Moirl strength theory does not consider the influence of the intermediate principal stress σ2. However, the influence of σ2 and its rising and falling changes on strength has been confirmed by experiments; after stress adjustment caused by unloading in engineering practice, the structural plane on the stress circle of σ1-σ3 is tensile and the normal stress is negative. Due to the constraint of the side boundary surface and σ2 compressive stress, in the state where the structural plane inclination intersects σ1, the larger the intersection angle, the more obvious the influence. Therefore, the tensile shear of θ=45°+φ/2 appears complex. Anisotropic characteristics.
The shear strain energy strength theory and the octahedral stress theory consider the influence of the three principal stresses in the three-dimensional space of the rock mass on the rock strength. The shear strain energy theory is based on the energy point of view, while the octahedral stress theory It is based on the stress point of view and studies the strength conditions of rocks from different angles, and the results obtained are consistent. These two items are commonly used yield criteria in the field of engineering science and technology. They are three-dimensional strength criteria established based on three-dimensional compressive stress or three-dimensional tensile stress, assuming that compression and tension have the same resistance strength. This book uses this theory to explore the actual situation. In the three-dimensional compressive stress state, the one-dimensional stress changes in reverse, forming the intensity characteristics of distortion energy.
2.2.3.1 Shear strain strength theory
The shear strain strength theory proposes that rock damage must overcome the interaction between the inherent shape of the rock and the basic particles of the rock strength from a physical point of view. force. When the rock's shear strain energy reaches the ultimate deformation energy when subjected to force failure under the action of three-dimensional stresses σ1, σ2, and σ3, it is the strength condition or failure criterion of shear strain energy. Rocks are subjected to three-dimensional stress to produce strain, which reflects the full strain energy. According to the functional principle, the expression of its work is
Reaction force strain rock mechanics application in engineering
Formula Among: γ is shear strain. Based on the aforementioned failure concept, this theory is only applicable to brittle failure; therefore γij=0, so the total deformation of the rock is:
According to the generalized Hooke’s theorem
Represented by three invariants σ1, σ2, σ3, the total variable energy can be obtained after sorting:
Reaction force strain rock mechanics application in engineering
Total variable energy , including two parts: volume deformation energy and deformation energy.
Volume deformation energy of rock
In the formula: is the average stress; ε is the volume strain εV=ε1+ε2+ε3
After finishing,
The application of reaction force strain rock mechanics in engineering
Full variable energy UC includes volumetric deformation energy UV and deformation energy Uτ, that is,
The application of reaction force strain rock mechanics in engineering Application
Deformation energy is shear energy, which is obtained by subtracting total deformation energy from volume deformation energy, that is,
reaction force strain application of rock mechanics in engineering
Jian De
In a uniaxial compression or tension test, σ2=σ3=0 means yield failure
σ1=σs σs is the yield strength
Substituting the above conditions into Equation 2.20, the deformation energy of the rock when yielding in the uniaxial compression or tensile test is obtained
The application of reaction force strain rock mechanics in engineering
Elucidates the unidirectional stress The shear energy at failure is considered to be the yield condition.
Application of reaction force strain rock mechanics in engineering
That is, equation (2.20) is equal to equation (2.21).
Then
The application of reaction force strain rock mechanics in engineering
This is the rock strength condition and criterion derived from the shear energy theory.
2.2.3.2 Octahedral stress theory
The octahedral stress theory believes that when the shear stress value on the octahedron reaches the critical value of the material, it will cause material yield failure.
Choose a closed octahedron whose axial coordinate system is parallel to the three-dimensional principal stress, so that the normals of the equal slopes on the eight quadrants are equal to the angles between the three coordinate axes, that is, the normals are equal to x, The angles α, β, and γ between the y and z coordinate axes are equal.
Let
Because the areas of the three principal stresses on the isoclinic plane are Scosα, Scosβ, and Scosγ
in Under the action of force P, according to the equilibrium conditions of the force
The application of reaction force strain rock mechanics in engineering
The equilibrium equation is obtained
The force acting on the inclined surface Resultant force
Reaction force strain rock mechanics application in engineering
The normal stress on the inclined surface is the projection of each component force Px, Py and Pz on the surface on the normal axis of the surface and, that is
Reaction force strain application of rock mechanics in engineering
Shear stress on the inclined plane
τ8 octahedral shear strength
Then
After sorting out:
This is the physical meaning of the Mises yield criterion made by Nadai in 1933. Mises (R.V.) believes that when the shear stress on the octahedron is equal to the material's unidirectional force until it yields, the shear stress on the octahedron reaches the ultimate value and yields.
