连续介质热力学中热传导的非线性与非局部模型
Nonlinear and nonlocal models of heat conduction in continuum thermodynamics
AI总结：
该研究在连续介质热力学框架下构建通用热流本构方案，基于经典热力学与Coleman-Noll方法，分别推导局部与弱非局部率型热传导模型，复现多种经典热导模型并推广出相关非线性推广模型。
摘要：
本文旨在建立连续介质热力学框架下构建一套通用本构方案，用以描述可变形介质中的热流行为。研究从经典热力学方法出发，在物质（拉格朗日）描述下定义率型本构方程，其中标准时间导数满足客观性原理。所有本构函数均需依赖一组共同的自变量，且与热力学规律保持一致。\n第二定律的表述采用通用非局部形式：熵产率由一个非负本构函数给定，额外熵通量满足无流动边界条件。随后基于Coleman-Noll方法推导热力学响应。在局部表述中，假设自由能势与熵产率函数依赖于温度、温度梯度、热流矢量及其时间导数。该方法得到的热流矢量率型本构方程，其本质上符合第二定律，且易于开展分析。由此可推导出一大类线性与非线性率型模型（例如Cattaneo-Maxwell型、类Jeffreys型、Green-Naghdi型、Quintanilla型以及类Burgers热导体模型）。在第二定律的（弱）非局部表述中，假设熵产率与熵额外通量矢量均依赖于温度、温度梯度、热流矢量及其空间梯度与时间导数。在这一（经典）热力学框架内，可得到非局部Guyer-Krumhansl模型以及由Cimmelli和Sellitto提出的若干非线性推广模型。
英文摘要：
The aim of this paper is to develop a general constitutive scheme within continuum thermodynamics to describe the behavior of heat flow in deformable media. Starting from a classical thermodynamic approach, the rate-type constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. All constitutive functions are required to depend on a common set of independent variables and to be consistent with thermodynamics. The statement of the Second Law is formulated in a general nonlocal form, where the entropy production rate is prescribed by a non-negative constitutive function and the extra entropy flux obeys a no-flow boundary condition. The thermodynamic response is then developed based on Coleman-Noll procedure. In the local formulation, the free energy potential and the rate of entropy production function are assumed to depend on temperature, temperature gradient and heat-flux vector along with their time derivatives. This approach results in rate-type constitutive equations for the heat-flux vector that are intrinsically consistent with the Second Law and easily amenable to analysis. A huge class of linear and nonlinear models of the rate type are recovered (e.g., Cattaneo-Maxwell's, Jeffreys-like, Green-Naghdi's, Quintanilla's and Burgers-like heat conductors). In the (weakly) nonlocal formulation of the second law, both the entropy production rate and an entropy extra-flux vector are assumed to depend on temperature, temperature gradient and heat-flux vector along with their spatial gradients and time derivatives. Within this (classical) thermodynamic framework the nonlocal Guyer-Krumhansl model and some nonlinear generalizations devised by Cimmelli and Sellitto are obtained
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