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Os materiais compósitos de matriz metálica são aplicados em diversos ramos da indústria devido a suas propriedades mecânicas, como alta resistência e leveza. Tais características se devem a associação de fibras de um material de alta resistência, como o aço, por exemplo, com uma matriz de material mais leve, como o alumínio. A matriz metálica pode ser porosa com o intuito de tornar o metal mais leve ou devido a não-linearidade física decorrente do processo de fabricação. Apesar dessa composição proporcionar um material resistente, a consideração de poros na matriz pode levar à formação de falhas no material. A partir disso, nota-se a importância do estudo de se considerar a porosidade através da modelagem multi-escala que consiste na representação do macrocontínuo por um Elemento de Volume Representativo (EVR) da micro-escala. Por meio de análises numéricas avalia-se a efetividade da inserção de uma inclusão em uma matriz porosa mostrando as potencialidades da modelagem proposta.

A 2D Boundary Element Method (BEM) Formulation to obtain the constitutive response of heterogeneous materials based on the Representative Volume Element (RVE) concept is developed to deal with dissipative phenomena over the domain and the phase debonding on the interface zone. Despite the formulation is written within a multiscale framework, in this work the analyses are performed only at microstructure level, but if a full coupled multiscale analysis is performed the computational cost becomes very expensive. Usually this kind of modelling is performed considering the Finite Element Method (FEM), but it is important to find other numerical solutions which are less expensive, but still accurate. To analyse metal matrix composites the RVEs are assumed to contain a ductile matrix, rigid inclusions and an interface zone where the phase debonding is modelled by a cohesive fracture model using embedded cohesive contact finite elements into the boundary element mesh. The RVE domain is discretized into cells to compute the domain integral written in terms of the inelastic forces, which have a part related to the plasticity phenomenon and another one referred to the cohesive-contact problem. The homogenized results are compared to a FEM model in order to validate the proposed formulation.

This work deals with parametric analyses of the structural behaviour of steel-fiber-reinforced concrete using a Damage Constitutive Model and homogenization technique. This approach is contemporary once Brazil has not a standard procedure for the design of this type of material. The present study performs a parametric analysis of rectangular beams subjected to flexion at four points, with height variations (h), reinforcement area (As) and span (L) are used to verify the influence of each parameter on the numerical study of the deformability of beams composed by this material. Fiber volume concentrations of 2.00% and 2.50% have been used on the modelling. It is observed that the obtained results were mainly affected by height and span. It is noticeable that the inclusion of steel fibers leads to an increase in the maximum load and ductility of the analyzed beams in this work. resumo: Este trabalho aborda uma análise numérica paramétrica do comportamento estrutural do concreto armado reforçado com fibras metálicas utilizando um modelo constitutivo de Mecânica do Dano e técnica de homogeneização. Essa abordagem ainda é recente, visto que o Brasil não possui nenhuma normatização em relação ao procedimento de cálculo desse tipo de material. O presente estudo realiza uma análise paramétrica de vigas de seção retangular submetidas à flexão em quatro pontos com variações de altura (h), área de aço (As) e o vão (L) de forma a verificar a influência de cada parâmetro no estudo numérico da deformabilidade de vigas compostas por esse material. Foram utilizados percentuais de 2,00% e 2,50% de fibras metálicas na modelagem. Observa-se que os resultados obtidos foram afetados principalmente pela altura e pelo vão da peça. Também se percebe que a inclusão de fibras proporciona aumento da carga máxima da estrutura além de promover uma boa ductilidade.

