MATERIALS SGIEGE& ENGINEERING ELSEVIER Materials Science and Engineering A387-389(2004)266-271 www.elsevier.com/locate/msea Dislocation density based modeling of work hardening in the context of integrative modeling of aluminum processing M.Goerdelera.*,M.Crumbacha,M.Schneidera,G.Gottsteina,L.Neumannb, H.Aretzb,R.Koppb a Institut fur Metallkunde und Metallphysik,RWTH Aachen.Koperniskusstrasse 14.52056 Aachen,Germany bInstitut fur Bildsame Formgebung.RWTH Aachen,Intestrasse 10.52056 Aachen.Germany Received 25 August 2003 Abstract We present a work hardening model with three types of dislocation densities as internal variables.The kinetic equation of state-used to calculate the flow stress-is extended to account for solution hardening by taking solute atoms into account as obstacles for the mobile dislocations.Thus,the model is able to describe the effects of alloy composition on the hardening behavior.Examples are presented on how the model can be used to calculate stress-strain curves for a range of temperatures,strain rates and compositions.Its application to through process modeling is shown.Owing to the use of dislocation densities as internal variables,the model delivers an output that can be directly used as input for subsequent modeling of recrystallization. 2004 Elsevier B.V.All rights reserved. Keywords:Dislocation densities;Flow stress modeling;FEM;Through process modeling;Aluminum alloys 1.Introduction published elsewhere [1,2],previous examples on previous applications can be found in [3-5].In this study,the main Knowledge of the flow stress response of a metallic ma- focus was placed on the coupling to other microstructural terial to imposed deformation conditions is a prerequisite models and the application for through process modeling. for modeling of forming processes.In most commercial FE Hence,only the essentials of the model concept will be codes simple empirical flow stress descriptions are used reiterated. which often give a remarkably good fit to measured data, Dislocation density based models were developed but do not allow any extrapolation beyond the range of ex- by numerous authors during recent decades [6-10].In perimentally examined deformation conditions and initial those models,usually the overall dislocation density material states.In addition.most of these flow stress formu- (one-parameter-models [6,7])or two separate dislocation las use the strain as state variable of the material,which is densities (two-parameter-models [8-10])were used as in- wrong,though.Besides,it renders it virtually impossible to ternal variables.In all cases,an application to a wide use such formulations in an integrative approach of model- range of temperatures and strain rates turned out to be ing material microstructure evolution during the processing difficult with a single set of material parameters.The chain up to the properties of the final product.Instead,it is 3-Internal-Variable-Model (3IVM)used here distinguishes mandatory to use true internal state variables of the material. three dislocation densities.As only cell or subgrain forming Here,a dislocation density based work hardening model materials are considered,immobile dislocation densities in is used for this purpose.Details of the model have been cell interiors and cell walls are treated separately.The third internal variable is the density of mobile dislocations which *Corresponding author.Tel:+49-241-80-2-68-77; carry the plastic strain.The influence of this variable on the fax:+49-241-80-22-301. reaction of dislocation density models to strain rate changes E-mail address:goerdeler@imm.rwth-aachen.de (M.Goerdeler) has recently been investigated in [14]. 0921-5093/$-see front matter 2004 Elsevier B.V.All rights reserved doi:10.1016f.msea.2003.12.086
Materials Science and Engineering A 387–389 (2004) 266–271 Dislocation density based modeling of work hardening in the context of integrative modeling of aluminum processing M. Goerdeler a,∗, M. Crumbach a, M. Schneider a, G. Gottstein a, L. Neumann b, H. Aretz b, R. Kopp b a Institut für Metallkunde und Metallphysik, RWTH Aachen, Koperniskusstrasse 14, 52056 Aachen, Germany b Institut für Bildsame Formgebung, RWTH Aachen, Intzestrasse 10, 52056 Aachen, Germany Received 25 August 2003 Abstract We present a work hardening model with three types of dislocation densities as internal variables. The kinetic equation of state—used to calculate the flow stress—is extended to account for solution hardening by taking solute atoms into account as obstacles for the mobile dislocations. Thus, the model is able to describe the effects of alloy composition