Thermomechanical oscillations in material flow during high-speed ...

10.1098/rsta.2000.0756

Thermomechanical oscillations in material flow during high-speed machining By M a t t h e w A. Davies a n d T i m o t h y J. B u r n s National Institute of Standards and Technology, Gaithersburg, MD 20899, USA This paper presents a nonlinear dynamics approach for predicting the transition from continuous to shear-localized chip formation in machining. Experiments and a simplified one-dimensional model of the flow both show that, as cutting speed is increased, a transition takes place from continuous to shear-localized chip formation in the flowfield of the material being cut. Initially, the process appears to be somewhat disordered. With further increases in cutting speed, the average spacing between shear bands increases monotonically, and the spacing becomes more regular and asymptotically approaches a limiting value that is determined by the cutting conditions and the properties of the workpiece material. Keywords: high-speed machining; Hopf bifurcation; relaxation oscillations; chaotic dynamics; plasticity; adiabatic shear bands

1. Introduction Within the past decade, there has been a marked increase in the industrial use of high-speed machining technology. This decade has seen the development of machining centres capable of spindle and slide speeds that are an order of magnitude higher than those available on conventional machining centres. The most dramatic applications of high-speed machining have been in the manufacture of aluminium components, where volumetric material removal rates can be extremely high, often approaching thousands of cubic centimetres per minute. In the aerospace industry, unprecedented cost savings, improved part accuracy, performance, and fatigue characteristics have been demonstrated using high-speed machining technology (Halley et al . 1999). Progress in high-speed machining of materials other than aluminium has been more limited. Currently, a number of the factors that limit the broader use of high-speed machining result directly from the state of the material flow at the tool–chip interface. These factors include thermally activated crater wear of the tool at the interface between the tool and the secondary shear zone, rapid flank wear in the zone of the interaction between the tool nose and the workpiece, and a poor understanding of the mechanisms that produce subsurface damage and induce stresses in machined parts. These difficulties are compounded by a marked lack of empirical knowledge about high-speed machining when compared with conventional machining operations. Because they cannot keep up with rapid improvements in machine technology, empirical databases such as Metcut (1980) are hopelessly out of date. Therefore, the primary method used by industry to select process parameters often involves costly and time-consuming trial-and-error prototyping. This method is expensive, and it may lead to suboptimal parameter choices. For these reasons, and because Phil. Trans. R. Soc. Lond. A (2001) 359, 821–846

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of increasing international competition, manufacturers are beginning to investigate more sophisticated approaches to the design and optimization of machining processes. In a number of industries, manufacturers have begun to evaluate the effectiveness of the use of large-scale finite-element software to assist them in the prototyping of high-speed machining processes. While considerable progress has been made in the development of predictive models for low-strain-rate manufacturing processes, there is currently a need for improved predictive capabilities for high-rate processes. The accuracy of any model of a large-scale manufacturing operation is limited by how well it can predict the most basic components of the operation. Of these components, chip morphology is certainly the most fundamental, because it provides a direct record of the complex nonlinear plastic material flow process that determines the cutting forces and the temperature along the tool–material interface, which in turn have a dominant effect on tool wear as well as surface finish and subsurface damage of the workpiece. The purpose of the present paper is to summarize the progress we have made to date on modelling the transition from continuous to shear-localized chip formation. In the next section, we review some of the relevant experimental observations and theoretical work that have led to the basic continuum mechanics model on which most of modern cutting theory is based. In § 3, we present the results of our experiments and finite-element simulations in order to elucidate the basic physical features of the transition to shear-localized chip formation. Based on these observations, we have developed a simplified dynamical-systems model of the cutting process; this is developed in § 4. In § 5, we describe the current numerical algorithm we have been using for computer simulations of the model. A presentation and discussion of selected results of numerical simulations of the model are given in § 6. We end the paper with further discussion and some concluding remarks in § 7. Preliminary versions of this work, which do not include the effects of strain hardening in the workpiece material and use a different numerical algorithm for computer simulations of the model, have been published in Davies et al . (1997, 1999) and Burns & Davies (1997).

2. The theory of chip formation One of the earliest scientific reports on the action of a machining tool was presented in 1881 by Mallock in the Proceedings of the Royal Society of London (Mallock 1881). Nearly coincident with the publication of Mallock’s article was the appointment of F. W. Taylor as foreman of the machine shop at the Midvale Steel Company, and his work over the next 25 years is summarized in his now famous systematic study of machining (Taylor 1907). Mallock is thus arguably the founder of the theoretical scientific study of the machining process, while Taylor is the inventor of the more practical concepts of machinability and tool life. The next period of rapid development occurred in the 1930s and 1940s. The study of machining mechanics was for the first time placed on a solid physical and mathematical foundation by the work of Piispanen (Piispanen 1937, 1948), and of Ernst & Merchant (Ernst 1938; Ernst & Merchant 1941; Merchant 1944). Progress continued in the studies of Merchant (1945a, b), Field & Merchant (1949), Cook et al . (1954), Lee (1954), and Rice (1961). More advanced ideas from plasticity, thermoplasticity, Phil. Trans. R. Soc. Lond. A (2001)

