Modeling and Simulation for Material Selection and Mechanical Design Part 7 pot

Figure 40 Relation between stresses at the center of ingot and casting speed as ‘‘mushy zone’’ which is a mixture of solid and liquid phases Figure 49 illustrates an example of spray coating system, which is developed to fabricate laminated plate In most cases of the spray coating process, after the spray layer is solidified completely, the other kind of material is poured onto the substrate so that the inner boundary grows and the interface of molten state moves toward spray layer direction In the spray coating process, solidifying material in the growing domain undergoes a variation of mechanical quantities, such as mass, momentum and energy as well as the change of material properties due to phase transformation from liquid to solid state The complicated interaction between temperature and inelastic deformation, in this case, is to be taken into consideration B Numerical Model and X-Ray Residual Stress Measurement 1 Modeling of Numerical Simulation To verify the validity of the theory and the procedure stated above, simulation of the fields of temperature and solidification mode and the stress Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 41 water Relation between stresses at the center of ingot and discharge of cooling distribution are performed over the course of the spray coating process of laminated plates with two layers The, assumption is made that both ends of laminated plate are constrained during the spray coating, and that the laminated plates are large enough, so that a growing model of two-dimensional finite element is proposed to interpret the experimental phenomena of the spray coating process and to compute the interfacial thermo-mechanical behavior between the impinging particles and the surface of substrate just beneath them A model of the spraying layer represented by a flat disk of 30 mm diameter and 0.5 mm thickness of initially uniform temperature which is put into contact with the substrate elements is shown in Fig 50 When the numerical analysis is started, the growth elements incorporated with the impinging particle from the spraying direction were put into the substrate In this analysis, the growth model is represented by an axisymmetrical problem Thermal flux entering and leaving each element as well as the latent heat liberated within the elements themselves during the solidification process is evaluated and the resulting element temperature is computed after each successive time increment To simulate the spray coating processes, it is assumed that the different materials are successively poured into the substrate, i.e., the different Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 42 View of (a) twin-roll casting system and the (b) model for simulation material is supplied after the material in the outer layer is completely solidified On the outer surface along spraying direction, the heat radiation boundary conditions was set on the initial step of the coating, and heat transfer condition was set on the cylinder surface of the axisymmetrical model, respectively C Properties and Coating Condition of Specimen Materials Laminated layers are deposited on a stainless steel (SUS304) substrate of 5 mm thickness In this work, the layer thickness and layer materials produced were: 0.50 mm for stainless steel The specified thickness was obtained when spraying was performed 10 times Table 2 shows the components of these wires which are generally used for carbon dioxide arc welding Table 3 shows the conditions of the spray coating process The thermo-physical properties of the wire materials used for temperature and stress calculation incorporated with solidification are shown in Tables 4 and 5, respectively [69] It is assumed that the properties of substrate are the same as those of the wire The heat transfer coefficient Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 43 Mesh of finite element model for air on the surface of the model is chosen to be h ¼ 2.78 Â 10À3 (cal=(mm2 sec deg)) The thermal radiation coefficient G ¼ 7.028 Â 10À7 (cal=(mm sec K)) is used for the model Depending on the inelastic constitutive model of Section 2, the inelastic strain rate can be given by a viscoplastic relationship Here, the viscosity which described the viscoplastic model of