When the unidirectional force reaches the limit state
The application of reaction force strain rock mechanics in engineering
In Equation 2.23
According to the criteria established by Mises
Then
So
Figure 2.9 Decomposition of the vector in the stress space coordinate system
Equation 2.24 and The shear energy intensity condition is the same as Equation 2.22. Obviously, the failure strength of rock is closely related to σ2.
2.2.3.3 The geometry of the yield surface
In order to make the yield criterion more visual, the yield criterion is often expressed as a geometric surface in the stress space. Figure 2.9 introduces the stress at any point in the rock. The state is represented by a vector in the stress space coordinate system. Decomposed into two components
Since ON is the normal of the triangular slope selected in the stress space coordinate system, and the angle with the three stress coordinate axes is equal, it can be clearly understood from the previous section
Reaction force strain rock mechanics application in engineering
From
Then
So
This is a cylinder with the normal ON as the axis Equation, this cylindrical surface is the Misex intensity surface. For simplicity, the cylindrical surface parallel to the ON normal line can be projected onto the inclined surface to form a circle. Since the angle between the inclined surface and the three principal stress axes is equal, σ1=σ2=σ3=σm. It has the characteristics of ON-direction hydrostatic stress, σm is the ON-direction hydrostatic stress, this circle is called π circle, the circumference is the yield strength trajectory line, and the radius R of the π circle is OS.
That is,
The application of reaction force strain rock mechanics in engineering
At the engineering site, according to the actual field conditions, the adjusted three-dimensional stress is obtained, and its The radius r of the stress circle is
Reaction force strain application of rock mechanics in engineering
If r If r>R, the rock will yield failure. Figure 2.10 is the π plane yield trajectory diagram, and the circle is the Misex yield trajectory line of the (2/3)1/2σt trajectory. The regular hexagonal trajectory is the maximum shear energy Treleska yielding trajectory. Quote from Figure 6.22(b) in Mr. Chen Ziguang’s book on rock mechanical properties and tectonic stress fields. Figure 2.10 Yield trajectory line on the π plane ① Mises yield trajectory line; ② Coulomb-Moore yield trajectory line; ③ Treleska yield trajectory line Treleska criterion max=(1/2)(σ1-σ2)=k, then 2k=(σ1-σ2), during uniaxial stretching, σ2=σ3=0, because σ1-σ2=σt,=σ1/2, so 2k=σt. In pure shear, k= projects the spatial rectangular coordinate system of the applied stress onto the π plane, and the three new coordinate axes obtained are represented by σ1', σ2', and σ3' respectively, and their angles to each other are 120°. The cosine of the angle between the space stress σ1, σ2, σ3 and the normal line of the π plane is cosα=(1/3)1/2, so the cosine of the angle with the π plane is cosβ=(2/3)1/2 Therefore Project σ1', σ2', σ3' respectively onto the xy axis on the π plane, then Application of reaction force strain rock mechanics in engineering In uniaxial stretching, because σ1-σ2=σt, so x=σt/21/2, which is a straight line on the π plane parallel to the σ2' axis. In the same way, the relational expressions of other axial domains xy can be obtained, and a regular hexagonal figure composed of three pairs of parallel lines can be obtained, and the radius of its circumscribed circle is [(2/3)1/2]σt. The unequal hexagon is the trajectory of Mohr's intensity criterion on the π plane. The well-known Mohr-Coulomb criterion is =σntanφ+c or (σ1-σ3)/2=[(σ3+σ2)/2] sinφ+c·cosφ. According to the aforementioned principles and method steps, it can be obtained on the π plane yield trajectory line. In the case of plane stress, that is, σ3=0, on the σ1-σ2 plane, the Mises yield criterion (2.24) is simplified to σ21-σ1σ2+σ22=σ2t, and its yield trajectory on the π plane is an elliptical equation Figure 2.11. On the σ1-σ2 plane, the Trieska criterion is: The application of reaction force strain rock mechanics in engineering That is, as shown in Figure 2.11 Six straight lines. Figure 2.11 Ellipse trajectory line Figure 2.12 τ8=f(σ8) ultimate strength surface The Naday strength condition is based on the shear stress on the octahedron. It is caused by the critical value, and the critical value of shear stress is a function of the normal stress on the octahedron, and its condition is τ8=f(σ8) Application of reaction force strain rock mechanics in engineering In the case of reaction force, σ8≈0, so τ8=0. At this time, the intensity surface in the space coordinate system is no longer cylindrical, but a cone. This figure is Drucker-Braque's work by Mises. In principle, a hydrostatic factor is considered to form the form τ=αI1+(J2)1/2, which is supported by the conical shape of the stress space coordinate system, where: Reaction force strain rock mechanics application in engineering< /p> The conical strength yield surface can better reflect the actual condition of the rock than the cylindrical yield surface. Based on different stress states, different failure mechanisms of rocks, and suitable strength criteria for application, the conical strength yield surface is divided into three zones: ① zone is basically a tensile fracture brittle zone; The ② zone is basically a shear failure zone, usually compressive shear, but tensile shear damage may also occur. In the shallow areas below the surface, there are both compressive shear structural surfaces and filled or unfilled tensile shear structural surfaces; ③ Zone It is basically a plastic flow zone, but there are also tension-shear-brittle fracture types. During earthquakes, some epicenters occur at tension fractures. This is the characteristic. 