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and materials science have since been introduced to analyse various chip-formation phenomena, including (1) continuous chips (Lee & Shaffer 1951; Shaw et al . 1953; Hill 1954; Roth 1975; Palmer & Oxley 1959; Ramalingham & Black 1973; Von Turkovich 1970); (2) chips with built-up edge (Ernst & Martellotti 1938); (3) shear-localized chips (Recht 1964; Shaw 1967; Komanduri & Brown 1981; Molinari & Dudzinski 1992); (4) chips with periodic fracture (Vyas & Shaw 1999); and (5) segmental chips (Rice 1961; Komanduri & Brown 1981). More recently, scientific computation has become a powerful new tool for understanding machining. This approach has evolved from the use of fairly basic finiteelement software (Athavale & Strenkowski 1999; Childs 1999) into the use of more sophisticated commercial simulation codes, which have drawn from ideas in armour penetration (Marusich & Ortiz 1995) and forming (Ozel & Altan 1999; Cerretti 1999). From the body of theoretical, experimental, and numerical research that has been developed for many different types of materials under widely varying cutting conditions, some common conclusions can be drawn. Chip formation in machining involves a highly nonlinear thermo-viscoplastic flow of material. Strains, strain rates, and temperatures are extreme, with large gradients in the deformation zones near the tool–chip interface. Strain rates as high as 103 –105 s−1 are common, and in rare cases they can exceed 106 s−1 in shear-localized flows. Strains may be of the order of 10, and temperatures may increase from ambient to a large fraction of a material’s melting temperature on time-scales of milliseconds to microseconds. The flow patterns are complex, and may be either steady-state or oscillatory in nature. Furthermore, the type of flow pattern encountered is a strong function of the machining parameters. Most practical machining operations involve a tool–workpiece interaction that has non-zero gradients with respect to each orthogonal coordinate direction. Consequently, the resulting deformations are truly three dimensional, and are generally so complicated that the only possible methods for studying them involve laboratory experiments or numerical simulations. However, even these approaches may not be feasible unless the geometry is simplified. This has led to the use of orthogonal cutting, in which the derivatives of the flow variables, i.e. temperature, stress, plastic strain and strain rate, etc., are small in the direction parallel to the cutting edge; see figure 3. This highly useful two-dimensional model provides important intuition, because it retains many of the basic features of more complex machining processes. In this paper, we will restrict our attention to chip formation in orthogonal cutting. Rosenhain & Sturney (1925) were among the first to classify the types of chips that formed using the cutting tools of their day. They identified three basic types: tear, shear, and flow. About a decade later, when the scientific understanding of machining was undergoing rapid development, this classification scheme was superceded by a scheme introduced by Ernst (1938). In this scheme, which persists to this day, Ernst also identified three basic chip types: discontinuous, continuous, and continuous with built-up edge. Photographs of each of these may be found in fig. 2 of Komanduri’s Phil. Trans. R. Soc. Lond. A (2001)

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stagnation zone (built-up edge)

v

v

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(c) 'stick−slip' region v

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Figure 1. The five different types of chips that can be produced in an orthogonal cutting operation: (a) continuous; (b) continuous with built-up edge; (c) discontinuous; (d) segmental; (e) serrated or shear-localized.

(1993) comprehensive review paper. Komanduri (see Komanduri 1993) and others have since added two additional chip types: shear-localized chips, which were first identified by Shaw (see Shaw 1967; Vyas & Shaw 1999), and first modelled by Recht (1964); and segmental chips, which were first identified by Rice (1961). We will follow this classification scheme of the five chip types (see figure 1), but our emphasis will be on the point of view that the material flow patterns associated with the different chip types correspond to dynamic equilibrium states of the nonlinear physical system. Thus, from a nonlinear dynamics point of view, continuous chip formation is a non-oscillatory flow pattern in which the stress, strain, strain rate, and temperature profiles remain constant in time, in a frame fixed with respect to the tool; see figure 1a. A continuous chip with built-up edge is the result of a dynamic equilibrium state of the system that contains a stagnation zone (figure 1b). The tendency for built-up edge formation is known to decrease with increasing cutting speed. Discontinuous chips result from a periodic rupture in the shear zone, as shown in figure 1c. According to Field & Merchant (1949), this rupture results from an oscillatory compressive stress on the shear zone, originating at the tool edge and propagating toward the free surface. Segmental chips (figure 1d) are characterized by chips that are continuously sheared, but with periodic variations in thickness. Komanduri & Brown (1981) assert that this type of chip results from a stick–slip oscillation and fracture (i.e. an interaction between the primary and secondary shear zones), and that it occurs only in certain speed ranges. An interpretation of Komanduri’s experimental results in the context of dynamical-systems might suggest that this type of chip exhibits the characteristics of a subcritical bifurcation. This interpretation is probably worth further investigation. Finally, shear-localized (also called serrated) chips (figure 1e) Phil. Trans. R. Soc. Lond. A (2001)


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are characterized by oscillatory profiles. Experiments show that, for most ductile metals, as the cutting speed increases monotonically in orthogonal cutting, a transition takes place from continuous to shear-localized chip formation in the flowfield of the material being cut. Initially, the process appears to be somewhat disordered. However, the average spacing between shear bands increases monotonically with cutting speed, and the spacing becomes more regular and asymptotically approaches a limiting value that is determined by the cutting conditions and the properties of the workpiece material; see figure 2. A unified approach to the problem would describe each type of chip as a stable dynamic equilibrium of the partial differential equations describing the elastic-plastic flow. In the present study, we will focus on modelling the transition from continuous to shear-localized chip formation in high-speed orthogonal machining. As Komanduri (1993) has pointed out, over 100 years ago Mallock (1881) performed experiments at the University of Cambridge on orthogonal cutting of metals. In his paper, Mallock made the following observation: ‘The tools do not act, properly speaking, by cutting but by shearing, and the shaving removed by them may be accurately described as a metallic slate.’ About 50 years later, before World War II, the Finnish scientist Piispanen (1937) independently made a similar observation. Piispanen developed an idealized cutting model that treated the work material as a sequence of parallel elements, inclined to the free surface by the shear angle φ. The action of the tool in this model is to cause successive ‘cards’, i.e. thin lamellae of material, to slide over their nearest undeformed neighbour. This work first became accessible to the non-Finnish speaking world in a paper published after the war (Piispanen 1948). The cutting model we will present in this paper retains a number of features of Piispanen’s purely mechanical ‘card model’, but its development has also been strongly influenced by the work of Recht (1964). Aware of Zener’s (1948) important research on fracture initiation due to localized shear, Recht argued that shear-localized ‘sawtooth’ chips are the result of the repeated formation of ‘adiabatic’ shear bands in the work material. These local plastic shear instabilities arise from a competition between the tendency of many materials to harden as they deform, and the opposing tendency of these materials to soften if local heating due to plastic dissipation is large enough. If a deformation process is sufficiently slow, then there is time for local thermal inhomogeneities to be smoothed out by heat conduction, and the process will be dominated by strain hardening and remain stable. However, if a material is deformed rapidly enough, as can occur in metal-cutting operations, then the time available for heat to be removed by conduction is too small, and the process is effectively adiabatic on the time-scales of interest. When this happens, the deformation process can become unstable locally, with the result that ‘catastrophic thermoplastic shear’ bands can form in the workpiece material. See figure 2 for a blow-up of two shear bands in a shear-localized chip of a hardened bearing steel (52100). By analysing the stability of a simple shearing deformation of a block of material, Recht also went on to develop a criterion for when such bands of localized plastic shear can be expected to occur. Thus, Recht provided the first explanation for the transition from continuous to shear-localized chip formation in metal cutting. However, not everyone agrees with his interpretation of this phenomenon. The topic is still being debated in the machining literature (Vyas & Shaw 1999). In the next section, we will present evidence, including laboratory experiments, theoretical modelling, and numerical simulation, Phil. Trans. R. Soc. Lond. A (2001)

