The classical theory of linear elastic fracture mechanics for a quasi-static crack in an infinite linear elastic body has two significant mathematical inconsistencies: it predicts unbounded crack-tip stresses and an elliptical crack opening profile. A new theory of fracture developed by Sendova and Walton, based on extending continuum mechanics to the nanoscale, corrects these erroneous effects. The fundamental attribute of this theory is the use of a dividing surface to describe the material interface. The dividing surface is endowed with an excess property, namely surface tension, which accounts for atomistic effects in the interfacial region. When the surface tension is taken to be a constant, Sendova and Walton show that the theory reduces the crack-tip stress from a square root to a logarithmic singularity and yields a finite angle opening profile. In addition, they show that if the surface tension depends on curvature, the theory completely removes the stress singularity at the crack-tip, for all but countably many values of the two surface tension parameters, and yields a cusp-like opening profile. In this work, we develop a numerical model using the finite element method for the Sendova-Walton fracture theory applied to the classical Griffith crack problem in the case of constant surface tension. We show that the numerical model behaves as predicted by the theory, yielding a reduced crack-tip singularity and a finite opening angle for all nonzero values of the constant surface tension. We also lay the groundwork for the numerical implementation of the curvature-dependent model by constructing an algorithm to determine the appropriate threshold values for the surface tension parameters that guarantee bounded crack-tip stresses. These values can then be directly applied to the forthcoming numerical model.

Une bonne tenue mécanique des structures du génie civil en béton armé sous chargements dynamiques sévères est primordiale pour la sécurité et nécessite une évaluation précise de leur comportement en présence de propagation dynamique de fissures. Dans ce travail, on se focalise sur la modélisation constitutive du béton assimilé à un matériau élastique-fragile endommageable. La localisation des déformations sera régie par un modèle d'endommagement à gradient où un champ scalaire réalise une description régularisée des phénomènes de rupture dynamique. La contribution de cette étude est à la fois théorique et numérique. On propose une formulation variationnelle des modèles d'endommagement à gradient en dynamique. Une définition rigoureuse de plusieurs taux de restitution d'énergie dans le modèle d'endommagement est donnée et on démontre que la propagation dynamique de fissures est régie par un critère de Griffith généralisé. On décrit ensuite une implémentation numérique efficace basée sur une discrétisation par éléments finis standards en espace et la méthode de Newmark en temps dans un cadre de calcul parallèle. Les résultats de simulation de plusieurs problèmes modèles sont discutés d'un point de vue numérique et physique. Les lois constitutives d'endommagement et les formulations d'asymétrie en traction et compression sont comparées par rapport à leur aptitude à modéliser la rupture fragile. Les propriétés spécifiques du modèle d'endommagement à gradient en dynamique sont analysées pour différentes phases de l'évolution de fissures : nucléation, initiation, propagation, arrêt, branchement et bifurcation. Des comparaisons avec les résultats expérimentaux sont aussi réalisées afin de valider le modèle et proposer des axes d'amélioration. / In civil engineering, mechanical integrity of the reinforced concrete structures under severe transient dynamic loading conditions is of paramount importance for safety and calls for an accurate assessment of structural behaviors in presence of dynamic crack propagation. In this work, we focus on the constitutive modeling of concrete regarded as an elastic-damage brittle material. The strain localization evolution is governed by a gradient-damage approach where a scalar field achieves a smeared description of dynamic fracture phenomena. The contribution of the present work is both theoretical and numerical. We propose a variationally consistent formulation of dynamic gradient damage models. A formal definition of several energy release rate concepts in the gradient damage model is given and we show that the dynamic crack tip equation of motion is governed by a generalized Griffith criterion. We then give an efficient numerical implementation of the model based on a standard finite-element spatial discretization and the Newmark time-stepping methods in a parallel computing framework. Simulation results of several problems are discussed both from a computational and physical point of view. Different damage constitutive laws and tension-compression asymmetry formulations are compared with respect to their aptitude to approximate brittle fracture. Specific properties of the dynamic gradient damage model are investigated for different phases of the crack evolution: nucleation, initiation, propagation, arrest, kinking and branching. Comparisons with experimental results are also performed in order to validate the model and indicate its further improvement.

[pt] O presente trabalho tem como objetivo a implementação e avaliação de um modelo constitutivo para solos reforçados com fibra (compósito). A principal característica do modelo constitutivo implementado é que cada material (matriz de solo e fibra) segue sua própria lei constitutiva e ao mesmo tempo interagem entre si. Utilizando um algoritmo explícito, são implementados os modelos Cam Clay Modificado e Lade-Kim para a matriz de solo, cuja verificação é feita mediante o programa PLAXIS 2D e curvas tensão-deformação obtidas da literatura, respectivamente. Posteriormente, é adicionado o comportamento da fibra no desenvolvimento das tensões no compósito e verificado mediante a comparação das curvas tensão-deformação com as apresentadas por Diambra et al. (2013). As linguagens de programação utilizadas neste trabalho foram duas, a primeira é a utilizada no programa MATLAB, onde os códigos dos modelos são verificados e validados em relação à um conjunto de ensaios triaxiais de areia reforçada com fibra. Posteriormente foi usada a linguagem de programação FORTRAN para incluir o modelo constitutivo para solo reforçado com fibras no programa de elementos finitos ABAQUS, através da sub-rotina UMAT. Porém, para a implementação na sub-rotina UMAT os códigos dos modelos implementados no MATLAB sofrem algumas modificações com a finalidade de que o ABAQUS consiga compilar e representar adequadamento o comportamento do modelo constitutivo, mediante a correta utilização de vetores e propriedades desta. Finalmente, são modelados ensaios triaxiais drenados para verificar que a implementação mediante a sub-rotina UMAT é satisfatória. / [en] The present work aims to implement and evaluate a constitutive model for fiber-reinforced soils (composite). The main characteristic of the constitutive model implemented is that each material (soil and fiber matrix) follows its own constitutive law and at the same time interact with each other. Using an explicit algorithm, the Cam Clay Modified and Lade-Kim models are implemented for the soil matrix, verified by the PLAXIS 2D software and stress-strain curves obtained from the literature, respectively. Later, it is included the behavior of the fiber in the development of the stresses in the composite and verified by the comparison of the stress-strain curves with those presented by Diambra et al. (2013). The programming languages used in this work were two, the first one is the one used in the MATLAB program, where the codes of the models are verified and validated in relation to a set of triaxial tests of fiber-reinforced sands. Later the programming language was converted into FORTRAN to include the constitutive model for fiber reinforced soil in the ABAQUS finite element software, through the UMAT subroutine. However, for the implementation in the UMAT subroutine the codes of the models implemented in MATLAB undergo some modifications in order that ABAQUS can compile and represent adequately the behavior of the constitutive model through the correct use of vectors and its properties. Finally, drained triaxial tests are modeled to verify that the implementation through the UMAT subroutine is satisfactory.