Metal Machining - Theory and Applications Episode 2 Part 2 pps

Judgement of saturation of the plastic flow is made either on the basis of the tool load reaching a maximum value or of conservation of volume – i.e. that the computed flow of material out of the plastic zone into the chip balances that of the work into the plastic zone. Reformation of the flow field supposes that the separation between nodes along a streamline is unchanged by reformation, but that the direction from one node to the next is altered to bring it more closely tangential to the calculated flow. For each flow line consisting of a node sequence j – 1, j, j + 1 . . ., the updated (x, y) coordinates of node j are given by u˘ x,j–1 + u˘ x,j u˘ y,j–1 + u˘ y,j x j = x j–1 + ————— L j , y j = y j–1 + ————— L j (7.10) ᐉ j ᐉ j The Iterative Convergence Method (ICM) 213 Fig. 7.12 Developed flow-chart of the iterative convergence method Childs Part 2 28:3:2000 3:15 pm Page 213 where (u˘ x,j , u˘ y,j ) are the calculated velocities at node j, ᐉ j is the resultant average velocity of nodes j–1 and j, and L j is the separation between nodes j–1 and j: ᐉ j = ( (u˘ x,j–1 + u˘ x,j ) 2 + (u˘ y,j–1 + u˘ y,j ) 2 ) 1 / 2 (7.11a) L j = ( (x j–1 – x j ) 2 + (y j–1 – y j ) 2 ) 1 / 2 (7.11b) The reformation using equations (7.10) and (7.11) is implemented from the beginning to the end of a flow line so that the coordinates (x j–1 ,y j–1 ) have already been revised. The equivalent plastic strain e — in each element is evaluated by the integration of its rate e ˘ — along the reformed flow lines: e — ˘ e — = ∫ e — ˘ dt = ∫ — d ᐉ (7.12) v˘ e where v˘ e the element velocity, obtained from the average of an element’s nodal velocities. Relations between flow stress, strain, strain rate and temperature are considered in Section 7.4. Figure 7.13 shows an ICM mesh for two-dimensional machining with a single point tool, in which the x- and y-axes are taken respectively parallel and perpendicular to the cutting direction, in a rectangular Cartesian coordinate system. The tool is assumed to be stationary and rigid, while the workpiece moves towards it at the specified cutting speed. 214 Finite element methods Fig. 7.13 Two-dimensional finite element assemblage with boundary conditions Childs Part 2 28:3:2000 3:15 pm Page 214 The mesh is highly refined in the primary and secondary shear zones, in line with the considerations of Chapter 6. The friction boundary at the tool–chip interface is treated as follows. For the nodes contacting the rake face, the conditions imposed on the finite element equation (equation 7.9(b)) with respect to the nodal force rate F ˘ and the nodal velocity u˘ are: dt F ˘ x′ = ( —— ) F ˘ y′ , u˘ y′ = 0 (7.13) ds n where x′ and y′ are the local coordinate systems parallel and perpendicular to the rake face (as shown in Figure 7.13) and (dt/ds n ) is the local slope of the friction characteristic curve (for example the inset in Figure 2.23) at the value of s n associated with the nodal force F y′ . In the course of the elastic–plastic analysis, loop I of Figure 7.12, the chip contact length may increase or decrease. A chip surface node in contact with the rake face is judged to leave contact if its F y′ force becomes tensile; and a node out of contact is judged to come into contact if its reformed y′ becomes negative (penetrates the tool). Thus, the ICM method auto- matically determines the chip-tool contact length as one aspect of determining the chip flow. The separation of material at the cutting edge is taken into account geometrically. The streamline at the cutting edge bifurcates both onto the rake face and onto the clearance surface of the work. In the ICM calculation, the relative displacement between the work near the cutting edge and the tool is only about 1/20 of the uncut chip thickness. A small crack imposed on the mesh, of that length, is sufficient to cope with separation without additional treatments, such as reconstruction of node and element sequences and special procedures to ensure a force balance at the crack tip. (This is not the case when the actual loading path of an element has to be followed, as in the analysis of unsteady or discontin- uous chip flows, to be considered in Section 7.3.3.) Finally, Figure 7.13 shows the boundary conditions for the temperature analysis (loop II). The forward and bottom surfaces of the work are fixed at room temperature. No heat is conducted across the chip and work exit surfaces (adiabatic condition), although there is of course convection. Heat loss by convection is allowed at those surfaces surrounded by atmosphere. Heat loss by radiation is negligible in the analysis. 