Mathematical Modeling of Impact of Possible Shock Dynamic Loads on the Modern Composite Materials

In this paper, we study the dynamic processes in materials reinforced with fibers, that can be represented as composite rods. There has been developed a mathematical model of wave propagation under the impact of a shock pulse in semi-infinite composite rods. It is believed that the considered composite rod consists of two layers formed by simpler rods of different isotropic materials with different mechanical properties. The cross sections of such rods are considered to be constant and identical. When such composite materials are impacted by dynamic loads, a significant part of the energy is dissipated due to the presence of friction forces between the contact surfaces of the rods. In this regard, we study the propagation of waves in an elastic fiber-rod, the layers of which interact according to Coulomb law of dry friction. The case of instantaneous excitation of rods by step pulses is investigated. The blow is applied to a rod made of a harder material. In the absence of slippage, the friction force gets a value not exceeding the absolute value of the limit. In the absence of slippage, the friction force takes a value not exceeding the absolute value of the limit. Let us consider the value of the friction force constant. Normal stresses and velocities satisfy the equations of motion and Hooke's law. The problem statement results in the solution of inhomogeneous wave equations by the method of characteristics in different domains, which are the lines of discontinuities of the solution. Solutions are found in all constructed domains. On the basis of the analysis of the obtained solution, qualitative conclusions are made and curves are constructed according to the obtained ratios. From the found analytical solution of the problem it is possible to obtain ratios for stresses and strain rates in composite rods and composite materials.


Introduction
Composites [1] in which the components are typically arranged at a certain frequency are of interest because of their ability to attenuate shock pulses [2]. The extreme complexity of the interaction of waves in the propagation of the shock pulse in the real composite material leads to the fact that modern theoretical methods of investigation of this problem are limited to strongly idealized models [3].
Modern pipeline transport systems are becoming increasingly in higher demand due to growing economic competition. Nowadays, pipelines may be located not just underground, where they may be subject to seismic shocks, but also in more aggressive and corrosive environments, for example, sea water. This necessitates the development of new materials and their mathematical models. The new mathematical models must take into account wide variety of materials used and complicated interactions between layers that form a composite material. The developers must consider the effects of friction conditions between layers, sliding of layers with respect to each other at macro-, micro, and nano-levels, and other intricate layer interaction effects while projecting and synthesizing new composite materials. Modeling of such materials allows reaching the increased dependability and service life of new technological systems, including pipeline transport systems. [4,5].
In this work , a mathematical model of dynamic processes in materials reinforced with fibers, which are e represented by composite rods, has been developed.. When such materials are subject to dynamic loads, a significant portion of energy is dissipated due to friction forces between the contact surfaces of the rods. In this regards, , the propagation of waves in the elastic fiberrod, layers of which interact according to the law of dry friction of the Coulomb [6][7][8], is consideredA semipointed composite rod consisting of two layers is examined. Each of the layers is considered to be an elastic rod of constant cross-section S. Part of the surfaces of these rods with perimeters of normal cross sections L interact with each other according to the law of dry friction of Coulomb. In the case when there is a slip between the rods, the tangential stresses on the side surface will be equal to fN where N is the lateral pressure on the rod and f is the coefficient of friction between the materials of the rods.
In the case, when there exist a relative movement between the rods on their surfaces, the limit friction force will act, the absolute value of which per unit length of the rods in the case of dry friction, is equal to fLN F f = [9]. This force always acts in the opposite direction to the velocity vector of the relative motion of the sections. In the absence of slippage, the friction force assumes a certain value not exceeding the absolute limit [10].

Equations and mathematics
Let us consider two resilient rods interacting with each other according to the law of dry friction. We believe that for the first rod the Young's modulus has a meaning and density for the second one -and in the case of motion and in the case of rest takes any value between -1 and +1.
The systems of equations (1) and (2) can be reduced to non-uniform wave equations.
In order to find a general solution to the non-uniform wave equation, it is necessary to use relationships along the characteristics of this equation.
Characteristics of non-uniform wave equations are lines: A discontinuity in the solution extends along these lines.
Along the characteristics there are relations: The solution to the Cauchy problem is sought in the form: for the first rod: for the second rod: The leading edge of the perturbation in the rod begins to spread as a shock. The condition for saving the movement quantity on the characteristic With regard to condition (5) this equation can be rewritten as: From conditions (5) and (7) we find a solution in the area I for the second rod, considering that The area of motion from the area of rest in this case is separated by the front Also solving by the method of characteristics, we find a solution in the area III: .
af at x a at x In the area II we solve the system (21), provided that on its boundary

Results and discussion
The proposed method found solutions within each of the eleven areas, which are constructed using the method of characteristics. A full range of solutions has been obtained throughout the quadrant Qualitative conclusions from the obtained solution of the problem can be obtained by considering the dependence and at fixed (Figs. 8, 9a, 9b).

Conclusion
The obtained analytical solution makes possible obtaining relations for stresses and strain rates in composite rods and materials. The results can be generalized to large number of layers with different properties and geometry. Such generalization will allow developing new mathematical models that are directly relevant to real technical products [11][12][13].
This work was carried out using equipment provided by the Center of Collective Use of MSUT "STANKIN".