# Eigendegradation Algorithm Applied to Visco-Plastic Weak Layers
## 2 Constitutive Model
### Eigenversion and Eigensoftening Algorithms  
Within the context of the OTM formulation, fracture can be modeled simply by eroding material points according to an energy-release or stress criteria, depending on whether the eigenerosion [47; 48; 49; 50] or the eigensoftening [38; 39; 51] algorithm are adopted, respectively. In both methodologies, if the material points failed, in order to approximate the presence of cracks, the material points can be extracted from the computation of stresses. However, the way to reach the zero stiffness state is different for each model: meanwhile, in the eigenerosion, the failure is instantaneous, in the eigensoftening, the material follows a softening curve, which depends on the material. It needs to be noticed that when a material point satisfies the failure condition, its contribution to the material stiffness matrix, as well as to the internal forces, is set to zero. However, it is necessary to maintain its contribution to the mass matrix in order to fulfill the mass conservation law. We only can discard it if an eroded material point is finally unconnected to any other node. In the following, we compute the energy-release rate attendant to the failure of the material point \(p\), which is the starting point of the aforementioned methodologies:  
\[G_{p,k+1} = \frac{Ce}{m_{p,k+1}}\sum_{x_{q,k+1}\in B_{e}(x_{p,k+1})}m_{q}W_{k }(F_{q,k+1}),\] \[m_{p,k+1} = \sum_{x_{q,k+1}\in B_{e}(x_{p,k+1})}m_{q}, \tag{10}\]  
where \(W_{k}(F_{q,k+1})\) is the free-energy density per unit mass at the material point \(x_{q,k+1}\), and \(B_{e}(x_{p,k+1})\) is the \(e\)-neighborhood of the material point, which is calculated as the volume of \(e\) size centered at the aforementioned \(x_{p,k+1}\). \(m_{p,k+1}\) is the mass of this neighborhood and \(C\) is a normalizing constant, which ranges from 1 to 2, in order to extend the influence from the first line of neighbors to the second [47; 48]. Everything is evaluated at the time step \(k+1\). A scheme of the \(e\)-neighborhood and how it is configured is plotted in Figure 1.  
The material point is failed when the value of \(G_{p,k+1}\) (Equation (10)) exceeds the critical energy release rate, \(G_{F}\). This parameter estimates the specific energy required to create a fracture surface per unit of area. In the proposed algorithm, every time step the eroded (failed) material-point set was updated, taking into account this criterion. [PERSON] et al. [47] have demonstrated that this approximation converges to Griffith fracture for linear elasticity when an infinitely fine discretization is presumed. Certainly, schemes that estimate the energy-release rate based on the energy of a single material point may suffer from an overestimation of the toughness of the material as well as mesh-dependency if non-local approaches are adopted. On the other hand, the implementation of the eigensoftening algorithm consists of adopting a strength criterion for crack initiation and a law which is capable of reproducing a proper reduction of strength of the material under study before the formation of a stress-free  
Figure 1: Scheme of a set of fractured material points as a fractured layer (black dots) and the \(e\)-neighborhood of the points neighbors of the material point at the crack tip (gray dots). crack, which is known as softening. This second process tends to accumulate less energy until the crack appears. When the tensile strength, \(f_{t}\), is achieved, a crack is conformed with zero opening displacement. Because of its cohesive behavior, once the opening displacement, \(w\), reaches a critical vale, \(w_{c}\), a stress-free crack is attained. The energy below the softening curve represents the fracture energy per unit of area in static conditions, \(G_{F}\), which is sketched in Figure 2[38]. For the eigensoftening calculation, Equation (10) can be rewritten in terms of the principal stresses at time \(t_{k+1}\), since this model employs the first principal stress as a failure criterion.