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Figure 2. SEM photograph showing: polished and etched chip cross sections from machining experiments in 52100 bearing steel, at cutting speeds of (a) 0.7 m s−1 , (b) 1.0 m s−1 , (c) 1.5 m s−1 , and (d) 4.3 m s−1 ; a plot of the mean serration spacing as a function of cutting speed; and a highly magnified view of two regions of localized shear.

in support of Recht’s hypothesis that sawtooth chips are the result of shear localization, and not of periodic fracture. Recht (1964) also emphasized the importance of tool-face pressure in producing shear-localized chips, and he made the following statement. ‘Obviously, the distance between slip zones is limited by geometry during machining; slip ceases as soon as a unit of material passes out of the stress field. At lower speeds, thermal gradients will be lower and the distance between slip zones would be expected to be smaller; this Phil. Trans. R. Soc. Lond. A (2001)


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α

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tool b

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Figure 3. Orthogonal cutting parameter definition.

distance approaches zero as the velocity decreases to the critical velocity at which the deformation becomes uniform.’ In what follows, we will develop a simplified onedimensional thermoplastic model that also exhibits this type of behaviour.

3. Experimental and numerical observations of thermoplastic oscillations In this section, we will present experimental evidence for the existence of thermoplastic instabilities in chip formation, the qualitative behaviour of the instabilities, and the thermoplastic oscillations that ensue. In addition, we will present the results of some finite-element computer simulations, using commercial software, that also demonstrate the development of thermoplastic oscillations in material flow, leading to the formation of shear-localized chips. This will help to motivate the development of the one-dimensional thermo-viscoplastic chip formation model we will present in § 4. (a) Experimental observations The transition to shear-banded chip formation has been demonstrated in a number of materials (Recht 1964; Komanduri et al . 1982; Davies et al . 1996, 1997). Here we show representative results from two materials, a hardened 52100 bearing steel (approximately 62 Rc), and electroplated nickel–phosphorus on a copper substrate. The steel cutting tests were done in an orthogonal configuration as detailed in Davies et al . (1996). The nickel–phosphorus tests used semi-orthogonal, interrupted cuts, as described in Davies et al . (1997). As shown in figure 3, the parameters are defined as follows: α is the rake angle; vr is the cutting speed; φ is the shear angle; and h is the uncut chip thickness. A two-axis diamond turning machine with an air-bearing spindle was used in the experiments, which were conducted for a variety of rake angles, uncut chip thicknesses, and cutting speeds. Chips were collected and examined in a scanning electron microscope (SEM). By examining the back surface of the chip (the side opposite the tool rake face), we were able to make quantitative measurements of the onset of serrated chip formation, as well as the dependence of average segment spacing Phil. Trans. R. Soc. Lond. A (2001)

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Figure 4. Back view of polished and etched chips from machining 52100 bearing steel at a number of parameter values (α = −27◦ , h = 30 µm).

(spatial frequency) on cutting speed. In addition, by polishing and etching the steel chip cross-sections with a solution of 2% nitric acid in ethyl alcohol, the qualitative characteristics of the material flow could be examined. Figures 2 and 4 show typical results for orthogonal machining of 52100 bearing steel with a rake angle α of −10◦ and an uncut chip thickness h of ca. 31 µm. Figure 2 shows polished and etched chip cross-sections at cutting speeds ranging from 0.7 m s−1 to 4.3 m s−1 . At low speeds, steady homogeneous shear is evident throughout the chip. As the speed is increased, the shear begins to localize, with regions of undeformed material becoming discernible at 1.5 m s−1 . For cutting speeds at and just above this critical speed, the segments form at irregular intervals (see the 1.5 m s−1 case). However, as the speed is increased, the segments become larger and their spacing becomes very regular. This variation in average segment spacing normalized to uncut chip thickness is plotted as a function of cutting speed in the figure. The shape of this curve is suggestive of a Hopf bifurcation occurring at a critical cutting speed between 1.0 m s−1 and 1.5 m s−1 . Experiments in cutting nickel–phosphorus showed similar behaviour. Figure 5 shows a qualitative comparison between two SEM micrographs of the serrated sides of two chips. Figure 5a shows a serrated chip of AISI 52100 cut at α = −27◦ , vr = 1.5 m s−1 and h = 31 µm, and figure 5b shows, at twice the magnification, a nickel–phosphorus chip, cut at α = −10◦ , vr = 2.1 m s−1 , and h = 16 µm. Both show easily discernible serrations of a similar spatial frequency when normalized to the respective chip thicknesses. In light of the dissimilarity of the microstructures of the two materials, the qualitative similarity between the chips is significant. The 52100 has a coarse-grain structure with hard carbides in a martensitic matrix and an average grain size of approximately a few tens of micrometres. By contrast, the nickel–phosphorus is amorphous. This provides motivation for modelling the materiPhil. Trans. R. Soc. Lond. A (2001)

