GNGTS 2018 - 37° Convegno Nazionale

766 GNGTS 2018 S essione 3.3 an extension of the PM-theory that allows plastic deformation during the first compression of the medium. We cast such elasto-plastic PM-model in matrix notation, following the discrete formulation of (Guyer et al. , 1997). We apply the model to two loading/unloading cycles of the Gulf of Mexico sand (Zimmer, 2003) and we compare it with the classical PM model. The proposed examples demonstrate that the elasto-plastic PM space is able to correctly model the plastic deformation, and this improves the prediction of the stress path and of the dynamic bulk modulus. The elasto-plastic Preisach-Mayergoyz model. Rocks have a hysteretic nonlinear elastic behaviour. The Preisach-Mayergoyz model represents a rock as a collection of hysterestic mesoscopic elastic units (HMEU). An example of HMEU is shown in Fig. 1a. Each unit is characterised by a pair of pressures ( P c , P o ) which are associated to a pair of equilibrium lengths ( l c , l o ), where l c < l o ). During compression, the unit has length l o until it reaches pressure P c at which it contracts to length l c , analogously, during unloading, the unit has length l c until it reaches pressure P o at which it expands to length l o . When P c ≠ P o , the unit behaves hysteretically. For consolidated rocks, we can imagine a unit as a small cube of rock with a crack that closes and opens under different pressure conditions. For unconsolidated rocks, the strain corresponds to deformation, slipping and rotation of grains, which causes closure of gaps in the interstitial pore space between grains. The density ρ of HMEU in the ( P c , P o ) space determines the macroscopic elastic behaviour of the rock. Fig. 1b shows an example of distribution of HMEU in the ( P c , P o ) space. In the figure, the blue circles represent the classic PM-space, whereas the red circles are the plastic part. During compression, the units progressively close from left to right. Analogously, during unloading the units progressively open from top to bottom. In the plastic formulation of the PM space that we propose, we allow P o to assume negative values (red circles). This choice has importan consequences. During compression, the units with negative P o contribute to the total strain, as these units close during the scanning of the PM space from left to right. Differently, during unloading, the space is scanned from top to bottom down to zero pressure, therefore the negative P o units do not contribute in strain or bulk modulus, they remain close after unloading and they produce an irreversible plastic strain. In our formulation, we follow the discretized representation of the PM space of Guyer et al. (1997), such that the density, ρ , in the PM space is represented as a matrix in which each entry spans a square interval Δ P × Δ P : (1) Note that in this representation the rock behaviour is defined only on Δ P intervals. In this notation, to account for plastic strain, it is sufficient to add an extra row at the bottom of the density matrix. This row corresponds to the contribution of negative P o (see Fig. 1c) to ρ , and it therefore characterizes the plastic behaviour of the medium during the first loading. After the first compression, the medium behaves elastically. As the current pressure, P , increases, all the units with P c ≤ P close. For convenience of the notation, Guyer et al. (1997) introduced a state matrix σ mn , which can assume 0 and 1 values and it indicates whether the (m,n) cell of the ρ matrix is open or closed. According to it, the current strain is: (2) where Δ ε = ( l o – l c )/ l o is the strain of a single unit, which is supposed to be the same for all units for simplicity. Note that the second summation starts from n = – 1, which is the index of the plastic row of ρ . The static bulk modulus for the loading and unloading paths are: