Hybrid analytical–numerical solution for the shear angle in orthogonal metal cutting — Part II: experimental verification

The hybrid analytical–finite element model described in Part I is applied to predict the shear angle for a range of cutting velocity, uncut chip thickness, and two tool orthogonal rake angles. Experimental results and an empirical equation are also presented for the influence of the cutting conditions and cutting tool geometry on the chip–tool contact length. It is shown that there is a linear dependence between the chip–tool contact length/uncut chip thickness ratio and chip thickness/uncut chip thickness ratio over the range of cutting conditions assumed. The increase of the shear angle with the tool orthogonal rake is mostly due to the reduction of the specific shear energy in the primary shear zone and the specific friction energy in the secondary shear zone accompanied by a reduction of the chip–tool contact zone. The uncut chip thickness and cutting velocity influence the shear angle through their effect on the interface temperature and hence on the material flow stress in the secondary shear zone. The change in both parameters does not change significantly the specific shear energy in the primary shear zone. The model results are compared with the experimental results for a work material 0.18% C steel. The agreement between the predicted and experimental results is seen to be exceptionally good.     

Crystal anisotropy-dependent shear angle variation in orthogonal cutting of single crystalline copper

Shear deformation that dominates elementary chip formation in metal cutting greatly relies on crystal anisotropy. In the present work we investigate the influence of crystallographic orientation on shear angle in ultra-precision orthogonal diamond cutting of single crystalline copper by joint crystal plasticity finite element simulations and in-situ experiments integrated in scanning electron microscope. In particular, the experimental cutting conditions including a straight cutting edge are the same with that used in the 2D finite element simulations. Both simulations and experiments demonstrate a well agreement in chip profile and shear angle, as well as their dependence on crystallography. A series of finite element simulations of orthogonal cutting along different cutting directions for a specific crystallographic orientation are further performed, and predicated values of shear angle are used to calibrate an extended analytical model of shear angle based on the Ernst–Merchant relationship.     

Thermal modeling of the metal cutting process: Part I — Temperature rise distribution due to shear plane heat source

The model of an oblique band heat source moving in the direction of cutting, first introduced by Hahn (Proceedings of First U.S. National Congress of Applied Mechanics 1951. p. 661–6) for an infinite medium in 1951 and subsequently modified by Chao and Trigger (Transactions of ASME 1953;75:109–20) in 1953 for a semi-infinite medium, is extended in this investigation by including appropriate image heat sources. It is used for the determination of the temperature rise distribution in the chip and the work material near the shear plane caused by the main shear plane heat source in orthogonal machining of a continuous chip. A new approach is taken in that the analysis is made in two separate parts, namely, the workmaterial side and the chip side of the shear plane and then combined. The workmaterial (or the chip) is extended past the shear plane as an imaginary region for continuity to determine the temperature distribution in the workmaterial (or the chip) near the shear plane. The imaginary regions are the regions either of the workmaterial that was cut by the cutting tool prior to this instance and became the chip or will be cut by the cutting tool prior to becoming the chip. An appropriate image heat source with the same intensity as the shear plane heat source is considered for each case. The temperature distributions in the chip and the workmaterial were determined separately by this method and combined to obtain isotherms of the total temperature distribution (and not merely the average temperatures). It appears that the significance of Hahn's ingenious idea and his general solution have not been fully appreciated; instead, an approximate approach involving heat partition between the chip and the work was frequently used (Trigger and Chao. Transactions of ASME 1951;73:57–68; Loewen and Shaw. Transactions of ASME 1954;71:217–31; Leone. Transactions of ASME 1954;76:121–5; Nakayama. Bulletin of the Faculty of Engineering National University of Yokohama, Yokohama, Japan, 1956;21:1–5; Boothroyd. Proceedings of the Institution of Mechanical Engineers (Lon) 1963;177(29):789–810). It may be noted that in utilizing Hahn's modified solution, it is not necessary to make an explicit a priori assumption regarding partitioning of heat between the workmaterial and the chip, as was common in most prior work. Instead, this information is provided as part of the solution. The results obtained with the exact analysis were compared with other methods using the experimental data available in the literature to point out some of the discrepancies in the simplified models. It may be pointed out that these models assume the temperature rise at the chip–tool interface to be nearly uniform and equals the average temperature rise in this volume. A comparison of the calculated temperature rise by these methods with the exact analysis indicates that the differences can be quite significant (50% or higher). It is hoped that future researchers would recognize the significance and the versatility of the exact analysis in determining the temperature distribution in the shear zone in metal cutting.     

