Quantum mechanical interpretation for the (odd?) fact that a strained metal wire increases its volume If a short nickel wire is strained within the range of validity of Hooke’s Law, its volume increases by 0.02 mm$^3$.
Paradoxically, Göritz* (p. 194) says that for entropy elasticity, volume consistency has always been presupposed.
Göritz doesn’t say anything about a temperature effect, but according to Le Chatelier’s Principle, of course, cooling is to be expected.
My question is, how this cooling would be explained with theoretical physics (quantum statistics). − But it is entangled by the additional question whether the volume increase plays any crucial role, which is confusing since I previously never heard of such an increase. 

*) D. Göritz, Messung der Volumenänderung beim uniaxialen Dehnen, Colloid Polym. Sci. 260, p. 193 (1982). − He does not say anything, how long this wire was, but from fig. 1 in the article, I would estimate it at 7 cm. 
 A: One way to understand the cooling effect from a thermodynamic perspective is to recall that reversible work adds no entropy. 
As you noted, uniaxial tensile loading of a solid increases its volume to some degree (because all solids have a Poisson's ratio of less than 0.5; therefore, lateral contraction never entirely offsets axial elongation, as it would in a liquid). The increase in volume increases the entropic configuration of the constitutive molecules (assuming that the solid is a typical ceramic, metal, or crosslinked polymer, not an elastomer, which I'll discuss later). But if the deformation occurs adiabatically (i.e., sufficiently insulated from the environment) and reversibly (i.e., at the unreachable limit of gradual loading), then the process is isentropic; no entropy is added or generated. 
Therefore, the entropy associated with the increased freedom of the molecules comes from the solid's temperature itself, and cooling occurs.
If easy heat transfer to the environment is allowed, then the solid stays at ambient temperature. Interestingly, in this case, it is also somewhat more compliant, representing the difference between stretching it at constant entropy or at constant temperature.
If the loading is not reversible (e.g., it is performed quickly), then the temperature might decrease, stay the same, or increase, depend on the magnitude of dissipative processes such as vibrations or shock waves that generate entropy and thus heat the object.
For an elastomer, the situation is different; uniaxial loading uncoils and unkinks long unorganized polymers, which decreases their entropy. For an ideal elastomer, stiffness is entirely entropy-based, not enthalpy-based as it generally is in metals and ceramics. Again, for an adiabatic process, the entropy has nowhere to go except to increase the material's temperature. For such a material, the temperature coefficient of thermal expansion is negative.
As you noted, Le Chatelier's Principle can indicate the direction of temperature change, but only under the condition of reversibility, and one must know the sign of the temperature coefficient of thermal expansion. So the formulation in terms of the entropy of the constitutive molecules is more general.