on the hardening behavior. Examples are presented on how the model can be used to calculate stress–strain curves for a range of temperatures, strain rates and compositions. Its application to through process modeling is shown. Owing to the use of dislocation densities as internal variables, the model delivers an output that can be directly used as input for subsequent modeling of recrystallization. © 2004 Elsevier B.V. All rights reserved. Keywords: Dislocation densities; Flow stress modeling; FEM; Through process modeling; Aluminum alloys 1. Introduction Knowledge of the flow stress response of a metallic material to imposed deformation conditions is a prerequisite for modeling of forming processes. In most commercial FE codes simple empirical flow stress descriptions are used which often give a remarkably good fit to measured data, but do not allow any extrapolation beyond the range of experimentally examined deformation conditions and initial material states. In addition, most of these flow stress formulas use the strain as state variable of the material, which is wrong, though. Besides, it renders it virtually impossible to use such formulations in an integrative approach of modeling material microstructure evolution during the processing chain up to the properties of the final product. Instead, it is mandatory to use true internal state variables of the material. Here, a dislocation density based work hardening model is used for this purpose. Details of the model have been ∗ Corresponding author. Tel.: +49-241-80-2-68-77; fax: +49-241-80-22-301. E-mail address: goerdeler@imm.rwth-aachen.de (M. Goerdeler). published elsewhere [1,2], previous examples on previous applications can be found in [3–5]. In this study, the main focus was placed on the coupling to other microstructural models and the application for through process modeling. Hence, only the essentials of the model concept will be reiterated. Dislocation density based models were developed by numerous authors during recent decades [6–10]. In those models, usually the overall dislocation density (one-parameter-models [6,7]) or two separate dislocation densities (two-parameter-models [8–10]) were used as internal variables. In all cases, an application to a wide range of temperatures and strain rates turned out to be difficult with a single set of material parameters. The 3-Internal-Variable-Model (3IVM) used here distinguishes three dislocation densities. As only cell or subgrain forming materials are considered, immobile dislocation densities in cell interiors and cell walls are treated separately. The third internal variable is the density of mobile dislocations which carry the plastic strain. The influence of this variable on the reaction of dislocation density models to strain rate changes has recently been investigated in [14]. 0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.12.086
M.Goerdeler et al./Materials Science and Engineering A 387-389 (2004)266-271 267 2.The model to their respective influence on the obstacle distance as de- tailed below.The jump width corresponds to the mean spac- We confine ourselves to cell forming metals and alloys, ing of obstacles.In former versions of the 3IVM only the which comprise most commercial aluminum,copper and fer- mean spacing of forest dislocations in cell walls and inte- rous alloys.A cellular arrangement of dislocations develops riors was used as obstacle spacing.In the current version during straining,composed of cell walls featuring a high also the spacing of solute atoms,in the specific case investi- dislocation density which enclose cell interiors with a much gated in this paper magnesium atoms,is taken into account. lower dislocation density.Upon loading dislocation sources The approach of linear reciprocal superposition of the differ- generate mobile dislocation loops,which carry the imposed ent obstacle spacings is adopted.The equation for the jump plastic strain by their movement and interact with the con- width then reads sidered microstructural constituents.These microstructural constituents include the immobile dislocations in the cell (3) walls and interiors,respectively,and solute atoms as well 入i. =√/Pi,w+ as second phase particles,like precipitates.This interaction The subscripts i,w denote the values obtained in cell interi- will eventually lead to immobilization of dislocations and ors and cell walls,respectively,and /F is the Friedel length. their storage either in cell interiors or cell walls.As the used as measure for the effective spacing of solute atoms. temperature of deformation is increased,these stored dis- With the overall obstacle spacing,the activation energy is locations become increasingly subjected to dynamic recov- calculated as ery effects,which reduce the storage rate and finally lead to /入dislocation steady state behavior under constant deformation conditions O=Odislocation 1/入solute (4) like temperature and strain rate. 1/ 1/入 For the sake of simplicity and speed of calculation,the This makes sure that the dominating obstacle type will also dislocation loops are considered to be of square shape and dominate the activation energy. the loop expansion is represented by the motion of one of its Combining Egs.