wire material (stainless steel) is shown in Fig 50 D X-Ray Residual Stress Measurement To measure the residual stresses in the spray coating process, CrKa characteristic x-rays were used Diffraction planes and angles were (2 1 1) and 2y0 ¼ 1568 for the stainless steel SUS304 Surface roughness of the coated layer was about 6.5 mm These values might be too large for x-ray stress measurement to provide reliable results However, the parallel beam method could give stress values with sufficient accuracy on such rough surface Before x-ray stress measurement, electropolishing conducted to remove an Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 44 Distribution of temperature in strip and roll oxide-film from the layers Stresses were measured parallel and perpendicular to the spray traveling direction The x-ray measuring conditions are shown in Table 6 The full width at the half-maximum method was used to determine peak positions We measured the residual stresses in a phase with 2 1 1 diffraction and g phase with 2 2 0 diffraction The stresses were obtained by the sin2 c method The c-diffractometer method was used for the measurement of residual stresses E Verification and Discussion of Simulation Results An example was used for simulating thermo-mechanical behavior and residual stress during the spray coating Figure 51 shows the variation of temperature distribution on the central element of spraying surface and outside surface of the layer From these results, we arrive at the temperature difference between the central element and the outside surface of the spraying layer due to the solidifying process The distribution of temperature on the total domain dependent on time is shown in Figs 52 and 53 The volume fraction of solid and the variation of the solidified thickness are depicted in Figs 54 and 55, respectively These figures, show that the temperature on the central element tends to decrease slowly by the latent heat generation Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 45 Temperature variations at center and surface of strip Figure 46 Distribution of solid fraction due to solidification and also by the heat supply by the successively poured material followed by the rapid temperature decrease at the end of solidification Difference in heat conductivity due to the temperature difference reveals the influence on the cooling rate and mode of solidification As for the results of stress analysis, residual stress is represented in the following figures, in which increasing stress reduces fluctuation or jump depending on the solidification and growing domain Distribution of residual stresses sr on the radial direction is shown by the lines in Fig 56 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 47 Distribution of stress sx The data are compared with the experimental results represented by the same condition of the process Relatively reasonable agreement between both values is seen even in the region with fluctuation of stresses on the interface boundary between the spraying layer and the substrate The distributions of residual stress sy and sz are shown in Figs 57 and 58 Here, the jump behavior of stresses on the interface is presented by these simulated results Thus, it is important to reveal the damage of the spraying layer based on the theory and numerical method VI DESIGNING OF FORGING PROCESS FOR CONTROL OF INTERFACIAL STRESS A Basic Description A typical industrial metal component may be manufactured by forging and heat treatment In the design of forging processes, information such Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 48 Dependence of constitutive relationship on stress distribution Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 59 Table 7 Outline of basic numerical model Material Properties of Heat Conduction Heat conductivity Die Workpiece 31 W=m K 35 W=m K Specific heat 470 J=kg K 800 J=kg K Heat transfer with ambient 20 W=m2 K 52 W=m2 K illustrate equivalent stress and axial residual stress distribution from the center to the surface in the middle of the rod after quenching, respectively In Fig 66, two cases are shown for axial residual stress distribution: one case is quenching without taking into account the residual stress from the forging process; and the other is quenching, taking account of Figure 60 Punch velocity of