Determination of the strength and elastic modulus of basalt based on point load test

Abstract

Point load test is an effective method for obtaining the mechanical parameters of the rock. However, the theoretical basis of the test is not clear at present, which greatly limits its acceptability and application. Therefore, it is essential to find the theoretical solution for point load test and its testing results. To calculate the uniaxial compressive strength, tensile strength and elastic modulus of rocks, this paper establishes the quantitative relationship between point load strength and rock mechanical parameters based on the theory of elasticity mechanics. The point load tester was used to test 120 cylindrical basalt specimens, the thickness range of basalt is from 18 to 60 mm. Comparing the calculated results with the experimental results, it is found that the prediction error range of the proposed method is from 4% to 8%. The approach provides a new way for predicting the mechanical parameters of the rock.

Graphical abstract

Keywords

point load test / mechanical parameters / basalt / elastic mechanics

1 Introduction

Point load test (PLT) is a widely used method for quickly obtaining the uniaxial compressive strength of rock because it has no strict requirements on the shape of the rock specimens and is much more convenient to operate onsite than other indoor testing equipment [1–3]. Broch and Franklin [4] proposed the PLT method (Broch–Franklin method) for rock strength test, which was recommended by the International Society for Rock Mechanics (ISRM) and amended in 1985 [5,6]. The American Society for Testing and Materials (ASTM) developed a test method for the dimensionally corrected PLT strength index, i.e., I_s(50) [7], and a large number of scholars carried out in-depth studies and a variety of empirical formulas and analyses were proposed [3,8–19] on I_s(50) with uniaxial compressive strength and tensile strength. In the study of mechanical properties in geotechnical engineering, numerous researchers such as Lan et al. [20], Bao et al. [21], Liu et al. [22], Khan et al. [23], and Le et al. [24] have conducted extensive work, providing diverse analytical approaches. Hiramatsu and Oka [8] investigated the internal stress variation rule of three specimens under point load. Peng [25] used axisymmetric finite element to solve the stress distribution. Wei et al. [26] established a series of displacement function. Chau and Wei [27] analyzed and gave an analytical solution by finite element method, and Wei et al. [28] obtained similar results through theory and experiment. In previous studies, the calculation formulas of the standard PLT vary with different experimental conditions and rock types and are not universal.

The elastic modulus (E) of rocks is usually obtained by uniaxial compression tests, which can’t be tested by PLT at present. Scholars have investigated it in combination with hardness and ultrasonic velocity parameters by regression analysis, artificial neural networks, and other methods [29–33]. However, the theoretical relationship between PLT indexes and rock strength and elastic modulus is not clear till now. Therefore, it is of great significance to establish the theoretical relationship between PLT results and rock mechanical parameters, so that to simplify the test method and improve the accuracy of the test results.

Due to the nonhomogeneity of rocks, their strength and deformation properties vary considerably with specimen size [34–38], this is the so called “size effect”. Several theoretical frameworks have been established to interpret the size effect observed in intact rocks including statistical [39], fracture energy [40] and fractality [41]. Masoumi et al. [38,42] proposed the Unified Size Effect Law (USEL) and improved it, and Quiñones et al. [43] and Zhai et al. [44] found good agreement between strength and elastic modulus with USEL and IUSEL. By incorporating the USEL into the original Hoek–Brown model while maintaining the parameter m_i unchanged, a size-dependent Hoek–Brown failure criterion was ultimately established [45]. Roshan et al. [46] proposed a generalized correlation method for predicting compressive strength (σ_c) of sedimentary rocks of different sizes, and Haeri et al. [47,48] pointed out that I_s(50) increases with model size and grain size no effect on compressive strength. Several studies have also analyzed size effects in intact rock through PLTs [49] and Brazilian tests [35,49,50], revealing that rocks generally follow a generalized size effect trend. However, prior research exhibits an absence of theoretical formulas incorporating size effects for PLT.