(1)and(2),the effective stresses tefr in segments.In our case,it is specifically an edge segment that cell walls and cell interiors can be calculated,depending on is considered.The percolating motion of the loop and its fi- strain rate,temperature and dislocation densities.To obtain nal halt are here considered as the motion of the observed the flow stress for cell walls or cell interiors,the athermal segment of the loop and its cease of movement after trav- stress due to long range dislocation interactions has to be elling a slip length L.The slip length is determined by the added spacing of obstacles which a dislocation might encounter on its way through the crystal Tx Teff.x +a(ibpx,x=i,w (5) where a is a constant of the order of one.The required 2.I.The kinetic equation of state external stress is then calculated as the weighted sum of the The kinetic equation of state relates the imposed deforma- stresses in the two volumes,multiplied with the polycrystal Taylor factor M: tion conditions and the actual values of the internal variables to the necessary external flow stress.The equation used in Gext M(fiti+fwtw) (6) the 3IVM is based on the Orowan equation and reads y=M=pmbu where fi and f are the volume fractions of cell interiors (1) and cell walls,respectively. In this equation,is the imposed strain rate,M the aver- Second phase particles can either be taken into account age polycrystal Taylor factor,y the glide rate,and pm is the as another obstacle type in the kinetic equation of state in density of mobile dislocations.The mean dislocation veloc- the case that they can be cut by dislocations or they can ity v in Eq.(1)is calculated under the assumption of stress be considered by adding an Orowan stress in Eq.(6)in the assisted and thermally activated overcoming of short range case of particles that dislocations have to circumvent.The obstacles: structure evolution equations for the dislocations densities and examples for the resulting evolution during deformation U=入vo exp 1 (2) can be found elsewhere [1. where is the jump width,vo the attack frequency,O the 2.2.Interfaces to other microstructure models and FEM effective activation energy for passing of obstacles,and V is the activation volume.The used activation energy is an ef- Several links of the 31VM to other microstructure models fective value,comprising all short range obstacles like the are obvious and necessary.The average polycrystal Taylor cutting of forest dislocations,dragging of jogs and also so- factor is required in the model,and its evolution during de- lute atoms.The influence of dislocation interaction and so- formation can be calculated from simple deformation texture lute effects on the activation energy is balanced according models like a standard Taylor model.The 3IVM can then
M. Goerdeler et al. / Materials Science and Engineering A 387–389 (2004) 266–271 267 2. The model We confine ourselves to cell forming metals and alloys, which comprise most commercial aluminum, copper and ferrous alloys. A cellular arrangement of dislocations develops during straining, composed of cell walls featuring a high dislocation density which enclose cell interiors with a much lower dislocation density. Upon loading dislocation sources generate mobile dislocation loops, which carry the imposed plastic strain by their movement and interact with the considered microstructural constituents. These microstructural constituents include the immobile dislocations in the cell walls and interiors, respectively, and solute atoms as well as second phase particles, like precipitates. This interaction will eventually lead to immobilization of dislocations and their storage either in cell interiors or cell walls. As the temperature of deformation is increased, these stored dislocations become increasingly subjected to dynamic recovery effects, which reduce the storage rate and finally lead to steady state behavior under constant deformation conditions like temperature and strain rate. For the sake of simplicity and speed of calculation, the dislocation loops are considered to be of square shape and the loop expansion is represented by the motion of one of its segments. In our case, it is specifically an edge segment that is considered. The percolating motion of the loop and its fi- nal halt are here considered as the motion of the observed segment of the loop and its cease of movement after travelling a slip length L. The slip length is determined by the spacing of obstacles which a dislocation might encounter on its way through the crystal. 