forging, holding, and ejecting Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 61 The rod in its (a) initial position and at the (b) end of ejecting stage Figure 62 Residual stress at the end of ejecting stage Figure 63 Cooling curve Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 64 Volume fraction of metallic structures the residual stress from the forging process Experimental results measured by the Sacks method are also shown in the same figure In both cases, the axial residual stress is in tension near the center and in compression near the surface Because of the effect of the residual stress from the forging process, less tension near the center and more compression near the surface were obtained as seen in Fig 66 The axial residual stress coming from the forging process does not make a big difference with the quenching calculation, Figure 65 Effective stress after quenching Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 1 Comparing with calculated results and experimental data for temperature, distortion and residual stress in these metallurgical processes, the metallo-thermo-mechanical theory and simulation method proposed in Section II are verified 2 Effects of cooling curves, distortion, and residual stresses on the occurrence of the phase transformation in quenching are proved by simulations of quenching and carburizing-quenching processes 3 It is important to identify the heat transfer coefficients of quenchants with respect to the quenching process are obtained from simulation results 4 The unified inelastic constitutive equation may describe the stress and deformation in the whole region of the solidifying process including liquid and solid state 5 In the simulation of continuous casting, the development of stresses from solidifying domain is presented On the other hand, the effects of distortion on solidification also shown to be an important factor 6 In the simulation of coating process, the jump behavior of stresses on the interface between substrate and spraying layer is shown Thus, it is important to reveal the damage of the spraying layer based on the metallo-thermo-mechanical theory and numerical method 7 The advantage of finite volume technique over the finite element method in the simulation of forging was shown From this point of view, the finite volume method is expected to be a powerful tool in the simulation of metal forming processes REFERENCES 1 Noyan, C.; Cohen, J.B Residual Stress, Springer Verlag, New York, 1987 2 Mura, T Residual stresses due to thermal treatments Res Rep Faculty Eng Meiji Univ 1995, 10, 14–27 3 Hanabusa, T Japanese Standard for X-ray Stress Measurement, Proceedings of 6th International Conference on Residual Stresses, Oxford, England, July 10–12, 2000; 181–188 4 Standard Method of X-ray Stress Measurement, JSMS, 1997 5 Bacon, G.E Neutron Diffraction; 3rd Ed Oxford University Press, Oxford, England, 1975 6 Pyzalla, A Determination of the residual stress state in components using neutron diffraction J Neutron Res 2000, 8, 187–213 7 Redner, S.; Perry, C.C Factors affecting the accuracy of residual stress measurements using the blind hole drilling method Proceedings of 7th International Conference on Experimental Stress Analysis, New York, August; 1982; 604–616 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Sakaida, Y.; Tanaka, K.; Shirakihara, K Local stress measurement of notched sapphire by Raman microspectroscopy Proceedings of 6th International Conference on Residual Stresses, IOM Communications, London, England, July 10–12, 2000; 311–318 Inoue, T.; Tanaka, K An elastic–plastic stress analysis of quenching when considering a transformation Int J Mech Sci 1975, 17, 361–367 Inoue, T.; Raniecki, B Determination of thermal-hardening stresses in steel by use of thermoplasticity theory J Mech Phys Solids 1978, 26, 187–212 Leblond, J.B.; Mottet, G.; Devaux, J.; Devaux, J.C Mathematical models of anisotropic phase transformation of steels and predicted plastic behavior Mater Sci Technol 1985, 1, 815–822 Inoue, T.; Wang, Z.G Coupling between stress, temperature and metallic structures during process involving phase transformations Mater Sci Technol., 1985, 1, 845–850 Gautier, E.; Simon, A Martensitic transformation kinetics during anisothermal creep and tensile tests Proceedings of International Conference on Martensitic