The Merchant's model of orthogonal cutting revisited: A new insight into the modeling of chip formation

The Merchant's model, as the most famous approach of orthogonal cutting, is widely used in introductive courses on machining. However, the shear angle predicted by the Merchant's model from the criterion of minimization of the cutting energy, does not generally agree with experimental data and numerical simulations. The aim of this paper is to elucidate the theoretical reason for which the Merchant's model fails to predict the correct orientation of the primary shear zone. It is shown that the principle of minimum of the cutting energy must be supplemented by a stability criterion of the chip morphology. A modified Merchant's formula is then obtained for the value of the shear angle.     

Thermal modeling of the metal cutting process — Part II: temperature rise distribution due to frictional heat source at the tool–chip interface

Heat partition and the temperature rise distribution in the moving chip as well as in the stationary tool due to frictional heat source at the chip–tool interface alone in metal cutting were determined analytically using functional analysis. An analytical model was developed that incorporates two modifications to the classical solutions of Jaeger's moving band (for the chip) and stationary rectangular (for the tool) heat sources for application to metal cutting. It takes into account appropriate boundaries (besides the tool–chip contact interface) and considers non-uniform distribution of the heat partition fraction along the tool–chip interface for the purpose of matching the temperature distribution both on the chip side and the tool side. Using the functional analysis approach, originally proposed by Chao and Trigger (Transactions of ASME, 1951; 73:57–68), a pair of functional expressions for the non-uniform heat partition fraction along the tool–chip interface — one for the moving band heat source (for the chip side) and the other for the stationary rectangular heat source (for the tool side) were developed. Using this analysis, the temperature rise distribution in the chip and the tool were determined for two cases of machining, namely, conventional machining of steel with a carbide tool at high Peclet number (N_Pe≈5–20) and ultraprecision machining of aluminum with a single-crystal diamond tool at low Peclet number (N_Pe–0.5). The calculated temperature rise distribution curves on the two sides of the tool–chip interface are found to be well matched for both cases. The analytical method developed was found to be much faster, easier to use, and more accurate than various numerical methods used earlier. Further, the model provides a better physical appreciation of the thermal aspects of the metal cutting process.     

Thermal modeling of the metal cutting process — Part III: temperature rise distribution due to the combined effects of shear plane heat source and the tool–chip interface frictional heat source

This paper is Part III of a 3-part series on the Thermal Modeling of the Metal Cutting Process. In Part I (Komanduri, Hou, International Journal of Mechanical Sciences 2000;42(9):1715–1752), the temperature rise distribution in the workmaterial and the chip due to shear plane heat source alone was presented using modified Hahn's moving oblique band heat source solution with appropriate image sources for the shear plane (Hahn, Proceedings of the First US National Congress of Applied Mechanics 1951. p. 661–6). In Part II (Komanduri, Hou, International Journal of Mechanical Sciences 2000;43(1):57–88), the temperature rise distribution due to the frictional heat source at the tool–chip interface alone is considered using the modified Jaeger's moving-band (in the chip) and stationary rectangular (in the tool) heat source solutions (Jaeger, Proceedings of the Royal Society of New SouthWales, 1942;76:203–24; Carlsaw, Jaeger. Conduction of heat in solids, Oxford, UK: Oxford University Press, 1959) with appropriate image sources and non-uniform distribution of heat intensity. The matching of the temperature rise distribution at the tool–chip contact interface for a moving-band (chip) and a stationary rectangular heat source (tool) was accomplished using functional analysis technique, originally proposed by Chao and Trigger (Transactions of ASME 1955;75:1107–21). This paper (Part III) deals with the temperature rise distribution in metal cutting due to the combined effect of shear plane heat source in the primary shear zone and frictional heat source at the tool–chip interface. The basic approach is similar to that presented in Parts I and II. The model was applied to two cases of metal cutting, namely, conventional machining of steel with a carbide tool at high Peclet numbers (≈5–20) using data from Chao and Trigger (Transactions of ASME 1955;75:1107–21) and ultraprecision machining of aluminum using a single-crystal diamond at low Peclet numbers (≈0.5) using data from Ueda et al. (Annals of CIRP1998;47(1):41–4). The analytical results were found to be in good agreement with the experimental results, thus validating the model. Using relevant computer programs developed for the analytical solutions, the computation of the temperature rise distributions in the workmaterial, the chip, and the tool were found. The analytical method was found to be much easier, faster, and more accurate to use than the numerical methods used (e.g., Dutt, Brewer, International Journal of Production Research 1964;4:91–114; Tay, Stevenson, de Vahl Davis, Proceedings of the Institution of Mechanical Engineers (London) 1974;188:627). The analytical model also provides a better physical understanding of the thermal process in metal cutting.     

