The stretched exponential function called Kohlrausch-Williams-Watts (KWW) equation is the same as a Weibull function. While KWW is derived and used for dielectric and glass transition relaxation, Weibull function is used for failure analysis. There is much literature on failure analysis, and this may be analogous to exfoliation of layered materials considered as a failure of the laminate structure.

The Kohlaursch function is also used for evaluating electrical lifetime tests of insulators (long term dielectric strength). This is analogous to the mechanical failure test. (As far as I know it has to do with the weakest link probability).

A physical picture of viscoelastic recovery via a stretched exponential function is provided by Fancey, K. S. (2001), "A latch-based Weibull model for polymeric creep and recovery", Journal of Polymer Engineering 21(6): 489-509. If all recovery modes could begin at the starting time then the process should be exponential. However, some modes cannot begin until other modes release free volume or steric inhibition, so they are delayed (the physical analogy of a latch opening) giving a stretched exponential response (Spoljaric, S., Goh, T. K., Blencowe, A., Qiao, G. G. and Shanks, R. A. (2011), "Thermal, Optical, and Static/Dynamic Mechanical Properties of Linear-core Crosslinked Star Polymer Blends", Macromolecular Chemistry and Physics 212: 1778–1790). Failure described by Weibull may be like this where a single event does not cause failure, but a coincidence of more than one or several events combining to give failure.

This description could be speculated to apply to exfoliation of layered materials where though all layers may be equally adhered, the layers cannot all exfoliate in a single thermally activated step, thus exponential. When some layers separate, other layers deeper within the structure are liberated to separate. This is not to imply that the layers peel sequentially from one or other face. There is potential for random exfoliations leaving few layer materials that are then more able to take part in the exfoliation resulting in a stretched exponential process instead of exponential.

Sure, 2 exponents for partial solutions were assymptotically matched in order to obtain General Solution for Non - Steady State Osmotic Phenomena. PDF file is attached.

I agree with the previous comments. The stretched exponential function works very well with most relaxation phenomena. This is because it actually allows a distribution in the relaxation times (it also has two parameters instead of only one...). For breakdown it is the result of the weakest link concept of Weilbull analysis.I attached the calculations of this width, as obtain with a recursive method with gnuplot (also explained on the wikipedia page of stretched exp).