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 flow268 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 dur￾ing each iteration. The local alloy composition, Taylor fac￾tor, 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 parti￾cles 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 cast￾ing (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 de￾formed 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 dis￾tribution during casting were simulated with various mod￾els [3,4]. The results were then passed onto the above men￾tioned ClaNG model, for the subsequent simulation of sec￾ond phase constituent dissolution and precipitation of dis￾persoids during homogenization at 480 ◦C for 12 h. The re￾sulting 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 tem￾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 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