To address these limitations, this paper proposes a theoretical formula based on the PLT method for determining the mechanical parameters of basalt, enabling rapid determination of rock strength and elastic modulus., It first analyzes the axial stress under PLT based on the elastic mechanics, introduces the Griffith strength theory to derive the solutions of uniaxial compressive strength and tensile strength. Secondly, the theoretical solution of rock elastic modulus was deduced from the axial failure displacement in the PLT according to the theory of elastic mechanics. Then, PLT are carried out on basalt collected from a project site in China, and the parameters σ_c, tensile strength (σ_t), and E of the rock are calculated by the proposed theoretical formula of PLT and the Broch–Franklin method. And the calculation results are also compared with the results of the commonly used test methods. Finally, the size effect of the parameters of the rock is investigated.

2 Experimental methods and observations

Basalt rock samples for PLT, Brazilian split and uniaxial compression tests are all collected from the same project site in China. 120 samples for PLT are drilled and processed into cylindrical specimens of different sizes, and 4 of the same basalt samples are prepared for the uniaxial compression tests by MTS815 Rock Mechanics Test System (MTS815) and Brazilian split tests with MTS, respectively.

2.1 Methods

MTS815 testing machine produced in the USA is used to carry out uniaxial compression tests as shown in Fig. 1(a). Specimens with a height-to-diameter ratio of 2:1 (φ50 mm × 100 mm) were placed in the servo-controlled press. Axial load was applied at a precisely controlled rate until failure occurred (n = 4 specimens). Brazilian splitting tests employed the same MTS815 system with six φ50 mm × 25 mm specimens, the test procedure was shown in Fig. 1(b). Each disc was horizontally positioned and subjected to diametral line loading at 0.001 mm/s until diametrical splitting occurred.

The PLT was accomplished using a portable device as shown in Fig. 1(c). The portable device was developed by the joint laboratory of Rock Innovation Company Zhejiang and Shaoxing University, which is verified by around 5000 sample testing and quality inspected by the government in China. The loading element is drived through an oil pump with electric servo till the rupture of the specimen, automatically recording the testing data with a data collector. The data will be transmitted wirelessly to the cloud platform–Statistical mechanics of rock masses for the calculation of parameters σ_c, σ_c (Broch–Franklin method), σ_t, and E.

2.2 Phenomenological analysis of point load test

Basalt specimens used for PLT are collected from a project site in China which has a cryptocrystalline structure and exhibits brittle damage characteristics. The specimen thickness D (mm), maximum displacement w (mm), and peak load P (kN) were automatically recorded by a self-developed PLT apparatus [51], and the rupture phenomenon of the specimen was observed.

2.2.1 Damage phenomenon of specimen

Rupture of rock experiments in the test shows the following characteristics as shown in Fig. 2(a).

1) A nearly circular crushing zone appears at the both loading points, showing plastic damage characteristics. The distance from the loading point to the boundary of the plastic zone can be called as the radius of the plastic zone, and along the loading axis it is expressed by z_c (mm).

2) The specimen very quickly breaks and the fracture surface shows a “magnetic lines” type of trajectory map, indicating the tensile tearing characteristics. The tearing trajectories start from the boundaries of the two plastic zones and meet each other in half way and axis-symmetrically.

2.2.2 The load–displacement curves

The load–axial displacement curves, i.e., P–w curves, recorded by the data collector in PLT on the basalt specimens are shown in Fig. 2(b). The curves are generally close to straight lines with slightly up-concave, indicating that the rupture of the rock is a sudden brittle damage after elastic deformation.

3 Derivation of the analytical solution

3.1 Theoretical solution of tensile and compressive strengths of rocks based on point load test

The stress distribution within the rock specimen under PLT condition can be analyzed with the theory of elastic mechanics, then the theoretical solutions for σ_c and tensile strength of rocks are able to be established through introducing the Griffith strength criterion.

3.1.1 Axial stresses under point load

Since the integral of the stress σ_z along any horizontal surface in Fig. 3(a) is in equilibrium with the point load P, it can be taken as a complete separator for stress analysis. The model of Fig. 3(a) is mirrored to form a PLT model with symmetric action P on the upper and lower horizontal surfaces, and the distance between the two action points is D, as shown in Fig. 3(b). Due to the stresses caused by load P reduce rapidly with the distance r, the error led by the combined PLT model than the solution in semi-infinite body will be acceptable, which is according to St. Venant’s principle.

