Procedure for calculation of rod piercing action

. The mechanical action of the rods is the most typical and diverse type of their interaction with the barrier. This variety is associated with those physical phenomena that occur during penetration of the barrier rod, for example, destruction of the penetrating rod, formation of a secondary stream of fragments from the barrier material, etc. The article is devoted to the development of a methodology for calculating the piercing action of a finite thickness barrier with a rod. The main stages of the barrier penetration process are considered. The model contains three stages: implementation, penetration and plug formation. Design dependencies at the main stages of penetration of the final thickness barrier by the rod are given. To compare theoretical developments, to assess their convergence with experience, the results of experimental studies are presented, which show satisfactory accuracy of calculations with experience.


Introduction
Before proceeding to a quantitative assessment of the penetration effect of rods, let's find out the influence of the main factors on the nature of the interaction of the rod with the barrier.
After the rod collides with an obstacle, strong compression waves are formed and begin to propagate in the materials of both bodies, the intensity of which, as is known, mainly depends on the degree of dynamism of the collision process [1,2].Unloading waves arising from the reflection of compression waves from the free lateral surfaces of the rod and propagating inside the rod material collide, creating thus, the high voltage zone [3,4].
The compression wave after hitting the rod moves deep into the barrier.If the barrier is characterized by a finite thickness δ and compression waves propagate in its material at the speed of sound c, then at the moment of time от t c   the compression wave reaches the rear surface of the barrier and, reflecting, turns into an unloading wave that will propagate in the opposite direction at some speed.The reflected wave and the counter compression wave interact with each other.This process is schematically shown in Fig. 1. [5,6] From the consideration of the scheme, it can be seen that the algebraic sum of stresses σ of compression and unloading waves at the considered point of the barrier (for example, in Fig. 1 at point A) at some point in time after the rod collides with the barrier becomes negative, i.e. tensile stresses act at the considered point.If the values of the tensile stresses exceed a certain value of the dynamic yield strength of the barrier material, cracks form in the barrier (in Fig. 1 the breakaway plane) [7,8].In this case, part of the amount of movement of the mass of the barrier material is reported to the breakaway part of the barrier, as a result of which the latter acquires some speed in the direction of impact, and a new section is exposed on the rear surface of the barrier.This process is repeated until the intensity of the stress waves is sufficiently high, and ends with the formation of a deflection cavity in the barrier material [9,10].Simultaneously with the formation of a recline cavity under the action of the rod, the depth of the crater in the barrier increases.At some point in time, the discharge wave reflected from the rear surface of the barrier reaches the moving crater floor, which causes an additional increase in the depth of the crater.This process continues until the pressure force from the rod exceeds a certain amount required to destroy the remaining thickness of the barrier [11,12].When this condition is met, a so-called "plug" is formed in the barrier, which subsequently moves along with the rod, overcoming mainly the force of shear stresses acting along the entire lateral surface.Thus, there is a through penetration of the barrier [13,14].
Taking into account the above, the expression for determining the maximum thickness of the penetration of the barrier h can be written as where h1 is the depth of the rod penetration into the barrier before the formation of the "plug"; δ1the height of the "plug" knocked out by the rod from the barrier; Δ -is the thickness of the breakaway.

