Abstract In the present article, the idea of using the variable-order fractional-derivative thermoviscoelastic constitutive laws in dynamic stress and vibration analysis of the engineering structures, the required implementation backgrounds, and the relevant numerical solution procedures are investigated for the first time. In this regard, dynamic 3D stress and displacement fields and radial/transverse vibrations of transversely graded viscoelastic spinning thick plates/discs exposed to sudden thermoelastic loads are investigated. Instead of using the approximate plate theories, the exact thermoviscoelasticity theory is employed in the development of the governing equations. Since the variable fractional order is dependent on the localized deformation rates, the resulting thermoviscoelastic integro-differential equations are nonlinear. These equations are solved by utilizing a combination of the second-order backward/central/forward finite difference discretization of the spatial and time domains, numerical evaluation and updating of the Caputo-type fractional derivatives, updating the growing number of terms of the governing equations, and Picard's iterations. Various edge conditions are considered. Finally, comprehensive sensitivity analyses and various 3D plots are presented and discussed regarding the effects of the variable fractional order of the constitutive law, time variations of the nonuniformly distributed transverse loads, and edge conditions on the distributions and damping of the resulting displacement and stress components.

中文翻译：

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N3D 非线性可变应变率相关阶次分数热粘弹性动态应力研究和厚横向渐变旋转环形板/盘的振动

摘要 本文首次研究了变阶分数阶导数热粘弹性本构定律在工程结构的动态应力和振动分析中的应用思路、所需的实施背景和相关的数值求解程序。在这方面，研究了暴露于突然热弹性载荷的横向渐变粘弹性旋转厚板/圆盘的动态 3D 应力和位移场以及径向/横向振动。在控制方程的开发中，不使用近似板理论，而是采用精确的热粘弹性理论。由于可变分数阶数取决于局部变形率，因此产生的热粘弹性积分微分方程是非线性的。这些方程是通过利用空间和时间域的二阶后向/中心/前向有限差​​分离散化、数值评估和更新 Caputo 型分数阶导数、更新控制方程的项数不断增加来求解的, 和 Picard 的迭代。考虑了各种边缘条件。最后，针对本构律的可变分数阶次、非均匀分布横向载荷的时间变化以及边缘条件对所得位移和应力分量的分布和阻尼的影响，提出并讨论了综合敏感性分析和各种 3D 图.