Metal Machining - Theory and Applications 2008 Part 12 pps

and side views of the stock to be removed by a two-flute square end mill 16 mm in diame- ter, rotating with a spindle speed of 350 rev/min. When the feed rate was set at 100 mm/min in a trial cut, the dimensional error varied with large amplitude, as shown in Figure 9.37(b). Then, using an equation similar to equation (9.44a), the feed was adjusted as follows: E critical f a,error = ——— f (9.44c) E siml where f a,error is the feed adjusted for the limit of dimensional error E critical , and E siml is the error simulated under the trial conditions. Figures 9.37(c), (d) and (e) show the adjusted feed rate, measured error, and simulated and measured cutting forces. The dimensional error is almost constant over the workpiece as expected. The simulated and measured cutting forces show good agreement Figure 9.38 shows the principle of a second use of the machining scenario, to diagnose faults in an operation. A fault may be excessive tool wear, tool breakage, chatter vibration, tangling of chips, incorrect workpiece positioning, incorrect tool geometry, workpiece geometry incorrectly pre-machined, incorrect tool preset, among others. In any case, it will cause the measured force variation with time to differ from the expected one. If a measured wave form differs from the expected one by more than a set amount, a fault hypothesis library is activated. It holds information on how different types of fault may be expected to change an expected pattern. A fault simulation routine modifies the expected pattern accordingly. This is compared with the measured pattern and a fault diagnosis is produced from the best match between measured and simulated alternatives. Model-based systems for simulation and control 323 Fig. 9.38 Diagnosis procedure for faulty machining states (Takata, 1993) Childs Part 3 31:3:2000 10:41 am Page 323 To demonstrate the system’s abilities, the workpiece shown in Figure 9.39 was prepared and machined instead of the intended workpiece shown in Figure 9.37(a). The diagnosis system detected the difference between the two workpieces when the centre of the end mill had travelled 37 mm from the left end. It diagnosed the force error as arising from too small an axial depth of cut and that this was due to an error in the workpiece shape. Details of the comparator algorithm are given in Takata (1993). 9.5.3 Conclusions A huge number of experiments have been carried out and many theoretical approaches have been developed to support machining technologies. Nevertheless, it is often felt that the available experimental and theoretical data are insufficient for determining the machin- ing conditions for a particular workpiece and operation. These days, partly because of a decrease in the number of experts and partly because of the demands of unmanned and highly flexible machining systems, machine tool systems are expected to have at least a little intelligence to assist decision making. For this purpose, expert systems for determining initial cutting conditions and cutting state monitoring tech- nologies are increasingly being implemented. Up to now, monitoring technologies in partic- ular have been intensively studied for maintaining trouble-free machining. Nowadays, they are regarded as indispensable in the development of intelligent machining systems. However, machining systems have not yet been equipped with effective functions for diag- nosing and settling machining troubles and revising cutting conditions by themselves. To develop such a system, prediction, control, design and monitoring of cutting processes should be integrated by sharing the same information on cutting states. A model-based system, with advanced process models, provides a way of enabling that integration. This integration will help the further development of autonomous and distributed machining systems with increased intelligence and flexibility. The theory of machining can contribute greatly to this. References Akaike, H. (1974) A new look at the statistical model identification. IEEE Trans. on Auto. Cont. 19, 716–722. ASM (1990) Classification and designation of carbon and low-alloy steels. In ASM Handbook 10th edn. Vol. 1, 140–194. Ohio: American Society of Metals. 