Zugl.: Freiberg (Sachsen), Techn. Univ., Diss., 2008

A lifetime prediction scheme is proposed based on the assumption that the lifetime (time to failure) of rocks under load is governed by the growth of microstructual defects (microcracks). The numerical approach is based on linear elastic fracture mechanics. The numerical calculation scheme is implemented as a cellular automat, where each cell contains a microcrack with length and orientation following certain distributions. The propagation of the microcrack is controlled by the Charles equation, based on subcritical crack growth. The zone inside the numerical model fails if the microcrack has reached the zone dimension or the stress intensity factor of the crack reached the fracture toughness. Macroscopic fractures are formed by these coalesced propagating microcracks, and finally lead to failure of the model. In the numerical approaches, elasto-plastic stress redistributions take place during the forming of the macroscopic fractures. Distinct microcrack propagation types have been programmed and applied to the proposed numerical models. These numerical models are studied under different loading conditions. Numerical results with excellent agreement with the analytical solutions are obtained with respective to predicted lifetime, important parameters for the microcracks, fracture pattern and damage evolution. Potential applications of the proposed numerical model schemes are investigated in some preliminary studies and simulation results are discussed. Finally, conclusions are drawn and possible improvements to the numerical approaches and extensions of the research work are given. / 本文认为微结构缺陷（微裂纹）的扩展决定了受力岩石的寿命（破坏时间）。基于此假设，提出了岩石寿命预测方法。利用线弹性断裂力学理论，通过FLAC进行了数值模拟。数值模型中每个单元定义一条初始裂纹，其长度与方向服从特定分布。基于亚临界裂纹扩展理论，由Charles方程决定微裂纹的扩展（速度）。如微裂纹发展至单元边界，或应力强度系数到达断裂韧度，则单元破坏。宏观裂纹由微裂纹所联合形成，并最终贯穿模型导致破坏。在形成宏观裂纹的过程中，发生弹塑性应力重分布。在数值模型中，编制了不同类型的微裂纹扩展方式，并在不同的受力条件下加以分析。数值模型的岩石寿命，裂纹形状，破坏方式以及一些重要的参数的数值模拟结果与解析解有较好的一致性。对本文所提出的数值模型的初步实际应用进行了分析，并讨论了计算结果。最后讨论了本文所提出的岩石寿命预测方法的可能改良与发展，并对进一步的研究工作给出建议。

The shear behaviour of rough rock joints was investigated by both laboratory testing and numerical simulation. The most powerful servo-controlled direct shear box apparatus in the world with normal forces up to 1000 kN, shear loading up to 800 kN and frequencies up to 40 Hz under full load was used to investigate the shear strength of schistose rock blocks with dimensions of up to 350 x 200 x 160 mm in length, width and height, respectively. The experiments were performed to study the behaviour of rough rock joints under constant normal load, constant normal stiffness and dynamic boundary conditions. The joint surface of rock specimen was scanned 3-dimensional at the initial stage before shearing by new 3D optical-scanning equipment. The 3D-scanner data were used to estimate the joint roughness coefficient (JRC) and to reconstruct rough surface of rock discontinuities in numerical models. Three dimensional numerical models were developed using FLAC3D to study the macro and micromechanical shear behaviour of the joints. Numerical simulation results were compared to experimental results. Three dimensional characteristic of the joint surface including micro-slope angle, aperture, contact area and normal stress distribution were determined and analyzed.