This work deals with numerical modeling of mechanical behavior in quasi-brittle materials, such as concrete. For this propose, a two-dimensional meso-scale model based on RVE existence is presented. The material is considered as a three-phase material consisting of interface zone (ITZ), matrix and inclusions - each constituent modeled by an independent constitutive model. The Representative Volume Element (RVE) consists of inclusions idealized as circular shapes symmetrically and non-symmetrically placed into the specimen. The interface zone is modeled by means of cohesive contact finite elements. The inclusion is modeled as linear elastic and matrix region is considered as elastoplastic material. Our main goal here is to show a computational homogenization-based approach as an alternative to complex macroscopic constitutive models for the mechanical behavior of the brittle materials using a finite element procedure within a purely kinematical multi-scale framework. Besides, the fundamental importance of the representing dissipative phenomena in the interface zone to model the complex microstructural responses of materials like concrete is focused in this work. A set of numerical examples, involving the microcracking processes, is provided in order to illustrate the performance of the proposed modeling. Resumo Este trabalho trata da modelagem numérica do comportamento mecânico em materiais quase-frágeis, tal como o concreto. Para este fim, um modelo 2D de escala mesoscópica baseado no conceito de Elemento de Volume Representativo (EVR) é apresentado. O material é considerado como composto por três fases consistindo de zona de interface, matriz e inclusões, onde cada constituinte é modelado independentemente. O EVR consiste de inclusões idealizadas como de forma circular dispostas de maneira simétrica e não simétrica. A zona de interface é modelada por meio de elementos finitos coesivos de contato. A inclusão é modelada como sendo um material elástico linear, já a matriz é considerada como material elastoplástico. Nosso principal objetivo é mostrar que uma formulação baseada na homogeneização computacional é uma alternativa aos modelos constitutivos macroscópicos complexos para o comportamento mecânico de matérias frágeis usando um procedimento baseado no Método dos Elementos Finitos no âmbito de uma teoria multi-escala. Além disso, o trabalho foca na fundamental importância em representar os fenômenos dissipativos na Zona de Transição para obter uma resposta microestrutural de um material complexo como o concreto. Uma série de exemplos envolvendo processos de microfissuração é apresentada de modo a ilustrar o desempenho da modelagem proposta.

RESUMO O presente trabalho trata da simulação do comportamento mecânico de microestruturas dúcteis reforçadas por inclusões elásticas e a repercussão de seu comportamento na resposta macromecânica de estruturas submetidas a carregamentos predominantes de cisalhamento. Para tanto, uma modelagem baseada em conceitos de homogeneização computacional utilizando um acoplamento direto entre escalas e do MEC/MEF é utilizado. Também, o fenômeno de descolamento de fase é levado em conta com o uso de modelo de fratura e contato onde a abertura da fratura por escorregamento é considerada. Por outro lado, a ductilidade da matriz metálica é representada pelo modelo de Von Mises. As análises numéricas são realizadas em duas etapas: uma primeira, em nível de microestrutura e, em seguida, é realizada a análise de uma chapa através do acoplamento direto entre as escalas. Contata-se coerência dos resultados encontrados nas micro e macro escalas com a utilização da modelagem proposta. ABSTRACT The present work deals with numerical analyses on mechanical behavior of ductile microstructures reinforced by elastic inclusions and the consequences upon micromechanical behavior of structures submitted to shear loadings. For this proposal, a proposed modeling based on computational homogenization technique using full coupling between scales and BEM/FEM is used. Also, the phase debonding is taken into account by using cohesive fracture and contact models where the opening sliding phenomenon is considered. On the other hand, the ductility of the metallic matrix is represented by Von Mises model. Initially, the numerical analyses are performed only at microstructural level and then a full coupled multi-scale analysis of a plate is considered, showing coherence between the micro and macro scale results.

espanolEste trabajo pretende presentar el desarrollo de una formulacion basada en Mecanica de Danos para tratar con medios elasticos que presentan comportamientos mecanicos diferentes en regimenes de tension y compresion, asi como algun grado de anisotropia cuando se producen procesos de dano. En primer lugar, se revisa una formulacion para medios elasticos bimodulares y anisotropicos. Con el fin de tener en cuenta los procesos de dano en el material, se proponen algunas mejoras asi como la definicion de un hiperplano para dividir el espacio de deformacion para tratar con materiales bimodulares. Ademas, los tensores anisotropicos de cuarto orden son solicitados por la formulacion dependiendo de la clase de anisotropia. Poco despues, se describen dos modelos constitutivos de danos escritos en concordancia con la formulacion propuesta en este trabajo. Finalmente, se muestra que la formulacion propuesta es una forma facil de escribir y comprender los complejos modelos de dano. EnglishThis work intends to present the development of a formulation based on Continuum Damage Mechanics to deal with elastic media introducing different mechanical behaviors in tension and compression regimes as well as some anisotropy degree when damage processes take place. First of all, a formulation for bimodular and anisotropic elastic media is reviewed. To take into account the damage processes in the material, some improvements are proposed as well as the definition of a hyperplane in order to divide the strain space to deal with bimodular materials. Besides, fourth-order anisotropic tensors are requested by the formulation depending on the class of anisotropy. Soon after, two models of constitutive damage written accordingly with the formulation proposed in this work are described. Finally, it is shown that the proposed formulation is an easy way to write and to understand complex constitutive damage models.

