A macroscopic model of electro-discharge machining

lm J M.cn 1~,ol Des Res ',o, 22 N,, 4 pp 337 33'9. 19S2 Printed m Greal Brnam

A MACROSCOPIC

I~b2u -35- S2 ~J4,.,~ 41- ~i3,,~

,,

MODEL OF ELECTRO-DISCHARGE MACHINING

J. A. McGEOtGH* and H. R~SXRSSEN÷ IRe, ell ed l;'r publi~ ati, m 22 J u n e 19821

Abstract- The shaping of metal_~ b.~ electrodischarge machining tEDM I, in v, hich sparks are discharged bet~een tool- and ~orkpiece-.electrodes. is described A model for the process is proposed, a ke3 feature being the replacement of the time-dependent electric field, x~hich a solution of the complete M a x , e l i ' s equations v, ould produce, b3 an electrostatic field. A macroscopic theor', is then formulated on the basis of experimental obser, ation~ that (il the electric field needed for the production o(sparks in the inter-electrode gap must exceed rome critical ~alue. (ii I sparks occur ox er electrode regions at ~hich the local field is highest and fiii) the rate of metal remo~ al ~s proportional to the energ~ transmitted b~ the sparks. The model is applied to the one-dimensional case in which plane parallel electrodes are used. Results from the model are discussed in terms of the relationships betv, een ~ear ratio and the velociD of the tool-electrode

N O M EN CL.-~-f U R E

4) (.'< y l |. t

hlx. t ) 9~',. t t E r~t

J tt D U P

electrostatic potenual (\'~ rectangular coordinate s.~stem applied ',ullage (VI time ',ariable tsl surface of cathode-tool surface of anode-~orkpiece electric field (V m l position vector for anode surface unit vector in x-direction unit vector in ?.-direction process parameter heaxiside unit step function cathode-toot speed ~m sl length scale (ml ratio of v,ear of tool to workpiece non-dimensional tool feed rate inter-electrode gap ~vidth Imm

Subscripts d ffrlt ( 5

condition relating to anode-workpiece m i n i m u m value (of field for spark to occur~ condition relaung to cathode tool-electrode stead)-stale condition

Superscripts *

non-dimensional quantiD

1 INTRODUCTION IN ELECTROD1SCHARGEmachining (EDM), t\~o metal electrodes, one being the tool of a predetermined shape, and the other being the x¥orkpiece, are immersed in a liquid dielectric, such as paraffin or light oil. A series of voltage pulses, usually of rectangular form. of magnitude about 80V and of frequency of the order of SHz-5kHz. is applied betv,een the t\vo electrodes, v,hich are

* Department of Engineering. Uni~ersit.~ of Aberdeen..Aberdeen. U K +Department of Applied Mathematics. University of Western Ontario, London. Omario. C a n a d a 333

334

J. A. McGEoLGHand H. RASMUSSEN

separated by a small gap, typically 0.01-0.5 ram. Localised breakdown of the dielectric occurs and sparks are generated across the inter-electrode gap, generally at regions where the local electric field is highest. Each spark erodes a tiny amount of metal from the surfaces of both electrodes. The high frequencies at which the voltage pulses are supplied, together with the forward movement (by means of a servomechanism) of the tool-electrode towards the workpiece, enables the sparking action to be eventually achieved along the entire length of the electrodes. The process can be controlled such that substantially more metal is removed from the workpiece than from the tool.electrode. As EDM proceeds, the image shape of the toolelectrode is gradually reproduced to fine accuracy on the workpiece. Since metal removal is effected by sparks and not by the mechanical action of the tool, the rate of machining is not limited by the hardness of the workpiece. These attractive features of EDM have led to its widespread industrial use, for example in the manufacture of dies and moulds. Most previous theoretical work has been concerned with microscopic metal removal arising from a single spark, the effects being modelled from heat conduction theory [1, 2, 3]. The scale-up of these models to simulate the macroscopic effects of metal shaping in EDM has normally been accepted as a formidable problem. Consequently, a theoretical treatment of the metal shaping problem in EDM remains effectively untackled. (Many of the difficulties have been discussed by Crookall and Moncrieff [4] in their proposals for predicting shape degeneration from erosion rates, obtained from standard data or simple tests.) A contribution to the solution of this problem for EDM forms the basis of this paper. A theory is developed which predicts how the workpiece is gradually spark-machined to yield a shape complementary to that of the tool-electrode. The model is solved for the onedimensional case of two plane parallel electrodes to demonstrate its usefulness in correlating the wear ratio of the two electrodes with the velocity of the tool-electrode. Its extension to cover two-dimensional electrode profiles is discussed. In order to explain the motivation behind the physical basis for the model, a brief description of the principles of EDM is first given.