Physical picture of the penetrating effect of the barrier
In the works devoted to the penetration of the barrier by the rod, three models of the penetration of the rod into the barrier are mainly used to obtain the dependence (1) on the penetration: hydrodynamic, aerodynamic and a model based on the constancy of specific energy [15,16].
The analysis of these dependencies shows that they have their advantages and disadvantages, which manifest themselves in certain conditions of collision.At the same time, the best convergence with experimental data in a wide range of impact speeds (from 500 to 3000 m/s) is given by formulas based on an aerodynamic model of breaking through an obstacle [16].
In this case, the expression for determining the resistance force of the barrier to the penetrating element can be represented as the following polynomial where ϑ is the velocity of the penetrating element in the barrier material; G, H, Nare coefficients characterizing the physical and mechanical properties of the barrier and the penetrating element.
Considering the components of the barrier resistance force in expression (2) from the point of view of their significance, we will make the following observations [17,18]: 1. Parameter G characterizes the strength of the barrier and does not depend on the penetration rate of the rod.
2. The parameter H takes into account the viscosity of the barrier material, which begins to manifest itself at speeds of collision of the rod and the barrier more than 1500 m / s.
3. The parameter N determines the inertial properties of the barrier particles involved in the movement of the penetrating element [10].
Since it is currently not entirely clear how to quantify the viscosity of the barrier material, in their calculations, most researchers in expression (2), instead of the first two terms, introduce the so-called dynamic hardness of the barrier material Hg, which takes into account to some extent the difference in the resistance of the barrier material during static and dynamic insertion of the element, multiplied by the area of the midsection S of the penetrating element.
Taking into account these remarks, expression (2) is reduced to a binomial law of resistance where c1 and c2 are coefficients depending on the shape of the penetrating body; ρ1the density of the barrier material; Sis the midsection of the penetrating element.
It should be noted that in expression (3), Hg refers to the specific work of the impact indentation of a steel ball, defined as a quotient of the division of the impact indentation work by the volume of the imprint.Having the same dimension as the standard hardness value [9], the Hg value has a different physical meaning and refers not to the deformation resistance on the sample surface, but to the deformation resistance of the displaced metal volume.It is believed that for most metals and alloys, Hg weakly depends on the velocity of the penetrating element [15].
To deduce the dependence of the maximum penetration thickness of the barrier h1 by the rod, consider the following problem.
Let a rod with a mass of m2, having a circle of radius R in cross section, penetrate an obstacle with a thickness δ, and the impact on the obstacle is made so that the axis of the rod is parallel to the surface of the obstacle, and the collision velocity vector is perpendicular to it (Fig. 2).The task is to determine the law of movement of the rod in an obstacle with known physical and mechanical properties, the resistance force of which to the penetrating body is expressed by the formula (3) without taking into account the coefficients c1 and c2, according to a given shape and mass of the rod, a known collision velocity ϑc.The penetrating rod is considered absolutely rigid, and its kinetic energy is spent only on the deformation of the medium and the release of particles of the barrier material.
The movement of the rod will be considered in a rectangular coordinate system x0y, the origin of which is at the point of initial contact of the rod with the barrier (Fig. 2), the 0y axis coincides with the collision vector, and the 0x axis forms a right angle with the longitudinal axis of the rod.
According to the above simplified penetration model, the process of interaction of a rod with a barrier of finite thickness without taking into account the "breakaway" can be conditionally divided into the following stages (Fig. 3) [19]: the first stage is penetration of the cutting edge of the rod with a variable contact surface and the release of particles of the barrier material (crater formation); the second stage is penetration of the rod with a constant contact area; the third stage is through penetration of the barrier with knocking out the cork.Taking into account the above, assuming that the displaced particles of the barrier medium move tangentially to the surface of the rod, at the first stage, the equation of motion of a rod of unit length in the material of the barrier has the form [12]   where m2 = ρ2πR 2 is the linear mass of the rod; here ρ2 is the density of the rod material; F is the resistance force of the barrier medium (3); φ is the angle between the axis 0y and the force Fв (Fig. 3).
To determine the dependence Fв, we assume that the barrier medium is incompressible.In this case, the velocity of the particles of the barrier material relative to point A located on the surface of the rod (Fig. 4) can be written as In turn, point A moves along the surface of the rod at a speed of Bearing in mind that during the insertion of the cutting part of the rod, the velocity of the medium is equal to the velocity of its ejection along the tangent to the surface of the rod, it is possible to find the velocity of the particles of the medium relative to the rod .
Taking into account formulas ( 6) and ( 7), the formula for determining ϑz will take the form Note that the emission of particles of the medium relative to the barrier itself occurs at a speed of ϑв, and where Substituting the expression for φ and ϑz in (9), after simple transformations we will have Assuming that the kinetic energy of the elementary mass of the medium dmc ejected by the rod is the work of the ejection phase of the Fв on the elementary displacement of the rod dy1, we write From Fig. 4, we can write an expression for determining the elementary mass of dmc, which has the form where ρ1 is the density of the barrier material; S(y1) is the area of the midsection of the penetrating rod of unit length at the current time.
Substituting in (11) the expression 1 c dm dy from (12) taking into account (10), we obtain a dependence for determining the ejection force Fв of the particles of the barrier medium where is the coefficient taking into account the shape of the cutting edge of the rod.
The expression for   1 S y can be easily obtained from the consideration of Fig. 4 in the form After substituting ( 13) into ( 4), taking into account the formula (3) and the expression for determining the angle φ, we write down the equation of motion of the rod at the first stage of its penetration into the barrier The initial penetration conditions can be written as To determine the dependency Integrating equations ( 14) and (15) numerically, it is possible to determine the law of motion of the rod  in the form at the first stage of its interaction with the barrier.
The implementation process at this stage is considered completed 1 10 y y R   at the moment of time .
At the second stage of penetration of the rod into the barrier , the law of movement of the rod of unit length is written as Considering that during the introduction of the rod before the formation of a plug in the barrier, its speed varies from ϑ10 to a certain value ϑ20, where ϑ10 is the speed of the rod at the end of the first stage, equation ( 16) is written as 20 20 2 . 1 From here we get the speed of the rod at the end of the second stage of its penetration into the barrier In this case, the rod will go into the barrier to a depth of It is obvious that the formation of a traffic jam (Fig. 5) can occur only after the reflection of the compression shock wave from the rear surface of the barrier.Assuming in formulas ( 17) and (18) 20 t с   and t10 = 0, we obtain the values of velocity ϑ20 and depth y20 at the moment when the compression wave touches the rear surface of the barrier 2 tg arc tg .Let's assume the assumption associated with the condition of the formation of a traffic jam.Let's assume that the formation of a plug will occur if the pressure force from the moving rod on the barrier exceeds the shear stress forces τg of the "unbroken" thickness of the barrier.In this case, the equation of motion of the rod together with the plug will have the form where 1 m  is the mass of a plug of unit length, knocked out by a rod from an obstacle;  -the perimeter of the traffic jam (fig.5); 3 g g    is the tangential stress at which the barrier collapses [13], here σg is the dynamic strength of the barrier.
Let's introduce a new variable z -y3. .
Note that in equation (20), the variable z characterizes the thickness of the plug being knocked out (Fig. 5), which varies from 0 to a certain maximum thickness of the plug δпр, which corresponds to the penetration rate ϑ3 of the rod from 0 to ϑ20.
Integrating expression (20) within the specified integration limits, we obtain a formula for determining the maximum thickness of the cork knocked out of the barrier by the rod, which is already penetrating at a different speed ϑ20 On the other hand, the maximum thickness of the cork knocked out of the barrier by the rod is related to the thickness of the pierced barrier δ (excluding the thickness of the breakaway) by the ratio δ1 = δ -y20, substituting into which the obtained dependencies for δ1 and y2, taking into account the expression for ϑ20, we will have Substituting into the formula (21) the value of the speed of movement of the rod ϑ10 at the end of the first stage of penetration, using the method of successive approximations from (22), it is possible to determine the maximum thickness of the barrier h1 (1) pierced by the rod (without taking into account the phenomenon of breakaway in the barrier).Note that in this case, the rod has zero velocity at the time of the formation of the plug.With an increase in the speed of the collision of the rod with the obstacle, all other things being equal (or with a decrease in the thickness of the barrier), it follows from equation ( 22) that the height of the knocked-out plug decreases, and the rod at the time of the formation of the plug already has a velocity ϑ2 not equal to zero.In this case, in equation ( 16), when the variable y changes from y10 to y20, the velocity changes from ϑ10 to some value ϑ20.With this in mind, we write equation ( 16) in the form 20 20 integrating which, it is possible to obtain a dependence for determining the speed of movement of the rod at the time of the formation of the plug where y20 is determined by the formula (19), and the parameter y10 in this case is equal to R.
Based on the results of experimental studies of the interaction of the rod and the barrier, it was shown in [7] that in a wide range of values of the collision velocity of the rod with the barrier, the difference between the exit velocities of the plug and the rod is very insignificant (about 50 m/s).Based on these results, we assume the inelastic nature of the collision of the rod with the plug being knocked out of the barrier (Fig. 5).At the same time, we assume that the middles of the plug being knocked out of the barrier and the rod are the same.Then, according to the law of conservation of the amount of motion, we can write from where we get the dependence for determining the output speed ϑ30 of the rod and the plug knocked out of the barrier in which ϑ20 is determined by the formula (23); Then, taking into account the remaining thickness of the barrier, we have the linear mass of the knocked-out plug of unit length, δ1, where is determined by the right side of equation ( 22) It should be noted that in real conditions, a cork knocked out of an obstacle by a rod is not always a monolithic body and may consist of several fragments of the material of the pierced barrier that differ in mass.Nevertheless, the obtained formula (24) allows us to estimate the retrograde action of the rod in the first approximation.
To calculate rods whose cross-section has the shape of an isosceles triangle characterized by a base l and an angle at the vertex 2φ, the following formulas were obtained: the law of movement of the rod at the first stage the speed of movement of the rod at the time of the formation of the plug   In the case of the introduction of a rod with a cutting edge of a flat cross-section shape into the barrier, the first stage of implementation described above was omitted from the calculations.