RESUMO A curva tensão-deformação do concreto é importante para o dimensionamento de estruturas, pois por meio dela é possível obter sua resistência à compressão, sua deformação de pico, o módulo de elasticidade e a tenacidade do concreto. Entretanto, essa é uma propriedade que não é comumente determinada nos ensaios de controle de qualidade do concreto, sendo normalmente controlada apenas a resistência à compressão. Contudo, quando se utiliza de métodos mais refinados para projeto de estruturas de concreto, como, por exemplo, a análise não linear física com o emprego do método dos elementos finitos, essa é uma propriedade fundamental para a análise da estrutura. Neste caso, a curva tensão-deformação à compressão deveria ser controlada em laboratório, mesmo quando se utiliza de curvas recomendadas em códigos de projeto. Deste modo, o objetivo deste artigo é apresentar uma metodologia para avaliação da curva completa tensão-deformação à compressão do concreto simples e reforçado com fibras de aço a partir de máquinas de ensaio de baixo custo, disponível na maioria dos laboratórios de controle de qualidade de concreto. Adicionalmente, é apresentada uma metodologia numérica para análise inversa da curva tensão-deformação à compressão com o objetivo de estimar parâmetros para modelagem de estruturas de concreto por meio da mecânica do Dano. ABSTRACT The stress-strain diagram of concrete is important for the design of concrete structures because compressive strength, peak deformation, modulus of elasticity and toughness can be obtained from this diagram. However, the stress-strain relationship is not commonly determined in concrete quality control trials, and usually, only compressive strength is determined. However, when more refined methods are used for designing concrete structures, such as physical nonlinear analysis with the finite element method, the stress-strain diagram is fundamental for structure analysis. In this case, the compression strain-strain curve should be determined in a laboratory, even if recommended curves from design codes are used. Thus, the objective of this paper is to propose a methodology to evaluate the compressive strain-strain diagram of plain and steel fiber reinforced concretes using low-cost test machines available in most concrete control laboratories. In addition, a numerical method is presented for inverse analysis of the strain-strain diagram, with the objective of estimating parameters for modeling concrete structures by means of damage mechanics.

No trabalho estudam-se alguns aspectos relativos à formulação teórica e à simulação numérica de modelos constitutivos para o concreto fundamentados na Mecânica do Dano Contínuo, incluindo-se os chamados métodos simplificados de análise estrutural. Inicialmente apresenta-se uma discussão sobre deformações permanentes e anisotropia induzidas pela evolução do dano. A resposta unilateral do concreto submetido a solicitações com inversão de sinal, também é comentada. Cada um dos fenômenos é ilustrado por respostas observadas experimentalmente. O modelo de dano proposto por Mazars para o concreto submetido a carregamento proporcionalmente crescente é então analisado. Em seguida, apresenta-se uma extensão do modelo considerando-se o aspecto unilateral no comportamento do concreto. Na sequência, analisa-se o modelo constitutivo proposto por La Borderie em seus aspectos de formulação e resposta numérica. O modelo é mais completo permitindo levar em conta todos os fenômenos discutidos anteriormente. Um outro aspecto considerado no trabalho é relativo à aplicação dos modelos estudados à análise de estruturas aporticadas. Em termos de discretização destacam-se a técnica de divisão dos elementos estruturais em estratos e os chamados modelos simplificados. Neste último caso, o modelo de Flórez-López é analisado e a simplificação consiste na definição prévia, sobre a estrutura discretizada através de elementos de viga e de coluna, de zonas de localização da plastificação e do dano; no limite com a evolução do processo de carregamento, aquelas zonas passam a se constituir em rótulas. Por último uma generalização do modelo anterior proposta por Álvares é estudada. Os resultados numéricos fornecidos pelos modelos são confrontados com os experimentais de vigas em concreto armado (biapoiadas e com diferentes taxas de armadura) e de um pórtico em concreto armado. In this work some aspects related to the theoretical formulation and numerical simulation of constitutive models for the concrete based on the Continuum Damage Mechanics are studied, including simplified methods of structural analysis. Initially a discussion about permanent strains and anisotropy induced by the evolution of the damage is presented. The unilateral response of the concrete submitted to loadings with signal inversion is commented as well. Each one of the phenomena is illustrated by experimental tests. The damage model proposed by Mazars for the concrete under proportional increasing load is then analysed. Afterwards, it is shown an extension of the model considering the unilateral aspect ofthe concrete behaviour. In the sequence the constitutive model proposed by La Borderie is presented in its aspects of formulation and numerical response. This model is more complete, allowing to take in to account all the phenomena discussed previously. Another aspect considered in the work is related to the application of the models studied to the analysis of framed structures. Regarding the discretization, the technique of division of the structural elements in layers and the so called simplified models are enhanced. In this last case, the model proposed by Flórez-López is analysed and the simplification consists on previous definition of yielding and damage zones over the assembly of beam and colunm elements, which compose the discretized structure; in the limit evolution of the loading process, these zones become hinges. Finally, a generalization of the previous model proposed by Álvares is studied. The numerical results supplied by the models are compared with the experimental results of reinforced concrete beams (simply supported and with different reinforced rates) and of a reinforced concrete frame.

