# Evaluation of the Implicit Gradient-Enhanced Regularization of a Damage-Plasticity Rock Model
## 1 Introduction  
The appropriate representation of complex failure modes of cohesive-frictional materials in numerical simulations is of great interest for many engineering applications, which include, among others, geotechnical applications characterized by material failure playing a dominant role in the overall structural response [1; 2]. A prominent example of material failure under highly confined stress states is given in the context of tunnel construction: During the construction of deep tunnels, high geostat stresses in the surrounding rock mass are redistributed in consequence of the excavation process [3; 4]. Depending on the quality of the surrounding rock mass, the installed supporting measures, and the type of excavation process, those stress redistributions can lead to damage in the rock mass, often accompanied by the transition from the rock mass as a continuum to rock blocks moving towards the tunnel center with localized shear bands indicating the sliding interfaces. It follows that an adequate representation of such failure phenomena by means of advanced constitutive models is of great importance and a prerequisite for the risk assessment of potential structural collapse. Constitutive models based on the combination of the theory of plasticity and continuum damage mechanics, simply denoted as damage-plasticity models, provide a powerful framework for describing inelastic deformations, hardening and softening material behavior, as well as stiffness degradation due to damage. They are well suited for the description of cohesive-frictional materials such as concrete, rock, and soils, since different types of material failure, e.g., cracking in tension, crushing in compression, or failure under mixed stress states, can be described in a realistic manner [5]. On the basis of damage-plasticity models, the material behavior is described in terms of mere continuumrelations, i.e., constitutive relations describing the behavior of an infinitesimal material point. However, softening material behavior, described in terms of continuum models, exhibits several theoretical deficiencies, as reported and summarized in [6]:  
* the softening process zone is infinitesimally small;
* at structural level, snapback behavior due to the infinitesimally small softening zone is observed;
* the dissipated energy during the failure process is zero due to the infinitesimal zone in which energy is dissipated. In a mathematical sense, those issues are related to the loss of ellipticity of the initial boundary value problem (IBVP) in static and quasi-static analyses. In finite element analyses, those deficiencies lead to mesh-dependent results, commonly referred to as a pathological mesh sensitivity. Failure patterns like cracks tend to localize into the smallest possible bandwidth, i.e., usually a single layer of finite elements, and upon mesh refinement the localization zone is decreasing to an arbitrarily small domain. Accordingly, the obtained results are not objective with respect to the finite element mesh, i.e., the results are sensitive with respect to the numerical discretization scheme. In the past decades, several techniques to overcome these deficiencies in numerical simulations have been proposed, for instance the crack band approach based on a mesh-adjusted softening modulus [7], nonlocal approaches of the integral type [8], models based on the Cosserat continuum [9], viscoplastic formulations [10], phase field models [11], and explicit and implicit gradient-enhanced formulations [12]. Among these approaches, models based on implicit gradient-enhanced formulations by now form a well-established branch in the literature due to their computational efficiency and numerical stability [12]. Numerous implicit gradient-enhanced damage and plasticity models have been proposed in recent years [13; 14; 15; 16; 17; 18; 19; 20; 21], and their performance has been demonstrated based on different examples of material failure.