基于表面弹性理论的开裂椭圆孔断裂力学研究

研究纳米尺度时开裂椭圆孔的III型断裂性能。基于表面弹性理论和保角映射技术，利用复势函数理论获得了缺陷(裂纹和椭圆孔)周围应力场和裂纹尖端应力强度因子的闭合解答。所得结果具有一般性，许多已有和新的解答可由本文退化的特殊情形得到。利用解析结果讨论了缺陷的绝对尺寸、椭圆孔的形状比以及裂纹的相对尺寸对应力强度因子的影响。结果表明：考虑表面效应且缺陷尺寸在纳米尺度时，应力强度因子具有显著的尺寸依赖效应；应力强度因子随椭圆孔形状比的变化规律受缺陷表面常数的影响；缺陷表面效应的影响取决于椭圆孔的形状比，非常大的形状比屏蔽了表面效应的影响；裂纹相对尺寸非常小时表面效应影响较弱，裂纹相对尺寸较大时表面效应较为明显。

Fracture mechanics of cracked elliptical hole based on surface elasticity theory

Mode III fracture characteristics of a nano-sized cracked elliptical hole is investigated. Based on the surface elasticity theory and the conformal mapping technique, closed form solutions of the stress fields around the defects (crack and elliptical hole) and the stress intensity factor at crack tip are presented by using the complex potential function. The present solution is fairly general such that many existing and new results can be regarded as special degenerated cases. By applying the analytical solutions to typical examples, effects of the size of defects, the shape ratio of elliptical hole and the relative size of crack on the stress intensity factor are discussed. The results show that the stress intensity factor is size-dependent significantly when considering the surface effect of the defects at the nanometer scale. The size effect of the defects decreases with the increase of the defects size, and gradually approaches the classical elastic theory. The variation of the stress intensity factor with shape ratio of the elliptical hole is related to the surface constant of the defects. With the increase of the elliptical hole shape ratio, the dimensionless stress intensity factor increases slightly and then decreases for the case of classical elastic theory and positive surface constant. When the surface constant is negative, the dimensionless stress intensity factor monotonously decreases to a stable value. The effect of the surface effect of defects depends on the shape ratio of the elliptical hole, and the very high shape ratio shields the contribution of the surface effect. With the relative size of the crack increases, the dimensionless stress intensity factor increases first to the maximum and then decreases. The surface effect is weak when the relative size of the crack is very small, whereas the surface effect is obvious when the relative size of the crack is large.