This work deals with unilateral effect of quasi-brittle materials, such as concrete. For this propose, a two-dimensional meso-scale model is presented. The material is considered as a three-phase material consisting of interface zone, matrix and inclusions - each constituent modeled by an appropriate constitutive model. The Representative Volume Element (RVE) consists of inclusions idealized as circular shapes randomly placed into the specimen. The interface zone is modeled by means of cohesive contact finite elements developed here in order to capture the effects of phase debonding and interface crack closure/opening. As an initial approximation, the inclusion is modeled as linear elastic as well as the matrix. Our main goal here is to show a computational homogenization-based approach as an alternative to complex macroscopic constitutive models for the mechanical behavior of the quasi-brittle materials using a finite element procedure within a purely kinematical multi-scale framework. A set of numerical examples, involving the microcracking processes, is provided. It illustrates the performance of the proposed model. In summary, the proposed homogenization-based model is found to be a suitable tool for the identification of macroscopic mechanical behavior of quasi-brittle materials dealing with unilateral effect.

This work deals with a damage model formulation taking into account the unilateral effect of the mechanical behaviour of brittle materials such as concrete. The material is assumed as an initial elastic isotropic medium presenting anisotropy, permanent strains and bimodularity induced by damage evolution. Two damage tensors governing the stiffness in tension or compression regimes are introduced. A new damage tensor in tension regimes is proposed in order to model the diffuse damage originated in prevails compression regimes. Accordingly with micromechanical theory, the constitutive model is validate when dealing with unilateral effect of brittle materials, Finally, the proposed model is applied in the analyses of reinforced concrete framed structures submitted to reversal loading. The numerical results have shown the good performance of the modelling and its potentialities to simulate practical problems in structural engineering.

A multi-scale modelling for analyzing the bending problem of plates composed of heterogeneous materials is presented. The macro-continuum is modelled by a non-linear formulation of the boundary element method (BEM) taking into account the consistent tangent operator (CTO). The micro-scale is represented by the RVE (representative volume element) being its equilibrium problem solved by a finite element formulation that takes into account the Hill-Mandel Principle of Macro-Homogeneity while the volume averaging hypothesis of the strain and stress tensors is used to make the micro-to-macro transition. Three different boundary conditions are imposed over the RVE: (i) linear displacements, (ii) periodic displacement fluctuation and (iii) uniform boundary tractions. The material behaviour is governed by the Von Mises elasto-plastic criterion although the proposed multi-scale model can be used with any other non-linear model. Numerical examples are presented to illustrate the main features and scope of the proposed formulation.