According to the Boussinesq solution [52] for a semi-infinite elastic body under concentrated load P, the stress distribution at a point M(ρ, φ, z) in the body can be expressed as Eq. (1)

(1)\{\begin{array}{rl}& {\sigma }_{\mathrm{z}}=\frac{3P}{2\pi }\frac{{\mathrm{z}}^3}{r^5},\\ & {\sigma }_r=-\frac{P}{2\pi r^2}[\frac{\left(1-2\nu \right)r}{r+\mathrm{z}}-\frac{3{\rho }^2\mathrm{z}}{r^3})],r=\sqrt{{\rho }^2+{\mathrm{z}}^2},\end{array}

where P is the magnitude of the concentrated load, r is the radius to point M, ρ is the horizontal element of r, φ is the azimuth angle, and z is the height. Taking positive in terms of pressure, σ_z is the compressive stress along the axis direction, σ_ρ is the tensile stress in any direction perpendicular to the axis, v is the Poisson’s ratio of the rock.

In the axis on which the load is acting, ρ = 0, r = z, the stresses are given by the following equations

(2)\{\begin{array}{rl}& {\sigma }_{\mathrm{z}}=\frac{3P}{2\pi }\frac{1}{{\mathrm{z}}^2},\\ & {\sigma }_{\rho }=-\frac{\left(1-2\nu \right)P}{4\pi }\frac{1}{{\mathrm{z}}^2}.\end{array}

Therefore, the stress distribution under the PLT model can be simply extended as follows

(3)\{\begin{array}{rl}& {\sigma }_{\mathrm{z}}=\frac{3P}{2\pi }\frac{k^2}{{\mathrm{z}}^2},\\ & {\sigma }_{\rho }=-\frac{(1-2\nu )P}{4\pi }\frac{k^2}{{\mathrm{z}}^2},k^2=1+\frac{1}{{(D/\mathrm{z}-1)}^2},\end{array}

where k is a dimensionless coefficient. The value of k is usually taken to be slightly more than 1.0 but less than \sqrt{2} as the thickness of the specimen is between D = 20 to 100 mm. Figure 3(c) demonstrates the k–D relationship curve for general cases.

The relationship between the stresses in Eq. (3) and the distance from the loading point z is shown in Fig. 3(d). Along the axis direction with increasing z, the absolute values of tensile and compressive stresses decrease steeply from an infinitely high value at the loading contact point (z = 0), then symmetrically increase in the vicinity of z = D, manifesting that stress concentration occurs near the loading points, in accordance with the St. Venant’s principle. Wei et al. [26] and Chau et al. [27] have also theoretically investigated the internal stress state and obtained similar distributions.

3.1.2 Theoretical solution of tensile and compressive strengths of rocks

Since the damage of rock by PLT conforms to Griffith rupture mode, the Griffith criterion is used to study the point load strength of rocks. Griffith criterion is

(4)\{\begin{array}{ll}{\sigma }_{\rho }=-{\sigma }_{\mathrm{t}},& 3{\sigma }_{\rho }+{\sigma }_{\mathrm{z}}⩽0,\\ {\displaystyle \frac{{({\sigma }_{\mathrm{z}}-{\sigma }_{\rho })}^2}{{\sigma }_{\mathrm{z}}+{\sigma }_{\rho }}}=8{\sigma }_{\mathrm{t}},& 3{\sigma }_{\rho }+{\sigma }_{\mathrm{z}}>0,\end{array}

where σ_t is tensile strength of rocks.

From Eq. (3), we can find

3{\sigma }_{\rho }+{\sigma }_{\mathrm{z}}=\frac{3P}{4\pi }\frac{k^2}{{\mathrm{z}}^2}\left(1+2\nu \right)>0.

Therefore, the second equation of Eq. (4) is used for this case.

On the other hand, from Eq. (3) we can obtain

(5)\{\begin{array}{rl}& {\sigma }_{\mathrm{z}}+{\sigma }_{\rho }=\frac{(5+2\nu )P}{4\pi }\frac{k^2}{{{\mathrm{z}}_{\mathrm{c}}}^2},\\ & {\sigma }_{\mathrm{z}}-{\sigma }_{\rho }=\frac{(7-2\nu )P}{4\pi }\frac{k^2}{{{\mathrm{z}}_{\mathrm{c}}}^2},\end{array}

where P is the ultimate load, z_c is the radius of the plastic zone of the rock specimen.

