A state-of-the-art anisotropic rock deformation model incorporating the development of mobilised shear strength

Currently rock deformation is estimated using the relationship between the deformation modulus Em and the stress-strain curve. There have been many studies conducted to estimate the value of Em. This Em is basically derived from conducting unconfined compression test, UCS. However, the actual stress condition of the rock in the ground is anisotropic stress condition where the rock mass is subjected to different confining and vertical pressures. In addition, there is still no empirical or semi-empirical framework that has been developed for the prediction of rock stress-strain response under anisotropic stress condition. Arock triaxial machine GCTS Triaxial RTX-3000 has been deployed to obtain the anisotropic stress-strain relationship for weathered granite grade II from Rawang, Selangor sampled at depth of 20 m and subjected to confining pressure of 2 MPa, 7.5 MPa and 14 MPa. The developed mobilised shear strength envelope within the specimen of 50 mm diameter and 100 mm height during the application of the deviator stress is interpreted from the stress-strain curves. These mobilised shear strength envelopes at various axial strains are the intrinsic property and unique for the rock. Once this property has been established then it is being used to predict the stress-strain relationship at any confining pressure. The predicted stress-strain curves are compared against the curves obtained from the tests. A very close prediction is achieved to substantiate the applicability of this rock deformation model. This is a state-of-the art rock deformation theory which characterise the deformation base on the applied load and the developed mobilised shear strength within the rock body.


Introduction
Progressive slow deformation in rock structure will result in catastrophic failures and therefore rock deformation modelling is important as it helps to predict future deformation for stability of any foundation or infrastructure. It is generally known that the strength of rock also plays an important role in the multi-peak deformation behavior of rock mass. Both strength and deformability of fractured rock masses are important factors in the design and construction of civil and mining structures [1]. The importance of obtaining not only the peak strength but also the complete stress-strain curve of rocks by conducting laboratory tests has also been recognized as it could affect rock stability [2]. Up to date, there are a competitive number of researchers focusing on rock strength deformation model. However, most of them had excluded the role of confining pressure in their rock deformation model like the model introduced by Cheng et al [3], Korneva et al [4] and Xue et al [5]. Some of the researchers perform numerical simulations to assess failure behavior and reconstruct rock deformation as conducted by Wang et al [6] and Li et al [7]. However, the design parameter and simulations is found to be troublesome due to its complexity. This paper introduces a new rock deformation model, which incorporate the build-up mobilised shear strength within the rock body during the application vertical loading. In fact this model is based on the interaction between role of strength and the applied load in characterizing the rock deformation. The prediction framework utilizes the non-linear shear strength envelopes introduced by Md Noor et al [8][9]. This model is based on the rock true stress-strain curves.

Laboratory rock triaxial test
The stress-strain curves of the test rock specimen are obtained using rock triaxial machine, GCTS Triaxial RTX-3000 which uses GCTS CATS software to run the test. The test specimens are in the form of cylindrical specimen of 50 mm diameter and 100 mm height. The test rock is weathered granite grade II, which obtained from site within Rawang by deep boring work at depth of 20 m. The placement of the test specimen in GCTS Triaxial machine is shown in Figure 1 while Figure 2 shows a schematic diagram of the overall set up. The procedure of triaxial testing followed the standard GCTS CATS software.  The RTX-3000 GCTS triaxial machine capable of applying different confining pressures which may be varied from 0 MPa to 150 MPa. As mention by Hoek and Brown [10], the maximum confining pressure can be applied is up to 50 % of maximum compressive strength. However in this study, the rock specimens were only subjected to three confining pressures which are 2 MPa, 7.5 MPa and 14 MPa. The application of deviator stress is at a constant rate of 0.01% strain. This is to suit the time of failure to be within 5 to 20 minutes. Figure 3 then depicts the stress-strain curves of the test rock at the different confining pressures.  Based on Figure 3, since the failure occurs at different axial strain which are 1.58%, 1.98% and 2.35% for confining pressure of 2 MPa, 7.5 MPa and 14 MPa respectively, then a normalize stress strain curves is needed in order to plot the development of the mobilised shear strength envelopes during the application of the deviator stress. The normalised stress-strain curves are obtained by multiplying the respective axial strains with respective normalise factor so that the peak failure deviator stresses is shifted to coincide with the maximum failure axial strain of 2.35%. The normalise conversion factors are 1.482 and 1.186 for the confining pressures of 2 MPa and 7.5 MPa respectively and they are calculated using Equation 1. The normalised stress-strain curves are as shown in Figure 5.  Figure 7. These mobilised shear strength envelopes are considered as the inherent property of the test rock, and will be utilized for the prediction of deviator stress under any confining pressure. Whenever the rock specimen is compressed by applying the deviator stress the rock mass automatically resists the compression through the build-up mobilised shear strength developed within the rock mass. Therefore, the mobilised shear strength is taken as the deformation resisting variable in this theoretical rock deformation model. The applied anisotropic stress to the specimen is defined by the Mohr circle. When the deviator stress is increased, the diameter of the Mohr circles grows bigger and the Mohr circle extends beyond the current mobilised shear strength envelope. This is a state of in-equilibrium and the rock mass automatically developed a higher mobilised shear strength. This is indicated by the mobilised shear strength envelope rotating anti-clockwise about the origin to mark the increase. Whenever the envelope reinstates the touching of the Mohr circle, then the state of equilibrium between the applied pressure and the developed mobilised shear strength is achieved. This process repeats continuously until the shear strength envelope at failure is arrived. Different rock would have a different distribution

