Dynamical modelling of spontaneous oscillation during nanostructuring burnishing Текст научной статьи по специальности «Физика»

УДК 62-294

Динамическое моделирование спонтанных колебаний в процессе наноструктурирующего выглаживания

Я.А. Ляшенко1 2, В.П. Кузнецов3' 4, М. Попов2' 5, В.Л. Попов2' 5> 6, В.Г. Горгоц4

1 Сумский государственный университет, Сумы, 40007, Украина 2 Берлинский технический университет, Берлин, 10623, Германия 3 Уральский федеральный университет им. первого Президента России Б.Н. Ельцина, Екатеринбург, 640002, Россия 4 Курганский государственный университет, Курган, 640669, Россия

5 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия

6 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия Проведено экспериментальное и теоретическое исследование процесса наноструктурирующего выглаживания при помощи

сферического индентора и показано, что в случае когда выглаживаемые поверхности содержат локализованные неровности (просверленное отверстие), возможно установление устойчивого предельного цикла автоколебаний, что приводит к волнистой остаточной деформации поверхности. Для расчета топографии поверхности после наноструктурирующего выглаживания предложена численная модель, которая показывает качественное совпадение с экспериментом. На основе модели рассчитано критическое значение затухания, при котором автоколебания подавляются.

Ключевые слова: выглаживание, нелинейная динамика, профиль поверхности, подавление колебаний

Dynamical modelling of spontaneous oscillation during nanostructuring burnishing

I.A. Lyashenko1, 2, V.P. Kuznetsov3, 4, M. Popov2, 5, V.L. Popov2, 5 6, and V.G. Gorgots4

1 Sumy State University, Sumy, 40007, Ukraine

2 Technische Universität Berlin, Berlin, 10623, Germany

3 Ural Federal University, Yekaterinburg, 640002, Russia

4 Kurgan State University, Kurgan, 640669, Russia 5 National Research Tomsk State University, Tomsk, 634050, Russia 6 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia

We study experimentally and theoretically the process of nanostructuring burnishing by a spherical indenter and show that if the smoothing surfaces contain a localized unevenness (a drilled hole), a stable limiting cycle of self-excited oscillations may occur, which results in wavy permanent deformation of the surface. We propose a numerical model for computing the topography of a surface that has been subjected to nanostructuring burnishing and show that it is in a good qualitative agreement with the experimental observations. Based on the model, the critical value of the damping is determined suppressing the oscillating mode. Keywords: burnishing, nonlinear dynamics, surface patterns, suppression of instabilities

1. Introduction

In recent years, the process of nanostructuring burnish-

ing has gained importance in engineering applications [1-

3]. The process creates a zone of severe plastic deformation in the surface layer, which leads to fragmentation and

change of the properties of the surface layer. In particular, the plasticity and hardness of the surface are greatly improved. Nanostructuring burnishing is a very promising method of surface finishing as it represents a relatively

simple way of enhancing the surface properties of mechanical parts subject to friction. In this method a hard indenter

makes several passes over the workpiece. The required number of passes is an important empirical parameter of the process. In every cycle of severe plastic deformation micro- and nanoparticles formed during the previous pass are ground up to some minimum size which corresponds to the minimum of effective potential [4, 5]. This is determined by the temperature, the level of plastic deformation, the sliding velocity cb and the material properties. Experiments show that several regimes of burnishing are possible. Ideally, the process runs in a stationary mode where the treatment leads to a smooth and hardened surface. Also pos-

© Lyashenko I.A., Kuznetsov V.P., Popov M., Popov V.L., Gorgots V.G., 2015

sible is the mode of damped oscillation where the surface of the workpiece is wavy at the beginning but is smoothed out during the process. But under some conditions self-sustaining oscillations occur which result in a surface with an uneven comb-like structure. This mode attracts the most research interest because it leads to the destruction of expensive workpieces and increased tool wear. In the present paper we investigate the self-oscillating and other modes of severe plastic deformation smoothing using a dynamic model of indenter motion.

2. Model

Figure 1 shows a schematic representation of the system for nanostucturing burnishing.

