Relationship Between Pre-failure and Post-failure ... - Springer Link

Rock Mech Rock Eng (2015) 48:121–141 DOI 10.1007/s00603-014-0549-1

ORIGINAL PAPER

Relationship Between Pre-failure and Post-failure Mechanical Properties of Rock Material of Different Origin ¨ ge Levent Tutluog˘lu • I˙brahim Ferid O Celal Karpuz

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Received: 19 September 2013 / Accepted: 18 January 2014 / Published online: 12 February 2014 Ó Springer-Verlag Wien 2014

Abstract Under compression, gathering data related to the post-failure part of the stress–strain curve requires stiff servo-controlled testing systems. In unconfined conditions, data related to the post-peak region of the intact rock parameters are not common as pre-peak and peak state parameters of stress–strain behavior. For problems involving rock in the failed state around structures, proper choice of plastic constitutive laws and post-failure parameters is important for the modeling of the failed state. The aim is to relate commonly used intact rock parameters of pre-failure (tangent modulus Ei and secant modulus Es) and peak strength (rci) states to parameters of the post-failure state under unconfined compression. Post-failure parameters are the drop modulus (Dpf), representing the slope of the falling portion in brittle state, residual strength (rcr), and dilatancy angle (w°). Complete stress–strain curves were generated for various intact rock of different origin. Seventy-three post-failure tests were conducted. Samples included in the testing program were chosen to represent rocks of different origin. Specimens of granite, rhyodacite, dunite, quartzite series, glauberite, argillite, marl, and lignite were used in the tests. The results from the pre-failure and peak state testing parts were processed and compared to the post-failure stress–strain parameters. For the estimation of post-failure parameters in terms of the pre-peak

¨ ge  C. Karpuz L. Tutluog˘lu (&)  ˙I. F. O Department of Mining Engineering, Middle East Technical ¨ niversiteler Mahallesi, Dumlupınar Bulvarı No:1, University, U 06800 C¸ankaya, Ankara, Turkey e-mail: [email protected] ¨ ge I˙. F. O e-mail: [email protected] C. Karpuz e-mail: [email protected]

and peak states, the functional relations were assessed. It was found that the drop modulus Dpf increases with rock strength rci, following a power function with an approximate power of two. With an exponential trend, the Dpf/Es ratio increases with decreasing Ei/rci ratio. Relations estimating the residual strength and dilatancy from the prepeak and peak state parameters are in logarithmic and exponential functional forms, respectively. Keywords Rock deformability  Rock strength  Post-failure  Stress–strain curve  Brittle state  Dilation

1 Introduction With the development of stiff testing machines in the 1960s and 1970s, it became possible to obtain information about the post-peak failure state parameters of rocks of especially highly brittle nature. Rock testing with these testing machines provided the complete stress–strain curves under compression. Then, it was possible to process and use the previously unknown information on the behavior of rocks under compression at the post-peak failure state. For some rock engineering problems, properties related to the post-peak state part of the stress–strain behavior are important. Hudson and Harrison (1997) pointed out that, in situ, the high stresses that can lead to the material entering the post-peak region either occur directly, as a result of excavation, or indirectly at the corners and edges of rock blocks which have been disturbed by the process of excavation. Estimates of the strength and deformability characteristics of rock masses are required for the analysis of underground excavations (Crowder and Bawden 2004). The formation of plastic or post-failure state regions around underground structures is sometimes unavoidable;

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design procedures and supporting systems are then to be modified considering the existence of these regions. In mining applications, the safe and optimum design of supporting pillars is not only based on the peak pillar strength, but also on the post-failure behavior, particularly in coal mining and the mining of valuable ores in deep levels of underground mines. Parameters related to the post-failure portion of the complete stress–strain behavior play a major role in such design efforts. The best approach for the estimation of parameters related to the post-failure part of the stress–strain behavior of supporting pillar elements is to conduct large-scale in-situ compression tests on such pillars (Jaiswal and Shrivastva 2009). In-situ large-scale complete stress–strain tests on relatively large rock blocks or on supporting structural pillars were conducted in the past by several researchers (Bieniawski 1968, 1969; Cook et al. 1971; Wagner 1974; Van Heerden 1975). However, large-scale tests were not always practical and economical, considering that such tests were difficult to set up, time consuming, and rather expensive. In the design of tunnels and other underground excavations, a common method of analyses nowadays involves the use of numerical modeling as an important part of the design procedure. The numerical modeling of critical-state structural problems for stress and deformation analyses requires the appropriate choice of constitutive laws for the stress–strain behavior. Introducing the constitutive laws to the models properly with accurate values of the related input parameters representing the elastic and post-failure states increases the quality of the results for an appropriate modeling of the rock mass surrounding the problem region. Numerical modeling programs commonly used in geoengineering applications require input parameters for the pre-failure state, peak failure state or yield state, and postfailure state response of the ground around the structure. The pre-failure state stress–strain response is normally expected to follow a linear elastic trend. The slope of the axial stress–axial strain curve for this trend yields the elastic modulus. If the stress–strain curve follows a linear fashion, significant differences are not observed between the tangent and secant moduli obtained from the slope of the curves. This is usually the case for relatively homogenous fine-grained hard rock types of igneous origin. However, the nonlinear stress–strain response may become dominant as the origin of rock samples changes to metamorphic and sedimentary. Also, inhomogeneous igneous rock may present nonlinear stress–strain behavior. Nonlinear behavior may become more dominant for rock types located at the weak rock side which is at the verge of transition of the origin to soil-like material. Then, the slope of the tangent taken around mid-levels of the stress–strain curve can be accepted to represent the modulus of linear elasticity or Young’s modulus. The slope of a

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line drawn from the zero load state to the load level of the peak failure state can be named as the secant modulus of the stress–strain response. The pre-failure nonlinear nature of deformation response is best represented in this modulus. The secant modulus defined this way is believed to involve the effects of irreversible changes in the internal structure of tested rock. Irreversible changes may include the nonlinearities caused by the initial closure of pores and cracks or microcracking and shearing close to the yield state of the rock. For peak failure state or yield analyses, numerical modeling programs ask the user to choose the appropriate yield function to represent the peak failure sate. Yield function choices like Mohr–Coulomb, Drucker–Prager, and Hoek–Brown are commonly available in the input modules of modeling programs. In a design process or modeling analysis focusing on the elastic and peak failure states only, factors of safety and displacement distributions are provided in the results based on the choice of the yield function. Stress and deformation modeling of the post-failure state requires parameters related to the constitutive laws of plasticity and strain softening. The residual strength is either directly requested or related strength parameters represented by the residual cohesion and internal friction angle, and the dilatancy angle is requested as an input (Phase2 of Rocscience Inc. 2012). The finite difference program FLAC requires the definition of the functional forms related to the decay of cohesion, friction angle, and dilatancy angle with increasing plastic strain and postfailure deformations around the structures (Itasca Consulting Group Inc. 2005). The plastic state is characterized by a dilatancy angle input requested in Plaxis program for the Mohr–Coulomb yield state (Plaxis 2010). Using stiff loading machines equipped with servohydraulic closed-loop electronic and hydraulic systems, testing for the quantification of post-failure parameters has been possible for almost 60 years. In conventional testing programs for rock engineering projects, the measurement of post-failure state parameters is not a common practice, as it is time and effort consuming, with the requirement of sophisticated testing systems of high standards. The aim of this study is to develop relations between pre-failure rock parameters that are conveniently available from simple testing and post-failure strength and deformability parameters of various intact rock of different origin. Unconfined compression tests are conducted on different rock types and complete stress–strain curves are test fitted until a clearly defined state of residual strength is reached. A stiff testing system equipped with servohydraulic closed-loop electronic and hydraulic systems is employed to capture the entire stress–strain behavior. Not all tests can be considered successful in this sense,

Relationship Between Pre-failure and Post-failure Mechanical Properties

considering the complex internal structure of the rock material. Depending on the origin of the rock specimen, obtaining a clear post-failure portion and a well-defined residual state is not always easy. An attempt to minimize this problem involved conducting a large number of tests and individual averaging of the post-failure properties within each rock type group. Then, using the average values of related parameters for eight rock type groups, plots of parameters or combinations of some parameters against each other are generated. Functions are fitted to the curves in the plots and parametric equations are proposed to relate pre-failure and peak-state parameters to postfailure state parameters.

