Di Paola M, Alotta G, Burlon A, Failla G
Department of Engineering (DI), University of Palermo, Viale delle Scienze Ed. 8, 90128 Palermo, Italy.
Department of Civil, Energy, Environmental and Materials Engineering (DICEAM), 'Mediterranean' University of Reggio Calabria, via Graziella, 89124 Reggio Calabria, Italy.
Philos Trans A Math Phys Eng Sci. 2020 May 29;378(2172):20190296. doi: 10.1098/rsta.2019.0296. Epub 2020 May 11.
This paper addresses nonlinear viscoelastic behaviour of fractional systems with variable time-dependent fractional order. In this case, the main challenge is that the Boltzmann linear superposition principle, i.e. the theoretical basis on which linear viscoelastic fractional operators are formulated, does not apply in standard form because the fractional order is not constant with time. Moving from this consideration, the paper proposes a novel approach where the system response is derived by a consistent application of the Boltzmann principle to an equivalent system, built at every time instant based on the fractional order at that instant and the response at all the previous ones. The approach is readily implementable in numerical form, to calculate either stress or strain responses of any fractional system where fractional order may change with time. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
本文探讨了分数阶随时间变化的分数系统的非线性粘弹性行为。在这种情况下，主要挑战在于玻尔兹曼线性叠加原理，即线性粘弹性分数阶算子所基于的理论基础，由于分数阶并非随时间恒定，因此不能以标准形式应用。基于此考虑，本文提出了一种新颖的方法，通过将玻尔兹曼原理一致地应用于一个等效系统来推导系统响应，该等效系统在每个时刻根据该时刻的分数阶以及之前所有时刻的响应构建而成。该方法易于以数值形式实现，用于计算分数阶可能随时间变化的任何分数系统的应力或应变响应。本文是“通过分数阶微积分进行先进材料建模：挑战与展望”主题特刊的一部分。
