Abstract
This paper presents results from, and theory to support the anisotropic testing of a series of laminated Coal Measure siltstones. It was found that laminations within the siltstone units created an anisotropic fabric with varying strength and stiffness properties in relation to the orientation of the planes of lamination. New results are presented illustrating changes in triaxial, unconfined compressive, elastic modulus and tensile strength for ten orientations of the lamination plane. A number of anisotropic failure criteria are reviewed and then compared to the test results allowing the selection of the optimum method of characterising the siltstone strength anisotropy. The influence of lamination plane orientation on the tensile strength and Elastic Modulus was also assessed and analysed in relation to theoretical models.
The mechanical properties of anisotropic rocks vary with the orientation of bedding planes, lamination planes, mineral cleavage and other anisotropic effects. A number of researchers such as Jaeger (1960); Walsh & Brace (1964); McClintock & Walsh (1963); Hobbs (1964); Donath 1964, 1972); McLamore & Gray (1967); Attewell & Sandford (1974); Hoek & Brown (1980) and more recently Amadei (1988, 1996); Yasar (1995); Sheorey (1997) and Chen et al. (1998) have investigated the deformation characteristics and failure of anisotropic rocks using experimental data and theory. One of the main factors which affects the anisotropy of UK Coal Measure rocks is lamination within the lithological unit. Laminations are attributed to a variation of mineralogy, grain size and orientation, density and extent of packing.
Previous experimental data on UK Coal Measures rocks indicated that the mechanical properties show significant differences between parallel and perpendicular loading to the stratification (Hassani 1980; Yasar 1995). For this investigation, a number of core specimens of Middle Coal Measures laminated siltstone were prepared at ten orientations to the lamination plane at 10° increments between 0° and 90° degrees. A series of uniaxial compressive, indirect tensile, triaxial and elastic deformation tests has been conducted at each angle. The test results indicated different elastic constants and strengths in the various loading directions relative to the lamination plane.
To determine the optimum method of characterizing the strength anisotropy of the siltstones the strength data obtained from the uniaxial and triaxial tests was compared to the following three different failure criterions developed for anisotropic materials:

Single Plane of Weakness criterion (Jaeger 1960), (theoretical)

Variable Friction Angle and Cohesive Strength criterion (Donath 1972) (empirical)

