In this paper, we introduce a new unified formula that governs two different definitions of fractional derivative; the classical Caputo definition and Caputo–Fabrizio's new definition. Hence, an analysis has been constructed for fractional viscothermoelastic, isotropic, and homogeneous nanobeams. The governing equations of the viscothermoelastic nanobeam have been constructed in the context of the non-Fourier heat conduction law with one relaxation of Lord and Shulman (L-S). Laplace transform has been applied, and its inversions have been calculated by using the Tzou method of approximation. The numerical results have been validated for a thermoelastic rectangular nanobeam of silicon as a case when it is subjected to ramp-type heating and simply supported. The fractional-order parameter based on the two types of fractional derivatives has significant impacts on all the studied functions except the temperature increment function. The results based on the two types of fractional derivatives are different, although they generate the same behavior of the thermomechanical waves. The ramp-time heat parameter has significant effects on all the studied functions.

在本文中，我们引入了一个新的统一公式，该公式控制分数阶导数的两种不同定义；经典的 Caputo 定义和 Caputo–Fabrizio 的新定义。因此，构建了分数粘热弹性、各向同性和均匀纳米束的分析。粘热弹性纳米梁的控制方程是在非傅立叶热传导定律的背景下构建的，其中有一个 Lord 和 Shulman (LS) 的松弛。应用了拉普拉斯变换，并使用 Tzou 近似法计算了其反演。当硅的热弹性矩形纳米梁受到斜坡式加热和简单支撑时，数值结果已经得到验证。基于两类分数阶导数的分数阶参数对除温度增量函数外的所有研究函数都有显着影响。基于两种类型的分数阶导数的结果是不同的，尽管它们产生相同的热机械波行为。斜坡时间热参数对所有研究函数都有显着影响。