Empirical equations between characteristic impedance and mechanical properties of rocks

Zong-Xin Zhng, De-Feng Hou, Adeyemi Aldejre

a Oulu Mining School, University of Oulu, Oulu, 90014, Finland

b College of Resources and Safety Engineering, China University of Mining and Technology, Beijing,100083, China

Keywords:Characteristic impedance Mechanical properties Rock mass Classification Strength

ABSTRACT Based on a great number of experimental data on various mechanical properties of rock in the literature,six empirical equations between the characteristic impedance (product of density and P-wave velocity)and mechanical properties of rock are proposed.These properties include uniaxial compressive strength,tensile strength, shear strength, mode I fracture toughness, Young’s modulus, and Poisson’s ratio. These empirical equations show that the values of the aforementioned properties increase with increase in characteristic impedance. It also implies that the characteristic impedance of rock may be considered as an index to represent the main properties of rock. In this sense, it is possible to consider using characteristic impedance to classify rock masses for studies in the future.

1. Introduction

Rock is a kind of solid which is much more complex than artificial materials,such as steels.Accordingly,it is difficult to represent the basic mechanical characteristics of rock by using a single parameter, although some rock properties are found to be correlated with one another. For example, the compressive strength,tensile strength and shear strength of rock are correlated with each other under uniaxial loading (e.g. Jaeger et al., 2007; Goodman,1989; Whittaker et al., 1992; Wang and Aladejare, 2016; Zhang,2016). However, the relationships among these three strengths are not clear in nature, e.g. the ratio of uniaxial compressive strength to uniaxial tensile strength often varies in a wide range of 8-15. One of main reasons for this variation is that rock strength measured in laboratory is not a deterministic parameter,since it is not directly related to the fragment size and the energy used(Zhang et al., 1999). In addition, it has been reported that rock fracture toughness is related to its tensile strength (e.g. Gunsallus and Kulhawy, 1984; Zhang, 2002; Jin et al., 2011), point load index (Gunsallus and Kulhawy, 1984), and compressive strength(Gunsallus and Kulhawy,1984),while point load index is related to the uniaxial compressive strength and tensile strength(e.g.Heidari et al., 2012; Singh et al., 2012; Wang and Aladejare, 2015).Furthermore, it was found that the fracture toughness could be used to evaluate the rock fragmentation in crushing,grinding,and blasting (e.g. Ouchterlony et al., 2004; Sanchidrián et al., 2007;Zhang, 2016).

The above studies indicate that fracture toughness may be a better rock property for evaluating rock fragmentation than rock strength.However,similar to rock strength,rock fracture toughness can be only determined in laboratory using small rock samples.Hence, fracture toughness is not suitable for evaluating rock mass dealing with geological structures,such as faults and joints.In other words, the commonly used rock properties, such as strengths and fracture toughness, could not be used to evaluate rock mass directly.However,in the field of mining and rock engineering,it is necessary to classify rock masses upon design and performance assessment of rock constructions. For example, one can select an optimum method of mining, if the local rocks and ore masses are classified and known very well.

At present, the methods used to characterize or classify rocks can be divided into two categories, i.e. geological classification system and engineering classification system.The former is mainly based on mineral content, texture, mineral size, chemical composition and origin (sedimentary, igneous or metamorphic) of rocks(e.g.Bieniawski,1976;Goodman,1989).The latter includes several sub-systems like rock quality designation (RQD) (Deere, 1967),tunneling quality index (Q) (Barton et al.,1974), rock mass rating system (RMR) (Bieniawski, 1973), geological strength index (GSI)(Hoek et al.,1995),rock mass index(RMI)(Palmstr?m,1996),etc.In addition to the aforementioned classifications,some other systems are also reported using sonic velocity, especially primary wave velocity(P-wave)of rock(e.g.Rawlings and Barton,1995;Zhao and Wu, 2000; Nourani et al., 2017; Chawre,2018).