2.2.3.4 Discussion on Distortion Energy The aforementioned theories are all three-dimensional compressive stress state with σ1>σ2>σ3, and also consider the tensile strength σt, but the apparent pressure and tension The theory of the same strength, maximum shear energy and octahedral stress is based on the yield criterion for studying the plastic flow of metals under triaxial compressive stress conditions. It is applied in geotechnical engineering and treats rocks as isotropic materials. The strength characteristics under three-dimensional compressive stress are consistent with the results of similar triaxial tests. The tensile condition in the original study was caused by the Poisson effect of the maximum principal compressive stress or the unloading rebound effect. The stress change was within the original three-dimensional compressive stress range, so it was basically within the range defined by the above theory. Therefore, in engineering research, according to various theoretical criteria, the equilibrium conditions obtained take into account the anisotropy of rocks and the uncertainty of design mechanical parameters, increase the safety factor, and carry out engineering processing and verification testing accordingly, but still Some situations that are difficult to predict occur. At this time, the complex factors of f value decay are often blamed. The coupling and superposition of the reverse stress of the temperature difference stress and the tensile stress are not taken into account. This is called the unnoticed hidden force. Due to this hidden force, The activity of energy causes brittle yielding and rupture of rock mass strength, and releases wave force, generating huge impact energy and causing huge disasters. This is the consequences of distorted energy in a three-dimensional state. The influence of distortion energy on shear energy intensity. Under the action of reverse stress, the three-dimensional stresses on the rock are σ1, σ2, and -σ3. According to rock mechanics convention, tensile stress is negative. The strains formed are corresponding to ε1, ε2, and -ε3. Now we study the distortion energy reflected by the maximum shear energy. The deformation energy is still solved according to the energy theory. First find the total variable energy From the generalized Hooke’s theorem then we get The volume variable energy The reason Then Shear energy i.e. The application of reaction force strain rock mechanics in engineering With Comparing Equation (2.25) with Equation (2.20), if -σ3 in Equation (2.25) is |σ3|, the two equations are completely consistent. When considering the reaction direction of σ3, which is the reaction force, it is Equation (2.25). The stress space coordinate system of equation (2.25) is an ellipse on the π plane, and the three-dimensional stress in its space is a deformed ellipsoid. Deep in the formation, due to the Poisson effect, a compression-induced tension state occurs When , it belongs to a standard deformable ellipsoid, but in the slope area near the surface, the reaction force acts in one direction and has no corresponding force couple, so it belongs to a distorted deformed ellipsoid, which is called a distorted deformed spindle. The body is more precise, and its ultimate strength surface is an elliptical paraboloid, or a distorted elliptical paraboloid. In the elliptical parabola, it is in a complete tension state. At this time, σ1=0, σ2=0, and -σ3=σt Then The reaction force strain Application of rock mechanics in engineering The strength condition is: Reaction force strain Application of rock mechanics in engineering With (1/2) [(σ1- σ2)2+(σ2+σ3)2+(σ1+σ3)2]≥σ2t is used as the criterion for yield brittle fracture. The shear energy of the three-dimensional stress ellipsoid of the one-way reaction force is solved according to the energy theory. The obtained Equation 2.26 is consistent with Equation 2.22 and is also consistent with Equation 2.24, indicating that the same effect can be obtained by studying the octahedral stress theory. The theories of Trieska and Mises focus on the yield extension and failure of metal materials under compressive stress conditions, so the ② and ③ zones in Figure (2.12) are different from those in experiments and engineering practice. The situation matches. However, under the action of reaction force, the spatial coordinate system of deep stress also presents a distorted ellipsoid state. High-temperature and high-pressure rock mass hardens due to unloading and pressure reduction, or hardens due to temperature difference tensile unloading caused by temperature changes, which increases the yield value, but the damage value remains unchanged. Its damage characteristics change from plastic fracture to Brittle fracture failure, so the octahedral ellipsoid cone strength limit surface is not zoned under reaction force and strain conditions. In areas with natural steep slopes and artificial steep slopes, the natural stress