One-dimensional localisation studied with a second grade model

Restoring Mesh Independency in FEM-DEM Multi-scale Modelling of Strain Localization Using Second Gradient Regularization

“…In this way, the kinematics of the media is enriched by its micro-scale introducing a local dependence on an internal length parameter. This characteristic length [1] regularizes the solution making strain localization mesh independent. A local formulation complies with the principle of local action so it states a stress-strain relationship in the same manner as a classical con-stitutive laws do, this makes the implementation of local regularization in classical models a straight forward procedure (Fig.…”

“…An example of nonlocal regularization in a FEMxDEM model is presented in [12]. Local formulations use a local relationship between stress and strains in the same manner as classical constitutive relations are defined.Second gradient model, as a particular case of the Germain theory [6] has been developed [4,21,1,14]. It has been extensively applied in geomechanics and engineering applications with satisfactory results [3,13,24].…”

Using Shear Strain Localisation to Model the Fracturing Around Gallery in Unsaturated Callovo-Oxfordian Claystone

“…An enhanced model, introducing an internal length scale, is thus needed to correctly model the post peak and localisation behaviour. Among the different regularization methods, the second gradient local model (Chambon et al 1998) is used. In the latter, the continuum is enriched with microstructure effects: the kinematics includes the classical ones (macro) and the microkinematics (Mindlin 1964).…”

“…The second gradient law gives Σ ijk as a function of the (micro) second gradient. It is a linear elastic law with isotropic linear relationship deduced from Mindlin (1964) and it depends only on one elastic parameter D. The shear band width is proportional to this elastic parameter (Chambon et al 1998) and Σ ijk has no link with p w .…”

Modelling the influence of strain localisation and viscosity on the behaviour of underground drifts drilled in claystone

“…11 is composed of only one elastic parameter D in its simplified version proposed in [32]. This elastic modulus represents the physical microstructure of the material and is linked to an internal length scale introduced to regularise the post-localisation material behaviour [33,34]. The shear…”