Thermomechanically consistent formulations of the standard linear solid using fractional derivatives
Thermomechanically consistent formulations of the standard linear solid using fractional derivatives
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使用分数导数的标准线性固体的热机械一致公式
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
A. Lion
中科院分区：
文献类型：
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作者：
A. Lion
我们研究了一个常用的分数推广的标准线性固体的热机械性能。它的数学结构来自于应力和应变之间的普通线性微分方程，当用0 < α，β < 1的分数阶导数代替一阶时间速率时。如果不进一步限制参数α和β，则模型会导致非物理行为。在谐波变形的情况下，耗散模量可以变为负值。这相当于负熵产生，违反了热力学第二定律。然后，我们提出了两个概括的标准线性固体，这是基于所谓的一致性分数流变元素。它具有一个非负的自由能和耗散率的任意变形过程，并符合热力学第二定律。所提出的推广的应力和应变之间的微分方程也包含不同阶的分数阶导数，但动态模量和松弛谱都是它们的参数的非负函数。不需要对材料参数进行限制。
We study the thermomechanical properties of a frequently used fractional generalisation of the standard linear solid. Its mathematical structure arises from an ordinary linear differential equation between stress and strain when replacing the first order time rates by fractional derivatives of the order 0 < α , β < 1. If the parameters α and β are not further restricted, the model leads to an unphysical behaviour. In the case of harmonic deformations the dissipation modulus can become negative. This corresponds to a negative entropy production and violates the second law of thermodynamics. Then we propose two generalisations of the standard linear solid which are based on a so-called thermodynamically consistent fractional rheological element. It possesses a non-negative free energy and rate of dissipation for arbitrary deformation processes and is compatible with the second law of thermodynamics. The differential equations between stress and strain of the proposed generalisations contain also fractional derivatives of different orders but both the dynamic moduli and the relaxation spectra are non-negative functions of their arguments. No restrictions on the material parameters are required.