Substituting Eq. (5) into the second equation of Eq. (4) yields

(6)\frac{{({\sigma }_{\mathrm{z}}-{\sigma }_{\rho })}^2}{{\sigma }_{\mathrm{z}}+{\sigma }_{\rho }}=\frac{{(7-2\nu )}^{2}P}{4\pi \left(5+2\nu \right)}\frac{k^2}{{{\mathrm{z}}_{\mathrm{c}}}^2}=8{\sigma }_t.

This leads to the formula for calculating the tensile strength of rock

(7){\sigma }_t=\frac{{(7-2\nu )}^{2}P}{32\pi \left(5+2\nu \right)}\frac{k^2}{{{\mathrm{z}}_{\mathrm{c}}}^2}.

It should be aware that, in Eq. (7), the tensile strength σ_t will be very sensitive to the plasticity radius z_c, but the latter is difficult to measured. Fortunately, we can easily obtain the axial displacement w in accord with the failure force P from the testing curve, i.e., Fig. 2(b). To establish a relationship between z_c and w seems to be a way to solve this problem.

Point load tests were conducted on different-sized specimens obtained from the same rock with identical tensile strength, and the failure displacement (w) was measured. At the same time, calculate the z_c values of the related specimens with Eq. (7), i.e.,

(8){\mathrm{z}}_{\mathrm{c}}=\frac{\left(7-2\nu \right)k}{4}\sqrt{\frac{P}{2\pi \left(5+2\nu \right){\sigma }_t}}.

Then the relationship between z_c and w can be established through regression analysis. This paper employs Solver tool to perform parameter fitting for these two parameters. Through PLT of 120 basalt specimens, the z_c–w relationship has been established as

(9){\mathrm{z}}_{\mathrm{c}}=10.74w^{0.407},

where w is the failure displacement of rocks.

Figure 4 shows the correlation curve between z_c and w. Substituting Eq. (9) into Eq. (7), we can obtain the formula for calculating the PLT tensile strength of rock.

From the second equation of Eq. (4), the Griffith’s σ_c of rock can be obtained

(10){\sigma }_{\mathrm{c}}=8{\sigma }_{\mathrm{t}}=\frac{{(7-2\nu )}^{2}P}{4\pi \left(5+2\nu \right)}\frac{k^2}{{{\mathrm{z}}_{\mathrm{c}}}^2},

where σ_c is compressive strength of rocks.

The scale factor in Eq. (10) can become the Griffith ratio.

3.2 Theoretical solution of rock elastic modulus based on point load test

The elastic modulus is one of the important indexes to evaluates the properties of rock in engineering practice, the PLT will be much more convenient than the current indoor laboratory. From the Boussinesq solution [52] for the model shown in Fig. 3(a), i.e., the semi-infinite body model, the axial displacement at any point M(ρ, φ, z) is

(11)\mathrm{\Delta }w=\frac{\left(1+\nu \right)\mathrm{\Delta }P}{2\pi Er}\left[2\left(1-\nu \right)+\frac{{\mathrm{z}}^2}{r^2}\right],

where E is the elastic modulus of the rock.

On the point load axis, the axial displacements at the upper and lower loading points are

(12)\mathrm{\Delta }{w}_{\mathrm{u}\mathrm{p}}=\mathrm{\Delta }{w}_{\mathrm{l}\mathrm{o}\mathrm{w}}=\frac{\left(1+\nu \right)\left(3-2\nu \right)\mathrm{\Delta }P}{2\pi E\mathrm{z}},

where z is the distance from the loading point for both the up and low semi-infinite model.