3D Strength criterion (Amadei 1988). (Theoretical)
Apparent elastic moduli of each of the siltstones were found for each orientation of the lamination plane angle to the loading direction. The siltstones were considered to represent a transversely isotropic medium and the elastic constants that characterize such a medium have been determined for each siltstone type.
The variation of tensile strength with lamination plane orientation was investigated by the means of the Brazilian disc test. The stress concentration factor for the centre of the specimen at the various loading directions was evaluated using FLAC, a commercially available geomechanical finite difference package.
Sample descriptions
Bulk samples of two distinct siltstone horizons (Type 1 and Type 2) were obtained from an opencast coal mine in Alfreton, Derbyshire. The siltstones were of Middle Coal Measure age and were taken from the proximity of the Deep Soft coal seam. Type 1 siltstone was taken from a 1.15 metre thick horizon in the immediate roof of the seam. Type 2 siltstone was taken from a 9.05 metre thick horizon in the immediate floor of the coal seam. Both siltstones showed no signs of weathering with the bulk samples obtained from the more uniformly laminated sections of each horizon.
Type 1 Siltstone
In hand specimen this siltstone horizon typically consisted of interlaminations of grey siltstone and dark grey mudstone. The laminations were generally planar with approximately 60% of the horizon consisting of siltstone laminae and 40% mudstone laminae. Typical thickness of both types of laminae was 1 mm. In thin section the siltstone laminae were shown to predominately comprise of euhedral grains of quartz with lesser amounts of biotite. The siltstone laminae were grain supported and had a silica cement. The mudstone laminae consisted of a dark clay material with lesser amounts of preferentially orientated mica and occasional euhedral quartz grains.
Type 2 Siltstone
In hand specimen this horizon consisted of a grey siltstone interlaminated with dark grey laminae of mudstone. The siltstone laminae to mudstone laminae ratio was approximately 4:1 with the typical thickness of the siltstone laminations being 4 mm and the mudstone lamination being 1 mm. In thin section the siltstone laminations consisted predominantly of euhedral quartz grains with a small amount of biotite. The siltstone laminae were grain supported and had a silica cement. The mudstone laminae comprised mainly of a dark coloured clay material.
Test methodology
Uniaxial and triaxial strength test
The bulk samples for both of the siltstone types were used for the preparation, to ISRM standards (ISRM 1981), of cylindrical test samples, with diameters of approximately 36 mm and lengths of 72 mm. The samples were prepared at 10° increments of the angle of the laminations to the long axis of the cylinder to provide a range of samples with lamination inclination angles varying between 0 and 90^{o}. A series of triaxial and uniaxial tests were then conducted on the samples at confining pressures of 0, 2, 4 and 8 MPa. The testing was undertaken using a RDP servocontrolled press at a constant loading rate such that failure occurred between 5 to 10 minutes of loading. A Hoek cell provided confinement at a constant pressure during the triaxial test and the load at failure was recorded. A total of 82 and 62 samples were tested for horizons 1 and 2 respectively. The results of the triaxial and uniaxial tests are included as Figure 1 .
Elastic modulus test
Samples were prepared in the same manner as for the unconfined compressive strength tests to provide a range of cylindrical test samples with the laminae orientated at 10^{o} increments to the axis of the sample. The sample, platen and dial gauge arrangement used in the elastic modulus test is illustrated in Figure 2 . The top platen incorporated a spherical seat to ensure even load distribution across the ends of the sample. Two dial gauges with digital readouts accurate to 0.01 mm allowed the measurement, between two brackets attached to the ends of the platens adjacent to the sample, of the vertical displacement of the sample. Monitoring of the deformation was undertaken at regular increments of load until the samples showed sign of failure where upon the dial gauges were removed and loading continued until failure.
Brazilian disc test
Indirect tensile tests were conducted using Brazilian discs as per the procedure outlined by the International Society for Rock Mechanics (ISRM 1981). The test samples were prepared into discs, approximately 36 mm in diameter and 18 mm in length, from cores oriented at 90° to the trend of the laminations. The samples were placed into a Brazilian disc test apparatus (ISRM 1981) with the arc of contact between the apparatus and the sample being approximately 10^{o}. The samples were orientated so that the direction of loading was at ten increments to the inclination of the lamination planes. Again the testing was undertaken using a RDP servocontrolled stiff press with a constant loading rate of approximately 200 N/sec with the load at failure being recorded.
Characterizing the strength of the siltstones
Anisotropic strength criterions
The test data obtained from the uniaxial and triaxial tests were compared to the following three different failure criterions developed for anisotropic materials

Single Plane of Weakness criterion (Jaeger 1960) (theoretical)

Variable Friction Angle and Cohesive Strength criterion (Donath 1972) (empirical)

3D Strength criterion (Amadei 1988) (theoretical)
Criterions 1 and 2 consider a 2D stress field with the affect of the intermediate principal stress being ignored. Criterion 1 was derived by theoretical considerations of anisotropic rock failure whilst Criterion 2 was derived from empirical observations. The third criterion takes into consideration the effect of the intermediate principal stress and the orientation of the weak plane within a 3D stress field, thus allowing a prediction of the rock response in more complex multiaxial states of stress. It should be noted that all these criterions assume shear failure of the rock samples.
The Single Plane of Weakness criterion (Jaeger 1960): This criterion describes an isotropic matrix, which is assumed to possess a plane of weakness. Within the theory the rock matrix strength is described by the Mohr–Coulomb Criterion with a constant friction angle and cohesive strength (Equation 1). (1)

where τ_{n}=shear strength of matrix

c=cohesive strength of matrix

σ_{n}=normal stress

ϕ=friction angle of matrix
Equation 1 may also be expressed in terms of the principal stresses (Equation 2) (Jaeger & Cook 1979): (2)

where σ_{1}=maximum principal stress magnitude

σ _{3}=Minimum principal stress magnitude
The strength of the weak plane is also represented by a Mohr–Coulomb criterion with a constant friction angle and cohesion. The orientation of the weak plane with respect to the principal stresses is defined by an angle (β) which represents the angle between the applied maximum principal stress (σ _{1}) and the inclination of the weak plane. Resolving the shear and normal stresses across the weak plane at failure into principal stresses generates a criterion in terms of the principal stresses for slip along the weak plane (Equation 3) (Jaeger & Cook 1979): (3)