2. PHYSICAL BASIS OF THE EDM PROCESS In section 1, the spark erosion mechanism was attributed to the electrical breakdown of the dielectric in the inter-electrode gap, caused by the application of voltage pulses. The breakdown arises from the acceleration towards the anode of both the electrons emitted from the cathode by the applied field and the stray electrons present in the dielectric in the gap. These electrons collide with neutral atoms of the dielectric, thereby creating positive ions and further electrons, which in turn are accelerated towards respectively the cathode and anode, If the multiplication of electrons by this process is sufficiently high, an avalanche of electrons and positive ions occurs. These eventually reach the electrodes and a current flows. The entire breakdown process is a localized event, occurring in a channel of radius approximately 10/2m. When the electrons and positive ions reach the anode and cathode, they give up their kinetic energy in the form of heat. Heat fluxes up to 101 - W m- 2 can be attained, so that even with sparks of very short duration (of the order of/~sec) the temperature of the electrodes can be raised locally to more than their normal boiling point. Owing to the evaporation of the dielectric, the pressure in the plasma channel rises rapidly. to values as high as 20 arm. Such great pressures prevent the evaporation of the superheated metal. However. at the end of the voltage pulse, when the voltage is removed, the pressure also drops suddenly and the superheated metal is evaporated explosively. Metal is thus removed from the electrodes. The relation between the amount of metal removed from the anode and cathode, depends on the respective contributions of the electrons and positive ions to the total current flow. The electron current predominates in the early stages of the discharge, since the positive ions. being roughly 104 times more massive than the electrons, are less easily mobilised than the electrons. Consequently. the erosion of the anode-workpiece should be greater initially than

A Macroscopic Model of Electro-dischargeMachining

335

~c

,t.a P C ,.I e -

FI(, 1 Workpiece and tool.electrode configuration for EDM.

that of the cathode-tool. As E D M proceeds the plasma channel increases in width and the current density across the inter-electrode gap decreases. With the fraction of the current due to the electrons diminishing, the contribution from the positive ions rises and proportionally more metal is then eroded from the cathode-tool. By applying a series of voltage pulses and by driving one electrode towards the other, the required spark discharge conditions can be maintained so that metal is eroded from the entire workpiece surface, with the latter gradually being shaped to a form complementary to that of the tool-electrode. 3, FORMULATION OF THE MACROSCOPIC MODEL The criteria for breakdown of the dielectric in the inter-electrode gap are as follov, s. la) A sufficiently high electric field must exist to give the electrons the energy required to cause ionization. (b) An adequate number of collisions must occur before the electrons reach the anode. Thus spark erosion can be considered to occur at those points where the instantaneous electric field exceeds some critical value. Although this critical field is unlikely to be constant generally, in this model, it will be assumed to be so. to a first approximation. Next, since a macroscopic model is the aim of this work. a relationship will be postulated between the spark density at a given point on the electrodes and an electrostatic field. This latter field can be interpreted as the time-average of the applied pulsed field which is actuall) used. [the time-interval for the average field being considered to be long compared with the duration of each pulse of the applied voltage or field). Thus in contrast with earlier analytic treatments of EDM which have been concerned with the influence of a single spark, the present work deals mainly with the average effects of a large number of sparks. In principle, the average field could be deduced by the solution of the complete Maxwell's equations for conditions betv,een the two electrodes, and then by averaging the resulting field over small time-intervals, for instance of the order of 0.1s. However. this procedure would require exceedingly expensive computing, as well as yielding far more accurate estimates of the field than warranted by the next part of the model. Thus instead, some approximate method for obtaining values for the field has to be sought. Nov, over time-intervals that are long compared with duration of the vohage pulses, say 0.1 s. a probe placed in the inter-electrode gap would not record a fluctuating field : instead a steady field would be registered. Thus the time-dependent electric field which a solution of the complete Maxwell equations would produce, may be replaced with the electrostatic field given by: 4~.,:, -'- O,, = 0

ll}

336

J.A. McGEot GHand H. R ¢S,~'SSEN

and 0 =0

at y = h t x , t)

0=)'

at ) ' = g ( x , t )

(2)

where d) is the electrostatic field. (x. y) is a rectangular coordinate system. V is the applied vohage, t is the time variable, v = h(x, t)is the surface of the cathode-tool, and y = g(x, t)is that of the anode-workpiece. The configuration of electrodes and the field conditions are shown in Fig, 1. Then the electric field is given by: E = V0

(3)

