Surface energy, cutting edge radius and material flow stress size effects in continuous chip formation of metals

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Abstract

Finite element simulations of continuous chip formation have been carried out over a range of uncut chip thicknesses from 0.002 to 0.25 mm and for tools with cutting edge radii from 5 to 160 μm, using material models for annealed or normalised medium carbon steels. Expressions are derived for the dependence of ploughing forces on edge radius, uncut chip thickness and chip shear plane angle and flow stress. Non-linearity of force variation with uncut chip thickness is demonstrated, in addition to what might result from ploughing, caused by variations of temperature and strain-rate and hence of work material shear flow stress. These ploughing and non-linearity effects quantitatively account for the well-known size effects associated with machining, certainly for the machining of steels. There is no need to include ductile fracture energy from forming fresh surfaces in models of continuous chip formation in order to account for the size effect.

Introduction

In metal cutting experiments in which cutting and thrust forces are measured over a range of uncut chip thickness, linear extrapolation of forces often leads to positive non-zero values at zero uncut chip thickness. This is known as the size effect. There is a growing literature, of increasing influence, that claims this is due to the ductile fracture energy required to form new surface as the chip is parted from the work piece by the cutting edge of the tool. The main purpose of this paper is to rebut this view, through finite element simulations of metal machining with tools of finite edge radius. These support the older view that the size effect, at least for the conditions of the simulations, can be fully accounted for by a tool's edge geometry and the strain, strain-rate and temperature dependence of the work material's flow stress.

In coming to this conclusion, the finite element simulations have been validated against previously published experimental data on the micro-machining of steels. The paper therefore also contributes to the modelling of micro-machining processes.

The remainder of this introduction summarises the competing views (surface energy, edge geometry and material flow stress dependence) of the causes of the size effect, with an outline of how these will be tested. It also summarises the previous experimental studies on which this entirely simulation-based paper builds.

The view that the size effect depends on the energy to form new surface was initiated in a paper by Atkins [1]. In a series of steps, for example [2], [3], it has evolved to a proposed method for measuring fracture toughness of ductile metals (and other materials) from machining tests [4]. Other researchers, particularly interested in micro-machining where size effects are likely to be most important, have started to evaluate the work [5].

In the initial paper [1] the work done by the cutting force was equated to the sum of the works of chip formation by plastic shear, sliding friction between chip and tool and the energy to form new surface. The first two terms reduce in proportion to uncut chip thickness, the last term does not. Hence the size effect was explained. Fig. 1a relates to the evolved form [4]. It conceptually separates the resultant force between the work and the chip into two parts. One is RS associated with chip shearing and friction work. The other is RE associated with new surface formation at the cutting edge. The figure is conceptual in its literal spatial separation of RS and RE: Atkins [3] describes them as uncoupled. Each in principal can have X and Y components that in metal cutting are more usually described as cutting and thrust components FC,S and FT,S and FC,E, FT,E. Then the total force components FC and FT are the sum of the shear/friction and edge components. In the standard way, FC,S and FT,S, when resolved on to the shear plane BC at shear angle ϕ to the cutting direction (ϕ is determined from the chip thickness ratio t/h and tool rake angle γ as in Eq. (1) equals the shear force on the plane according to Eq. (2). k is the plastic shear flow stress in the shear plane region. In [1], [2], [3], [4], FT,E is taken to be zero and FC,E is identified with GC, the energy per unit area to form new surface. Its dimensions of energy per area are the same as force per unit width. Then, after writing FC,S as (FC  GC) and FT,S as FT, substituting into Eq. (2), dividing through by cos ϕ and re-arranging, Eq. (3) is obtained. This, with changed notation, is [4]’s Eq. (3) too. If cutting and thrust forces and chip thickness ratio have been measured in experiments over a range of uncut chip thicknesses, (FC  FT tan ϕ) can be plotted against (h/sin 2ϕ). The shear flow stress k can be obtained from the slope and GC from the intercept.th=cos(ϕγ)sinϕFC,ScosϕFT,Ssinϕ=khsinϕFCFTtanϕ=2khsin2ϕ+GCThis procedure has been followed in [4] for classic data in the literature on the machining of a free-cutting low carbon steel SAE 1112 over a range of rake angles and at uncut chip thicknesses from 0.05 to 0.25 mm [6]. Table 1 records the extracted values of GC (as well as of ϕ referred to later). GC varies with rake angle in the range ≈40–15 kJ/m2. As kJ/m2 is equivalent to N/mm, these values are equivalent to 40–15 N/mm. These are the size of contribution to the total cutting force per unit chip width that are ascribed to energy to form new surfaces. Data was also presented in [4], again based on results in [6], for an aluminium alloy: GC was found to be in the range ≈5–10 kJ/m2.

