Brazilian test: stress field and tensile strength of anisotropic rocks using an analytical solution

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Abstract

Tensile strength of rock is among the most important parameters influencing rock deformability, rock crushing and blasting results. To calculate the tensile strength from the indirect tensile (Brazilian) test, one must know the principal tensile stress, in particular at the rock disc center, where a crack initiates. This stress can be assessed by an analytical solution. A study of this solution for anisotropic (transversely isotropic) rock is presented.

The solution is given explicitly. The key expansion coefficients are obtained from a complex-valued 2×2 matrix equation. The convergence of the solution is greatly improved by a new procedure. It is shown that the dimensionless stress field depends only on two intrinsic parameters, E′/E and b. The stress at the center of the disc is given in charts as a function of these parameters (and the angle θb between the direction of applied force and the plane of transverse isotropy). Furthermore, a new, reasonably accurate, approximate formula for the principal tension at the disc center, (0,0), is derived from the analytical solution:σpt(0,0)≅PπRL(E/E′4)cos(2θb)cos(4θb)4(b−1),whereb=EE′21G′2ν′E′. The elastic parameters of rock in two perpendicular directions were measured in the laboratory. The result of the stress analysis was applied in calculating the indirect tensile strength of gneiss, which has a well-defined foliation plane (transversely isotropic). When the results were compared with the tensile strength of rock obtained by using a conventional formula that assumes isotropic material, there was a significant difference. Moreover, good agreement was observed for the tensile strength calculated from the stress charts and the proposed formula, when compared with other published stress charts.

Introduction

Indirect tensile (Brazilian) testing of rock cores is an easy and common method for determining the tensile strength of rock. Tensile strength is calculated in this test by using an equation, which assumes isotropic material properties. Since many rock types (e.g. metamorphic and sedimentary) are anisotropic, and in particular transversely isotropic, it is necessary to find a method for determining the tensile strength of these types of rocks from the Brazilian test. Many researchers [1], [2], [3] have studied the tensile strength of anisotropic rocks by using the Brazilian test with the equation for isotropic material.

A comprehensive analytical solution for an anisotropic disc subjected to the Brazilian test was presented by Amadei et al. [4]. It is based on a solution method given by Lekhnitskii [5]. The calculation of stress and tensile strength of the anisotropic rock sample requires that the principal elastic constants E, E′, ν, ν′ and G′ be determined (ν/E=ν′/E′). Chen [6] presents in diagrams the principal stress at the center of a rock disc as a function of the three dimensionless parameters E/E′, ν′ and E/G′.

The original objectives of this work were to study the analytical solution of Lekhnitskii, Amadei et al. and Chen, and to apply the results of the stress analysis to calculate the tensile strength of anisotropic rock material from laboratory tests.

The study resulted in a few improvements of the solution method. The convergence of the series solution was improved greatly by a new procedure. The stress, strain and displacement can be calculated, even at the disc periphery, with a high degree of accuracy and without any problems of convergence. The key expansion coefficients involve the solution of 4×4 matrix equations (Eq. (24)), involving the real and imaginary parts of Am and Bm). This equation is reduced to a simpler complex-valued 2×2 matrix (Eq. (25)). The analytical solution shows that the dimensionless stress field depends on only two intrinsic parameters (and on the angle θb between the direction of the applied force and the direction normal to the plane of transverse isotropy). The stress at the center of the disc may be given in the form of charts. Finally, it was found that from the analytical solution, new, quite good approximate formulas could be derived for the principal tension and compression at the disc center (, ). The tensile stress σpt (0,0) at the center of the disc can be obtained from either the charts or the approximate formula, Eq. (36).

Section snippets

The Brazilian test

In the Brazilian (indirect tensile) test, a disc of material is subjected to two opposing normal strip loads at the disc periphery (Fig. 1 left). The applied load is P (N). The rather thin disc has a radius R and thickness L. In the standard method of ISRM [7], the tensile strength of rocks, σt, is calculated from the equation (diameter D=2R):σt=2PπDL=PπRL.Eq. (1) is based on the theory of elasticity for isotropic media. The formula gives the tensile stress perpendicular to the loaded diameter

Problem to be solved

We consider plane stress in a transversely isotropic disc. The basic equations to fulfill are force balance, the relation between strain and displacement, and Hooke's law:σxx+τxyy=0,τxyx+σyy=0,εx=ux,εy=vy,γxy=uy+vx,εx=σxEν′σyE′,εy=σyE′ν′σxE′,γxy=τxyG′.It should be noted that ν (=νE/E′) does not appear directly in the above equations. The four elastic parameters that occur in the equations are E, E′, ν′ and G′. The transversely isotropic disc has prescribed forces along its

Analytical solution

The analytical solution for isotropic case is well established for any boundary forces; see [11], [12]. The solution was extended to the case of an anisotropic disc by the Russian school following Muskhelishvili [13]. Lekhnitskii [5] gives a complete solution, which has been used by Chen et al. [14] to analyze transversely isotropic discs. A good survey of the Brazilian test and the analytical solution is given in [15].