324 Process selection, improvement and control Fig. 9.39 Diagnosis of machining a different workpiece (Takata, 1993) Childs Part 3 31:3:2000 10:41 am Page 324 Barr, A. and Feigenbaum, E. A. (eds) (1981) The Handbook of Artificial Intelligence Vol. I. Los Altos: William Kaufmann. Barr, A. and Feigenbaum, E. A. (eds) (1982) The Handbook of Artificial Intelligence Vol. II. Los Altos: William Kaufmann. Bedini, R. and Pinotti, P. C. (1982) Experiments on adaptive constrained control of a CNC lathe. Trans. ASME. J. Eng. Ind. 104, 139–149. Blum, T. and Inasaki, I. (1990) A study on acoustic emission from the orthogonal cutting process. Trans. ASME J. Eng. Ind. 112, 203–211. Boothroyd, G. and Knight, W. A. (1989) Fundamentals of Machining and Machine Tools. New York: Marcel Dekker. British Standard (1991) BS 970: Part 1: Wrought Steels for Mechanical and Allied Engineering Purposes. London: British Standards Institution. Budak, E., Altintas, Y. and Armarego, E. J. A. (1996) Prediction of milling force coefficients from orthogonal cutting data. Trans. ASME. J. Manufact. Sci. Eng. 118, 216–224. Byrne, G., Dornfeld, D., Inasaki, I., Ketteler, G., Konig, W. and Teti, R. (1995) Tool condition monitoring (TCM) – the status of research and industrial application. Annals CIRP 44(2), 541–567. Carpenter, G. and Gossberg, S. (1987) ART 2: self-organisation of stable category recognition codes for analog input patterns. Appl. Opt. 26, 4919–4930. Centner, R. M. and Idelsohn, J. M. (1964) Adaptive controller for a metal cutting process. IEEE Trans. Appl. Ind. 83, 154–161. Chen, S. J., Hinduja, S. and Barrow, G. (1989) Automatic tool selection for rough turning operation. Int. J. Machine Tools Manufact. 29, 535–553. Chen, Y. B., Sha, J. L. and Wu, S. M. (1990) Diagnosis of the tapping process by information measure and probability voting approach. Trans. ASME J. Eng. Ind. 112, 319–325. Chen, Y., Hui, A. and Du, R. (1995) A fuzzy expert system for the design of machining operations. Int. J. Machine Tools Manufact. 35, 1605–1621. Cheng, D. K. (1972) Analysis of Linear Systems. Reading: Addison-Wesley. Chryssolouris, G., Domroese, M. and Beaulieu, P. (1992) Sensor synthesis for control of manufac- turing processes. Trans ASME J. Eng. Ind. 114, 158–174. Dan, L. and Mathew, J. (1990) Tool wear and failure monitoring techniques for turning – a review. Int. J. Machine Tools Manufact. 30, 579–598. Daubechies, I. (1988) Orthonormal bases of compactly supported wavelets. Comm. Pure Applied Math. 41, 909–996. Diniz, A. E., Liu, J. J. and Dornfeld, D. A. (1992) Correlating tool life, tool wear and surface rough- ness by monitoring acoustic emission in finish turning. Wear 152, 395–407. Dornfeld, D. A. (1990) Neural network sensor fusion for tool condition monitoring. Annals of the CIRP 39, 101–104. Ghasempoor, A., Moore, T. N. and Jeswiet, J. (1998) On-line wear estimation using neural networks. Proc. I. Mech. E. Lond. 212 Pt.B, 105–112. Giusti, F., Santochi, M. and Dini, G. (1986) COATS: An expert module for optimal tool selection. Annals of the CIRP 35, 337–340. Gong, W., Obikawa, T. and Shirakashi, T. (1997) Monitoring of tool wear states in turning based on wavelet analysis. JSME Int. J. (Series C) 40, 447–453. International Standard (1991) ISO 1832 3rd edn.: Indexable Inserts for Cutting Tools – Designation . Geneva: International Organization for Standardization. International Standard (1995) ISO 5608: Turning and Copying Tool Holders and Cartridges for Indexable Inserts – Designation. Geneva: International Organization for Standardization. Kasashima, N., Mori, K. and Ruiz, G. H. (1994) Diagnosing cutting tool conditions in milling using wavelet transform. In Usui, E. (ed.), Advancement of Intelligent Production. Amsterdam: Elsevier Science BV, 339–344. References 325 Childs Part 3 31:3:2000 10:41 am Page 325 Kline, W. A. and DeVor, R. E. (1983) The effect of runout on cutting geometry and forces in end milling. Int. J. Mach. Tool Des. Res. 23, 123–140. Kline, W A., DeVor, R. E. and Lindberg, J. R. (1982) The prediction of cutting forces in end milling with application to cornering cuts. Int. J. Mach. Tool Des. Res. 22, 7–22. Ko, T. J. and Cho, D. W. (1994) Tool wear monitoring in diamond turning by fuzzy pattern recogni- tion. Trans ASME J. Eng. Ind. 116, 225–232. Koornwinder, T. H. (ed) (1993) Wavelets: An Elementary Treatment of Theory and Application. Singapore: World Scientific. Konig, W., Langhammer, K. and Schemmel, H. U. (1972) Correlation between cutting force compo- nents and tool wear. Annals of the CIRP 21, 19–20. Koren, Y., Ko, T. R., Ulsoy, A. G. and Danai, K. (1991) Flank wear estimation under varying cutting conditions. Trans ASME J. Dyn. Sys. Meas. Control 113, 300–307. Leem, C. S., Dornfeld, D. A. and Dreyfus, S. E. (1995) A customised neural network for sensor fusion in on-line monitoring of cutting tool wear. Trans ASME J. Eng. Ind. 117, 152–159. Liu, T. I. and Anantharaman, K. S. (1994) Intelligent classification and measurement of drill wear. Trans ASME J. Eng. Ind. 116, 392–397. Matsushima, K. and Sata, T. (1974) On-line control of the cutting state by the pattern recognition technique. Annals of the CIRP 23, 151–152. Micheletti, G. F., Koenig, W. and Victor, H. R. (1976) In process tool wear sensors for cutting oper- ations. Annals of the CIRP 25, 483–496. Moriwaki, T. and Tobito, M. (1990) A new approach to automatic detection of life of coated tool based on acoustic emission measurement. Trans. ASME J. Eng. Ind. 112, 212–218. Moriwaki, T. and Mori, Y. (1993) Recognition of cutting state based on neural network sensor fusion. J. Japan Soc. Prec. Eng. 59, 779–784. Moriwaki, T, Shamoto, E., Kou, D. and Sugihara, K. (1995) Milling state recognition based on dynamic cutting force model (1st report). Trans. Japan Soc. Mech. Eng. 61-C, 2592–2598. Niu, Y. M., Wong, Y. S., Hong, G. S. and Liu, T. I. (1998) Multi-category classification of tool condi- tions using wavelet packets and ART2 network. Trans. ASME J. Manufact. Sci. Eng. 120, 807–816. Obikawa, T. and Matsumura, T. (1994) Visualization, adaptation and optimization of machining technology. J. Japan Soc. Prec. Eng. 60, 641–646. Obikawa, T., Liu, W. N. and Shirakashi, T. (1995) Prediction of cutting temperature and flank wear rate for practical use using personal computer. Proc. 7th Int. Manufac. Conf. China, Harbin, 2, 55–60. Obikawa, T., Kaseda, C., Matsumura, T., Gong, W. G. and Shirakashi, T. (1996) Tool wear monitor- ing for optimizing cutting conditions. J. Mat. Proc. Tech. 62, 374–379. Oraby, S. E. and Hayhurst, D. R. (1991) Development of models for tool wear force relationships in metal cutting. Int. J. Mech. Sci. 33, 125–138. Paul, B. and Mirandy, L. (1976) An improved fracture criterion for three-dimensional stress states. Trans. ASME, J. Eng. Mat. Tech. 98, 159–163. Rangwala, S. and Dornfeld, D. (1990) Sensor integration using neural networks for intelligent tool condition monitoring. Trans. ASME J. Eng. Ind. 112, 219–228. Rosenblatt, F. (1961) Principles of Neurodynamics: Perceptron and the Theory of Brain Mechanics . Washington DC: Spartan Books. Ryabov, O., Mori, K., Kasashima, N and Uehara, K. (1996) An in-process direct monitoring method for milling tool failures using a laser sensor. Annals of the CIRP 45(1), 97–100. Sata, T., Matsushima, K., Nagakura, T. and Kono, E. (1973) Learning and recognition of the cutting states by the spectrum analysis. Annals of the CIRP 22, 41–42. Shaw, M. C. (1984) Metal Cutting Principles. New York: Oxford University Press. Shinozuka, J. (1998) Analytical prediction of cutting performance of grooved rake face tools. PhD Thesis. Tokyo: Tokyo Institute of Technology. 