基于时间分数阶双相滞模型的超快激光非傅里叶导热研究

Time-fractional dual-phase-lag modeling of non-Fourier heat conduction under ultrafast laser irradiation

摘要

在微纳尺度下，超快激光的热传导常表现出热响应滞后、非局部扩散和记忆效应等非傅里叶特性，传统傅里叶模型难以准确描述。为此，文章构建了基于时间分数阶导数的双相滞(time-fractional dual-phase-lag，TFDPL)热传导模型，在经典双相滞理论中引入分数阶算子以刻画材料对热流历史的记忆效应，提升对微尺度非局部热输运行为的表征能力；通过Laplace变换获得解析解，并采用Stehfest数值逆变换恢复时域温度场，探究分数阶阶数α、克努森数Kn及滞后时间比值b等参数对温度响应的影响。结果表明:α显著影响热扩散速度与温度峰值，体现系统时间非局部性；Kn决定空间尺度效应并调控α的敏感性；b与α具有相对独立的调节作用。TFDPL模型可有效描述超快激光作用下的非傅里叶热行为，为微纳热加工的热管理建模提供理论参考。

Abstract

At micro- and nanoscale dimensions, heat conduction induced by ultrafast laser irradiation exhibits pronounced non-Fourier characteristics, such as thermal response delay, nonlocal diffusion, and memory effects, which cannot be accurately described by the classical Fourier model. To address this limitation, a time-fractional dual-phase-lag (TFDPL) heat conduction model is developed by introducing a fractional-order time derivative into the conventional dual-phase-lag framework to characterize the material's memory effect on the temporal history of heat flux and enhance the description of microscale nonlocal heat transport. Analytical solutions are obtained using the Laplace transform, and the time domain temperature field is reconstructed through the Stehfest numerical inversion method. The influences of key parameters, including the fractional order α, Knudsen number (Kn), and lag-time ratio b—on the temperature response are systematically investigated. The results show that α significantly affects heat diffusion rate and peak temperature, highlighting the temporal nonlocality of the system; Kn governs spatial scale effects and modulates the sensitivity of the thermal response to α, b and α exhibit relatively independent regulatory effects on thermal behavior. Overall, the TFDPL model effectively captures the non-Fourier heat conduction behavior under ultrafast laser heating and provides a theoretical reference for thermal modeling and management in micro/nanoscale laser processing.

关键词

超快激光 / 分数阶导数 / TFDPL / 时间记忆效应

Key words

ultrafast laser / fractional derivative / TFDPL / temporal memory effect

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基金资助

山东省高等学校青创科技支持计划项目(2019KJH012)

济南市科研带头人工作室项目(2019GXRC056)

山东省自然科学基金项目(ZR2017BEE052)