Onset of Crack Initiation in Uniaxial and Triaxial Compression Tests of Dolomite Samples

Abstract The paper presents results of laboratory investigation and analysis of crack initiation threshold identification of dolomite samples. First, selected methods for determining crack initiation thresholds are briefly described with special attention paid to four methods: crack volume strain method [14], change in Poisson’s ratio [8], lateral strain response method [16], and dilatancy method [4]. The investigation performed on dolomite samples shows that for the uniaxial and conventional triaxial compression tests, the above mentioned methods give quite similar values, except for the crack volume strain method. Crack initiation threshold determined by this method has a distinctively lower value than that obtained by the other methods. The aim of the present paper was to review and assess these methods for identifying crack initiation threshold based on laboratory tests of dolomite samples.


INTRODUCTION
Existing, natural fractures and flaws in the rock sample and created under loading conditions play crucial role in the brittle damage process in uniaxial and triaxial compression tests.Micro and macro fracture process is a random phenomenon, identified as a development of micro cracks, which arranging into groups, causes a formation of macro crack or a certain zone of cracks [15], [18].Fracture process can be observed as a nonlinear effect visible on stress-strain characteristics in the unconfined and confined tests [2]- [4], [10].These effects are caused by initial porosity as well as inhomogeneity.Brace et al. [4] and Kwaśniewski [11], [13] show that in the stress-strain characteristics (Fig. 1) one can find thresholds and stages describing: • crack closure, observed in the axial σ 1 or differential (σ 1 -σ 3 ) stress-axial strain ε 1 relation at the beginning of loading process up to its linearity, • crack initiation or onset of dilatancy (OD), observed in the volumetric strain ε V characteristic (threshold of relative dilatancy), • unstable crack growth (TD) understood as threshold of absolute dilatancy, when the volumetric strain ε V has a maximum value, This phenomenon is visible in both unconfined and confined compression tests.Both thresholds play im-portant role in the interpretation of rock behavior and define loading conditions for the stable and unstable crack propagation.Both thresholds are useful for the identification of rock parameters for constitutive models [5], [6], they can be interpreted as precursors of earthquakes, or mining-induced rock bursts [12], [13], and as reported by Andersson et al. [1], can be used to investigate the onset of spalling (cracking) in a fractured rock mass.and failure stress (σ F ). Compaction has a positive sign (after Kwaśniewski [11], [13]) Identification of unstable crack growth threshold (threshold of absolute dilatancy) makes no difficulty, is unique and precise, but identification of the onset of dilatancy is usually not easy, especially in porous, initially cracked rocks.In last years various authors have proposed different methods for crack initiation threshold identification [16], for example: Martin and Chandler [14] -crack volume strain method; Diederichs [8] -the change in Poisson's ratio as an indicator for establishing crack initiation threshold; Nicksiar and Martin [16] lateral strain response method (LSR), Eberhardt et al. [9] and Diederichs et al. [7] -acoustic emission method.
In this paper, results of the first three methods are compared to the threshold of relative dilatancy obtained for dolomite samples under uniaxial and conventional triaxial loading conditions.

CRACK INITIATION THRESHOLD DETERMINATION 2.1. ONSET OF DILATANCY METHOD
In the case of brittle rocks dilatancy refers to the development of the volume change during the inelastic deformation, under applied deviatoric stress [17].In brittle rocks it is caused mainly by microcracking but other mechanisms and models are also possible [11].
Crack initiation threshold is visible on the axial (or differential) stress-volumetric strain curve (Fig. 2) when it diverges from the straight line [4].In practice small deviation of the stress-volumetric strain curve from the straight line can make some difficulties to define one point determining the threshold of relative dilatancy (OD).