Comparison of calculation results with experimental studies
Calculations were carried out according to the above formulas sequentially in stages: introduction, penetration of the rod and the formation of a plug.A fragment (part) of the calculations is given in Tables 1-3.To solve the equation of motion of the rod (15) during the introduction of its cutting edge (stage I), the fourth-order Runge-Kutta method was used.The final condition for integration at the first stage is the cessation of the increase in the area of the midsection embedded in the barrier of the rod.The transition from the second (movement of the rod in a semi-infinite medium) to the third stage, associated with knocking the plug out of the barrier, was carried out by solving equation ( 22) by iteration method.To assess the convergence of the developed mathematical model with experimental data, special experimental studies were carried out.The experimental part of the research was carried out according to the methodology [19].
Some of the experiments were carried out using an X-ray machine.The target environment of the research is shown in Fig. 6.
The speed of collision of the rod with the obstacle was recorded by the chronometric method according to the flight time to the target wire frame and the X-ray response time.
The values of the depth of penetration of the rod 1 i h into the semi-infinite barrier were measured using a conventional depth gauge from the surface of the barrier at several points along the length of the cut on the barrier.The average depth of penetration of the rod into the barrier was determined by the formula where m is the number of measurements.The results of some experiments are given in Tables 1-3.
The evaluation results indicate that the error h in the studied range of collision velocities does not exceed 15%.
The value of the speed of the plug and the rod at the exit from the barrier was determined by the formula (24), and the mass of the plug knocked out by the rod from the barrier was calculated by the formula (25).
Consideration of the experimental results allows us to conclude that the penetrating ability of the rod at the speeds under consideration depends on the shape of its cross-section: rods with a triangular shape penetrate deeper into the barrier than with a flat one.The output speed of the rod after breaking through the barrier depends on its thickness.

