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（5 ） Strength criteria for isotropic rock material（1）Types of strength criterionA peak strength criterion is a relation between stress components which will permit the peak strengths developed under various stress combinations to be Similarly, a residual strength criterion may be used to predict residual strengths under varying stress In the same way, a yield criterion is a relation between stress components which is satisfied at the onset of permanent Given that effective stresses control the stress-strain behaviour of rocks, strength and yield criteria are best written in effective stress However, around most mining excavations, the pore-water will be low, if not zero, and so For this reason it is common in mining rock mechanics to use total stresses in the majority of cases and to use effective stress criteria only in special The data presented in the preceding sections indicate that the general form of the peak strength criterion should be （8）This is sometimes written in terms of the shear, and normal stresses, on a particular plane in the specimen:（9）Because the available data indicate that the intermediate principal stress, has less influence on peak strength than the minor principal stress, all of the criteria used in practice are reduced to the form (10)2 Coulomb’s shear strength criterionIn one of the classic paper of rock and of engineering science, Coulomb(1977) postulated that the shear strengths of rock and of soil are made up of two part – a constant cohesion and a normal stress-dependent frictional (Actually, Coulomb presented his ideas and calculations in terms of forces; the differential concept of stress that we use today was not introduced until the ) Thus, the shear strength that can be developed on a plane such as ab in figure 22 is（11）Where c=cohesion and Ф= angle of internal Applying the stress transformation equation to the case shown in figure 22 givesAnd Substitution for and s = τ in equation 11 and rearranging gives the limiting stress condition on any plane defined by β as（12） There will be a critical plane on which the available shear strength will be first reaches as б1 is The Mohr circle construction of Figure 4023a given the orientation of this critical plane as （13）This result may also be obtained by putting d(s-τ)/dβ = 0 For the critical plane, sin2β = cosФ, cos2β = -sinФ, and equation 12 reduces to （14）This linear relation between and the peak value of is shown in Figure Note that the slope of this envelope is related to Ф by the equation（15）And that the uniaxial compressive strength is related to c and Ф by （16） If the Coulomb shown in Figure 23b is extrapolated to = 0, it will intersect the axis at an apparent value of uniaxial strength of the material given by （17）The measurement of the uniaxial tensile strength of rock is fraught with However, when it is satisfactorily measured, it takes values that are generally lower than those predicted value of uniaxial tensile stress, = Although it is widely used, Coulomb’s criterion is not a particularly satisfactory peak strength criterion for rock The reasons for this are:(a) It implies that a major shear fracture exist at peak Observations such as those made by Wawersik and Fairhurst(1970) show that is not always the (b) It implies a direction of shear failure which does not always agree with experimental (c) Experimental peak strength envelopes are generally non- They can be considered linear only over limited ranges of or For these reasons, other peak strength criteria are preferred for intact However, the Coulomb criterion can provide a good representation of residual strength conditions, and more particularly, of the shear strength of discontinuities in rock (section 7)3 Griffith crack theoryIn another of the classic papers of engineering science, Griffith (1921) postulated that fracture of brittle materials, such as steel and glass, is initial at tensile stress concentrations at the tips of minute, thin cracks (now referred to as Griffith based his determination of the conditions under which a crack would extend on his energy instability concept: A crack will extend only when the total potential energy of the system of applied forces and material decreases or remains constant with an increase in crack ROCK STRENGTH AND DEFORMABILITY For the case in which the potential energy of the applied forces is taken to be constant throughout, the criterion for crack extension may be written (19)Where c is a crack length parameter, We is the elastic energy stored around the crack and Wd is the surface energy of the crack Griffith (1921) applied this theory to the extension of an elliptical crack of initial length 2c that is perpendicular to the direction of loading of a plate of unit thickness subjected to a uniaxial tensile stress, б He found that the crack will extend when （20）Where α is the surface energy per unit area of the crack surfaces (associated with the rupturing of atomic bonds when the crack is formed), and E is the Young’s modulus of the uncracked It is important to note that it is the surface energy, α, which is the fundamental material property involved Experimental studies show that, for rock, a preexisting crack does not extend as a single pair of crack surface, but a fracture zone containing large numbers of very small cracks develops ahead of the propagating crack 9FIGURE 25) In this case, it is preferable to treat α as an apparent surface energy to distinguish it from the surface energy which may have a significantly smaller It is difficult, if not impossible, to correlate the results of different types of direct and indirect tensile test on rock using the average tensile stress in the fracture zone as the basic material For this reason, measurement of the ‘tensile strength’ of rock has not been discussed in this However, Hardy(1973) was to obtain good correlation between the results of a rang of tests involving tensile fracture when the apparent surface energy was used as the unifying material Griffith (1924) extended his theory to the case of applied compressive Neglecting the influence of friction on the cracks which will close under compression, and assuming the elliptical crack will propagate from the points of maximum tensile stress concentration (P IN Figure 26), Griffith obtained the following criterion for crack extension in plane compression:（20）Where is the uniaxial tensile strength of the uncracked material (a positive number) This criterion can also be expressed in terms of the shear stress, τ , and the normal stress, acting on the plane containing the major axis of the crack:（21） The envelopes given by equations and 21 are shown in Figure Note that this theory predicts that the uniaxial compressive compressive stress at crack extension will always be eight times the uniaxial tensile