2.1. The kinetic equation of state The kinetic equation of state relates the imposed deformation conditions and the actual values of the internal variables to the necessary external flow stress. The equation used in the 3IVM is based on the Orowan equation and reads γ˙ = ε˙M = ρmbv (1) In this equation, ε˙ is the imposed strain rate, M the average polycrystal Taylor factor, γ˙ the glide rate, and ρm is the density of mobile dislocations. The mean dislocation velocity v in Eq. (1) is calculated under the assumption of stress assisted and thermally activated overcoming of short range obstacles: v = λυ0 exp − Q kBT sinh τeffV kBT (2) where λ is the jump width, v0 the attack frequency, Q the effective activation energy for passing of obstacles, and V is the activation volume. The used activation energy is an effective value, comprising all short range obstacles like the cutting of forest dislocations, dragging of jogs and also solute atoms. The influence of dislocation interaction and solute effects on the activation energy is balanced according to their respective influence on the obstacle distance as detailed below. The jump width corresponds to the mean spacing of obstacles. In former versions of the 3IVM only the mean spacing of forest dislocations in cell walls and interiors was used as obstacle spacing. In the current version also the spacing of solute atoms, in the specific case investigated in this paper magnesium atoms, is taken into account. The approach of linear reciprocal superposition of the different obstacle spacings is adopted. The equation for the jump width then reads 1 λi,w = √ρi,w + 1 lF (3) The subscripts i, w denote the values obtained in cell interiors and cell walls, respectively, and lF is the Friedel length, used as measure for the effective spacing of solute atoms. With the overall obstacle spacing, the activation energy is calculated as Q = Qdislocation 1/λdislocation 1/λ + Qsolute 1/λsolute 1/λ (4) This makes sure that the dominating obstacle type will also dominate the activation energy. Combining Eqs. (1) and (2), the effective stresses τeff in cell walls and cell interiors can be calculated, depending on strain rate, temperature and dislocation densities. To obtain the flow stress for cell walls or cell interiors, the athermal stress due to long range dislocation interactions has to be added τx = τeff,x + αGb√ρx, x = i, w (5) where α is a constant of the order of one. The required external stress is then calculated as the weighted sum of the stresses in the two volumes, multiplied with the polycrystal Taylor factor M: σext = M(fiτi + fwτw) (6) where fi and fw are the volume fractions of cell interiors and cell walls, respectively. Second phase particles can either be taken into account as another obstacle type in the kinetic equation of state in the case that they can be cut by dislocations or they can be considered by adding an Orowan stress in Eq. (6) in the case of particles that dislocations have to circumvent. The structure evolution equations for the dislocations densities and examples for the resulting evolution during deformation can be found elsewhere [1]. 2.2. Interfaces to other microstructure models and FEM Several links of the 3IVM to other microstructure models are obvious and necessary. The average polycrystal Taylor factor is required in the model, and its evolution during deformation can be calculated from simple deformation texture models like a standard Taylor model. The 3IVM can then