Prediction of stress-strain response
Since the mobilised shear strength envelopes obtained in Figure 7 is the inherent properties of the test rock, then this property is valid for any stress range. Therefore this mobilise shear strength envelopes can be applied to predict the stress strain response of the test rock at any confining pressure. Thus in order to check the validity of this assumption, the stress-strain response at confining pressure of 2 MPa, 7.5 MPa, and 14 MPa are predicted by drawing the Mohr circles with the volume of V3 as the minor principle stress which is equals to the applied confining pressure. The Mohr circles are drawn to touch the respective mobilised shear strength envelopes and the diameter of the Mohr circle represent the corresponding deviator stress as shown in Figure 8a, 8b and 8c respectively.  Table 1. In order to compare the prediction of the stress-strain curves to the actual experimental results, the normalised strain need to be multiply by the inverse factor to represent the corresponding stress to the respective axial strain. The inverse factors are 0.675 and 0.843 for the confining pressures of 2 MPa and 7.5 MPa respectively and they are calculated using Equation 2. The complete prediction of stress-strain values are presented in Table 2. Figure 9 then depicts the predicted stress-strain curves plotted superimposed on the actual laboratory stress-strain curves.   The predicted stress and strain magnitudes for confining pressures of 2 MPa, 7.5 MPa and 14 MPa are as being presented in Table 2 and have been plotted as stress-stress data points in Figure 9. The line graphs are the stress-strain curves obtained from the triaxial tests at confining pressures of 2 MPa, 7.5 MPa and 14 MPa. A very close predicted stress-strain response has been achieved in comparison with the graphs obtained from the laboratory triaxial tests. This is the advantage of incorporating both deformation driving and resisting variables in the deformation framework. Besides, the framework is derived from the actual stress-strain behaviour. The procedure deployed by this rock deformation framework for predicting the rock stress-strain response is summarized as follows; i. Conducting rock triaxial tests for at least three different confining pressures to determine the test rock stress-strain curves. ii.
Plot the rock stress-strain curves. iii.
Determine the maximum axial strain at failure from the stress-strain curves. iv.
Obtain normalised stress-strain curves by multiplying the respective axial strains with respective normalise factor so that all peak failure deviator stresses is shifted to coincide with the maximum failure axial strain. ( refer Figure 5) v.
Decide the increment of axial strain from 0% to failure, so that the mobilised shear strength envelopes could be drawn based on this increment. vi.
Draw the Mohr circles corresponding to each axial strain under consideration. vii.
Draw the mobilised shear strength envelopes corresponding to each axial strain under consideration. viii.
Once all of the mobilised shear strength have been drawn, the Mohr circles need to be removed. These envelopes are then considered as the inherent property of the test rock, and will be utilized for the prediction of deviator stress for any confining pressure. (Figure 7) ix.
A mobilised shear strength envelope is representing a certain value of axial strain. x.
To predict the magnitude of deviator stress corresponding to a certain axial strain a Mohr circle is drawn to touch the corresponding mobilised shear strength envelope with the minor principle stress, V3 representing the applied confining pressure. (Figure 8a, 8b, 8c) xi.
The resulted diameter of the Mohr circle represents the deviator stress corresponding to the axial strain considered. xii.
This process is repeated for all considered mobilised shear strength envelopes and the relationship between the deviator stress and the corresponding axial strain can be plotted to represent the stress-strain relationship for the confining pressure under consideration. xiii.
This process can be repeated for other confining pressure and the stress-strain curve can be determined. xiv.
The normalised strain values need to be converted to actual axial strain by multiplying with the normalised inverse factor as obtained in Equation 2. (Data as in Table 2) xv.
Plot the graph using the inversed axial strain versus the predicted deviator stress. In this manner the predicted stress-strain curves can be obtained. (refer Figure 9)

Conclusions
In this paper, a state-of-the-art theoretical method for anisotropic rock deformation model has been introduced. The model applies the rock true stress-strain curves to derive the development of mobilised shear strength whenever the rock is subjected to anisotropic compression. The relationship between the position of the mobilised shear strength envelopes and the corresponding axial strain is considered as a unique rock mass inherent property and this property is used to predict the rock stressstrain response under anisotropic stress condition. The position of the mobilised shear strength envelopes is identified base on the mobilised minimum friction angle. This inherent mobilised shear strength essentially increases when the rock is subjected to anisotropic compression and this can be seen as the envelope rotates upwards towards the shear strength envelope at failure. By this manner the mobilised shear strength is incorporated in characterising the rock deformation with respect to the applied stress.
The following conclusions can be drawn from this study; i. Currently there is no theoretical framework to predict rock stress-strain response under anisotropic stress condition. ii. This anisotropic rock deformation model is the first rock theoretical framework that can predict the rock stress-strain response. iii.
The predicted stress-strain response has close agreement compared to the laboratory stressstrain curves as shown in Figure 9. This substantiates the validity of this rock volume change framework. iv.
The applicability of this theoretical framework for other rocks needs to be tested.