‘Coupled process’ implies that one process affects the initiation and progress of the others and vice versa. The deformation and damage behaviors of rock under loading process change the fluid flow field within it, and lead to altering in permeable characteristics; on the other side inner fluid flow leads to altering in pore pressure and effective stress of rock matrix and flow by influencing stress strain behavior of rock. Therefore, responses of rock to natural or man-made perturbations cannot be predicted with confidence by considering each process independently. As far as hydro-mechanical behavior of rock is concerned, the researchers have always been making efforts to develop the model which can represent the permeable characteristics as well as stress-strain behaviors during the entire damage process. A brittle low porous granite was chosen as the study object in this thesis, the aim is to establish a corresponding constitutive law including the relation between permeability evolution and mechanical deformation as well as the rock failure behavior under hydro-mechanical coupled conditions based on own hydro-mechanical coupled lab tests. The main research works of this thesis are as follows: 1. The fluid flow and mechanical theoretical models have been reviewed and the theoretical methods to solve hydro-mechanical coupled problems of porous medium such as flow equations, elasto-plastic constitutive law, and Biot coupled control equations have been summarized. 2. A series of laboratory tests have been conducted on the granite from Erzgebirge–Vogtland region within the Saxothuringian segment of Central Europe, including: permeability measurements, ultrasonic wave speed measurements, Brazilian tests, uniaxial and triaxial compression tests. A hydro-mechanical coupled testing system has been designed and used to conduct drained, undrained triaxial compression tests and permeability evolution measurements during complete loading process. A set of physical and mechanical parameters were obtained. 3. Based on analyzing the complete stress-strain curves obtained from triaxial compression tests and Hoek-Brown failure criterion, a modified elemental elasto-plastic constitutive law was developed which can represent strength degradation and volume dilation considering the influence of confining pressure. 4. The mechanism of HM-coupled behavior according to the Biot theory of elastic porous medium is summarized. A trilinear evolution rule for Biot’s coefficient based on the laboratory observations was deduced to eliminate the error in predicting rock strength caused by constant Biot’s coefficient. 5. The permeability evolution of low porous rock during the failure process was described based on literature data and own measurements, a general rule for the permeability evolution was developed for the laboratory scale, a strong linear relation between permeability and volumetrical strain was observed and a linear function was extracted to predict permeability evolution during loading process based on own measurements. 6. By combining modified constitutive law, the trilinear Biot’s coefficient evolution model and the linear relationship between permeability and volumetrical strain, a fully hydro-mechanical coupled numerical simulation scheme was developed and implemented in FLAC3D. A series of numerical simulations of triaxial compression test considering the hydro-mechanical coupling were performed with FLAC3D. And a good agreement was found between the numerical simulation results and the laboratory measurements under 20 MPa confining pressure and 10 MPa fluid pressure, the feasibility of this fully hydro-mechanical coupled model was proven.

The shear behaviour of rough rock joints was investigated by both laboratory testing and numerical simulation. The most powerful servo-controlled direct shear box apparatus in the world with normal forces up to 1000 kN, shear loading up to 800 kN and frequencies up to 40 Hz under full load was used to investigate the shear strength of schistose rock blocks with dimensions of up to 350 x 200 x 160 mm in length, width and height, respectively. The experiments were performed to study the behaviour of rough rock joints under constant normal load, constant normal stiffness and dynamic boundary conditions. The joint surface of rock specimen was scanned 3-dimensional at the initial stage before shearing by new 3D optical-scanning equipment. The 3D-scanner data were used to estimate the joint roughness coefficient (JRC) and to reconstruct rough surface of rock discontinuities in numerical models. Three dimensional numerical models were developed using FLAC3D to study the macro and micromechanical shear behaviour of the joints. Numerical simulation results were compared to experimental results. Three dimensional characteristic of the joint surface including micro-slope angle, aperture, contact area and normal stress distribution were determined and analyzed.

The anisotropic behavior of a rock mass with persistent and planar joint sets is mainly governed by the geometrical and mechanical characteristics of the joints. The aim of the study is to develop a continuum-based approach for simulation of multi jointed geomaterials. There are two available numerical techniques for the strain-stress analysis of rock masses: continuum-based methods and discontinuum based methods. Joints are simulated explicitly in discontinuous methodology. This technique provides a more accurate description for the behavior of a rock mass. However, in some projects, the explicit definition becomes impractical, especially with increasing number of joints. Besides, the calculation efficiency will be significant reduced as the number of joints increases within the model. Considering the above mentioned shortcomings of the discontinuous method, the continuum-based approach is widely used in rock mechanics. Within the continuum methods, the discontinuities are regarded as smeared cracks in an implicit manner and all the joint parameters are incorporated into the equivalent constitutive equations. A new equivalent continuum model, called multi-joint model, is developed for jointed rock masses which may contain up to three arbitrary persistent joint sets. The Mohr-Coulomb yield criterion is used to check failure of the intact rock and the joints. The proposed model has solved the issue of multiple plasticity surfaces involved in this approach combined with multiple failure mechanisms. The multi-joint model is implemented into FLAC and is verified against the distinct element method (UDEC), analytical solutions, and experimental data. Uniaxial compression tests with artificial rock-like material (gypsum) are carried out in the laboratory in order to verify the developed constitutive model and to investigate the behavior of jointed specimen. Samples with two crossing joints covering more than 20 angle configurations and two different property sets were prepared and tested. Simulation results are in good agreement with experimental observations. The developed model is applied to two potential practical applications: the stability analysis of a slope and a tunnel under different stress conditions. Finally, the main achievements of the whole PhD study are summarized and future research work is proposed.