7.3.2 ICM simulation examples The following is an example of the application of the ICM scheme to the two-dimensional machining of an 18%Mn–5%Cr high-hardness steel (Maekawa et al., 1988). The cutting conditions used were a cutting speed of 30 m/min, an uncut chip thickness of 0.3 mm, unit cutting width, a P20 grade carbide tool with a zero rake angle and dry cutting. Figure 7.14 shows the predicted chip shape and nodal displacement vectors. Material separation at the tool tip and chip curl are successfully simulated. Figure 7.15 gives the distribution of equivalent plastic strain rate, showing where severe plastic deformation takes place. The deformation concentrates at the so-called shear plane, but is widely distributed around that plane. The secondary plastic zone is also clearly visible along the rake face, although the deformation is not as severe as in the primary zone. These features are reflected in the temperature distribution in the chip and workpiece, as shown in Figure 7.16. A maximum temperature of more than 800˚C appears on the rake face at up to two feed distances from the tool tip. The Iterative Convergence Method (ICM) 215 Childs Part 2 28:3:2000 3:15 pm Page 215 216 Finite element methods Fig. 7.14 Chip shape and velocity vectors in machining high manganese steel: cutting speed = 30 m/min, undeformed chip thickness = 0.3 mm, width of cut =1 mm, rake angle = 0º, no coolant Fig. 7.15 Distribution of equivalent plastic strain rate, showing concentration of plastic deformation: cutting condi- tions as Figure 7.14 Childs Part 2 28:3:2000 3:15 pm Page 216 Experimental verification has also been performed. Figure 7.17 compares the predicted and measured specific cutting forces under the same conditions (but varying speed). The observed force–velocity characteristics are well simulated. Similar agreement was confirmed in other quantities such as chip curl, rake temperature, stresses on the rake face and tool wear. For tool wear, a diffusive wear law as described in equation (4.1) was assumed (Maekawa et al., 1988). The calculation time for the ICM method depends both on the computer hardware and on the number of finite elements. In the above case, it takes only a few minutes from ICM execution to graphical presentations, using a recent high-specification PC (Pentium II, 400 MHz CPU) and an assemblage of 390 nodes and 780 triangular elements. However, a pre- processor to prepare the finite element assemblage and a post-processor to handle a large amount of data for visualization are required. Further ICM steady flow examples will be presented in Chapter 8, together with the finite element analysis of unsteady and discontinuous chip formation. The latter requires more consideration of the chip separation criterion. 7.3.3 A treatment of unsteady chip flows As has been written above, the ICM scheme cannot be applied to the analysis of non- steady metal machining. The iteration around an incremental small strain plastic loading The Iterative Convergence Method (ICM) 217 Fig. 7.16 Isotherms near the cutting tip, cutting conditions as Figure 7.14 Childs Part 2 28:3:2000 3:15 pm Page 217 path closely coupled with a steady state temperature calculation (Figure 7.12) must be replaced by an incremental large strain and deformation analysis, coupled with a non- steady state temperature calculation (Appendix 2.4.4.), along the actual material loading path. Movement of the tool relative to the work over distances much greater than the feed, or uncut chip thickness, requires a way of reforming the nodes at the feed depth, as they approach the cutting edge, to form the work clearance surface and the chip surface in contact with the rake face. In addition, if the unsteady flow being treated involves fracture within the primary shear zone, a fracture criterion and a way of handling crack propagation are also needed. All these potentially require more comput- ing power. The examples of unsteady flow in Chapter 8.2 deal with these complications in the following ways (Obikawa and Usui, 1996; Obikawa et al., 1997). Computational