268 M.Goerdeler et al./Materials Science and Engineering A 387-389 (2004)266-271 experimental data 200 —0.45%Mg300°10s4 180 一0.45%Mg400°10s1 160 =2.50%Mg400°10s1 140 -4.50%Mg400°10s1 120 fitted data 100 ··0.45%Mg300°10s1 ssajs enJL 80 s■ 0.45%Mg400°10s1 predicted data -2.50%Mg40010s -4.50%Mg40010s1 20 0.0 0.10.2 0.30.40.5 0.6 0.7 0.8 True strain [1] Fig.1.Predicted flow curves of alloys with different Mg content in comparison with experimental data. calculate the flow stress and dislocation density evolution All other alloy contents and the processing route were the taking the strain-dependent Taylor factor into account. same for these alloys.Flow curves were measured in uniaxial The 3IVM as well as an FC Taylor model have also been compression at four different strain rates and three different implemented into the implicit FE software LARSTRAN.When temperatures for each alloy(examples shown as solid lines in using this interactive modeling scheme,the FE code calls Fig.1).The parameters of the 3IVM were then optimized for the 31VM routine at each Gauss point of the FE grid dur- the set of flow curves of the alloy with the lowest magnesium ing each iteration.The local alloy composition,Taylor fac- content(0.46%,dotted curves in Fig.1).Subsequently only tor,temperature,strain rate and length of the time step are the alloy composition was changed in the parameter set to transferred to the 3IVM by the FE code.In turn,the 3IVM predict flow curves of the other alloys (2.5 and 4.5%Mg calculates the dislocation density evolution and the resulting content,dashed curves in Fig.1).The result of this first flow stress.The Taylor factor is updated incrementally by a attempt to include solute hardening effects into the 3IVM Taylor model in each element [3,5]. is very promising,the flow curves of the alloys with higher Another microstructure model closely linked to the 31VM solute content are very well predicted(see also [12]). is the Classical Nucleation and Growth(ClaNG)model,a A small slab of the commercial aluminum alloy AA2024 statistical model for the evolution of second phase parti- (Al-4%Cu-1%Mg)and a binary model alloy containing cles during the processing of aluminum alloys [11].This only Al and Cu were cast and homogenized in the same model allows to calculate the precipitation of new particles way.The evolution of microstructure,solute and phase dis- (dispersoids)during thermomechanical treatment as well as tribution during casting were simulated with various mod- the dissolution of second phase particles remnant from cast- els [3,4].The results were then passed onto the above men- ing(constituents).The results of the ClaNG model,second tioned ClaNG model,for the subsequent simulation of sec- phase particle size and volume fraction and the new matrix ond phase constituent dissolution and precipitation of dis- solute concentrations obviously influence the results of the persoids during homogenization at 480C for 12 h.The re- 3IVM as explained in the previous section. sulting microstructure information was taken as input for the The results of the 31VM in terms of dislocation densities 3IVM. can be utilized as input to model recrystallization of the de- The necessary tuning of the material constants in the formed material.For high calculation speed such a detailed 3IVM was done with compression samples machined from analysis usually is not done inline in a FE calculation but as the model alloy and tested at different strain rates and tem- post-processing of the FE data. peratures.Using the microstructure information available from the models,the 3IVM was then optimized for the mea- sured compression test data of the model alloy,giving the 3.Applications results shown in Fig.2. For the hot rolling of the alloy AA2024,the results of the The flow stress model formulation for the influence of ClaNG model for the homogenization process of this alloy solutes was tested for a field of flow curves measured on a were fed into the 3IVM and the flow stress behavior of this set of three model alloys with different magnesium contents. alloy was predicted.To exemplify the changes of the flow
268 M. Goerdeler et al. / Materials Science and Engineering A 387–389 (2004) 266–271 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 20 40 60 80 100 120 140 160 180 200 experimental data 0.45% Mg 300˚ 10s-1 0.45% Mg 400˚ 10s-1 2.50% Mg 400˚ 10s-1 4.50% Mg 400˚ 10s-1 fitted data 0.45% Mg 300˚ 10s-1 0.45% Mg 400˚ 10s-1 predicted data 2.50% Mg 400˚ 10s-1 4.50% Mg 400˚ 10s-1 True stress [MPa] True strain [1] Fig. 1. Predicted flow curves of alloys with different Mg content in comparison with experimental data. calculate the flow stress and dislocation density evolution taking the strain-dependent Taylor factor into account. The 3IVM as well as an FC Taylor model have also been implemented into the implicit FE software Larstran. When using this interactive modeling scheme, the FE code calls the 3IVM routine at each Gauss point of the FE grid during each iteration. The local alloy composition, Taylor factor, temperature, strain rate and length of the time step are transferred to the 3IVM by the FE code. In turn, the 3IVM calculates the dislocation density evolution and the resulting