High-level radioactive waste (HLW) poses threat to the biosphere. Geological disposal is accepted as a safe way for HLW disposal. Waste canisters made of Sintered Silicon Carbide (SSiC) are proposed and geomechanical safety aspects relating to such SSiC canisters are investigated. First part of the thesis reviews the state-of-the-art and demands for HLW disposal. The reason for considering Silicon Carbide (SiC) as canister material is explained. Especially in terms of corrosion and lifetime, ceramics and especially SiC is superior to metals or concrete. The only concern is its brittle behavior. The second part of the thesis presents results on static and dynamic mechanical properties of SiC in general and in particular for SSiC based on literature review and own lab tests. Although strength values for SiC and especially SSiC are very high, the extreme brittle behavior has to be considered in case of impact or point-like loading. The third and most extensive part of the thesis part contains numerical simulations, which consider most critical potential loading situations during transport and installation of the canisters underground. Both, pure elastic continuum and DEM based models are used considering the following loading situations (critical scenarios): Freefall of canister during transport or installation (FF), Impact by falling rock block at disposal site (RF), Point loading due to accidental insertion of small stone below the canister (PL), Anisotropic earth pressure loading after disposal (EP). Coating to protect the canisters against damage is investigated and preliminary parameters in terms of stiffness and thickness are recommended.

‘Coupled process’ implies that one process affects the initiation and progress of the others and vice versa. The deformation and damage behaviors of rock under loading process change the fluid flow field within it, and lead to altering in permeable characteristics; on the other side inner fluid flow leads to altering in pore pressure and effective stress of rock matrix and flow by influencing stress strain behavior of rock. Therefore, responses of rock to natural or man-made perturbations cannot be predicted with confidence by considering each process independently. As far as hydro-mechanical behavior of rock is concerned, the researchers have always been making efforts to develop the model which can represent the permeable characteristics as well as stress-strain behaviors during the entire damage process. A brittle low porous granite was chosen as the study object in this thesis, the aim is to establish a corresponding constitutive law including the relation between permeability evolution and mechanical deformation as well as the rock failure behavior under hydro-mechanical coupled conditions based on own hydro-mechanical coupled lab tests. The main research works of this thesis are as follows: 1. The fluid flow and mechanical theoretical models have been reviewed and the theoretical methods to solve hydro-mechanical coupled problems of porous medium such as flow equations, elasto-plastic constitutive law, and Biot coupled control equations have been summarized. 2. A series of laboratory tests have been conducted on the granite from Erzgebirge–Vogtland region within the Saxothuringian segment of Central Europe, including: permeability measurements, ultrasonic wave speed measurements, Brazilian tests, uniaxial and triaxial compression tests. A hydro-mechanical coupled testing system has been designed and used to conduct drained, undrained triaxial compression tests and permeability evolution measurements during complete loading process. A set of physical and mechanical parameters were obtained. 3. Based on analyzing the complete stress-strain curves obtained from triaxial compression tests and Hoek-Brown failure criterion, a modified elemental elasto-plastic constitutive law was developed which can represent strength degradation and volume dilation considering the influence of confining pressure. 4. The mechanism of HM-coupled behavior according to the Biot theory of elastic porous medium is summarized. A trilinear evolution rule for Biot’s coefficient based on the laboratory observations was deduced to eliminate the error in predicting rock strength caused by constant Biot’s coefficient. 5. The permeability evolution of low porous rock during the failure process was described based on literature data and own measurements, a general rule for the permeability evolution was developed for the laboratory scale, a strong linear relation between permeability and volumetrical strain was observed and a linear function was extracted to predict permeability evolution during loading process based on own measurements. 6. By combining modified constitutive law, the trilinear Biot’s coefficient evolution model and the linear relationship between permeability and volumetrical strain, a fully hydro-mechanical coupled numerical simulation scheme was developed and implemented in FLAC3D. A series of numerical simulations of triaxial compression test considering the hydro-mechanical coupling were performed with FLAC3D. And a good agreement was found between the numerical simulation results and the laboratory measurements under 20 MPa confining pressure and 10 MPa fluid pressure, the feasibility of this fully hydro-mechanical coupled model was proven.

