Issue 53

A. Kostina et alii, Frattura ed Integrità Strutturale, 53 (2020) 394-405; DOI: 10.3221/IGF-ESIS.53.30 395 prevents its failure under external loads, i.e. no cracks occur in the wall. At this thickness value, the ice wall is in the ultimate stress-strain state, i.e., in the state where the stresses at a given instant of time do not exceed the frozen soil strength at the same instant. The most common methods for calculating of ice wall thickness are those proposed by Domke [1] and Lame-Gadolin [1]. The Lame-Godolin formula was obtained in a linear elastic approximation and did not prove effectiveness at soil depths of more than 50 m. The Domke equation takes into account the plastic deformations of frozen soil, the onset of which is determined according to the criterion for the difference between the largest and smallest principal normal stresses. Because the strength of saturated soils increases with an increase in the mechanical pressure up to a certain value, the application of the Mohr-Coulomb and Druker-Prager criteria makes it possible to calculate plastic strain with higher accuracy and to take into account the fact that the frozen soil strength increases with decreasing temperature [2]. The triggering of high hydrostatic pressure leads to melting and failure of the ice contained in the pores. This has given impetus to different modifications of these criteria [3]. The thermo-hydro-dynamical models of soil freezing which take into account plastic strains are presented in [4-7]. In [4-6], the so-called Barcelona Basic Model [8] is used to describe the plastic deformations of saturated frozen soils. According to this model, the stress-strain state of soils is determined by analyzing two state variables – effective stress and a parameter that describes the moisture migration effect. For the non-frozen soil, the Barcelona Basic Model is simplified and reduced to the modified Cam-Clay model. The thermo-hydro-dynamical models based on this approach can be employed to consider the influence of moisture migration and mechanical pressure on the plastic deformation of frozen soils. In [6], the Barcelona Basic Model is generalized to the case of large deformations. Another way [7] has been proposed to derive the constitutive relations for describing the elastoplastic behavior of frozen soils in the framework of a thermodynamic approach. It has been suggested in [9] that the yield surface defined in terms of a double clay hardening model can be used to assess plastic strains. Elastic and plastic strains characterize the instantaneous response of the material to the applied loads. However, due to the pronounced rheological behavior of saturated soils under long-term loads, their deformation increases and strength reduces. This feature of soils is of prime importance for shaft sinking where the loads from the surrounding rocks and ground water are taken over by the lateral surface of a mine during the entire time required for lining construction. In order to calculate the optimal thickness of an ice wall (  ) and to take into account the rheological behavior of frozen soils, S.S.Vyalov suggested the following relations [10, 11]:     1 1 1 1 1 ' 1 ,                              m m m p m p h E a k A t T a (1)                                            1 2 1 1 2 , m m p m p E a A t T a (2)   3   s p ph E t (3)                                             2 0 1 2 0 45 /2 1 0 45 / 2 1 1 1 2 45 / 2 tg p p tg E a c t tg (4) where  is the optimal ice wall thickness, a is the cylinder inner radius, k’ is the coefficient dependent on soil compaction conditions, m,  A(t p ,  ) are the parameters characterizing the rheological behavior of the soil, t p is the loading period , p is the load applied to the ice wall, h is the height of unfixed part of mine shaft,  is the maximum radial movement of the frozen