Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 51475120) and the Key Program of the National Natural Science Foundation of China (Grant No. U1537201).
Received Date:29 October 2018
Accepted Date:01 December 2018
Available Online:01 February 2019
Published Online:20 February 2019
Abstract: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. Keywords:thermoviscoelastic wave/ numerical simulation/ alloy melt/ Kelvin medium/ attenuation coefficient
式中, ${V_{{\rm{P}} - \rm Ph}}$为平面波的相速度; $\omega $为波动圆频率; $\eta {\rm{ = }}{\lambda '}{\rm{ + }}2{\mu '}$, 其中${\lambda '},{\mu '}$是介质的粘滞拉梅系数; $M = \lambda + 2\mu $, 其中$\lambda ,\mu $为介质的弹性拉梅系数. 频率为20 kHz, 速度振幅相同的振动在600 ℃的均匀熔体中的传播过程如图2所示. 由于粘滞系数的存在, Kelvin热粘弹波的阶跃被抹平了, 如图2中的波前所示, 弹性波的波前梯度变化比较清晰, 而粘弹波的波前被拉得很宽. 在相同波长和相同温度场条件下, 热粘弹波的衰减非常快, 而在完全弹性介质中无衰减. 从图2中还可以看出, 热粘弹波场中的温度变化比位移超前约${{\text{π}}/2}$. 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 Figure2. Distribution of displacement and temperature difference of thermoviscoelastic wave in elastic medium and Kelvin medium.
24.3.热粘弹波在变温均匀介质中的传播 -->
4.3.热粘弹波在变温均匀介质中的传播
频率为20 kHz, 速度振幅相同的振动在不同温度的均匀熔体中的传播过程如图3所示. 热粘弹波在高温介质中的波长小, 衰减快, 传播距离短. 当熔体完全处于液相时, 波衰减最快, 传播距离只有0.1 m, 而在固液区传播距离大于2 m. 波在介质中的衰减主要有热损失、粘滞衰减和散射. 本文所建立的模型并未考虑介质的微观结构, 因此在均匀介质中不产生散射, 这里的衰减主要是粘滞衰减和热损失. 从该结果可以推断, 超声在液相中的粘滞衰减和热损失要大于高温固相. 图 3 热粘弹波在变温均匀介质中的位移和温差的分布 Figure3. Distribution of displacement and temperature difference of thermoviscoelastic wave in homogeneous medium with variable temperature.
24.4.热粘弹波在变温非均匀介质中的传播 -->
4.4.热粘弹波在变温非均匀介质中的传播
在变温非均匀介质中波传播的数值模拟, 需要将各个节点的计算参数按实时的温度值给出, 这个温度是在不断变化的, 但由于波的传播速度较大, 引起介质的温度变化也较小, 因此可以将变温介质看成是瞬态恒温介质来给计算参数赋值. 介质中的一维温度场为550 ℃到650 ℃的线性分布, 振源位于550 ℃的一端. 如图4所示, 正弦热粘弹性波在非均匀介质中传播时, 波动影响区域的质点位移都随时间成正弦变化, 但在空间上, 波的振幅随传播距离的增加而衰减. 当介质处于压缩状态时, 介质中的温度升高; 处于拉伸状态时, 温度降低. 虽然热粘弹波会对温度场造成影响, 但图4表明, 由热损失和粘滞衰减引起的熔体温度变化非常小, 几乎可以忽略不计. 许多研究表明, 熔体中施加超声后会对熔体的温度场造成非常大的影响, 使熔体温度场变均匀, 金属凝固时间变短[22]. 图4的结果表明, 熔体温度场的剧烈变化很难由熔体对超声的吸收引起. 该结果间接证明了, 超声对熔体温度场的改变主要由声流对熔体的搅拌作用引起. 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 Figure4. Propagation of thermoviscoelastic waves in inhomogeneous medium: (a) Displacement changes with time and space; (b) temperature difference changes with time and space.
不均匀的温度场对波动会产生非常大的影响, 在如图5所示的线性温度场中, 热粘弹波在初始时传播速度快, 衰减较小. 热粘弹波在非均匀温度场中传播相当于在层状介质中传播, 每层的波阻抗不同, 因此波在遇到界面时会发生折射和反射, 在传播过程中不断遇到界面, 散射衰减加重, 传播距离缩短. 图 5 热粘弹波在不同的均匀温度场中的位移分布 Figure5. Distribution of displacement of thermoviscoelastic wave in different uniform temperature field.
如图6所示, 当波从合金熔体的高温区向低温区传播时, 非均匀温度场的分布对波传播的规律有非常大的影响, 波入射处的温度场梯度越小, 波的衰减越小, 传播的距离越远; 当波从合金熔体的高温区向低温区传播时, 温度场的不同分布对波传播几乎没有影响. 造成此现象的原因和波在不同温度区的衰减系数不同有关. 波在高温区的衰减系数比低温区大很多, 所以在高温区, 温度场的分布对波造成的影响不如衰减系数的影响大, 波还未来得及对不同分布的温度场产生响应, 波已经完全衰减了, 所以不同温度场下位移和温差的分布差异较小. 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 Figure6. 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.