Bsu se ofthe potential boundary conditions [equation (21] and equation (3), we now stud',' conditions for metal removal from the anode. For convenience in the development of the model, this electrode will be assumed to be the workpiece. However it will be clear that the theor 3 also allows for interchange of the potential values [equation (2)] if conditions in which the cathode is the workpiece are to be modelled. A relationship between the electric field [equation (3)] and the metal removal rate from the electrodes is now postulated, on the basis of the experimental obsen, ations that the electric field is greater than the critical field, and that sparks occur at points on the electrodes where the electric field is greatest. Thus we can interpret

IE[ = fv ,l obtained by solving equations (1) and (2). along the anode-workpiece as a measure of the spark density over a large time interval, Experimental evidence is also available which shows that the amount of metal removed from the electrodes is generally proportional to the pulse energy [5]. We can then postulate that the rate of change of an)' position along the anode workpiece surface is proportional to the energy transmitted by the sparks, i.e. proportional to the square of the field strength. The position vector for the anode surface may be written in the form: r(t) = x(t)f -.-

y(t)j

(4)

where [ and j are the respective unit vectors in the x- and )-directions. Then the above postulate implies that: --

~

drdt

Ea

M,(E~ - E ~ i t ) ' - H ( E . - E~,i,)-E-~

(5}

where _E+ = electric field at the anode surface. E° -= l_E.!

E,j, = minimum value of E,, at which a spark occurs

H(E. -

Em,)

=

Heaxiside's unit step function

= {10 for E . _>E.,, for E~ < Ecrit M . is a process parameter determined from experiments.

The quantity H i E . - E ~ , ) i s used to model the condition that at points where E. < E.~,. no metal removal takes place. The equation for the anode surface may now be written in the form: y ( t ) - g[x(tl, r] = O. On differentiation with respect to t, we obtain: dy dt

dx g~ ~ - g' = 0.

(6)

A Macroscopic Model of Electro-discharge Machining

337

No~ from equations 14} and (6I. dx dr

-

-

dy and ~ = F.cb~..

=

F,~.,

where {E a -

F, = M,

E,m,t 2

HfE~ - Ecrit)

E.

(7)

and E° =

+ ,f.ot'

:

The subscript a indicates that qS~and ~,. are evaluated at the anode surface. Then equation (6) can be ~'ritten as: o,

=

F.(¢,.~

-

0~%.~i.

(8t

From the initial shape of the anode surface .v = Oo(x). the initial condition for equation (8) is deduced: (9)

g(x, O~ = go(x~.

A similar equation is obtained for the cathode-tool. If the coordinate system (x. y) is supposed to be fixed on the anode, and the cathode is moved at a constant speed, u, along the y axis towards the anode, the equation for the change of the anode surface becomes : h~ = F¢(49~.., - hx4),,.~l + u

(101

where F, = ?,I

( E , . - Ecrit)" E

H(E

- E~r.)

and E, = 10 ., +

-'.

The subscript c indicates that ~b~ and 0 , are evaluated at the cathode surface. The initial condition for equation (101 is: h(x, O) = holxl

t12t

where v = ho(x I is the initial shape of the cathode-tool. I t may be noted that the critical field strength Ec, , has been assumed to have the same ~alue at the surfaces of both anode- and tool-electrodes. By this assumption. E , , is implied to depend only upon the properties of the dielectric. This does not mean that the properties of the two electrodes themselves have no effect : their influence is taken into account through the anode and cathode metal process parameters M, and 3.1,. However. if two widely different materials are used for the two electrodes, two values for Em,--one for the anode and the other for the cathode---could be needed. In this case. the additional condition--that the applied field strength has not had to exceed both values for Ecrit before sparks occur - - would have to be added to the model postulated in this section.) In order that the governing equation can be written in non-dimensional form. a length scale D is defined : the minimum distance between the two electrodes when the electric field at the same point at the electrode surface is just equal to the critical field. Thus the nondimensional variables can be defined by: x* = x D , t* =

y* = y D . ~ *

t Mfl03

=~'I

Ec*i, = Emt D l'.

338

J.A. McGEotGH and H. RASMUSSEN

Then the model can be written in the non-dimensional form" 4b:., + 4~.~ = 0

(E.

-

g' -

h, = :~

(13)

¢b = 0

at v = h(x, t)

4) = 1

at y = g ( x , t)

114)

E.~i,) 2 E.

H(E~ -

( E , - Era, l: E--

Eel,)(ch,.~

-

g~cb~.~)

-- Ecrit) (C~., -- hxO.~.,) + U

(15) (16)

where the superscript * has been dropped. The parameter Mc M. is the ratio of wear of tool to that of the workpiece. The significance of various typical values for ~ is discussed in the next section. U is the non-dimensional feed-rate of the tool: D2 U = u - -

M.V 2 "

4. O N E - D I M E N S I O N A L

C A S E OF" P L A N E .

