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Research on viscoelastic behavior and rheological constitutive parameters of metallic glasses based on fractional-differential rheological model

Xu Fu Li Ke-Feng Deng Xu-Hui Zhang Ping Long Zhi-Lin

Research on viscoelastic behavior and rheological constitutive parameters of metallic glasses based on fractional-differential rheological model

Xu Fu, Li Ke-Feng, Deng Xu-Hui, Zhang Ping, Long Zhi-Lin
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  • Metallic glasses offer novel physical, chemical and mechanical properties and have promising potential applications. Recently, exploring the structure and deformation mechanism of metallic glasses according to the rheological mechanical behavior in the nominal elastic region has been the object of intensive research. Physically the mechanical analogues of fractional elements can be represented by self-similarity spring-dashpot fractal networks. In light of the fractal distribution features of the structural heterogeneities, a fractional differential rheological model is introduced to explore the viscoelastic a behavior of metallic glasses in this paper. To investigate the viscoelastic deformation mechanism, carefully designed nanoindentation tests at ambient temperature are proposed in this study. Three kinds of metallic glasses with different Poisson's ratio and glass transition temperature, which have the chemical compositions of Pd40Cu30Ni10P20, Zr48Cu34Pd2Al8Ag8, and (Fe0.432Co0.288B0.192Si0.048Nb0.04) 96Cr4 are selected as the model materials. Experimental and theoretical results clearly indicate that in the nominal elastic region, these metallic glasses exhibit linear viscoelasticity, implying a loading rate-dependent character. Based on the fractional calculus and Riemann-Liouville definition, experimental results are analyzed by the fractional-differential and integer order rheology models respectively. According to the stability of the fitting parameters, here we show that the fractional-differential Kelvin model, which consists of a spring and a fractional dashpot element, can fit the experimental viscoelastic deformation data of the investigated metallic glasses better than that with integer order rheological model. The extracted elastic modulis E1 of the spring in the fractional-differential Kelvin model are comparable to those of samples measured by traditional methods. Such a similarity can be well explained by the mechanical analogue of fractal model proposed for describing the distribution features of the structural heterogeneities in metallic glasses. The rheological parameters obtained here including viscosity index A and fractional order are capable of reflecting the rheological features and the flowing tendency of the above-mentioned metallic glasses. It is found that there exists a clear relationship between the rheological parameters and the reduced glass transition temperature as well as Poisson's ratio, which is helpful for understanding the correlation between plasticity and Poisson's ratio from the micro-structural point of view. The current work provides a rheological model-structure-property relation that may be applicable to a wide range of glassy materials.
      Corresponding author: Xu Fu, xufu@xtu.edu.cn;longzl@xtu.edu.cn ; Long Zhi-Lin, xufu@xtu.edu.cn;longzl@xtu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 51471139, 51401176, 21376199) and the Natural Science Foundation of Hunan Province, China (Grant No. 14JJ3078).
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    [2]

    Liu Y H, Wang D, Nakajima K, Zhang W, Hirata A, Nishi T, Inoue A, Chen M W 2011 Phys. Rev. Lett. 106 125504

    [3]

    Wagner H, Bedorf D, Kchemann S, Schwabe M, Zhang B, Arnold W, Samwer K 2011 Nat. Mater. 10 439

    [4]

    Wang J G, Zhao D Q, Pan M X, Shek C H, Wang W H 2009 Appl. Phys. Lett. 94 031904

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    Huang B, Bai H Y, Wen P, Ding D W, Zhao D Q, Pan M X, Wang W H 2013 J. Appl. Phys. 114 113508

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    Dmowski W, Iwashita T, Chuang C P, Almer J, Egami T 2010 Phys. Rev. Lett. 105 205502

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    Ye J C, Lu J, Liu C T, Wang Q, Yang Y 2010 Nat. Mater. 9 619

    [10]

    Yang Y, Zeng J F, Ye J C, Lu J 2010 Appl. Phys. Lett. 97 261905

    [11]

    Huo L S, Ma J, Ke H B, Bai H Y, Zhao D Q, Wang W H 2012 J. Appl. Phys. 111 113522

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    Park K W, Lee C M, Wakeda M, Shibutani Y, Falk M L, Lee J C 2008 Acta Mater. 56 5440

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    Ke H B, Wen P, Peng H L, Wang W H, Greer A L 2011 Scripta Mater. 64 966