2 Characteristics of the Complete Stress–Strain Response The deformability behavior of rock material can be represented by stress–strain curves. The complete stress–strain curve involves the coverage of deformation response regarding the post-failure state as well. Goodman (1989) discussed mechanisms involved at different stages of a typical complete stress–strain curve for a rock specimen under uniaxial loading. The axial stress versus axial strain response is illustrated in Fig. 1 in terms of the stress deviator and corresponding strain. Several researchers (Goodman 1989; Jaeger et al. 2007; Vermeer and de Borst 1984) commented on the characteristics and nonlinear mechanisms of deformation response regarding different regions of the complete stress– strain curve of a rock sample under compressive loading. Five regions and one peak failure state point are identified in Fig. 1. The mechanisms dominating different regions and the peak state are summarized below. The curve is slightly concave upwards in region I (Fig. 1). Open fissures, cracks, pores, and other defects σ 1,deviatoric

C III Cracking B II

IV macrocracking, brittle

Elastic

A I

V Residual state

Crack closure axial strain

Fig. 1 Illustration of a typical complete stress–strain curve under uniaxial loading (modified from Goodman 1989)

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begin to close; these are the first evidences of nonlinearity in the curves. The rock sample seems to be more deformable, indicated by the low slope of the stress–strain curve. However, loading–unloading shows no significant global indication of irreversible changes in the internal structure. Region II in Fig. 1 commonly shows characteristics of a nearly linear portion as an indication of linear elastic behavior. Loading–unloading results in no significant irreversible changes that are reflected as permanent strain in the horizontal axis. The tangent modulus of elasticity is commonly defined and computed from the slopes of tangents assigned to certain levels of the curves in this region. In this work, the symbol Ei represents the tangent modulus of elasticity of the intact rock computed at 50 % of the peak stress level of the curves. Region III (Fig. 1) is around stress levels above 50 % of the maximum and covers the curves up to the peak failure state or yield point. Major crack formation with a stable propagation state is characteristic of this region. Cracks grow to a finite length with stress increments and stop there, and this is repeated at the next increment of axial stress. Irreversible changes in the internal structure of the rock occur within the region. Pre-peak plasticity or the ductile deformation state associated with significant nonlinearity in the curves starts here. The rock can sustain further permanent deformation without losing load-carrying capacity. This state is the beginning of dilatancy, which is the volume increase dominated by the lateral expansion of the sample. The peak failure state is at the yield point where the slope of the curve decreases to zero. This maximum stress point is marked as point C in Fig. 1. At this point, unstable crack propagation starts, and cracks intersect each other and start forming a major failure plane reaching the boundaries of the specimen. The testing system should be stiffer than the specimen. If the testing system is not stiff, the test will be terminated by a violent explosion of the brittle samples at this point. In uniaxial testing, this point corresponds to the unconfined or uniaxial compressive strength (UCS or rci) of the rock on the stress scale. Different names like post-failure state, plastic state, post-peak state, and strain-softening or -hardening region exist for the description of region IV of Fig. 1. In this state, the slope of the curve is negative for the softening behavior. The material loses its ability to resist or sustain load with increasing deformation or strain. Region IV is characteristic of brittle behavior in which material loses its load-resisting ability with increasing deformation. This part of the stress–strain curve is usually obscured by the instability of the machine–specimen system (violent failure) and mechanisms controlling this are discussed by Jaeger et al. (2007). Here, cracks propagate to the

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124 Fig. 2 Linearized sketches of different parts of stress–strain curves for several post-failure modes

A

Stress (σ)

Brittle Failure Strain Softening Failure Perfectly Plastic Failure Pre-failure region

B

Strain Hardening Failure

Post-failure region

Strain (ε) boundaries of the sample, and the system of intersecting and coalescing cracks in an unstable manner form a fault or failure surface. By Zheng et al. (2005) and Tiwari and Rao (2006), brittle failure was identified to exhibit an abrupt post-peak drop in stress on the stress–strain curve. In the brittle failure type, when a point in a stress space is loaded from its initial elastic state to the peak strength, the stress will drop abruptly to the residual strength. In the strainsoftening or -weakening failure type, a sudden decrease of strength is not observed, as illustrated in Fig. 2. A gradual decrease of strength until the residual state with a finite slope is typical for this failure type (Zheng et al. 2005; Tiwari and Rao 2006). Here, rock material volume starts to increase at a higher rate. Dilatancy develops in the negative sense corresponding to expansion of the rock sample being tested. As a result of the large lateral expansion compared to axial shortening, the material volume can be more than the original volume compared at the beginning of the loading. In Region V, rock material reaches its residual state (marked as B in Fig. 2) and deformations on existing cracks continues under a constant level of axial stress. An exactly constant stress level corresponding to the residual strength state rcr may not always be attained clearly. However, the rate of stress fall of the softening part decreases here, and a tendency to reach the residual state is observed following the flattening of the curve. Portions of stress–strain curves can be idealized in the form of lines. The purpose of such representation is to identify and simplify the related characteristic parameters of that particular portion. Idealizations can be observed in the work of other researchers, such as Vermeer and de Borst (1984), Crowder and Bawden (2004), Zhao and Cai (2010), and Alejano et al. (2009). In Fig. 2, linearized representations can be seen for the pre-peak and various post-peak material behaviors.

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3 Background Related to the Pre- and Post-failure Behavior of Rocks In addition to conventional stress–strain testing in the laboratory, there are other simple tests for the estimation of the pre-failure deformability and yield strength properties of rock. Parametric expressions for the estimation of these parameters are commonly available in the literature. Proposed expressions are based generally on the results of index tests like point load strength index test, Schmidt hammer rebound test, sound velocity test, and impact strength test. Kahraman (2001) compared the correlations of the UCS with the results of predictions based on indirect index tests conducted by numerous researchers. Yagiz (2009) proposed correlations for the estimation of the elastic modulus and UCS based on the results of Schmidt hardness tests on some sedimentary rock types. Although there is considerably more work on index tests for the correlation of strength and pre-failure deformability parameters, there is a lack of effort for the estimation of post-failure deformability parameters from the pre-failure parameters. Palchik and Hatzor (2004) determined the UCS, point load strength, and indirect tensile (Brazilian) strength of a porous chalk formation and studied how porosity influenced the magnitudes of these properties as well as the relationship between these mechanical properties and porosity. Palchik (2013) used 60 carbonate rocks to investigate the connection between porosity, elastic constants, and stress–strain curve parameters and the type of volumetric strain curve. Relations were found between crack damage stress, elastic modulus, modulus ratio (E/rc), porosity, and maximum total volumetric strain. Reported results are not so common for the estimation of post-failure parameters. Research on the post-failure

