Modern rheology and electric induction: multivalued index of refraction, splitting of eigenvalues and fatigue

Abstract

The modern constitutive equations of the rheology of analeptic media and of the polarization in dielectric media are discussed in the time and frequency domains. It is seen that the most appropriate form of the constitutive equations of these media is based on the convolution of the stress and strain, and of the electric field and induction respectively, with appropriate functions representing the memory of the medium to previous stress or induction. The most successful memory forma]ism in representing the observed phenomena is that of the derivative of fractional order which allows a variety of problems to be solved by means of the Laplace Trans- form. The consequent index of refraction, generally, is a set valued function which implies that a monochromatic wave generated in the medium is split into a set of waves with the same frequency but different wavelength which interfere and have a quasi periodically varying amplitude. The eigenfunctions of these media, associated to the ordinarily used principal values of the index of retraction, are split and so are the tree modes. The fatigue of anelastic media is studied using the new constitutive equations containing fractional derivatives; a phenomenological method is introduced to compute the number of cycles which gives fatigue for a strain or stress with given amplitude and frequency. The number of cycles which may cause fatigue is proportional to the amplitude and frequency of the applied strain or stress, or inversely proportional to the rate of stress or strain. A criterion is also established to see whether an applied cyclic stress or strain may cause fatigue. This criterion is tested with experimental data on steel and rocks verifying that, in general, it is consistent with the data although thc data are not always in the linear range. It is tentatively seen that fatigue docs not seem one of the causes of the release of the elastic energy stored in the Earth.