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    Caron A, Kawashima A, Fecht H J, Louzguine-Luzguin D V, Inoue A 2011 Appl. Phys. Lett. 99 171907

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    Fujita T, Wang Z, Liu Y H, Sheng H, Wang W H, Chen M W 2012 Acta Mater. 60 3741

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    Huo L S, Zeng J F, Wang W H, Liu C T, Yang Y 2013 Acta Mater. 61 4329

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    Jiao W, Wen P, Peng H L, Bai H Y, Sun B A, Wang W H 2013 Appl. Phys. Lett. 102 101903

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    Ke H B, Liu C T, Yang Y 2014 Sci. China Tech. Sci. 58 47

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    Wang W H 2013 Prog. Phys. 33 177 (in Chinese) [汪卫华 2013 物理学进展 33 177]

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    Yu H B, Wang W H, Bai H Y, Wu Y, Chen M W 2010 Phys. Rev. B 81 220201

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    Wang W H 2012 Nat. Mater. 11 275

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    Ma D, Stoica A, Wang X L, Lu Z, Clausen B, Brown D 2012 Phys. Rev. Lett. 108 085501

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    Zhao L, Ma C L, Fu M W, Zeng X R 2012 Intermetallics 30 65

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    Wang W H. 2012 Prog. Mater. Sci. 57 487

    [30]

    Liao G K, Long Z L, Xu F, Liu W, Zhang Z Y, Yang M 2015 Acta Phys. Sin. 64 136101 (in Chinese) [廖光开, 龙志林, 许福, 刘为, 张志洋, 杨妙 2015 物理学报 64 136101]

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    Wang W H 2011 J. Appl. Phys. 110 053521

    [32]

    Gao M, Liu S T, Wang Z, Wang W H 2012 Mod. Phys. 24 10 (in Chinese) [高萌, 刘诗彤, 王峥, 汪卫华 2012 现代物理知识 24 10]

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    Heymans N, Bauwens J C 1994 Rheol. Acta 33 210

    [35]

    Zhu K Q, Hu K X, Yang D 2007 Proceedings of the 5th International Conference on Fluid Mechanics Shanghai, China, August 15-19, 2007 p506

    [36]

    Zhang C Y 2006 Viscoelastic Fracture Mechanics (Beijing: Science Press) p23

    [37]

    Ma D, Stoica A D, Wang X L 2009 Nat. Mater. 8 30

    [38]

    Cheng Y Q, Ma E 2011 Prog. Mater. Sci. 56 379

    [39]

    Peng H L, Li M Z, Sun B A, Wang W H 2012 J. Appl. Phys. 112 023516

    [40]

    Sun B A, Wang W H 2010 Physics 39 628 (in Chinese) [孙保安, 汪卫华 2010 物理 39 628]

    [41]

    Ruan L L, Qu S L, Guo Z Y 2010 Chin. Phys. B 19 034204

    [42]

    Sun B A, Yu H B, Jiao W, Bai H Y, Zhao D Q, Wang W H 2010 Phys. Rev. Lett. 105 035501

    [43]

    Sun B A, Pauly S, Tan J, Stoica M, Wang W H, Khn U, Eckert J 2012 Acta Mater. 60 4160

    [44]

    Jiang M Q, Meng J X, Gao J B, Wang X L, Rouxel T, Keryvin V, Ling Z, Dai L H 2010 Intermetallics 18 2468

    [45]

    Xu F 2012 Ph. D. Dissertation (Xiangtan: Xiangtan University) (in Chinese) [许福 2012 博士学位论文 (湘潭:湘潭大学)]

    [46]

    Long Z L, Shao Y, Xie G Q, Zhang P, Shen B L, Inoue A 2008 J. Alloy. Compd 462 52

    [47]

    Zhang Q S, Zhang W, Inoue A 2007 Mater. Trans. 48 3031

    [48]

    Inoue A, Nishiyama N, Masumoto T 1996 Mater. Trans. JIM 37 181

    [49]

    Radok J R M 1957 Q. Appl. Math. 15 198

    [50]

    Lee E H, Radok J R M 1960 J. Appl. Mech. 27 438

    [51]

    Ting T C T 1966 J. Appl. Mech. 33 845

    [52]

    Podlubny I 1999 Fractional Differential Equations (New York: Academic Press) p62

    [53]

