Possibilities of the numerical solution of the dislocation evolution equation for stochastic variables

• Autorzy: Milenin Ivan , Rauch Łukasz , Szeliga Danuta , Pietrzyk Maciej

Warianty tytułu

PL

Możliwości numerycznego rozwiązania równanie opisującego ewolucję dyslokacji dla zmiennych stochastycznych

• Języki publikacji: EN

Abstrakty

EN

The model describing evolution of dislocation population based on fundamental works of Kocks, Estrin and Mecking (KEM) is a useful tool in modelling of metallic materials processing. In combination with the Sandstrom and Lagneborg approach it can predict changes of the dislocation density accounting for hardening, recovery and recrystallization. Numerical solutions of a one-parameter model (average dislocation density), as well as for two types of dislocations and three types of dislocation are described in the literature. All these solutions were performed for deterministic variables. On the other hand, an advanced modelling of materials requires often an information about distribution of parameters. This is the case when uncertainty of the model has to be evaluated or when an information about distribution of product properties is needed. The latter is crucial when deterioration of local formability is caused by sharp gradients of properties. Thus, the investigation of possibilities of numerical solution for the KEM model with stochastic variables was the main objective of the present work. Evolution equation was written for the distribution function and solution was performed using Monte Carlo method. Analysis of the results with respect to the reliability and computing costs was performed. The conclusions towards selection of the best approach were formulated.

PL

Model opisujący ewolucję populacji dyslokacji wykorzystujący fundamentalne prace Kocksa, Estrina i Meckinga (KEM model) jest użytecznym narzędziem w modelowaniu przetwórstwa materiałów metalicznych. W połączeniu z modelem Sandstroma i Lagneborga możliwe jest przewidywanie zmian gęstości dyslokacji uwzględniając zjawiska umocnienia, zdrowienia i rekrystalizacji. Numeryczne rozwiązania dla jednoparametrowego modelu (średniej gęstości dyslokacji), jak i dla dwóch lub trzech rozdajów dyslokacji, jest opisane w literaturze. Te rozwiązania zostały przeprowadzone dla zmiennych deterministycznych. Z drugiej strony zaawansowane modelowanie materiałów wymaga informacji o rozkładzie parametrów. Ma to miejsce np., kiedy potrzebna jest ocena niepewności wyników lub informacja o funkcji rozkładu własności materiału. To ostatnie jest ważne, kiedy obniżenie lokalnej odporności mateiału na pękanie jest powodowane przez ostre gradienty własności. Stąd celem niniejszej pracy była ocena możliwości numerycznego rozwiązania dla modelu KEM ze zmiennymi losowymi. Równanie ewolucji dyslokacji zapisano dla funkcji rozkładu prawdopodobieństwa i przeprowadzono rozwiązanie wykorzystując metodę Monte Carlo. Przeprowadzono analizę wyników w aspekcie ich dokładności oraz oceniając koszty obliczeń. Sformułowane zostały wnioski sugerujące dobór najlepszych parametrów modelu numerycznego.

Słowa kluczowe

EN

evolution of dislocation; stochastic variable; Monte Carlo method

PL

zmienna stochastyczna; metoda Monte Carlo

• Wydawca: Wydawnictwa AGH
• Czasopismo: Computer Methods in Materials Science
• Rocznik: 2019
• Tom: Vol. 19, No. 4
• Strony: 169--173
• Opis fizyczny: Bibliogr. 41 poz., rys.