326 Process selection, improvement and control Childs Part 3 31:3:2000 10:41 am Page 326 Shinozuka, J., Obikawa, T. and Shirakashi, T. (1994) Cutting performance of tools with curved rake face. In Usui, E. (ed), Advancement of Intelligent Production. Amsterdam: Elsevier Science BV, 379–384. Shirakashi, T., Ihara, T., Kanazawa, K. and Usui, E. (1987) Analytical prediction of chipping occur- rence of carbide tool in interrupted turning operation with temperature rise (1st report). J. Japan Soc. Prec. Eng. 53, 1589–1595. Shirakashi, T., Obikawa, T. and Shinozuka, J. (1998) Development of intelligent milling head. In Chen, D. et al. (eds), Progress of Cutting and Grinding. Beijing: International Academic Publishers, pp. 344–349. Shirakashi, T., Obikawa, T., Shinozuka, J. and Yoshino, M. (1999) Development of reaction-control- lable milling head and its cutting performance (2nd Report). Trans. Japan Soc. Mech. Eng. 65-C, 1223–1228. SITC (1987) Application of expert system to preparation of MC processing. In Technological Development of Collaborative Research. Kawaguchi: Saitama Industrial Technology Center, p. 27. Spence, A. D. and Altintas, Y. (1994) A solid modeller based milling process simulation and plan- ning system. Trans. ASME J. Eng. Ind. 116, 61–69. Stephenson, D. A. (1991) Assessment of steady-state metal cutting temperature models based on simultaneous infrared and thermocouple data. Trans. ASME, J. Eng. Ind. 113, 121–128. Stephenson, D. A. and Agapiou, J. S. (1997) Metal Cutting Theory and Practice. New York: Marcel Dekker, p. 601. Takata, S. (1993) Generation of a machining scenario and its applications to intelligent machining operation. Annals of the CIRP 42(1), 531–534. Tansel, I. N. (1992) Modelling 3-D dynamics with neural networks. Int. J. Mach. Tools Manufact. 32, 829–853. Tansel, I. N., Mekdeci, C., Rodriguez, O. and Uragun, B. (1993) Monitoring drill conditions with wavelet based encoding and neutral networks. Int. J. Mach. Tools Manufact. 33, 559–575. Tansel, I. N., Mekdeci, C. and McLaughlin, C. (1995) Detection of tool failure in end milling with wavelet transformations and neural networks (WT-NN). Int. J. Mach. Tools and Manufact. 35, 1137–1147. Tarng, Y. S., Hseih, Y. W. and Hwang, S. T. (1994) Sensing tool breakage in face milling with a neural network. Int. J. Mach. Tools and Manufact. 34, 341–350. Tarng, Y. S., Cheng, C. I. and Kao, J. Y. (1995) Modeling of three-dimensional numerically controlled end milling operation. Int. J. Mach. Tools and Manufact. 35, 939–950. Tlusty, J. and Andrews, G. C. (1983) A critical review of sensors for unmanned machining. Annals of the CIRP 32, 563–572. Tlusty, J. (1985) Machine dynamics. In King, R. I. (ed), Handbook of High Speed Machining Technology. New York: Chapman and Hall, pp. 48–153. Tonshoff, H. K., Wulfsberg, J. P., Kals, H. J. J., Konig, W. and van Luttervelt, C. A., (1988) Developments and trends in monitoring and control of machining processes. Annals of the CIRP 37(2), 611–622. Trigger, K. J. and Chao, B. T. (1956) The mechanism of crater wear of cemented carbide tools. Trans. ASME. 78, 1119–1126. Usui, E., Shirakashi, T. and Kitagawa, T. (1978) Analytical prediction of three dimensional cutting process. Trans. ASME., J. Eng. Ind. 100, 236–243. Usui, E., Ihara, T. and Shirakashi, T. (1979) Probabilistic stress-criterion of brittle fracture of carbide tool materials. Bull. Japan