    Zhang W M 2006 Ph. D. Dissertation (Xiangtan: Xiangtan University) (in Chinese) [张为民 2006 博士学位论文 (湘潭:湘潭大学)]

    [54]

    Wang J Q, Wang W H, Yu H B, Bai H Y 2009 Appl. Phys. Lett. 94 121904

    [55]

    Baricco M, Baser TA, Das J, Eckert J 2009 J. Alloy. Compd 483 125

    [56]

    Jiang M Q, Dai L H 2007 Phys. Rev. B 76 054204

    [57]

    Wang Z, Wen P, Huo L S, Bai H Y, Wang W H 2012 Appl. Phys. Lett. 101 121906

    [58]

    Johnson W L, Samwer K 2005 Phys. Rev. Lett. 95 195501

    [59]

    Turnbull D, Cohen M H 1961 J. Chem. Phys. 34 120

    [60]

    Miracle D B 2004 Nat. Mater. 3 697

    [61]

    Zhan X L, Zhang X N, Wang D Y, Lu L 2009 Eng. Mech. 26 187 (in Chinese) [詹小丽, 张肖宁, 王端宜, 卢亮 2009 工程力学 26 187]

    [62]

    Zhou H W, Wang C P, Han B B, Duan Z Q 2011 Int. J. Rock. Mech. Min. 48 116

    [63]

    Zhou H W, Wang C P, Duan Z Q, Zhang M, Liu J F 2012 Sci. Sin-Phys. Mech. Astron. 42 310 (in Chinese) [周宏伟, 王春萍, 段志强, 张淼, 刘建锋 2012 中国科学: 物理学力学天文学 42 310]

    [64]

    Wang D P, Zhao D Q, Ding D W, Bai H Y, Wang W H 2014 J. Appl. Phys. 115 123507

  • [1]

    Poulsen H F, Wert J A, Neuefeind J, Honkimki V, Daymond M 2004 Nat. Mater. 4 33

    [2]

    Liu Y H, Wang D, Nakajima K, Zhang W, Hirata A, Nishi T, Inoue A, Chen M W 2011 Phys. Rev. Lett. 106 125504

    [3]

    Wagner H, Bedorf D, Kchemann S, Schwabe M, Zhang B, Arnold W, Samwer K 2011 Nat. Mater. 10 439

    [4]

    Wang J G, Zhao D Q, Pan M X, Shek C H, Wang W H 2009 Appl. Phys. Lett. 94 031904

    [5]

    Hirata A, Guan P F, Fujita T, Hirotsu Y, Inoue A, Yavari A R, Sakurai T, Chen M W 2011 Nat. Mater. 10 28

    [6]

    Yang Y, Zeng J F, Volland A, Blandin J J, Gravier S, Liu C T 2012 Acta Mater. 60 5260

    [7]

    Huang B, Bai H Y, Wen P, Ding D W, Zhao D Q, Pan M X, Wang W H 2013 J. Appl. Phys. 114 113508

    [8]

    Dmowski W, Iwashita T, Chuang C P, Almer J, Egami T 2010 Phys. Rev. Lett. 105 205502

    [9]

    Ye J C, Lu J, Liu C T, Wang Q, Yang Y 2010 Nat. Mater. 9 619

    [10]

    Yang Y, Zeng J F, Ye J C, Lu J 2010 Appl. Phys. Lett. 97 261905

    [11]

    Huo L S, Ma J, Ke H B, Bai H Y, Zhao D Q, Wang W H 2012 J. Appl. Phys. 111 113522

    [12]

    Park K W, Lee C M, Wakeda M, Shibutani Y, Falk M L, Lee J C 2008 Acta Mater. 56 5440

    [13]

    Ke H B, Wen P, Peng H L, Wang W H, Greer A L 2011 Scripta Mater. 64 966

    [14]

    Caron A, Kawashima A, Fecht H J, Louzguine-Luzguin D V, Inoue A 2011 Appl. Phys. Lett. 99 171907

    [15]

    Fujita T, Wang Z, Liu Y H, Sheng H, Wang W H, Chen M W 2012 Acta Mater. 60 3741

    [16]

    Huo L S, Zeng J F, Wang W H, Liu C T, Yang Y 2013 Acta Mater. 61 4329

    [17]

    Jiao W, Wen P, Peng H L, Bai H Y, Sun B A, Wang W H 2013 Appl. Phys. Lett. 102 101903

    [18]

