The bending fracture of an ultra-thin plate containing a thickness-through crack with consideration of surface elasticity is studied in this paper. Based on the Kirchhoff plate theory incorporating surface elasticity, a governing equation is derived for static bending of nanoplates with surface elasticity. The fracture problem of an infinite isotropic elastic nanoplate with a thickness-through crack is presented and solved when the plate is subjected to uniform bending moment, twisting moment, and out-of-plane tearing load, respectively. The Fourier integral transform is applied to solve associated mixed boundary value problems. A hypersingular integral equation is deduced and an exact solution is determined for each case. Complete elastic fields at any position of the nanoplate are given explicitly in terms of elementary functions and the expressions for the intensity factors of stress, shear force, and bending/twisting moment at the crack tips are obtained in closed form. The intensity factors of stress and moment exhibit a square-root singularity, while the effective shear force intensity factors have an $r^{-3/2}$ singularity near the crack tips, r being the distance from the crack tip. Results are presented graphically to show that the stress intensity factors (SIFs) are related to the bulk and surface material properties. When neglecting surface elasticity, the obtained size-dependent SIFs reduce to the well-known classical counterparts.

本文研究了考虑表面弹性的含厚度贯穿裂纹超薄板的弯曲断裂问题。基于结合表面弹性的基尔霍夫板理论，推导出具有表面弹性的纳米板静态弯曲的控制方程。提出并解决了具有厚度贯穿裂纹的无限各向同性弹性纳米板在分别承受均匀弯矩、扭转力矩和面外撕裂载荷时的断裂问题。傅里叶积分变换用于解决相关的混合边值问题。推导出超奇异积分方程，并为每种情况确定精确解。根据基本函数明确给出了纳米板任意位置的完整弹性场，并以闭合形式获得了裂纹尖端的应力、剪切力和弯曲/扭转力矩强度因子的表达式。应力和力矩的强度因子呈现平方根奇异性，而有效剪力强度因子具有$r^{——3/2}$裂纹尖端附近的奇点，r是裂纹尖端的距离。结果以图形方式显示，以表明应力强度因子 (SIF) 与本体和表面材料特性有关。当忽略表面弹性时，获得的与尺寸相关的 SIF 减少到众所周知的经典对应物。