This dissertation evaluates the possibilities and limitations of rock type identification in rock cutting with conical picks. For this, machine learning in conjunction with features derived from high frequency cutting force measurements is used. On the basis of linear cutting experiments, it is shown that boundary layers can be identified with a precision of less than 3.7 cm when using the developed programme routine. It is further shown that rocks weakened by cracks can be well identified and that anisotropic rock behaviour may be problematic to the classification success. In a case study, it is shown that the supervised algorithms artificial neural network and distributed random forest perform relatively well while unsupervised k-means clustering provides limited accuracies for complex situations. The 3d-results are visualised in a web app. The results suggest that a possible rock classification system can achieve good results—that are robust to changes in the cutting parameters when using the proposed evaluation methods.:1 Introduction...1 2 Cutting Excavation with Conical Picks...5 2.1 Cutting Process...8 2.1.2 Cutting Parameters...11 2.1.3 Influences of Rock Mechanical Properties...17 2.1.4 Influences of the Rock Mass...23 2.2 Ratios of Cutting Force Components...24 3 State of the Art...29 3.1 Data Analysis in Rock Cutting Research...29 3.2 Rock Classification Systems...32 3.2.1 MWC – Measure-While-Cutting...32 3.2.2 MWD – Measuring-While-Drilling...34 3.2.3 Automated Profiling During Cutting...35 3.2.4 Wear Monitoring...36 3.3 Machine learning for Rock Classification...36 4 Problem Statement and Justification of Topic...38 5 Material and Methods...40 5.1 Rock Cutting Equipment...40 5.2 Software & PC...42 5.3 Samples and Rock Cutting Parameters...43 5.3.1 Sample Sites...43 5.3.2 Experiment CO – Zoned Concrete...45 5.3.3 Experiment GN – Anisotropic Rock Gneiss...47 5.3.4 Experiment GR – Uncracked and Cracked Granite...49 5.3.5 Case Study PB and FBA – Lead-Zinc and Fluorite-Barite Ores...50 5.4 Data Processing...53 5.5 Force Component Ratio Calculation...54 5.6 Procedural Selection of Features...57 5.7 Image-Based Referencing and Rock Boundary Modelling...60 5.8 Block Modelling and Gridding...61 5.9 Correlation Analysis...63 5.10 Regression Analysis of Effect...64 5.11 Machine Learning...65 5.11.2 K-Means Algorithm...66 5.11.3 Artificial Neural Networks...67 5.11.4 Distributed Random Forest...70 5.11.5 Classification Success...72 5.11.6 Boundary Layer Recognition Precision...73 5.12 Machine Learning Case Study...74 6 Results...75 6.1 CO – Zoned Concrete...75 6.1.1 Descriptive Statistics...75 6.1.2 Procedural Evaluation...76 6.1.3 Correlation of the Covariates...78 6.1.4 K-Means Cluster Analysis...79 6.2 GN – Foliated Gneiss...85 6.2.1 Cutting Forces...86 6.2.2 Regression Analysis of Effect...88 6.2.3 Details Irregular Behaviour...90 6.2.4 Interpretation of Anisotropic Behaviour...92 6.2.5 Force Component Ratios...92 6.2.6 Summary and Interpretations of Results...93 6.3 CR – Cracked Granite...94 6.3.1 Force Component Results...94 6.3.2 Spatial Analysis...97 6.3.3 Error Analysis...99 6.3.4 Summary...100 6.4 Case Study...100 6.4.1 Feature Distribution in Block Models...101 6.4.2 Distributed Random Forest...105 6.4.3 Artificial Neural Network...107 6.4.4 K-Means...110 6.4.5 Training Data Required...112 7 Discussion...114 7.1 Critical Discussion of Experimental Results...114 7.1.1 Experiment CO...114 7.1.2 Experiment GN...115 7.1.3 Experiment GR...116 7.1.4 Case Study...116 7.1.5 Additional Outcomes...117 7.2 Comparison of Machine Learning Algorithms...118 7.2.1 K-Means...118 7.2.2 Artificial Neural Networks and Distributed Random Forest...119 7.2.3 Summary...120 7.3 Considerations Towards Sensor System...121 7.3.1 Force