Out-of-plane curvature of real submarine slopes imposes limitations on applicability of existing planar criteria for catastrophic growth of slip surfaces. In this paper, the growth of an initially weakened zone in three-dimensional (3D) convex and concave slopes is investigated using the process zone approach. The geometry of the problem is presented in a curvilinear coordinate system for which the governing equations for the three-dimensional slip surface growth are derived. Solution of these equations for an axisymmetric problem is obtained both analytically and numerically (using a finite differences scheme) and benchmarked against Coupled Eulerian-Lagrangian finite element simulations. It is shown that the application of the planar slope solution to conical slopes constitutes an overestimation of the slope's stability. The closed form criteria for an unstable 3D slip surface growth in both convex and concave slopes are proposed and validated by fitting numerical results for various sizes and aspect ratios of the initially pre-softened zone.

中文翻译：

---

3D 锥形斜坡中滑动面的增长

真实海底斜坡的平面外曲率限制了现有平面标准对滑面灾难性增长的适用性。在本文中，使用过程区方法研究了三维 (3D) 凸面和凹面斜坡中初始弱化区的生长。问题的几何结构在曲线坐标系中提出，由此推导出了三维滑动面增长的控制方程。轴对称问题的这些方程的解是通过解析和数值（使用有限差分方案）获得的，并以耦合欧拉-拉格朗日有限元模拟为基准。结果表明，将平面边坡解应用于锥形边坡构成了对边坡稳定性的高估。