裂纹偏折与分岔是工程实践中无法避免的力学现象，是断裂力学的经典问题之一。裂纹可能由于动态扩展、结构非均匀性、局部应力状态变化等因素导致偏折或分岔。在工程实践中，裂纹的偏折与分岔不仅在安全设计方面有很大指导意义，在能源开采领域的价值也在日益凸显，在飞速发展的水力压裂领域有着广阔的应用前景。

为了准确预测裂纹的扩展路径，通常需要获取裂纹尖端的应力场或者应力强度因子。现有的针对偏折或分岔裂纹应力强度因子的研究，都只局限于偏折或分岔部分趋于无穷小的情况，而对于任意尺寸的偏折或分岔裂纹则无法给出准确的解答。基于这一研究背景，本文通过平面弹性理论以及复分析的方法，深入探讨了裂纹的形状以及边界条件对裂纹尖端应力场和应力强度因子的影响，并取得了如下的创新性成果：

1. 通过平面弹性理论和Schwarz-Christoffel映射，将原平面上的偏折裂纹外部区域映射到新复平面平面上的单位圆外部，大大简化了对应边界值问题的求解难度，得到了无限大平面中偏折裂纹对应的复变应力函数，并基于此计算出了裂纹尖端的弹性应力场以及应力强度因子的解析解。
2. 通过求解偏折裂纹尖端的应力强度因子以及应变能释放率，基于最大应变能释放率准则及裂纹与界面交互准则分析了偏折裂纹在各向同性介质以及含有异质界面的介质中发生二次偏折的行为，并讨论了裂纹的形状及异质界面对二次偏折造成的影响，发现了偏折裂纹在发生二次偏折时更容易产生z字型裂纹的现象。
3. 考虑了边界值问题中求解奇异积分方程时积分函数在积分路径上存在间断点的影响，将借助复分析的求解方法推广至适用于无限大平面中含有任意多分岔裂纹的边界值问题，求解出了Laurent级数形式的复变应力函数，并得到了裂纹尖端应力强度因子的半解析解。
4. 以常见的三分岔裂纹和四分岔裂纹为例，通过求解其裂纹尖端的应力强度因子以及应变能释放率，在能量准则下讨论了多分岔裂纹的形状尺寸以及外部加载形式对于裂纹发生进一步扩展的影响。
5. 以前文建立的偏折裂纹尖端应力强度因子的理论求解框架为基础，我们分析了页岩内部偏折型压裂裂纹在弱区的扩展路径选择问题。我们建立了页岩压裂的基本力学模型，并分析了压裂裂纹所处的应力状态。我们还发展了快速估算非自由表面偏折裂纹尖端应力强度因子的摄动方法，并给出了其一阶渐近解。利用这一结果以及前文的应力强度因子的求解方法，我们成功获得了压裂裂纹尖端的应力强度因子，并配合相应的扩展准则，分析了偏折裂纹的形状以及与弱区的相对位置对裂纹扩展路径选择的影响。

建立偏折或分岔裂纹尖端应力场及应力强度因子的理论求解模型，并配合相应的裂纹扩展判定准则，将有助于更进一步预测裂纹的扩展路径，分析裂纹尖端的不稳定性，并为研究裂纹扩展及裂纹网系的形成提供基本的理论工具。

Crack kinking and branching is inevitable in engineering practice, and also one of the classic problems in fracture mechanics. When cracks are subjected to dynamic or non-uniform stress field, or propagate in heterogeneous media with internal discontinuities, they may kink or branch. In engineering practice, this phenomenon is not only of great significance in safety design, but also of increasing value in energy harvesting. The latter is evidently seen in the course of rapid progress of hydraulic fracking.

In order to accurately predict the propagation path of cracks, stress fields or the stress intensity factors of the crack tips are often desired. Existing researches on the stress intensity factors of kinked or branched crack are aimed on infinitesimal kink or branches. A general method to tackle the stress intensity factors of kinked or branched cracks with arbitrary sizes is still lacking. Based on this research background, this thesis explored the influence of crack shape and boundary condition on the stress fields and stress intensity factors at the crack tips by employing theory of planar elasticity and complex analysis, and has achieved the following innovative results:

1. By employing the theory of planar elasticity and Schwarz-Christoffel mapping, the outer region of a kinked crack on the original plane is mapped into the exterior of the unit circle on a new complex plane. This significantly simplifies the difficulty of solving the corresponding boundary value problem. As a result, the complex stress functions for a kinked crack in an infinite plane are obtained, which are then used to calculate the analytical solutions for the stress fields and the stress intensity factors at the crack tips.
2. By solving for the stress intensity factors and strain energy release rates at the tip of a kinked crack, this study analyzed the behavior of secondary deflection in both isotropic media and media containing heterogeneous interfaces. Based on the maximum strain energy release rate criterion and the criterion for crack-interface interaction, we examined the influence of crack shape and heterogeneous interfaces on secondary deflection. It was found that a kinked crack is more likely to form a zig-zag trajectory when undergoing secondary deflection.
3. Considering the influence of discontinuities in the integrand along the integration path when solving singular integral equations in the boundary value problems, the method using complex analysis has been extended to apply to boundary value problems in an infinite plane containing an arbitrary branched crack. The solution yields a Laurent series form of the complex stress functions and provides semi-analytical solutions for the stress intensity factors at the crack tips.
4. Taking three-branched and four-branched cracks for instance, this study solved for the stress intensity factors and strain energy release rates at the tips of these cracks. Under energy-based criterion, we discussed the influence of the shape and size of multi-branched cracks as well as external loading conditions on the further propagation of cracks.
5. Based on the theoretical framework we established for accurately calculating the stress intensity factors of kinked cracks, we conducted an analysis on the propagation path prediction of kinked hydraulic cracks facing weak zones in shale formation. We formulated a fundamental mechanical model for shale fracking and examining the stress status surrounding the crack. Additionally, we established a perturbation method for rapidly estimating the stress intensity factors of a kinked crack with surface tractions. Utilizing the above result, we successfully obtained the stress intensity factors of the hydraulic cracks. In accordance with the corresponding propagation criteria, we analyzed the influence of the shape of a kinked hydraulic crack and its relative position to the weak zone on the crack propagation path.

The theoretical framework for solving the stress fields and stress intensity factors at the crack tips of kinked or branched cracks, coupled with the suitable criteria for crack propagation, will contribute to further prediction of crack propagation path and analysis of crack tip stabilities. We hence expect that it can serve as a convenient tool to analyze the crack propagation and crack network formation.