灾变破坏是岩石类非均匀脆性材料破坏的一种典型形式，其发生时伴随着大量能量的突然释放，具有极大的破坏性。大地震的发生往往也属于这种破坏类型。在灾变破坏点附近，许多物理量（如应变、位移、声发射事件数、能量释放率等）呈现出加速的演化行为，并且其变化率在破坏点处遵循幂律规律发散到无穷大。这种演化曾被类比于相变的临界现象来加以研究。但不同于经典的临界现象，灾变破坏的幂律指数呈现出许多新的特点。此外，确定临界幂律奇异性指数的特征，对于准确地建立灾变预测方法至关重要。

本文通过大理岩和花岗岩试样的准静态单轴压缩实验，发现灾变岩样的响应函数R = du/dU在灾变破坏前呈现出临界幂律奇异性，其中u是试样的整体压缩量，U是试验机输出的位移。实验发现，R的临界幂律奇异性指数在−1和−1/2之间分布。在理论上阐明了响应函数R的临界幂律奇异性是灾变破坏能量准则的必然结果，而临界幂律奇异性指数的分布范围可由幂函数近似来导出。

基于临界幂律奇异性指数的分布范围，将响应函数和加载过程关联起来，提出了一个灾变预测方法，来实时的计算破坏点的上界及相应的预测区间。随着加载的进行和破坏点的临近，预测点逐渐向破坏点移动，包含真实破坏点的预测区间也逐渐缩小。

通过数字散斑相关方法测量了试样的表面位移场和应变场，采用条带分析方法研究了位移场和应变场的时空演化，观察到了灾变破坏前试样表面的应变局部化现象。实验发现，所有条带的平均位移分量构造的局部响应函数在灾变破坏前均出现临界幂律奇异性，临界幂指数介于−1和−1/2之间。破裂带附近垂直于破裂带方向的条带平均正应变构造的局部响应函数也呈现出临界幂律奇异性，临界幂指数接近−1。这种临界幂律奇异性具有跨尺度的特征，即对于包含破裂带的不同大小的平均窗口而言均会出现临界幂律奇异性，但临界幂指数随着平均窗口的增大而增大。这表明宏观物理量的临界幂律奇异性来源于局部化区域内响应量的加速演化。利用条带平均量构造的局部响应函数对灾变破坏点进行了实时预测，结果表明，随着加载的进行，预测值逐渐减小并向真实的灾变破坏点靠拢。靠近破裂带的、面积较小的计算区域，得到的实时预测值较小；远离破裂带的、面积较大的区域，得到的实时预测值较大。由此可以为灾变破坏的区域预测提供基础。

利用汶川地震震前4期GPS非连续观测站数据和同震的位移数据重构了龙门山破裂带附近的应变增量场，结果表明，在汶川地震前垂直破裂带方向的压应变增量分量呈现局部化现象。利用GPS连续观测站构成的三角网格，计算了垂直破裂带方向的响应函数，结果表明，汶川地震前响应函数具有临界幂律奇异性。临界幂指数接近−1。这说明汶川地震具有幂律前兆。基于此，将建立的实时预测方法应用于对汶川地震发震时间的后验性预测，实时地给出了发震时间的上界及预测区间。随着采样的进行，预测值逐渐向真实值靠拢，包含真实发震时间的预测区间也逐渐缩小。

Catastrophic rupture is a typical form of failure in rock-like heterogeneous brittle materials. The occurrence of large earthquakes often belong to this type of rupture. Near the rupture point, many physical quantities (such as strain, displacement, number of AE events, energy release rate, etc.) show the accelerated evolution, and there rate diverge to infinity according to a power-law relationship at the rupture point. This evolution has been studied by analogy with the critical phenomena of phase transitions. But unlike the classic critical phenomenon, the critical power-law exponent of catastrophic rupture presents many new features. In addition, determining the characteristics of the critical power-law singularity exponent is crucial for accurately establishing a prediction method for catastrophe.

Based on the quasi-static uniaxial compression experiments of marble and granite samples, it is found that the response function, R = du/dU, shows a critical power-law singularity before catastrophic rupture, and the power-law exponent of R is distributed between −1 and −1/2, where u is the deformation of a sample and U is the output displacement of the testing machine. It is clarified that this singularity of R is the inevitable result of the energy criterion for catastrophic rupture, and the distribution range of the power-law exponent can be analyzed by a power function approximation.

Based on the range of the critical power-law exponent, and associating the response function R with the loading process, a real-time prediction method for catastrophic rupture is proposed to calculate the upper bound of the rupture point and the corresponding prediction interval in real time. As the loading progresses and the rupture point approaches, the prediction point gradually moves to the actual rupture point, and the prediction interval including the actual rupture point also gradually narrows.

The surface displacement field and strain field of samples are measured by the digital speckle correlation method. The spatio-temporal evolution of the displacement field and the strain field is studied by a band analysis method, and the strain localization on the surface of samples before catastrophic rupture is observed. It is found that the local response function calculated by the mean displacement components of all bands have a critical power-law singularity before catastrophic rupture. The critical power-law exponent is between −1 and −1/2. The local response function calculated by the mean normal strain perpendicular to the rupture surface of the bands near the rupture surface also exhibits a critical power-law singularity with a critical power-law exponent close to −1. This kind of critical power-law singularity is a trans-scale character, that is, the critical power-law singularity will appear for the calculation window with different sizes containing the rupture surface, but the critical power-law exponent increases with the increasing size of the calculation window. This indicates that the critical power-law singularity of the macroscopic physical quantity originates from the accelerated evolution of the response quantities within the localized region. The local response function calculated by the band mean quantities is used to predict the catastrophic rupture point in real time. It is shown that as the loading progresses, the predicted values ​​gradually decrease and move closer to the actual catastrophic rupture point. The smaller area of ​​the calculation area near the rupture surface results in a smaller real-time prediction value; the larger area away from the rupture surface has a larger real-time prediction value. This can provide a basis for regional predictions of catastrophic rupture.

By using the non-continuous GPS data observed before the Wenchuan M_S8.0 earthquake (May 12, 2012) and the co-seismic displacements data, the strain increment field around the Longmenshan fault is reconstructed. It is shown that before the Wenchuan earthquake, the compressive strain increment which perpendicular to the fault exhibits localization evolution. By using the triangular mesh generated by GPS continuous observation stations, the response function in the direction perpendicular to the fault is calculated. It is shown that the response function of exhibits a critical power-law singularity before the Wenchuan earthquake, and the critical power-law exponent is close to −1. This shows that the Wenchuan earthquake has a power-law precursor. Based on this, the established real-time prediction method is applied to a posteriori prediction of the rupture time of the Wenchuan earthquake, and the upper bound and the prediction interval containing the actual earthquake time is given in real time. As the sampling progresses, the predicted value gradually converges to the true value, and the prediction interval including the real rupture time also gradually narrows.