why do we know that?  How, if at all, does it relate to tension? If different than tension, what is tension in general?  How do we know that is the proper and fully general definition?  Is it reasonable in light of known and acceptable physics?  How does negative pressure relate to the classical energy conditions?  Are any deviations or clashes with classical energy conditions justifiable or acceptable?
 A: You could probably get a negative pressure in polymer physics, so you could view a big block of rubber as behaving this way.
Basically: negative pressures happen when an increase in volume causes a decrease in entropy. Polymers might be a good example because you have these molecules which "want" to be tangled up and kinked ("want" in the sense of "it is entropically favorable for..."). When you increase the volume of such a system by stretching it, it generally decreases the entropy, so you are opposing an entropic force which wants the system to return back to its "resting" size. 
A: Pressure is the (outwardly directed) force normal to any area. This definition most naturally fits hydrostatic pressure, e.g. in gases and liquids. In ideal media, this kind of pressure is never negative.
In real media, that is not necessarily true. The most obvious example occurs at the boundary of just about any liquid: There a negative pressure acts on the molecules at the surface. However, nobody uses the phrase "negative pressure" for it. The common way to call it is surface tension. Every other occurrence of negative pressures, created by attractive rather than repulsive forces in a medium, are treated likewise: They are tensions.
The example you gave, negative "pressure" in a solid is such an example: Engineers quantify the maximum of it that a material can take as ultimate tensile strength. However, pressure does not really describe the situation for solids very well, because forces acting at a surface need not necessarily be normal to that surface. A better concept than the (scalar) pressure is the stress tensor that can capture this force's direction and its variation depending on the orientation of the surface it acts on.
A: The Mie–Gruneisen equation of state for solids http://en.wikipedia.org/wiki/Mie%E2%80%93Gruneisen_equation_of_state is a model that combines the thermal pressure components and "cold" components of the pressure where the latter is derived thermodynamically from a model intermolecular potential. It has the form
$p = p_T(\rho_0,T) + p_c(\rho_0,\chi)$, where $T$ is the temperature, $\chi=1-\rho_0/\rho$, $\rho_0$ is the initial density, and $\chi$ can be negative which corresponds to rarefaction.
A: The stress-energy tensor that general relativity uses includes a three by three matrix that signifies pressure and stress. It's not clear if you count things that are under tension in one direction and pressure in another as positive or negative pressure, but if the matrix is negative definite, meaning that the object is being pulled apart to some degree in every direction, then it's under negative pressure. If it's a negative number times the identity matrix, then it's under that much negative pressure.
A: The absolute pressure of a liquid can also be negative, and, it often is in turbomachinery problems. Consider a volume of water at atmospheric pressure that you draw from and put through a venturi. The stagnation pressure is thus $p_{atm}$, and the pressure in the venturi will be:
$$p_{atm} - .5 \rho U^2$$
If this is water, then $\rho = 1000$ and this equation will be negative if $U>14$ m/s. If the water is clean (free of microbubbles, contaminants, and micro-organisms), then the water will pass through the venturi with that negative pressure. If, however, there are weaknesses (nuclei) in the water, then the water will cavitate and form vapour.
Clean water can stay as a liquid even at ridiculous pressures below -30 MPa. See here, for example. Or for an experiment with a venturi as described above, see here.