    Xue R J, Wang D P, Zhu Z G, Ding D W, Zhang B, Wang W H 2013 J. Appl. Phys. 114 123514

    [19]

    Zhu Z G, Wen P, Wang D P, Xue R J, Zhao D Q, Wang W H 2013 J. Appl. Phys. 114 083512

    [20]

    Ke H B, Liu C T, Yang Y 2014 Sci. China Tech. Sci. 58 47

    [21]

    Huang B, Bai H Y, Wang W H 2014 J. Appl. Phys. 115 153505

    [22]

    Wang W H 2013 Prog. Phys. 33 177 (in Chinese) [汪卫华 2013 物理学进展 33 177]

    [23]

    Yu H B, Wang W H, Bai H Y, Wu Y, Chen M W 2010 Phys. Rev. B 81 220201

    [24]

    Liu S T, Wang Z, Peng H L, Yu H B, Wang W H 2012 Scripta Mater. 67 4

    [25]

    Argon A S 1979 Acta Metall. 27 47

    [26]

    Wang W H 2012 Nat. Mater. 11 275

    [27]

    Ma D, Stoica A, Wang X L, Lu Z, Clausen B, Brown D 2012 Phys. Rev. Lett. 108 085501

    [28]

    Zhao L, Ma C L, Fu M W, Zeng X R 2012 Intermetallics 30 65

    [29]

    Wang W H. 2012 Prog. Mater. Sci. 57 487

    [30]

    Liao G K, Long Z L, Xu F, Liu W, Zhang Z Y, Yang M 2015 Acta Phys. Sin. 64 136101 (in Chinese) [廖光开, 龙志林, 许福, 刘为, 张志洋, 杨妙 2015 物理学报 64 136101]

    [31]

    Wang W H 2011 J. Appl. Phys. 110 053521

    [32]

    Gao M, Liu S T, Wang Z, Wang W H 2012 Mod. Phys. 24 10 (in Chinese) [高萌, 刘诗彤, 王峥, 汪卫华 2012 现代物理知识 24 10]

    [33]

    Schiessel H, Blumen A 1993 J. Phys. A-Math. Gen. 26 5057

    [34]

    Heymans N, Bauwens J C 1994 Rheol. Acta 33 210

    [35]

    Zhu K Q, Hu K X, Yang D 2007 Proceedings of the 5th International Conference on Fluid Mechanics Shanghai, China, August 15-19, 2007 p506

    [36]

    Zhang C Y 2006 Viscoelastic Fracture Mechanics (Beijing: Science Press) p23

    [37]

    Ma D, Stoica A D, Wang X L 2009 Nat. Mater. 8 30

    [38]

    Cheng Y Q, Ma E 2011 Prog. Mater. Sci. 56 379

    [39]

    Peng H L, Li M Z, Sun B A, Wang W H 2012 J. Appl. Phys. 112 023516

    [40]

    Sun B A, Wang W H 2010 Physics 39 628 (in Chinese) [孙保安, 汪卫华 2010 物理 39 628]

    [41]

    Ruan L L, Qu S L, Guo Z Y 2010 Chin. Phys. B 19 034204

    [42]

    Sun B A, Yu H B, Jiao W, Bai H Y, Zhao D Q, Wang W H 2010 Phys. Rev. Lett. 105 035501

    [43]

    Sun B A, Pauly S, Tan J, Stoica M, Wang W H, Khn U, Eckert J 2012 Acta Mater. 60 4160

    [44]

    Jiang M Q, Meng J X, Gao J B, Wang X L, Rouxel T, Keryvin V, Ling Z, Dai L H 2010 Intermetallics 18 2468

    [45]

    Xu F 2012 Ph. D. Dissertation (Xiangtan: Xiangtan University) (in Chinese) [许福 2012 博士学位论文 (湘潭:湘潭大学)]

    [46]

    Long Z L, Shao Y, Xie G Q, Zhang P, Shen B L, Inoue A 2008 J. Alloy. Compd 462 52

    [47]

    Zhang Q S, Zhang W, Inoue A 2007 Mater. Trans. 48 3031

    [48]

    Inoue A, Nishiyama N, Masumoto T 1996 Mater. Trans. JIM 37 181

    [49]

    Radok J R M 1957 Q. Appl. Math. 15 198

    [50]

    Lee E H, Radok J R M 1960 J. Appl. Mech. 27 438

    [51]

    Ting T C T 1966 J. Appl. Mech. 33 845

    [52]