Transformation; 1986; 502–507 Gautier, E.; Simon, A Analysis of the kinetics of pearlitic transformation under stress or after plastic deformation Proceedings of International Conference on Phase Transformations, Cambridge, England, July 6–10, The Institute of Materials: 1987; Lorimer, G.W., Ed; 451–454 Inoue, T Metallo-thermo-mechanical coupling—analysis of quenching, welding and continuous casting processes Bergund Huttenmannische Monatshefte 1987, 132, 63–75 Inoue, T In Inelastic Constitutive Relationships and Applications to Some Thermomechanical Processes Involving Phase Transformations, Thermal Stresses; Hetnarski, R.B., Ed.; Elsevier Science: Holland, 1989; Vol 3, Ju, D.Y.; Inoue, T.; Yoshihara, N Solidification mode and viscoplastic stresses in aluminum cylinder during vertical semi-continuous direct chill casting process Advanced Technology of Plasticity, 1990, 2, 1095–1100, JSTP, Tokyo, Japan Denis, S.; Farias, D.; Simon, A Mathematical model coupling phase transformation kinetics in steel ISIJ Int 1992, 32, 316–325 Inoue, T.; Wang, Z.G Coupling between stress, temperature and metallic structures during process involving phase transformations Mater Sci Technol 1985, 1, 845–850 Inoue, T.; Yamaguchi, T.; Wang, Z.G Stresses and phase transformations occuring in quenching of carburized steel gear wheel Mater Sci Technol 1985, 1, 872–876 Wang, Z.G.; Inoue, T Viscoplastic constitutive relation incorporating phase transformation- application to the welding Mater Sci Technol 1985, 1, 899–903 Inoue, T.; Ju, D.Y In Thermo-mechanical Simulation of Some Types of Steady Continuous Casting Processes, Advances in Continuum Mechanics; Bruller, O.; Mannl, V.; Najar, J., Eds.; Springer-Verlag: Herderberg, 1991; 389–406 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Fletcher, A.J.; Nasseri, M Effect of plate orientation on quenching characteristics Mater Sci Technol 1995, 11, 375–381 Ju, D.Y.; Inoue, T In Simulation of Solidification and Thermo-mechanical Behavior in Continuous Casting Process, Modeling and Simulation in Metallurgical Engineering and Materials Science; Yu, Z.; Xiao, Z., Eds.; Metallurgical Industry Press: Beijing, 1996; 351–356 Ju, D.Y.; Sahashi, M.; Omori, T.; Inoue, T Residual stresses and distortion of a ring in quenching–tempering process based on metallo-thermo-mechanics Proceedings of the 2nd International Conference on Quenching and the Control of Distortion, ASM International: Ohio, 1996; 249–257 Dwoling, W.; Pattok, T.; Ferguson, B.L.; Shick, D.; Helen, Y GU; Howes, M Development of a carburizing and quenching simulation tool: program overview Proceedings of 2nd International Conference on Quenching and Control of Distortion; Totten, G.E., Howers, M.A.H., Sjostrom, S., Funatani, K., Eds.; ASM International: Ohio, 1996; 329–340 Inden, G.; Neuman, P Simulation of diffusion controlled phase transformation in steel Steel Res 1996, 67, 401–407 Aubry, C.; Denis, S.; Achambault, P.; Simon, A.; Ruckstuhl, F.; Miege, B Prediction of microstructure and quenching residual stresses in case hardened pieces including selftempering effects Proceedings of 4th ASM Heat Treatment and Surface Engineering Conference in Europe; 1998; 273–282 Ju, D.Y.; Nishida, M.; Hanabusa, T Simulation of the thermo-mechanical behavior and residual stresses in the spray coating process J Mater Process Technol 1999, 92–93, 243–250 Ju, D.Y Simulation of thermo-metallo-mechanical behavior and interfacial stress of metal matrix composite under thermalshock process J Composite Struct 2000 30, 113–118 Ding, P.; Ju, D.Y.; Inoue, T.; de Vries, E Numerical simulation of forging and subsequent heat treatment of a rod by a finite volume method Acta Metallurgica Sinica (English Letters) 2000, 13 (1), 270–280 Inoue, T.; Ju, D.Y.; Arimoto, K Metallo-thermo-mechanical simulation of quenching process—theory and implementation of computer code ‘‘Hearts’’, Proceedings of 1st International Conference on Quenching and Control of Distortion; ASM International: Ohio, 1992; 205–212 Inoue, T.; Arimoto, K Development and implementation of CAE system ‘‘HEARTS’’ for heat treatment simulation based on metallo-thermo-mechanics J Mater Eng Perform 1997, 6 (1), 51–60 Bowen, R.M Theory of mixture In Continuum Physics; Eringen, A.C., Ed.; Academic Press: New York, 1976; Vol 3 Wray, P.J.; Holmes, M.F Plastic deformation