Lekhnitskii's solution is elaborated and presented in detail in [16]. A

Improved convergence

It is a well-known experience that the convergence of Fourier series may be slow. The convergence of series (23) for the Φ-functions and their derivatives may also be quite slow, in particular at and near the boundary of the disc. Lekhnitskii [5] stresses the importance of this problem. The normal stress component at the periphery of the disc, shown in Fig. 4, was calculated with terms up to m=9 and 15 (the even terms are zero). The load at θ=π/2 becomes sharper as the number of terms

Summary of the general solution behavior

The solution for any boundary force is obtained from the formulas above in the following way. The Fourier coefficients of the boundary forces are first calculated (Eq. (5)). The Fourier coefficients of the integrals of the boundary forces are given by Eq. (19). The constants in Eq. (25) are obtained from , , , , and the expansion coefficients Am and Bm are determined from Eq. (25). The stress field is given by , . The functions Φ1(z) and Φ2(z) are defined by , , and their derivatives by , . The

Parameters of the solution

The proposed solution is used for calculating the tensile strength of a material subjected to the Brazilian test. The solution's dependence on the material parameters is thus of great importance. We are particularly interested in the minimum number of independent parameters and the scale factors. We consider the case of two opposing normal point forces. The parameters are then:E,E′,ν′,G′,R,L,P,θb(0⩽θb⩽π/2).

It should be noted that ν does not appear in the equations or in the solutions. There are

Symmetry due to interchange of axes

Whether there are symmetries in the solution is an important question. There is actually a particular intrinsic symmetry. Let us consider what happens when the axes x and y are interchanged:x→y,y→x.The stress field is reversed, but the stress balance expressed in Eq. (2) is of course still valid. This is also true for the (renamed) displacement, Eq. (2), and compatibility relation Eq. (24). Hooke's law, Eq. (3), remains to be fulfilled. The modulus of elasticity, E and E′, must also be

Principal stresses at the center of the disc

The principal tension σpt(0,0) at the center of the disc is of particular interest. We are also interested in the principal compression σpc(0,0) and the ratio ηtc=σpt(0,0)/[−σpc(0,0)] at the disc center. The principal stresses at (0,0) have E′/E, b and θb as parameters. Fig. 10, left, shows the variations of σpt(0,0) with b for θb=0 for a few E′/E. Fig. 10, right, shows the variation with E′/E (=Eq) for θb=π/6 for a few b.

It is valuable to have explicit expressions for the principal stresses at

Principal stresses around the center of the disc

The tensile stress at the center of the disc causes the failure. The variation of the stress field near the center is also of interest. Fig. 11 shows the principal tensile stress (left) and the principal compressive stress (right) in the area –0.5R<x<0.5R and –0.5R<y<0.5R for the reference example, Eq. (29), with θb=π/4.

The variation of the stress around the center of the disc is quite smooth in Fig. 11. The ratio of principal tensile to compressive stress has a maximum at the center. The

Diagrams for the tensile stress at the center of the disc

The principal tensile stress at the center of a disc due to normal point loads at the disc periphery may be written in the following way:σpt(0,0)=PπRLσpt0(E′/E,b,θb).The dimensionless principal tensile stress at the center,σpt0, depends on the dimensionless parameters E′/E and b, and the angle θb. The scale factor is P/(πRL). Fig. 12 shows the function σpt0(E′/E,b,θb) in charts for different angles θb. Each chart shows the variation of the stress with the parameters E′/E and b for a particular

Summary and conclusions

The tensile strength of rock is a key parameter that influences rock deformability, rock crushing and blasting results. The indirect tensile, or Brazilian, test of rock cores is a common method for determining the tensile strength. Calculating the tensile strength from the Brazilian test requires knowledge of the principal tensile stress, in particular at the rock core center, where cracks start. This stress can be assessed by a well-known analytical solution for an isotropic material. There

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