Conclusions
1) Based on the conducted comparisons of experimental and calculated data, it can be argued that the above model of the penetration of an obstacle by a rod, which is based on the twoterm law of resistance of the barrier material, on the whole satisfactorily reflects the penetration process in the range of the studied collision velocities (from 2200 to 3000 m/s).
2) Analysis of the results of calculations and experimental data given in Tables 1-3 shows that in most cases the estimated depth of the rod penetration into the barrier is slightly greater than the values of the depth of penetration that were obtained in the experiment.This circumstance, apparently, is explained by the "triggering" of the rod material when it penetrates into the barrier at a collision speed of 2400 m/s and higher.
3) The same comparison of the results on the evaluation of the maximum penetration thickness of the barrier shows that not taking into account the thickness of the breakaway in the calculations leads to underestimated results compared to experimental data of the calculated maximum thickness of the penetration of the barrier.At the same time, the value of the relative error h does not exceed 10%.
4) The penetrating power of the rod at the speeds under consideration depends on the shape of its cross-section: rods with a triangular shape penetrate deeper into the barrier than with a flat one.The output speed of the rod after breaking through the barrier depends on its thickness.

Fig. 2 .
Fig. 2. The initial position of the rod relative to the barrier

Fig. 3 .
Fig. 3. Model of rod movement in a barrier of finite thickness

Fig. 4 .
Fig. 4. Penetration of the cutting edge of the rod into the barrier

Fig. 5 .
Fig. 5.The scheme of knocking out the cork  the linear mass of the rod of unit length.

Table 2 .
Values of the depth of penetration of the h1 steel rod into the KVK-2 steel barrier with a thickness of 30 mm

Table 3 .
Values of the depth of penetration of the h1 copper rod into the D-16T duralumin barrier with a thickness of 60 mm