intensity is reduced by using meshes less refined than that shown in Figure 7.13, despite a possible loss of detail in the secondary shear region at high cutting speeds (Figure 6.12). Figure 7.18 shows the four-node quadrilateral finite element meshes used in plane strain condi- tions, similar to those in Figures 7.6 and 7.7. Hydrostatic pressure variations in large strain elastic–plastic analyses are dealt with easier using four-node quadrilateral than three-node triangular elements (Nagtegaal et al., 1974): more detail of large strain plasticity is summarized in Obikawa and Usui (1996). Details of node separation at the cutting edge and the propagation of a ductile primary shear fracture are shown respectively in the lower and upper parts of Figure 7.19. Node separation A geometrical criterion is used for node separation. A node i reforms to two nodes i and i′ once its distance from the cutting edge becomes less than 1/20 of the element’s side length 218 Finite element methods Fig. 7.17 Comparison of predicted specific forces with experiment for the same feed and rake angle as Figure 7.14, but with varying cutting speed Childs Part 2 28:3:2000 3:15 pm Page 218 The Iterative Convergence Method (ICM) 219 Fig. 7.18 (a) Coarse (b) fine finite element mesh Fig. 7.19 (a) Separation of nodes within a fracturing chip; and (b) release of nodal forces at the cutting edge Childs Part 2 28:3:2000 3:15 pm Page 219 (about 5 mm in the examples to be considered) and once the previously separated node has come into contact with the rake face. To avoid a sudden change in nodal forces, which can cause the computation to become unstable, the forces F i and F i′ acting on the separated nodes are not relaxed to zero immediately. Instead their components in the cutting direc- tion are reduced step-by-step, under the constraint that both nodes move parallel to the cutting direction, to reach zero as i reaches the rake face (when the friction boundary condition takes over its movement). As in the ICM method, the small artificial crack at the cutting edge introduced by this procedure does not significantly alter the machining para- meters. Fracture initiation and crack growth Shear fracture is proposed to occur if the equivalent strain exceeds an amount depending on the size of the hydrostatic pressure p (positive in compression) relative to the equiva- lent stress s — , and on the absolute temperature T and equivalent strain rate e — ˘ : p e — > e — 0 + a — + f(T, e — ˘ ) (7.14) s — where f(T, e — ˘ ) causes the critical strain to increase with increasing temperature and reduc- ing strain rate, as considered further in Chapter 8. The upper part of Figure 7.19 shows the method of treating crack propagation, for the case of crack initiation at the cutting edge (a crack may alternatively initiate at the free surface end of the primary shear zone). If the strain at node I exceeds the limit of equa- tion (7.14), an actual crack is assumed to propagate in the direction of the maximum shear stress t m to a point P. If point P is closer to node J than to K, a nominal crack is assumed to form along IJ, but if (as shown) P is closer to K, the nominal crack contin- ues along JK to K. If the fracture limit is still exceeded at P, the actual crack continues to propagate in the direction of t m there, to Q; and so on to R, until the fracture criterion is no longer satisfied. The nominal crack growth, for the example shown, follows the path IJKLMN. 7.4 Material flow stress modelling for finite element analyses Flow stress, friction and, as considered in the previous section, fracture behaviour of metals, are all required as inputs to finite element analyses. This final section of this chap- ter concentrates on the flow stress dependence on strain, strain rate and temperature. The reason is that most of what is known about friction in metal cutting has already been intro- duced in Chapter 2; and there is insufficient information about the application of ductile shear fracture criteria to machining to enable a sensible review to be made. Only on flow stress behaviour is there more currently to be written. The topic of flow stress dependence on strain, strain-rate and temperature has been introduced in Section 6.3. There, flow stress was related to strain by a power law, with the constant of proportionality and power law exponent both