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The quantitative calculation of mechanical wave propagation in the solidification of metal is difficult because of the complicated structure of melt in the solidification process. In this study, the Kelvin model is used to describe the viscoelastic characteristics of alloy melt, and the thermoviscoelastic wave equations are established in conjunction with continuity equation, momentum equation, energy equation considering thermo-mechanical effect and constitutive equation considering thermal stress. The difference equation of thermoviscoelastic wave in the variable non-uniform temperature field is built by using the implicit finite difference method of second order in space and first order in time, and taking into account the variable temperature and non-uniformity of melt. The difference equation is solved numerically by taking the thermodynamic parameters of ZL203A alloy in solid-liquid region varying with temperature as calculation parameters, and the propagation law of thermoviscoelastic wave in the inhomogeneous alloy melt with varying temperature is obtained. By comparing the propagation law of the wave with same wavelength in elastic medium with that in viscoelastic medium, it is found that the thermoviscoelastic wave attenuates seriously, and the thermoelastic wave has no attenuation. The step of displacement of thermoviscoelastic wave will be smoothed rapidly during propagation. The comparison among propagations of thermoviscoelastic wave in homogeneous medium at different temperatures shows that the wavelength of thermoviscoelastic wave in high temperature melt is smaller and the attenuation is more serious than in low temperature melt. The propagation of thermoviscoelastic wave in inhomogeneous medium is equivalent to the propagation in layered medium with different impedance, which makes the attenuation more serious than in homogeneous medium. When the thermoviscoelastic wave propagates from the low temperature region to the high one, the distribution of inhomogeneous temperature field has a great influence on the propagation of wave. The smaller the slope of the temperature field at the incidence of the wave, the smaller the attenuation of the wave is, and the farther the propagation distance is, conversely, when the wave propagates from the high temperature region to the low one, the distribution of the inhomogeneous temperature field has little influence on the propagation of wave. The calculation results of attenuation coefficients of thermoviscoelastic wave with different frequencies at different temperatures show that the attenuation coefficients of thermoviscoelastic waves in alloy melt are bigger in the high temperature medium than in the low temperature medium, and it increases linearly with the frequency increasing. The attenuation coefficient first increases and then decreases with temperature increasing, and reaches a maximum value at the coherency temperature.
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Keywords:
- thermoviscoelastic wave /
- numerical simulation /
- alloy melt /
- Kelvin medium /
- attenuation coefficient
[1] Guo F, Luo P, Qiu B, Li K 2008 Met. Mat. Metall. Eng. 1 015
[2] Kotadia H R, Qian M, Eskin D G, Das A 2017 Mater. Design 6 65
[3] Chen R R, Zheng D S, Guo J J, Fu H Z 2016 Mater. Sci. Eng. A 653 23
Google Scholar
[4] Eskin G I, Eskin D G 2014 Ultrasonic Treatment of Light Alloy Melts (Boca Raton: CRC Press)
[5] Wang R J, Wu S P, Chen W 2018 T. Nonferr. Metal. Soc. 28 1514
Google Scholar
[6] Zhang M, Li X, Li Z H, Xu X 2011 Special Casting and Nonferrous Alloys 2 034
[7] Wu S P, Wang R J, Chen W 2017 Acta Metall. Sin. 54 247
[8] Wang R J, Wu S P, Chen W 2019 Simulation 95 3
Google Scholar
[9] Kong W, Cang D 2012 Simulation 88 694
Google Scholar
[10] Jiang R P, Li X Q, Ju Z Y, Zhang M 2014 J. South China Univ. Techno.: Nat. Sci. Ed. 4 014
[11] Shao Z W, Le Q, Zhang Z Q, Cui J Z 2011 T. Nonferr. Metal. Soc. 21 2476
Google Scholar
[12] Mccarthy M F, Moodie T B, Sawatzky R P 1988 Q. Appl. Math. 46 539
Google Scholar
[13] Nowinski J L, Boley B 1980 J. Appl .Mech 47
[14] Bruno B A, Jerome H W 2012 Theory of Thermal Stresses [https://www.ebookmall.com/ebook/theory-of-thermal-stresses/bruno-a-boley/9780486695792]
[15] 范绪箕, 陈国光 1982 力学进展 12 339
Fan X J, Chen G G 1982 Adv. Mech. 12 339
[16] Takeuti Y, Tanigawa Y 1979 Int. J. Numer. Meth. Eng. 14 987
Google Scholar
[17] 刘驰, 李庆春 1988 铸造 9 28
Liu C, Li Q C 1988 Foundry 9 28
[18] Fraizier E , Nadal M H , Oltra R 2002 Ultrasonics 40 543
Google Scholar
[19] Jarzynski J 1963 Proc. Phys. Soc. 81 745
Google Scholar
[20] Bansal R 2002 J. Phys. C: Solid State Phys. 6 1204
[21] 牛滨华, 孙春岩 2007 半无限空间各向同性黏弹性介质与地震波传播(北京: 地质出版社) 第107页
Niu B H, Sun C Y 2007 Half-space homogeneous isotropic viscoelastic medium and seismic wave propagation (Beijing: Geological Publishing House) p107 (in Chinese)
[22] Li J W, Momono T, Ying F U, Jia Z, Yu Y T 2007 T. Nonferr. Metal. Soc. 17 691
Google Scholar
[23] Lavender J D 1972 Non-Destructive Testing 52 107
[24] Magnin B, Maenner L, Katgerman Laurens, Engler S 1996 Mater. Sci. Forum 217 1209
[25] Sivkov G, Yagodin D, Kofanov S, Gornov O, Volodin S, Bykov V, Dahlborg U 2007 J. Non-Cryst. Solids 353 3274
Google Scholar
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图 1 数值模拟中采用的计算参数
Figure 1. Parameters used in numerical simulation.