of austenitic iron at intermediate strain rates Metall Trans 1975, A-6, 1189–1196 Wray, P.J On set of recrystallization during the tensile deformation of austenitic iron at intermediate strain rates Metall Trans 1975, 6A, 1197–1203 Pugh, C.E.; Robinson, D.N Some trends in constitutive equation model development for high temperature behavior of first-reactor structural alloys Nucl Eng Des 1978, 42, 269–278 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 38 Inoue, T.; Nagaki, S A constitutive modeling of thermo-viscoelastic-plastic materials J Therm Stresses 1978, 1, 53–61 39 Kujawski, D.; Mroz, Z A viscoplastic material model and its application to cyclic loading Acta Mech 1980, 36, 213–230 40 Matsui, H.; Ju, D.Y.; Inoue, T Inelastic behavior and unified constitutive equations of SUS304 at high temperature J Soc Mater Sci (JSMS) 1992, 41, 1153–1159 41 Ju, D.Y.; Inoue, T.; Matsui, H Visco-plastic Behaviour of SUS304 Stainless Steel at Ultra-high Temperature, Advances in Engineering Plasticity and its Application; Lee, D., Ed.; 1993; 521–528 42 Perzyna, P The constitutive equations for rate sensitive plastic materials Arch Mech Stos 1968, 20, 499–512 43 Perzyna, P Thermodynamic Theory of Viscoplasticity, in Advance of Applied Mechanics; Academic Press: New York, 1971; vol 11, 313–354 44 Sorimachi, K.; Brimacombe, J.K Improvements in mathematical modeling of stresses in continuous casting of steel Iron Steelmaking 1977, 4, 240–245 45 Chan, K.S.; Bodner, S.R.; Walker, K.P.; Lindholm, U.S A survey of unified constitutive theories Proceedings of 2nd Symposium on Nonlinear Constitutive Relations for High Temperature Applications, Cleveland; 1984; 108–112 46 Bodner, S.R.; Partom, Y Constitutive equations for elastic-viscoplastic strain hardening materials J Appl Mech., Trans ASME 1975, 42, 385–389 47 Bodner, S.R.; Merzer, A Viscoplastic constitutive equations for copper with strain rate history and temperature effects J Appl Mech., Trans ASME 1978, 100, 388–394 48 Chaboche, J.L.; Rousseler, G On the plastic and viscoplastic constitutive equation (part 1) Trans ASME, PVT 1983, 105, 153–158 49 Chabache, J.L.; Rousselie, G On the plastic and viscoplastic constitutive equations (part 2) Trans ASME, PVT 1983, 105, 158–164 50 Ackermann, P.; kurtz, W.; Heinemann, W In situ tensile testing of solidifying aluminium and Al–Mg shells Mater Sci Eng 1985, 75, 79–86 51 Suzuki, T.; Tacke, K.H.; Schwerdtfeger, K Influence of solidification structure on creep at high temperatures Metall Trans 1988, 19, 2857–2859 52 Inoue, T et al., Benchmark Project on the application of inelastic constitutive relations in plasticity-creep interaction condition to structural analysis and the prediction of fatigue-creep life, part-I Proceedings of Subcommittee Inelastic Analysis of High Temperature Materials, JSMS 1991; Vol.1, 1–5 53 Magee, C.L Nucleation of Martensite; ASM: New York, 1968 54 Johnson, W.A.; Mehl, R.F Reaction kinetics in processes of nucleation and growth Trans AIME 1939, 135, 416–458 55 Boley, A; Weiner, J.H Theory of Thermal Stresses; Wiley: New York, 1960 56 Rubenstein, L.I The Stefan Problem; American Mathematical Society: Tennessee, 1971 57 Flemings, M.C Behavior of metal alloys in the semisolid state Metal Trans A 1991, 22A, 957–980 58 Battle, T.P.; Pehlks, R.D Mathematical of microsegregation in binary metallic alloys Metal Trans 1990, 21B, 357–375 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 59 Zienkiewicz, O.C The Finite Element Method; McGraw-Hill: New York, 1977 60 Peirce, D.; Shih, D.F.; Needlemen, A A tangent modulus for rate dependent solid Int J Computer Struct 1984, 18, 875–887 61 Williams, J.R.; Lewis, R.W.; Morgan, K An elastic–viscoplastic thermal stress model with applications to the continuous casting of metals Int J Num Meth Eng 1979, 14, 1–9 62 Ju, D.Y.; Inoue, T Metallo-mechanical simulation of centrifugal casting process of multi-layer roll J Mater Sci Res Int., 1996, 2 (1), 18–25 63 Ju, D.Y.; Ichitani, K.; Nakamura, E.; Mukai, R Simulation and experimental verification of residual stresses and distortion in quenching process with stirring, heat treatment and surface engineering AIM 1998, 2, 283–291 64 Narazaki, M.; Ju, D.Y Simulation of distortion during quenching of steel effect of heat transfer in quenching Proceedings of 18th ASM Heat Treating Society Conference