CRACK VOLUMETRIC STRAIN METHOD (CVS)
Martin and Chandler [14] proposed that crack initiation could be determined using a plot of crack volumetric strain versus axial strain (Fig. 3).Crack volumetric strain ε Vcr is calculated as a difference of the elastic volumetric strain ε Vel and volumetric strain ε V determined in the test, where ε 1 , ε 2 -axial and lateral strain, σ 1 , σ 2 = σ 3 -axial and confining stress, E, v -Young's modulus and Poisson's ratio.Special care should be taken when Poisson's ratio and Young's modulus are determined for this method [9].Crack volumetric strain is calculated on the basis of these two elastic constants and is strongly sensitive to its value.This is probably why this method does not give objective values.

CHANGE OF POISSON'S RATIO METHOD (PR)
Diederichs [8] proposed a method of crack initiation threshold identification based on change of Poisson's ratio.The onset of crack initiation can be identified by analysis of the relation of Poisson's ratio, evaluated locally, to the log of the axial stress (Fig. 4).

LATERAL STRAIN RESPONSE METHOD (LSR)
Following the Nicksiar and Martin [16] the methodology for identification of crack initiation threshold with the LSR method can be described in points: 1. Determine the threshold of absolute dilatancy TD (Fig. 2).
2. Determine the linear lateral strain reference line and plot the axial stress-lateral strain curve (Fig. 5a).
3. Calculate the difference in lateral strain between the loading and linear reference line (ΔLSR).

RESULTS OF THE CRACK INITIATION TRESHOLD IDENTIFICATION FOR DOLOMITE SAMPLES
Results of the ultimate strength σ F and thresholds of the relative OD and absolute TD dilatancy, for confined and unconfined tests are presented in Fig. 6.
In the conventional triaxial compression tests effect of coffining pressure on the ultimate strength σ F is clearly visible (Fig. 6a).Ultimate strength of dolomite samples increases, following the rise of the confining pressure value.The threshold of absolute dilatancy (TD) is also dependent on confining pressure similar to the ultimate strength.The values of the dif-ferential stress corresponding to the threshold of absolute dilatancy (TD) normalized by ultimate strength σ F (Fig. 6b) are close to 1 for uniaxial compression tests and 0.9 for confined tests.This means that in the case of uniaxial compression samples failure occurs immediately after absolute dilatancy threshold is reached.Their relative dilatancy threshold is identified as about 0.6 value of ultimate strength in both confined and unconfined tests and has constant value.
For the uniaxial and triaxial compression tests crack initiation threshold determined by all the methods (Fig. 7), excluding results obtained by crack volume strain method (CVS), are in the range 0.4-0.8 of ultimate strength.Results obtained by dilatancy method (OD) and the lateral strain response method It should be noted that crack volumetric strain is calculated on the basis of two elastic constants Poisson's ratio and Young's modulus determined for a sample [9].Their identification, especially Poison's ratio is quite problematic and subjective.This is probably the reason why this method does not give objective values.

CONCLUSIONS
The paper presents results of the application of four methods of crack initiation threshold determination in uniaxial and triaxial compression tests of dolomite samples.
Identification of unstable crack growth threshold (threshold of absolute dilatancy) usually makes no difficulty and is precise.In the case of the onset of crack initiation (threshold of relative dilatancy) identification is not easy, especially in porous, initially cracked rocks.Results of crack initiation threshold obtained by dilatacy method (OD) and the lateral strain response (LSR) method have similar values, but the LSR method eliminates user's judgment.It could be greatly advantageous for investigation of porous and cracked rocks.

Fig. 2 .
Fig. 2. Axial stress-volumetric strain curve with the threshold of relative (OD) and absolute (TD) dilatancy and failure stress σ F for dolomite sample (uniaxial compression case)

Fig. 5 .
Fig. 5. Methodology of identification of the onset of crack initiation with LSR method (uniaxial compression case)

Fig. 6 .Fig. 7 .
Fig. 6.Effect of confining pressure on the ultimate strength σ F and the threshold of relative (OD) and absolute dilatancy (TD)