Soc. Prec. Eng. 13, 189–194. Usui, E. Shirakashi, T. and Kitagawa, T. (1984) Analytical prediction of cutting tool wear. Wear 100, 129–151. Zimmermann, H. J. (1976) Description and optimization of fuzzy systems, Int. J. General Systems 2, 209–215. Zimmermann, H. J. (1991) Fuzzy Set Theory and Applications. Boston: Kluwer Academic Publishers. References 327 Childs Part 3 31:3:2000 10:41 am Page 327 Appendix 1 Metals’ plasticity, and its finite element formulation This appendix supports Chapters 2 and 6 and subsequent chapters. More complete descrip- tions of plasticity mechanics can be found in any of the excellent texts from the early works of Hill (1950) and Prager and Hodge (1951), through books such as by Thomsen et al. (1965) and Johnson and Mellor (1973), to more recent finite element oriented work (Kobayashi et al. 1989). Section A1.1 answers the questions, initially in terms of principal stresses and strains (Figure A1.1) concerning (i) what combinations of principal stresses s 1 , s 2 , and s 3 will cause yielding of a metal; (ii) if a metal has yielded, and the stress state is changed to cause further plastic strain increments de 1 ,de 2 , and de 3 , what are the relations between the strain increments and the stresses; and (iii) what is the work rate in a plastic field? Extension of the answers to non-principal stress state descriptions is briefly introduced. In Section A1.1, elastic components of deformation are ignored. Any anisotropy of flow, such as is impor- tant for example in sheet metal forming analysis, is also ignored. To analyse flow in any particular application, the yielding and flow laws (constitutive laws) are combined with equilibrium and compatibility equations and boundary condi- tions. If the flow is in plane strain conditions and when a metal’s elastic responses and work hardening can be ignored, the equilibrium and compatibility equations take a partic- ularly simple form if they are referred to maximum shear stress directions. The analysis of flow in this case is known as slip-line field theory and is introduced in Section A1.2. Apart from the circumstances of slip-line field theory, the simultaneous solution of Fig. A1.1 (a) Principal stresses and (b) principal strain increments Childs Part 3 31:3:2000 10:41 am Page 328 constitutive, equilibrium and compatibility equations is difficult. Finite element approxi- mations are needed to solve metal machining problems. Further analysis of stress, needed to support finite element methods, is found in Section A1.3. Section A1.4 extends the constitutive laws to include elastic deformation, and manipulates both rigid–plastic and elastic–plastic laws to forms suitable for numerical analysis. Section A1.5 considers finite element methods in particular. A1.1 Yielding and flow under triaxial stresses: initial concepts A1.1.1 Yielding and the description of stress The principal stresses acting on a metal may be written as the sum of a hydrostatic (or mean) part s m and a deviation from the mean, or deviatoric part, which will be written as s ′: s m =(s 1 + s 2 + s 3 )/3 s 1 ′ = s 1 – s m ≡ 2s 1 /3 – (s 2 + s 3 )/3 s 2 ′ = s 2 – s m ≡ 2s 2 /3 – (s 3 + s 1 )/3 } (A1.1) s 3 ′ = s 3 – s m ≡ 2s 3 /3 – (s 1 + s 2 )/3 Hydrostatic stress plays no part in the yielding of cast or wrought metals, if they have no porosity. (They are incompressible; any hydrostatic volume change is elastic and is recovered on unloading.) An acceptable yield criterion must be a function only of the deviatoric stresses. Inspection of equation (A1.1) shows that the sum (s 1 ′ + s 2 ′ + s 3 ′) is always zero: yielding cannot be a function of this. However, the resultant deviatoric stress s r ′: s r ′ =(s 1 ′ 2 + s 2 ′ 2 + s 3 ′ 2 ) ½ (A1.2) has been found by experiment to form a suitable yield function. That yielding occurs when s r ′ reaches