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Figure 5. Qualitative comparison of (a) a serrated AISI 52100 steel chip, cut at α = −27◦ , h = 31 µm, and vr = 1.5 m s−1 ; (b) a serrated nickel–phosphorus chip, cut at α = −10◦ , h = 16 µm, and vr = 2.1 m s−1 .

mean normalized serration spacing

2.0 (i) 1.6

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Figure 6. Plots of normalized mean segment spacing as a function of cutting speed from several experiments on both 52100 steel and nickel–phosphorus. (i) AISI 52100, rake is −27◦ , h = 31 µm; (ii) Ni–Ph, rake is −10◦ , h = 16 µm; (iii) AISI 52100, rake is −10◦ , h = 31 µm; (iv) AISI 52100, rake is −27◦ , h = 16 µm.

als as continua. The normalized average segment spacings from several experiments on both the 52100 steel and the nickel–phosphorus at different parameter values are given in figure 6. As has already been discussed briefly, another significant finding of the experiments is an irregularity in the shear band pattern immediately following the bifurcation. This disorder can be quantified using the chip cross-sections. These cross-sections were digitally photographed, and Matlab (MathWorks 1999) image-processing routines were used to isolate the free edge. This free edge was then treated as a signal that recorded the dynamics of the chip-formation process. As an example, consider the two chip profiles shown in figure 7. Both were taken from chips cut with α = −27◦ and an uncut chip thickness of h = 31 µm. However, the chip that generated the profile shown in figure 7a was generated at a cutting speed of 1.5 m s−1 , while the chip that generated the profile shown in figure 7b was generated at a cutting speed of 4.3 m s−1 . We believe that this irregularity can be explained, for the first time, by correctly modelling the dynamics of the chip-formation process. This will be done in the next Phil. Trans. R. Soc. Lond. A (2001)

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x (µm) Figure 7. Profiles of chip cross-sections showing (a) disordered serrated chip, cut at 1.5 m s−1 , and (b) periodic serrated chip, cut at 4.3 m s−1 .

section. To help motivate this development, we first present the results of some finiteelement simulations of orthogonal cutting. (b) Numerical results Numerical simulations of orthogonal cutting of 1045 steel were carried out using the finite-element analysis (FEA) code AdvantEdge, of Third Wave Systems (1999). Figures 8–11 present a sequence of frames from a calculation with a tool rake angle of −25◦ , a depth of cut of 0.25 mm, and a cutting speed of 8.0 m s−1 , that nicely illustrates the repetitive shear band formation process. As the tool moves forward, a high-strain-rate, high-temperature zone remains present on the rake-face–chip interface along the line PQ in frame 1 (figure 8). This boundary layer is the so-called secondary shear zone. However, the deformation in the bulk of the material is cyclic. This cyclic pattern arises as follows: (1) before the formation of a localized shear band, the tooltip indents the cold material ahead of it, and stresses build up along the line AB (frame 1); Phil. Trans. R. Soc. Lond. A (2001)


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temperature (ºC) 997.78 922.56 847.34 772.13 696.91 621.69 546.47 471.25 396.03 320.81 245.60 170.38 95.16

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Figure 8. Finite-element simulation of chip formation in orthogonal cutting of AISI 1045 steel with a rake angle of −25◦ , a depth of cut of h = 0.25 mm, and a cutting speed of 8.0 m s−1 . Contours show the thermal field at a selected time after the cut-in phase. Frame 1.

(2) when the loads reach the cold yield point of the material, shear begins in the bulk material along a line AC (frame 2); (3) most of the plastic work is dissipated as heat and the material begins to heat up locally (frame 3); (4) the heating causes thermal softening that further concentrates the deformation zone (frame 4); (5) if the rate of convection and heat generation, which are generally both proportional to the cutting speed, are large compared to the heat conduction, the newly formed deformation zone falls behind the tooltip, and is carried away in the chip, as shown in frames 3 and 4. After the zone is carried away in the flow, the stage is set for the formation of another zone by the same sequence of events (frame 4). This sequence is the basis for the simplified mathematical model presented in the next section.

4. One-dimensional cutting model What may be called the conventional model of orthogonal machining was developed by Merchant (1944), independently of and roughly during the same time period as Phil. Trans. R. Soc. Lond. A (2001)

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M. A. Davies and T. J. Burns temperature (ºC) 997.78 922.56 847.34 772.13 696.91 621.69 546.47 471.25 396.03 320.81 245.60 170.38 95.16

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the closely related model of Piispanen (1937, 1948). This purely mechanical model is still in use today, because of its practical utility. Among the main assumptions of this model are the following (see Shaw 1984): cutting of a continuous chip with no built-up edge takes place in plane strain by a perfectly sharp tool that is wider than the workpiece and which does not contact the workpiece along the clearance face; the cutting surface is an infinitely thin plane that extends to the free surface of the workpiece material from the cutting edge of the tool; the cutting speed vr and the depth of cut h remain constant; the process is isothermal; there is no strain hardening in the workpiece; and the normal and shear stresses along the tool and the shear plane remain uniform. As Shaw points out, this latter condition arises from Merchant’s strength of materials approach to the problem. We make a number of modifications to the conventional model. Besides the pioneering work of Piispanen, Ernst, Merchant, and Recht, our thinking has been influenced by the recent application of ideas from nonlinear dynamics to problems in metal deformation (Kubin et al . 1982, 1984; Moon 1998, 1999), and to problems in chemical oscillations (Gray & Scott 1990; Scott 1991; Milik et al . 1998). We assume that the cutting region is a thin three-dimensional region of a material that hardens with increasing strain and softens with increasing temperature, and we allow stress gradients to evolve in the material. We also allow for the production of heat by the dissipation of plastic work, and the conduction of heat by Fourier’s law. Specifically, our orthogonal cutting model is based on the following list of assumptions. Here, a tilde over a variable denotes a dimensional quantity, and the same variable without a tilde denotes the corresponding dimensionless quantity. The nonPhil. Trans. R. Soc. Lond. A (2001)


