1. compressive strength
The compressive strength of rock is the ultimate failure value of rock specimen under uniaxial pressure, which is numerically equal to the maximum compressive stress during failure. The compressive strength of rock is usually measured by pressure test with a press in the laboratory. Specimens are generally cylindrical (core drilling) or cubic (the cross-sectional dimensions of specimens are processed with rocks, and the diameter of cylindrical specimens is D = 5 cm, and some D = 7cm；; Cubic column test partner, using 5cm×5cm or 7cm×7cm). The height h of the sample shall meet the following conditions:
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Where d is the cross-sectional diameter of the sample; A is the cross sectional area of the sample.
Test results press type to calculate compressive strength:
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Where: Rc is the uniaxial compressive strength of rock; Pc is the axial pressure applied when the rock sample is destroyed; S is the cross-sectional area of the rock sample.
2. Shear strength
The shear strength of rock refers to the ultimate strength of rock to resist shear failure or sliding, which is expressed by the ultimate stress when rock shears or slides. Shear strength of rock is one of the most important engineering mechanical properties of rock, which is often more meaningful than compressive strength and tensile strength of rock. The mechanical indexes of rock shear strength are cohesion C and internal friction angle φ, which are measured by various rock shear experiments. Under the action of vertical pressure P, shear stress T is applied horizontally until the rock sample is sheared. At this time, the normal stress σ and shear stress τ on the shear plane are respectively
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Where: p and t are respectively the maximum vertical pressure and the maximum horizontal shear force applied when the specimen starts to slide along the existing shear plane; S is the shear surface area.
In order to be close to the engineering practice, the shear strength of rocks can be further divided into three types, namely, shear strength, shear strength and shear strength.
(1) shear strength
Shear strength Under the action of vertical pressure P, shear stress T is applied in the horizontal direction until the sample is sheared. At this time, according to the Mohr-Coulomb strength theory, the shear strength τf of rock is
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(2) Shear strength
Shear strength means that when a rock specimen has a pre-existing shear surface (joint surface or fracture surface), a shear force T is applied in the horizontal direction under the action of vertical pressure P until the specimen slides. At this point, the shear strength of rock τf is
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(3) Shear strength
Shear strength refers to the application of shear force T in the horizontal direction without vertical pressure until the rock sample is sheared. At this time, there is no normal stress on the shear plane, only the shear stress T, and then the shear stress.
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Where: t is the maximum horizontal shear force applied when shearing rock samples; S is the area of the pre-existing shear plane. According to Mohr strength theory, shear strength is defined as
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Rock shear strength test and formula can also be used to determine the shear strength of weak structural plane in rock mass.
3. Fracture criterion The so-called fracture criterion is the condition of rock fracture. Assuming that the rock breaks under the stress state of (σ 1, σ2, σ3), we call the σ 1 = f (σ2, σ3) relationship as the fracture criterion. Several common failure forms and fracture criteria are discussed in detail below. The reservoir rock is underground, and the principal stress is generally compressive, and shear fracture mainly occurs, so there are many discussions about shear fracture. However, under the condition of hydraulic fracturing, the pore pressure in the rock is large enough, and tensile fracturing is also possible.
(1) Coulomb molar fracture criterion
This is the most widely used strength theory in rock mechanics. It is believed that when the shear stress on a certain surface exceeds the ultimate shear stress that it can bear, the rock will be destroyed. 178 1 year, the French physicist Coulomb solved the balance problem by using the proportional relationship between friction and normal pressure when an object slides, and obtained Coulomb's law of friction. The experimental results of rock fracture can be expressed by a simple relation similar to the friction formula, which is the so-called Coulomb fracture criterion:
If the normal stress σ and shear stress τ on a plane inside the rock meet the condition τ = c+μ σ, the plane will break, where c is called cohesion or cohesive strength of the rock; μ is called internal friction coefficient, which is often called μ = tan φ in engineering, and φ is called internal friction angle. Figure 3-7 shows the Coulomb fracture criterion diagram. When the shear force τ increases to a certain extent, the rock breaks. If the normal stress σ is large, the internal friction will increase, and greater shear force τ is needed to break the rock.
