The elasticity characteristics belong in the group of the important parameters of solids because they are often used in the analytical solution of the problems of deformation and failure. The elasticity values are used in all engineering calculations and the design of components, sections and whole structures. In the development and application of specific materials and high-strength components produced from them, it is necessary to consider the required rigidity and also the probability that a certain amount of energy will build up in the system during service. Actual solids are characterised by the scattering of mechanical energy in them, i.e. internal friction. This representation of the anelasticity of materials and their transition from elastic to anelastic, microplastic or even plastic response to external loading are the consequences of the effects of external loading and the activity of various mechanisms and sources of scattering of mechanical energy in the material which may be characterised by relaxation, dislocational, mechanical and magneto-mechanical hysteresis. These mechanisms result in changes of the structure-sensitive properties of materials and also the structure-sensitive component of the Young modulus which is still regarded erroneously in a number of publications as a material constant. The external factors, such as mechanical loading, temperature, the effect of the magnetic field, different frequency of the changes of loading, etc., lead to changes of the nature and mechanisms of the processes of scattering of mechanical energy in materials. The relationships between the changes of internal friction and the defect of the Young modulus with...