Including the Liu Dai Memorial Symposium; ASM International: Ohio, 1998; 629–638 65 Narazaki, M.; Ju, D.Y Influence of transformation plasticity on quenching distortion of carbon steel Proceedings of the 3rd International Conference on Quenching and Control of Distortion, Prague, March; 24–26 ASM International: Ohio, 1999; 405–415 66 Ju, D.Y Analysis of residual stresses during quenching process of large steel shaft J Saitama Institute Technol 1994, (3), 17–21 67 Ju, D.Y Computer prediction of thermo-mechanical behavior and residual stresses during induction hardening of notched cylinder J Mater Sci Forum, Trans Tech Publications 2000, 347–349, 352–357 68 Ju, D.Y.; Narazaki, M Simulation and experimental verification of residual stresses and distortion during quenching of steel Proceedings of 20th ASM Heat Treating Society Conference Including the Prof J.B Cohen Memorial Symposium; ASM International: Ohio, 2000; 441–447 69 Ju, D.Y.; Narasaki, M.; Kamisugi, H Computer predictions and experimental verification of residual stresses and distortion in carburizing–quenching of steel J Shanghai Jiaotong University 2000, E-5(1), 165–172 70 Ju, D.Y Computer prediction of residual stresses and distortion in carburizing–quenching of gear Proceedings of 6th International Conference on Residual Stresses; IOM Communications: London, England, 2000; Vol 1, 550–556 71 Ju, D.Y.; Liu, C.C Numerical modeling and simulation of carburized and nitrided quenching Proceedings of International Conference on Advances in Materials and Processing Technologies, Ed J M Torralba, Universidad Calos III De Madrid, Spain, Madrid, Sept 18–21; 2001, Vol 3, 1025–1032 72 Japanese Industrial Standard, Heat Treating Oils, JIS K 2242–1980 Japanese Standards Association: Tokyo, Japan, Analysis of Quenching Processes Using Lumped-Heat-Capacity Method, 1980 73 Narazaki, M.; Kogawara, M.; Shirayori, A.; Fuchizawa, S Analysis of Quenching Process Using Lumped-Heat-Capacity Method Proceedings of the 6th International Seminar of IFHT, Kyongju, Korea; ASM International: Ohio, 1997; 428–435 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 74 Industrial Quenching Oil—Determination of Cooling Characteristics—Nickel– Alloy Probe Test Method, International Standard, ISO 9950 1995 (E) 75 Narazaki, M.; Hiratsuka, H.; Shirayori, A.; Fuchizawa, S Examination of Methods for Obtaining Heat Transfer Coefficients by Quenching Small Probes Proceedings of the Asian Conference on Heat Treatment of Materials, Beijing, 1998; 269–274 76 Ju, D.Y.; Inoue, T A Thermomechanical model incorporating moving liquid= solid interface and its application to solidification process Proceedings of the 6th International Conference on Mechanical Behaviour of Materials, JSMS, July 28, Kyoto, 1991; Vol 5, 119–120 77 Ju, D.Y.; Oshika, Y.; Inoue, T Simulation of solidification and temperature in the centrifugal casting process (in Japanese) J Soc Mater Sci (JSMS) 1991, 40, 12–18 78 Ju, D.Y.; Takemura, S.; Inoue, T Analysis of coupled mode of solidification and stresses the centrifugal casting process J Soc Mater Sci (JSMS) 1992, 41 (464), 751–757 79 Sham, T.L.; Chow, H.W A finite element method for an incremental viscoplasticity theory based on overstress Int J Comp Mech 1989, 34, 143–156 80 Grill, A.; Brimacombe, J.K.; Weinberg, F Mathematical analysis of stresses in continuous casting of steel Iron Steelmaking 1976, 1, 38–47 81 Miyazawa, K.; Szekely, J A mathematical model of the splat coolong process using the twin-roll technology Metall Trans 1981, 12A, 1047–1057 82 Ju, D.Y.; Inoue, T.; Yoshihara, N Simulation of vertical semi-continuous direct chill casting process of cylindrical ingot of aluminum alloy (in Japanese).Trans Jpn Soc Mech Eng (JSME) 1989, 55 (513), 1236– 1243 83 Ju, D.Y.; Inoue, T Simulation of solidification and Evaluation of residual stresses during centrifugal casting Proceedings of the 3rd International Conference on Residual Stresses, Elsevier Science: Holland, 1991; Vol 1, 220–225 84 Ju, D.Y.; Inoue, T Simulation of solidification and viscoplastic deformation in the twin roll continuous casting process (in Japanese), Trans Jpn Soc Mech Eng (JSME) 1991, 57, 1147–1154 85 Gassot, H.; Junquera, T.; Ji, V.; Jeandin, M.; Guipont, V.; Coddet, C.; Verdy, C.; Grandsire, L A Comparative Study of Mechanical Properties and Residual Stress