being functions of strain rate and temperature (equations (6.10) and (6.14)). Comparisons were made between flow stress data deduced from machining tests and high strain-rate compression tests (Figure 6.11). Those compression tests were carried out in a high speed hammer press, driven by 220 Finite element methods Childs Part 2 28:3:2000 3:15 pm Page 220 compressed air, on material brought to temperature (up to 1100˚C) by pre-heating in a furnace (Oxley, 1989; from Oyane et al., 1967). Pre-heating in a furnace allows a material’s microstructure to come into thermal equi- librium. This differs from the conditions experienced in metal machining. There, metal is heated and passes through the deforming region in the order of milliseconds. The microstructures of chips, in the hot secondary shear region, appear heavily cold worked and not largely recovered or recrystallized. For steels, traces of austenitization and quenching are hardly ever seen, even though secondary shear temperatures are calcu- lated to be high enough for that to occur for longer heating times. The ideal mechanical testing of metals for machining applications involves high heating rates as well as strain rates. 7.4.1 High heating-rate and strain-rate mechanical testing Such testing has been developed by Shirakashi et al. (1983). A Hopkinson bar creates strain rates up to 2000 s –1 in a cylindrical sample of metal (6 mm diameter by 10 mm long). Induction heating and a quench tank heat and cool the sample within a 5 s cycle. A stopping ring limits the strain per cycle to 0.05: multiple cycling allows the effect of strain path (varying strain rate and temperature along the path) on flow stress to be studied. Figure 7.20 shows the principle of the test, with a measured temperature/time result of heating a 0.15%C steel to 600˚C. Subsidiary tests show that a single sample can be heated for up to 90 s at temperatures up to 680˚C before thermal annealing or age hardening modifies the flow stress generated by straining. Thus, 20 cycles, each taking 5 s, developing a strain up to 1, can be achieved Material flow stress modelling 221 Fig. 7.20 Principle of the rapid heating and quenching high strain rate test (after Shirakashi et al. , 1983) Childs Part 2 28:3:2000 3:15 pm Page 221 before the time at which temperature degrades the results. Phase transformation prevents useful testing above 720˚C. Even testing at strains up to 1, strain rates up to 2000 and temperatures up to ≈ 700˚C (for steels) does not reach metal cutting secondary shear conditions, but it is the closest yet achieved. With this equipment, the flow stresses of a range of carbon and low alloy steels have been measured. Varying both strain rate and temperature along a strain path has been observed to influence the stress/strain curve. An empirical equation to represent this has been developed: e ˘ — M e ˘ — m e ˘ — –m/NN s — = A ( —— ) e aT ( —— )( ∫ strain path e –aT/N ( —— ) de — ) (7.15a) 1000 1000 1000 When straining takes place at constant strain rate and temperature, it reduces to: e ˘ — M s — = A ( —— ) e — N (7.15b) 1000 where A, M and N may all vary with temperature. Measured values are given in Appendix 4.3. Figure 7.21 gives example results for a low alloy steel (the 0.36C-Cr-Mo-Ni material of Table A4.4). The Hopkinson bar equipment has established different laws for non-ferrous face centred cubic metals such as aluminium and a-brass. A much greater strain rate path effect and no temperature path effect has been observed (Usui and Shirakashi, 1982). At temper- atures, T˚C, up to about 300˚C (higher temperature data would be useful but is not reported) B —— e ˘ — M e ˘ — mN s — = A ( e – T+273 )( —— )( ∫ strain path ( —— ) de — ) (7.16a) 1000 1000 which, at constant strain rate, simplifies to the form 222 Finite element methods Fig. 7.21 Flow stress behaviour of a low alloy steel: dashed line at 20ºC and a strain rate of 10 –3 s –1 ; solid lines at strain rate of 10 3 s –1 and temperatures as marked Childs Part 2 28:3:2000 3:15 pm Page 222 [...]