a critical value is now known as the von Mises yield criterion. The magnitude of the critical value can be related to the yield stress Y in a simple tension test. In simple tension, two of the principal stresses, say s 2 and s 3 , are zero. Substituting these and s 1 = Y into equations (A1.1) for the deviatoric stresses and then these into equation (A1.2) gives for the yield criterion s r ′ = Y ǰ˭˭˭ 2/3 (A1.3a) Alternatively, the critical value may be related to the yield stress k in a simple shear test, in which for example s 1 = – s 2 = k and s 3 = 0. By substituting these values in equations (A1.1) and (A1.2), s r ′ = k ǰ˭˭ 2 (A1.3b) That the yield stress in tension is √3 times that in shear is just one consequence of the von Mises yield criterion. It is customary to introduce a quantity known as the equivalent stress, s – , equal to √(3/2) times the resultant deviatoric stress. The critical value of the equivalent stress for yielding to occur is then identical to the yield stress in simple tension. The von Mises yield crite- rion becomes Yielding and flow under triaxial stresses 329 Childs Part 3 31:3:2000 10:41 am Page 329 s – ≡ ǰ˭˭˭3/2 s r ′ = Y (A1.4) s – ≡ ǰ˭˭˭ 3/2 s r ′ = k ǰ˭˭ 3 } The equivalent stress and the yield criterion may be represented in a number of differ- ent ways. Figure A1.2(a) is a geometrical view of a state of stress P in principal stress space, origin O. The vector OP is the resultant stress s r . It has principal components (s 1 , s 2 , s 3 ). Alternatively, it has components OO′ and O′P along and perpendicular to the hydrostatic line s 1 = s 2 = s 3 . This line has direction cosines 1/√3 with the principal axes, so OO′ = s m √3. OP is s r ′. By vector addition s r ′ 2 = s r 2 –3s m 2 =(s 1 2 + s 2 2 + s 3 2 ) – 3s m 2 (A1.5) After substituting for s m from equation (A1.1), 3s r ′ 2 =(s 1 – s 2 ) 2 +(s 2 – s 3 ) 2 +(s 3 – s 1 ) 2 (A1.6) The yield criterion may be restated in terms of the principal stresses: 1 s – 2 =— [ (s 1 – s 2 ) 2 +(s 2 – s 3 ) 2 +(s 3 – s 1 ) 2 ] = Y 2 or 3k 2 (A1.7) 2 330 Appendix 1 Fig. A1.2 Geometrical representations of principal stresses and yielding Childs Part 3 31:3:2000 10:41 am Page 330 The yield criterion, equation (A1.3) or (A1.7), can be represented (Figure A1.2(b)) by the cylinder, s r ′ = constant. For a material to yield, its stress state must be raised to lie on the surface of the cylinder. A simpler diagram (Figure A1.2(c)) is produced by projecting the stress state on to the deviatoric plane: that is the plane perpendicular to s m through the point O′. The principal deviatoric stress directions have direction cosines √(2/3) with their projections in the deviatoric plane. Figure A1.2(c) shows the projected deviatoric stress components as well as the resultant deviatoric stress. Yield occurs when the resultant devi- atoric stress lies on the yield locus of radius k√2. A1.1.2. Plastic flow rules and equivalent strain Suppose that material has been loaded to a plastic state P (Figure A1.3(a)) and is further loaded to P* to cause more yielding, so that the yield locus expands by work hardening to a new radius s r ′*: what further plastic principal strain increments (de 1 ,de 2 ,de 3 ) then occur? It is found (Figure A1.3(b)) that the strain increments are in proportion to the deviatoric stresses. A resultant strain increment de r , is defined analogously to s r ′ as de r = (de 1 2 + de 2 2 + de 3 2 ) ½ (A1.8) de r is parallel to s r ′. It is as if the change of deviatoric stress, ds r ′ in Figure A1.3(a) has a component tangential to the yield locus that causes no strain and one normal to the locus which is responsible for the plastic strain. In fact, the tangential component causes elastic strain, but this is neglected until Section A1.4. The proportionalities between de r and s r ′ may be written de 1 =cs 1 ′;de 