flow stress. The Taylor factor is updated incrementally by a Taylor model in each element [3,5]. Another microstructure model closely linked to the 3IVM is the Classical Nucleation and Growth (ClaNG) model, a statistical model for the evolution of second phase particles during the processing of aluminum alloys [11]. This model allows to calculate the precipitation of new particles (dispersoids) during thermomechanical treatment as well as the dissolution of second phase particles remnant from casting (constituents). The results of the ClaNG model, second phase particle size and volume fraction and the new matrix solute concentrations obviously influence the results of the 3IVM as explained in the previous section. The results of the 3IVM in terms of dislocation densities can be utilized as input to model recrystallization of the deformed material. For high calculation speed such a detailed analysis usually is not done inline in a FE calculation but as post-processing of the FE data. 3. Applications The flow stress model formulation for the influence of solutes was tested for a field of flow curves measured on a set of three model alloys with different magnesium contents. All other alloy contents and the processing route were the same for these alloys. Flow curves were measured in uniaxial compression at four different strain rates and three different temperatures for each alloy (examples shown as solid lines in Fig. 1). The parameters of the 3IVM were then optimized for the set of flow curves of the alloy with the lowest magnesium content (0.46%, dotted curves in Fig. 1). Subsequently only the alloy composition was changed in the parameter set to predict flow curves of the other alloys (2.5 and 4.5% Mg content, dashed curves in Fig. 1). The result of this first attempt to include solute hardening effects into the 3IVM is very promising, the flow curves of the alloys with higher solute content are very well predicted (see also [12]). A small slab of the commercial aluminum alloy AA2024 (Al–4%Cu–1%Mg) and a binary model alloy containing only Al and Cu were cast and homogenized in the same way. The evolution of microstructure, solute and phase distribution during casting were simulated with various models [3,4]. The results were then passed onto the above mentioned ClaNG model, for the subsequent simulation of second phase constituent dissolution and precipitation of dispersoids during homogenization at 480 ◦C for 12 h. The resulting microstructure information was taken as input for the 3IVM. The necessary tuning of the material constants in the 3IVM was done with compression samples machined from the model alloy and tested at different strain rates and temperatures. Using the microstructure information available from the models, the 3IVM was then optimized for the measured compression test data of the model alloy, giving the results shown in Fig. 2. For the hot rolling of the alloy AA2024, the results of the ClaNG model for the homogenization process of this alloy were fed into the 3IVM and the flow stress behavior of this alloy was predicted. To exemplify the changes of the flow
M.Goerdeler et al./Materials Science and Engineering A 387-389 (2004)266-271 269 180 160 140 120 ledW]ssans 100 50 0090000909g000004-6o 60 AlCu4,experimental,400C,1 s-1 AlCu4,optimised,400C,1 s-1 40 AlCu4,experimental,400C,10 s-1 AICu4,optimised,400C,10 s-1 20 AA2024,predicted,400C,1 s-1 --AA2024,predicted,400C,10 s-1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Strain [1] Fig.2.Measured flow curves and optimized 3IVM results for the Al-Cu model alloy and predicted flow curves for alloy AA2024 stress resulting from the changed alloy composition some of a small slab (e.g.rapid quenching of the sample surface compression test flow curves were calculated for this alloy in contact with the cold work rolls)a satisfying result is ob- too,which are also shown in Fig.2.But these calculated tained.It is emphasized that the numerical result of this case AA2024 compression flow curves were not checked against study is a true prediction. measurements,instead the 3IVM results where directly used Obviously this agreement of the final macroscopic results in the interactive FE simulation of the hot rolling process. does not guarantee the correct prediction of local quantities The results of this macroscopic FE computation where But the results of the ClaNG model have been experimen- then directly compared to the measured values during pro- tally validated for several alloys and annealing cycles [11] cessing of the material.Fig.3 shows the results for the case And the capability of the 31VM to take changes in the so- of the rolling force needed in a rolling pass of the AA2024 lute content into account has already been shown,too [12]. material.After smoothing of the spikes in the simulation re- Though dislocation densities have not been measured for sults which are due to numerical instabilities in the FE cal- this case study,the calculated densities where checked to be culation and correcting for the estimated experimental errors in a sensible range for a rolling pass at high temperature and due to the inhomogeneous process of laboratory hot rolling not too high strain and strain rate,around 2 x 1013m-2. 