19

Compression waves in a rod

"It is theoretically possible to solve any problem of vibrations or of strength propagation in an elastic body, if one adds related boundary conditions to the equations

But practically the exact solutions have not been obtained even for a simplest case of vibration of a finite-length cylinder, though in this particular case one can construct the solutions that give the results very near to those true, when the length of cylinder is large relatively to its diameter. This problem was studied first by Pochhammer on the basis of his equations of elasticity [8] and independently of him by Chrec [9]" [2, p. 58].

We see that the system (7)- (9) reduces to (2)- (4) in fact. Hence, if in a general problem its formulation part was questionable, the questions automatically transfer to the particular case of the wave propagation in rods. In this paper we give the results of studying of this issue.

2. Features of transversal deformation corresponding to that longitudinal

When considering the elastic properties of some finite-size model, we usually abstract from its substructure in case of linear modelling, averaging the model properties and the describing parameters. But with it we must note that such averaging is allowable only in limits where it does not distort the physical essence of the described processes. In case of Poisson coefficient, "as the experience shows, with the rod extension its length increases by l , while the width diminishes by  b = b - b₁ . The relative longitudinal deformation is

… The experiments show that for most materials ₁  is 3- 4 times less than " [10, p. 38].

However there exists also some changed definition of this coefficient on which the investigators base much more often. "Poisson coefficient is the absolute value of the ratio of value of relative transversal deformation _yx =  _y /_x or _zx =  _z /_x , where _x , _y and _z are the deformations as related axes (with the specimen extension in the x its cross section narrows)" [11, p. 245]. It follows from the second definition that must be always a positive value, irrespectively, does the specimen narrowing correspond to its extension or not. And this is very important, as in describing the mathematical model we cannot substitute a sign at the deformation coefficient in a mathematical expression by an additional verbal determination in brackets. If just the narrowing corresponds to the extension of the specimen, the sign of   must be certainly negative. But determining the Poisson coefficient through the module of ratios of deformation tensors causes the fact that for today a whole research trend exists studying the possibility of Poisson coefficient to be negative for definite materials [12].

But the main difference of the above definitions is that in the second definition there was made some generalisation by way of substitution of the value b   used in the first definition by some abstract value _y, _z. At the first sight this all is quite natural. Actually, with the longitudinal deformation _x of the specimen some deformations in y and z appear. But the feature of process of accompanying deformation is that b  is rigidly related to l   and conditioned by it. It reveals first of all in, the specimen diameter variation b   premises the accompanying deformation to be directed in the radius-vector from the longitudinal axis of the specimen to its border. And this is typical both for the entire specimen and for any its part. If choosing any elementary region of the studied specimen, the related transversal deformation in it will remain this direction in radius-vector, but not in limits of the picked out elementary region. As opposite to this, the tensors _y and _z describe the deformation directed in y and z correspondingly, but not in the radius-vector from the axis of the picked out region to the periphery. It makes the substitution of b  into _y and _z   incorrect in the very essence of modelling of the process of accompanying deformation.

The gap of interrelation between the longitudinal and corresponding transversal deformation causes not only the appearance of three independent differential equations like (2) and (3) in mathematical modelling, but also the contradictions with the initial Poisson coefficient conditioned by this. Actually, it follows from the independence of equations that not only the propagation velocities of longitudinal and accompanying transversal wave will be different, but their delay phases will be different too. In its turn, it follows from this that at some distance from the source the longitudinal and transversal vibrations will become anti-phase, and then the thickening will correspond to the longitudinal extension of an elementary cross-section of a specimen, and thinning - to the compression, which clearly contradicts both the initial definition of Poisson coefficient and experimental data. With it, in accord with the second definition, independently of the above synchronism or anti-phaseness of processes,   will remain the positive value not reacting to such changes violating the physical essence of the process of solid body deformation.