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Combined with the microstructure evolution in amorphous alloys under the external load, a fractional order viscoelastic constitutive model is first derived by replacing a Newtonian dashpot in the classical Zener model with the fractional derivative Abel dashpot. Based on the Hertzian theory and the fractional order viscoelastic constitutive model, a relationship between displacement and load (or time) for an instrumental nanoindentation test with a spherical indenter is then proposed. Finally, a series of nanoindentation test data for an Fe-base bulk amorphous alloy are employed to verify the derived model, and its viscoelastic behavior in the apparent elastic region is analyzed in detail. Results show that the fractional order rheological model has higher fitting accuracy than that of the integer order model, and the fitting parameters of the proposed model are more suitable to reflect the effect of the loading rate on the viscoelastic behavior in the alloy studied. Variation of the above-mentioned fitting parameters exhibits a strong correlation with the microstructure evolution during the loading of this Fe-base amorphous alloy.
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Keywords:
- bulk amorphous alloys /
- fractional order rheological model /
- nanoindentation /
- viscoelastic behavior
[1] Liu L, Sun M, Chen Q, Liu B, Qiu C L 2006 Acta Phys. Sin. 55 1930 (in Chinese) [柳林, 孙民, 谌祺, 刘兵, 邱春雷 2006 55 1930]
[2] Yang L, Guo G Q 2010 Chin. Phys. B 19 126101
[3] Imran M, Hussain F, Rashid M, Cai Y Q, Ahmad S A 2013 Chin. Phys. B 22 096101
[4] Xu F, Ding Y H, Deng X H, Zhang P, Long Z L 2014 Phys. B 450 84
[5] Liao G K, Long Z L, Yang M, Chen S M, Zou P 2014 Chin. J. Nonfe. Met. 24 2762 (in Chinese) [廖光开, 龙志林, 杨妙, 陈舒敏, 邹萍 2014 中国有色金属学报 2 4 2762]
[6] Lu Z, Jiao W, Wang W H, Bai H Y 2014 Phys. Rev. Lett. 113 045501
[7] Wang W H 2011 Physics 40 701 (in Chinese) [汪卫华 2011 物理 40 701]
[8] Yang Y, Zeng J F, Volland A, Blandin J J, Gravier S, Liu C T 2012 Acta Mater. 60 5260
[9] Peng H L, Li M Z, Sun B A, Wang W H 2012 J. Appl. Phys. 112 023516
[10] Yang Y, Zeng J F, Ye J C, Lu J 2010 Appl. Phys. Lett. 97 261905
[11] Ye J C, Lu J, Liu C T, Wang Q, Yang Y 2010 Nat. Mater. 9 619
[12] Huo L S, MA J, Ke H B 2012 J. Appl. Phys. 111 113522
[13] Huo L S, Zeng J F, Wang W H, Liu C T, Yang Y 2013 Acta Mater. 61 4329
[14] Peng J, Long Z L, Wei H Q, Li X A, Zhang Z C 2009 Acta Phys. Sin. 58 4059 (in Chinese) [彭建, 龙志林, 危洪清, 李乡安, 张志纯 2009 58 4059]
[15] Chen W, Sun H G, Li X C 2010 The use of fractional derivative modeling in mechanics and engineering problems (Beijing: Science Press) p12 (in Chinese) [陈文, 孙洪广, 李西成 2010 力学与工程问题中的分数阶导数建模 (北京:科学出版社)] p12
[16] Yin D S, Ren J J, He C L, Chen W 2007 Chin. J. Rock Mech. Eng. 26 1899 (in Chinese) [殷德顺, 任俊娟, 和成亮, 陈文 2007 岩石力学与工程学报 26 1899]
[17] Chen H S, Li M M, Kang Y G, Zhang S L 2008 Chem. J. Chin. Univ. 29 1271 (in Chinese) [陈宏善, 李明明, 康永刚, 张素玲 2008 高等学校化学学报 29 1271]
[18] 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)]
[19] Mandelbrot B B 1982 The Fractal Geometry of Nature (New York: W H Freman) p14-19
[20] Ma D, Stoica A D, Wang X L 2009 Nat. Mater. 8 30
