Issue 53

A. Kostina et alii, Frattura ed Integrità Strutturale, 53 (2020) 394-405; DOI: 10.3221/IGF-ESIS.53.30 404 cannot be used in this case to calculate the ice wall thickness at a depth of 500 m because the base of power function becomes negative. For chalk Fig.6(b), there are two similar data blocks. The first group includes Eqns. (4) and (20), and another – Eqns. (2), (3) and (22). Eqn. (1) yields the maximum ice wall thickness reserve at all loads examined here. The maximum difference between Eqns. (4) and (20) is equal to 0.242 m and is achieved at the load of 1.881 М P а . The ice wall thicknesses which enter the second group have similar values up to the load of 1.881 М P а . At loads more than 2.25 М Pa, Eqn. (2) gives values that exceed those calculated by Eqns. (3) and (22). Analogous results are obtained for clay (Fig. 6c). Two groups of equations provide almost similar values for ice wall thickness: (4), (20) and (2), (22). The maximum difference between Eqns. (4) and (20) corresponds to the load of 0.678 М P а and is equal to 0.35m. Eqns. (2) and (22) yield close thickness values at the load less than 2.301 MPa , and Eqn. (2) cannot be used to calculate the ice wall thickness at the load exceeding this value. The maximum margin of safety is obtained using Eqn. (3). On the basis of the obtained results, some conclusions were drawn regarding the applicability of the analyzed relations to ice wall thickness calculations. To perform strength calculations, it is desirable that Eqn. (20) would be used for all soils under study because Eqn. (3) yields excess thickness, and Eqn. (4) is obtained on the assumption of the ultimate state of the ice wall, and therefore it should be used with some prescribed margin of safety. In ultimate displacement calculations for chalk, it is advisable to use Eqn. (22) because Eqn. (1) yields the excess thickness for all loads under consideration. At loads less than 2.25 MPa (this value corresponds to a depth of 300 m), Eqn. (2) can be applied. In ultimate displacements calculations for clay up to the load of 2.301 М P а (this value corresponds to a depth of 300m), relations (2) and (22) can be used. In ice wall thickness calculations for frozen sand at the load no less than 4.889 М P а (this value corresponds to a depth of 300 m), it is desirable that relation (22) is used. At loads less than 3.222 MPa (this value corresponds to a depth of 200 m), Eqn. (2) can also be applied. At high loads that exceed 4.899 for sand and 2.301 MPa for clay (these values correspond to a depth of 300 m), the ice wall thicknesses calculated by Eqns. (1)-(4), (20) and (22) exhibit great scatter, which generates a need for further investigations aimed at modifying the existing constitutive equations. This necessity stems from the fact that at great loads the hydrostatic pressure can have a significant effect on the stress-strain state of rocks. C ONCLUSION e have performed a theoretical study to evaluate the applicability of analytical Vyalov’s formulas to the calculation of the ultimate stress state of an ice-rock cylinder of unlimited and finite heights. The numerical results showed that the design values of the ice wall thickness obtained by the formula for a cylinder of limited height have a margin of safety for all considered rocks (sand, chalk, and clay). This is associated with the fact that the internal friction is ignored, which increases the strength. The application of the Coulomb–Mohr criterion made it possible to model the deformation of the ice-rock cylinder. A comparison of the thicknesses obtained numerically with the results found by Eqn. (3) showed that the greatest thickness reserve was observed for clay and sand. Using the results of numerical simulation of the stress-strain state of the ice wall, we suggested two modifications of Eqn. (4). The first variant involves the use of the piecewise-linear relation and enables describing the simulation results both qualitatively and quantitatively. The second is a simple variant and it can be used for making rapid estimation for the designed ice-wall thickness. A comprehensive comparative analysis of six formulas (four Vyalov’s formulas and two modifications) has revealed that in strength calculations it is advisable to use Eqn. (20) for all soils examined here because Eqn. (3) causes the excess margin of thickness to occur, and Eqn. (4) must be used with some prescribed margin of safety. In ultimate displacement calculations for chalk, Eqn. (22) or (2) should be used at the load not exceeding 2.25 MPa, which corresponds to the soil depth of 300 m. In creep calculations for clay at the load of 2.301 MPa, which corresponds to the soil depth of 300 m, relations (2) and (22) can be used. In order to calculate the ice-wall made of frozen sand at the load less than 4.889 MPa (up to the soil depth of 300 m), it is advisable to use relation (22) and relation (2) at the load less than 3.222 MPa (up to the soil depth of 200 m). At high loads that exceed 4.899 MPa for sand and 2.301 MPa for clay there is a need for further investigations so that the existing constitutive equations describing the stress-strain state of rocks can be modified. W