Forming a Single Plastic Imprint of the Indenter under Ultrasonic Surface Hardening

The results of the experimental investigation of forming the geometry of a single imprint appearing on the machined surface during a single tool stroke in ultrasonic surface hardening are given. The equations of regression for calculating the geometrical parameters of the imprint depending on the machining modes such as the static force and the diameter of the deforming element are obtained. Unlike the existing models of forming the imprint, the appearance of the material pressed out around the tool during its penetration in the surface layer of the machined detail is taken into account. The pressed-out metal looks like a hill. The equations obtained allow us to determine the height of the hill as well as the depth and diameter of the imprint taking into consideration the hill dimension. The calculation of these parameters is necessary to find out the boundary conditions of starting the appearance of the plastic flow wave under ultrasonic plastic deforming of metals and alloys. These conditions determine the criterion of forming waviness and present the restrictions of the limiting values of feeding and speed in machining the cylindrical surfaces.


Introduction
Nowadays more and more interest in ultrasonic plastic deforming as hardening and finishing operations of details is taken [1][2][3][4][5][6][7][8][9]. Due to this fact, the questions of the technological quality assurance of the surface under the conditions of discrete deformation impact are considered to be vital. It is known that waviness appears on the machined surface under certain conditions in ultrasonic plastic deforming of natural steels. Paper [10] presents the theoretical investigations on finding out the conditions of appearing waves of plastic flow of metal under the discrete impact by the ultrasonic tool. This condition is a system of inequalities as restrictions determining the limiting values of the speed and feeding under which the waves appear. where V-the speed of the detail rotation, m per minute; dpl -the diameter of the plastic imprint with the pressed-out metal round the tool during its penetration in the surface, mm, fthe frequency of ultrasonic oscillations of the tool, Hz; Sthe feeding, mm per rotation. (Fig. 1).
Conversely, the dependency of the plastic imprint diameter on the mode parameters of machining is not presented in the paper. Traditionally the oscillation amplitude (A), static force (Fst) for pressing the ultrasonic tool to the detail and the diameter of the deforming element (DC) are used as the machining modes influencing the imprint parameters in ultrasonic plastic deforming. The purpose of the paper is estimating the influence of the technological machining modes mentioned above on the geometrical parameters of the imprint. To exclude the overlapping of the imprints one on another, it is necessary to determine the corresponding modes in machining. Kinematic parameters of machining such as the speed and the feeding affect the location of the imprint centers, and deformation parameters (the amplitude and the frequency of ultrasonic oscillations, the indenter diameter, the static force of pressing the tool to the detail) influence the imprint dimension. According to the deformation model described in Paper [11] the imprint diameter can be determined by the following formula: where h1(t) and h2(t)the functions which describe the depth change of the tool penetration in the respective phase of its direct and reverse movement in the material, Dc-the diameter of the spherical part of the indenter, hmaxthe maximum depth of the indenter penetration, t0the time during which the contact of the deforming element with the detail surface takes place, t1the time of the maximum penetration, t2the time of completing the contact between the deforming tool and the machined surface.
Calculating the imprint dimensions, it is necessary to take into account the tool sliding during its contact with the machined surface, as a result of which the imprint takes the form of ellipse. Paper [12] traces the correlation between the imprint parameters and geometrical, kinematic and force parameters of ultrasonic plastic deforming. According to the paper mentioned, in machining the cylindrical surface the ellipse dimensions depend on the indenter geometry (the diameter of the spherical part, r), the time of contact between the detail and the tool (Δt), the speed of the detail rotation (Vd), the depth of the deformer penetration (h) as well as elastic and plastic properties of the machined surface (Fig. 2). In this case the dimensions of the ellipse semi-axes are determined by the following equations: where the parameters r, h, hy, h are calculated by the deformation model [11]. The calculations showed that the value of the large ellipse axis (2b) did not exceed two diameters of the imprint (dim) at a maximum possible speed of the detail rotation, taking into consideration the real conditions of machining. а b Fig. 2. The scheme of contacting between the ultrasonic tool and the cylindrical surface (ain the longitudinal section, bin the cross section) Because the model used does not take into account the appearance of the pressed-out material around the imprint, we can consider that it is necessary to provide the distance of more than 2dimbetween the imprints in order to ensure the absence of overlapping the imprints. As the dimension of the pressed-out material is small in comparison with the imprint diameter, the distance in 3dim will be sufficient. The distance between the imprints in the feeding direction must be not less than 2dim. Thus, to obtain separate imprints on the machined surface in ultrasonic plastic deforming, the condition mentioned above can be presented in the following way: where lvthe distance between imprints in the direction of the speed of the detail rotation, lS-the distance between the imprints in the direction of feeding.
The distance between the imprints in the direction of the speed is presented in Paper [12] as follows: where fthe frequency of ultrasonic oscillations, Hz.
The value of feeding for one revolution is the distance lS. Thus, the restriction for ultrasonic plastic deforming modes, which provide the formation of separate imprints on the machined surface, can be presented by the following system of inequalities through simple transformations:

Methods of the Experiment
Cylindrical samples with a diameter of 50 mm made from steel were chosen for carrying out the experiment. The samples were preliminary turned to obtain the initial roughness of the surface Ra 2.5.
The sample machining with the successive change in each parameter (the static forcefrom 50 to 250 N and the deformer diameterfrom 5 to 20 mm) was provided on the modes chosen according to the restriction (Eq.6). Unchanged parameters in machining were the frequency of ultrasonic oscillations (22 kHz), the amplitude (20 micrometers), the speed of the detail rotation (280 m per min) and the feeding (1.3 mm per rotation).
The typical picture of the machined surface obtained by microscope Nikon MM-400 is presented in Fig. 3. Separate imprints in the form of ellipse, evenly distributed on the surface at a distance lv and lS in the directions V and S respectively, can be seen in the picture.
The profile diagrams (5 for each mode) were taken from each surface after machining, after that the geometrical parameters of the imprint were determined. The typical profile diagram with the geometrical parameters is presented in Fig. 4.

Results and Discussion
According to the experimental results the equations of regression for calculating dpl, hpl, him were obtained under different mode parameters.
The diameter of the plastic imprint (dpl) depending on the static force (Fst) and the deformer diameter is determined according to one of the Eq.8-10. dpl = -0.0051Fst 2

Conclusions
The obtained equations (8-10) describe the dependence of the geometrical parameters of the imprint on the static force and the indenter diameter during ultrasonic plastic deforming taking into account the appearance of the pressed-out material. A high validity of approximation allows us to use these dependences in engineering calculations. The established interconnections between the geometrical parameters of the imprint and machining modes give the possibility to determine the boundary condition of forming the waviness during ultrasonic plastic deforming.