The stated goal of this work is to develop a numerical explicit finite element method able to effectively address the strain localization process, discrete fracture and fragmentation process in quasi-brittle materials, involving highly nonlinear processes, such as, contact, non-linear geometric and material behavior. The description of the quasi-brittle material is based in the concepts of classical fracture mechanics plasticity theory and more specifically in the plastic damage model and its fundamental variable, the plastic damage variable, using traditional Mohr-Coulomb and Rankine models. Three types of algorithms were needed to reach our goal. A search contact algorithm, a combined penalty and Lagrange multipliers algorithm and adaptive local refinement algorithm. The search algorithms is necessary to determine the pairs of potential contacts with a reasonable computational cost; the second for establishing and compute the normal contact forces and the third one to allow not only a local refinement but also draw geometric modeling of discrete fracture. Moreover, given the inability of the irreducible linear elements formulation to capture collapse mechanisms, ultimate loads and strain localization processes, we opted to develop an explicit mixed finite element displacements and strain formulation in solid mechanics stabilized with variational sub-scales method, particularly using Orthogonal Sub-scales Method (OSS). This stabilized formulation avoids the LBB condition ( inf-sup condition) and allow to use elements with independent and equal interpolation order for displacements and strain fields. This formulation is one of the main original contribution of this work. Our methodology, called MEX-FEM, and developed to address problems in small and large deformations, not only it comes to enhance the traditional explicit time integration scheme, but also can get an improved stresses and strains field in a finite element mesh and also a better description of the strain localization virtually free of mesh-bias dependence without using tracking algorithms and accurate solutions in nearly incompressible problems.These numerical tools developed were integrated as a whole for our main goal: the discrete modeling fracture and fragmentation of quasi-brittle materials using finite element method.

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