The methods for classifying rocks have played an important role in rock mechanics and rock engineering, but they have certain drawbacks. For example, the geological classification systems contain little information on mechanical behavior of rock. The engineering classification systems like RQD, Q, RMI and RMR often require sample collection, tests of intact rock properties, and extensive field investigations for identifying the frequency and nature of the discontinuities.The classification systems using sonic velocity are more convenient to implement, compared to other systems, but they have a significant variability in the measured values of rock properties for a given velocity(e.g.Butel et al.,2014;Karakus et al.,2005).Therefore,to address the above drawbacks,a classification system that is simple, reliable and easy to use is needed. Unlike rock mass, intact rock refers to the unfractured block between discontinuities in a typical rock mass. Intact rock may range from a few millimeters to several meters in size(Hudson and Harrison,1997). The properties of intact rock are governed by the physical properties of the composed materials and the way that they are bonded to each other.The properties which may be used in describing the intact rock include petrological name,color,texture,grain size, minor lithological characteristics, density, porosity,strength,hardness,and deformability.The International Society for Rock Mechanics and Rock Engineering(ISRM)(Ulusay and Hudson,2007)has provided suggested methods for determination of all the properties of intact rock.

Zhang (2016) suggested using characteristic impedance to evaluate a rock mass and classify rocks, because the characteristic impedance of rock could represent the geological structures of the rock mass,e.g.joints,faults,bedding,and mineral composition,to a certain extent.In this paper,the characteristic impedance of rock is defined as the product of the P-wave velocity and the density of the rock.Since the characteristic impedance can be determined by nondestructive methods, it can be used to study various problems related to stress waves in rock mechanics(e.g.Yang et al.,2010;Xia and Yao, 2015). Xiao et al. (2015) suggested that the characteristic impedance could be taken as a measure of the resistance to the momentum transfer of rock. Some studies (Roy, 2005; Fan et al.,2018) also found that the characteristic impedance could characterize the energy transfer property of materials.

Based on the information available in the literature, this paper summarized the experimental data of mechanical properties and characteristic impedance of intact rocks.The relationships between the mechanical properties and the characteristic impedance of rock were investigated. The experimental data were collected from publications that could be assessed,and various types of rocks were involved. The summarized results were mainly focused on the relations between characteristic impedance (“impedance” for short)and mechanical properties. Finally, the implications of these relations in the rock engineering were discussed.

The experimental data in this study are those publically available in the field of rock mechanics and engineering. In addition,only the data whose sample preparation and testing were carried out according to ISRM standards were considered in this paper.Besides, the experimental data used in this study were those with rock conditions of grade I(fresh rock),grade II(slightly weathered rock), and grade III (moderately weathered rock). Grade IV and above were generally referred to as soil (Ehlen, 2002) and not considered in this study. Furthermore, statistical methods of rock data, i.e. mean, standard deviation, and ranges of values, were not used;instead, the experimental results were utilized in this study.Artificial materials, i.e. concrete and other structurally modified materials, are not considered herein.

2. Relations between characteristic impedance and mechanical properties

In the following sections, the experimental data of mechanical properties of dry rock samples were obtained at laboratory-room temperatures. In addition, the P-wave velocities used to determine the impedance of rock were measured under dry-rock condition.

2.1. Uniaxial compressive strength

Previous studies found that the uniaxial compressive strength of rock is related to either the density or the sonic velocity of rock(e.g.Lama and Vutukuri, 1978; Tuˇgrul and Zarif, 1999; Yasar and Erdogan, 2004; Khandelwal and Singh, 2009; Diamantis et al.,2009; Yagiz, 2011). Considering that the impedance of rock is also related to both the density and the sonic velocity of rock,we tried to correlate the impedance with the uniaxial compressive strength.

In this context, 409 experimental data measured from various types of rocks are presented in Fig. 1 (Kahraman et al., 2000;Moradian and Behnia,2009;Aalizad and Rashidinejad,2012;Servet et al., 2014; Wyering et al., 2014; Momeni et al., 2015; ˙Ince and Fener, 2016; Madhubabu et al., 2016; Majeed and Bakar, 2016;Brisevac et al., 2017; Mahanta et al., 2017; Ali and Othman, 2018;Khajevand and Fereidooni, 2018). It can be found that the uniaxial compressive strength σcand impedance of rock Z can be correlated(see Fig.1) as

where a1= 6.14.

