The application of polynomial reproducing methods has been explored in the context of linear and non linear problems. Of specific interest is the application of a recently developed reproducing scheme, referred to as the error reproducing kernel method (ERKM), which uses non-uniform rational B-splines (NURBS) to construct the basis functions, an aspect that potentially helps bring in locall support, convex approximation and variation diminishing properties in the functional approximation. Polynomial reproducing methods have been applied to solve problems coming under the class of a simplified theory called Cosserat theory. Structures such as a rod which have special geometric properties can be modeled with the aid of such simplified theories. It has been observed that the application of mesh-free methods to solve the aforementioned problems has the advantage that large deformations and exact cross-sectional deformations in a rod could be captured exactly by modeling the rod just in one dimension without the problem of distortion of elements or element locking which would have had some effect if the problem were to be solved using mesh based methods. Polynomial reproducing methods have been applied to problems in fracture mechanics to study the propagation of crack in a structure. As it is often desirable to limit the use of the polynomial reproducing methods to some parts of the domain where their unique advantages such as fast convergence, good accuracy, smooth derivatives, and trivial adaptivity are beneficial, a coupling procedure has been adopted with the objective of using the advantages of both FEM and polynomial reproducing methods. Exploration of SMW (Sherman-Morrison-Woodbury) in the context of polynomial reproducing methods has been done which would assist in calculating the inverse of a perturbed matrix (stiffness matrix in our case). This would to a great extent reduce the cost of computation. In this thesis, as a first step attempts have been made to apply Mesh free cosserat theory to one dimensional problems. The idea was to bring out the advantages and limitations of mesh free cosserat theory and then extend it to 2D problems.

[pt] Esta dissertação apresenta resultados de simulação física e numérica do comportamento mecânico de cavidades circulares em meios contínuos. Na simulação numérica foi possível reproduzir o comportamento tensão-deformação registrado nos ensaios. O comportamento mecânico do contínuo foi abordado de duas formas: i) Teoria clássica e ii) Modelo generalizado de Cosserat. A segunda abordagem, por dispor de um grau de liberdade extra, permite a reprodução numérica de algumas feições observadas ao redor das cavidades circulares em testes de laboratório de maneira mais realística. A teoria clássica de contínuo foi associada somente ao modelo constitutivo de Mohr-Coulomb. Já para Cosserat, foram utilizados dois modelos constitutivos: Mohr-Coulomb e Bogdanova-Lippmann Modificado. A motivação para apresentar contínuo generalizado neste trabalho é que o mesmo inclui a parcela referente ao comportamento das partículas. Em todos os testes foram utilizadas amostras do arenito Botucatu, obtidas em São Paulo e Paraná. Para caracterização mecânica deste material foram realizados ensaios uniaxiais, triaxiais e brasileiros. Já a simulação física do comportamento de cavidades circulares foi analisada segundo duas geometrias: cúbica (com aplicação de estado de tensão biaxial) e cilíndrica (TWC – Thick Walled Cylinder). O acompanhamento da ruptura das cavidades cilíndricas foi feito de forma visual (amostras cúbicas) e com monitoramento tomográfico em tempo real (amostras cilíndricas). Com base na observação experimental da ruptura das cavidades cilíndricas e nas simulações numéricas considerando o contínuo clássico e de Cosserat, foi possível verificar que, ambas as abordagens possibilitaram a reprodução das feições observadas. / [en] This work seeks to realize physical and numerical simulation of the mechanical behavior of the wellbore stability for continuum environment.The Continunm s mechanical behavior is approach by two ways: i) Classic Continuum Theory and ii) Cosserat Continuum. On the second approach, the theory allows an extra degree of freedom, which plays an important rule on instabilities and bifurcation problems; this allows a more realistic numerical simulation of the failure mechanism observed on circular cavity. The Classic Continuum Theory is associated to a Mohr-Coulomb constitutive model. On the other hand for Cosserat Theory s applied tow constitutive models: Mohr-Coulomb and Modified Bogdanova-Lippmann.The generalized continuum takes in account the microstructure of the material.It s used on all tests Botucatu s specimens, which were acquired at São Paulo and Paraná. For characterize the rock s behavior it s realized triaxial, uniaxial and brazilian tests. Then the physical simulation of the circular cavity s behavior was analyzed for two geometries: cubic samples (biaxial stress) and cylindric samples (TWC – Thick Walled Cylinder). The failure mechanism of circular cavity was followed visually (cubic samples) and with CT X-Ray in real time (cylindric samples).From the experimental observations of the failure mechanism of circular cavity and numerical simulations, with Classic Continuum and Cosserat, was possible to verify that both approaches reproduce the behavior of the rocks observed on experimental data.