In the current practice, the one-dimensional nonlocal constitutive relations are always employed to model beam-type structures, regardless of the inherent three-dimensional interactions between atoms, resulting in inaccurately predicted nonlocal structural behaviors. To improve modeling, the present work first reveals and establishes how the nonlocal interactions in beams’ width and height directions substantially affect the bending behaviors of nanobeams. Based on the new concept of a general three-dimensional nonlocal constitutive relation, a three-dimensional nonlocal Euler–Bernoulli beam model is developed for bending analysis. Closed-form solutions for the deflections of simply supported and cantilevered beams are derived. The cross-section effect on the nonlocal stress, the bending moment and the deflection is explored. Moreover, an effective calibration method is proposed for the determination of the intrinsic characteristic length based on molecular dynamics simulations. It has been found that the actually nonlocal physical dimension of beam-type structures is not in agreement with that from the one-dimensional geometric direction of classical beam mechanics, and the beams’ width and height must be taken into consideration, instead of just employing the beam’s length, a practice which has been somewhat blindly undertaken in the current practice without indeed much theoretical support and understanding. A beam problem at nanoscale has to be viewed and treated as a “three-dimensional” geometric and physical problem due to its geometric dimension in length and its physical dimensions in both the thickness and the width. When its intrinsic characteristic length is comparable to its width or thickness, a rigidity-softening effect of a beam can be observed. Further, it is established that the nonlocal effect is more sensitive in the thickness direction, as compared with the width direction.

在当前实践中，一维非局部本构关系始终被用来对梁型结构进行建模，而与原子之间固有的三维相互作用无关，从而导致预测的非局部结构行为不准确。为了改进建模，本工作首先揭示并确定了光束宽度和高度方向上的非局部相互作用如何实质上影响纳米束的弯曲行为。基于一般的三维非局部本构关系的新概念，开发了三维非局部Euler–Bernoulli梁模型用于弯曲分析。推导了简单支撑和悬臂梁挠度的封闭形式解。探讨了截面对非局部应力，弯矩和挠度的影响。此外，提出了一种基于分子动力学模拟确定内在特征长度的有效校准方法。已经发现，梁型结构的实际非局部物理尺寸与经典梁力学的一维几何方向上的尺寸不一致，必须考虑梁的宽度和高度，而不仅仅是采用梁的长度，这种做法在目前的实践中有些盲目地进行，实际上并没有太多的理论支持和理解。由于其在长度上的几何尺寸以及在厚度和宽度上的物理尺寸，因此必须将纳米级的光束问题视为“三维”几何和物理问题。当其固有特征长度与其宽度或厚度相当时，可以观察到梁的刚度软化效果。此外，可以确定的是，与宽度方向相比，非局部效应在厚度方向上更加敏感。