 Original Paper
 Open Access
Elastic waves in fractured rocks under periodic compression
 Elena L. Kossovich^{1}Email authorView ORCID ID profile,
 Alexey V. Talonov^{1, 2} and
 Viktoria L. Savatorova^{3}
https://doi.org/10.1186/s4071201700808
© The Author(s) 2017
Received: 10 September 2016
Accepted: 15 May 2017
Published: 5 June 2017
Abstract
Background
One of the current problems in studying the mechanical properties and behavior of structurally inhomogeneous media with cracks is the characterization of acoustic wave propagation. This is especially important in Geomechanics and prognosis of earthquakes.
Methods
In this work, the authors propose an approach that could simplify characterization of wave propagation in medium with cracks. It is based on homogenization procedure performed at a set of equations characterizing acoustic wave propagation in media weakened by fractures under condition of external distributed loading. Such kind of loading in most cases is close to the real one in case of consideration of Geomechanics problems.
Results
On the basis of the proposed homogenization technique, we performed characterization of elastic properties and plane acoustic waves propagation in a preloaded linear elastic medium weakened by a large amount of cracks. We have investigated two special cases of loading: uniaxial compression and complex compression. We have also studied how the wavespeeds depend on averaged concentration and distribution of craks.
Conclusions
Effective elastic properties were theoretically characterized for fractured media under external loading. The results revealed high dependency of the longitudinal wave propagation speed on the relation between stresses reasoned by an external loading.
Keywords
 Fractured medium
 Acoustic wave speed
 Effective elastic properties
 Homogenization
 Periodically distributed loading
 Compression
Background
Studying of acoustic wave propagation in structurally inhomogeneous media weakened by a large amount of fractures nowadays is a current problem for characterization of Earth shell, earthquakes prognosis, and mining. A complex hierarchical structure of geomaterials’ inhomogeneities became a reason for consideration of simplified models of acoustic wave propagation in such materials.
In the current work, authors propose characterization of plane wave propagation in preloaded linear elastic inhomogeneous media weakened by a large amount of isolated fractures. The existence of cracks makes the material to be highly heterogeneous with multiscale structure. Theoretical study of wave propagation in bodies and media with fractures was presented in many research papers (Schoenberg and Sayers 1995; Stiller and Wagner 1979; Schubnel and Gueguen 2003; Kachanov 1980). For example, in (Stiller and Wagner 1979), in order to obtain elastic wave velocities in fractured preloaded medium, the effective elastic moduli were used obtained for the case of uniaxial compression, and alteration of wave characteristics of the aforementioned medium was explained by fracturing effects. In Schoenberg and Sayers (1995); Schubnel and Gueguen (2003), plane wave propagation was studied in elastic media weakened by open fractures; this included problems on effects of anisotropy of open crack distribution at wave propagation speeds and decay characteristics. Numerical studies were performed by Zhang (2005); Chung et al. (2016) and other. In this research, the authors applied a linearslip displacementdiscontinuity model where fracture is assumed to have a vanishing width across which the tractions are taken to be continuous; however, displacements can be discontinuous. The novelty of our approach is the fact that we take into account the friction between the crack faces. Analysis of deformation properties of brittle materials revealed (Talonov and Tulinov 1988) that the characteristics of media weakened by fractures with edges interacting under compression stress fields depend on the stressstrain state at the stage prior to crack growth initiation. Therefore, the elastic medium weakened by fractures with interacting edges becomes anisotropic even at homogeneous and isotropic fracture distribution. In accordance with all the mentioned above, here, we propose investigation of wave propagation features in media weakened by fractures under condition of external periodically distributed loading.
Methods
Here, t is the time, x _{ k } is a coordinate (k=1,2,3), u _{ i } is a component of a displacement vector (i=1,2,3), and ρ is the density of medium. Also, we assume that the stress tensor components arising due to wave propagation in the medium are small in comparison with the external stress.
where \(\varepsilon _{ik}^{0}\) are the components of strain tensor for a solid elastic material without cracks, n _{ i } are the components of vectors normal to the faces of cracks, F(Z) is the function of set of parameters Z such as characteristic size of crack R and its orientation, and U _{ i } are the components of the jump of displacement averaged over the faces of cracks.
where δ _{ lk } is the Kronecker symbol.
The components of tensor A for the case of an axillary symmetric stress and isolated randomly oriented closed pennyshaped cracks with Coulomb friction are presented in Appendix 1.
On the basis of the problem stated above, we present an approach that is useful for characterization of effective elastic properties and acoustic wave propagation in the considered media with randomly distributed fractures under the condition of external loading.
Results and discussion
Elastic property characterization
Unlike the components of tensor S, components of tensor A depend on the external stress and, as a result, under the condition of external load effects, the initially isotropic media may become anisotropic. Along with this, the deformational properties of the media with closed fractures may also be altered at change of the external stress field. Regarding these facts, we consider a case when, under the conditions of a complex stressed state (σ _{33}<σ _{22}=σ _{11}≤0), one of the components of the external stress, for example, σ _{33}, will be a periodical function with respect to coordinates. Then, in accordance with the relations presented in Appendix 1, the deformational characteristics of the media with isolated closed fractures are also periodical functions of coordinates. In order to characterize wave propagation in the media with periodically distributed elastic characteristics, a method could be used developed in works SanchezPalencia (1980); Chen and Fish (2000); Andrianov et al. (2008); Bakhvalov and Panasenko (2012). Authors of (Chen and Fish 2000; Andrianov et al. 2008) applied this technique to homogenization of wave propagation in periodic composite materials.
In the current work, authors propose to utilize a technique to homogenization of wave propagation for characterization of the inhomogeneous external stress field effects at wave properties of the medium weakened by a large amoung of distributed microfractures.
The components of tensor C for the case of an axillary symmetric stress and isolated randomly oriented closed pennyshaped cracks are presented in Appendix 2.
Acoustic wave propagation characterization