In 1882, Moore introduced the Mohr circle to represent the stress state inside the material (Timoshenko, 1970), which can directly represent the fracture criterion. Figure 3-8 shows the Mohr circle in the limit equilibrium state.
Figure 3-7 Schematic Diagram of Coulomb Criterion
Figure 3-8 Mohr Circle in Limit Equilibrium State
First, consider the plane problem. As shown in fig. 3-9a, take any element in the rock mass, and let the two principal stresses acting on the tiny element be σ 1 and σ 3 (σ 1 >: σ3), and there are normal stress σ and shear stress τ on the mn plane that forms an arbitrary angle α with the maximum principal stress σ 1 in the microcell. In order to establish the relationship between σ, τ and σ 1, σ3, the microprism abc is taken as the isolator, as shown in Figure 3-9b.
Figure 3-9 Coulomb Molar Circle
According to the static equilibrium condition, each force is projected in the horizontal and vertical directions respectively.
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The above two equations are simultaneous, and the stress on the mn plane can be obtained by the following equation
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The relationship between σ, τ and σ 1, σ3 can be expressed by Coulomb-Mohr stress circle, as shown in Figure 3-9c. In σ τ rectangular coordinate system, OB and OC are taken along σ axis to represent σ3 and σ 1 respectively according to a certain proportion. With D as the center and (σ 1σ3) as the diameter of the circle, the DA line is obtained by rotating the DC counterclockwise by 2α angle, which intersects with the circle at point A. From the formula (3- 17), it can be seen that the abscissa of point A in the figure is the normal stress σ on the mn plane, and the ordinate is the shear stress τ. So Coulomb-Mohr circle can represent the stress state of a point in the rock, and the coordinates of each point on the circumference are the normal stress and shear stress of the point on the corresponding plane. In this way, Mohr circle can not only give the specific value of shear stress τ and normal stress σ when fracture occurs, but also show the direction of fracture.
In 1900, Moore proposed that when the shear stress τ and the normal stress σ on the plane satisfy a certain number of functional relations, that is
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Fracture will occur along this surface, which is the Mohr fracture criterion, in which the form of function f is related to the rock type. In this way, Moore generalized Coulomb criterion. Because Coulomb criterion represents a straight line on σ τ plane, Mohr criterion represents a curve on σ τ plane. This curve is usually called the fracture line, and some books call it the strength line. Another contribution of Moore is to extend Coulomb Mohr circle to three dimensions. On the τ σ plane, the Mohr circle has a diameter of (σ 1σ3), and it breaks when the fracture line AB is tangent to the great circle. The angle between the fracture surface and the direction of the maximum principal stress σ 1 is (π/2β), and the magnitude of the intermediate principal stress σ2 has no influence on the fracture occurrence conditions and the orientation of the fracture surface. Using three-dimensional Mohr circle, the normal stress and tangential stress on any plane in rock can be obtained. According to the included angle φ between the plane under study and the maximum stress direction and the included angle θ between the plane and the minimum principal stress direction, the radius with an included angle of 2φ (φ = 30, 2φ = 60 in this example) is made in the small circle formed by σ 1 and σ2, and the radius with an included angle of 2θ (2θ in this example) is made in the small circle formed by σ3 and σ2.
Figure 3- 10 3D Mohr circle
When τ = f (σ) is a straight line, it is consistent with Coulomb criterion and is called Coulomb-Mohr criterion or Coulomb-Mohr intensity line. Experiments show that when the rock is weak, its strength curve is approximately parabolic, and the Mohr fracture criterion table is τ 2 = σt (σ+σt), where σ t is the uniaxial tensile strength of the rock, and when τ 2 ≥σ t (σ+σ t), the rock will fracture. When the rock is hard, the strength curve is similar to hyperbola, which can be expressed as τ 2 = (σ+σ t) 2tan η+(σ+σ t) σ t, and its failure criterion is τ 2 ≥ (σ+σ t) 2tan η+(σ+σ t) σ t, where σc is uniaxial compressive strength.