Cutting edge radius plays no part in this analysis. In fact, it was not recorded in [6] what was the cutting edge radius. However the edge preparation method was given and, in an earlier paper from the same group, that edge preparation was described as giving a feather edge [7]. In yet another paper [8] on machining SAE 1113 steel, these authors also described that the edge preparation, either ‘keen’ or ‘with feather edge removed’ caused both cutting and thrust forces to be altered by about 30 N/mm. That paper also included micrographs of chip roots from which it can be seen that edge shape was irregular over dimensions of ≈20 μm. This value is assumed for cutting edge radius in later discussion in this paper (Section 4). And it is clear that the authors of [6], [7], [8] recognised the importance of edge preparation in influencing the size effect.

The earlier and still held view is that the size effect is due to the cutting edge not being perfectly sharp. Fig. 1b shows the interfaces AOB1 and AOB2 between a chip and two tools of different edge radius. The portion AO is common to both and has a resultant force ROA acting across it. The portions OB1 and OB2 differ between the tools, with resultant forces ROB1 and ROB2 acting across them. The vector difference between ROB2 and ROB1 leads to a change in force with changing edge radius. At a greater level of detail, there is flow stagnation at some point P (P1 or P2) in Fig. 1b. Over the portions AOP1 or AOP2 of the chip/tool interface, sliding of the work material over the tool is towards A (i.e. into the chip) but over the portions P1B1 and P2B2 it is towards B (i.e. towards the cut surface). The force acting over the portion PB is known as the ploughing force. The contribution of ploughing forces to the size effect is particularly considered in this paper.

The approach taken is based on the ideas that led to Eq. (3). Instead of supposing FC,E = GC, FT,E = 0, it is more generally supposed that FC,E and FT,E may also contain edge radius contributions, as indicated in Eq. (4), where fC and fT are functions of rβ, and perhaps depend on h as well, to be determined. Then Eq. (5) is obtained after proceeding in the same way that led to Eq. (3). If the left-hand side is plotted against (h/sin 2ϕ), k and GC can be obtained from the slope and intercept. In this paper, dependencies of fC and fT on rβ and h are obtained from finite element simulations of steel machining. It is shown that for the circumstances of the simulations, it is these dependencies and not surface energy that are responsible for the size effect.FC,E=GC+fC(rβ,h);FT,E=fT(rβ,h)[FCFTtanϕ][fC(rβ,h)fT(rβ,h)tanϕ]=2khsin2ϕ+GCThere is an implication in the application of both Eqs. (3), (5) that a straight line sufficiently describes the relation between their left-hand sides and (h/sin 2ϕ), i.e. that k is a constant, independent of h and ϕ. There is a well-established metal machining literature on the influence on chip formation of strain, strain-rate and temperature dependence of work material flow stress. In the context of the size effect, the increasing strain-rate and reducing temperature expected in the shear plane as h is reduced (strain-rate  h−1, Peclet number  h) have been proposed as contributing to the size effect, principally by Kopalinsky, Larsen-Basse and Oxley, as reviewed in [9]. These would lead to k increasing as h is reduced. A plot of the left-hand side of Eq. (3) or (5) against (h/sin 2ϕ) would therefore have an increasing slope as h reduces. Linear extrapolation of a set of data to zero h would then give a positive intercept. Variability of k, as well as the sharpness of cutting edges, is found to have a significant influence on the size effect in this paper's simulations.

The original work in this paper is entirely finite element simulation work. It calls on two types of previously published experimental work to support it. The first is work that generally supports the view outlined in Section 1.2 that the size effect depends on edge sharpness. The second is a particular recent work on the micro-machining of steels over a range of edge sharpness and h values which supports both this and that the size effect depends on the range of h values from which it is estimated. This paper's finite element simulations are validated against this particular work.

From a combined consideration of Fig. 1a and b, from the point of view of Section 1.2, the force RS might be expected to be proportional to h and RE to rβ. Then the specific cutting force (that is FC/h) would be expected to increase significantly, i.e. show a size effect, as h/rβ reduces below some typical value. This has been found to be the case over all experimental ranges of h. Experiments on a mild steel with carbide tools, rβ = 10, 50 and 100 μm, and h from 70 to 300 μm (i.e. general engineering conditions, overlapping the same conditions as [6]) show specific force increasing with reducing h for h/rβ < 5 [10]. In micro-machining experiments on a medium carbon steel with carbide tools, rβ = 5 and 50 μm, and h from 1 to 80 μm [11], considered further in Section 1.3.2, the specific cutting force also increases with reducing h as h/rβ reduces below ≈5. A similar critical range of h/rβ has been found in precision cutting experiments on tellurium copper, with a single crystal diamond (SCD) tool of cutting edge radius rβ = 0.25 μm and h from 0.01 to 20 μm [12].

The direct influence of a small value of rβ (as occurs with SCD tools) on the estimated force at zero h is explicitly seen in [12]. The cutting force extrapolated to zero h is < 0.1 N/mm. In machining experiments on an aluminium alloy with a SCD tool, rβ = 0.06–0.1 μm, the zero intercept cutting force has been found to be <0.4 N/mm [13]. These values are small compared to the >5 N/mm values observed in the previously referenced results with cemented carbide tools [4], [6] and give upper limits to the estimates of GC from these experiments.