A study on ultra–high-speed cutting of aluminium alloy:: Formation of welded metal on the secondary cutting edge of the tool and its effects on the quality of finished surface

To investigate the mechanism of ultra–high-speed cutting for aluminium alloy, cutting experiments by using a machine tool equipped with an active magnetic bearing spindle were performed over a range of cutting speeds from 20 to 260 m/s. On the whole, the finished surface tends to improve with an increase in cutting speed. However, the formation of welded metal on the tool edge in the speed range from 100 to 200 m/s promotes the material side flow on the finished surface, which causes the surface roughness to increase.     

Development of a Wittrick–Williams algorithm for the spectral element model of elastic–piezoelectric two-layer active beams

In this paper, a Wittrick–Williams algorithm is developed for the elastic–piezoelectric two-layer active beams. The exact dynamic stiffness matrix (or spectral element matrix) is used for the development. This algorithm may help calculate all the required natural frequencies, which lie below any chosen frequency, without the possibility of missing any due to close grouping or due to the sign change of the determinant of spectral element matrix via infinity instead of via zero. The uniform and partially patched active beams are considered as the illustrative examples to confirm the present algorithm.     

CASINO: A new monte carlo code in C language for electron beam interaction —part I: Description of the program

This paper is a guide to the ANSI standard C code of CASINO program which is a single scattering Monte CArlo SImulation of electroN trajectory in sOlid specially designed for low-beam interaction in a bulk and thin foil. CASINO can be used either on a DOS-based PC or on a UNIX-based workstation. This program uses tabulated Mott elastic cross sections and experimentally determined stopping powers. Function pointers are used for the most essential routine so that different physical models can easily be implemented. CASINO can be used to generate all of the recorded signals (x-rays, secondary, and backscattered) in a scanning electron microscope either as a point analysis, as a linescan, or as an image format, for all the accelerated voltages (0.1–30 kV). As an example of application, it was found that a 20 nm Guinier-Preston Mg₂Si in a light aluminum matrix can, theoretically, be imaged with a microchannel backscattered detector at 5 keV with a beam spot size of 5 nm.     

Three-parameter approach for elastic–plastic fracture of the semi-elliptical surface crack under tension

Systematic three-dimensional elastic–plastic finite element analyses are carried out for a semi-elliptical surface crack in plates under tension. Various aspect ratios (a/c) of three-dimensional fields are analyzed near the semi-elliptical surface crack front. It is shown that the developed J–Q annulus can effectively describe the influence of the in-plane stress parameters as the radial distances (r/(J/σ₀)) are relatively small, while the approach can hardly characterize it very well with the increase of r/(J/σ₀) and strain hardening exponent n. In order to characterize the important stress parameters well, such as the equivalent stress σ_e, the hydrostatic stress σ_m and the stress triaxiality R_σ, the three-parameter J–Q_T–T_z approach is proposed based on the numerical analysis as well as a critical discussion on the previous studies. By introducing the out-of-plane stress constraint factor T_z and the Q_T term, which is determined by matching the finite element analysis results, the J–Q_T–T_z solution can predict the corresponding three-dimensional stress state parameters and the equivalent strain effectively in the whole plastic zone. Furthermore, it is exciting to find that the values of J-integral are independent of n under small-scale yielding condition when the stress-free boundary conditions at the side and back surfaces of the plate have negligible effect on the stress state along the crack front, and the normalized J tends to a same value when φ equals about 31.5° for different a/c and n. Finally, the empirical formula of T_z and the stress components are provided to predict the stress state parameters effectively.