Abstract Transient thermoelastic interactions between materials and the moving heat sources, i.e. Laser additive manufacturing, Laser-assisted thermotherapy, high speed sliding and rolling contacts, are becoming increasingly important. In this work, a unified fractional thermoelastic theory is developed, and applied to study transient responses caused by a moving heat source. Theoretically, new insights on fractional thermoelasticity are provided by introducing new definitions of fractional derivative, i.e. Caputo-Fabrizio, Atangana–Baleanu and Tempered-Caputo type. Numerically, a semi-infinite medium subjected to a source of heat moving with constant velocity is considered within the present model under two different sets of boundary conditions: stress free and temperature given for the first, displacement fixed and thermally adiabatic for the second. Analytical solutions to all responses are firstly formulated in Laplace domain, and then transformed into time domain through numerical method. The numerical results show that Caputo-Fabrizio and Atangana–Baleanu type models predict smaller transient responses than Caputo type theory, while Tempered-Caputo model may give larger results by increasing the tempered parameter. Meanwhile, the effect of fractional order, tempered parameter of Tempered-Caputo model, and the velocity of heat source on all responses is discussed in detail. The time history of responses shows that: for long-term process, the exponential function of TC definition will make sense, and the temperature from TC model is greatly different from that of C model. This work may provide comprehensive understanding for thermoelastic interactions due to moving heat source, and open up possibly wide applications of such new fractional derivatives.

中文翻译：

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使用新分数阶导数的 Cattaneo 型热弹性分数阶理论

摘要 材料与运动热源之间的瞬态热弹性相互作用，即激光增材制造、激光辅助热疗、高速滑动和滚动接触变得越来越重要。在这项工作中，开发了统一的分数热弹性理论，并将其应用于研究由移动热源引起的瞬态响应。从理论上讲，通过引入分数导数的新定义，即 Caputo-Fabrizio、Atangana-Baleanu 和 Tempered-Caputo 类型，提供了对分数热弹性的新见解。在数值上，在本模型中，在两组不同的边界条件下考虑了受热源以恒定速度运动的半无限介质：无应力和给定的温度，第二个位移固定且热绝热。所有响应的解析解首先在拉普拉斯域中制定，然后通过数值方法转化为时域。数值结果表明，Caputo-Fabrizio 和 Atangana-Baleanu 型模型预测的瞬态响应比 Caputo 型理论更小，而 Tempered-Caputo 模型可以通过增加回火参数给出更大的结果。同时详细讨论了分数阶次、Tempered-Caputo模型的回火参数和热源速度对所有响应的影响。响应的时程表明：对于长期过程，TC定义的指数函数是有意义的，TC模型的温度与C模型的温度差异很大。