Determination of the characteristic length for the second gradient dilation model associated to a Druckerprager behavior: application to the clays of Sisteron

Document available in extended abstract form only. In order to describe rocks behavior with constitutive models able to represent the degradation of the material, a regularization method is used to limit the influence of the spatial discretization. The recent development of the regularization method called the second gradient of dilation in the finite element software Code-Aster (developed by EDF-R and D) allows an objective numerical description of coupled problem for dilating geo-materials in post localized calculations. In the second gradient of dilation method, an additional parameter must be determined with regard to the more classical models. This coefficient sets the thickness of numerical shear bands that appear in post localized states. It also rules the slope of the stress-strain curve which is directly connected to the energy dissipated during the degradation of the material. The aim of the work performed on clays extracted from the EDF site of Sisteron in France was to determine this coefficient for the material taking into account the experimental data available. The idea is to establish a link between the numerical shear bands, which result from the diffusion of the plastic strain in the mesh, and the experimental induced structure that appears for non-zero confining pressures in tri-axial tests. Experimental tri-axial compression tests have been carried out in 2011 by the LaEGO on macroscopic samples (76*38 mm) from the clay of Sisteron. These tests have been achieved for different confining pressures (0, 2, 5 and 10 MPa). The constitutive model used for this approach is the associated Drucker-Prager model with a linear softening evolution. This model was chosen for its simplicity and the small number of parameters governing the slope after the peak strength. Obviously, this model can reproduce only roughly the behavior of the material but the goal is essentially to define a methodology which could be reproduced with more complex models like the L and K model. The methodology followed throughout this work is described in the following. - The first step is the classical approach without regularization method on a material point, in order to settle the numerical constitutive model on experimental results and determine a set of parameters subsequently used as input data for the non-local approach. - Then, a characteristic length (bandwidth) is chosen, according to the size of the sample and the spatial discretization of the mesh. In our case, this length is taken equal to 1 cm with about 6 mesh elements in the bandwidth. - A theoretical indicator is then used to determine an analytical ratio representing the coefficient of regularization (coefficient to determine) out of the characteristic length squared. This theoretical factor is extracted from the resolution of a 1D analytical problem of a dilatant shear band using the tangent stiffness matrix of the problem. Ratios are calculated for different slopes after the peak strength and according to the chosen characteristic length, the coefficient of regularization to use in non-local computations can be deduced. - Non-local computations are conducted with the corresponding values of that coefficient. Numerical results can be confronted to experimental data in the stress-strain curve and shear bands can be observed. The methodology initiated in this work seems to give first good results. It allows for a given material and a given mesh to determine the coefficient of regularization taking into account the behavior of the material considered. However, the methodology used must be reproduced with different slopes and different characteristic lengths to determine which unique couple better fits the experimental data. The prospects for this work concern the transfer of the experience acquired: - on more complex constitutive models (as L and K model for Callovo-Oxfordian clays); - taking into account the non-uniqueness of bifurcated solutions obtained; - to underground applications on the research laboratory of Andra. (authors)