In our model, a hard indenter of mass m and radius R, modeling the tool, is pressed with a fixed force Fb against a rotating cylinder modeling the workpiece. The indenter is mounted in a linear guide that prevents it from moving in the x-direction. Due to high normal pressure, the indenter will cause elastic and plastic deformations of the near-surface volumes of the cylinder and let a plastically deformed track behind. In this study, we will not investigate in detail the processes of plastic deformation itself as well as changes of mechanical properties of the workpiece but will concentrate only on the dynamics of the tool and the macroscopic shape of the surface produced in the process of burnishing. During the entire process, the indenter slowly moves from one edge of the cylinder to the other, thus scanning the entire surface, possibly more than once. Due to the repeated overrunning of the same surface parts, small heterogeneities which appeared during previous runs can be either damped or amplified further, which finally can lead to formation of an unacceptable surface pattern. The conditions for development of such surface patterns are the main subject of the present study.

Let us denote the relative velocity of the surfaces of the indenter and the cylinder as To suppress the normal oscillations of the tool, a damping element with the damping constant n is introduced. When the indenter is in contact with the surface, it is subject to a reaction force Fc. The equation of motion for the vertical motion of the in-denter has the form

m^y + = Fb - Fc. (1)

dt2 dt

In this equation we neglected the frictional force as our simulations have shown that it does not change qualitatively the dynamics of the system. Thus, while the level of the frictional force may be very essential for the process of nano-structuring [6] it does not affect the formation of surface patterns.

To be able to solve the equation (1), the contact force Fc has to be specified. In elastoplastic indentation, we generally have to differentiate between the indentation and the relaxation phases of an "indentation cycle". However, as in our case the indentation occurs during sliding, only the indentation phase of the cycle is relevant (we may think the process at each time as indentation of the "fresh" material.) We therefore suggest approximating the contact force by its value during the indentation phase of an indentation cycle. If we assume the workpiece material to be an elastoplastic medium with indentation hardness Gc, we can formulate the following approximate governing equations determining the contact force [7]. First, application of the normal force Fc would lead to a region of permanent plastic deformation with the radius apl, which is connected with the normal force and hardness by the equation

Fc = napi g^ (2)

We can introduce the "plastic indentation depth" dpl according to

api = (3)

When the indenter passes over the surface it leaves behind a plastically deformed track of depth dpl. Equation (3) is only correct for contacts with linearly elastic or viscoelas-tic media [7]. However, as semiquantitative estimation, it can be used for any types of media. The actual contact radius a during the action of the normal force will be larger because of additional elastic indentation del. The relation between the contact force and the elastic part of the indentation depth can be obtained assuming constant contact stiffness k = 2aE* corresponding to the given contact radius a

[7]:

Fc = 2aE* del, (4)

where E * is the effective elastic modulus

1

1-

-v2

1—v2

(5)

Fig. 1. Schematic representation of the dynamical model for nano-structuring burnishing

where Ex and E2 are the Young's moduli of contacting bodies, and vx and v2, their Poisson's ratios. Finally, for

the relation between the „total contact radius" and the „total indentation depth" we will use the Hertz approximation

a R(dpi + dei). (6)

From Eqs. (2)-(6) the following dependence of Fc on the total indentation depth d = del + dpl can be derived:

2nRac E * d^Rd

-c---(7)

tcRGc + 2 E *4Rd

Given this explicit expression for the force, the contact stiffness is easily derived:

dFc = nR2 qc E* d (3nRac + 4 E *y[Rd)

~dd " =

In the case of y < 0 (when the indenter bumps off the sur-

F =■

K =

(8)

(nRac + 2 E *JRd f^Rd ' The dependencies of the elastic and plastic parts of the indentation on contact force are shown in Fig. 2, a, and the contact stiffness as function of contact force in Fig. 2, b. The elastic component is predominant at low forces while the plastic component comes to dominate beyond the critical force of

For =

_3 p2 _3

n R a c

8( E * )2

(9)

Note that at the critical force Fc = Fccr the following holds:

del = dpl ="

n2 Ra

(10)

nRac 8( E * )2 Let us take a closer look on the very first pass of the indenter. We assume that the surface is initially smooth and given by ysurf = 0. Positive coordinates of the indentor tip, y > 0, correspond therefore to indentation into the surface, while the indentation depth is just equal to the vertical coordinate of the indenter:

y = del + dpl + ysurf (11)

If we now substitute the reaction force (7) into Eq. (1), we obtain a differential equation for only one unknown variable y that can be solved numerically if the initial conditions are known. Once the total indentation y = del + dpl is determined, the elastic and plastic part of it, del and dpl, can be found separately using the relationships (2)-(6):

dpl = ' del = y - dpl d2)

face) we have dp

■ 0. The described procedure pro-

nRac

2 E *y[Ry'

vides the dependencies for y(t), del(t), dpl(t). Given the known value of the linear sliding velocity , the spatial dependence of the depth dpl (x) of the plastically deformed track left behind by the indenter can be obtained by setting t = x/Pb . The new reference shape of the surface (in the center of the track) is given by ysurf = dpl (x) and becomes smaller in the lateral direction according to the tool shape.

