基于序参量梯度的改进相场模型及对大尺度体系马氏体相变的模拟研究
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摘要
基于序参量变化在界面区域远大于非界面区域的特性,构造了全局修正函数,建立了适用于可调尺度体系马氏体相变模拟的相场模型。在不改变界面能密度的情况下,通过调整界面区域的体积自由能密度差与梯度能系数,有效增大了原相场模型中的界面宽度,实现了大尺度下的高效模拟,并能很好地表征马氏体相变。结果表明,改进后的相场模型能很好地解决原相场模型在大尺度体系模拟时存在的如生长速率过快、位向关系不合理及组织形貌杂乱无序等问题,模拟结果与实验结果符合较好。
The materials design and fabrication based on predicting microstructure have been drawn increasing attention from scientists and engineers. Martensite microstructure, which is well observed in many materials, has significant influence on physical and mechanical properties of the materials. Some experimental studies have been launched to understand the featured microstructure and its evolution in martensitic transformations(MT). Meantime, numerical approaches are often employed to assist the experimental studies due to the complex and nonlinear nature of MT. The phase field method is one of the most powerful tools in predicting microstructure. Due to the diffuse-interface description, phase field method can be used to simulate arbitrary morphologies without tracking the interface. As a consequence, the interface must contain enough elements to obtain reasonable results by using finite element method. On the other hand, the width of the interface is several orders smaller than its real counterpart. More compu-tational resources are required to resolve the phase field variables at the interface with the system size increased. Therefore, the simulation is restricted in smaller system even with state-of-the-art computer power. For arbitrary model formulations, the interfacial energy depends on the interfacial width and other specific properties of materials. However, the phase field models of martensitic transformation do not have enough degrees of freedom to increase the interfacial width without changing the interfacial energy. In the present study, a scalable phase field model by introducing a global modified function is constructed to study MT, the modified function takes into account the inhomogeneous nature of order parameter gradient across the interfacial region. Through adjusting the free energy density and gradient coefficient,meanwhile keeping the interfacial energy density unchanged, the interfacial width and system size are increased, yet the MT feature can be fully characterized. The simulation results show that the modified phase field model can well solve the drawbacks such as fast growth rate of martensite, artificial orientation relationship between the variants of martensite, and disordered martensite microstructure in large scale system.
The materials design and fabrication based on predicting microstructure have been drawn increasing attention from scientists and engineers. Martensite microstructure, which is well observed in many materials, has significant influence on physical and mechanical properties of the materials. Some experimental studies have been launched to understand the featured microstructure and its evolution in martensitic transformations(MT). Meantime, numerical approaches are often employed to assist the experimental studies due to the complex and nonlinear nature of MT. The phase field method is one of the most powerful tools in predicting microstructure. Due to the diffuse-interface description, phase field method can be used to simulate arbitrary morphologies without tracking the interface. As a consequence, the interface must contain enough elements to obtain reasonable results by using finite element method. On the other hand, the width of the interface is several orders smaller than its real counterpart. More compu-tational resources are required to resolve the phase field variables at the interface with the system size increased. Therefore, the simulation is restricted in smaller system even with state-of-the-art computer power. For arbitrary model formulations, the interfacial energy depends on the interfacial width and other specific properties of materials. However, the phase field models of martensitic transformation do not have enough degrees of freedom to increase the interfacial width without changing the interfacial energy. In the present study, a scalable phase field model by introducing a global modified function is constructed to study MT, the modified function takes into account the inhomogeneous nature of order parameter gradient across the interfacial region. Through adjusting the free energy density and gradient coefficient,meanwhile keeping the interfacial energy density unchanged, the interfacial width and system size are increased, yet the MT feature can be fully characterized. The simulation results show that the modified phase field model can well solve the drawbacks such as fast growth rate of martensite, artificial orientation relationship between the variants of martensite, and disordered martensite microstructure in large scale system.