THIS WORK DEALS WITH NUMERICAL SIMULATION OF THE MECHANICAL BEHAV-IOR OF MATERIALS COMPOSED OF HETEROGENEOUS DUCTILE MICROSTRUCTURES USING A MULTI-SCALE APPROACH CONSIDERING PLASTICITY PROCESSES AND PHASE DEBONDING. DUE TO FEW STUDIES ABOUT YIELD SURFACES OF METAL MATRIX COMPOSITES (MMC) WITH WEAK INTERFACES PRESENTED IN THE LITERATURE, THE MAJOR GOAL OF THIS WORK IS TO PROPOSE YIELD SURFACES FOR METAL MATRIX COMPOSITES REINFORCED BY RIGID INCLUSIONS. THE YIELD SURFACES ARE OBTAINED FOR REPRESENTATIVE VOLUME ELEMENTS (RVES) OF MATERIALS PRESENTING PERFECTLY BONDED INCLUSIONS AND PHASE DEBONDING IN THE INTERFACE ZONE. THE MATRIX IS CONSIDERED AN IDEALLY PLASTIC MATERIAL GOVERNED BY VON MISES MODEL, WHEREBY THE INTERFACE ZONE IS MODELED BY MEANS CONTACT AND FRACTURE CONSTITUTIVE MODELS INCORPORATED IN A FINITE ELEMENT DEVELOPED FOR THIS PROPOSAL. ALSO, RVES CONTAINING DIFFERENT VOLUME RATIOS OF VOIDS FOLLOWING DIFFERENT DISTRIBUTIONS ARE ANALYZED. CONSIDERING THE PHASE DEBONDING, FOR COMPRESSIVE LOADINGS THE RVE BEHAVES LIKE RVE WITH PERFECTLY BONDED INCLUSIONS WHEREAS FOR TENSION LOADINGS THE RVE PRESENTS A BEHAVIOR QUITE SIMILAR TO THE ONE WITH VOIDS. IN THE OTHER HAND, THE CONCENTRATION OF VOIDS IN THE RVE DECREASES ITS MECHANICAL STRENGTH.

RESUMO Neste trabalho apresentam-se análises de flexão de placas compostas por materiais heterogêneos através de uma abordagem multi-escala. O macro-contínuo, representado neste trabalho pela placa, é modelado por uma formulação não-linear do Método dos Elementos de Contorno (MEC), que leva em conta o operador tangente consistente (CTO). A micro-escala é representada pelo EVR (Elemento de Volume Representativo), sendo seu problema de equilíbrio definido em termos de flutuação dos deslocamentos e solucionado através do Método dos Elementos Finitos (MEF), onde a hipótese de média volumétrica das tensões e deformações é adotada para se fazer a passagem do micro-contínuo para o macro-contínuo. A cada ponto do macro-contínuo, onde se necessita conhecer as tensões e o tensor constitutivo deve estar associado um EVR, onde se podem definir inclusões e/ou vazios no interior de uma matriz a fim de representar a micro-estrutura de um material heterogêneo. Nos exemplos numéricos são considerados diferentes EVRs com inclusões elásticas dentro de uma matriz, onde os modelos de Von Mises ou Mohr Coulomb são adotados, a fim de governar o comportamento do seu material. Consideram-se diferentes frações volumétricas para as inclusões a fim de verificar a influência na resposta homogeneizada da microestrutura e, consequentemente, no comportamento mecânico do macro-contínuo. Para solucionar o problema de equilíbrio do EVR devem-se adotar condições de contorno em termos de flutuações dos deslocamentos, que nos exemplos analisados no presente trabalho serão consideradas como periódicas. ABSTRACT Analyses of the bending problem of plates composed of heterogeneous materials are performed considering a multi-scale modelling. The macro-continuum, represented in this paper by the plate, is modelled by a nonlinear formulation of the boundary element method (BEM) taking into account the consistent tangent operator (CTO) in the iterative procedure required to solve the plate equilibrium problem. The micro-scale is represented by the RVE (Representative Volume Element) being its equilibrium problem solved in terms of dis-placements fluctuations by a Finite Element Formulation (FEM), where the volume averaging hypothesis of strain and stress tensors is used to make the micro-to-macro transition. To each point of the macro-continuum where the stresses and constitutive tensor have to be computed, a RVE must be assigned, where inclusions and voids can be defined inside the matrix in order to represent the microstructure of a heterogeneous material. In the numerical examples are considered different RVEs with elastic inclusions, while for the matrix the Von Mises or Mohr-Coulomb criteria can be adopted to govern its material behavior. Different volume fractions have been adopted for the inclusions in order to verify its influence on the homogenized response of the microstructure as well as on the mechanical behavior of the macrostructure. To solve the RVE equilibrium problem, boundary conditions in terms of displacement fluctuations have to be imposed, which for the numerical examples presented in this paper will be adopted periodic.