On the second as well as subsequent passes the plastically deformed area will influence the evolution of the parameters of Eq. (1), in which the change surface profile has to be substituted in (11). During the time of one revolution of the cylinder the indenter is replaced in the lateral direction so that it will be placed not exactly at the center of the track. We will take this into account by using an "attenuation factor" describing how far the indenter has been replaced in the lateral direction.

3. Simulation results and discussion

We have numerically simulated the burnishing process by integrating the Eq. (1) using the 4-th order Runge-Kutta method. On the first pass, as described above, y = del + dpl holds. On the second and subsequent passes, it was assumed that the indenter is not moving over fresh surface y = 0, but over 30 % of the depth of the already produced track. For example, for the second pass the surface profile was defined accordingly to ysurf = Adpl(x). The parameter A = 0.3 indicates that the indenter moves at 30 % of the depth of the track depth in the middle of the track. In a similar way, the surface profile was redefined for further runs of the in-denter.

Figure 3 shows the results of simulation of N= 50 passes of the indenter. Here the indenter is moving along the abscissa and returns to the initial position on the abscissa after every pass, while also moving up by a fixed distance on the ordinate.

The evolution of the system significantly depends on the initial conditions y(0), y(0). For the top image of Fig. 3, when the indenter returns to the position x = 0 time is "re-

Fig. 2. Dependence of the elastic del (1) and plastic dpl (2) indentation depths and the total indentation depth d = del + dpl (3) on the contact force Fc given R = 0.003 m, ac = 1.5 • 109 Pa, E* = 1011 Pa (a); dependence of the contact stiffness K (8) on the contact force Fc (7) (b)

set" and the initial conditionsy(0) = 15 • 10 , y(0) = 0 are chosen. Therefore, at the beginning of each new cycle, before the indenter starts moving along the track from the previous pass, the dependences of dpl(x) are similar and represent a decaying oscillation. During the first pass, the stationary coordinate y0 is finally achieved which is determined by the equation

y0-

2Fb nRo„

y0 +

n2 R V2

yo '

4R(E * )2

-= 0. (13)

With the chosen parameters of the model (m = 0.008 kg, R = 0.003 m, gc = 1.5 • 109Pa, n = 60 kg • m/s, E* = 1011 Pa, Fb = 120 N) the stationary coordinate y0 is determined to be

36-2/3 \ A + 2^6Fb(E* )2

y =■

no„

with

An2a2( E* )2 R

(14)

A =

-

9nR<x. +

81R2tc3CTC -48Fb(E*)2

n ac (E )

2/3

(15)

With the given parameters y0 ~ 11.692 ^m, the depth of plastic indentation is dpl ~ 8.49 ^m.

Note that in the upper part of the image there is an area of stationary smoothing without oscillation. However, the depth of indentation is larger here, as the depth of the deformation from previous passes is added in. This case is, in some sense, equivalent to a defect at the beginning of the motion x = 0 that changes the coordinate y and velocity y

Fig. 3. Simulation results with parameters m = 0.008 kg, R = = 0.003 m, gc = 1.5 • 109 Pa, n = 60 kg • m/s, E* = 10" Pa, Fb = = 120 N. The equation (1) was integrated with a time step At = = 10-9 s. One pass has the duration tmax = 0.004 s. The depth of the deformation is shown in grey scale and ranges from 0 to 30 |im, the darker regions corresponding to a deeper indentation

(the initial values for the pass) of the indenter when it passes over the defect. A hole bored into the workpiece may have a similar effect. The upper image of Fig. 3 is consistent with the surface structure obtained experimentally on a cylinder with a small hole that was treated with nanostructuring smoothing (Fig. 4). In the simulation we observe the same main peculiarities as in the experiment [8]: there is an area of vibrational excitation of the indenter, an area of non-decaying oscillation, an area of decaying oscillation, an area that was smoothed out after decay of the oscillations and an area with amplifying oscillation.

