A Slip-line Field for Ploughing during Orthogonal Cutting

Under normal machining conditions, the cutting forces are primarily due to the bulk shearing of the workpiece material in a narrow zone called the shear zone. However, under finishing conditions, when the uncut chip thickness is of the order of the cutting edge radius, a ploughing component of the forces becomes significant as compared to the shear forces. Predicting forces under these conditions requires an estimate of ploughing. A slip-line field is developed to model the ploughing components of the cutting force. The field is based on other slip-line fields developed for a rigid wedge sliding on a half-space and for negative rake angle orthogonal cutting. It incorporates the observed phenomena of a small stable build-up of material adhered to the edge and a raised prow of material formed ahead of the edge. The model shows how ploughing forces are related to cutter edge radius a larger edge causing larger ploughing forces. A series of experiments were run on 6061-T6 aluminum using tools with different edge radii including some exaggerated in size and different levels of uncut chip thickness. Resulting force measurements match well to predictions using the proposed slip-line field. The results show great promise for understanding and quantifying the effects of edge radius and worn tool on cutting forces. INTRODUCTION The first studies on the phenomenon of ploughing in metal cutting (Albrecht 1960) were undertaken in an attempt to explain why the apparent coefficient of friction between the rake face of the tool and the chip varied with rake angle. A finite radius on the cutting edge was believed to be responsible by contributing additional forces as some material was directed downward below the edge and pressed into the workpiece. Although the discovery of a secondary shear zone at the tool-chip interface changed the view of friction at that interface, ploughing and its importance in cutting became the subject of a number of investigations (Palmer and Yeo, 1963; Johnson, 1967; Moneim and Scrutton, 1974; Heginbotham and Gogia, 1961) and continue to be studied (Rubenstein, 1990; Zhang et al., 1991; Sawar and Thompson, 1981; Parthimos et al., 1993; Wu, 1988; Endres et al., 1995; Elanayar and Shin, 1994) as researchers attempt to gain a complete understanding of the mechanisms of the cutting process. For example, a size effect on cutting forces has long been observed, in which the ratio of forces to the area of cut increases as uncut chip thickness decreases. While this effect can be explained in part by a varying shear angle, ploughing is also believed to play a role. Wu (Wu, 1988) and others (Endres, et al., 1995; Elbestawi, et al., 1991; Sisson and Kegg, 1969; Albrecht, 1965) believe that because of its phase lag with respect to uncut chip thickness variation due to dynamic vibrations, the ploughing component of cutting forces is directly related to cutting process damping and machine-tool stability issues. Since the ploughed material eventually becomes a part of the machined workpiece surface, understanding the phenomenon of ploughing has been linked (Johnson, 1967; Haslam and Rubenstein, 1970; Thomsen, et al., 1953) to the study of the properties of the machined surface. Furthermore, the indenting nature of the ploughing process bears resemblance to the contact of a worn flank surface, and many researchers (Elanayar and Shin, 1994; Usui, et al., 1984) have studied the ploughing component of forces as part of an attempt to predict forces on a worn tool. Several physical models of the ploughing mechanism have been proposed, but very little verification of those models has taken place because of the difficulty in measuring ploughing forces and separating them from the total cutting force. Rubenstein and others (Rubenstein, 1990; Haslam and Rubenstein, 1970; Connolly and Rubenstein, 1968) describe an extrusion-recovery mechanism in which a thin layer of material initially at a depth h (see Figure 1) above the level of the bottom of the cutting edge is extruded below the edge and recovers back to its original level. The material is thought to have a separation point S on the cutting edge at a critical rake angle as, estimated to be around 70 . Others (Wu, 1988; Endres et al., 1995; Sisson and Kegg, 1969) have taken a similar view, suggesting empirical calibration of the penetration depth h and predicting ploughing forces proportional to the elastically displaced volume of the extruded material. Elanayar and Shin (Elanayar and Shin, 1994) also propose an empirical fit to h and model the mechanism as elastic indentation rather than extrusion. Unfortunately, little evidence has been presented to support such a description of the physical scenario at tool tip, while some evidence has been shown to contradict it (Sarwar and Thompson, 1981; Waldorf et al., 1996) . In a study using plasticity