The flexural wave propagation in a microbeam is studied based upon the nonlocal strain gradient model with the spatial and time fractional order differentials in the present work. To capture the dispersive behaviour induced by the inherent nanoscale heterogeneity, the stress gradient elasticity and the strain gradient elasticity are often used to model the mechanical behaviour. The present model incorporates the two models and introduces the fractional order derivatives which can be understood as a generalization of integral order nonlocal strain gradient model. The Laplacian operator in the constitutive equation is replaced with the symmetric Caputo fractional differential in the present model. To illustrate the flexibility of the present model, the flexural wave propagation in a microbeam is studied. The fractional order in the present model as a new material parameter can be adjusted appropriately to describe the dispersive properties of the flexural waves. The numerical results based on the new nonlocal strain gradient elasticity with fractional order derivatives are provided for both Euler–Bernoulli beam and Timoshenko beam. The comparisons with the integer order nonlocal strain gradient model and the molecular dynamic simulation are performed to validate the flexibility of the fractional order nonlocal strain gradient model.

在当前工作中，基于具有空间和时间分数阶差分的非局部应变梯度模型，研究了微束中的弯曲波传播。为了捕获由固有的纳米级异质性引起的色散行为，通常使用应力梯度弹性和应变梯度弹性来模拟机械行为。本模型结合了两个模型，并引入了分数阶导数，可以将其理解为积分阶非局部应变梯度模型的推广。在本模型中，本构方程中的拉普拉斯算子被对称的Caputo分数微分代替。为了说明本模型的灵活性，研究了微束中的弯曲波传播。可以适当调整本模型中作为新材料参数的分数阶，以描述弯曲波的色散特性。欧拉–伯努利梁和蒂莫申科梁都基于基于新的具有分数阶导数的非局部应变梯度弹性的数值结果。与整数阶非局部应变梯度模型进行比较，并进行了分子动力学模拟，以验证分数阶非局部应变梯度模型的灵活性。