Relationship Between Pre-failure and Post-failure Mechanical Properties

behavior of rocks usually concentrates on the success of tests on certain brittle rock types; the purpose is to trace the post-failure portion of the stress–strain curve in the brittle region until a clearly defined residual state is observed. Interpretation of the volumetric strain behavior and the deformation mechanisms dominating the sudden release of energy in the post-failure region is the aim in general. Success to obtain post-failure portion and clear residual state in tests on highly brittle rock is usually guaranteed by the use of a confining pressure. Abdullah and Amin (2008) conducted compression tests on sandstones with a conventional and a servo-controlled stiff compression machine. They concluded that a violent failure was not the intrinsic characteristic of a rock, but, rather, due to the rapid release of strain energy in the loading parts of the compression testing system when the sample reached its peak strength. Stiff compression machines were found to be necessary in order to obtain the post-failure curve of the rocks. Li et al. (1998) developed model simulations of nonlinearities of deformation behavior. Nonlinearities caused by crack closure, propagation, and friction on crack surfaces were reflected as changes in Young’s modulus in the pre-failure part of the stress–strain response. An apparent modulus similar to the secant modulus used in the next sections of this work was claimed to be one of the quantitative measures for the crack-based nonlinear deformation response. It is important to review the work and discuss previous efforts trying to relate the pre-peak microstructure and properties to the peak and post-peak state characteristics of stress–strain behavior. Establishing a realistic microstructure for rock with grains, bonding, particles, and pore spaces is a difficult task. To simulate the pre-peak internal microstructure properly, different modeling techniques have dominated the research in this area. Numerical simulations by discrete element method (DEM) models illustrated effectively how the micromechanical parameters might impact the compressive strength of a rock (Cundall and Strack 1979). The bonded-particle model (BPM) by Potyondy and Cundall (2004) reproduced features of rock behavior, including elasticity, fracturing, acoustic emission, and damage accumulation producing material anisotropy, hysteresis, dilation, post-peak softening, and strength increase with confinement. These were emergent properties of the model that were invoked from a relatively simple set of microproperties. A material-genesis procedure and microproperties to represent Lac du Bonnet granite were presented. In Scho¨pfer et al. (2009), the stress drop after failure decreased and the material became less brittle with an increasing number of non-bonded contacts in the microstructure generated by DEM. It was noted that the Young’s

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modulus, strength, and stress drop could be increased by increasing both the particle stiffness and the bond strength without significantly modifying the friction angle and the UCS/tensile strength ratio. By Wang and Tonon (2009), a DEM model was successfully calibrated to match the experimental triaxial test data of Lac du Bonnet granite. Based on microcrack evolution, three distinct stages of the stress–strain curve could be well represented by the accumulated number of contact failures and the mode of contact failures. Recent work continues to concentrate on the micromechanical modeling techniques to establish links between the pre-peak, peak, and post-peak states. The pore-emanated crack model by Baud et al. (2013) showed the relation between the UCS and the initial level of damage and/ or the crack density in a rock. It was demonstrated that pore size as a microstructural parameter strongly influenced this relation. In the three-dimensional particle flow code (PFC3D) model of Ding et al. (2013), the effect of model scale and particle size distribution on the macroscopic mechanical properties, UCS, Young’s modulus, and Poisson’s ratio was emphasized. Joseph and Barron (2003) assumed a second-degree polynomial functional form to express a conceptual apparent friction in terms of the post-peak strain. This concept, with its associated quadratic form for the falling portion of the stress–strain curve, related the post-failure apparent cohesive and frictional characteristics of the rock to the strain. They used triaxial, direct shear, and simple tilt tests to construct peak, residual, and base strength criteria envelopes for a limited number of rock types, like mudstone, siltstone, and coal. Then, they developed a strainsoftening constitutive relationship that described the postfailure stress–strain curve for rock. However, the form of the softening curve is inherently limited to a quadratic form based on the initial assumption. Vermeer and de Borst (1984) emphasized the need for a non-associated plasticity theory for the softening behavior of sand, concrete, and rock. The theory was based on the dilatancy concept, which was described as the change in volume with shear distortion of an element in the material. The dilatancy angle w was accepted to be a suitable parameter for characterizing a dilatant material. For cemented granular materials, the degradation of cohesion was postulated to follow increasing inelastic deformation; plastic deformations tended to localize in shear bands even before the peak strength was reached. For the softening behavior, a non-associated flow potential was suggested by Vermeer and de Borst (1984). The non-associated flow rule potential function resembles the yield function, but it involves a dilatancy angle instead of a friction angle. A plastically volume-preserving material of zero dilatancy gives a different response upon loading than a material

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which exhibits plastic dilation controlled by w. They commented that, for granular soils, rocks, and concrete, the dilatancy angle was significantly smaller than the friction angle; the dilatancy angle was at least 20° smaller than the internal friction angle. Medhurst and Brown (1998) reported the complete stress–strain test results of triaxial compression experiments on relatively large scale coal samples of varying diameter with a range of dimensions from 61 to 300 mm. The volumetric strain behavior was found to be independent of scale after some threshold sample size. Interpretation of the results in the post-failure region was carried out in terms of the gradient of the principal plastic strain increment vector dep1/dep3. Lateral expansion was treated negatively. The volumetric strain versus axial strain responses illustrated decreases in the sample dilation with increasing confining pressure. Plastic volumetric strain increments were inversely proportional to the confining pressure. Gradient values of the plastic strain increment vector varied from approximately -0.2 to -1. At low confining pressures, lateral expansion strain increments were about five times higher than axial strain increments, corresponding approximately to the lower end -0.2 of the gradient. The value dep1/dep3 = -1 represented a physical limit at which a perfectly plastic state and zero volume change occurred. Alejano and Alonso (2005) reinterpreted the post-failure test results of the previous work of several researchers, including the tests results of Medhurst and Brown (1998). High lateral strain increments at low confining pressures above were found to indicate high dilatancy angles, from 50° to 55°. This finding was not completely in agreement with some of the results of Vermeer and de Borst (1984). Alejano and Alonso (2005) showed that the dilatancy angle decreased with increasing confining pressure. The peak dilatancy angle value decreased with increasing plastic strain. They proposed a model to estimate the peak dilatancy angle for a given confining stress level and dilatancy angle decay in line with plasticity. Arzu´a and Alejano (2013) conducted servo-controlled triaxial compression tests on three types of granites. From the stress–strain plots, the drop modulus of the post-failure portion and dilatancy angles were computed. The dilatancy angle was represented as a function of the confining pressure and plastic strain. The UCS of the granites were in the range 75–145 MPa, with dilatancy angles between 50° and 60°. The authors reported that, under uniaxial compression with the absence of confinement, there was a lack of findings due to the extremely brittle nature of the rock samples in the press: the rock crushed in an explosive manner, splitting the sample into fragments. Zhao and Cai (2010) presented literature and observations on rock dilatancy and mentioned the importance of

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the true implementation of dilatancy to the models. They proposed a dilatancy model as a function of confining stress and plastic strain. Again, the higher the confining stress, the lower the dilatancy angle. With increasing plastic strain, the dilatancy angle decayed. However, peak dilatancy angle values greater than 70° were reported under low confining stress levels. The proposed dilatancy angle model was subjected to numerical analysis and verification was done. So far, it has been discussed that the non-associated flow rule is more appropriate for characterizing the softening behavior of rocks and some other materials like sand and concrete. The dilatancy angle is an important parameter in the plastic constitutive laws to relate strain and stress increments. The dilatancy angle can attain quite high values; values can even be greater than the internal friction angle of the material. This highly dilatant behavior is more obvious under low confining stress or under unconfined conditions. Due to the brittle nature of rocks, the determination of dilatancy is difficult under unconfined loading conditions. Work related to dilatancy under unconfined conditions is rare. The estimation of dilatancy is important for investigating the stability of the free faces of excavations like tunnels and slopes. Free faces like the tunnel face, roof, sidewall, and slope faces are either under unconfined conditions or there exists a low confinement due to the action of supporting elements. Therefore, for the estimation of dilatancy, correlation of the pre-peak and peak state parameters is provided in the following sections.

4 Parameters Used for Different States of the Stress– Strain Response Stress–strain test data up to the peak failure state is processed to extract two different slopes. The slope of the tangent of the stress–strain curve at 50 % of the peak failure stress is accepted to represent linear elastic characteristics of the deformation response of the sample. This provides Ei, which is accepted as the elastic modulus of the intact rock sample. On the stress–strain curves, the slope of a line drawn from the point at which the sample takes the load to the point at which it reaches the peak failure state usually illustrates some nonlinearities. These nonlinearities are believed to result from deformations like the closure of pore spaces and cracking in the sample. Such occurrences are irreversible and are believed to be related to the ductile state on the curves. The slope Es for this case is named here as the modulus of deformation of the sample. Es is thought to include inherently the characteristics of nonlinear parts of the stress–strain curve for a particular rock type.

