The universality of nonclassical nonlinearity and applications to NDE

In the field of Materials Science many patterns have emerged which are remarkably similar although they refer to completely different phenomenologies. This is hardly surprising, since often the 'background' mathematics is the same. We shall call them 'universalities', in the sense that they refer to a 'transversal' generality (not to a uniformly general behaviour within a given class of phenomena). The inverse may also be true, i.e. we conjecture that for each mathematical 'niche' some related effect may be found in nature (whether in physics, chemistry, biology or even sociological sciences), if only one looks hard enough. One of such universalities concerns the presence of nonclassical nonlinearity, which has been discovered in a variety of different materials and experimental conditions. We present here a unified approach to its different manifestations (such as fast and slow dynamics), and try to explain it in the framework of a recently proposed "Interaction box formalism". We also review its currently proposed applications in the field of NDE.