Eq. (1) indicates that the uniaxial compressive strength increases with increase in impedance. Interestingly, previous experiments showed that the uniaxial compressive strength of rock depends on the porosity of rock (e.g. Lama and Vutukuri, 1978;Palchik,1999;Chen and Hu,2003;Ludovico-Marques et al.,2012),indicating that the porosity and the impedance may be correlated.Note that rock cores in the work of Majeed and Bakar(2016) were drilled perpendicular to the beddings (see Fig.1).

2.2. Tensile strength

The tensile strength data considered in this study are those measured by Brazilian test method. Previous studies discovered relationship between the P-wave velocity and tensile strength of various rocks (e.g. Ersoy and Atici, 2007; Khandelwal and Singh,2009). Because the P-wave velocity is related to the impedance, a correlation between the tensile strength and impedance of rock may exist.By using 168 experimental data(Kahraman et al.,2000,2008; Aalizad and Rashidinejad, 2012; Xiao et al., 2015; ˙Ince and Fener, 2016; Majeed and Bakar, 2016; Sajid et al., 2016; Mahanta et al., 2017; Naseri and Khanlari, 2017; Su and Momayez, 2017;Khajevand and Fereidooni, 2018), we found that the tensile strength σtand the impedance are related(see Fig. 2) as

where a2= 0.6.

Fig.1. Uniaxial compressive strength vs. characteristic impedance of rock.

Eq.(2)indicates that the tensile strength is directly proportional to the impedance.Note that the P-wave velocity in Kahraman et al.(2008)was measured using indirect method,and the rock samples were defective or with beddings in the studies of Naseri and Khanlari (2017) and Majeed and Bakar (2016). All such data are marked with“*”in Fig.2.Obviously,the data points with“*”show a larger scattering, suggesting that the indirect measurement methods and defective samples may be the reason for data scattering.

2.3. Shear strength

Only ten experimental data on the shear strength and impedance of rock were found in the work of Ersoy and Atici(2007).Based on the ten datasets, a relation between shear strength σsand impedance of rock was found (see Fig. 3) as

where a3=1.24. This equation indicates that the shear strength is directly proportional to the impedance. Since the experimental data of shear strength are few,more data are needed to verify this relation in the future.

2.4. Mode I fracture toughness

Previous studies showed that the mode I fracture toughness of rock is related to the following parameters:(1)P-wave velocity(e.g.Huang and Wang,1985; Alber, 2008), (2) density (e.g. Brown and Reddish, 1997; Alber, 2008), (3) tensile strength (e.g. Zhang,2002; Alber, 2008), and (4) compressive strength and point-load index (e.g. Gunsallus and Kulhawy, 1984). This implies that the fracture toughness of the rock might be related to the impedance.

By using 37 experimental data(Alber,2008;Zhang et al.,2017a),we found that mode I fracture toughness KIcis related to the impedance of rock (see Fig. 4) as

where a4= 0.005 and b4= 0.02. This equation indicates that the mode I fracture toughness increases with increasing impedance of rock. Note that only static fracture toughness was selected from Zhang et al. (2017a).

Previous studies indicated that rock fracture toughness was influenced by many factors such as mineral composition,grain size,crack geometry (Huang and Wang,1985), applied stress level and crack geometry(Gunsallus and Kulhawy,1984),and anisotropy and cleavage dominance (Brown and Reddish,1997).