PARALLEL

ELECTRODES

Significant features of the model can be illustrated b) considering EDM in which a flat cathode-tool is driven towards an anode-workpiece, the surface of which is plane and parallel to the former electrode. The field is now given by: t7- v t7 - g

d) = - -

where it is assumed that /7 > g. At the start of EDM, the i n t e r - e l e c t r o d e gap width is considered to be unit). Then equations (151 and (16t b e c o m e dA=

_

t7 1 - E )2 - g

dt and

(

dh _1 d--f = :~ t7 ,j

E

-

U

,

If the gap width p is defined as

p=h-

g

then the rate of change of gap width is obtained as

dr

d t = ( 1 + ~)

-E

- U

(17)

with the initial condition p(0)---- 1. The solution of this equation gives the gap p as a function of machining time. The resulting expression is complicated, and yields little additional information. However. some particular solutions to equation (17) are readily tractable, and provide a useful insight into EDM.

A Macroscopic Model of Electro-discharge Machining

339

From equation (17). the steady-state machining gap p~ is given by (1 + :~)

- E

- U = 0.

(181

This equation can be re-arranged to express the wear ratio as U :~-(~_E)2,

1.

(19,

That is. for constant spark gap. the wear ratio varies directly with the electrode velocity, This theoretical result has been confirmed from experiments in which a copper tool-electrode was used in the spark erosion of a mild steel workpiece in paraffin. For a sparking gap of approximately 0.I22 mm, the wear ratio was found to rise linearly from about one to three per cent as the tool velocity was increased respectively from approximately 0.007 to 0.113 mm,'sec [6]. Similar results were obtained for higher wear ratios ranging from seven to fifteen per cent which corresponded to electrode velocities between 0.001 and 0.026 rams.

CONCLUSIONS

The EDM process may now be analysed theoretically by the macroscopic model described in this paper. A significant feature of the work is the reasoning for replacing the usual timedependent field by a stead) field. Analysis of the complicated and random nature of the multiple spark action in EDM is thereby greatly eased. As a result, macroscopic effects such as tool wear can be investigated and practical conditions such as "fast" and "slo~'" EDM assessed. It is clear that various aspects of the model need checking through experiment. On the other hand this model now provides guidance on the kinds of experiment which are necessary to elucidate the understanding of the EDM shaping problem, in contrast to the microscopic and metallurgical effects which have hitherto been the main objects of stud~. With the formulation of this macroscopic model, its extension to two-dimensional electrode profiles is desirable. Numerical methods of solution will be required. The handling of practical two-dimensional shapes will pro~e formidable, since their geometry coupled with the smallness of the inter-electrode gap will give rise to ver~ large derivatives of the electrostatic potential. Convergent numerical routines will be difficult to devise without resorting to an exceedingly large number of grid points. It seems likely that a combination of procedures will be necessary to resolve the problem. In the vicinity of the smallest electrode gap. perturbation techniques might have to be used. with numerical methods being employed in the remaining region betv,een the electrodes.

Acknowledqements--The authors" collaborative ~ork is funded b~ a NATO research grant. The) v,ish to express their thanks for the support and encouragement of Professor E. Hansen and the staff of the Labora~orx of Apphed Mathematical Physics. Technical Universit) of Denmark. ~here the5 started their v,ork for this paper C)ne of them (H.R.) is grateful for financial support from Sr,'F. during his period of stud~ leaxe in Denmark. Mr I M Crichton and Mr Z. Sadollah are thanked for some valuable assistance

REFERENCES [1] F. \.~,~ DLWCKand R. S,,o~.',s. Pro~.. Inr C o ~ on Production Engmeerinq. Tokjo. Japan, pp. 46-50. Japan Society of Precision Engineering t 19'7,4 ) [2] R. Syop,s and F. \.~,s D1jc~ C.I.RP. General Assembly. 71P3. Poland {1971 I. [3] A. ERDEXand B. K ~t:a ~,xoGtt', Proc. 21st Int. Con~ Machine Tool Design7 and Research. pp. 351-358 119~0) [4] J. R. CROOk,EL and A, J. R. ~'I~)NCRIEFF.Pro~. Inst. n~'th Engrs 187. 51 11973). [5] M L. Jlis\v.~l. Wear 55, 153 (19"79). [6] Z. S~,DOLI.AI-I.Data collected from Handbook..~,h~(hinimi D,a~ for E,*ro_spark Pulse Gener,~t,,r Typ, 5".4 produced by Agemaspark Ltd.. October 11978