A generalized non-local stress–strain gradient theory is presented using fractional calculus. The proposed theory includes as a special case: the classical theory; the non-local strain gradient theory; the Eringen non-local theory; the strain gradient theory; the general Eringen non-local theory; and the general strain gradient theory. This new formulation is therefore more comprehensive and more complete to model physical phenomena. Its application has been shown in free vibration, buckling and bending of simply supported (S–S) nano-beams. The non-linear governing equations have been solved by the Galerkin method. Furthermore the effects of different (additional) model parameters like: the length scale parameter; the non-local parameter; and different orders (integer and non-integer) of strain and stress gradients have been shown.

使用分数演算提出了一种广义的非局部应力应变梯度理论。所提出的理论作为一个特例包括：古典理论；非局部应变梯度理论；艾林根非本地理论；应变梯度理论 一般的艾林根非局部理论；以及一般的应变梯度理论。因此，此新公式对于物理现象的建模更加全面和完整。在简单支撑的（S–S）纳米光束的自由振动，屈曲和弯曲中已经证明了其应用。非线性控制方程已通过Galerkin方法求解。此外，不同（附加）模型参数的影响，例如：长度比例参数；非本地参数；并已显示出应变和应力梯度的不同阶（整数和非整数）。