where β=angle between maximum principal stress σ_{1} and normal to slip plane

c_{p}=cohesive strength of weak plane

ϕ_{p}=friction angle of weak plane
From Equation 3 it can be seen that the differential stress at failure (σ_{1}−σ_{3}) varies with β for a constant σ_{3} and ϕ_{p} and tends to infinity as β→90° and as β→ϕ_{p}. Between these two extremes failure can occur along the plane of weakness at a stress governed by Equation 3 for a range of β orientations with the lowest strength occurring when: (4)
For low and high values of β, failure cannot occur by slip along the plane of weakness and failure occurs through the rock matrix at stress conditions determined by Equation 1.
Variable friction angle and cohesion criterion:
This criterion assumes that material failure at a constant angle of lamination inclination is defined by the Mohr–Coulomb criterion. The variable friction angle and cohesion criterion states that the cohesion and friction angle of the rock material varies as a function of the inclination angle β (Equations 5 & 6). (5) (6)

where A, B, C and D are material constants

β′=angle between maximum principal stress σ_{1} and the normal to weak plane when the shear strength is at a minimum.

c_{β}=cohesion at angle β

ϕ_{β}=friction angle at angle β
To evaluate the Variable friction angle and Cohesion criterion, it is necessary to perform a series of compression tests at variations of β to determine the values of c_{β} and the ϕ_{β} for each orientation which are then used to evaluate the constants A, B, C and D. The fracture strength of the material as a function of the orientation can then be calculated using Equation 7. (7)
3D Strength criterion (Amadei 1988)
Within this criterion the affect of the magnitude and orientation of the three principal stresses on rock failure is taken into consideration. The rock material is assumed to comprise of a weak plane or planes set in a rock matrix, which exhibits isotropic strength properties.
The rock matrix’s strength is defined by the nonlinear empirical Hoek–Brown failure criterion (Equation 8). (8)

Where σ_{c}=unconfined compressive strength

m_{i}=material constant
The shear strength of the weak plane is represented by a Mohr–Coulomb criterion with a constant friction angle and cohesion (Equation 9) (9)