... shear flow stress k and temperature T (°C) Childs Part 2 28:3 :20 00 3:16 pm Page 22 8 22 8 Applications of finite element analysis Fig 8.1 continued Childs Part 2 28:3 :20 00 3:16 pm Page 22 9 Simulation of BUE formation 22 9 velocities along the flow lines (following the method of Johnson and Kudo, 19 62) The grey area represents the plastic deformation zone The predicted normal stress st and friction stress... (1989) Mechanics of Machining, Chicester: Ellis Horwood Oyane, M., Takashima, F., Osakada, K and Tanaka, H (1967) The behaviour of some steels under dynamic compression In: 10th Japan Congress on Testing Materials, pp 72 76 Childs Part 2 28:3 :20 00 3:15 pm Page 22 5 References 22 5 Rakotomolala, R., Joyot, P and Touratier, M (1993) Arbitrary Lagrangian-Eulerian thermomechanical finite-element modelof material... temperature dependence of strain hardening, through the reduction of a metal s elastic shear modulus, G, with temperature: Childs Part 2 28:3 :20 00 3:15 pm Page 22 4 22 4 Finite element methods s— = (C1 + C5e— n)(GT/G293) + C2 exp[(–C3 + C41ne˘)T] — (7.19) where, for steels (GT/G293) ≈ 1.13 – 0.000445T A question arises about extrapolation of these, and other power law equations, to strains much greater than 1... m/min: (a) and (c) as in Figures 8.1 and 8 .2 but (b) distribution of γm and hydrostatic pressure p Childs Part 2 28:3 :20 00 3:16 pm Page 23 2 23 2 Applications of finite element analysis Fig 8.3 continued generate the nucleus of a BUE If nucleation occurs, debris stuck to the rake face will have enough hardness to resist loading by the chip body Figure 8.4 is a quick-stop observation showing that separation... Zerilli, F J and Armstrong, R W (1997) Dislocation mechanics based analysis of materials dynamics behaviour J de Physique IV 7(C8), 637–648 Childs Part 2 28:3 :20 00 3:15 pm Page 22 6 8 Applications of finite element analysis In this chapter, a number of special topics are considered as examples of applications to which finite element methods have already contributed Built-up edge (BUE) and serrated... Annals CIRP 40(1), 61–64 Ueda, K., Manabe, K and Nozaki, S (1996) Rigid-plastic FEM of three-dimensional cutting mechanism (2nd report) – simulation of plain milling process J Japan Soc Prec Eng 62( 4), 526 –531 Usui, E and Shirakashi, T (19 82) Mechanics of machining – from descriptive to predictive theory ASME Publication PED 7, 13–35 Usui, E., Shirakashi, T and Obikawa, T (1977) Simulation analysis of... and Shirakashi, T (1981) Simulation analysis of built-up edge formation in machining of low carbon steel Bull Japan Soc Prec Eng 15(4), 23 7 24 2 Zienkiewicz, O C (1971) The Finite Eelement Method in Engineering Science 2nd edn Ch 18 London: McGraw-Hill Zerilli, F J and Armstrong, R W (1987) Dislocation-mechanics based constitutive relations for material dynamics calculations J Appl Phys 61, 1816–1 825 ... Trent, E M (1991) Metal Cutting, 3rd edn Oxford: Butterworth Heinemann Ueda, K and Sugita, T (1983) Application of fracture mechanics in micro-cutting of engineering ceramics Annals CIRP 32( 1), 83–86 Ueda, K and Manabe, K (1993) Rigid-plastic FEM analysis of three-dimensional deformation field in chip formation process Annals CIRP 42( 1), 35–38 Ueda, K., Sugita, T and Hiraga, H (1991) A J-integral approach...Childs Part 2 28:3 :20 00 3:15 pm Page 22 3 Material flow stress modelling 22 3 s— = A B – —— T +27 3 e e˘— —— 1000 ( )( ) M* e— N (7.16b) Coefficients in these equations, with data for other alloys too, are also given in Appendix 4.3 When flow stress data from these Hopkinson bar tests are used in machining simulations in which the predicted temperatures... fundamentally-based equations for metal machining applications will be developed over the coming years Eventually the goal of relating flow behaviour to a metal s composition and microstructure will be reached However today, the empirical forms outlined in Section 7.3.1 are the best validated that are available References Ceretti, E., Fallbohmer, P., Wu, W T and Altan, T (1996) Application of 2- D FEM to . pp. 72 76. 22 4 Finite element methods Childs Part 2 28:3 :20 00 3:15 pm Page 22 4 Rakotomolala, R., Joyot, P. and Touratier, M. (1993) Arbitrary Lagrangian-Eulerian thermomechan- ical finite-element. maximum shear strain γ m and strain rate γ ˘ m and (c) distributions of shear flow stress k and temperature T (°C) Childs Part 2 28:3 :20 00 3:15 pm Page 22 7 22 8 Applications of finite element. line at 20 ºC and a strain rate of 10 –3 s –1 ; solid lines at strain rate of 10 3 s –1 and temperatures as marked Childs Part 2 28:3 :20 00 3:15 pm Page 22 2 B —— e ˘ — M* s — = A ( e – T +27 3 )( —— ) e — N (7.16b) 1000 Coefficients