Relationship Between Pre-failure and Post-failure Mechanical Properties

In earth sciences, soil mechanics, and rock mechanics, applications different symbols like qu, C0, and rc are commonly used for the unconfined compressive strength or intact rock UCS. The rock mass strength is commonly expressed by rcm. To emphasize the difference between the strength of the rock material and the rock mass, the symbol rci is adopted here for the UCS of the intact rock. The drop modulus Dpf is defined here to characterize the slope of the falling portion of the post-failure part of the stress–strain curve. M was used to denote the drop modulus in Alejano et al. (2009). Different symbols like Epp (Joseph and Barron 2003) and Epost (Jaiswal and Shrivastva 2009) were used for the post-failure drop modulus. Dpf normally has a negative slope value, since it represents the slope of the falling portion of the stress–strain curve in the postfailure strain-softening region. For simplicity, Dpf is included as an absolute value in the interpretation of the results. Dilation starts with nonlinearities in the pre-failure state with microcracking in the sample. Pre-failure cracking is irreversible and believed to be related to the ductile state on the curves. Material dilation is at its highest rate in the peak and post-peak failure states, and dilation is represented by a dilatancy angle w. The falling portion ends at the residual state region. rcr represents the residual compressive strength. This state is theoretically expected to tend to an elastic perfectly plastic material behavior, which is characterized by constant stress with increasing strain and zero volume change (dilatancy angle w = 0). After determination of the parameters described above, relations among the pre-peak, peak, and post-peak state parameters are investigated. Owing to the difficulties of

Glauberite Quartzite-

Cayirhan, Ankara

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post-failure tests on unconfined specimens, the number of fully successful experiments varies for particular groups of rocks. A fully successful test involves the clear identification of all of the above parameters. In some tests, parameters such as Dpf, rcr, and w may not be obtained clearly. The average value of a parameter is calculated first within each individual rock group. Plots are generated with these average values in order to investigate any possible relation of the pre-peak and peak-state parameters to the post-peak state parameters.

5 Rock Sample Groups The experimental work is designed to cover a wide range of rock types with various strength, deformability, and geological properties. Depending on the origin, rock specimens are categorized under eight major groups. Rock samples are transferred to the laboratory from different parts of Turkey. The locations of the rock groups and samples taken are shown in Fig. 3. 5.1 Rhyodacite Group The rock samples in this group are from a gold mine planned to be situated in Sındırgı, Balıkesir province of western Turkey. Banyard (2010) described this rock unit as having crystal-rich ignimbrite faces with abundant rock fragments in fine ash matrix. The rhyodacite unit among the lavas of various compositions (Oygu¨r 1997) is categorized as an igneous rock type. The effect of alteration was clearly observed on samples; they were rated as moderately to highly weathered.

Argillite Granite

Kure, Kastamonu

series- Istanbul

Erzurum

Lignite Marl Rhyodacite

Kangal, Sivas and

Bigadic, Balikesir

Tufanbeyli, Adana

Sindirgi, Balikesir

Dunite Aladag, Adana

Fig. 3 Locations of different rock groups on a map sketch of Turkey

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5.2 Glauberite Group Samples of the glauberite group are from a sodium sulfate (NaS) mine located at Cayırhan district near to Ankara, Turkey. Evaporate glauberite is a member of the upper Miocene Kirmir Formation of the Beypazari Basin. The deposition in this basin is formed by sedimentation (Ortı´ et al. 2002). Chemical alteration is a characteristic nature of such deposition. However, the samples tested were in an unweathered condition. 5.3 Granite Group Granite samples in this group consist of quartz, hornblende, and plagioclase feldspar. The granite or granitic rocks in the region are in the form of porphyry intrusions (Bozkus¸ 1992). The related region is located at the upper northeast corner of Turkey and samples belong to a state highway project between Artvin and Erzurum. Samples tested were described as being in a moderately weathered condition. 5.4 Quartzite Series Group Quartzite series samples are from the Kurtkoy formation of the Anatolian section of Istanbul. They belong to foundation units of high-rise condominium-type residential construction projects. The rock material in this series shows wide variations in origin and texture, depending upon the depth at which the samples were taken. Indication of the transition of origin from sedimentary to metamorphic is commonly observed. Magmatic and mafic fragments dominantly including feldspar are typical. Color and appearance vary in the form of dark to light tones of magenta and purple, and greenish ash is characteristic for the samples in this group (Ozer 2008). The samples tested in this group were in a slightly weathered condition. 5.5 Dunite Group Samples are taken from a chromite mine in Aladag˘, Adana province, in southern Turkey. Some samples of the dunite group are described to be associated with layered chromite of variable degree of serpentinization (Parlak et al. 2002). The condition of the dunite samples tested varied from slightly weathered to moderately weathered. 5.6 Argillite Group Samples are taken from Asikoy underground mine of Eti Bakir Company located in Ku¨re, Kastamonu province, in northern Turkey. Argillite tested in this work can be considered as a metamorphic rock unit with a high amount of slightly metamorphosed clay. Copper deposits found in

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Ku¨re appear as stockwork-disseminated ore at the upper levels of spillites and as massive lenses between the spillites and argillites (C¸ag˘atay et al. 1982). Rock mass dominated by argillite is heavily fractured and highly weathered in the mine site. 5.7 Marl Group Samples are taken from Simav open pit of Bigadic¸ Borate Mine in Balikesir, Turkey. Marl group is associated with limestone and a high percentage of claystone and mudstone laminations and intercalations (Helvacı and Alaca 1991). Marl samples were in a slightly weathered state during testing. 5.8 Lignite Group Samples of lignite are from the Etyemez and Kangal regions of Sivas, and the Tufanbeyli region of Adana in the mideastern part of Turkey. Samples consist of low-calorie lignite coal having a high content of soft saturated clay of high plasticity. They are from Sivas and Adana provinces of the same coal basin, and their appearances and mechanical properties are very close to each other. Details of the geology of this basin can be found in S¸ en and Sarac¸ (2000), Erik (2010), and Ege and Tonbul (2003). Lignite samples maintained their undisturbed state of high moisture content for a long time, including the testing period.

6 Experimental Work Experiments to generate complete stress–strain curves followed the International Society for Rock Mechanics (ISRM) suggestions (Fairhurst and Hudson 1999) regarding the instrumentation, testing procedures, and data processing. For the rock sample groups, the number of experiments was three for rhyodacite, nine for glauberite, six for granite, 22 for quartzite series, 12 for dunite, three for argillite, 15 for marl, and three for lignite. The number of tests was only three for rhyodacite, argillite, and lignite groups. The relatively low number of tests for these groups was due to the lack of properly sized core samples. Negative effects of alteration and weathering reduced the chance of gathering core samples of acceptable dimensions. Samples of the rhyodacite group were from the shallow exploration boreholes of a planned surface gold mine. Most core samples were fragmented into smaller pieces due to alteration and weathering. Lignite core samples of the required dimensions for proper testing were rare in the borehole sections through the coal-bearing strata. This was because lignite pieces were clayey soillike, being very soft and saturated.