Where τ_{n}= shear stress at failure along weak plane

σ_{n}=normal stress at failure along weak plane

c_{p}=cohesion of weak plane

ϕ_{p}=friction of weak plane
A dip angle and a dip direction relative to the axis of the 3 principal stresses define the orientation of the weak plane. The dip angle of the weak plane, β, is defined as the angle of inclination of the weak plane from the plane containing the σ_{2}− σ_{3} principal stress vectors. The dipdirection angle, α, of the weak plane is found by projecting the dip direction onto the σ_{2}− σ_{3} plane and measuring the angle formed by the σ_{2} axis and the projected line. The upward unit vector parallel to the pole of the weak plane then has direction cosines S_{1}, S_{2} and S_{3} such that S_{1}, S_{2} and S_{3} are parallel to the σ_{1,} σ_{2} and σ_{3} stress vectors respectively and: (10) (11) (12)
The state of stress across the joint can be represented by a stress vector, with normal and shear components σ_{n}, τ_{n} such that (13) (14)
Under triaxial compression an axisymetric stress condition exists i.e. σ_{1} > σ_{2}=σ_{3}. In such a stress condition the dip direction; α, of the weak plane is taken to be 45^{o} as the weak plane will experience both shear and normal stress.
Thus the shear and normal components of the stress vector become: (15) (16)
Letting Ff=τ_{n}− (σ_{n} + c_{p}/tanϕ_{p}).tanϕ then the condition Ff=0 represents a state of limit equilibrium. Slip along the joint occurs when Ff is >0 whilst no slip occurs at negative values of Ff. This relationship at limit equilibrium may also be expressed in terms of the principal stress components by squaring Equation 9 and substituting Equations 15 & 16 into it. This joint failure criterion in terms of principal stress is given below as Equation 17: (17)
Characterizing the siltstone strength anisotropy
The three criterions were applied to the triaxial and uniaxial test data in order to evaluate the optimum method of characterizing the strength anisotropy of the two siltstones. The application of the test data to the criterions is summarized as follows:
Single plane of weakness criterion
The rock matrix strength was determined by calculating the mean σ_{1} for each confinement stress at the 0^{o} and 10^{o} β angles, where it was assumed failure would be entirely through the rock matrix. A nonlinear least squares regression analysis was then undertaken on the averaged σ_{1} and σ_{3} data to obtain a relationship between σ_{1} and σ_{3}. The corresponding Mohr–Coulomb failure envelope and friction and cohesion parameters were derived from this relationship using Balmer’s equations as per the procedure detailed by Hoek et al. (1995).
The friction and cohesion values of the weak plane were determined by averaging the σ_{1} data for each confinement at the 70^{o} beta angle, where it was assumed that failure would be entirely along the plane of weakness. From the σ_{3} and averaged σ_{1} data the friction and cohesion value was then calculated using the same procedure as for the rock matrix. The friction and cohesion values of the rock matrix and plane of weakness calculated for the two siltstone types are given in Table 1 .
Variable friction angle and cohesion criterion
To determine the material constants A, B, C and D the σ_{1} data for each β orientation at each confinement was averaged. A nonlinear least squares regression analysis was undertaken to obtain a relationship between the averaged σ_{1} and σ_{3} for the individual β angles. The corresponding Mohr–Coulomb failure envelope and friction and cohesion values were then derived for each β angle using Balmer’s Equations (Hoek et al.1995). The calculated friction and cohesion values for each β angle were then used within Equations 5 & 6 and the A, B, C and D constants were determined by linear regression. The determined constants for the two siltstone types are given in Table 2 .
3D Anisotropic failure criterion
The Hoek–Brown material constant for the rock matrix strength was determined by calculating the mean σ_{1} for each confinement stress at the 0 and 10^{o} β angle, where it was assumed failure would be entirely through the rock matrix. Rearranging Equation 8 gives Equation 18. (18)
Equation 18 allows the determination, by linear regression, of the Hoek–Brown material constant m_{i} value for the rock matrix.
Using Equations 13 & 14 the shear and normal stress on the weak plane was calculated for the test data obtained at β angles of 60^{o} and 70^{o} where it was assumed failure would be along the weak plane. The friction and cohesion value of the weakness plane was then found by linear regression (Fig. 3 ). The m_{i} constants for the rock matrix and the friction and cohesion for the weakness planes of the two siltstone types are given in Table 3 . The Mohr Circles and failure envelope for the rock matrix is shown as Figure 4 .
Discussion
Figures 5 & 6 illustrate the application of the three failure criterions to test data for the Type 1 and Type 2 siltstone respectively. The theoretical Single Plane of Weakness failure criterion and 3D Failure criterion can be seen in the Figures to produce a similar predictionfor the axial stress at failure. In general they predict that the strength reduction associated with slip along the weakness plane occurs over a β range of approximately 35^{o} with a minimum strength at approximately β=70^{o}. Typically outside this range, where the two criterions predict failure through the rock matrix, they over estimated the strength of the siltstones. The exception to this observation is for Type 1 siltstone tested at 4 MPa and 8 MPa confining pressures where these two criterions can be seen to produce a reasonable fit to the test data.
