EDM is one of the most popular non-conventional machining processes, which involves the use of a large number of spark discharges in order to remove material from the workpiece surface. The discharges occur as a result of a sufficient voltage difference between an electrode, acting as cathode and the workpiece acting as anode, which gives rise to a plasma channel of high energy density[1]. Subsequently, high temperatures occur in the gap between anode and cathode, melting or ablating the workpiece material[2, 3]. Thus, this process is particularly effective during processing of hard-to-cut materials, such as hard steels, titanium or nickel alloys, as it does not involve mechanical contact between a tool and the workpiece and it can process every electrically conductive material, regardless its mechanical strength[4]. Various parameters can affect the efficiency of EDM, such as the pulse-on current, pulse-on time, machining voltage, duty factor, thermo-physical properties of the workpiece material etc. Due to the large number of parameters, researchers often conduct a considerable amount of experiments for each material and perform optimization studies[5].
EDM process has been extensively studied during the last decades, as it has been proven important for the construction of various parts for the automotive, aerospace and biomedical industries[6]. Although this process can be successfully carried out in industrial practice, its application for advanced or novel materials is not straightforward and requires optimization based on both the efficiency of the process, indicated by Material Removal Rate(MRR) and Tool Wear Ratio (TWR),and on the surface integrity, indicated by surface roughness or modification of the microstructure[5]. However, due to obvious difficulties, both in directly monitoring the process and determining the procedure of the creation of heat affected zone (HAZ) and white layer (WL), reliable simulation models are required in order to provide a deeper understanding of the complex phenomena occurring in this process. In fact, modeling of EDM process is not a trivial task, as the occurrence of complex thermo-physical phenomena should be taken into consideration. Although simulation models for EDM have been developed for several decades, there are issues which remain unresolved still nowadays and thus EDM remains not entirely understood or explained by a universal model. Furthermore, it is inevitable to adopt reasonable simplifications for the models in order to perform simulations in a feasible timeframe, given the limitations of the available computational power, but the accuracy of the simulations should be ensured by optimizing the computational model parameters based on experimental observations.
As the thermal effects are dominant in EDM, most simulation models focus on modeling the effect of heat input into the workpiece by using a heat source with a Gaussian distribution[7, 8]. This heat source represents a single spark and heat transfer is mainly considered to take place by conduction[9, 10]. Then, the calculated MRR resulting from the action of the spark is compared to the experimental[7, 11]. Other important issues with the models are the determination of energy distribution between electrode and workpiece, which cannot be considered constant, but dependent on process parameters [12–15] and the determination of the PFE, which is relevant to the percentage of molten material removed from the workpiece by every spark. For PFE, the simplification that it is equal to 100% is definitely unacceptable[7, 15], as it contradicts the formation of WL, which is evident by many experimental works.
Especially for the plasma column, although some authors neglected the variability of its spatial dimension, most authors take into account the plasma channel radius by semi-empirical formulas or even adopt time-dependent relations, as it was shown that the variation of the plasma channel radius can directly affect the simulation results, especially for shorter discharge times[16]. Semi-empirical formulas usually correlate the radius with discharge power and pulse-on time [2, 17, 18] or pulse-on time and discharge current, but formulas using gap voltage exist as well [19], although other parameters, such as properties of the electrode and dielectric, polarity and workpiece material can also affect it [14]. These formulas are developed based on the experimental results after fitting procedure, while direct measurements of the plasma column dimension e.g., using high speed cameras and spectrometers are rare [20, 21]. These formulas are considerably popular, can be used for a specific range of conditions and can provide fairly good approximation of the dimensions of the plasma column, something that is fundamental for the application of heat source in the model. However, phenomena such as the rapid expansion of the plasma column during the first microseconds after its creation can only be taken into consideration by time-dependent formulas.
Regarding the modeling of plasma column expansion, apart from the obvious difficulties in observing this phenomenon directly, there is also a lack of established models for plasma dynamics describing the different states of plasma column during EDM, i.e., creation, expansion and collapse [22]. Only a few authors have adopted comprehensive approaches relevant to the time evolution of plasma column. For example, Mujumdar et al. [22] attempted to create a model for plasma column taking into account chemical phenomena, apart from electrical and thermal ones. Using this model, they directly predicted heat flux to the workpiece and electrode under various conditions. Eubank et al. [23] developed a model for the calculation of plasma radius, temperature and pressure evolution using a fluid mechanics equation, an energy balance equation, a radiation equation and an equation of state, whereas thermo-physical properties were calculated by taking into consideration some fundamental reactions occurring during the process. Pandey and Jilani [24] adopted an iterative procedure for the determination of plasma channel radius based on the assumption that cathode spot temperature is constant during the pulse-on time and equal to the boiling point temperature of the electrode material. Results were obtained for various electrode materials and this approach was considered sufficiently accurate compared to experimental data. Zhang et al. [25] determined a formula for the spark column expansion, based on a differential equation describing spark volume increase and also took into account variable heat transfer time, depending on the distance of points from the center of the spark. Chu et al. [26] presented a comprehensive methodology for the calculation of plasma column radius, temperature and pressure including various components, such as a model for breakdown in the dielectric medium, which takes into consideration the nucleation of bubbles and production of discharge and models for the initial stages of plasma column formation and expansion stage until the end of each discharge. Dhanik and Joshi [27] also developed a model for plasma column, including the nucleation and growth of bubbles at the first stage and appropriate fluid dynamics and heat transfer models for the heating stage.