Relationship Between Pre-failure and Post-failure Mechanical Properties

Stress–strain tests are conducted under the uniaxial compression conditions without any lateral confinement. With careful displacement-controlled load application, complete stress–strain curves can be traced successfully to reach a well-defined residual state. Parameters like Es, Ei, and Dpf can be recovered from the slopes of the related parts of the curve. rci is measured as the peak of the axial stress on the curve. Following the strain-softening region, the residual strength rcr can be approximately estimated by observing the flattening trend along the post-failure part of the stress–strain curve; the slope of the descending part is quite high in the softening region, but the rate of falling trend decreases towards the plastic residual state. Lateral strain data are also recorded to observe the lateral circumferential expansion around the specimen. Axial and lateral diametric strain data are used in the interpretation of the data for the estimation of the dilatancy angle w. A 2,800-kN MTS Model 815 Servohydraulic Testing System is used for compressive loading in the tests. This system is designed for high precision in advanced applications such as computerized displacement controlled testing to successfully trace the post-failure part of the stress–strain curves. The system is equipped with a 200-kHz data acquisition system IOtech DaqBook 2000X in order to condition and transfer the A/D or D/A signals in and out of the load and displacement transducers and PCs. An external load cell with a capacity of 500 kN is used for the load measurements. Displacement controlled testing is necessary for detecting the post-failure part of the stress–strain curve. Displacement controlled testing is conducted with an applied displacement rate of 0.005–0.0005 mm/s. The choice of a fast or slow rate depends on the preliminary observations and estimates of the strength and stiffness characteristics of the rock samples. Slower rates are necessary in order to detect post-failure regions of stress–strain curves, especially for brittle rock types. Two vertically aligned strain gage type linear displacement transducers having a maximum range of 1 cm is positioned between the upper and lower loading platens to detect length changes of the sample. For lateral strain measurements, a circumferential chain type sensor arrangement is attached surrounding the sample lateral boundary. This arrangement responds to the change of diameter of the cylindrical core specimen. From the lateral deformation data, the lateral diametric strain is computed and included in the processing of data for the computation of the volumetric strain and dilatancy angle. Analog load and deformation data signals are converted to digital form through the data acquisition system and transferred to the PC. Data are then processed to generate plots of axial stress versus axial and lateral strains.

129

7 Interpretation of Stress–Strain Curves to Determine the Parameters Reaching a well-defined residual state is not always possible, considering the unconfined state of lateral boundaries of the sample. Interruption of an experiment because of complete splitting following uncontrolled loss of strength is the common cause. When a rock sample reaches its residual state, relatively large deformations on existing macrocracks are thought to proceed under an almost constant level of axial stress. It becomes practically impossible to detect large deformations at this state due to the limitation of the circumferential strain measurement apparatus. The constant stress level corresponding to the residual strength state rcr cannot be clearly identified for all tests. The rate of stress drop associated with the softening part decreases usually in this state. Then, a tendency to reach the residual state is sensed accordingly following the flattening trend of the curve. The residual state decision is made based on the observation of this tendency. Nonlinear parts are common along the stress–strain curves, especially for the weak sample groups. Idealization of the slopes and the residual parts in the form of lines is necessary. The purpose of such simplifications is to identify and quantify the related characteristic parameters of that particular portion. Based on the discussions above, some typical examples related to the interpretation of different parts of the curves are presented in Figs. 4, 5, and 6. In Fig. 4, the procedure for the linear approximation and averaging of the slopes is illustrated for a sample selected from the marl group. On this stress–strain curve, nonlinearities of the pre-failure part are reflected in Es and the average Dpf is relatively well defined. The interpretation procedure for identifying rcr is shown in the figure. For some tests, the slope of the curve for the softening part shows fluctuations. A line needs to be fitted to the curve to represent the average, as shown in Fig. 5, which is the result of a stress–strain test on an argillite group sample. Figure 6 presents a typical complete stress–strain curve result of a test on a lignite sample of highly plastic nature. Although a transition to the residual state is not observed clearly, it is still possible to identify the Dpf and rcr values. Due to the plastic nature of the lignite group, Dpf is very low and the residual strength is almost equal to the peak strength. In some of the tests, no residual state data can be obtained. Figure 7 shows a case where a clear residual state cannot be identified. For such cases, the residual compressive strength values are not included in the processing of the data and presentation of the results.

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130 Fig. 4 Example interpretation of a stress–strain test on a sample of the marl group

25 σ ci

axial stress, σ axial , (MPa)

20

15

Dpf Ei

10

σ cr

Es 5

0 0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

axial strain, ε axial

Fig. 5 Example stress–strain curve of a test on a sample of the argillite group

40 σ ci 35

axial stress, σaxial, (MPa)

30 25

σ cr

Dpf

20 15

Ei

10 Es 5 0 0

0.001

0.002

0.003

0.004

0.005

0.006

axial strain, εaxial

Dilatancy response is controlled by the dilatancy angle. Average dilation angles are calculated by interpreting the lateral strain–axial strain results of the unconfined compression tests on different rock groups. In the computation of the dilatancy angle, an expression suggested by Vermeer and de Borst (1984) is used. This expression can be used for calculation of the dilatancy angle under uniaxial or triaxial compressive loading conditions:

123



e_pv w ¼ arcsin 2e_p1 þ e_pv

 ð1Þ

In Eq. 1, the superscript p refers to the plastic or postfailure state parts of the strain components. The symbol e_pv represents the plastic volumetric strain rate and e_p1 is the plastic axial principal strain rate. The dot above the strain components implies the material time derivative. Viscous

Relationship Between Pre-failure and Post-failure Mechanical Properties Fig. 6 Approximation of the drop modulus and residual state; stress–strain test result on a lignite sample of Etyemez

131

0.8

σ ci

Dpf

0.7

axial stress, σ axial, (MPa)

0.6 σ cr

0.5 0.4 Ei

0.3 0.2 Es 0.1 0 0

Fig. 7 Stress–strain test on a sample of the granite group; a residual state is not achieved

0.002

0.004

0.006 axial strain, ε axial

80

0.008

0.01

0.012

σ ci

70

axial stress, σ axial, (MPa)

60 50 Dpf

40 Ei 30 20 Es 10 0 0.000

0.001

0.002

effects are not significant in this work; so, the dotted rate derivative type notation is not necessary. The stress interval Dr is used for the evaluation of the average slopes of the falling portion in the post-peak part of the stress–strain curve. Related strain increments are Depv for the volumetric strain component and Dep1 for the axial principal strain component in this part of the curve.

0.003 0.004 0.005 axial strain, ε axial

0.006

0.007

0.008

In terms of plastic strain increments, Eq. 1 can be modified as:   Depv w ¼ arcsin ð2Þ 2Dep1 þ Depv The principal plastic strain components Dep2 and Dep3 result from the lateral deformation response of the core samples.

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132 Fig. 8 Stress–strain curve of a dunite sample; the lateral diametric strain is on the left of the graph

Axial Stress MPa

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.005

-0.003

-0.001

0.001

ε 3 diametric strain

0.003

0.005

ε 1 axial strain

Fig. 9 Idealized stress–strain plot for the evaluation of slopes

Volumetric strain in terms of the principal strain components can be expressed as: Depv ¼ Dep1 þ Dep2 þ Dep3

ð3Þ

For an unconfined cylindrical sample, Dep2 ¼ Dep3 and the lateral diametric deformation can be included as 2Dep3 in Depv of Eq. 2. In this way, Eq. 2 is simplified as:  p  De1 þ 2Dep3 w ¼ arcsin ð4Þ Dep1 þ 2Dep3 In order to determine the dilatancy response of the tested rock samples, a plot of the lateral (diametric) strain versus axial stress is needed. As an example, a typical lateral diametric strain curve from a test on a relatively weak sample of the dunite group is presented. The lateral diametric strain plot is on the left of the vertical axis, which is the axial stress scale of Fig. 8.