The Variable friction angle and cohesion criterion produced a failure envelope with a continuous change in strength in relation to the β orientation. As can be seen in Figures 5 & 6 the test data also showed a continuous change in strength in relation to the angle of inclination of the lamination planes with a minimum strength occurring at an β angle of 60^{o} to 70^{o}. In general the Variable Friction and Cohesion criterion provided a reasonable fit to the test data.
Analysis of the test data indicated a more complex failure mechanism of the siltstones than simple shear failure either through the rock matrix or along planes of weakness as predicted by the theoretical criteria. Observations of the samples after testing showed that the samples failed in some cases by shear both through the rock matrix and along the plane of weakness at intermediate β angles i.e. where 10^{o}<β<60^{o}. Tensile splitting along the plane of weakness, not shear failure through the rock matrix as predicted by the theories, was observed to occur in failed samples with β=90^{o}. These observations would explain the substantial lower strengths for the siltstones compared to that predicted by the theoretical criteria. At higher confining pressures the theoretical failure criterions did however offer a reasonable prediction of the test data for Type 1 siltstone. This seems to indicate that there is a change in the mechanism of failure of this siltstone with increased confinement.
From the results of this investigation it can be concluded that for a situation insitu where σ_{2}=σ_{3} then the strength of the two siltstone types would be best characterized by the variable friction and cohesion criterion. However where σ_{2} > σ_{3} then it is predicted, by setting σ_{2} > σ_{3} within the 3D anisotropic criterion, that the relative orientation of the lamination planes within the 3D stress field becomes important. For the situation where the intermediate principal stress is not equal to the minimum principal stress it is considered that the 3D anisotropic failure criterion would provide the optimum method of characterizing the siltstones strength.
Characterizing the anisotropic stiffness of the siltstones
Effect of anisotropy on the apparent elastic modulus
The axial deformation of the laminated siltstone as recorded in the elastic moduli tests was found to be dependent on the angle of inclination formed between the lamination planes and the direction of loading. Thus the stressstrain relationship at an individual angle of inclination of the lamination planes provides an apparent elastic modulus for that inclination angle. The recorded test data for each of the tested samples was utilized to construct stressstrain curves. The apparent elastic moduli for each of the tested samples has been determined from the linear portion of the stressstrain graphs constructed from the recorded test data for each of the tested samples. Figure 7 illustrates the change in the apparent elastic modulus with respect to β orientation for the two siltstone types. The scatter in data is large, however the data exhibits a general decline in the apparent modulus from β=0° to β=90°. The change in apparent modulus with lamination orientation can be considered as being a manifestation of the samples having different elastic properties perpendicular and parallel to the planes of lamination
Elastic properties and anisotropy
In a layered medium, with the layers having different elastic properties, the deformation behaviour of the medium prior to failure will be dependant on the elastic properties, thickness and spatial distribution of the individual layers, and also the orientation of the stress field relative to the plane of layering. For a finely layered material the layering may be considered ubiquitous throughout the material. The material as a whole can then be approximated as an elastically, transversely isotropic material with different elastic properties perpendicular and parallel to the layering, which are assumed to be the planes of elastic isotropy. The advantage of this assumption is that only five unique elastic constants are required to determine the elastic deformation of the material without the requirement of determining the thickness, spatial distribution and elastic properties of the individual layers. The five constants are namely the elastic moduli in the plane of layering and in a direction normal to it (E and E’ respectively), the Poisson’s ratios characterizing thelateral strain in the plane of stratification and in a direction normal to it (ν’ and ν) and G’ which is the shear modulus in planes perpendicular to the plane of stratification.
Characterizing the siltstones as a transversely isotropic elastic media
For the purpose of providing a model of the elastic behaviour of the siltstones at different inclinations of the applied stress field the siltstones were considered as representing a transversely elastic media. This assumption was considered reasonable, as several lamination planes would have influenced the deformation response of the samples upon loading.
For each of the two siltstone types the elastic modulus within the planes perpendicular to the plane of stratification (E’) was determined by averaging the apparent elastic moduli for β =0^{o} and 10^{o} orientations where it was considered that that E and G’ would have little effect.
The elastic modulus within the plane of stratification (E) was considered to be the elastic modulus calculated for β=90^{o} where it was considered that that E’ and G’ would have little effect.
From the theory of elasticity for transversely isotropic media and assuming uniform stresses and strains in the specimens the strain parallel to the direction of applied stress can be related to the applied stress by Equation 19 (Amadei 1992). (19)