In the middle image of Fig. 3, time is not reset to zero and the initial conditions are only set once y(0) = 15 • 10-6, y (0) = 0. Upon return to the position x = 0 the integration of Eq. (1) continues without interruption, but taking into account the existing plastically deformed area from the previous pass. The vertical coordinate is not adjusted and is simply taken fromy, y of the previous time cycle. In this case the left and the right border of the image are identical, which represents a single motion without discontinuities. The observed surface pattern is similar to that in the upper part of Fig. 4 where the indenter no longer touches the hole while passing over the surface. Finally, the lower image in Fig. 3 combines both approaches: in the first 25 passes the initial conditions changed, as in the upper image of Fig. 3, and for next 25 passes the initial conditions do not change, as in the middle image of Fig. 3. The overall behavior in this case is qualitatively similar to the experimental result presented in Fig. 4, which includes a wedge of smooth surface between an area of nonsmoothness and auto-oscillation.

It is also possible, instead of introducing "defects" through the initial conditions, to include them in the initial profile of the surface. Figure 5 shows a simulation result with this approach, for a surface with a spherical depression. The white spot in the middle of the defect indicates that it lies outside this range. It can be seen from the image that the area around the defect is deformed and that oscillating mode results from passing over it. This is in agreement with the experimental data shown in Fig. 4.

The overall appearance of the presented images is critically dependent on the relationships between the parameters of the model. A key role is played by the shape of the

Fig. 4. A specimen after nanostructuring smoothing. A small hole was bored in the center part (black spot)

Fig. 5. Result of modelling the smoothing of a surface with a spherical depression

wear track formed by the first pass of the indenter. Remember that on the first pass y = del + dpl. At the given value of the smoothing force Fb = 120N, a contact stiffness of K « « 11.7 -106 N/m follows from Fig. 2, b. Considering this, equation (1) can be replaced by the following linearized approximation:

d2y n dy K Fb —f+——+—y = —-.

dt2 m dt m m The stationary value of the indentation depth obtained from Eq. (16) on the first pass is equal to y0 « Fb/K « 10.26 ^m and differs from the exact value given by Eq. (14) by only about 12 %. According to Eq. (16), the indenter oscillates with the angular frequency

(16)

ю =

K n2

m 4m

2

(17)

For large enough damping constant

n>nc =24Km, (18)

an overdamped motion without oscillation takes place. This is precisely the condition that ensures the technologically desirable mode of stationary smoothing. With the given parameters the critical value is nc « 612 kg - m/s. This number is approximate and significantly exceeds the capabilities of current systems. It is however of critical importance, since it enables stationary smoothing when working with parts having holes or other defects.

4. Conclusion

In the present work we investigated the process of nano-structuring smoothing using a dynamical nonlinear model that describes the indenter motion and takes the elastic and plastic deformation of the surface layer into account. We have shown that in a wide range of parameters there is an oscillatory mode that produces a corrugated surface. The model allows us to determine the shape of the surface as depending on the main technological parameters, such as the radius of the spherical surface of the indenter, hardness

of the treated material, its elastic modulus, indenter mass, smoothing force and damping coefficient. It was shown that the simulation results qualitatively agree with experimental smoothing of a cylinder with a defective surface. In particular, both in the simulation and in the experiment we see the same regions: the region of amplified oscillation, region of non-damping oscillation, region of decaying oscillation, region of smoothed surface after oscillation decay, etc. The simulations were performed with similar values to those used in the experiment. This may allow us to predict the behavior of a real system under changing parameters. Also discussed were the mechanisms and conditions of the undesirable autooscillating burnishing mode and conditions of suppression of this regime.

Acknowledgements

This work was supported by the Federal Ministry of Economics and Technology (Germany) under the contract 03EFT9BE55, the Ministry of Education and Science of the Russian Federation, and Deutsche Forschungsgemeinschaft (DFG). This work has been carried out within the framework of State R&D Task No. 01201461774.

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Поступила в редакцию 25.08.2014 г.

Сведения об авторах

Ляшенко Яков Александрович, к.ф.-м.н., доц. Сумского государственного университета, nabla04@ukr.net

Кузнецов Виктор Павлович, д.т.н., проф. УФУ, внс КГУ, wpkuzn@mail.ru

Попов Михаил, нс Берлинского технического университета, mikhail.popov@tu-berlin.de

Попов Валентин Леонидович, д.ф.-м.н., зав. каф. Берлинского технического университета, проф. ТГУ, проф. ТПУ, v.popov@tu-berlin.de Горгоц Владимир Георгиевич, к.т.н., доц. КГУ, gorgoc@gmail.com