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temperature (ºC) 997.78 922.56 847.34 772.13 696.91 621.69 546.47 471.25 396.03 320.81 245.60 170.38 95.16

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dimensionalizations are obtained by using the uncut chip thickness h as the lengthscale and the reciprocal of the nominal plastic strain rate, Φ0 , defined below, as the time-scale, together with some natural scalings that are also defined below. (1) The tool is perfectly sharp, rigid, and thermally insulated from the workpiece material. (2) The cutting takes place in a thin region of concentrated shear, bounded by planes, that extends from the nose of the tool to the free surface of the chip; material that lies ahead of this region is unstressed, but it can conduct heat; see figure 3. (3) The tool distributes a load in the workpiece material in a neighbourhood of the primary shear zone, over a contact length L (figure 3); this takes into account Recht’s (1964) discussion of the importance of tool pressure on shear-localized chip formation. (4) Initially, the compressive stress follows σ ˜ Hooke’s law, so that it is proportional to the compressive strain; this leads to an evolution equation, in which the rate of change of the compressive stress is proportional to the difference in the velocity of the tool in the direction of shear and the local velocity of the Phil. Trans. R. Soc. Lond. A (2001)

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M. A. Davies and T. J. Burns temperature (ºC) 997.78 922.56 847.34 772.13 696.91 621.69 546.47 471.25 396.03 320.81 245.60 170.38 95.16

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Figure 11. Frame 4.

workpiece material, divided by a scale length δ, vs − v˜ ∂σ ˜ =M ; δ ∂ t˜

(4.1)

here, M is the appropriate elastic modulus of the workpiece material. (5) The local deformation of the material by the tool is one dimensional; the tool compresses the material along the contact length L in the direction parallel to the planes bounding the zone of concentrated shear, and this deformation is elastic-perfectly plastic (see equation (4.2); here, 0 < β 1, so that the exponential term on the right-hand side is small except when σ ≈ 1). (6) The local compression eventually causes a larger-scale shearing deformation; this is also one-dimensional, and takes place on planar surfaces parallel to concentrated shear region; the momentum of this sheared material can be neglected; see equation (4.3). (7) The total shear strain rate, ∂v/∂y, is equal to the plastic strain rate, ∂εp /∂t, and this in turn determines the velocity v (see equations (4.4) and (4.6)) and displacement u (see equation (4.7)) in the direction parallel to planes bounding the cutting region. Phil. Trans. R. Soc. Lond. A (2001)


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(8) The plastic flow behaviour of the workpiece material exhibits strain hardening, thermal softening, and strain-rate sensitivity. In the present work, we have chosen a constitutive model of Arrhenius type, equation (4.10). While this commonly used phenomenological form is not essential, its use here serves to highlight the similarity of our high-speed metal-cutting model to mathematical models of open chemical reactors (Gray & Scott 1990; Scott 1991; Bebernes & Eberly 1989). (9) All of the plastic work is converted into heat, i.e. the Taylor–Quinney coefficient is set equal to one (Taylor & Quinney 1934); this heat is transferred by conduction, only in the direction normal to the planes bounding the concentrated shear region; see equation (4.5). The resulting mathematical model can be written in dimensionless form, as follows. σ−1 ∂σ −v , (4.2) = Ψ 1 − exp ∂t β ∂τ = ζσ (y > 0), (4.3) ∂y ∂v (4.4) = Φ(τ, εp , θ), ∂y ∂2θ ∂θ = Λ 2 + Qτ Φ(τ, εp , θ), (4.5) ∂t ∂y ∂εp = Φ(τ, εp , θ), (4.6) ∂t ∂u = v. (4.7) ∂t The dimensionless groups of parameters are defined by  √ ζ = 3 sin φ[1 + µ tan(α − φ)],      K   , Λ=  2  ρcΦ0 h    τy    Q= ,   ρcθ0 Mh  Ψ=√ cos(α − φ),    3τy δ     vs   Φ0 = ,   h    cos α   . vs = vr  cos(α − φ)

(4.8)

Here, σ is the compressive stress; τ is the shear stress; εp is the plastic strain; θ is the temperature; v is the material velocity; Φ is the plastic strain rate; u is the displacement; φ is the shear angle; α is the rake angle of the tool; µ is the coefficient of friction at the tool–work material interface; Φ0 = vs /h is the nominal plastic strain rate, where vs is the nominal shear speed, which in turn is determined by the cutting speed vr , the shear and rake angles, and the uncut chip thickness h; K, ρ, c and τy Phil. Trans. R. Soc. Lond. A (2001)

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are respectively the thermal conductivity, density, heat capacity, and equivalent flow stress (in shear) of the workpiece material; M is the elastic modulus; and δ is the length-scale for local compression. The specific parameter values that we have chosen for the present study are discussed at the end of the next section. In dimensionless variables, the constitutive model for plastic flow is assumed to be given by (4.9) τ = (1 + εp /εy )n + (1 + θ) ln(1 + ∂εp /∂t)m − νθ, ˜p /∂ t˜, and θ = (θ˜ − θ0 )/θ0 . Thus, the initial where τ = τ˜/τy , t = Φ0 t˜, ∂εp /∂t = Φ−1 0 ∂ε temperature is normalized to zero, and the yield stress at the initial temperature is normalized to one. The parameters εy , n, m, and ν are the yield strain (corresponding to τy ), strain-hardening, strain-rate-sensitivity, and the thermal-softening, respectively. Typically, 0 < n 1, and 0 < m 1, while ν is positive and of the order of magnitude of one. Solving equation (4.9) for the plastic strain rate leads to the following Arrhenius-type of equation, τ − F (εp , θ) ∂εp − 1, (4.10) = Φ(τ, εp , θ) = exp ∂t m(1 + θ) where the plastic flow surface F is defined by F (εp , θ) = (1 + εp /εy )n − νθ.