Twórcy

autor

Milenin Ivan

• AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland

autor

Rauch Łukasz

• AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland

autor

Szeliga Danuta

• AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland

autor

Pietrzyk Maciej

• AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland

Bibliografia

• Bako, B., Groma, I., 1999, Stochastic O(N) algorithm for dislocation dynamics, Modelling and Simulation in Materials Science and Engineering, 7(2) doi:10.1088/0965-0393/7/2/004.
• Chattopadhyay, A., Aifantis, E.C., 2016, Stochastically forced dislocation density distribution in plastic deformation, Physical Review E, 94(2), doi:10.1103/PhysRevE.94.022139.
• Dini, H., Svoboda, A., Andersson, N.-E., Ghassemali, E., 2018, Optimization and validation of a dislocation density based constitutive model for as-cast Mg-9%Al-1%Zn, Materials Science & Engineering A, 710, 17-26.
• Estrin, Y., 1996, Dislocation density related constitutive modelling, in: Unified constitutive laws of plastic deformation, eds, Krausz, A.S., Krausz, K., Academic Press. Estrin, Y., Mecking, H., 1984, A unified phenomenological description of work hardening and creep based on oneparameter models, Acta Metallurgica, 32, 57-70.
• Gao, Z., Feng, J., Wang, Z., Niu, J., Sommitsch, C., 2019, Dislocation density-based modeling of dynamic recrystallized microstructure and process in friction stir spot welding of AA6082, Metals, 9, 672; doi:10.3390/met9060672.
• Guo, R., Wu, J., 2018, Dislocation density based model for Al-Cu-Mg alloy during quenching with considering the quench-induced precipitates, Journal of Alloys and Compounds, 741, 432-441.
• He, X., Yao, Y., 2017, A dislocation density based viscoplastic constitutive model for lead free solder under drop impact, International Journal of Solids and Structures, 120, 236-244.
• Huang, M.X., Rivera-Diaz-del-Castillo, P., Bouaziz, O., Zwaag, S., 2009, Modelling plastic deformation of metals over a wide range of strain rates using irreversible thermodynamics, IOP Conference Series: Materials Science and Engineering, 3, doi.org/10.1088/1757-899X/3/1/012006.
• Huang, G.R., Huang, J.C., Tsai, W.Y., 2016, Origin of sample size effect: Stochastic dislocation formation in crystalline metals at small scales, Scientific Reports, 6, doi.org/10.1038/srep39242
• Huang, C., Deng, J., Wang, S., Liu, L.A., 2017, physical-based constitutive model to describe the strain-hardening and dynamic recovery behaviors of 5754 aluminum alloy, Materials Science and Engineering, 699(24), 106-113.
• Hellinger, E., 1909, Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen, Journal für die reine und angewandte Mathematik, 136, 210-271.
• Levitas, V.I., Roy, A.M., Dean, Preston, L., 2013, Multiple twinning and variant-variant transformations in martensite: Phase-field approach, Physical Review B, 88, 054113.
• Mecking, H., Kocks, U.F., 1981, Kinetics of flow and strainhardening, Acta Metallurgica, 29, 1865-1875.
• Nastac, L., 2018, 3D Stochastic modelling of microstructure evolution during solidification of alloy 718, Proc. 9th Int. Symp. on Superalloy 718 and Derivatives, Energy, Aerospace and Industrial Applications, Springer, TMS, 379-387.
• Nastac, L., Zhang, D., 2014, 3D Stochastic Modeling of microstructure evolution during the solidification of dendritic alloys, in: eds, Bernard, D. Buffière, J.Y., Pollock, T., Friis, H., Rollett, P.A., Uchic, M. , Proc. 2nd Int. Congress on 3D Materials Science, Annecy, The Minerals, Metals & Materials Society.
• Nieto-Fuentes, J.C., Rittel, D., Osovski, S., 2018, Dislocation based constitutive model, International Journal of Plasticity, 108, 55-69.
• Ordon, J., Kuziak, R., Pietrzyk, M., 2000, History dependent constitutive law for austenitic steels, Proc. Int. Conf. Metal Forming 2000, eds, Pietrzyk, M., Kusiak, J., Majta, J., Hartley, P., Pillinger, I., Publ. A. Balkema, Krakow, 747-753.
• Pham, M.-S., Iadicola, M., Creuziger, A., Hu, L., Rollett, A.D., 2015, Thermally-activated constitutive model including dislocation interactions, aging and recovery for strain path dependence of solid solution strengthened alloys: Application to AA5754-O, International Journal of Plasticity, 75, 226-243.
• Poletti, M.C., Bureau, R., Loidolt, P., Simon, P., Mitsche, S., Spuller, M., 2018, Microstructure Evolution in a 6082 Aluminium Alloy during Thermomechanical Treatment, Materials, 11(8), doi.org/10.3390/ma11081319.