2 =cs 2 ′;de 3 =cs 3 ′ (A1.9) where the constant c depends on the material’s work hardening rate. By substituting equa- tions (A1.9) into (A1.8), c = de r /s r ′. To simplify the description of work hardening, an equivalent strain increment de – is Yielding and flow under triaxial stresses 331 Fig. A1.3 (a) A plastic stress increment, P to P*; (b) the resulting strain increment; and (c) the linking work-hardening relationship Childs Part 3 31:3:2000 10:41 am Page 331 introduced, proportional to de r , just as s – has been introduced proportional to s r ′. de – is defined as de – = ǰ˭˭˭ 2/3 de r (A1.10) Then, in a simple tension test (in which de 2 = de 3 = – 0.5de 1 ), de – =de 1 . A plot of equiv- alent stress against equivalent strain (Figure A1.3(c)), gives the work hardening of the material along any loading path. H′ is the work hardening rate ds – /de – . de r and s r ′ in the expression for c may be replaced by ǰ˭˭˭ 3/2 de – and ǰ˭˭˭ 2/3s – , and de – by ds – /H′, to give 3 ds – c = ——— (A1.11) 2 H ′s – Equations (A1.9) and (A1.11) are known as the Levy–Mises flow laws. A1.1.3 Extended yield and flow rules, and the plastic work rate The yield criterion must be able to be formulated in any set of non-principal axes, with equation (A1.7) as a special case. Consider the expression (s x – s y ) 2 + (s y – s z ) 2 + (s z – s x ) 2 + 6(t 2 xy + t 2 yz + t 2 zx ) = 6k 2 or 2Y 2 (A1.12) When the shear stresses t are zero, it is identical to equation (A1.7). When the direct stresses are zero, the factor 6 cancels out and the equation states that yielding occurs when the resultant shear stress reaches k. Equation (A1.12) thus is possible as an expression for the yield criterion generalized to non-principal stress axes. It is established more rigor- ously in Section (A1.3). Similarly, the Levy–Mises flow rules may be written more generally as de x de y de z de xy de yz de zx 3 de – 3 ds – —— = —— = —— = —— = —— = —— = ——— or ——— s′ x s′ y s′ z t xy t yz t zx 2 s – x 2 H ′s – (A1.13) Care must be taken to interpret the shear strains. de xy = de yx = 1/2(∂u/∂y + ∂v/∂x), for exam- ple, where u and v have the usual meanings as displacement increments in the x and y direc- tions respectively. This differs from the definition g = (∂u/∂y + ∂v/∂x) by a factor of 2. Finally, the work increment dU per unit volume in a plastic flow field is dU = s xx de xx + s yy de yy + s zz de zz + 2(s xy de xy + s yz de yz + s zx de zx ) ≡ s – de – + s m (de xx +de yy + de zz ) (A1.14) but because the material is incompressible, the last term is zero: the work increment per unit volume is simply s – de – . A1.2 The special case of perfectly plastic material in plane strain Section A1.1 is concerned with a plastic material’s constitutive laws. Material within a plastically flowing region is also subjected to equilibrium and compatibility (volume conservation) conditions, for example in Cartesian coordinates 332 Appendix 1 Childs Part 3 31:3:2000 10:41 am Page 332 [...]... Kobayashi, S., Oh, S-I and Altan, T (1989) Metal Forming and the Finite Element Method New York: Oxford University Press Kudo, H (1965) Some new slip-line solutions for two-dimentional steady state machining Int J Mech Sci 7, 43–55 Johnson, W and Mellor, P B (1973) Engineering Plasticity London: van Nostrand Johnson, W., Sowerby, R and Venter, R D (1982) Plane-Strain Slip-Line Fields for MetalDeformation... k and l, between the left and right-hand sides of the equation, implies that it represents all nine equations for the components of s* The meaning of the equation is unchanged by substituting another pair of letter suffixes, say m and n, for i and j: suffixes such as i and j, repeated on the same side of an equation, are called dummy suffixes and are said to be interchangeable Suffixes such as k and. .. vertices B and C Checking for