(4.11)

It is assumed here that τ 0. Whether or not plastic flow takes place at a given location in the workpiece material at a given time is then determined by the following criterion (Wallace 1981): ∂εp 0, if τ < F (εp , θ); = (4.12) ∂t Φ(τ, εp , θ), otherwise. For metals, the yield stress is several orders of magnitude smaller than any of the elastic moduli, so that Ψ 1. Using parameter values typical of iron alloys, we find that the dimensionless heat production coefficient Q is of order of one, while at the cutting speeds of interest here, the dimensionless heat transfer coefficient Λ is an order of magnitude smaller. We also note that Λ decreases with increasing cutting speed as well as with increasing chip thickness, whereas Q depends only on material parameters and the initial temperature. Thus, at higher cutting speeds, heat is conducted less rapidly away from the cutting region, so there is more pronounced thermal softening, which weakens the workpiece material in the cutting region to additional deformation. As will be shown below, this provides a mechanism, which depends on the cutting parameters, for the onset of oscillations in the stress and temperature fields in the workpiece material at the tooltip.

5. Numerical algorithm The model system of equations (4.2)–(4.7) consists of a coupled set of three partial differential equations, one parabolic and two elliptic, and four ordinary differential equations. Furthermore, it involves several time and spatial scales. Due to the complexity of the system, so far we have only been able to analyse its evolution by means of computer simulations. (However, in the case n = 0, i.e. when strain-hardening Phil. Trans. R. Soc. Lond. A (2001)


θ1

η1

tool

τ1

σ1

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y

u1

Figure 12. The discretization of our orthogonal cutting model looks like Piispanen’s ‘deck of cards’ model.

effects are ignored, we have been able make some lumped-parameter assumptions and derive a more tractable ordinary-differential-equations model (see Burns & Davies 1997).) In this section, we will discuss briefly the numerical algorithm we are currently using to study the evolution of the system, and in the following section, we will present the results of some simulations that we have performed using the algorithm. An unconventional one-dimensional reaction–diffusion equation, equation (4.5), governs the balance of energy. The problem is complicated by the fact that the tool is moving, and the behaviour of the material in the region of contact between the tool and the workpiece can change with time. Depending on the location of the tool and the local state of the workpiece material, the local compressive stress σ, whose evolution is governed by equation (4.2), can be either zero or positive at a given location in the material. As a result, the shear stress τ , which is determined by the elliptic equation (4.3), can also be either zero or positive. Thus, the source term in equation (4.5) can be either zero or positive as well. We convert the continuous model into a system of index-1 differential-algebraic equations (Brenan et al . 1996), by discretizing the spatial region of interest using an equally spaced grid. As in the early ‘card model’ of the cutting process developed by Piispanen (1937), this discretization can be viewed as approximating the deformation by a discrete set of elements, or lamellae, that resembles a deck of cards; see figure 12. A major new feature in our model is that thermal communication is allowed between the elements, which, we will show below, has an important influence on whether or not the cutting process produces continuous or segmented chips. Another new feature is that the elements are also coupled mechanically, in the sense that local compression of several adjacent elements by the tool leads to ‘global’, i.e. larger-scale shear of the workpiece material by means of plastic deformation. Initially, τ , v, θ, εp , σ, and u are set equal to zero, and the tool is located at one end of the grid. The evolution equations are solved on a set of equal timeintervals, where each interval is the time required for the tool to move across one spatial zone at the given cutting speed. At the start of each time interval, the shear stress τ is computed using equation (4.3), and the average temperature θ¯ and plastic strain ε¯p are computed for each zone, using the current values of σ, θ, and εp . Then, the plastic strain rate ∂εp /∂t is computed using the updated values of τ, ε¯p , and θ¯ in the constitutive equation (4.9). Using equation (4.6), the material velocity v is Phil. Trans. R. Soc. Lond. A (2001)

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then updated. The temperature θ and the local workpiece displacement u are then computed using equation (4.5) and equation (4.7), and a cycle is complete. The tool is then advanced by one zone, and the next cycle is computed. Because Ψ is large, the local compressive stress σ = σ ˜ /σy evolves on a much faster time-scale than do the temperature θ, plastic strain ε p , and displacement u. Here, √ σy = 3 τy is the equivalent (von Mises) flow stress of the workpiece material in tension. Since initially σ = 0 and v = 0, the exponential term in equation (4.2) is small, and σ increases linearly on the fast dimensionless time-scale Ψ t. Thus, σ rapidly approaches one, i.e. σ ˜ ≈ σy , and ceases to evolve. As the ‘local’ compressive stress builds up over several zones however, the ‘global’ shear stress τ increases near the shear zone, until it is large enough that the plastic strain rate Φ becomes significant, and plastic shear flow sets in locally. When this happens, the workpiece material begins to move away from the tool, causing the local compressive stress σ in equation (4.2) to decrease. Once v 1, the material is assumed to have separated from the tool, and is no longer considered to be part of the stress problem, although it is still allowed to conduct heat away from the shearing region. One of the biggest difficulties in modelling the response of materials at high strain rates is the lack of good experimental data. In the present case, we have no data for necessary coefficients like the strain-hardening parameter n, the strainrate-sensitivity parameter m, or the thermal-softening parameter ν, for either of the materials discussed in § 2. What we have done for the computer simulations of orthogonal machining of 52100 bearing steel that we will present in the next section is to use the algorithm just described with material parameters for the structural steel HY-100 (Burns 1990), which are based on some careful experimental studies of Marchand & Duffy (1988), with two exceptions. First, we have set the flow stress, τy , equal to 1.5 GPa, the value for the 52100 steel, which is about twice that for the HY-100 steel. This material constant enters the simulations through the dimensionless group Q in (4.8). Second, we have changed the value of the strain-hardening parameter n from Marchand & Duffy’s value of 0.107 to n = 0.015. Marchand and Duffy state that the larger value is valid only for plastic strains in the limited range 0.05 εp 0.20. By contrast, in high-speed machining, an order of magnitude of ten for εp is not unusual. In addition, we have set M equal to Young’s modulus for iron, and δ = h/10; while these choices are really educated guesses, we have found that computer simulations of the model are relatively insensitive to the values of these parameters, providing that the dimensionless group Ψ 1. Numerical values for the remaining parameters necessary to specify all of the dimensionless groups used in the simulations may be found in Davies et al . (1997).