• Roters, F., Raabe, D., Gottstein, G., 2000, Work hardening in heterogeneous alloys – a microstructural approach based on three internal state variables, Acta Materialia, 48, 4181-4189.
• Sandstrom, R., Lagneborg, R., 1975, A model for hot working occurring by recrystallization, Acta Metallurgica, 23, 387-398.
• Schacht, K., Motaman, A.H., Prahl, U., Bleck, W., 2017, A unified dislocation density-dependent physical-based constitutive model for cold metal forming, AIP Conference Proceedings, 1896(1), 160020.
• Szeliga, D., Chang, Y., Bleck, W., Pietrzyk, M., 2019, Evaluation of using distribution functions for mean field modelling of multiphase steels, Procedia Manufaturing, 27, 72-77.
• Yadav, S.D., El-Tahawy, M., Kalacska, S., Domankova, M, Yubero, D.C., Poletti, C., 2017, Characterizing dislocation configurations and their evolution during creep of a new 12% Cr steel, Materials Characterization, 134, 387-397.
• Zamani, M., Dini, H., Svoboda, A., Lindgren, L.-E., Seifeddine, S., Andersson, N.E., Jarfors, A.E.W., 2017, A dislocation density based constitutive model for as-cast Al-Si alloys: Effect of temperature and microstructure, International Journal of Mechanical Sciences, 121, 164-170.
• Hellinger, E., 1909, Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen, Journal für die reine und angewandte Mathematik, 136, 210-271.
• Levitas, V.I., Roy, A.M., Dean, Preston, L., 2013, Multiple twinning and variant-variant transformations in martensite: Phase-field approach, Physical Review B, 88, 054113.
• Mecking, H., Kocks, U.F., 1981, Kinetics of flow and strainhardening, Acta Metallurgica, 29, 1865-1875.
• Nastac, L., 2018, 3D Stochastic modelling of microstructure evolution during solidification of alloy 718, Proc. 9th Int. Symp. on Superalloy 718 and Derivatives, Energy, Aerospace and Industrial Applications, Springer, TMS, 379-387.
• Nastac, L., Zhang, D., 2014, 3D Stochastic Modeling of microstructure evolution during the solidification of dendritic alloys, in: eds, Bernard, D. Buffière, J.Y., Pollock, T., Friis, H., Rollett, P.A., Uchic, M. , Proc. 2nd Int. Congress on 3D Materials Science, Annecy, The Minerals, Metals & Materials Society.
• Nieto-Fuentes, J.C., Rittel, D., Osovski, S., 2018, Dislocation based constitutive model, International Journal of Plasticity, 108, 55-69.
• Ordon, J., Kuziak, R., Pietrzyk, M., 2000, History dependent constitutive law for austenitic steels, Proc. Int. Conf. Metal Forming 2000, eds, Pietrzyk, M., Kusiak, J., Majta, J., Hartley, P., Pillinger, I., Publ. A. Balkema, Krakow, 747-753.
• Pham, M.-S., Iadicola, M., Creuziger, A., Hu, L., Rollett, A.D., 2015, Thermally-activated constitutive model including dislocation interactions, aging and recovery for strain path dependence of solid solution strengthened alloys: Application to AA5754-O, International Journal of Plasticity, 75, 226-243.
• Poletti, M.C., Bureau, R., Loidolt, P., Simon, P., Mitsche, S., Spuller, M., 2018, Microstructure Evolution in a 6082 Aluminium Alloy during Thermomechanical Treatment, Materials, 11(8), doi.org/10.3390/ma11081319.
• Roters, F., Raabe, D., Gottstein, G., 2000, Work hardening in heterogeneous alloys – a microstructural approach based on three internal state variables, Acta Materialia, 48, 4181-4189.
• Sandstrom, R., Lagneborg, R., 1975, A model for hot working occurring by recrystallization, Acta Metallurgica, 23, 387-398.
• Schacht, K., Motaman, A.H., Prahl, U., Bleck, W., 2017, A unified dislocation density-dependent physical-based constitutive model for cold metal forming, AIP Conference Proceedings, 1896(1), 160020.
• Szeliga, D., Chang, Y., Bleck, W., Pietrzyk, M., 2019, Evaluation of using distribution functions for mean field modelling of multiphase steels, Procedia Manufaturing, 27, 72-77.
• Yadav, S.D., El-Tahawy, M., Kalacska, S., Domankova, M, Yubero, D.C., Poletti, C., 2017, Characterizing dislocation configurations and their evolution during creep of a new 12% Cr steel, Materials Characterization, 134, 387-397.
• Zamani, M., Dini, H., Svoboda, A., Lindgren, L.-E., Seifeddine, S., Andersson, N.E., Jarfors, A.E.W., 2017, A dislocation density based constitutive model for as-cast Al-Si alloys: Effect of temperature and microstructure, International Journal of Mechanical Sciences, 121, 164-170.
• Zhuang, Z., Liu, Z., Cui, Y., 2019, Dislocation based crystalplasticity: theory and computation at micron and submicron scale, Elsevier, Academic Press, London, 91-119.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).