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圆孔扩张理论作为一种相对成熟的理论工具已经广泛的运用于岩土工程中的各类问题,但是对于一些初始孔形并非圆孔的扩孔问题,圆孔扩张理论并不适用。基于保角变换的方法将原物理平面上初始非圆形孔洞映射到像平面上的单位圆,将原物理平面上由于孔洞扩张所产生的位移边界条件转换到像平面上,利用平面复变弹性理论,得到初始非圆形孔洞孔扩张的弹性理论解。通过基于该文的方法计算圆孔扩张并且与现有的圆孔扩张的弹性解进行对比分析,验证该文解的可靠性。续而,针对初始孔形为正方形的孔洞,采用该文方法详细地分析了正方形孔扩张的力学特性。研究结果表明,该文解在退化到圆孔扩张问题后与现有的圆孔扩张弹性解完全一致;正方形孔在扩张过程中角点处出现应力集中现象,尤其在极角为30°~45°时应力急剧增加。在离孔心的距离大于4倍的a0/2(a0为初始正方形孔边长)处,其应力分布状态就接近于圆孔扩张,即孔洞非轴对称的影响只在离孔心一定的范围内,当大于这个范围时,这种影响就可以忽略不计。
As a relatively mature theoretical tool, the hole expansion theory has been widely used in geotechnical engineering problems, but for some hole expansion problems that are not circular holes, the hole expansion theory is not suitable. Based on the conformal mapping method, the original non-circular holes on the original physical plane are mapped to the unit circle on the image plane, and the displacement boundary conditions due to the hole expansion on the original physical plane are transformed onto the image plane. Theory, the elastic theoretical solution of the initial non-circular hole expansion is obtained. By calculating the expansion of the circular hole based on the method and comparing with the existing elastic solution of the circular hole expansion, the reliability of the solution is verified. In addition, aiming at the holes whose initial hole shape is square, the mechanical properties of square hole expansion are analyzed in detail by using the method. The results show that this solution is completely consistent with the existing elastic expansion of the circular hole after degenerated to the circular hole expansion problem. The square hole has stress concentration at the corner during the expansion process, especially when the polar angle is 30 ° -45 ° sharp increase in stress. At a distance of more than 4 times from the hole center a0 / 2 (a0 is the length of the initial square hole side), the stress distribution is close to the expansion of the hole, that is, the effect of non-axisymmetric hole is only within a certain range Within this range, the effect is negligible.