6. Results of computer simulations Figure 13 summarizes the results of two computer simulations of our one-dimensional orthogonal cutting model for cutting speeds of vr = 1.0 m s−1 and vr = 4.0 m s−1 . For clarity, the initial transient cut-in behaviour is not included. In the upper four sets of plots (a)–(d), all of the dependent variables are plotted against the dimensionless spatial coordinate at the same time. The tooltip location is denoted by a dashed vertical line. In (e), the stress and temperature at the tooltip, denoted by τ ∗ and θ∗ , respectively, are plotted against dimensionless time, scaled so that the current time agrees with the tool position. It is clear that a transition takes Phil. Trans. R. Soc. Lond. A (2001)

Thermomechanical oscillations vr = 1.0 m s−1 (a)

(b)

(c)

(d)

(e)

vr = 4.0 m s−1

1.0

1.0

0.5

0.5

0

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2

10

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0

15

15

10

10

5

5

0

0

1

1

0 30

839

34

38

42

0 30

34

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Figure 13. Dimensionless dependent variables of orthogonal cutting model; in (a)–(d), the horizontal axis is the dimensionless spatial coordinate, all fields are plotted at a fixed time, and the location of the tooltip is indicated by a dashed vertical line. In (a), the solid curve is the flow surface, F (4.11), the crosses are the local compressive stress, σ, and the open circles are the shear stress, τ . The plastic strain, εp , is plotted in (b). In (c), the solid curves are the velocity, v, and the dashed curves are the temperature, θ. Displacement u is plotted in (d). In (e), stress, τ ∗ (solid curves), and temperature, θ∗ (dashed curves), at the tooltip, are plotted as functions of scaled dimensionless time. The slower cutting speed of vr = 1.0 m s−1 (left column) corresponds to continuous chip formation at the tooltip, a steady, non-oscillatory dynamic equilibrium state of the system. The faster cutting speed of vr = 4.0 m s−1 (right column) corresponds to periodic serrated chip formation, an oscillatory dynamic equilibrium state of the system.

place from continuous, steady-state chip formation to extremely regular periodic serrated chip formation in the cutting process. Another interesting feature of these two simulations is that, at the lower cutting speed (left column of plots), the maximum temperature in the workpiece material is located very close to the tooltip; see (c). Thus, as uncut material enters the cutting region, the tool ‘sees’ material that has been preheated by conductive heat transfer, and the cutting takes place near the tip of the tool, as long as the cutting is not too fast. At the four times higher cutting speed depicted in the second column of plots, however, much less heat is transferred to the uncut material ahead of the tool, Phil. Trans. R. Soc. Lond. A (2001)

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M. A. Davies and T. J. Burns τ* 1.0

2

(a) 0.5

1

0 1.0

0 2

(b) 0.5

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0 2

(c) 0.5

1

0 1.0

0 2

(d) 0.5

1

0

10

20 time

30

40

0

θ*

10

20 time

30

40

Figure 14. Behaviour with respect to increasing cutting speed, vs , of dimensionless shear stress, τ ∗ , and temperature, θ∗ , at the tooltip as functions of dimensionless time. A transition takes place from constant stress and temperature to oscillatory behaviour. At first, the oscillations are irregular, but they quickly become periodic. (a) vr = 1.2 m s−1 ; (b) vr = 1.4 m s−1 ; (c) vr = 1.6 m s−1 ; (d) vr = 1.8 m s−1 .

so that this material has a higher flow stress. The result is that, after a sufficiently large gradient in shear stress has been set up in the material by the local compressive stresses, cutting takes place initially at the tooltip. According to the model, once this begins to happen, the material locally becomes hotter, with the result that the shear stress decreases, the deformation becomes unstable, and an adiabatic shear band forms in a thin region of material. Meanwhile, the tool continues to move into cooler material, and the shear stress is no longer above the local flow surface, so plastic deformation ceases to take place at the tooltip. However, there is a gradient in the shear stress, and the lower stress further up the face of the tool is sufficient to continue cutting the hotter material that is initially cut by the tooltip. Eventually, the shear stress behind the tooltip is too small anywhere for plastic deformation to continue, so by equation (4.2), the material unloads. The tool continues to move and to compress new material, until the shear stress once again builds up sufficiently for rapid, unstable plastic flow to take place. However, as indicated in the next series of plots, figure 14, this transition to cyclic serrated chip formation is more complicated Phil. Trans. R. Soc. Lond. A (2001)


