热力学问题的变式表述

S. A. Lurie, P. Belov, A. V. Volkov
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引用次数: 0

摘要

本文提出利用四维时空连续体建立变分热力学连续体模型。为了确定可逆过程中的物理常数,提出了物理上合理的假设。这些假设包括互补剪应力、动量对速度的经典依赖性以及热流潜能(广义麦克斯韦-卡塔尼奥定律)。杜哈梅尔-诺伊曼定律被假定为经典定律。在所考虑的模型中,广义麦克斯韦-卡塔尼奥定律和杜哈梅尔-诺伊曼定律并非从现象学角度引入。它们是通过从温度、热流和压力的构成方程中排除热势,从相容方程中推导出来的。耗散通道被视为最简单的非可积分变分形式,在参数变化中呈线性。因此,对 L.I. Sedov 原理进行概括的变分原理应运而生。它是虚功原理的结果,被称为可逆热力学过程的拉格朗日变化与耗散通道代数和之间的差异。研究证明,对于具有二阶微分方程的经典热力学过程,只能存在六个耗散通道。其中两个决定了非耦合系统中的耗散--在运动方程和热平衡方程中。其余四个通道定义了耗散热力学耦合问题中的耦合效应。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Variational formulation of thermomechanical problems
This article proposes that a 4D space-time continuum is used for building variational thermomechanical continuum models. In order to identify physical constants in reversible processes, physically justified hypotheses were formulated. They are the hypotheses of complementary shear stress, classical dependence of momentum on velocity, and heat flow potentiality (generalized Maxwell–Cattaneo law). The Duhamel–Neumann law was assumed to be classical. In the considered model, the generalized Maxwell–Cattaneo and Duhamel–Neumann laws were not introduced phenomenologically. They were derived from the compatibility equations by excluding thermal potential from the constitutive equations for temperature, heat flow, and pressure. Dissipation channels were considered as the simplest non-integrable variational forms, which are linear in the variations of arguments. As a result, a variational principle that generalizes L.I. Sedov’s principle was developed. It is a consequence of the virtual work principle and termed as the difference between the variation of the Lagrangian of reversible thermomechanical processes and the algebraic sum of dissipation channels. It was proved that for the classical thermomechanical processes, with second-order differential equations, there can only exist six dissipation channels. Two of them determine dissipation in an uncoupled system – in the equations of motion and heat balance. The remaining four channels define coupling effects in coupled problems of dissipative thermomechanics.
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