图 2 热粘弹波在弹性介质和Kelvin介质中的位移和温差的分布
Figure 2. Distribution of displacement and temperature difference of thermoviscoelastic wave in elastic medium and Kelvin medium.
图 3 热粘弹波在变温均匀介质中的位移和温差的分布
Figure 3. Distribution of displacement and temperature difference of thermoviscoelastic wave in homogeneous medium with variable temperature.
图 4 热粘弹波在非均匀介质中的传播 (a)位移随时间和空间的变化；(b)温差随时间和空间的变化
Figure 4. Propagation of thermoviscoelastic waves in inhomogeneous medium: (a) Displacement changes with time and space; (b) temperature difference changes with time and space.
图 5 热粘弹波在不同的均匀温度场中的位移分布
Figure 5. Distribution of displacement of thermoviscoelastic wave in different uniform temperature field.
图 6 热粘弹波在不同的非均匀温度场中的位移和温差的分布 (a)波从低温区域向高温区域传播；(b)波从高温区域向低温区域传播
Figure 6. Distribution of displacement and temperature difference of thermoviscoelastic wave in different inhomogeneous temperature field: (a) Propagation from low temperature region to high temperature region; (b) propagation from high temperature region to low temperature region.
图 7 不同温度场下热粘弹波的衰减系数与频率的关系
Figure 7. Relationship between attenuation coefficient and frequency of thermoviscoelastic wave in different temperature fields.
图 8 不同频率下热粘弹波的衰减系数与温度的关系
Figure 8. Relationship between attenuation coefficient of thermoviscoelastic wave and temperature in different frequency.
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[1] Guo F, Luo P, Qiu B, Li K 2008 Met. Mat. Metall. Eng. 1 015
[2] Kotadia H R, Qian M, Eskin D G, Das A 2017 Mater. Design 6 65
[3] Chen R R, Zheng D S, Guo J J, Fu H Z 2016 Mater. Sci. Eng. A 653 23
Google Scholar
[4] Eskin G I, Eskin D G 2014 Ultrasonic Treatment of Light Alloy Melts (Boca Raton: CRC Press)
[5] Wang R J, Wu S P, Chen W 2018 T. Nonferr. Metal. Soc. 28 1514
Google Scholar
[6] Zhang M, Li X, Li Z H, Xu X 2011 Special Casting and Nonferrous Alloys 2 034
[7] Wu S P, Wang R J, Chen W 2017 Acta Metall. Sin. 54 247
[8] Wang R J, Wu S P, Chen W 2019 Simulation 95 3
Google Scholar
[9] Kong W, Cang D 2012 Simulation 88 694
Google Scholar
[10] Jiang R P, Li X Q, Ju Z Y, Zhang M 2014 J. South China Univ. Techno.: Nat. Sci. Ed. 4 014
[11] Shao Z W, Le Q, Zhang Z Q, Cui J Z 2011 T. Nonferr. Metal. Soc. 21 2476
Google Scholar
[12] Mccarthy M F, Moodie T B, Sawatzky R P 1988 Q. Appl. Math. 46 539
Google Scholar
[13] Nowinski J L, Boley B 1980 J. Appl .Mech 47
[14] Bruno B A, Jerome H W 2012 Theory of Thermal Stresses [https://www.ebookmall.com/ebook/theory-of-thermal-stresses/bruno-a-boley/9780486695792]
[15] 范绪箕, 陈国光 1982 力学进展 12 339
Fan X J, Chen G G 1982 Adv. Mech. 12 339
[16] Takeuti Y, Tanigawa Y 1979 Int. J. Numer. Meth. Eng. 14 987
Google Scholar
[17] 刘驰, 李庆春 1988 铸造 9 28
Liu C, Li Q C 1988 Foundry 9 28
[18] Fraizier E , Nadal M H , Oltra R 2002 Ultrasonics 40 543
Google Scholar
[19] Jarzynski J 1963 Proc. Phys. Soc. 81 745
Google Scholar
[20] Bansal R 2002 J. Phys. C: Solid State Phys. 6 1204
[21] 牛滨华, 孙春岩 2007 半无限空间各向同性黏弹性介质与地震波传播(北京: 地质出版社) 第107页
Niu B H, Sun C Y 2007 Half-space homogeneous isotropic viscoelastic medium and seismic wave propagation (Beijing: Geological Publishing House) p107 (in Chinese)
[22] Li J W, Momono T, Ying F U, Jia Z, Yu Y T 2007 T. Nonferr. Metal. Soc. 17 691
Google Scholar
[23] Lavender J D 1972 Non-Destructive Testing 52 107
[24] Magnin B, Maenner L, Katgerman Laurens, Engler S 1996 Mater. Sci. Forum 217 1209
[25] Sivkov G, Yagodin D, Kofanov S, Gornov O, Volodin S, Bykov V, Dahlborg U 2007 J. Non-Cryst. Solids 353 3274
Google Scholar
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