overstressing is introduced in another context in Appendix A5 The overstressing limits developed in Appendix 5 (Hill, 1954) apply here too A1.2.5 Machining slip-line fields Figure A1.8 collects a range of slip-line fields, and their velocity diagrams (due to Lee and Shaffer, 1951, Kudo, 1965, and Dewhurst, 1978), which describe steady state chip formation by a flat-faced cutting... OABC, the faces of which are normal to the x, y, z and x* directions Writing the direction cosines of x* with x, y and z as ax*x, ax*y and ax*z, with similar quantities ay*x, ay*y, ay*z and az*x, az*y, az*z for the direction cosines of y* and z* with x, y and z, first, by geometry, the ratios of the areas OAC, OAB and OBC to ABC are respectively ax*x, ax*y and ax*z Then, from force equilibrium on the tetrahedron,... curvilinear maximum shear stress lines is known as the slip-line field Determining the shape of the net for any application and then the stresses and velocities in the field is achieved through slip-line field theory This theory is now outlined A1.2.1 Constitutive laws for a non-hardening material in plane strain When the strain in one direction, say the z-direction, is zero, from the flow rules (equation... plane strain 337 Fig A1.7 (a) A possible machining process with (b) a partial velocity diagram and (c) an illustration of a velocity discontinuity across a slip-line seen that the velocity of H is part of the rigid-body rotation: if the boundary CD is not a slip-line, it cannot accommodate velocity changes that must occur in a plastic field If the boundary is a slip-line, a point H can only be joined to... Pergamon Press Lee, E H and Shaffer, B W (1951) The theory of plasticity applied to a problem of machining Trans ASME J Appl Mech 18, 405–413 Osakada, K., Nakano, N and Mori, K (1982) Finite element method for rigid plastic analysis of metal forming Int J Mech Sci 24, 459–468 Prager, W and Hodge, P G (1951) The Theory of Perfectly Plastic Solids New York: Wiley Thompsen, E G., Yang, C T and Kobayashi, S (1965)... the cutting edge O and the remainder DA of the primary shear region is a single plane With this field, the slip-lines intersect the rake face at a constant angle, so that it describes constant friction stress conditions The fan angle y can take a Childs Part 3 31:3:2000 10:41 am Page 339 Perfectly plastic material in plane strain 339 Fig A1.8 Metal machining slip-line fields (left) and their velocity... the a-line intersects the tool face Similarly, a and b slip-lines meet a free surface at 45˚ (tf/k = 0) Because there is no normal stress on a free surface, p = ± k there, depending on the direction of k Fig A1.5 (a) Nets satisfying internal force equilibrium and (b) slip-lines meeting a friction boundary Childs Part 3 31:3:2000 10:41 am Page 336 336 Appendix 1 A1.2.3 Velocity relations in a slip-line... Suppose that any boundary such as CD is not a slip-line Then any point such as H inside the plastic region can be joined to the boundary in two places by two slip-lines, for example to F and G by HF and HG Figure A1.7(b) is the velocity diagram The velocities vF and vG of points F and G are determined from the rigid body rotation of the chip to be wOF and wOG, where w is the angular velocity of the chip . the machin- ing conditions for a particular workpiece and operation. These days, partly because of a decrease in the number of experts and partly because of the demands of unmanned and highly. T., Shinozuka, J. and Yoshino, M. (1999) Development of reaction-control- lable milling head and its cutting performance (2nd Report). Trans. Japan Soc. Mech. Eng. 65-C, 122 3 122 8. SITC (1987). based on simultaneous infrared and thermocouple data. Trans. ASME, J. Eng. Ind. 113, 121 128 . Stephenson, D. A. and Agapiou, J. S. (1997) Metal Cutting Theory and Practice. New York: Marcel Dekker,