field potential has been adjusted so that the maximum principal stress is parallel to the slope inclination, the intermediate principal stress is basically parallel to the slope direction, and the minimum principal stress is perpendicular to the slope direction. With the slope surface, σ3 is generally consistent with the direction of the potential energy released by the slope, but due to the influence of local factors such as terrain and lithology, it can form a certain difference. At this time, the slope stability affects the most deterrent position, which is incompatible with the new stress. The spatial coordinate system is slightly different, and the impact on slope stability is determined by the amount of deviatoric stress. In the octahedral stress theory, the stress intensity is related to the shear stress, and the octahedral shear stress is related to the second invariant of the deviatoric stress tensor. Mises suggested using the second invariant of the stress tensor to represent the yield criterion (Kf) That is, When the deviator stress τσd ≥ Kf, yield deformation occurs. Application of reaction force strain rock mechanics in engineering There is a certain relationship between the shear stress on the octahedron and the second invariant of the stress deviator, that is, τ28=(2/3) J2 Then Based on geomorphological and geohistorical research and regional geostress test data, the macroscopic geostress field in the region and the adjusted state and corresponding values ??of geostress in local areas can be understood , you can use Equations (2.27) and Equations (2.28) to find the stress state (τ8/σd) at the specific location of the project, and compare it with the mechanical properties of the rock mass (τs/Rt) in this state, then you can determine the edge of the project. The stability of the slope and the most likely type of deformation failure. Rt is the tensile strength of rock, and s is the shear strength value of rock. When τ8 When τ8≥s, σd We can make the judgment in a simpler way. According to the geological stress ellipsoid analysis method τmax=1/2(σ1-σ3), during uniaxial stretching, τmax=(1/2)σt, That is, τmax/σt=0.5; in a three-dimensional stress state, τmax/σt=0.5-(σ3/2σ1). When the ratio of max/σt increases, the shear deformation increases, and when the ratio decreases, the possibility of brittle fracture increases. Natural and artificial slopes are both in a three-dimensional state. Therefore, in general, τmax/σt<0.5 increases the brittle fracture characteristics of rocks. However, due to unloading rebound, which involves the stress sign changing from positive to negative, τmax/σt>0.5 increases controlled shear deformation. On the slope of the field engineering site, these two deformation types can be displayed at the same time. The Mohr's circle under the three-dimensional stress state can also be used, and under the action of the reaction force, the stress can be adjusted, and many new Mohr's circles can be drawn based on the continuous adjustment of the stress, as shown in Equation (2.27 ) and equation (2.28) to find the change trajectories of σd and τ8 under the action of reverse stress. When the three-dimensional stress is in a compressive stress state, its deviatoric stress σd and octahedral shear stress 8 Reaction force strain rock mechanics applied in engineering The compressive stress increases and the unloading decreases , find the trajectories of σd and τ8 according to Equation (2.29) and Equation (2.30). When the unloading σ3 becomes tensile stress, the trajectory change line of σd and τ8 points can be calculated according to Equation (2.27) and Equation (2.28). Under the action of thrust, due to the Poisson effect, a critical state of tensile brittle fracture is also formed in the deep part. . Now, the indoor true triaxial test results of the 18th layer of calcareous mudstone in the Longtan Project are used to study the position of the deviatoric stress on the three-dimensional stress circle. The trend of the 18th layer of calcareous mudstone is N80°W, inclined to NE, and the inclination angle is about 60°. In the 18th layer of calcareous mudstone at the dam base, there is a parallel rock layer trend, inclined to NE, and the inclination angle is close to the upright cryptocleavage development. The core is drilled vertically and the indoor triaxial test is performed. The angle between the core axis and the rock formation is 30°, and the angle between the core axis and the cleavage plane is 20°. The test confining pressure is σ3=1MPa, σ2=5MPa, σ1=5.3σ3+17.3=22.6MPa. The three-dimensional stress circle is shown in Figure 2.13. From equation (2.29), σd=11.43MPa is obtained. From equation (2.30), 8=9.33MPa is obtained. , the M point on the diagram is determined by the values ??of σd and 8. This point is located within the maximum Mohr circle and outside the two small Mohr circles. Therefore, only the maximum Mohr stress circle of the stress ellipsoid in the stress change represents the three-dimensional The change of the stress circle can be obtained as the change of point M. Figure 2.13 Three-dimensional stress map of the true triaxial test results of the 18th layer of the Longtan Dam site In an engineering area, due to the differences in topography and lithology, the stress ellipsoid The distribution situation is different everywhere; the distribution of the stress ellipsoid in the