引文
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[11]Yamanaka A,Takaki T,Tomita Y.Elastoplastic phase-field simulation of self-and plastic accommodations in cubic→tetragonal martensitic transformation[J].Mater.Sci.Eng.,2008,A491:378
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[14]Man J,Zhang J H,Rong Y H.Three-dimensional phase field study on strain self-accommodation in martenstic transformation[J].Acta Metall.Sin.,2010,46:775(满蛟,张骥华,戎咏华.马氏体相变中应变自协调效应的三维相场研究[J].金属学报,2010,46:775)
[15]Ke C B,Ma X,Zhang X P.Phase field simulation of effects of pores on B2-R phase transformation in Ni Ti shape memory alloy[J].Acta Metall.Sin.,2011,47:129(柯常波,马骁,张新平.孔隙对Ni Ti形状记忆合金中B2-R相变影响的相场模拟[J].金属学报,2011,47:129)
[16]She H,Liu Y,Wang B.Phase field simulation of heterogeneous cubic→tetragonal martensite nucleation[J].Int.J.Solids Struct.,2013,50:1187
[17]Shen C,Chen Q,Wen Y H,et al.Increasing length scale of quantitative phase field modeling of growth-dominant or coarseningdominant process[J].Scr.Mater.,2004,50:1023
[18]Artemev A,Jin Y,Khachaturyan A G.Three-dimensional phase field model of proper martensitic transformation[J].Acta Mater.,2001,49:1165
[19]Zhong Y,Zhu T.Phase-field modeling of martensitic microstructure in Ni Ti shape memory alloys[J].Acta Mater.,2014,75:337
[20]Cahn J W,Hilliard J E.Free energy of a nonuniform system.I.Interfacial free energy[J].J.Chem.Phys.,1958,28:258
[21]Gunton J D,Miguel M,Sahni P S.Phase Transitions and Critical Phenomena[M].New York:Academic,1983:267
[22]Moelans N,Blanpain B,Wollants P.An introduction to phase-field modeling of microstructure evolution[J].Calphad,2008,32:268
[23]Bhattacharya K,Kohn R V.Symmetry,texture and the recoverable strain of shape-memory polycrystals[J].Acta Mater.,1996,44:529
[24]Wagner M F X,Windl W.Lattice stability,elastic constants and macroscopic moduli of Ni Ti martensites from first principles[J].Acta Mater.,2008,56:6232
[25]Jin Y M,Artemev A,Khachaturyan A G.Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal:Simulation of z′martensite in Au Cd alloys[J].Acta Mater.,2001,49:2309
[26]Bhattacharya K.Microstructure of Martensite:Why it Forms and How it Gives Rise to the Shape-Memory Effect[M].Oxford:Oxford University Press,2003:46
[27]Fukuda T,Saburi T,Doi K,et al.Nucleation and self-accommodation of the R-phase in Ti-Ni alloys[J].Mater.Trans.,1992,33:271
[2]Waitz T,Kazykhanov V,Karnthaler H P.Martensitic phase transformations in nanocrystalline Ni Ti studied by TEM[J].Acta Mater.,2004,52:137
[3]Zhang Z Q,Dong L M,Yang Y,et al.Influences of quenching temperature on the microstructure and deformation behaviors of TC16titanium alloy[J].Acta Metall.Sin.,2011,47:1257(张志强,董利民,杨洋等.淬火温度对TC16钛合金显微组织及变形行为的影响[J].金属学报,2011,47:1257)
[4]Kinney C C,Pytlewski K R,Khachaturyan A G,et al.The microstructure of lath martensite in quenched 9Ni steel[J].Acta Mater.2014,69:372
[5]Zhang S H,Wang P,Li D Z,et al.Investigation of trip effect in ZG06Cr13Ni4Mo martensitic stainless steel by in situ synchrotron high energy X-ray diffraction[J].Acta Metall.Sin.,2015,51:1306(张盛华,王培,李殿中等.ZG06Cr13Ni4Mo马氏体不锈钢中TRIP效应的同步辐射高能X射线原位研究[J].金属学报,201551:1306)