“Slow” coordinate x, giving the general location of the point

“Fast” coordinate ξ∈Y, the location of the point within the periodic cell
Coordinates x={x _{1},x _{2},x _{3}} and ξ={ξ _{1},ξ _{2},ξ _{3}} are related by ξ=x/α (α≪1). All the quantities will, in general, be functions of coordinates x and ξ.
Assume that, according to the aforementioned conclusions, the expansions (9) are Yperiodic in ξ.
We substitute series (9) into the problem (8) and collect all factors at identical powers of α getting recurrent chain of equations.
where Y is volume of a periodic cell Y.
Homogenization of the recurrent chain of equations gives us the following results:
Later, \(u_{i}^{(0)}\) will be defined as a solution of the homogenized macroscopic equation.
In Eq. (16), N _{ ijl }(ξ) are ξperiodic solutions of the cell problem.
where \(m_{3}=\cos \theta, m_{1}=\sin \theta \cos \phi, m_{2}=\sin \theta \sin \phi \) (see Fig. 1 for angles ϕ and θ), \(\overrightarrow {k}\) is the wave vector, and \(k=\left \vert \overrightarrow {k}\right \vert =\frac {2\pi }{\lambda }\) is the wavenumber.
In order to investigate the inhomogeneity of external stress field distribution effects on the acoustic wave speed, we considered a case of σ _{33} distribution as shown in Fig. 2. Stress σ _{33} continuously varied in the interval between σ ^{∗}−σ _{0} and σ ^{∗}+σ _{0} with period l.
Results on variation of the longitudinal wave speed with respect to changing of values of σ ^{∗} and unaltered σ _{0} are shown in Fig. 3. It is obvious that the average value of σ _{33} significantly effects the change in the longitudinal wave velocity. At the same time, at the change of the amplitude σ _{0} of variation of σ _{33}, the longitudinal wave speed changes only in quite narrow range of σ _{11}.
Conclusions
We have considered the propagation of plane waves in a preloaded linear elastic medium weakened by a large number of cracks with periodic function of fracturing degree. The cracks were considered to be closed pennyshaped, isolated, randomly oriented. The model incorporates Coulomb friction between the faces of cracks. A homogenization technique is used to obtain a macroscopic equation for the case of plane wave propagation in effectively elastic media weakened by cracks. We have investigated lowfrequency case where wavelength exceeds the characteristic size of heterogeneities. For two special cases of loading (uniaxial compression and complex compression with σ _{33}<σ _{11}=σ _{22}≤0), we have characterized (theoretically) the effective elastic properties of the media. In terms of plane wave propagation through the fractured media, we mostly concentrated at speeds of longitudinal waves. We have studied how wave speeds depend on averaged concentration and distribution of cracks and changing of external load.
Appendix 1: evaluating components of tensor A

Case 1: uniaxial compression with σ _{33}≠0, σ _{11}=σ _{22}=0. In this case, α _{1} and α _{2} do not depend on stress and can be evaluated as (see Talonov and Tulinov 1988)$$ \alpha_{1}=\frac{\pi}{2},\;\alpha_{2}=\beta=\arctan\mu. $$(27)

Case 2: a complex stress with σ _{33}≠0, σ _{11}=σ _{22}≠0. In this case, α _{1} and α _{2} depend on stress and can be written as (see (Talonov and Tulinov 1988))$$ \alpha_{1}=\frac{\beta}{2}+\frac{1}{2}\arcsin\left(\sin \beta\cdot\frac{\gamma+1}{\gamma1}\right), $$(28)$$ \alpha_{2}=\frac{\pi}{2}+\beta\alpha_{1}=\frac{\pi}{2}+\frac{\beta}{2}\frac{1}{2}\arcsin\left(\sin \beta\cdot\frac{\gamma+1}{\gamma1}\right), $$(29)where$$ \beta=\arctan \mu,\; \gamma=\frac{\sigma_{33}}{\sigma_{11}},\;\gamma>\frac{1+\sin\beta}{1\sin\beta}. $$(30)
Appendix 2: nonzero dimensionless averaged components of tensor C
Declarations
Funding
The authors gratefully acknowledge the financial support of the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISiS”(K42014085).
Authors’ contributions
AV and VL worked on the problem statement, model derivation, and development of homogenization technique. EL developed the software for obtaining the graphical results of the manuscript and performed the numerical experiments for the characterization of the longitudinal wave speed velocity variation. AV, VL, and EL participated in the analysis and discussion of results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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