Vectors and Data Acquisition Rate...121 7.3.2 Sensor Types...122 7.3.3 Computation Speed...123 8 Summary and Outlook...125 References...128 Annex A Fields of Application of Conical Tools...145 Annex B Supplements Cutting and Rock Parameters...149 Annex C Details Topic-Analysis Rock Cutting Publications...155 Annex D Details Patent Analysis...157 Annex E Details Rock Cutting Unit HSX-1000-50...161 Annex F Details Used Pick...162 Annex G Error Analysis Cutting Experiments...163 Annex H Details Photographic Modelling...166 Annex I Laser Offset...168 Annex J Supplements Experiment CO...169 Annex K Supplements Experiment GN...187 Annex L Supplements Experiment GR...191 Annex M Preliminary Artificial Neural Network Training...195 Annex N Supplements Case Study (CD)...201 Annex O R-Codes (CD)...203 Annex P Supplements Rock Mechanical Tests (CD)...204 / Die Dissertation evaluiert Möglichkeiten und Grenzen der Gebirgserkennung bei der schneidenden Gewinnung von Festgesteinen mit Rundschaftmeißeln unter Nutzung maschinellen Lernens – in Verbindung mit aus hochaufgelösten Schnittkraftmessungen abgeleiteten Kennwerten. Es wird auf linearen Schneidversuchen aufbauend gezeigt, dass Schichtgrenzen mit Genauigkeiten unter 3,7 cm identifiziert werden können. Ferner wird gezeigt, dass durch Risse geschwächte Gesteine gut identifiziert werden können und dass anisotropes Gesteinsverhalten möglicherweise problematisch auf den Klassifizierungserfolg wirkt. In einer Fallstudie wird gezeigt, dass die überwachten Algorithmen Künstliches Neurales Netz und Distributed Random Forest teils sehr gute Ergebnisse erzielen und unüberwachtes k-means-Clustering begrenzte Genauigkeiten für komplexe Situationen liefert. Die Ergebnisse werden in einer Web-App visualisiert. Aus den Ergebnissen wird abgeleitet, dass ein mögliches Sensorsystem mit den vorgeschlagenen Auswerteroutinen gute Ergebnisse erzielen kann, die gleichzeitig robust gegen Änderungen der Schneidparameter sind.:1 Introduction...1 2 Cutting Excavation with Conical Picks...5 2.1 Cutting Process...8 2.1.2 Cutting Parameters...11 2.1.3 Influences of Rock Mechanical Properties...17 2.1.4 Influences of the Rock Mass...23 2.2 Ratios of Cutting Force Components...24 3 State of the Art...29 3.1 Data Analysis in Rock Cutting Research...29 3.2 Rock Classification Systems...32 3.2.1 MWC – Measure-While-Cutting...32 3.2.2 MWD – Measuring-While-Drilling...34 3.2.3 Automated Profiling During Cutting...35 3.2.4 Wear Monitoring...36 3.3 Machine learning for Rock Classification...36 4 Problem Statement and Justification of Topic...38 5 Material and Methods...40 5.1 Rock Cutting Equipment...40 5.2 Software & PC...42 5.3 Samples and Rock Cutting Parameters...43 5.3.1 Sample Sites...43 5.3.2 Experiment CO – Zoned Concrete...45 5.3.3 Experiment GN – Anisotropic Rock Gneiss...47 5.3.4 Experiment GR – Uncracked and Cracked Granite...49 5.3.5 Case Study PB and FBA – Lead-Zinc and Fluorite-Barite Ores...50 5.4 Data Processing...53 5.5 Force Component Ratio Calculation...54 5.6 Procedural Selection of Features...57 5.7 Image-Based Referencing and Rock Boundary Modelling...60 5.8 Block Modelling and Gridding...61 5.9 Correlation Analysis...63 5.10 Regression Analysis of Effect...64 5.11 Machine Learning...65 5.11.2 K-Means Algorithm...66 5.11.3 Artificial Neural Networks...67 5.11.4 Distributed Random Forest...70 5.11.5 Classification Success...72 5.11.6 Boundary Layer Recognition Precision...73 5.12 Machine Learning Case Study...74 6 Results...75 6.1 CO – Zoned Concrete...75 6.1.1 Descriptive Statistics...75 6.1.2 Procedural Evaluation...76 6.1.3 Correlation of the Covariates...78 6.1.4 K-Means Cluster Analysis...79 6.2 GN – Foliated Gneiss...85 6.2.1 Cutting Forces...86 6.2.2 Regression Analysis of Effect...88 6.2.3 