For the PLT shown in Fig. 3(b), we generally fix the low loading point at the base of the device, and measure the total axial displacement of the specimen under the load. Reasonably considering the loading point z = 0 moves synchronously with the displacement to half of the total displacement, i.e., \mathrm{z}={\displaystyle \frac{\mathrm{\Delta }w}{2}} for both sides of the model, and \mathrm{\Delta }{w}_{\mathrm{u}\mathrm{p}}=\mathrm{\Delta }{w}_{\mathrm{l}\mathrm{o}\mathrm{w}}, the total displacement will be

(13)\begin{array}{rl}\mathrm{\Delta }w=& \mathrm{\Delta }{w}_{\mathrm{u}\mathrm{p}}+\mathrm{\Delta }{w}_{\mathrm{l}\mathrm{o}\mathrm{w}}=\frac{\left(1+\nu \right)\left(3-2\nu \right)\mathrm{\Delta }P}{2\pi E}\left(\frac{1}{{\mathrm{z}}_{\mathrm{u}\mathrm{p}}}+\frac{1}{{\mathrm{z}}_{\mathrm{l}\mathrm{o}\mathrm{w}}}\right)\\ =& \frac{\left(1+\nu \right)\left(3-2\nu \right)\mathrm{\Delta }P}{2\pi E}\left(\frac{1}{\mathrm{\Delta }w/2}+\frac{1}{\mathrm{\Delta }w/2}\right)\\ =& \frac{2\left(1+\nu \right)\left(3-2\nu \right)\mathrm{\Delta }P}{\pi E\mathrm{\Delta }w}.\end{array}

For the elastic deformation and brittle failure cases, the P–w curve is close to straight line, taking \mathrm{\Delta }P = P and \mathrm{\Delta }w = w, then the Eq. (13) can be written as

(14)E=\frac{2\left(1+\nu \right)\left(3-2\nu \right)}{\pi }\frac{P}{w^2}.

This is the formula for rock elastic modulus under PLT.

4 Validation and analysis

For validating the proposed method in this paper, 120 PLTs of basalt specimens are tested with the self-developed PLT apparatus [51] as mentioned above; meanwhile, 4 sets of the same basalt samples are tested with the MTS 815 testing machine produced in USA for the indexes σ_c and E through compressive tests, and σ_t by the MTS-Brazilian disc splitting test. All the testing results are shown in Fig. 5.

The parameters σ_t, σ_c, and E are calculated by Eqs. (7), (10), and (14), respectively, based on the testing data. The calculation of σ_c using the Broch−Franklin method employs the following formula recommended by ISRM [6] and Chinese national standards [53–54],

(15){\sigma }_{\mathrm{c}}=22.82\cdot {\left[\frac{P_{\mathrm{c}}}{D_{\mathrm{e}}^2}{\left(\frac{D_{\mathrm{e}}}{50}\right)}^{0.45}\right]}^{0.75},

where D_e is the equivalent thickness of the specimen, P_c is the ultimate load.

For the comparison of the results, 9 samples with different sizes are collected out of the 120 tests as representatives. The results calculated through theoretical formulas and Broch–Franklin method are listed in Table 1. The PLT average results are 121.68 MPa, 147.01 MPa, 15.21 MPa, and 49.93 GPa for σ_c (theoretical method), σ_c (Broch−Franklin method), σ_t, and E. The paper also conducted uniaxial compressive tests and Brazilian split tests on the same rock specimen using the MTS815. The obtained uniaxial compressive strength, Brazilian split strength, and elastic modulus were 131.00 MPa, 14.90 MPa, and 52.05 GPa, respectively. Thes e reference values were then compared with and validated against the calculated results from the PLTs. The relative error of the PLT results compared to the reference values are −7.12%, 12.22%, 2.08%, and −4.07% for σ_c (theoretical method), σ_c (Broch–Franklin method), σ_t, and E. In this paper, the theoretical formula is verified by calculating the relative error, which is obtained by multiplying the ratio of the absolute error to the MTS test value by 100%.

The errors for different indexes in Table 1 indicate that theoretical solutions for σ_c, σ_t, and E of rock by PLT are acceptable and the results slightly less than the the commonly used method.