123

An idealized example stress–strain curve is presented in Fig. 9. Slope computations are for a constant interval of stress difference Dr in the softening parts of the curve. This interval is used as an average stress drop for the evaluation of average slopes in the entire softening parts of the axial and lateral strain curves. The slope Ddia pf is defined for the falling portion of the axial stress–lateral strain curve as in Fig. 9. Slopes of softening parts of the axial stress versus axial and lateral strain curves can be inserted into Eq. 4 as: 0 1 1 1 Dr þ 2 Dr p De B Dep 3 C w ¼ arcsin@ 11 ð5Þ A  Dr þ 2 Dr1 De

p 1

De

p 3

Considering that slope evaluations are for a constant stress interval, the dilatancy angle is computed by using Dpf and Ddia pf as:

Relationship Between Pre-failure and Post-failure Mechanical Properties

0 w ¼ arcsin@

1 Dpf

þ 2 D1dia pf

 D1pf þ 2 D1dia

1 A

ð6Þ

pf

For plane strain problems, strain perpendicular to the plane of structural problems like long tunnels and slope or dam sections is assumed to be zero. The dilation angle computed by the above expression can be used to estimate the lateral expansion coefficient of plane strain (2D) plastic potential functions. The out-of-plane plastic lateral strain component Dep2 is zero for such 2D plane strain problems. The Mohr–Coulomb yield function can be expressed as: f ¼ r1  rci  N/ r3

ð8Þ

Similar to the Mohr–Coulomb yield function, a plastic potential function can be defined as: g ¼ r1  C  Nw r3

ð9Þ

C is a constant corresponding to the unconfined strength of the yielding sample at different stages of the post-failure state. Similar to the functional forms in Eq. 8, Nw in terms of the dilatancy angle can be represented as: Nw ¼ ð1 þ sin wÞ=ðð1  sin wÞ ¼ tan2 ð45 þ w=2Þ

ð10Þ

In the evaluation of the test data, a dilatancy angle w is computed first from the general expression involving all three plastic principle strain components in Eq. 6; this angle is substituted into the expression in Eq. 10 and parameter Nw is computed. Nw is also included in the results table. The reason for this is explained below. Detailed description and application of the plastic flow rule can be found in a number of references, such as Vermeer and de Borst (1984), Alejano and Alonso (2005), and Arzu´a and Alejano (2013). Based on the flow rule, differentiation of the potential function with respect to the principal stresses yields the relations between stress and strain increments in the post-failure part. The dilatancy parameter is symbolized as Kw by Alejano and Alonso (2005) or Nw by the Itasca Consulting Group, Inc. (2005). The resulting dilatancy parameter Nw applicable to the plane strain type 2D modeling problems controls the ratio of lateral strain over axial strain increments as follows: Nw ¼ 

Dep3 Dep1

perpendicular to the problem plane is kept at zero. The conservation of energy requires the transfer of energy in terms of additional deformations of the roof, wall, and face boundaries of the 2D problem. This results in higher lateral expansions and deformations along the structure boundaries. For design applications, the situation can be regarded to be on the safe side. Nw is presented in the results to give an idea of how high the lateral strain increments can be during the modeling of softening around structure boundaries like the tunnel roof and wall.

8 Results and Discussion

ð7Þ

This is a 2D yield function in terms of two of the principal stresses, r1 and r3. The friction factor N/ depends on the internal friction angle / in the following way: N/ ¼ ð1 þ sin /Þ=ð1  sin /Þ ¼ tan2 ð45 þ /=2Þ

133

ð11Þ

In summary, a general w is computed from tests on 3D cylindrical samples. In plane strain type modeling, strain

The results of a total of 73 complete stress–strain tests on different rock groups are presented in Table 1. Clearly identified entries for some of the parameters in the rock type groups vary, due to the difficulties associated mostly with the post-failure parts. The number of successful tests for the evaluation of an average for a particular parameter is indicated in parentheses located under the average value of that parameter. Under unconfined compression testing, difficulties like reaching a clear residual compressive strength state may not be possible in all tests. This is caused by the interruption of the experiment following splitting, intense lateral bulging, or uncontrolled breakage of the core specimens. Rhyodacite and granite are at the highly brittle end of the post-peak stress–strain responses of the different sample groups. Dpf is quite high for these two groups. After obtaining a falling portion sufficient to identify Dpf, the tests were terminated by uncontrolled and violent splitting and/or cracking. This violent breakage also threw and disconnected the axial and circumferential measurement arrangements. Therefore, th residual state was not caught in any of the tests on three rhyodacite samples. Healthy lateral expansion data to obtain w was achieved in only one test. Similar problems were observed for tests on granite samples; only one residual state value and two values of w were found. To obtain lateral expansion data and w, mounting the circumferential measurement kit to lignite cores was not attempted, since these samples were very soft with muddy appearance, and their stiffness was so low that the range limits of the circumferential kit were exceeded, even in the pre-peak state. The peak and residual state stress levels are close on the stress–strain curves of the tests on lignite. It is rather difficult to distinguish and clearly identify these states and the related slopes of the post-failure portion of the curves. With the soft and highly plastic characteristic appearance and nature of the lignite samples, lignite is located at the lower limit for the parameters entered in the data

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134 Table 1 Complete stress–strain test results and average values of the pre-failure, peak, and post-failure state parameters Sample group Rhyodacite Glauberite Granite Quartzite series Dunite Argillite Marl Lignite

Ei (GPa)

Es (GPa)

rci (MPa)

Dpf (GPa)

rcr (GPa) (–)

11.75 ± 1.67

10.10 ± 1.76

54.24 ± 7.04

55.92 ± 23.63

(3)

(3)

(3)

(3)

6.26 ± 1.09

4.34 ± 0.91

11.83 ± 4.58

4.25 ± 1.91

W (°)

Nw = tan2(45 ? w/2)

65.43

21.09

(1) 4.03 ± 0.71

57.27 ± 4.80

(9)

(9)

(9)

(9)

(3)

(9)

23.79 ± 2.35

18.84 ± 3.18

89.62 ± 18.87

71.24 ± 12.30

8.22

67.73

(6)

(6)

(6)

(6)

(1)

(2)

15.07 ± 5.50

11.72 ± 5.47

43.50 ± 15.38

34.31 ± 15.08

11.63 ± 3.89

62.85 ± 8.91

(22)

(22)

(22)

(22)

(7)

(12)

7.60 ± 5.55

5.59 ± 4.15

17.12 ± 11.44

14.60 ± 12.85

4.05 ± 3.49

58.82 ± 9.45

11.60 25.81 17.15 12.84

(12)

(12)

(12)

(12)

(7)

(12)

12.40 ± 4.35 (3)

8.45 ± 4.08 (3)

34.36 ± 3.92 (3)

19.90 ± 17.40 (3)

11.54 ± 2.90 (3)

62.98 ± 4.32 (3)

17.32 26.78

9.07 ± 4.20

6.78 ± 3.03

29.95 ± 13.88

20.62 ± 17.17

4.02 ± 1.42

68.13 ± 3.08

(15)

(15)

(15)

(15)

(14)

(15)

0.91 ± 0.64

0.11 ± 0.06

1.16 ± 0.54

0.012 ± 0.01

1.10 ± 0.45

(–)

(3)

(3)

(3)

(3)

(3)

(–)

The number of tests for individual rock sample groups is indicated in parentheses under the average values

processing. Stress–strain curves with the yield stress levels close to those of the soils are typically observed for the lignite group. The characteristics of brittle state with high stiffness, high unconfined compressive strength, and high drop modulus are not well defined for tests on lignite. This situation causes difficulties in the accurate estimation of some of the pre-failure and post-failure state parameters. In general, there are eight basic rock groups in the data processing, including lignite. For the parametric analyses in the next section, the lignite entry is omitted in plots related to the dilatancy analyses, since the lateral deformation of lignite samples is too high, exceeding the detection range of the circumferential strain measurement kit. This kit is designed for the testing of rock material, whereas lignite is highly deformable, like soil. Unless otherwise indicated, the averaged data points for the related parameters of the groups encompass all samples, including lignite. 8.1 Relations Between the Pre-failure State and Peakstate Parameters Stress–strain test data up to the peak failure state are processed to extract two different slopes corresponding to Ei and Es. Pre-peak nonlinearities are believed to result from deformations like the closure of pore spaces and cracking in loaded samples. Such occurrences are irreversible and are believed to be related to the pre-failure ductile state of the rock sample. The slope Es is thought to better represent the characteristics of pre-peak nonlinearities of the stress– strain curves.

