静动载荷作用下岩石工程结构多裂纹扩展的理论研究

Most of failures in rock engineering structures, either rock slopes or underground tunnels/chambers, start with fissure initiation, growth, coalescence up to cutting-through. For the rock engineering structures, the physical laws and the mathematical models associated with the static and dynamic response are both different, with the need to be studied respectively. In the setting of continuum, 1) a criterion for crack initiation/extension will be formulated, tackling the issues of whether a crack is initiated or extended, the direction in which it develops, and whether the initiation/extension is branched; 2) a mixed complementarity problem arising from static growth of multiple cracks is posed, together with a projection-contraction algorithm. By solving the problem, both the equilibrium condition and the crack propagation criterion are satisfied; and 3) an optimal control model for describing dynamic growth of multiple cracks is built, along with an optimization algorithm for this model, which overcomes the indeterminacy of cracking velocity. All these studies are implemented in the context of the numerical manifold method. The corresponding codes are developed with no mesh dependence. The laboratory tests are conducted for tensile and compression-shear extension, respectively, aiming at the verification of the physical and mathematical models. Finally, The full process response analysis of some typical high rock slopes and in-depth tunnels in hydropower station is carried out. The studies are of significance in both theory and practice in understanding the mechanism of failure of rock engineering structures and perfecting the numerical manifold method.

岩石工程结构的失稳破坏，无论是边坡还是地下硐室，大都起自裂纹的萌生、发育、合并直至贯通。静、动载荷作用下的多裂纹扩展所遵守的物理法则和数学模型都不相同，需分别进行研究。本项研究在连续介质框架下，1）建立基于强度理论的裂纹萌生和扩展准则，该准则回答裂纹是否萌生或扩展、向何处扩展，以及是否会产生分叉扩展；2）提出多裂纹静态扩展的混合互补问题并设计投影-收缩算法，以便同时满足裂纹静态扩展时的平衡条件和裂纹扩展准则；3）建立多裂纹动态扩展的最优控制模型和优化算法，解决多裂纹动态扩展速度的不确定问题。选用数值流形方法作为实施工具，研制无网格依赖性的多裂纹静、动态扩展分析程序。进行多裂纹张拉扩展和压剪扩展的室内测试，并对数理模型进行验证。最后，针对典型水电工程高陡边坡和深埋隧道进行全程响应分析。研究成果对于深入揭示岩石工程结构的变形失稳机制、完善数值流形方法具有理论和实际意义。

主要研究内容：在连续介质框架下，1）建立基于强度理论的裂纹萌生和扩展准则，该准则回答裂纹是否萌生或扩展、向何处扩展，以及是否会产生分叉扩展；2）提出多裂纹静态扩展的混合互补问题并设计投影-收缩算法，以便同时满足裂纹静态扩展时的平衡条件和裂纹扩展准则；3）建立多裂纹动态扩展的最优控制模型和优化算法，解决多裂纹动态扩展速度的不确定问题。.重要结果：1）提出了适用于基于单位分解法的对角化质量矩阵的一般方案；2）解决了基于单位分解法的本质边界和界面连续性条件的施加问题；3）构建了外问题的数值流形法；4）解决动态裂纹扩展的自由度继承问题。.关键数据：据不完全统计，在项目的资助下，在国际主流期刊上发表SCI论文约30篇，4篇进入ESI，SCI引用次数超500，是国际上在这一领域发文最多和影响最大的研究小组。.科学意义：前三个研究结果解决了计算力学中多个悬而未决的难题：1）我们所提出的高阶单元质量矩阵的对角化方案有严格的数学力学基础，打破了权威人士关于高阶单元质量矩阵对角化的论断：无规可循，纯粹是“tricky（魔术）”；2）基于单位分解法的本质边界条件和界面连续性条件的施加实现了严格满足Galerkin法的解的构造，至少对于二维问题而言是一个终极解决方案（石根华评语）；3）外边值问题的NMM可以精确地再现解在趋于无穷式的渐进行为，为最终动力学问题的彻底解决探明了一个途径。