Shabgard et al. [28] used a differential equation for the calculation of plasma column radius, which took into consideration the contribution of electric field created by charged particles to the acceleration of plasma channel, the contribution of magnetic field created by the movement of the charged particles, the contributions of internal and external pressure of plasma channel, as well as the surface tension in the interface between plasma channel and dielectric. Especially for magnetic field-assisted EDM process, Shabgard et al. [29] also used a slightly modified model of their original one, including the effect of the additional magnetic field in the differential equation of plasma expansion. Gholipoor et al. [30] compared the results of the model presented by Shabgard et al. [28] and showed that the calculated plasma radius values are slightly higher to those of semi-empirical models.
However, in the majority of relevant works, due to computational power limits, a simpler approach was followed using a single or a piecewise function to model the time dependence of plasma column radius. In specific, the following types of functions have been already proposed in the relevant literature: linear, power law, combined power law with semi-empirical and piecewise. For the linear function, it can be noted that only a few works have used it, such as the work of Singh and Ghosh [31], who analyzed plasma formation by a fluid mechanics equation, in order to establish a thermo-electric model for the calculation of the electrostatic force on the electrode and the stress distribution on the workpiece during a discharge. In their work, plasma radius was assumed to vary linearly with time for small discharge durations.
On the other hand, the power law function is by far the most popular in the relevant literature. In these works, the assumption made is relevant to the continuous expansion of plasma column radius during the discharge time [24]. Izquierdo et al. [32] argued that modeling of plasma radius with a power law function including a constant term is in line with the common assumption that the expansion of plasma channel is abrupt for a few microseconds and then stabilizes. Similar approaches were adopted by Shao and Rajurkar [10] and Guo et al. [11]. Kliuev et al.[33] used a similar function with the addition of a second constant term which could be determined by an optimization procedure. Schneider et al. [34] also employed a function with a second constant term, which was dependent on maximum spark radius, initial radius and discharge duration. In every case where power law was adopted, the exponent has values below 1.0. In many of these works [10, 11] the exponent of the power law function is 0.75; however, it is worth noting that an exponent with a value around 0.2 has also been suggested in the works of Revaz [35] and Perez [36] based on various experimental observations, in contrast to the works which used larger exponents. More specifically, Revaz et al. [35]commented that the exponent value of 0.75 corresponds only at the very beginning of the discharge and determined that the most suitable value for the exponent was close to 0.2.
Assarzadeh and Ghoreishi [37, 38] noted that the discharge channel radius is dependent on various factors and utilized a combined semi-empirical/power law relation for the growth of discharge channel radius, including an additional term dependent on discharge current. In this relation, time was varied between 0 and Ton value, after which the plasma channel is assumed to collapse and this relation was considered accurate up to a specific spark energy level. Similar approaches were used by Guo et al. [39] and Xie et al. [40]. Vishwakarma et al. [41] also introduced a relation which included other terms rather than time, such as the discharge length and discharge power, as well as empirical constants. The latter approach was also mentioned by Kumar et al. [14] and Kansal et al. [42].
Rajhi et al. [43] conducted a comprehensive comparison regarding the use of constant equivalent plasma column radius functions, time-dependent radius function and time-dependent radius function with a semi-empirical term.The models which employed a constant radius, even if different semi-empirical functions were used, showed similar results both regarding the highest temperature and the time when it was reached, as well as similar temperature variation in respect to time. For the time-dependent models, although the time to reach the maximum temperature was similar, significant differences were obtained for the temperature profiles. Regarding crater morphology, it was shown that both crater diameter and depth varied for different radius models and especially for crater diameter, considerably different results could be obtained. One model showed shallow crater morphology, whereas the others predicted a more hemispherically-shaped one. Finally, MRR prediction was also found to be dependent on the radius model, with some models overestimating MRR and others underestimating its values, whereas the accuracy of some of the models was also limited only to cases with low or high values of discharge current.
Liu and Guo [44, 45] developed a plasma radius growth model with a piecewise function employing two empirical constants. The first expression, which was a power law function in respect to time, was considered valid up to a certain critical time and then the plasma radius was considered constant. The estimation of empirical constants and critical time was based on a considerable amount of experimental data [45]. This model reflects some experimental observations, which indicated that plasma is not continuously expanding during the entire pulse duration [45]. Li et al. [46] used also a piecewise function, but the function employed for the expansion stage was linear in respect to time.
Natsu et al. [21] performed observations with high-speed camera and confirmed that the expansion of plasma is completed after a few microseconds after the breakdown of the dielectric. Thus, they proposed a piecewise model, termed as First Stage Expansion Model (FSEM) and compared it to a power law model which was inferior to the FSEM regarding the crater dimensions. Izquierdo et al. [47] also compared power law functions to piecewise ones (FSEM), including a linear part for an extremely brief time period and a constant part. The simulation results were compared to experimental ones regarding MRR and surface roughness indicators. At first, it was shown that a FSEM model with an optimal value for the final plasma radius can provide an excellent estimation for MRR and acceptable error for surface roughness. Thus, they considered necessary to include the first stage of rapid expansion in their model. However, the use of an improper final value for plasma radius leads to a much higher error level for FSAE models, compared to power law ones.
From the analysis of works relevant to the modeling of plasma column radius in the relevant literature, it can be observed that there is a lack of a comprehensive comparison of the results produced between different plasma radius expansion functions for various different conditions based on experimental validation of both crater dimensions and microstructure observations. Moreover, the variation of PFE values in respect to different plasma radius expansion functions has also not been considered yet in the literature. Thus, in the present work, a numerical model for the prediction of crater morphology, HAZ and WL formation during EDM is employed in order to comprehensively investigate the effect of the plasma column radius expansion on the results of the EDM simulations. Three different types of functions are compared, namely constant, linear and power law functions and their effect on the geometry of produced craters, energy absorption coefficient and PFE is analyzed, based also on actual microscope observations of craters produced during EDM of 60CrMoV18-5 steel under various conditions.