Distributions of Copper Coatings Obtained by Different Thermal Spray Processes, Surface Modification Technologies; IOM Communications, London, England, 2000; 16–23 86 Ju, Y.; Nishida, M.; Hanabusa, T Simulation of the thermo-mechanical behavior and residual stresses in the spray coating process J Mater Process Technol., 1999, 92–93, 243–250 87 Kirara, S Evaluation for the influence of press speed on the working load and die temperature The Proceedings of the 44th Japanese Joint Conference for the Technology of Plasticity (in Japanese); J JSTP 1993; 6–9 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 88 Kato, T.; Akai, M.; Tozawa, Y Thermal analysis of cold upsetting J JSTP (in Japanese) 1987, 28 (319), 791–798 89 Kennedy, K.F.; Lahoti, G.D Review of flow shess date Battle columbus laboratories, 1981 90 Nakanishi, K.; Nonoyama, F.; Sawamura, M.; Danno, A Evaluation of interface heat transfer coefficient for thermal analysis in forging J JSTP (in Japanese) 1996, 37 (421), 207–212 91 Isogawa, S.; Mori, I.; Tozawa, Y Determination of basic data for numerical simulation—analysis of multi-stage warm forging sequence for austenitic stainless steel I J JSTP (in Japanese) 1997, 38 (436), 84–89 92 Ding, P.; Ju, D.Y.; Inoue, T.; de Vries, E Numerical Simulation of forging and subsequent heat treatment of a rod by a finite volume method Third International Conference on Physical and Numerical Simulation of Materials and Hot Working (ICPNS’99), 1999; 270–280 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 4 Modeling and Simulation of Mechanical Behavior Essam El-Magd Aachen University, Aachen, Germany With the rapid increase of the capacity and speed of computers, work stations, and even personal computers, numerical methods can now be applied to solve easily many complex engineering problems, for example, in the fields of metal forming, strength of materials, and reliability studies of parts, components or systems Some of the conventional methods of stress analysis, such as photo-elasticity, have nearly disappeared Also, the interest in analytical methods like the elementary theory of plasticity and the slip line theory is decreasing to some extent with the increasing accuracy of the numerical methods At first, great effort had to be done in developing the numerical methods themselves to insure stability, convergence, and accuracy of the computation process In the mean time, many powerful codes that carry over the major part of this responsibility are commercially available Especially for engineers, the main field of activities changed in another direction, namely to the development of adequate constitutive equations that describe well the behavior of the material in the macroscopic, microscopic, and even in the atomistic scale From the practical point of view, it is not urgently required to increase numerical accuracy by some 0.1%, while the material data for plastic deformation processes deviate by more than 2% from reality or when the data for creep or fatigue life scatter by a factor between 1 and 2 2 There are still several important improvements to be done in the numerical codes, for example, in order to achieve an accurate consideration of large deformations However, the determination and implementation of adequate material data seem to be one of the most important tasks Due Copyright 2004 by Marcel Dekker, Inc All Rights Reserved to lack of material data, the computations were carried out in the past mostly assuming an elastic material behavior using the common values of the modulus of elasticity and the Poisson ratio, even if it was well known that the material behavior is inelastic under the conditions considered Complex procedures were developed in order to estimate the inelastic behavior using the elastic computational results This is changing monotonically towards the consideration of the inelastic material behavior with current codes The formulation of the material law is decisive for the experimental effort and costs needed to determine its parameters Empirical relations may be helpful when only few variables of the process are considered and when these variables vary within limited ranges Otherwise, physically founded material models may be more suitable, as they a priori define the tendency of the relations to be determined However, the experimental determination