Calculation of lateral stresses for uniaxial compression of geomaterial samples

A method is proposed for calculating the transverce stresses during uniaxial compression of geomaterial samples from the measured load and mutually perpendicular deformations. Analytical expressions connecting the indicated parameters are obtained. The dependences of the change in the calculated transverse stresses on time are plotted for various values of Poisson’s ratio. The difference in transverse stresses demonstrates a much greater sensitivity to mechanical stress than each transverse stress separately. Sharp changes in the values of the difference in transverse stresses are observed, which coincide with bursts of AE activity.


Introduction
The similarity of processes occurring in complex geological structures of the earth's crust and deformation processes observed in laboratory experiments on loaded rock samples [1][2][3][4][5][6] creates a prerequisite for studying processes occurring in the depths of the earth's crust by physical modeling on rock samples. Modeling of inelastic deformation, structure formation and fracture by testing samples of geomaterials on rheological presses has already demonstrated its effectiveness for a number of problems in seismology and tectonophysics [7][8][9][10].
In experiments on the destruction of geomaterials, installations with uniaxial compression are traditionally used. In this case, in addition to measuring acoustic emission (AE) and axial load, as a rule, measurements are made of the deformations of the sample under study in three mutually perpendicular directions. In order to have a complete picture of the stress-strain state, it is necessary, in addition to the principal stress, to obtain also the values of the transverse stresses.
The purpose of this work is to calculate the transverse stresses during uniaxial compression of the sample.
1.1. The condition for the occurrence of transverse stresses. Uniaxial loading of a rectangular specimen is considered (Fig. 1). The sample is positioned in a Cartesian coordinate system centered in the middle. Three mutually perpendicular directions can always be drawn through any point of the deformable body, the shifts between which are taken to be zero. Figure 1. Schematic arrangement of a rectangular solid body in accordance with the Cartesian coordinate system. When a load is applied to a sample of square cross-section, mechanical stresses arise in it, the mathematical description of which is given, respectively, by the tensors of deformations and stresses: The relationship between the components of strain and stress tensors (1) is described by the generalized Hooke's law in the form: where cijkl is the fourth-order stiffness tensor, N and M are the number of rows and columns of the deformation tensor in the Cartesian coordinate system. The stress tensor σ is usually represented as the sum of the spherical stress tensor σs and the stress deviator σd: In what follows, only the components of the spherical tensor will be considered. The stress spherical tensor is called the mean pressure at a point and characterizes all-round uniform compression or tension: After transforming and simplifying the components of the stiffness tensor cijkl in accordance with the Voigt notation, equation (2) can be represented in matrix form: i.e., lateral stresses are absent or negligible. The condition for the occurrence of transverse deformations: − > 0 (7) The fulfillment of condition (7) allows deriving from the system of equations (5) the design formulas for transverse stresses: The transverse stresses obtained from formulas (8) and (9) are functions of time: If condition (7) is met, the difference in lateral deformations can be converted to the difference in the occurring transverse stresses, which carry important information about the mechanical processes during compression of the sample. Differences in transverse deformations and stresses can be converted to a more convenient form: The relationship between the differences in stresses and strains comes from equation (5) taking into account transformations (11) and (12):   The installation for loading rock samples is a gravity-lever press with a linearly increasing load, the maximum value of which is 250 kN (Fig. 2). The uniaxial compression experiment was carried out using water leakage [11]. The location of the strain gauges corresponds to the Cartesian coordinate system.

Results
The samples were made from natural sandstone from the Kegety deposit, Kyrgyzstan, and had the shape of a parallelepiped with dimensions of 24.5 × 25.5 × 61 mm. Poisson's ratio of sandstone is in the range 3.3 ÷ 7.5. The Young's modulus required for calculations was taken equal to 0.37 MPa, and was considered constant throughout the experiment. Linear differential transformers with the following ranges of registered linear displacements were used as sensors recording longitudinal and lateral deformations: ± 0.127 mm; ± 0.254 mm and ± 1.27 mm. During the uniaxial compression experiment, the load on the sample varied from 36.30 kN to 87.24 kN.
In Fig. 3 shows     Figure 4. Deformation diagram of a sandstone sample. The calculation of the values of transverse deformations was carried out on the basis of expressions (8) and (9). In this case, not one value of Poisson's ratio was used, but the interval from 3.3 to 7.5 with a step of 0.53. In Fig. 5 and 6 show the dependences of the obtained transverse stresses along the x and y axes, respectively, for different values of Poisson's ratio.  Poisson's ratio. The difference in transverse stresses demonstrates a much greater sensitivity to mechanical stress than each transverse stress separately. Sharp changes in the values of ∆σxy are observed, which coincide with bursts of AE activity. The AE activity, in turn, demonstrates the rate of defect formation in the sample under study. It can be argued that an increase in transverse stresses leads to the formation of new defects, which, in turn, is accompanied by an increase in the AE activity. When approaching the moment of destruction, starting from 9.5 × 10 4 seconds of observation, the curves intersect at one point, which corresponds to the moment of destruction. Further bursts of AE activity and a sharp slope of the difference in transverse stresses from 11.5 × 10 4 to 12.02 × 10 4 s indicate the final destruction of the sample.

Conclusion
Analytical expressions have been obtained that make it possible to calculate the transverse stresses during uniaxial compression of the sample from the measured mutually perpendicular deformations and the value of the axial load. The obtained transverse stresses make it possible to obtain a complete picture of the stress-strain state of the sample under study. Changes in transverse stresses in time are plotted. The results of experimental data processing demonstrate an increase in transverse stresses with increasing load. The deformation along the x axis has a large steepness in relation to the deformation of the y axis, intersects with it at the point corresponding to the onset of destruction of the sample. This is due to factors such as the location of the main crack and roughness of the sample surface. Comparison of the difference between the transverse stresses and the activity of acoustic emission demonstrates the general picture of the behavior of the parameters: there are sharp changes in the values of ∆σxy, mass formation of defects, accompanied by bursts of AE activity.