500 400 300 200 100 ---Experiment -Simulation (Texture+3IVM) 0 0 50 100 150 200 250 roll path [mm] Fig 3.Measured and predicted rolling force for a hot rolling pass of aluminum alloy AA2024.The horizontal lines indicate the estimated average values after correction for numerical instabilities in the model results and typical experimental uncertainties in laboratory hot rolling of aluminum alloys
M. Goerdeler et al. / Materials Science and Engineering A 387–389 (2004) 266–271 269 Fig. 2. Measured flow curves and optimized 3IVM results for the Al–Cu model alloy and predicted flow curves for alloy AA2024. stress resulting from the changed alloy composition some compression test flow curves were calculated for this alloy too, which are also shown in Fig. 2. But these calculated AA2024 compression flow curves were not checked against measurements, instead the 3IVM results where directly used in the interactive FE simulation of the hot rolling process. The results of this macroscopic FE computation where then directly compared to the measured values during processing of the material. Fig. 3 shows the results for the case of the rolling force needed in a rolling pass of the AA2024 material. After smoothing of the spikes in the simulation results which are due to numerical instabilities in the FE calculation and correcting for the estimated experimental errors due to the inhomogeneous process of laboratory hot rolling Fig. 3. Measured and predicted rolling force for a hot rolling pass of aluminum alloy AA2024. The horizontal lines indicate the estimated average values after correction for numerical instabilities in the model results and typical experimental uncertainties in laboratory hot rolling of aluminum alloys. of a small slab (e.g. rapid quenching of the sample surface in contact with the cold work rolls) a satisfying result is obtained. It is emphasized that the numerical result of this case study is a true prediction. Obviously this agreement of the final macroscopic results does not guarantee the correct prediction of local quantities. But the results of the ClaNG model have been experimentally validated for several alloys and annealing cycles [11]. And the capability of the 3IVM to take changes in the solute content into account has already been shown, too [12]. Though dislocation densities have not been measured for this case study, the calculated densities where checked to be in a sensible range for a rolling pass at high temperature and not too high strain and strain rate, around 2 × 1013 m−2
270 M.Goerdeler et al./Materials Science and Engineering A 387-389 (2004)266-271 P1 Centre (S=0) EXP f(g)max =6.6 GIA f(g)max =9.4 4 02=45 9=65 9=90 a sections of the ODF,contour levels:1.2/2/4/7 Surface (S=1) EXP f(g)max =4.3 GIA f(g)max =5.2 945 9=65 902=90° sections of the ODF,contour levels:1.2/2/4 Fig.4.Measured (marked EXP)and predicted(marked GIA)crystallographic texture in the hot rolled AA2024,next to the surface (s =1)and close to center (s =0). Also the texture of the processed material was predicted of real prediction of the mechanical properties of an alu- simultaneously with a very satisfying result,even for the minum alloy during the production process.The use of the difficult to predict sample surface,as shown in Fig.4.Details 31VM in FE calculations results in a spatially resolved dis- on the texture calculation also for the subsequent annealing location density information in the deformed metallic ma- can be found elsewhere [13]. terial,which may serve as input for a subsequent modeling of the recrystallization of the formed products. 4.Summary and conclusions Acknowledgements A microstructural flow stress model based on the consid- Financial support by the Deutsche Forschungsgemein- eration of three dislocation densities as internal state vari- ables was presented.The model consists of a kinetic equa- schaft(DFG)in the Collaborative Research Center 370"In- tion of state and evolution laws for the dislocation densities. tegral Modeling of Materials"is gratefully acknowledged. An essential feature of the 3IVM is its ease of use in FE codes and in integrative modeling schemes.The model was used for predictions of the effects of chemical composition References on the flow stress in an integrative approach to the model- ing of the production process of an aluminum alloy.Good [1]F.Roters,D.Raabe,G.Gottstein,Acta Mater.48(2000)4181 agreement with experimental results was obtained in a case [2]M.Goerdeler,G.Gottstein,Mater.Sci.Eng.A 309-310 (2001)377
