# Eigendegradation Algorithm Applied to Visco-Plastic Weak Layers
## 1 Introduction  
The failure of geomaterials has been a topic of study since the early work of [PERSON] back in 1877. The work of [PERSON] and [PERSON] [1] is an example of the theoretical derivation of these problems. Failures can be classified in two types depending on the state of the material: diffuse and localized. Diffuse failures are associated with the instability of loose and saturated materials (e.g., [2; 3]) under monotonic and dynamic loads. In the latter, the increase of the pore pressure can cause the liquefaction of the material, when its effective stress vanishes as a result of this increase of pore pressure while the total stress remains constant. This phenomenon can produce landslides, involving the displacement of large volumes of material. The numerical modeling of the initiation of diffuse failure has been developed in finite element based codes to reproduce failures in mine tailing dams [4; 5], landslides [6; 7] or the study of the response of loose saturated, granular material under seismic action [8; 9]. Recently, these phenomena have been studied with meshfree techniques, allowing to evaluate the transition from solid to fluidized materials with SPH techniques with one (e.g., [3; 10; 11; 12; 13; 14]) and two phases [15; 16], Material Point Method, MPM, (e.g., [17; 18; 19; 20]) or Discrete Element Methods, DEM, for the liquefaction on granular materials (e.g., [21; 22]). In contrast, localized failures are associated with the concentration of strains in a narrow zone and a limited region, which produces a discontinuity in the deformation as well as in the strain rate. This failure is related to a weak discontinuity, which is also known as a weak layer. The progress of the failure of overconsolidated clays that present a softening behavior has been extensively studied. It consists of the reduction of the shear strength from the peak to the residual value. Early work by [PERSON] [23] provided the theoretical basis for understanding the physics and mathematics of the problem. The formation and propagation of shear bands have been studied through theoretical and lab-scale models(e.g., [24; 25]). Numerically, the localization of shear deformations associated with softening behaviors, which is related to negative slopes in the loading-displacement curve, traditionally depends on the mesh discretization (e.g., [26]. The employment of visco-plastic models has allowed the regularization of the problem (e.g., [27]). Visco-plastic formulations are common in the literature to evaluate the rate-dependent failure of geomaterials, shear bands, creep and stress relaxation. In general, such constitutive formulations include empirical models focused on the evaluation of creep and stress relaxation for soft clays [28; 29], rheological models for a wide range of deformations (e.g., [11; 30; 31]) and general stress-strain-time constitutive models (e.g., [32; 33]). Elasto-plastic constitutive models incorporating rate dependence are based mainly on the over-stress theory [34] and on the concept of a non-stationary flow surface (e.g., [35; 36]). The over-stress theory assumes there is no viscous strains occurring inside the static yield surface (elastic region). However, the inelastic region is rate dependent. This dependency is defined by an over-stress factor. This theory has been implemented in finite element and meshfree methods to study different geotechnical problems (e.g., [31; 37]). In this paper, we propose a methodology for evaluating the progression of a shear band or a weak layer through a phenomenological visco-plastic model, taking the benefits of a non-local failure approach. The proposed algorithm is based on the same principles of both eigenerosion and eigensoftening approaches, which have been widely employed through material point-based frameworks in order to assess the fracture evolution in quasi-brittle materials ([38; 39]) once specific energy (eigenerosion) or stress (eigensoftening) thresholds are reached when the material point fails.