2.5. Young’s modulus

Previous experimental studies indicated that the Young’s modulus is related to the P-wave velocity of rock (e.g. Yasar and Erdogan, 2004; Kahraman and Alber, 2006; Ersoy and Atici, 2007;Khandelwal and Singh, 2009; Yagiz, 2011; Kurtulus et al., 2012;Wen et al., 2019). In addition, the Young’s modulus and the electrical resistivity were also correlated (Su and Momayez, 2017). By using 361 experimental data on various igneous,metamorphic and sedimentary rocks(Kahraman et al.,2000;Singh and Dubey,2000;Moradian and Behnia, 2009; Aalizad and Rashidinejad, 2012;Coggan et al., 2013; Siratovich et al., 2014; Najibi et al., 2015;Madhubabu et al.,2016;Sajid et al.,2016;ündül,2016;Awang et al.,2017; Brisevac et al.,2017;Mahanta et al.,2017; Su and Momayez,2017; Zhang et al.,2017a,b; Ali and Othman,2018; Talukdar et al.,2018; Wen et al.,2019),a relation between the Young’s modulus E and impedance of rock was determined (see Fig. 5) as

Fig. 2. Tensile strength vs. characteristic impedance of rock.

where a5= 0.17 and b5= 0.8.Eq.(5)indicates that the Young’s modulus increases with increase in the impedance. Note that rock samples in Siratovich et al. (2014) had micro-cracks, and bedding planes in Su and Momayez (2017). The granite samples were hydrothermally treated before testing (Coggan et al., 2013).In addition, the rock samples in Singh and Dubey (2000) were cubical rather than cylindrical. All these may influence the testing results.

2.6. Poisson’s ratio

It was found that the Poisson’s ratio and P-wave velocity of rocks are correlated (Khandelwal and Singh, 2009; Wen et al., 2019). By using 117 collected experimental data (Mahanta et al., 2017; Wen et al.,2019; Siratovich et al.,2014; Madhubabu et al.,2016; ündül,2016;Singh and Dubey,2000),the Poisson’s ratio ν was found to be related to the impedance of the rock(see Fig. 6) and written as where a6= 0.021.Eq.(6)shows that the Poisson’s ratio increases with the increase in impedance. It was found that the Poisson’s ratio was strongly affected by the presence of cracks and anisotropy of rocks(Ide,1936;Ring,1983).Note that in Fig.6,the rock samples from Siratovich et al.(2014)had cracks and the samples from Singh and Dubey (2000) were cubical.

Fig. 3. Shear strength vs. characteristic impedance of rock.

Fig. 4. Mode I fracture toughness vs. characteristic impedance of rock.

Fig. 5. Young’s modulus vs. characteristic impedance of rock.

3. Discussion

3.1. Definition of characteristic impedance

Under one-dimensional (1D) shock wave loading, the characteristic impedance Zshockof a material is defined as (e.g. Cooper,1996; Zhang, 2016):

where ρ is the density and D is the shock wave velocity of the material. This characteristic impedance plays an important role in solving problems of shock waves.For example,when a shock wave travels from low-impedance material to high-impedance one, the shock pressure in the latter is greater than that in the former. In other words, the shock wave is strengthened. In contrast, when a shock wave travels from high-impedance material to lowimpedance one, the pressure induced in the latter is less than that in the original shock wave, that is, the shock pressure is weakened(Zhang,2016).Note that since the velocity of shock wave depends on that of the particle in the shock wave,the impedance of a material under shock wave loading is a variable parameter rather than a constant.

Under the 1D elastic stress waves,the characteristic impedance Z of a material is defined as (e.g. Kolsky,1963; Zhang, 2016):

where c is the sonic velocity of the material.Unlike the impedance under shock wave loading, the impedance of a material under elastic waves is constant since the sonic velocity of a material is constant. Under the 1D elastic waves, we have

where v is the particle velocity, and σ is the stress. Eq. (9) can be rewritten as

Eq.(10)indicates that the characteristic impedance of a material refers to the stress caused by particle velocity per unit in the material.This can be also found from the unit of ρc,i.e.(N／m2)／(m／s),where N／m2is the unit of stress and m／s is the unit of velocity. If one material has higher impedance, the stress caused by particle velocity per unit in the material will be greater, and vice versa.Thus, it is possible that the impedance defined in Eq. (8) can be taken as a parameter to classify rocks, although its definition is based on 1D elastic wave theory. The first reason is that the impedance is a product of the sonic velocity and the density rather than a single parameter, such as sonic velocity or density, suggesting that the impedance may be able to represent rocks better.The second one is that, in general, the structure of rock affects the sonic velocity;mineral components influence the density.The third is that both the sonic velocity and density of rock can be determined by non-destructive methods,even in the field.In particular,the sonic velocity of a rock mass can be easily determined by means of seismic systems in underground mines, tunnels and other underground spaces.

