分数阶广义热弹理论及其在压电/电磁介质和生物材料中的应用

With the progress of science and technology and the in-depth understanding of thermo-elastic properties of materials, the thermoelasticity theories based on integer calculus fails to describe thermoelastic response accurately. As a result, fractional calculus is being used in thermo-elastic problems. In view of the imperfection of fractional generalized thermoelasticity and lacking of accurate and efficient numerical analysis method, fractional generalized thermoelasticity will be introduced for multi-coupling (e.g. electric, thermal, magnetic and elastic fields) piezoelectric/electromagnetic media, and for porous medium considering micromorphic (micropolar and microstretch) effect. To understand the fractional generalized thermoelasticity deeply, the basic properties of proposed theories, such as uniqueness, reciprocal theorems and variational principle will be discussed. The efficient and accurate numerical analysis method for two/three-dimensional and multi-freedom (multi-field coupling) problems of fractional generalized thermoelasticity will be proposed, serving as an effective approach for numerical investigation. The effect of fractional order on thermoelastic response will be estimated, together with the inverse analysis based on parameter controlling, in which the fractional order will be deduced directly. Finally, the thermoelastic response of biological materials is considered by combining the proposed models with biomechanics, exploring the mechanism of un-comfort even pain people feel in an extreme environment and providing academic guidance for extensive application of thermal therapies.

随着科技进步及对材料热弹性能认识的深入，基于整数阶微积分的热弹理论已无法满足准确描述材料热弹响应的要求，分数阶微积分已被用于建立热弹耦合理论。针对分数阶广义热弹理论不够完善和准确高效求解方法缺乏的现状，本项目将建立压电/电磁介质（电、热、磁、弹多场耦合）和多孔介质（微极、微伸效应）的分数阶广义热弹理论，讨论所建理论的基本特性，如唯一性、互易定理和变分原理等，加深对分数阶广义热弹理论的理解；发展分数阶广义热弹问题高效准确的数值求解方法，为高维(二维以上)及多自由度（多场耦合）复杂问题的研究提供有效手段；研究分数阶次对热弹响应的影响并开展分数阶广义热弹问题参数控制反问题的研究，获得针对具体材料的分数阶次，为建立准确分数阶热传导方程提供依据；将所建分数阶理论与生物力学结合，研究生物材料（皮肤等）的热弹性响应，揭示极端环境下皮肤不适感机理，为热疗法的进一步改善和广泛应用提供理论指导。

随着科技进步及对材料热弹性能认识的深入，基于整数阶微积分的热弹耦合理论已无法满足准确描述材料瞬态热弹响应的要求。本项目根据材料热传导特性，建立了形式统一、物理意义明确的分数阶热弹耦合理论，发展了基于Laplace积分变换的分数阶广义热弹问题高效准确的半解析求解方法，为分数阶热弹问题的研究提供了有效手段；利用所建分数阶热弹耦合模型研究了生物介质（皮肤）、层合结构等的热弹响应，获得了模型的分数阶次、材料物性参数、层合结构界面热/机械阻抗等对响应的影响，为医疗中热疗法的改善和广泛应用提供理论指导。结果表明：所建热弹性模型随着分数阶次的减小，温度、位移和应力的最大幅值逐渐减小，分数阶次减小可描述热传播性能的减弱；界面热阻抗使界面处的温度出现阶跃，且热阻抗越大，温度阶跃越大；界面弹性阻抗越大，界面温差越大，热传播越远。基于G-N II型热弹性理论研究了双层模型皮肤的热弹响应，发现G-N II型热弹性模型可消除Pennes生物传热方程不合理的温度响应；在界面附近温度出现很大的跳跃，导致界面附近产生大的变形和损伤。根据瞬态热弹问题影响空间尺度微小的特点，项目将非局部效应引入热弹性理论，建立了非局部热弹性理论，发现考虑非局部效应可消除热波波前处温度的不连续，温度分布更加合理。同时，研究了广义热弹扩散问题等，有效推进了广义热弹耦合理论的发展。