材料力学中梁内单根纵向纤维弯曲变形研究
The Bending Deformation Analysis of a Beam Longitudinal Fiber in Mechanics of Materials

众所周知，根据单向受力假设，在纯弯矩作用下只在梁横截面上存在正应力，而不存在横向正应力。因此，可能会出现这样的疑问:对于梁上横向尺寸趋于零的纵向纤维，在这样的应力状态下为何会发生弯曲？本文将简要讨论这个问题。首先，梁中任意薄层的弯曲应力梯度会产生弯矩，然后得到该弯矩引起的弯曲变形，最后令薄层厚度趋于零，薄层应力梯度引起的弯曲变形曲率与一般梁理论推导的弯曲梁的曲率完全一致，最终证实薄层弯曲的原因是应力梯度。本文可以为学生深入理解梁弯曲与应力梯度提供一个有趣的视角。
It is well-known that only normal stress keeps along the cross-section of a beam under pure bending moment, without transverse normal stress according to the famous Kirchhoff assumption. Consequently, the doubt may come out: Why does the beam bend under such a load? This paper will present a brief discussion to answer this question. Firstly, the gradient of the bending stress along the transverse direction of the beam results in bending moment; and then, this moment will cause the bending deformation of the beam; finally, this paper proves that the curvature of the bending deformation arising from the stress gradient just mentioned is exactly consistent with that of bending beam derived by general theory of beam. This evidence conclusively confirms the key point of this study, and can be an interesting view for students to deeply understand the analysis of bending beam, as well as the methodology of mechanics.