Using the total energy balance equation and principle of the frame indifference, a fundamental set of relations of the local gradient continuum model of elastic solids is formulated. The model is based on taking account of non-convective and non-diffusive mass flux related to the changes in the material microstructure. Linear stationary governing equations of the local gradient theory and corresponding boundary conditions are also derived by variational principle. In order to investigate the size-dependent behavior of nano-scale structures, this model is combined with the Bernoulli–Euler beam theory. Deflection of the cantilever beam subjected to the end-point loading under the plane stress conditions is evaluated and compared to the corresponding ones provided by the classical theory and by the strain gradient theory. It is shown that the beam deflection within the local gradient theory is smaller than that predicted by the classical Bernoulli–Euler beam theory. This work may be of special interest for designing the devices utilizing the micro/nano-beam elements.

利用总能量平衡方程和框架无差异原理，建立了弹性固体局部梯度连续体模型的基本关系集。该模型基于考虑与材料微观结构变化有关的非对流和非扩散质量通量。利用变分原理推导了局部梯度理论的线性平稳控制方程及相应的边界条件。为了研究纳米级结构的尺寸依赖性行为，该模型与Bernoulli–Euler束理论相结合。评估了在平面应力条件下承受端点载荷的悬臂梁的挠度，并将其与经典理论和应变梯度理论提供的相应挠度进行了比较。结果表明，局部梯度理论中的光束偏转小于经典的伯努利-欧拉光束理论所预测的偏转。对于利用微/纳米束元件设计设备，这项工作可能会特别有意义。