These previous studies generally support the view of the importance of edge radius in determining the estimated forces at zero h. They are not considered in the papers [1], [2], [3], [4] arguing for the importance of surface energy in influencing the size effect.

Fig. 2 is constructed from specific cutting and thrust force data in [11], [14]. It shows the effect of cutting edge radius (5 and 50 μm) on cutting and thrust force variation with uncut chip thickness (h from 0.09 to 0.02 mm in Fig. 2a, from 0.01 to 0.001 mm in Fig. 2b) for dry turning of an AISI 1045 steel at cutting speeds from 50 to 200 m/min (results in [14] demonstrate that the influence of speed is small in this range).

As a general comment, the h values in Fig. 2a overlap the range from 0.05 to 0.25 mm in [6] from which the GC values in Table 1 are obtained. Cutting edge radius is seen to influence the forces. These results further support the view that edge radius would have significantly affected the tool forces reported in [6].

Further it can be seen that extrapolation to zero h of forces in Fig. 2a gives larger zero intercepts than in Fig. 2b. This supports the view that the extent of the linear extrapolation affects the size effect estimation. These conclusions are sufficiently clear that they are not obscured by the rather large scatter in the experimental data. Neither do they depend on the detail of the regression line fits that have been included simply to guide the reader's viewing. For example, without the regression lines, zero force intercepts from 25 to 150 N/mm would be deduced from Fig. 2a, and from ≈5 to 20 N/mm from Fig. 2b.

But the main purpose of including Fig. 2 is to provide the benchmark against which the finite element simulations to be reported are validated.

The primary purpose of this paper is quantitatively to re-assess the possible contributions of surface energy, finite edge radius and flow stress variation to the overall cutting and thrust forces in metal cutting, particularly how they may contribute to a size effect. Finite element simulations are reported that show the general influence of cutting edge radius on chip flow and forces and which in particular lead to expressions for fC and fT (Eqs. (4), (5)). These are the topics of Sections 2 The simulations, 3 Simulation results.

Material properties appropriate for medium carbon steels have been used in the simulations, including flow stress dependence on strain, strain-rate and temperature, so that they may be directly relevant to the results in [11] and also relevant to those in [4] for a low carbon steel, based on [6], after compensating for different levels of plastic flow stress. The relative contributions of edge radius, variation of shear stress with h and surface energy contributions to size effects are then evaluated, in Section 4, through considering the relative sizes and variations of the terms in Eq. (5).

As stated at the start of this introduction, the use of micro-machining experiments to validate the finite element simulations also contributes to developments in modelling micro-machining. This is briefly discussed, with the main findings of the paper, in Section 5.

Section snippets

The simulations

The commercial elastic-plastic finite element software AdvantEdge-2D™, bespoke for metal cutting modelling, has been used, as has been detailed in previous work [15], [16], [17].

Simulation results

The series 1 tests demonstrated that additional strain is associated with flow round the cutting edge. Fig. 6 shows the chip formed for h = 0.25 mm, rβ = 10 μm. In the general view (Fig. 6a), equivalent strain contours are 0.25, 0.75 and 1.0 (the bulk strain in the chip was ≈1.0). In the detail view (Fig. 6b), the strain contours are 0.5, 1.0 and 2.0. At the cut surface and chip/tool interface, strains are in excess of 2.0. In fact they were ≈3.0.

It was observed that some aspects of the chip

Experimental comparisons

The series 3 simulations for rβ = 5 and 50 μm are compared with the experimental results from Fig. 2 [11] in Fig. 14. The simulated results fall within the scatter of the experimental observations except for the variation of thrust force with uncut chip thickness when rβ = 50 μm (Fig. 14d). In that case the simulated force is 10% below the experimental range for h = 0.05 mm, falling to 30% below at h = 0.1 mm.

It is satisfying to note that the finite element code and material property model that have given

Discussion

The initial and main reason for carrying out the simulations described in this paper was to assess the recent claims [1], [2], [3], [4] that failure to consider the ductile fracture energy associated with creating new work and chip surfaces around the cutting edge in metal cutting chip formation is a major error of most metal cutting analyses.

The initial simulations of steel machining, assuming zero chip/tool friction in order to focus on deformation associated with flow round the cutting edge,

Conclusion

The claim of recent papers [1], [2], [3], [4], that are gaining influence, that it is necessary to include ductile fracture energy from forming fresh surfaces in models of continuous chip formation in order to account for the size effect in metal cutting (the prediction of positive cutting forces at zero uncut chip thickness from linear interpolation of results at larger uncut chip thicknesses) is shown to be wrong, at least for the machining of steels that is the subject of this paper. It is

Acknowledgements

I wish to acknowledge discussions with A.G. Atkins. He read an early draft of this paper and made fruitful and constructive comments despite the clear difference of opinions between us. I also wish to thank J. Kotschenreuther for making available thrust force results from his thesis [14].

References (19)

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