123

Both Ei and Es stand for the stiffness of a sample. If the sample strength is zero, its stiffness is expected to be zero as well. This is a mechanical constraint that should be considered in the fitting process for developing a parametric relation and associated functional form as a result of statistical data processing efforts. The best-fit functions may not always be meaningful regarding the mechanical considerations. The functional form fitted for Ei versus rci is given in Fig. 10. Fitting a functional form as a power law appears to be quite successful in providing the relation between Ei versus rci related to the elastic deformation part. The fitted expression is in the functional form as: Ei ¼ A ðrci Þb

ð12Þ

The units of Ei and rci are in MPa in Eq. 12 of the related figure. As rci tends to zero, Ei tends to zero as well, indicating the achievement of a correct functional form here. Linear and logarithmic functional fitting efforts result in significantly lower R2 values. A power law form of functional fit is presented in Fig. 11 for the relation of Es versus rci. The functional form of the fitted expression is: Es ¼ A ðrci Þb

ð13Þ

The units of Es and rci are in MPa. This is, again, a nonlinear power function form, which results in Es being zero when rci is zero. As indicated by the coefficient A of Eqs. 12 and 13 given in Figs. 10 and 11, Es is lower than Ei. This is

Relationship Between Pre-failure and Post-failure Mechanical Properties Fig. 10 Ei versus rci plot

135

25 20 Ei ( 103 MPa)

rhyodacite

Ei = 888(σci)0.72 R² = 0.97

glauberite granite

15

quartzite-series dunite

10

argillite

5

marl lignite

0 0

Fig. 11 Es versus rci plot

20

40 60 σ ci (MPa)

80

100

30 rhyodacite

Es = 131(σci)1.17 R² = 0.95

Es (x10 3 MPa )

25

glauberite granite

20

quartzite-series

15

dunite

10

argillite marl

5

lignite

0 0

expected considering the pre-peak nonlinearities in the curves. The fitting quality for Es is slightly lower than that for Ei. Power b is slightly different in the expressions of Eqs. 12 and 13 related to Figs. 10 and 11. The power of the fitted function is around 0.7 for Ei versus rci in Fig. 10, compared to the fitting power of around 1.2 for the Es–rci relation in Fig. 11. According to this power value comparison of the fitted functions, relation of Es to rci is realized with a slightly higher power. This is, again, attributed to a better coverage of pre-peak nonlinearities in Es . 8.2 Analyses of Results to Estimate the Drop Modulus of the Post-failure State The descending or post-failure portion of the stress–strain curves usually exhibits a nonlinear nature. However, this problem can be solved by fitting a line representing the average linear nature of the curved portion between the peak failure state and the residual state. A magnitude-wise high value of Dpf here can be accepted to represent a high degree of brittleness. With statistical processing of the test data and parametric studies, it is possible to estimate the post-failure

20

40 60 σci (MPa)

80

100

state parameters like the drop modulus from the pre-failure state parameters like Ei, Es, and rci. The drop modulus can be estimated from the results of relatively simple standard deformability tests. With conventional testing, it is relatively simple to compute Ei and Es. It is also possible to estimate Dpf from the results of relatively simple unconfined compressive strength tests. Using the curve fitted in the Fig. 12, Dpf can be estimated in terms of Ei computed from the pre-failure state of a relatively simple standard deformability test. The parametric expression for this estimation is Dpf = A(Ei)b. Both Ei and Dpf are in GPa in this equation. The fitting function indicates a power law type relation between Ei and Dpf. The fitted functional form also satisfies the mechanically expected requirements. The drop modulus is increasingly higher in magnitude than the tangent elastic modulus. Towards the high stiffness side of the trend curve with Ei reaching values of 20 GPa or higher, Dpf becomes four times or much higher in magnitude than Ei. In Fig. 13, Ei is replaced by the secant modulus Es: quality of fit is better than that of the fit for Dpf versus Ei. The fitting function again indicates a power law type relation between Dpf and Es. The form of the equation is

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136

Dpf (GPa)

Fig. 12 Dpf versus Ei plot

100 90 80 70 60 50 40 30 20 10 0

rhyodacite

Dpf = 0.025(Ei)2.79 R² = 0.95

granite quartzite-series dunite argillite marl lignite

0

Dpf (GPa)

Fig. 13 Variation of the drop modulus with the secant modulus of deformation

5

Dpf ( 103 MPa)

20

25

rhyodacite

Dpf = 0.58(Es)1.72 R² = 0.98

glauberite granite quartzite-series dunite argillite marl lignite

5

180 160 140 120 100 80 60 40 20 0

10 Es (GPa)

15

20

rhyodacite glauberite

Dpf =16.4(σci)2.04

granite

R² = 0.96

quartzite-series dunite argillite marl lignite

0

Dpf = A(Es)b. Compared to the previous analysis, the strength of the fitting power is reduced from 2.79 to 1.72. An interesting result is that Dpf of the post-failure state is slightly better related to Es of the pre-failure state than Ei. This is expected considering that the secant modulus Es is the slope of a line extending from the zero-load state to the yield point, including the effect of all slope variations and nonlinearities associated with the pre-failure part.

123

10 15 Ei (GPa)

100 90 80 70 60 50 40 30 20 10 0 0

Fig. 14 Variation of the drop modulus with intact rock strength

glauberite

20

40 60 σci (MPa)

80

100

As in Fig. 14, Dpf can be estimated from a relatively simple standard UCS test. The drop modulus increases with increasing intact rock strength, following an almost quadratic power function. The degree of brittleness, which is believed to be reflected by high Dpf, increases with increasing UCS of the intact rock. The granite group result seems to be more deviated from the general trend. This is thought to be due to very high brittleness of this group,

Relationship Between Pre-failure and Post-failure Mechanical Properties Fig. 15 Dpf/Ei versus Ei/rci plot

137

8

rhyodacite

7

glauberite

Dpf /Ei

6

granite

61.17e-0.010(Ei /σci)

Dpf/Ei = R² = 0.90

5

quartzite-series

4

dunite

3

argillite

2

marl

1

lignite

0 0

200

400

600

800

1000

Ei /σci Fig. 16 Variation of Dpf/Es with the modulus ratio Ei/rci

7

rhyodacite

6

glauberite

Dpf /Es

5

26.03e-0.007(Ei/σci)

granite

Dpf /Es = R² = 0.95

4

quartzite-series dunite

3

argillite

2

marl

1

lignite

0 0

200

400

600

800

1000

E i /σci

reflected by the highest average Dpf. The residual state is reached in only one test on this group. In most tests, the reliability of slope determination for Dpf is low and approximations like in Fig. 7 are needed; these approximations are believed to decrease the quality of the Dpf results used in the data processing. The fitted function is in the form Dpf = 16.4 (rci)2.04. Dpf and rci are in MPa in the fitted expression. Plots with scales in dimensionless forms of the drop modulus, tangent elastic modulus, secant deformation modulus, and rci can be practically useful for the estimation of the drop modulus from a single combined test type aimed to determine Ei, Es, and the intact rock UCS. Plots with normalizations like Dpf/Ei or Dpf/Es versus the modulus ratio Ei/rci will be interesting to analyze. Processing the data in this fashion, functional forms with the best fits are adopted. For the extreme ends of the functions, the validity ranges of the normalized parameters within the limits of the testing program are discussed. Figure 15 shows the fitted curve for the plot of Dpf/Ei versus Ei/rci. The form of the equation is Dpf/Ei = Aexp(b) with constants A = 61.17 and b = -0.010(Ei/rci). The drop

modulus/elastic modulus ratio (Dpf/Ei) varies in an approximate range of 0.01–7 for Ei/rci approximately between 800 and 200. Dpf/Ei increases with decreasing Ei/ rci ratio; at this side, either the rock material is less stiff or it has quite high rci. For a highly brittle rock, which has a very high rci, the Ei/rci ratio tends to zero and the predicted Dpf becomes extremely large. However, a strong supportive argument about this prediction is not fair to force here, since the range of results for Ei/rci was between 217 and 785 in this testing program. The plot in Fig. 16 is in a dimensionless form in terms of Dpf/Es as the vertical scale and Ei/rci as the horizontal scale. This plot can be used to estimate Dpf from the results of simple deformability tests, in which samples are loaded until failure to determine the yield points too. For relating Dpf/Es to Ei/rci, the following equation is proposed: Dpf =Es ¼ A expðbÞ

ð14Þ

For the Dpf/Es versus Ei/rci trend, an exponential function with A = 26.03 and b = -0.007 (Ei/rci) seems to be more successful in predictions than the previous one in terms of Ei. Es is, again, more effective in reflecting the pre-peak

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138 Fig. 17 Variation of residual strength with intact rock strength

1.2 glauberite

1.0

granite

(

σcr / σci

0.8

cr

ci)

= -0.2ln( ci) + 0.93 R² = 0.88

quartzite-series

0.6

dunite

0.4

argillite marl

0.2

lignite

0.0 0

internal structural characteristics to the post-failure softening process. The validity ranges are Ei/rci = 217–785 and Dpf/Es = 5.54–0.10, respectively. Dpf/Es increases with decreasing Ei/rci ratio. At this side, the modulus ratio is low; the rock material has lower stiffness and/or quite high rci. At the low strength side, Dpf/Es tends to zero as the Ei/rci ratio becomes large. This means that the peak and residual states are almost coincident for such rocks, since the drop modulus tends to zero.

