Abstract The relaxation time model is introduced to analyse the thermo-mechanical problem for multi-layered media in this paper. The Fourier heat conduction equation is extended by introducing a relaxation time based on the Lord-Shulman model. For multi-layered media, the analytical layer-element method is employed to establish the global stiffness matrix equations by assembling the correlative layer-elements in the Laplace-Hankel domain. Then, solutions for displacement and temperature in the time domain are obtained by solving the global matrix equations and implementing a numerical inversion procedure. For numerical implementations, computational overflows are avoided by eliminations of positive exponentials in the stiffness matrix, and numerical instabilities in special cases due to singularity are removed by replacing singular layer-elements with well-conditioned ones. Comparisons with the existing solutions and the FEM analysis demonstrate the accuracy and efficiency of the proposed method. Finally, a series of numerical examples are presented to discuss the effects of transient load type, stratification, relaxation time and the surface heat transfer coefficient.

中文翻译：

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基于 Lord-Shulman 模型的多层介质热力学行为

摘要 本文引入弛豫时间模型来分析多层介质的热力学问题。通过引入基于 Lord-Shulman 模型的弛豫时间扩展了傅立叶热传导方程。对于多层介质，通过在Laplace-Hankel域中组装相关层单元，采用解析层单元方法建立全局刚度矩阵方程。然后，通过求解全局矩阵方程并实施数值反演程序，获得时域中位移和温度的解。对于数值实现，通过消除刚度矩阵中的正指数来避免计算溢出，通过用条件良好的层元素替换奇异层元素，消除了由于奇异性引起的特殊情况下的数值不稳定性。与现有解决方案和有限元分析的比较证明了所提出方法的准确性和效率。最后，给出了一系列数值例子来讨论瞬态载荷类型、分层、松弛时间和表面传热系数的影响。