Details Irregular Behaviour...90 6.2.4 Interpretation of Anisotropic Behaviour...92 6.2.5 Force Component Ratios...92 6.2.6 Summary and Interpretations of Results...93 6.3 CR – Cracked Granite...94 6.3.1 Force Component Results...94 6.3.2 Spatial Analysis...97 6.3.3 Error Analysis...99 6.3.4 Summary...100 6.4 Case Study...100 6.4.1 Feature Distribution in Block Models...101 6.4.2 Distributed Random Forest...105 6.4.3 Artificial Neural Network...107 6.4.4 K-Means...110 6.4.5 Training Data Required...112 7 Discussion...114 7.1 Critical Discussion of Experimental Results...114 7.1.1 Experiment CO...114 7.1.2 Experiment GN...115 7.1.3 Experiment GR...116 7.1.4 Case Study...116 7.1.5 Additional Outcomes...117 7.2 Comparison of Machine Learning Algorithms...118 7.2.1 K-Means...118 7.2.2 Artificial Neural Networks and Distributed Random Forest...119 7.2.3 Summary...120 7.3 Considerations Towards Sensor System...121 7.3.1 Force Vectors and Data Acquisition Rate...121 7.3.2 Sensor Types...122 7.3.3 Computation Speed...123 8 Summary and Outlook...125 References...128 Annex A Fields of Application of Conical Tools...145 Annex B Supplements Cutting and Rock Parameters...149 Annex C Details Topic-Analysis Rock Cutting Publications...155 Annex D Details Patent Analysis...157 Annex E Details Rock Cutting Unit HSX-1000-50...161 Annex F Details Used Pick...162 Annex G Error Analysis Cutting Experiments...163 Annex H Details Photographic Modelling...166 Annex I Laser Offset...168 Annex J Supplements Experiment CO...169 Annex K Supplements Experiment GN...187 Annex L Supplements Experiment GR...191 Annex M Preliminary Artificial Neural Network Training...195 Annex N Supplements Case Study (CD)...201 Annex O R-Codes (CD)...203 Annex P Supplements Rock Mechanical Tests (CD)...204

A lifetime prediction scheme is proposed based on the assumption that the lifetime (time to failure) of rocks under load is governed by the growth of microstructual defects (microcracks). The numerical approach is based on linear elastic fracture mechanics. The numerical calculation scheme is implemented as a cellular automat, where each cell contains a microcrack with length and orientation following certain distributions. The propagation of the microcrack is controlled by the Charles equation, based on subcritical crack growth. The zone inside the numerical model fails if the microcrack has reached the zone dimension or the stress intensity factor of the crack reached the fracture toughness. Macroscopic fractures are formed by these coalesced propagating microcracks, and finally lead to failure of the model. In the numerical approaches, elasto-plastic stress redistributions take place during the forming of the macroscopic fractures. Distinct microcrack propagation types have been programmed and applied to the proposed numerical models. These numerical models are studied under different loading conditions. Numerical results with excellent agreement with the analytical solutions are obtained with respective to predicted lifetime, important parameters for the microcracks, fracture pattern and damage evolution. Potential applications of the proposed numerical model schemes are investigated in some preliminary studies and simulation results are discussed. Finally, conclusions are drawn and possible improvements to the numerical approaches and extensions of the research work are given. / 本文认为微结构缺陷（微裂纹）的扩展决定了受力岩石的寿命（破坏时间）。基于此假设，提出了岩石寿命预测方法。利用线弹性断裂力学理论，通过FLAC进行了数值模拟。数值模型中每个单元定义一条初始裂纹，其长度与方向服从特定分布。基于亚临界裂纹扩展理论，由Charles方程决定微裂纹的扩展（速度）。如微裂纹发展至单元边界，或应力强度系数到达断裂韧度，则单元破坏。宏观裂纹由微裂纹所联合形成，并最终贯穿模型导致破坏。在形成宏观裂纹的过程中，发生弹塑性应力重分布。在数值模型中，编制了不同类型的微裂纹扩展方式，并在不同的受力条件下加以分析。数值模型的岩石寿命，裂纹形状，破坏方式以及一些重要的参数的数值模拟结果与解析解有较好的一致性。对本文所提出的数值模型的初步实际应用进行了分析，并讨论了计算结果。最后讨论了本文所提出的岩石寿命预测方法的可能改良与发展，并对进一步的研究工作给出建议。