[21] Zhang C Y 2006 Viscoelastic Fracture Mechanics (Beijing: Science Press) p16
[22] Koeller R C 1984 J. Appl. Mech. 51 299
[23] Crothers D S F, Holland D, Kalmykov Y P, Coffey W T 2004 J. Mol. Liq. 114 27
[24] Kai Diethelm 2002 J. Math. Anal. Appl. 265 229
[25] Lee E H, Radok J R M 1960 J. Appl. Mech. 27 438
[26] Wang Z F, Zhang G Z, Liu G 2008 J. Chem. Eng. Chin. Univ. 22 351 (in Chinese) [王志方, 张国忠, 刘刚 2008 高校化学工程学报 22 351]
[27] Long Z L, Shao Y, Xie G Q, Zhang P, Shen B L, Inoue A 2008 J. Allo. Comp. 462 52
[28] Wang W H 2014 Sci. Sin-Phys. Mech. Astron. 44 396 (in Chinese) [汪卫华 2014 中国科学: 物理学 力学 天文学 44 396]
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[1] Liu L, Sun M, Chen Q, Liu B, Qiu C L 2006 Acta Phys. Sin. 55 1930 (in Chinese) [柳林, 孙民, 谌祺, 刘兵, 邱春雷 2006 55 1930]
[2] Yang L, Guo G Q 2010 Chin. Phys. B 19 126101
[3] Imran M, Hussain F, Rashid M, Cai Y Q, Ahmad S A 2013 Chin. Phys. B 22 096101
[4] Xu F, Ding Y H, Deng X H, Zhang P, Long Z L 2014 Phys. B 450 84
[5] Liao G K, Long Z L, Yang M, Chen S M, Zou P 2014 Chin. J. Nonfe. Met. 24 2762 (in Chinese) [廖光开, 龙志林, 杨妙, 陈舒敏, 邹萍 2014 中国有色金属学报 2 4 2762]
[6] Lu Z, Jiao W, Wang W H, Bai H Y 2014 Phys. Rev. Lett. 113 045501
[7] Wang W H 2011 Physics 40 701 (in Chinese) [汪卫华 2011 物理 40 701]
[8] Yang Y, Zeng J F, Volland A, Blandin J J, Gravier S, Liu C T 2012 Acta Mater. 60 5260
[9] Peng H L, Li M Z, Sun B A, Wang W H 2012 J. Appl. Phys. 112 023516
[10] Yang Y, Zeng J F, Ye J C, Lu J 2010 Appl. Phys. Lett. 97 261905
[11] Ye J C, Lu J, Liu C T, Wang Q, Yang Y 2010 Nat. Mater. 9 619
[12] Huo L S, MA J, Ke H B 2012 J. Appl. Phys. 111 113522
[13] Huo L S, Zeng J F, Wang W H, Liu C T, Yang Y 2013 Acta Mater. 61 4329
[14] Peng J, Long Z L, Wei H Q, Li X A, Zhang Z C 2009 Acta Phys. Sin. 58 4059 (in Chinese) [彭建, 龙志林, 危洪清, 李乡安, 张志纯 2009 58 4059]
[15] Chen W, Sun H G, Li X C 2010 The use of fractional derivative modeling in mechanics and engineering problems (Beijing: Science Press) p12 (in Chinese) [陈文, 孙洪广, 李西成 2010 力学与工程问题中的分数阶导数建模 (北京:科学出版社)] p12
[16] Yin D S, Ren J J, He C L, Chen W 2007 Chin. J. Rock Mech. Eng. 26 1899 (in Chinese) [殷德顺, 任俊娟, 和成亮, 陈文 2007 岩石力学与工程学报 26 1899]
[17] Chen H S, Li M M, Kang Y G, Zhang S L 2008 Chem. J. Chin. Univ. 29 1271 (in Chinese) [陈宏善, 李明明, 康永刚, 张素玲 2008 高等学校化学学报 29 1271]
[18] 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)]
[19] Mandelbrot B B 1982 The Fractal Geometry of Nature (New York: W H Freman) p14-19
[20] Ma D, Stoica A D, Wang X L 2009 Nat. Mater. 8 30
[21] Zhang C Y 2006 Viscoelastic Fracture Mechanics (Beijing: Science Press) p16
[22] Koeller R C 1984 J. Appl. Mech. 51 299
[23] Crothers D S F, Holland D, Kalmykov Y P, Coffey W T 2004 J. Mol. Liq. 114 27
[24] Kai Diethelm 2002 J. Math. Anal. Appl. 265 229
[25] Lee E H, Radok J R M 1960 J. Appl. Mech. 27 438
[26] Wang Z F, Zhang G Z, Liu G 2008 J. Chem. Eng. Chin. Univ. 22 351 (in Chinese) [王志方, 张国忠, 刘刚 2008 高校化学工程学报 22 351]
[27] Long Z L, Shao Y, Xie G Q, Zhang P, Shen B L, Inoue A 2008 J. Allo. Comp. 462 52
[28] Wang W H 2014 Sci. Sin-Phys. Mech. Astron. 44 396 (in Chinese) [汪卫华 2014 中国科学: 物理学 力学 天文学 44 396]
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