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S.B. Karavashkin and O.N. Karavashkina

To make more accurate the physics of processes of the transversal deformation accompanying to that longitudinal, we have to take into account the results of theorem proved in [13]. In that paper we showed that in case of equal coefficients of longitudinal and transversal stiffness, the direction of vibrations of an elastic lumped system will always coincide with the direction of the external force action. This statement was proved in general form and in examples of linear and closed-loop elastic systems. It follows from this that should the constraints within an elastic body be strictly isotropic and should this remain in the process of averaging for the macro-parameters of an isotropic body, then the accompanying transversal deformation as an thickening/thinning of the specimen would be zero, the same as Poisson coefficient determining it.

Supposing the effect of accompanying transversal deformation present, we in this way suppose beforehand the interior constraints of model to be non-isotropic, and particularly that their longitudinal and transversal stiffnesses at the level of molecular structure are inequal. Only in this case, at the condition of the boundaries and bends present in the microstructure of model, we should expect the transversal component accompanying the longitudinal transformation to appear.

With it at the molecular level of processes modelling, the velocities of longitudinal and transversal excitation will be different without the above paradox. In considering the processes at the level of molecular structure, we leave the concepts of cross-sections, material continuity and so on, i.e. of parameters characterising the averaging, and operate with the crystal, amorphous structures, long molecules etc. But these are already lumped models, and oscillations in them reveal in changing not the cross-sections but the molecules mutual location. And in this case it will be unimportant, how large the value of oscillation frequency is. If the constraints were non-isotropic and not all oriented strictly with the direction of dynamical affection, then, according to the theorem proved in [13], any longitudinal deformation will be accompanied by the transversal. And the transversal deformation will be accompanied by that longitudinal. The type of this accompanying transformation will be quite predictable for definite constraints, and should between the separate conventional elastic lines in a solid body the additional constraints be absent, or should the pattern of bends in these systems be strictly synchronous but not obeying the statistic regularities, then in dynamical processes the local thinning with the compression would be quite regular phenomenon. And for stationary deformations the regularity of the specimen diameter thinning would remain with the specimen lengthening. The reason is that at these conditions of the wave synchronous propagation along all elastic chains of a specimen, the difference in longitudinal and transversal stiffnesses would really lead to the difference of corresponding delay phases, and consequently, to the formation of sections with the anti-phase oscillations. But for it not only the difference between the longitudinal and transversal stiffnesses of elastic lines must be equal for all chains, but also the bends location in these chains must be the same. But if this last obeys the statistic regularity, the mutual influence of chains will average both the propagation velocities of longitudinal and transversal waves and related delay phases.

As we see, the delay phase of wave propagation for the whole rod results some stochastic averaging phases of separate substructures, given their mutual influence, when the transversal shift of one elastic chain of the substructure will effect on the longitudinal and transversal shift of the neighbouring structure, and so on. Thus at the macro-level of quasi-isotropic solid body we see a clear correspondence between the specimen compression and thickening, in full accordance with the first definition of Poisson coefficient. And this correspondence is true both for stationary and dynamical processes in a solid body.

As a consequence, the phase delays of longitudinal deformation and accompanying transversal deformation will be strictly equal, the same as the propagation velocities of corresponding waves.

The said does not mean that in models of such type the propagation velocity of transversal (shear) waves will be also equal to the longitudinal waves velocity. The presence of incomplete isotropic model just expects that these velocities will be inequal. But the pattern of shear waves will be determined by the parameters of transversal deformation, differing, as we revealed above, from the pattern of accompanying transversal deformation. The direction of deformation for the waves of such kind can be described with a full reason by tensors _y or _z . With it in the rod the longitudinal waves corresponding to those transversal will arise, and we can introduce for them a coefficient alike Poisson. The velocity of these accompanying waves will correspond to the transversal wave velocity, and these waves will reveal by the thinning/thickening of cross-sections being perpendicular to the wave propagation.

In other words, with the transversal (shear) rod deformations there will arise also, now longitudinal, waves of thinning/thickening of a model, and they will be now strictly conditioned by the parameters of transversal deformation, and the velocity of their propagation will also correspond to these last.

Proceeding from the carried out analysis, we can conclude that in the studying of longitudinal wave in a rod, due to the strict interrelation between the longitudinal and corresponding transversal deformation, we cannot write the modelling system of equations as an independent system of equations in axes, but we can determine the rod longitudinal deformation, and on their basis, using (11), we can determine the shape of surface transversal deformation accompanying this longitudinal. Such approach is validated also by the fact that in practical measurement of the modulus of specimen longitudinal elasticity we always obtain its value, basing on the accompanying transversal deformation. It means, if we determine the longitudinal vibrations of the specimen, basing on the value of longitudinal deformation modulus, then in frames of linearity of the rod elasticity modulus we will automatically take into account its cross-section variation arising due to the accompanying transversal deformation.