(2) Griffith strength theory
According to Mohr strength theory, a material is regarded as a complete and continuous homogeneous medium, but in fact, there are many tiny cracks or fissures in any material. Under the action of stress, there will be great stress concentration around these cracks (especially at the crack ends), and sometimes the local stress can reach 100 times of the additional stress, so the material failure mainly depends on the stress state around the internal cracks, and the material failure often starts from the crack ends and leads to complete failure through crack propagation. 1920, Griffith's classic paper made a breakthrough in the study of fracture mechanics. Griffith considered isolated cracks in solids under stress, and according to the basic energy theory of classical mechanics and thermodynamics, he proposed that
Crack diffusion theory. Under the action of external force, when the elastic potential energy gathered by the internal stress concentration of the material is greater than the work done to make it expand along the crack, the material will crack along the crack. As shown in figure 3- 1 1, there is a crack with a length of l in the material, and a crack with a length of Δ l is generated under the action of elastic potential energy u, and the released elastic potential energy is Δ u, then the energy release rate (energy gradient, also called crack propagation p)G is
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When the crack propagation length is δ L, the increased surface energy δ S is
Schematic diagram of crack propagation in Figure 3- 1 1
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Where γ is the surface energy per unit area (unit line length). Assuming that r is the surface energy increase rate or crack propagation resistance, there are
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It can be seen that only when G≥R can the crack propagate. Therefore, G≥R is the energy criterion of crack propagation.
Let's study the stress criterion of crack propagation.
If the crack propagation direction is selected as X-axis, then the Y-axis is perpendicular to the crack surface, and the stress at the crack tip is σx, σy and τxy. The tangential stress σb around the crack ellipse can be expressed by Inglis formula in elasticity (Ling Xianchang et al., 2002), and the maximum shear stress at the crack tip can be obtained by the following formula.
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Where m = b/a is the ratio of the major axis to the minor axis of the cracked ellipse. It must be noted that the crack is an elongated ellipse, and the shear stress at the end of the crack is along the Y axis. Thus, in σ y >; In the case of 0, the negative σb value can only be obtained by using the negative sign in formula (3-22), that is, it is tensile stress. When the stress is greater than σt (uniaxial tensile strength of rock), a new fracture will appear at the crack end, which will make the crack expand. Using principal stresses σ 1, σ2, σ3 to represent σx, σy, and τxy, the expression of fracture angle β (the angle between fracture surface and σ 1) can be obtained.
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This requires (σ1-σ 3)/2 (σ1+σ 3) ≤1,that is, σ 1+3σ 3 ≥ 0. If σ 1+3σ 3 ≥ 0 is satisfied, the strength criterion can be expressed by σy and τxy, or τ 2xy ≥ 4σ t (σ t-σ y). When σ 1 and σ3 are used, it is (σ1-σ 3) 2/(σ1+σ 3) ≥-8 σ t, and the negative sign appears here because the tensile stress in rock mechanics is negative, which makes the rock crack. In order to meet the above fracture conditions, σ 1 is quite different from σ3. When σ 3 = 0, that is, under uniaxial stress, Cos2β = 1/2, so there is 2β = 60, so the fracture angle β = 30; When σ 3 1/2, so β; 0, then (σ1-σ 3)/2 (σ1+σ 3) <1/2, β > 30, if σ 1 and σ3 are both large and the rock strength is small, then cos2β→0, that is β→ 45.
If the condition σ 1+3σ3 ≥ 0 is not met, it means that the rock is in a tensile stress environment. When σ 3 ≤-σ t, the rock cracks along the plane perpendicular to σ 3.
If liquid is pumped into the borehole of intact rock under a certain pressure, once the liquid pressure in the borehole is greater than the local stress field, the borehole wall rock will bear tensile stress, which is equal to or greater than the tensile strength of the rock, and tensile fracture will occur. This tensile fracture surface must pass through the axis of maximum principal stress and be perpendicular to the axis of minimum principal stress.