    Podlubny I 1999 Fractional Differential Equations (New York: Academic Press) p62

    [53]

    Zhang W M 2006 Ph. D. Dissertation (Xiangtan: Xiangtan University) (in Chinese) [张为民 2006 博士学位论文 (湘潭:湘潭大学)]

    [54]

    Wang J Q, Wang W H, Yu H B, Bai H Y 2009 Appl. Phys. Lett. 94 121904

    [55]

    Baricco M, Baser TA, Das J, Eckert J 2009 J. Alloy. Compd 483 125

    [56]

    Jiang M Q, Dai L H 2007 Phys. Rev. B 76 054204

    [57]

    Wang Z, Wen P, Huo L S, Bai H Y, Wang W H 2012 Appl. Phys. Lett. 101 121906

    [58]

    Johnson W L, Samwer K 2005 Phys. Rev. Lett. 95 195501

    [59]

    Turnbull D, Cohen M H 1961 J. Chem. Phys. 34 120

    [60]

    Miracle D B 2004 Nat. Mater. 3 697

    [61]

    Zhan X L, Zhang X N, Wang D Y, Lu L 2009 Eng. Mech. 26 187 (in Chinese) [詹小丽, 张肖宁, 王端宜, 卢亮 2009 工程力学 26 187]

    [62]

    Zhou H W, Wang C P, Han B B, Duan Z Q 2011 Int. J. Rock. Mech. Min. 48 116

    [63]

    Zhou H W, Wang C P, Duan Z Q, Zhang M, Liu J F 2012 Sci. Sin-Phys. Mech. Astron. 42 310 (in Chinese) [周宏伟, 王春萍, 段志强, 张淼, 刘建锋 2012 中国科学: 物理学力学天文学 42 310]

    [64]

    Wang D P, Zhao D Q, Ding D W, Bai H Y, Wang W H 2014 J. Appl. Phys. 115 123507

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  • Received Date:  12 August 2015
  • Accepted Date:  16 October 2015
  • Published Online:  05 February 2016

Research on viscoelastic behavior and rheological constitutive parameters of metallic glasses based on fractional-differential rheological model

Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 51471139, 51401176, 21376199) and the Natural Science Foundation of Hunan Province, China (Grant No. 14JJ3078).

Abstract: Metallic glasses offer novel physical, chemical and mechanical properties and have promising potential applications. Recently, exploring the structure and deformation mechanism of metallic glasses according to the rheological mechanical behavior in the nominal elastic region has been the object of intensive research. Physically the mechanical analogues of fractional elements can be represented by self-similarity spring-dashpot fractal networks. In light of the fractal distribution features of the structural heterogeneities, a fractional differential rheological model is introduced to explore the viscoelastic a behavior of metallic glasses in this paper. To investigate the viscoelastic deformation mechanism, carefully designed nanoindentation tests at ambient temperature are proposed in this study. Three kinds of metallic glasses with different Poisson's ratio and glass transition temperature, which have the chemical compositions of Pd40Cu30Ni10P20, Zr48Cu34Pd2Al8Ag8, and (Fe0.432Co0.288B0.192Si0.048Nb0.04) 96Cr4 are selected as the model materials. Experimental and theoretical results clearly indicate that in the nominal elastic region, these metallic glasses exhibit linear viscoelasticity, implying a loading rate-dependent character. Based on the fractional calculus and Riemann-Liouville definition, experimental results are analyzed by the fractional-differential and integer order rheology models respectively. According to the stability of the fitting parameters, here we show that the fractional-differential Kelvin model, which consists of a spring and a fractional dashpot element, can fit the experimental viscoelastic deformation data of the investigated metallic glasses better than that with integer order rheological model. The extracted elastic modulis E1 of the spring in the fractional-differential Kelvin model are comparable to those of samples measured by traditional methods. Such a similarity can be well explained by the mechanical analogue of fractal model proposed for describing the distribution features of the structural heterogeneities in metallic glasses. The rheological parameters obtained here including viscosity index A and fractional order are capable of reflecting the rheological features and the flowing tendency of the above-mentioned metallic glasses. It is found that there exists a clear relationship between the rheological parameters and the reduced glass transition temperature as well as Poisson's ratio, which is helpful for understanding the correlation between plasticity and Poisson's ratio from the micro-structural point of view. The current work provides a rheological model-structure-property relation that may be applicable to a wide range of glassy materials.

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