The thermal relaxation responses of graded materials satisfied power law
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摘要: 梯度材料热弛豫响应行为的理论研究对于热分析具有重要意义。结合Cattaneo-Vernotte线性双曲型热传导方程，推导得到幂律梯度材料的一维双曲型非傅里叶热传导方程。通过积分变换法，解得频域内温度场的贝塞尔级数形式解，随后利用极点留数法，得到时间域内温度场的第一类解析解。在第一类解析解的基础上，由简化欧拉方程解得第二类解析解。结合拉普拉斯数值逆变换方法，验证了解析解的准确性。以高温阻热梯度材料Mo-ZrC的应用为例，讨论了一般温度边界条件以及温度脉冲载荷作用下幂律梯度材料的热弛豫响应行为。分析发现，在所研究范围内温度场具备波动和传导衰减的双重特性。响应时间和温度幅值随着热弛豫时间系数的增大而增大，温度场分布和单元波形与梯度结构相关。
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关键词:
- 热弛豫响应 /
- Mo-ZrC梯度材料 /
- 贝塞尔级数 /
- 拉普拉斯变换
Abstract: To study the thermal relaxation behavior of the graded medium satisfied the power law, the one-dimensional hyperbolic non-Fourier heat conduction equation of the graded material which satisfied the power law was derived from the Cattaneo-Vernotte linear hyperbolic heat transfer equation with the thermal relaxation coefficient and graded exponent induced. The equation was first treated dimensionless. Based on the Laplace transformation, the new heat conduction equation was found to conform to the general form of the Bessel equation called the Lommel equation in the frequency domain, and the Bessel series solution of the temperature field in the frequency domain was obtained. With the asymptotic expansion of the Bessel series, the simplified expression of the temperature field in the frequency domain containing trigonometric function was obtained. The inverse Laplace transformation of the temperature field in the frequency domain was employed to get the first analytical solution of the temperature field in the time domain. Besides the first analytical solution, the new heat conduction equation in the frequency domain was simplified to the Euler equation, and the second kind of analytical solution was obtained by the pole residue method. The second analytical solution exhibits similar fluctuation attenuation and diffusion features, and both the waveform and response time are sensitive to the relaxation time coefficient. However, the second kind of analytical solution differs from the first kind of solution in terms of waveform elements which are highly related to the graded structure. The accuracy of the analytical result is verified by numerical calculation. Taking Mo-ZrC graded composite as an example, the thermal relaxation behavior of graded material satisfied power law under the first kind of temperature boundary and temperature pulse loading are discussed in detail. The temperature field shows both fluctuation attenuation and conduction characteristics. With the increase of the thermal relaxation coefficient, the response time and temperature wave amplitude increase, the unit waveform develops from a trapezoidal wave to a rectangular wave, and the oscillation approaching to the boundary shows an obvious bias. -
图 2 贝塞尔级数渐近展开可行性验证
Figure 2. Feasibility verification of asymptotic expansion of Bessel equation
图 3 拉普拉斯逆变换数值计算结果和第一类解析结果对比
Figure 3. Comparison of numerical results of inverse Laplace transformation and the results of the first analytic solution
图 4 第一类温度函数的二维分布图像
Figure 4. Two-dimensional distribution image of temperature function of the first analytic solution
图 5 第一类解在不同位置处的温度-时间曲线
Figure 5. Temperature-time curves at different positions of the first analytic solution
图 6 第一类解在不同时刻的温度分布曲线
Figure 6. Temperature-position curves at different times of the first analytic solution
图 7 第一类解在不同弛豫系数下的温度-时间曲线以及弛豫系数与响应时间、温度极值关系
Figure 7. The temperature-time curves corresponding to different relaxation coefficients and the relationship between relaxation coefficient and response time and maximum temperature of the first analytic solution
图 8 不同温度脉冲对应的温度-时间曲线
Figure 8. Temperature-time curves corresponding to the temperature pulses of different widths
图 9 拉普拉斯逆变换数值计算结果和第二类解析结果对比
Figure 9. Comparison of numerical results of inverse Laplace transformation and the results of the second analytic solution
图 10 第二类温度函数的二维分布
Figure 10. Two-dimensional distribution of temperature function of the second analytic solution
图 11 第二类解在不同位置处的温度-时间曲线
Figure 11. Temperature-time curves at different positions of the second analytic solution
图 12 第二类解在不同时刻的温度分布曲线
Figure 12. Temperature-position curves at different times of the second analytic solution
图 13 第二类解在不同弛豫系数下的温度时间曲线及弛豫系数与响应时间、温度极值关系
Figure 13. The temperature-time curves corresponding to different relaxation coefficients and the relationship between relaxation coefficient and response time and maximum temperature of the second analytic solution
表 1 Mo和ZrC的热力学参数
Table 1. Thermomechanical properties of Mo and ZrC
材料 密度/(kg·m−3) 比热容/(J·kg−1·K−1) 导热系数/(W·m−1·K−1) 熔点/℃ 杨氏模量/GPa 泊松比 Mo 10200 230 150 3400 279 0.32 ZrC 6510 310 10 2620 390 0.191 
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