of each parameter of these models according to its exact physical meaning may require such a great effort that this procedure remains restricted to academic research activities For the practical application, a compromise is gaining increasing interest, according to which the functions are taken from physical and microstructure-mechanical models, but their parameters are determined by curve fitting of the experimental data In the following, some examples are represented for modeling and simulation of the material behavior during plastic deformation, low cycle fatigue, creep, and impact strength I PLASTIC BEHAVIOR Flow curves represent the relationship between the true stress and the true stain during plastic deformation at constant strain rate and temperature They are usually determined in tensile, compression, or torsion tests When a cylindrical rod of an initial length L0 and initial cross-sectional area A0 is loaded by a force F, its dimensions change to L and A If the force is further increased by an increment dF, the length increases by dL and the area decreases by dA The corresponding increments of the engineering stress and strain are defined by dS ¼ dF=A0 and de ¼ dL=L0, while the increments of the true stress and strain increments are defined by ds ¼ dF=A and de ¼ dL=L At an arbitrary time point during the test, the stresses and strains are given by s ¼ F=A0 ; e ¼ ðL À L0 Þ=L0 s ¼ F=A; e ¼ lnðL=L0 Þ Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð1Þ If the deformation is uniformly distributed along the bar and volume constancy can be assumed, the true and the conventional stresses and strains are related by s ¼ Sð1 þ eÞ and e ¼ lnð1 þ eÞ In addition to the physical relevance, the use of true stresses and strains allows for: (a) a P simple addition of the strains of different deformation steps e ¼ ei , (b) a simple formulation of the plastic volume constancy by exx þ eyy þ ezz ¼ 0, and (c) equal absolute values in tension and compression if specimen length is increased or decreased by the same factor Under service conditions, engineering materials are usually subjected to a multiaxial stress state On the other hand, material data are determined in laboratory tests under almost uniaxial loading For comparison, an equivalent uniaxial stress is to be define for the multiaxial case In the case of isotropic incompressible materials, the equivalent stress is given according to von Mises by seq sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ðsxx À syy Þ2 þ ðsyy À szz Þ2 þ ðszz À sxx Þ2 ¼ þ 3 t2 þ t 2 þ t2 xy yz zx 2 ð2Þ No plastic deformation takes place, as long as this equivalent stress is lower than the flow stress sY of the material When the loads are so increased that the equivalent stress reaches the flow stress, the material starts to yield For the strain state, which is represented by the plastic strain tensor ep , an equivalent strain increment is defined by ij dep eq rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1h i 2h p 2 ðdexx Þ þ ðdep Þ2 þ ðdep Þ2 þ ðdgp Þ2 þ ðdgp Þ2 þ ðdgp Þ2 ¼ zz zx yy xy yz 3 3 ð3Þ where gij ¼ 2eij with i 6¼ j The plastic strains ep , which are the irreversible response of the mateij rial to the applied stresses, depend not only on the current values of the stresses, but also on the total loading history For isotropic incompressible materials, the plastic strain increments are proportional to the deviatoric stresses, i.e., the normal stresses reduced by their mean value ðsxx þ syy þ szz Þ=3 and the shear stresses unchanged, or in short form Sij ¼ sij À ð1=3Þdij skk If the material follows the Mises yield criterion, the plastic strain increments are given by dep ¼ ij 3 dep eq Sij 2 seq Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð4Þ ... on Physical and Numerical Simulation of Materials and Hot Working (ICPNS’99), 1999; 270 –280 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 4 Modeling and Simulation of Mechanical Behavior... England, 2000; Vol 1, 550–556 71 Ju, D.Y.; Liu, C.C Numerical modeling and simulation of carburized and nitrided quenching Proceedings of International Conference on Advances in Materials and. .. Inoue, T In Simulation of Solidification and Thermo -mechanical Behavior in Continuous Casting Process, Modeling and Simulation in Metallurgical Engineering and Materials Science; Yu, Z.; Xiao, Z.,