methods (Johnson, 1967)], a slip-line field was developed for restricted contact tools predicting sublayer plastic flow and even a chip exiting from the rear contact of the edge with the workpiece. This, however, has not generally been observed during orthogonal cutting. Still other models (Palmer and Yeo, 1963; Moneim and Scrutton, 1974; Zhang et al., 1991; Sarwar and Thompson, 1981) predict plastic flow beneath the edge and postulate a stable build-up of material adhered to the edge. Each of these theories/models has used plasticity methods and slip-line fields to model the flow, but the experimental difficulties of isolating the ploughing force components has resulted in very little verification of the models. Consequently, no consensus exists as to a reasonable method for predicting the ploughing forces or for decomposing a measured force into shearing and ploughing components. This paper presents a new slip-line model of the ploughing process in orthogonal cutting. The model is based on the assumption of a stable build-up adhered to the finite radius cutting edge. Instead of material separating at a stagnation point on the edge, the flow is diverted at the extreme edge of the build-up. Experimental results are compared to the theoretical predictions to show the success of the model. It is anticipated that the model will serve as a starting point for the development of a predictive algorithm for ploughing forces allowing for simple decomposition of measured forces into components due to shearing and ploughing. THE SLIP-LINE MODEL The ploughing process is complex because it involves both a sliding/indenting action of the tool edge on the workpiece and an interaction with the primary plastic deformation zone associated with the bulk shearing of the workpiece. As such, the proposed slip-line field representing ploughing derives partly from previous fields developed for wedge sliding and chip cutting mechanisms. Black, Kopalinsky, and Oxley (Black et al., 1993) present a review of fourteen years of research improving upon a model, originally proposed by Challen and Oxley (Challen and Oxley, 1979) and based on (Avitzur and Zhu, 1985), for asperity deformation and wedge sliding as a description of friction and wear processes. The result is a wave model for the sliding wedge (see Figure 2) in which material ahead of the wedge is compressed and pushed up to meet the wedge face, resulting in a moving wave of workpiece material ahead of the wedge. Considerable evidence has accumulated in support of the model (Black et al., 1993) [which has primarily been used to determine frictional relationships (see [Avitzur and Zhu, 1985]). The same compression of material and resulting stresses must also be present for the sliding contact of a cutting edge, except that the wave is not clearly seen since it is so small and is generally removed as part of the chip. A slip-line field proposed by Abebe and Appl (Abebe and Appl, 1981) for cutting with a large negative rake angle, however, includes a raised prow of material joining the uncut work surface to the chip and suggests the moving wave concept during cutting. It has long been known that the intersection of the chip and the uncut surface is not a sharp corner but rather a curved surface (see, for example [Palmer and Yeo, 1963]), but it is frequently approximated to be sharp. The raised prow is therefore realistic and matches the moving wave result of the sliding wedge model. The model in (Abebe and Appl, 1981) also includes a stable build-up region as Kita (1982) observed for cutting with negative rake angles simulating grinding. The proposed slip-line field for ploughing is shown in Figure 3 for a tool of edge radius re and rake angle a equal to zero. The field resembles one developed by Shi and Ramalingam (Shi and Ramalingam, 1991) for cutting with a flank wear land. Heavy arrows in the figure show the flow of material (starting with the workpiece velocity V) both into the chip and below the edge. Because of a velocity discontinuity along the lower boundary of the field, the workpiece material is seen to raise up before reaching the edge as is consistent with the wave model for sliding described above. The stable build-up region is shown shaded with point A at the point of material separation. AB is a type I slip-line and essentially represents the traditional shear plane inclined at the shear angle f. The prow is shown inclined at an angle r with respect to the uncut workpiece surface. The vertical distance between the uncut workpiece surface and the machined workpiece surface is the uncut chip thickness tc. The remainder of the field below AB can be determined from frictional and geometric considerations . Due to the high normal stress on the lower interface between the tool (build-up region) and the work CA, a constant frictional stress τ is assumed proportional to the material shear flow stress k