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τ*

θ*

0.62

0.66

(a) 0.60

0.62

0.62

0.66

(b) 0.60

0.62

0.66 0.7 (c) 0.62 0.6 0.58 0.5 0.66

0.7

(d) 0.62

0.6

0.58 10

15

20

25 y

30

35

0.5 10

15

20

25

30

35

y

Figure 15. Complex behaviour of dimensionless shear stress τ ∗ and temperature θ∗ at the tooltip, as the steady-state behaviour of continuous chip formation loses stability to a dynamic periodic oscillatory state of the system; both mixed-mode and apparently chaotic oscillations occur in the system. (a) vr = 1.39 m s−1 ; (b) vr = 1.4 m s−1 ; (c) vr = 1.425 m s−1 ; (d) vr = 1.45 m s−1 .

than the kind of behaviour associated with a typical supercritical Hopf bifurcation. This is a new feature that we discovered upon adding a strain-hardening effect to our model. Initially, the transition to stable periodic behaviour involves what appear to be, at least approximately, mixed-mode oscillations consisting of a repeating unit consisting of both (relatively) large and small amplitudes; see figure 15, where the stress and temperature at the tooltip are plotted on a smaller scale. The range of speeds over which this behaviour occurs is quite narrow. Just before the cutting speed is increased to 1.425 m s−1 , however, the system starts to exhibit what appear to be chaotic oscillations, just as have been observed in orthogonal cutting experiments (see figure 2). This ‘chaotic window’ persists with increasing cutting speed up to ca. 1.8 m s−1 . At this speed, a pattern of periodic oscillations sets up in the timeseries. As the cutting speed is increased further, the oscillations remain periodic but evolve into large-amplitude oscillations with two distinct time-scales, as is characteristic of relaxation oscillations. The distance between the shear-localized regions gradually increases up until a cutting speed of ca. 6.0 m s−1 , at which point the spacing appears to reach an asymptotic limit of about twice the depth of cut. The spacing given in figure 17 is based on the sequence of four computations summarized in figPhil. Trans. R. Soc. Lond. A (2001)

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M. A. Davies and T. J. Burns τ* 1.0

2

(a) 0.5

1

0 1.0

0 2

(b) 0.5

1

0 1.0

0 2

(c) 0.5

1

0 1.0

0 2

(d) 0.5

1

0

10

20 time

30

40

0

θ*

10

20 time

30

40

Figure 16. Dimensionless shear stress τ ∗ and temperature θ∗ at the tooltip as the periodic oscillations grow in amplitude and decrease in frequency with increasing cutting speed, becoming relaxation oscillations at higher speeds. (a) vr = 2.0 m s−1 ; (b) vr = 4.0 m s−1 ; (c) vr = 8.0 m s−1 ; (d) vr = 16.0 m s−1 .

ure 16. In addition to reproducing the behaviour that is observed in metal-cutting experiments (figure 2), the model’s behaviour is remarkably similar to what has been observed in numerical studies of some chemical reactor models (see Gray & Scott 1990; Scott 1991; Milik et al . 1998). The steady-state loses stability as a control parameter is adjusted monotonically, and after a complex but rapid transition that involves both mixed-mode and chaotic time-series, an exchange of stability with a periodic dynamic equilibrium state of the system takes place. By plotting the displacement of the workpiece material as it moves up the face of the tool, the chip shapes at various cutting speeds can be visualized, as in figure 18.

7. Discussion and concluding remarks We have taken a nonlinear dynamics approach to modelling the problem of shearlocalized chip formation during metal cutting. We have presented a new model that includes a mechanism for oscillations in the material flow to occur. Furthermore, we have presented numerical results that strongly support the hypothesis that the rapid transition from continuous to segmented chip formation is due to a supercritical Phil. Trans. R. Soc. Lond. A (2001)


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normalized serration spacing

2.0

1.6

1.2

0.8

0.4

0

4

8 cutting speed (m s−1)

12

Figure 17. Plots of periodic segment spacing as a function of cutting speed, from the computer simulations of the one-dimensional model given in figure 16. Irregular spacing at lower speeds is not included. The spacing asymptotes at approximately twice the depth of cut at the higher cutting speeds.

(a)

(b)

(c)

(d)

Figure 18. Plots of displacement at final time, from simulations of the one-dimensional chip model at different cutting speeds: (a) vr = 1.2 m s−1 , continuous chip; (b) vr = 1.6 m s−1 , irregular segmental chip; (c) vr = 2.0 m s−1 , periodic segmental chip; (d) vr = 4.0 m s−1 , periodic segmental chip.

bifurcation in the flow behaviour of the workpiece material as it is deformed by the cutting tool. This bifurcation occurs because, as the cutting speed is increased, the tool eventually ‘outruns’ the thermal front in the workpiece. The thermal front results from the large temperature gradient that occurs in a thin concentrated shear layer in the material, due to the very high-strain-rate plastic working that is associated with high-speed metal cutting. For the first time, we believe that we have explained the irregularity in the chip profiles that is observed during the transition from continuous to shear-localized chip formation. We have done this by correctly modelling the dynamics of the chipformation process, as described in § 4. The results presented here support the hypothesis that the irregular motions are the result of an instability that is due to a competiPhil. Trans. R. Soc. Lond. A (2001)

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tion between two strongly nonlinear effects, strain hardening and thermal softening. Without strain hardening, we have found that the instability is more like a highly stiff supercritical Hopf bifurcation that produces only periodic oscillations, which rapidly grow into relaxation oscillations (Davies et al . 1997; Burns et al . 1999). We acknowledge the important contributions that Chris Evans of the NIST Manufacturing Engineering Laboratory has made to this effort since its inception. In addition, M.A.D. acknowledges the partial support of the National Research Council Postdoctoral Fellowship Program. Certain commercial products are identified in this paper in order to document computational experiments adequately. Identification of such products does not constitute endorsement by NIST, nor does it imply that these are the most suitable products for the task. US Government work in the public domain in the United States.

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