PSI - Issue 32

A.A. Baryakh et al. / Procedia Structural Integrity 32 (2021) 109–116 A.A. Baryakh et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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element method in displacements (Zienkiewicz et al. 2013) with a discretisation of the considered area into triangular elements of the first order. The solution to the elastoplastic problem was based on the initial stress method (Fadeev 1987). The mathematical modelling methodology has been described in Baryakh et al. (2012, 2017). The localisation of plastic deformations in the physical sense was interpreted as the destruction of rocks of the waterblocking stratum: in the compression area, it was due to the development of shear cracks, and in the tension area, it resulted from the opening mode fracture. Based on these facts, we analysed the risk of discontinuity within the waterblocking strata due to mining operations. To analyse the deformation of the underworked mass ovver time, a rheological approach was used, based on a mathematical description of the graphs of the subsidence increase at the earth’s surface (Baryakh et al. 200 5). Within the linear theory of hereditary viscoelasticity, it was assumed that the vertical displacements of the daylight surface change over time, according to the relation: ൌ Ͳ ͳ ൅ Ͳ ൌ Ͳ ͳ ൅ (4) where Ͳ are elastic displacements related to the formation of the worked-out space; ሺ − ሻ is the influence function of stress ሺ ሻ at the instant of on deformation at the instant of , and Φ ሺ ሻ is the creep function. Since the real graphs of the increase in subsidence are determined by the difference between the total and elastic displacements, then: ൌ − Ͳ ൌ Ͳ (5) In this case, the creep function Φ ሺ ሻ is determined by the actual or predicted graphs of the subsidence increase at the earth’s surface. The implementation of relations (4) - (5) can be carried out based on a modification of the known rheological analysis – the method of alternating elastic moduli (Amusin 1973). The seam strain modulus at the time of mining can be determined by the expression: ൌ ൅ ൌ ͳ − (6) where ǡ are the chamber and pillar widths, respectively, is the recovery efficiency, and is the rock deformation module. It can be easily shown mathematically that in the zone of complete undermining, the dependence of vertical displacements over time will be determined by formulas (4), (5), if the alternating module of the worked-out layer is represented by the relation (Baryakh et al. 2005): ൌ ͳ൅ሺͳ − ሻ ሺ ሻ (7) When mining several layers, the expression (7) with the corresponding parameters (recovery efficiency, rock deformation module) and creep function applies to each of them. Currently, the creep function is determined using the actual graphs of the subsidence increase of the earth’s surface according to the relation: ሺ ሻ ൌ ሺ ሻ Ͳ (8) To implement this rheological approach for a specific field, information about the displacement progress over time must be known.