Where ϵ=strain parallel to the direction of applied stress

σ_{1}=applied stress
Rearranging Equation 19 gives Equation 20 which gives the shear modulus, G’, as a function of ϵ, σ_{1}, E’, E, β and v’. (20)
Individual G’ values for each orientation of β was calculated using Equation 20 with the value of E and E’ being taken from Table 4 . The G’ values were then averaged to determine the characteristic shear modulus for each siltstone type (Table 4). From previous testing on similar Coal Measure siltstones the Poisson’s ratio within the plane of lamination (υ’) was determined to be typically 0.16 (Branch 1987).
Equation 20 may also be used to predict the apparent elastic moduli for each β angle from the determined elastic constants. The predicted apparent elastic moduli have been plotted on the same axis as the measured values (Fig. 7).
Characterising the tensile strength of the siltstones
The tensile strength of a rock tested by the Brazilian disc indirect method is determined using Equation 21. (21)

Where:

σ_{t}=tensile strength

W_{f}=applied load at failure

D=diameter of specimen

t=thickness of specimen

q_{xx}=stress concentration factor at the centre of the sample
The stress concentration factor, q_{xx,} is related to the stress in the x direction (σ_{xx}) assuming the load, w, is applied in the y direction (Equation 22). (22)
The stress concentration factor for an isotropic rock is generally approximated to 2, however for the laminated siltstones the stress concentration at the centre of the sample will be influenced by the elastic properties and orientation of the lamination planes.
To provide a method of estimating the stress concentration factor, the siltstones were considered as representing a transversely isotropic elastic media. However for such a media, q_{xx,} is a complex function of the elastic constants E, E’, G’ and v’ and the orientation angle β cannot be easily solved (Chen et al. 1998). To approximate the value of σ_{xx} within the centre of the samples, during the Brazilian disc test, a commercially available geomechanical finite difference program known as FLAC V3.3 (Itasca 1995) was utilized. A circular grid was constructed and FLAC’s transversely isotropic constitutive model was utilized for the modelling the two siltstones. A plane stress analysis was undertaken with a load applied over an angle of 10^{o} at the top and bottom of the model thus simulating the application of the load during the test. Ten models were constructed for each siltstone type with the elastic properties for each set of models being selected from Table 4. Each model represented the Brazilian disc at one of the 10^{o} increments of the lamination plane orientation incremented at 10^{o} intervals from β=0^{o} to 90^{o}. The individual models were run until the internal stresses had stabilized. The simulated σ_{xx} values determined from this modelling were used within Equation 22 to calculate the stress concentration factor for each angle of lamination inclination (Fig. 8 ). The tensile strengths for the two siltstone types were then calculated from the test data and the stress concentration factors using Equation 21.
The calculated tensile strengths were plotted against inclination angle and are shown as Figure 9 . A linear regression line has been fitted to both graphs, which indicate a linear reduction in tensile strength with increasing β angle for both types of siltstone.
Conclusions
This study has shown that the strength and stiffness properties of two laminated siltstones of Middle Coal Measure age was dependant on the inclination of the lamination planes relative to the direction of applied stress.
Uniaxial and triaxial testing on samples with various angles of lamination inclination indicate that the minimum strength of the siltstone is obtained at a lamination inclination angle of between 60^{o} to 70^{o} whilst the maximum strength is obtained at an inclination angle of 0^{o}. It was shown that theoretical failure criteria developed for anisotropic rock overestimate the strength of the siltstones over a wide range of lamination inclination angles. This is attributed to more complex failure mechanisms than that considered by the theoretical failure criteria. It was observed during the testing that at angles of inclination between 10^{o} and 60^{o} often the rock failed both along laminations and through the rock matrix whilst at inclination angles of 90^{o} the rock failed by tensile splitting along the lamination plane. These mechanisms are not considered in the theoretical criteria were the rock fails either through the rock matrix or along the planes of weakness. The best fit to the test data was provided by the empirical variable friction and cohesion criterion. It is therefore recommended that this criterion should be used to characterize the siltstone strength when the intermediate principal stress and minor principal stress are approximately equal.
The apparent Elastic Modulus of the two types of siltstone was found to vary with the orientation of the plane of anisotropy. The material was considered as a transverse isotropic elastic medium and the Elastic constants E, E’ and G’ calculated for each siltstone type show that elastic moduli for directions perpendicular and parallel to the lamination planes are different.
The tensile strength of the two siltstones was determined from the indirect Brazilian discs, with the stress concentration factor at the centre of the sample being derived by finite difference modelling. The tensile strength of the siltstones was found to decrease as a linear function of lamination inclination.
This paper introduces the preliminary phase of a more detailed programme of research involving the detailed testing of a wider range of coal measure rocks. Triaxial testing and measurement of all directional deformation moduli properties are proposed and the anisotropic strength and elastic theories applied to a more comprehensive and detailed data sets of all major UK Coal Measure lithologies.
 © 2002 The Geological Society of London