Issue 53

A. Kostina et alii, Frattura ed Integrità Strutturale, 53 (2020) 394-405; DOI: 10.3221/IGF-ESIS.53.30 397    σ 0 (5)   :   σ C ε ε p (6)   1 2         ε u u T (7) 0        n n F tg c (8)     ε σ  p F (9) where  ={  r ,   ,  z ,  rz } is the stress tensor,  ={  r ,   ,  z ,  rz } is the strain tensor, С is the tensor of elastic constants which reduces in the isotropic case to two elastic constants (E ′ – Young’s modulus,  – Poisson’s ratio), u ={ u r , u z } is the displacement vector,  p ={  p r ,  p  ,  p z ,  p rz } is the plastic strain tensor, F is the Mohr-Coulomb yield criterion,    n is the maximum tangential stress in the area with a normal n ,  n is the normal stress operating in the same area,  is the indefinite multiplier determined using Prager’s compatibility conditions, and the dots over the symbols denote time derivatives. The Prager compatibility conditions are written as 0, 0    F , (or 0   F (10) 0, 0     F , (and 0   F ) (11) The system of Eqns. (5)-(11) is supplemented with boundary conditions that correspond to the calculation scheme from Fig.1. 1 0   z u (12) 2     n σ P (13) 3 '     n σ P (14) where the vectors P ={ p ,0} and P ′ ={ p ′ ,0} correspond to the calculated loads obtained by the formulas:   r h p p p (15) 0 0 2 90 90 ' 2 2 2                        r m p g h tg c tg (16)     h w gw p g h (17) ' '     m p g h (18) In formulas (15) - (18), the following notation is used:  m =2000 kg/m 3 – average density of the material, h ′ – bed rock density,  w =1000 kg/m 3 – water density, g =10 m/s 2 – gravitational constant, h gw =1.5 m – groundwater depth. The load p h is considered only for saturated rocks.