DESTRUCTION OF A STRUCTURAL HOMOGENEOUS BAR OF A FINITE LENGTH UNDER SOME LENGTH- WISE DIRECTED IMPACT LOADING

R . S . G u l i y e v Candidate of Physical and Mathematical Sciences* E-mail: a.seyfullayev@yahoo.com A . I . S e y f u l l a y e v Candidate of Physical and Mathematical Sciences* E-mail: a.seyfullayev@yahoo.com A . O . Y u z b a s h i y e v a Candidate of Physical and Mathematical Sciences* E-mail: afa7803@rambler.ru *Azerbaijan National Academy of Sciences Institute of Mathematics and Mechanics B. Vaxabzade St., 9, Baku, Azerbaijan, AZ1143 У даній роботі характеристичним методом досліджено руйнування структурного неоднорідного стрижня, який підданий удару в його лівому кінці в напрямку його осі, і чисельним шляхом знайдені залежності перших і повних моментів руйнування після різних механічних характеристик. Знаходження аналітичного рішення, що характеризує дане дослідження, вельми складно, і тому задача вирішена приблизно. Результати показують , що вплив пропорцій механічних характеристик частин стержня на процес руйнування є значним Ключові слова: полімер, поздовжній удар, нелінійна спадковість, пошкодження, заліковування дефектів , тривала міцність


Introduction
The structures in most of constructions produced and used in modern technology are exposed to impact loadings.Nowadays wide use of structures made of polymers and composite materials requires studying of lasting firmness and destruction problems of such materials while they are exposed to an impact.In [1] the results show that the spreading of a non-linear hereditary-elastic wave in a semi-infinite bar remains unchangeable, and for this case it is made use of the non-linear hereditary dependence with weak-singular cores.In [2,3] a mutual connection of processes of deformation and accumulation of damages are pointed out and it has been grounded on the fact that the damaging operator must enter not only the general determinant dependence but also the destruction criterion too.According to these facts in the mentioned works a one-dimensional model of damaging of a hereditary-elastic material is given.In [4] a deformation model of a hereditary medium is used by entering the hereditary operators characterizing the accumulation process of damages.The results in [4] are obtained for regular viscosity and damaging operators.In [5] the spreading process of a one-dimensional non-linear hereditary-elastic wave in a homogeneous bar and a pieces-wise homogeneous bar, and the process of restoration of defects are studied.For this case, viscosity and damaging operators are taken in a singular form.Woks [6 -8] are the one-dimensional research works which consider the influence of viscosity on the spreading of lengthwise wave.The research works of last year's [5, 9 -13] are devoted to the destruction problem and they have been provided in more of high level and quality in comparison with the previous works.
In this article the process of a scattering destruction in a piece-wise homogeneous bar of a finite length being exposed to an one-stage small velocity impact, is investigated.What is news in this work is that for the first time the problem is considered in the complex form in distinction from the previous research works provided in this subject area, i.e. the processes of viscosity, formation and accumulation of damages, restoration of defects, hereditary deformability of material, as wells as the factors of changing of mechanical quality (the instantaneous Young's modulus) of material of the destructed domain, pliancy of the obstacle (soil), are all taken into account what allows us to get the more real solution of the problem.For the description of viscosity and damaging processes in polymer and composite materials the singular type hereditary cores found by the experiment are used in the this study.

The problem of destruction of structures
In the problem the considered bar is given having two pieces and the material of one piece of the bar is linear-elastic while the material the other one is damaging non-linear hereditary-elastic.On the left end the bar is exposed to a lengthwise impact, and its right end has contact with a pliant obstacle.A certain quantity which the piece of the bar possesses is marked at the down index of the quantity with a corresponding figure.In the contact of the different homogeneous pieces the rigid sticking condition, continuity condition of speed and forces are accepted as here σ is stress, υ is speed, t is a time variable, 1 l is a length of the first piece of the bar.
The initial conditions are zero in The boundary conditions are given as here x is a space variable, u is displacement, ε is deformation, 0 υ is a stable velocity characterizing the impact, * t is a stable points to the time interval of the impact loading, k is a bed coefficient of soil, l is a length of the bar and H is Heavyside function.
The investigation for the second piece of the bar is done on the base of Suvorova's one-dimensional model for the damaging solid body [2,3].The general determinant relation is as in [2]: here φ -the non-linear instantaneous deformation function, so that ( ) ( ) M -the hereditary type integral operators characterizing accordingly the processes of viscosity and damaging, so that Φ is a function of restoration of defects, so that L and M are however weak-singular Abel cores, so that where E is an instantaneous Young's modulus, γ is a nonlinearity parameter of material, 0 ε is an elastic deformation limit and * ε is a deformation limit determining the restoration of the defects.The scattering destruction criteria are in the following form [4]: here * σ -lengthwise firmness limit.The measureless quantities are included in the following form: here 0 E -the instantaneous Young's modulus appropriate to the non-destructed domains of the bar, ρ is density and 0 c is a spreading speed of the elastic wave, so that A relative quantity is included in the following form: The motion and correspondingly combinatorial relations are After the passage into the measureless quantities in the general determinant relations and differentiating them by time and applying the combinatorial equations in the later obtained relations, it is found in [11] here c ( ) ε -the spreading speed of the non-linear hereditaryelastic wave, so that d ( ) c( ) .d φ ε ε = ε Relations (6) found as a system of the quasi-linear hyperbolic type partial integraldifferential equations of first-order.Deformation is described in the following form in [1]: here, ψ is the inverse function of φ .

Solution of the mathematical problem
The entering of the integral operator relations (1) makes the analytical solution of the problem difficult.Therefore, the problem is solved numerically by the characteristic method with the first-order evident finite differences scheme in the rectangular Lagrange netting fulfilling the Courant-Rees-Isacson's stability condition for the evident scheme [14].The approximate relations are calculated with help of computer.The influences of the proportions of the elastic deformation limits of the pieces of the bar and of the spreading speeds of the elastic wave in them at the 0 t -first respectively the p tcomplete destruction moments of the damaging piece of the bar are studied.When the computation has been done, the values of the number of the quantities have been accepted, in general case, in the following form:

Conclusions
The following results have been obtained.1.The increase of the proportion of elastic deformation limits of the pieces of the bar makes that the occurring of the first scattering destruction of its damaging piece is rapid until the value of this proportion is equal to 1 and after this point it delays weakly; the occurring of the complete scattering destruction is rapid.When this proportion is equal to 1, namely when 01 02 ε = ε , the first destruction occurs more rapidly (Fig. 1, a).
2. The increase of the proportion of the spreading speeds of elastic wave in the pieces of the bar makes that the occurring of both the first and complete scattering destructions of its damaging piece is rapid until the definite moment of time and after this it delays.When this proportion is equal to unit, namely 01 02 c c = , the first and complete destructions occur most rapidly (Fig. 1, b

Fig. 1 .
Fig. 1.Dependences of the 0 t -first and the p t -complete destruction moments of the damaging piece of the bar upon: a -the 01 02 ε ε -proportion of the elastic deformation limits of its pieces when 0 6,0 υ = ; b -the 01 02 c c -proportion of the spreading speeds of the elastic wave in its pieces when 7,0