5 Discussion of size effect

Size effect of rock parameters is a commonly concerned issue, which not only reflects the essential mechanical properties of rocks related to size, but also involves the transformation of test results from different specimen sizes to the recognized standard. This effect on the tensile strength of brittle materials has been investigated by Weibull [55]. The Weibull’s size effect infers that the strength of material decreases with the specimen volume, but the dispersity of the results reduces much more rapidly, i.e.,

(16){\sigma }_{\mathrm{c}\mathrm{m}}\sim {\sigma }_0{V}^{-\frac{1}{m}},S^2\sim {\sigma }_0^2{V}^{-\frac{2}{m}},

where σ_cm, σ₀, m, and V are the strength mod, scale constant, dispersity of strength and the volume of specimens, respectively, and S² is the variance of the results.

In fact, this kind of effect is also reflected from the previous analysis and the Eqs. (7), (10), and (14) for σ_t, σ_c, and E, respectively. To reveal these characteristics, we make the following mathematical processes.

Doing the regression analysis on {\displaystyle \frac{k^{2}P}{{{\mathrm{z}}_{\mathrm{c}}}^2}}-D and {\displaystyle \frac{P}{w^2}}-D on the basis of the 120 tests mentioned above, as in Figs. 6(a) and 6(b), we obtain the following statistic relationship

(17)\{\begin{array}{rl}& \frac{k^{2}P}{{{\mathrm{z}}_{\mathrm{c}}}^2}=211.09{\mathrm{e}}^{-0.0002D},\\ & \frac{P}{w^2}=36.33{\mathrm{e}}^{-0.0096D},\end{array}

then the Eqs. (7), (10), and (14) can be re-written as

(18)\{\begin{array}{rl}& {\sigma }_{\mathrm{c}}=\frac{52.77{(7-2\nu )}^2}{\pi (5+2\nu )}{e}^{-0.0002D},\\ & {\sigma }_{\mathrm{t}}=\frac{6.60{(7-2\nu )}^2}{\pi (5+2\nu )}{e}^{-0.0002D},\\ & E=72.66e^{-0.0096D}.\end{array}

These are the formulas reflecting the size effect (negative exponential reduction) for the parameters obtained by PLTs. These formulas indicates that there are but not remarkable size effect for both the strength and elastic modulus, and it is slightly stronger for the latter than the former.

On the other hand, the transformation can be made from the testing results of variable specimen size D to the standard size (D = 50 mm), through the following ratios (or called the “size effect coefficient” in Fig. 6(c)).

(19)\{\begin{array}{rl}& {\sigma }_{\mathrm{c}50}={\sigma }_{\mathrm{c}D}{e}^{0.0002D-0.01},\\ & {\sigma }_{\mathrm{t}50}={\sigma }_{\mathrm{t}D}{e}^{0.0002D-0.01},\\ & E_{50}=E_D{e}^{0.0096D-0.48}.\end{array}

Based on the theory of elasticity, this study employs the Boussinesq solution, an analytical solution for stresses and displacements in a semi-infinite elastic medium under concentrated loads, and thus is applicable to elastic stress analysis of intact hard rocks. The insignificant size effect observed in basalt, characterized by such dimensional stability, stems from the statistically uniform distribution characteristics of microdefects within its structure, specifically, the absence of microcrack networks with preferred orientations and the lack of coarse mineral particles that dominate fracture behavior. These intrinsic features collectively result in a significant reduction in the dependence of macroscopic mechanical responses on specimen size.

6 Conclusions

Theoretical formula for basalt mechanical parameters based on PLT method are proposed in this paper, this formula enables rapid determination of rock strength and elastic modulus, and the comparison of the results have been made between the theoretical formula and Broch–Franklin method and that by MTS815 testing machine. It is found that the prediction error range of the proposed method is from 4% to 8%, which indicates the theoretical method is acceptable.

The size effect is an important factor affecting the testing result. However, it is important to note that the observed phenomenon should be interpreted as an inherent property of the rock, rather than a limitation of the PLT method or its theoretical solution. People can find the size effect law in the way tried in this paper and transform the result to satisfy the standard requirement.

While the PLT method tends to produce results with a wider statistical distribution compared to the high stability of benchtop systems like the MTS, it provides considerable practical benefits. The portability of our self-developed device allows for rapid and straightforward on-site testing. Consequently, this accessibility supports a statistical approach to reliability, where testing a larger number of specimens compensates for individual variability and leads to robust conclusions.

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