Fig. 6. Poisson’s ratio vs. characteristic impedance of rock.

In general, because the P-wave velocity cpand the shear wave velocity csare theoretically related to each other(e.g.Zhang,2016),

Although the characteristic impedance of rock is a product of the P-wave velocity and the density,P-wave velocity itself depends on several factors,i.e.joints,faults,bedding,and mineral composition.Therefore, the empirical equations proposed in this study can be used to provide a fast and approximate estimate when there are limits to perform robust testing for determination of impedance,especially at the early stage of small-to medium-scale rock engineering projects.

3.2. General relation between impedance and mechanical properties

Empirical equations between impedance and uniaxial compressive strength, tensile strength and shear strength are linear,implying that these three strengths are correlated with each other.This is consistent with the empirical equations among these three rock strengths, as mentioned at the beginning of this paper.Theoretically, it is necessary to investigate why the strengths and the impedance of rock are related.

In theory,the velocity of a sonic wave(P-or S-wave)is related to the Young’s modulus and Poisson’s ratio (e.g. Kolsky,1963; Zhang,2016), as follows:

According to these equations, it can be found that impedance,either ρcpor ρcs, is related to both Young’s modulus and Poisson’s ratio through a complicated mathematical function,indicating that it is necessary to further understand the relationships between impedance and Young’s modulus/Poisson’s ratio.

It is worth noticing that the experimental data collected for shear strength, fracture toughness, and Poisson’s ratio of rock are not sufficient. Thus, more experimental data are needed for these properties. Furthermore, the reason that the impedance is related to fracture toughness needs further investigations.

3.3. Applications

Impedance can be used to classify small-scale rocks, i.e. rock samples used in laboratory tests,since impedance is well related to the mechanical properties of small-scale rocks in this study. Technically,it is possible to measure the density and sonic velocity of the rock mass by non-destructive methods in the field. Thus, it is possible to determine the impedance of rock masses in a mine, a tunnel, or an underground opening. As a result, it is possible to classify rock masses by impedance.

If a similar relation exists between the impedance and the strength of in situ rock mass, the latter can be estimated. If so, an optimum design for rock drilling, rock blasting, rock support, and mining production could be realized. It could also be possible to predict rockbursts in rocks if the rockbursts are found to be correlated with the impedance of the rock mass. However, one issue concerning the above discussions is how to know and determine the strength of in situ rock mass. This is worthy of studying in the future.

4. Conclusions

Based on a great number of experimental data collected from previous studies, the following conclusions can be drawn:

(1) The impedance of rock is related to the main mechanical properties of rock.Therefore,the impedance can be taken as an index measurement to characterize and classify small-size rocks.

(2) The mechanical properties (i.e. uniaxial compressive strength, tensile strength, shear strength, mode I fracture toughness, Young’s modulus, and Poisson’s ratio) of rocks increase with the increase in impedance.

(3) The relations obtained can be used to estimate mechanical properties of rock,if the impedance is known.To avoid errors in estimate of rock properties from the relations proposed in this study, it is recommended that the empirical equations are used to estimate rock properties within the range of which they have been developed.

(4) It is possible to classify rock masses by means of their impedances.However,to confirm this,a great number of largesize experiments, theoretical studies, field measurements and validations are needed.

Declaration of competing interest

The authors declare that they have no known competing fniancial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This study was carried out in the University of Oulu.The support from China Scholarship Council(CSC)(Grant No.201706430058)to the second author is acknowledged.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2020.05.006.