[6]Chen L Q.Phase-field models for microstructure evolution[J].Annu.Rev.Mater.Res.,2002,32:113
[7]Wang Y,Khachaturyan A G.Three-dimensional field model and computer modeling of martensitic transformations[J].Acta Mater.1997,45:759
[8]Khachaturyan A G.Theory of Structural Transformations in Solids[M].New York:Wiley,1983:198
[9]Levitas V I,Preston D L.Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations.I.Austenite?martensite[J].Phys.Rev.,2002,66B:134206
[10]Levitas V I,Preston D L.Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations.II.Multivariant phase transformations and stress space analysis[J].Phys.Rev.,2002,66B:134207
[11]Yamanaka A,Takaki T,Tomita Y.Elastoplastic phase-field simulation of self-and plastic accommodations in cubic→tetragonal martensitic transformation[J].Mater.Sci.Eng.,2008,A491:378
[12]Yamanaka A,Takaki T,Tomita Y.Elastoplastic phase-field simulation of martensitic transformation with plastic deformation in polycrystal[J].Int.J.Mech.Sci.,2010,52:245
[13]Idesman A V,Cho J Y,Levitas V I.Finite element modeling of dynamics of martensitic phase transitions[J].Appl.Phys.Lett.,2008,93:043102.
[14]Man J,Zhang J H,Rong Y H.Three-dimensional phase field study on strain self-accommodation in martenstic transformation[J].Acta Metall.Sin.,2010,46:775(满蛟,张骥华,戎咏华.马氏体相变中应变自协调效应的三维相场研究[J].金属学报,2010,46:775)
[15]Ke C B,Ma X,Zhang X P.Phase field simulation of effects of pores on B2-R phase transformation in Ni Ti shape memory alloy[J].Acta Metall.Sin.,2011,47:129(柯常波,马骁,张新平.孔隙对Ni Ti形状记忆合金中B2-R相变影响的相场模拟[J].金属学报,2011,47:129)
[16]She H,Liu Y,Wang B.Phase field simulation of heterogeneous cubic→tetragonal martensite nucleation[J].Int.J.Solids Struct.,2013,50:1187
[17]Shen C,Chen Q,Wen Y H,et al.Increasing length scale of quantitative phase field modeling of growth-dominant or coarseningdominant process[J].Scr.Mater.,2004,50:1023
[18]Artemev A,Jin Y,Khachaturyan A G.Three-dimensional phase field model of proper martensitic transformation[J].Acta Mater.,2001,49:1165
[19]Zhong Y,Zhu T.Phase-field modeling of martensitic microstructure in Ni Ti shape memory alloys[J].Acta Mater.,2014,75:337
[20]Cahn J W,Hilliard J E.Free energy of a nonuniform system.I.Interfacial free energy[J].J.Chem.Phys.,1958,28:258
[21]Gunton J D,Miguel M,Sahni P S.Phase Transitions and Critical Phenomena[M].New York:Academic,1983:267
[22]Moelans N,Blanpain B,Wollants P.An introduction to phase-field modeling of microstructure evolution[J].Calphad,2008,32:268
[23]Bhattacharya K,Kohn R V.Symmetry,texture and the recoverable strain of shape-memory polycrystals[J].Acta Mater.,1996,44:529
[24]Wagner M F X,Windl W.Lattice stability,elastic constants and macroscopic moduli of Ni Ti martensites from first principles[J].Acta Mater.,2008,56:6232
[25]Jin Y M,Artemev A,Khachaturyan A G.Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal:Simulation of z′martensite in Au Cd alloys[J].Acta Mater.,2001,49:2309
[26]Bhattacharya K.Microstructure of Martensite:Why it Forms and How it Gives Rise to the Shape-Memory Effect[M].Oxford:Oxford University Press,2003:46
[27]Fukuda T,Saburi T,Doi K,et al.Nucleation and self-accommodation of the R-phase in Ti-Ni alloys[J].Mater.Trans.,1992,33:271