surface and deeper underground parts is controlled by the regional tectonic stress field and large faults. However, when studying the changes in the stress evolution of the three-dimensional stress circle, there is no strict directional restriction, and the change in the principal stress direction of the stress ellipsoid gradually changes from shallow to deep (Figure 2.14). In the 800m deep hole in Maoping in the Three Gorges Dam area, from shallow to deep, the direction of the maximum principal stress gradually changes from NE to NWW. Therefore, the changes in Mohr's maximum stress circle can be treated as a unified and continuous process in the σ-τ coordinate system. Study the specific changes of M point under the action of reaction force. Figure 2.14 The direction of the maximum horizontal principal stress changes with depth (according to the Crustal Stress Institute of the National Geological Service, 1989) Figure 2.15, circle A in the figure is uniaxial tension Circle, B is the uniaxial compression rupture circle, C1 is the maximum Mohr rupture diagram of the three-dimensional compressive strength, C2-C7 are the intensity circles where the three-dimensional stress changes according to the stress intensity formula, and based on the circles A, B, and C, the Mohr intensity is calculated envelope. Circle D is the maximum Mohr's circle of the three-dimensional stress state at a depth of 2000m, E is the maximum Mohr's circle of the three-dimensional stress state of the surface layer, and the circles Dre and Ere are Mohr's circles after the three-dimensional stress changes after being acted upon by the reaction force. At point M in the circle diagram, only More, MEre1 and MEre2 are calculated using Equations (2.27) and Equations (2.28). The remaining three-dimensional stresses are all in a compressive stress state and are calculated using Equations (2.29) and Equations (2.30). Point M in the C1-C7 stress circle increases as the stress increases, but they are all within the Mohr envelope and have not reached the yield and rupture limit. At a depth of 2000m, the Mohr stress circle is much smaller than the Mohr envelope. However, when the reservoir is impounded and the reservoir water seeps down along the tensile fracture, the following reaction forces are formed. One is the tension of the reservoir water on the fault of about 20MPa, and the other is the temperature difference stress generated by the reservoir water seeping along the fracture. Longtan The annual average temperature in the region is about 20°C, and the geothermal gradient adopts the general 30°C/km. The ground temperature at a depth of 2000m is 80°C. The reservoir water seeps down to 2000m. According to the temperature of the hot springs in the area, it is considered that the temperature rises to 40°C. The rock The thermal stress coefficient is 0.4MPa/℃, then the total reaction force is 36MPa. After offsetting the maximum horizontal principal stress in the vertical fault direction of 31MPa, there is still a tensile stress of 5MPa. During the stress change, the original σ1 remains unchanged, and σ2 becomes σ3, the original σ3 becomes σ2, and the maximum Mohr circle and point M of the new stress state are both larger than the Mohr envelope, indicating that the rock mass fractures and triggers reservoir earthquakes. According to the ultrastructural transmission electron microscope study of fault rock samples in the Longtanba area by the Institute of Geology of the Chinese Academy of Sciences, it was found that when the temperature is 350-400°C, the differential stress (σ1-σ3) when the rock layer fractures and fails is 50-70MPa. When the depth of reservoir water reaches 2000m, the difference between the reaction force and the maximum principal stress is more than 50MPa, which is consistent with the stress field conditions that caused rupture due to structural changes in history. It can prove that the situation identified in Figures 2 and 15 is correct. Reasonable. Because the ground temperature at 2000m is 80°C, which is a medium-low temperature state, the differential stress is selected to be 50MPa. According to the transmission electron microscope ultrastructural analysis of sheared granular rock samples from the gently inclined tensile normal faults in the Huangwashi landslide area of ??the Yangtze River by the School of Environment of China University of Geosciences, it was found that in a low-temperature and low-pressure environment, the differential stress that causes shear failure of the rock mass is 30 ~ 50MPa. Under the stress environment of Longtan Land, due to the influence of the reaction force caused by daily temperature changes, the Ere1 circle formed has exceeded the Mohr envelope -σ3<σ, but the MEre1 point is on the envelope. Within, it shows the lateral restriction of σ2, and σ1-|σ3|=15MPa<30MPa, so there will be no yield failure; the Ere2 circle, under the influence of annual temperature changes, the reaction force can reach 10MPa, MEre2 is already located in the Mohr package Outside the contact line, the yield strength of the rock mass has been reached. Since σ1-|σ3|=20MPa<30MPa, it can be regarded as being in a critical state, but the superposition of other reaction forces should be prevented. The stress conditions in Figure 2.15 are listed in Table 2.4. Figure 2.15 Change rules of the intersection point M of Mohr’s circles σd and τ8