270 M. Goerdeler et al. / Materials Science and Engineering A 387–389 (2004) 266–271 Fig. 4. Measured (marked EXP) and predicted (marked GIA) crystallographic texture in the hot rolled AA2024, next to the surface (s = 1) and close to center (s = 0). Also the texture of the processed material was predicted simultaneously with a very satisfying result, even for the difficult to predict sample surface, as shown in Fig. 4. Details on the texture calculation also for the subsequent annealing can be found elsewhere [13]. 4. Summary and conclusions A microstructural flow stress model based on the consideration of three dislocation densities as internal state variables was presented. The model consists of a kinetic equation of state and evolution laws for the dislocation densities. An essential feature of the 3IVM is its ease of use in FE codes and in integrative modeling schemes. The model was used for predictions of the effects of chemical composition on the flow stress in an integrative approach to the modeling of the production process of an aluminum alloy. Good agreement with experimental results was obtained in a case of real prediction of the mechanical properties of an aluminum alloy during the production process. The use of the 3IVM in FE calculations results in a spatially resolved dislocation density information in the deformed metallic material, which may serve as input for a subsequent modeling of the recrystallization of the formed products. Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft (DFG) in the Collaborative Research Center 370 “Integral Modeling of Materials” is gratefully acknowledged. References [1] F. Roters, D. Raabe, G. Gottstein, Acta Mater. 48 (2000) 4181. [2] M. Goerdeler, G. Gottstein, Mater. Sci. Eng. A 309–310 (2001) 377
M.Goerdeler et al./Materials Science and Engineering A 387-389 (2004)266-271 271 [3]L.Neumann,H.Aretz,R.Kopp,M.Goerdeler,M.Crumbach,G.[10]Y.Estrin,L.S.Toth,A.Molinari,Y.Brechet,Acta Mater.46(1998) Gottstein,Z.Metallk.94 (2003)593. 5509. [4]L.Neumann,R.Kopp,A.Ludwig,M.Wu,A.Buhrig-Polaczek,M. [11]M.Schneider,G.Gottstein,L.Lochte,J.Hirsch,Mater.Sci.Forum Schneider,M.Crumbach,G.Gottstein,Model.Simul.Mater.Sci. 396-402(2002)345. Eng.12(2004)519. [12]M.Goerdeler,G.Gottstein,in:G.Gottstein,D.Molodov (Eds.), [5]H.Aretz,R.Luce,M.Wolske,R.Kopp,M.Goerdeler,V.Marx,G. Proceedings of the First Joint International Conference on Re- Pomana,G.Gottstein,Model.Simul.Mater.Sci.Eng.8(2000)881. crystallisation and Grain Growth,Springer-Verlag,Berlin,2001, [6]U.F.Kocks,J.Eng.Mater.Technol.98(1976)72. p.987. [7]H.Mecking,U.F.Kocks,Acta Metall.29 (1981)1865. [13]M.Crumbach,M.Goerdeler,G.Gottstein,L.Neumann,A.Aretz, [8]H.Mughrabi,Acta Metall.31 (1983)1367. R.Kopp,Model.Simul.Mater.Sci.Eng.12(2004)51. [9]M.Zehetbauer,V.Seumer,Acta Metall.Mater.41 (1993)577. [14]F.Roters,Phys.Status Solidi B 240 (2003)68
M. Goerdeler et al. / Materials Science and Engineering A 387–389 (2004) 266–271 271 [3] L. Neumann, H. Aretz, R. Kopp, M. Goerdeler, M. Crumbach, G. Gottstein, Z. Metallk. 94 (2003) 593. [4] L. Neumann, R. Kopp, A. Ludwig, M. Wu, A. Bührig-Polaczek, M. Schneider, M. Crumbach, G. Gottstein, Model. Simul. Mater. Sci. Eng. 12 (2004) 519. [5] H. Aretz, R. Luce, M. Wolske, R. Kopp, M. Goerdeler, V. Marx, G. Pomana, G. Gottstein, Model. Simul. Mater. Sci. Eng. 8 (2000) 881. [6] U.F. Kocks, J. Eng. Mater. Technol. 98 (1976) 72. [7] H. Mecking, U.F. Kocks, Acta Metall. 29 (1981) 1865. [8] H. Mughrabi, Acta Metall. 31 (1983) 1367. [9] M. Zehetbauer, V. Seumer, Acta Metall. Mater. 41 (1993) 577. [10] Y. Estrin, L.S. Tóth, A. Molinari, Y. Bréchet, Acta Mater. 46 (1998) 5509. [11] M. Schneider, G. Gottstein, L. Löchte, J. Hirsch, Mater. Sci. Forum 396–402 (2002) 345. [12] M. Goerdeler, G. Gottstein, in: G. Gottstein, D. Molodov (Eds.), Proceedings of the First Joint International Conference on Recrystallisation and Grain Growth, Springer-Verlag, Berlin, 2001, p. 987. [13] M. Crumbach, M. Goerdeler, G. Gottstein, L. Neumann, A. Aretz, R. Kopp, Model. Simul. Mater. Sci. Eng. 12 (2004) 51. [14] F. Roters, Phys. Status Solidi B 240 (2003) 68
