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 引用本文: 王现辉, 李方琳, 刘宇建, 陈会涛, 禹建功. 板中热弹波传播: 一种改进的勒让德多项式方法[J]. 力学学报, 2020, 52(5): 1277-1285.
Wang Xianhui, Li Fanglin, Liu Yujian, Chen Huitao, Yu Jiangong. THERMOELASTIC WAVE PROPAGATION IN PLATES: AN IMPROVED LEGENDRE POLYNOMIAL APPROACH[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1277-1285.
 Citation: Wang Xianhui, Li Fanglin, Liu Yujian, Chen Huitao, Yu Jiangong. THERMOELASTIC WAVE PROPAGATION IN PLATES: AN IMPROVED LEGENDRE POLYNOMIAL APPROACH[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1277-1285.

## THERMOELASTIC WAVE PROPAGATION IN PLATES: AN IMPROVED LEGENDRE POLYNOMIAL APPROACH

• 摘要: 近年来, 超声导波因其衰减小, 传播距离远和信号覆盖范围广, 成为无损检测领域快速发展的方向之一. 然而, 基于超声导波的高温在线检测和激光超声技术却发展缓慢, 其关键在于热弹耦合波动方程求解难度大、传播与衰减特性研究困难. 作为一种有效的求解方法, 勒让德正交多项式方法已广泛应用于导波传播问题, 但该方法在求解热弹导波传播时存在两个不足, 限制其进一步的发展和应用. 这两个缺陷是: (1)求解过程中大量积分的存在, 致使计算效率低下; (2)仅能处理等热边界条件的热弹导波传播. 针对两项不足之处, 提出一种改进的勒让德正交多项式方法, 以求解分数阶热弹板中的导波传播. 推导求解方法中积分的解析表达式, 以提高计算效率; 引入温度梯度展开式, 发展适合勒让德多项式级数的绝热边界条件处理方法. 与已有文献结果对比表明改进方法的正确性; 与已有方法的计算时间对比说明改进方法的高效性. 最后将改进的方法用于求解分数阶热弹板中的导波传播, 研究分数阶次对频散、衰减曲线和应力、位移、温度分布等的影响.

Abstract: In recent years, the research of thermoelastic coupled wave has greatly promoted the development of high temperature online detection and laser ultrasonic technology. For its small attenuation, long propagation distance and wide signal coverage, ultrasonic guided wave has become one of the rapid development directions in the field of nondestructive testing. However, the development of guided wave high temperature on-line detection and laser ultrasonic guided wave technology is slow. The key lies in the difficulty in solving the coupled thermoelastic wave equation and the difficulty in studying the propagation and attenuation characteristics. As an effective method, Legendre polynomial approach has been widely used to solve the problem of guided wave propagation since 1999. But there are two shortcomings in this method, which limit its further development and application. Two defects are: (1) Due to the Legendre polynomial and its derivative in integral kernel function, the integrals in the solution process leads to low calculation efficiency; (2) Only the thermoelastic guided wave propagation with isothermal boundary conditions can be treated. In order to solve these two defects, an improved Legendre polynomial method is proposed to solve the fractional thermoelastic guided waves in plates. The analytical integral instead of numerical integration in the available conventional Legendre polynomial approach, which greatly improves the calculation efficiency. A new treatment of the adiabatic boundary condition for the Legendre polynomial is developed by introducing the temperature gradient expansion based on the rectangular window function. Compared with the available data shows the validity of the improved method. Comparison with the CPU time between two approaches indicates the higher efficiency of the presented approach. Finally, the phase velocity dispersion curves, attenuation curves, the stress, displacement and temperature distributions for a plate with different fractional orders are analysed. The fractional order has weak influence on the elastic mode velocity, but it has considerable influence on the elastic mode attenuation.

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