20

40 60 σci (MPa)

80

100

argument of rocks with a UCS larger than 105 MPa having negative residual strength is implied. The rci range of the rock groups tested is between 1.2 and 89.6 MPa. The concept of involving the residual strength state in the stability analyses is believed to be practically applicable for low- to medium-strength rock types. Due to the violent failure nature in the post-failure state, speaking of a residual state for high-strength and highly brittle rock units in unconfined conditions may not be meaningful in design procedures.

8.3 Analyses of Results to Estimate the Residual Strength

8.4 Analyses of Results to Estimate the Dilatancy

The residual strength can be estimated in the form of a ratio of the residual to peak uniaxial compressive, as seen in Fig. 17. In Fig. 17, the horizontal scale is rci in MPa. The range of the vertical scale, which is the ratio rcr/rci, varies approximately from 1 to 0. A ratio of 1 is thought to represent a plastic end, where the peak and residual strength values are almost identical. The lower end of the ratio converging to zero is believed to correspond to a highly brittle state. For a highly brittle state, reaching a definite residual strength state may not always be possible, considering the unconfined testing conditions and violent splitting of the samples. Using Fig. 17, the rcr/rci ratio can be estimated based on a simple UCS test result. No result entry is present for the rhyodacite group, as the residual strength state was not reached in any of the tests in this group. A logarithmic function fitted the results in an expression like rcr/rci = A 9 ln(rci) ? b. Coefficient A is -0.2 and constant b is 0.93. It should be noted that other functional forms result in quite low R2 values compared to 0.88 in the figure. The fitted functional form indicates a corresponding value of rci = 0.7 MPa for a ratio of rcr/rci = 1. On the brittle end for a rci around 105 MPa, the rcr/rci ratio becomes zero. Range considerations of the testing program should be reminded here once again. Otherwise, a misleading

Dilatancy evaluations are based on the results of tests on seven rock groups. In the results presented here, the lignite group is not included. The relation of the dilation angle in ° to the Ei/rci ratio is shown in Fig. 18. The exponential function fitted as w(°) = Aexp(b) has a coefficient A = 77.1 and power b = -6 9 10-4 (Ei/rci). A lower limit value around 54° is predicted for Ei/rci = 600, which corresponds to a low rci. The predicted value is approximately w = 69° for the high rci side. The dilatancy angle range is quite narrow. The data point entries in Fig. 18 represent the group averages of w. Here, the number of tests to estimate the dilatancy is quite high, and, interestingly, the dilatancy angle for all tests lies between 43° and 78°. This situation is consistent with the findings of investigators, as discussed in the background section. The relation of the dilatancy angle to Es/rci is shown in Fig. 19. Slightly better fitting quality for the dilatancy angle evaluation is obtained from the plots in terms of Es/ rci. This is expected, since nonlinearities are inherently included in Es, as discussed previously. A wider range in terms of the 2D plane strain dilatancy parameter Nw can be imposed on the estimations. The relation of Nw to Es/rci is shown in Fig. 20. Nw is around 5 for Es/rci = 600 and 25 for Es/rci = 200.

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Relationship Between Pre-failure and Post-failure Mechanical Properties Fig. 18 Dilatancy angle (w) versus Ei/rci plot

139

70 rhyodacite

68 glauberite

66 granite

64 quartzite-series

62 dunite

60

=

77.1e-6E-04(Ei

ci)

argillite

R² = 0.82

58

marl

56 0

Fig. 19 Dilatancy angle (w) versus Es/rci plot

100

200

300 Ei ci

400

500

600

70 rhyodacite

68 glauberite

66 granite

64 quartzite-series

62 dunite

60

=

58

80.9e-9E-04(Es

ci)

argillite

R² = 0.85

marl

56 0

Fig. 20 Plane strain dilatancy parameter Nw versus Es/rci plot

100

200 Es / ci

300

400

30 rhyodacite

N =tan2(45+ /2)

25

glauberite

20

granite

15

quartzite-series

N = 58.2e-0.004(Es R² = 0.81

10

dunite

ci )

argillite

5

marl

0 0

The exponential form fitted yields Nw = Aexp(b) with A = 58.2 and b = -0.004 (Es/rci).

9 Conclusion Laboratory testing to obtain stress–strain behavior under uniaxial loading was conducted on core samples of rock

100

200 Es / ci

300

400

groups of different origin. Parametric expressions were proposed to relate pre-failure deformability (Ei and Es) and peak-state intact strength (rci) to characteristic parameters of the post-failure state of the stress–strain curve for intact rock under uniaxial loading. The tangent modulus of elasticity Ei and secant modulus of deformation Es corresponding to the pre-failure stiffness of the rock samples were related to the intact rock strength

123

L. Tutluog˘lu et al.

140

rci by power laws of power b around 0.7 and 1.2, respectively. Stiffness represented by Ei and Es increased with increasing unconfined strength. Following power laws, the drop modulus (Dpf) was found to increase with increasing Ei and Es. Power b was around 2.8 and 1.7 for Ei and Es, respectively. The fitting quality was better for the data processed in terms of Es. Following an almost quadratic power function, the degree of brittleness believed to be characterized by high Dpf was concluded to increase with increasing uniaxial compressive strength (UCS) of the intact rock. Exponential functional forms related dimensionless Dpf/ Ei and Dpf/Es to the modulus ratio Ei/rci with reasonable fit qualities. The fitting quality was noticeably better for Dpf/ Es versus the Ei/rci relation. Both Dpf/Ei and Dpf/Es increased with decreasing Ei/rci ratio, which was interpreted as a rock type with high rci and low modulus ratio. A logarithmic relationship was proposed to estimate the residual strength. The ratio rcr/rci decreased with increasing intact rock compressive strength. rcr/rci was estimated to be around one for low-strength rock types. It was around zero at the brittle extreme of a rci of around 105 MPa. The dilatancy angle for all unconfined compression tests in this work lied between 43° and 78°. Slightly better fitting quality for the dilatancy angle data was observed in terms of Es/rci compared to Ei/rci. This result is interesting, considering that the secant modulus of deformation inherently involves the effects of pre-peak state nonlinearities. From a standard deformability and compressive strength test, the tangent modulus of elasticity, the secant modulus of deformation of the stress–strain curve between zero and the peak load state, and the unconfined compressive strength can be determined rather easily and practically. These properties, together with the proposed parametric expressions here, can be used to estimate post-failure parameters like the drop modulus of the strain-softening part, residual strength, and dilatancy angle. In the numerical modeling work, these post-failure properties are important for accurate simulation of the failure process in and around geo-structures. With the use of the guiding expressions proposed, input for plasticity parts of the programs is expected to be more realistic, and the modeling results in terms of stresses and deformations are believed to be more representative. For similar efforts in the future, it is recommended to increase the variety of rock types in the low- to mediumstrength range. Concentrating on rock types in this range is more meaningful. The failure of rock and reaching the residual state occur in a rather controlled manner for such rock types. In fact, stability control measures and support applications make sense for excavations in such rock types. For highly brittle rock of high strength, the aim must be to keep load levels low enough and stay away from the peak

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strength state of the rock. Local support application effort is not going to solve the problems of unstable collapse of structures in such rock units.

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