Este trabalho trata da formulação de leis constitutivas para meios elásticos, que uma vez danificados passam a apresentar diferentes comportamentos em tração e em compressão e certo grau de anisotropia. Inicialmente é revista uma formulação para meios elásticos anisótropos e bimodulares, sendo a mesma, então, estendida para incorporar os casos de meios elásticos com anisotropia e bimodularidade induzidas pelo dano. Seguindo os conceitos da mecânica do dano contínuo e a extensão mencionada da formulação, propõe-se um modelo constitutivo para o concreto assumindo por hipótese fundamental, a equivalência de energia entre meio danificado real e meio contínuo equivalente. Tal hipótese garante a simetria do tensor constitutivo e a sua consistência termodinâmica. De acordo com o modelo proposto, o material é considerado como um meio elástico inicialmente isótropo que passa a apresentar anisotropia induzida pela evolução do dano. Além disso, a danificação pode também induzir uma resposta bimodular no material, isto é, respostas elásticas diferentes para estados de tensão de tração ou de compressão predominantes. Nesse sentido, dois tensores de dano governando as rigidezes em regimes predominantes de tração ou de compressão são introduzidos. Sugere-se então, um critério afim de caracterizar os estados dominantes. As deformações permanentes induzidas pelo dano são, de uma forma geral, desconsideradas. No entanto, propõe-se uma versão unidimensional do modelo que permite a sua consideração. Por outro lado, os critérios para a ativação inicial dos processos de danificação e de sua posterior evolução são escritos em termos de densidade de energia de deformação. Os parâmetros do modelo podem ser identificados mediante experimentos, onde estados de tensão uniaxial e biaxial são induzidos. Também propõem-se leis de evolução de dano com base nos resultados experimentais. A boa coerência do modelo é ilustrada comparando-se uma série de respostas experimentais e numéricas no concreto relativas a estados de tensão uni, bi e triaxiais. Por fim, o modelo é empregado em análises unidimensionais e planas de vigas e pórtico em concreto armado com o objetivo de mostrar a sua potencialidade. This thesis deals with the formulation of constitutive laws for elastic media that start to present different behaviours in tension and compression and some anisotropy degree when damaged. Initially a formulation for bimodular and anisotropic elastic media is reviewed; then it is extended to take into account the bimodularity and anisotropy induced by damage. Following the concepts of the continuum damage mechanics and the formulation extension, a constitutive model is proposed here by exploring the fundamental hypothesis of energy equivalence between real and continuous medium. Such hypothesis guarantees symmetry to the constitutive tensor and its thermodynamic consistency. According to the proposed modeling, the material is assumed as an initial elastic isotropic medium presenting anisotropy induced by damage evolution. Moreover, damage can also induce a bimodular response in the material, i.e., distinct elastic responses whether traction or compression stress states prevail. To take into account the bimodularity conveniently, two damage tensors governing the rigidity in traction or compression regimes are introduced. A criterion is proposed in order to characterize the dominant states. In general, the permanent strains induced by damage are disregarded. However, a one-dimensional version of the model that allows considering permanent strains is proposed. On the other hand, damage criteria indicating the initial and further evolution of damage are expressed in terms of strain energy densities. The model parameters can be identified from experiments where one-axial and two-axial stress states are induced. Damage evolution laws are also proposed on the basis of experimental results. The good performance of the model is illustrated by comparing a variety of numerical and experimental responses, from one to three-axial stress states. Finally, the model is used in one-dimensional and plane analysis of reinforced concrete beams and frame in order to show its potentiality.

In this work, a homogenization-based approach considering elastoplastic behavior and fracture process is proposed in order to investigate the micromechanical behavior of ductile porous materials. The material microstructure is represented by a representative volume element composed of elastic inclusions or porous placed inside the matrix. For modeling the interface zone, a contact finite element is developed as well as a cohesive fracture model in order to capture the effects of phase debonding, interface crack closure or opening, and the cracking process inside the matrix. Besides, the matrix behavior is governed by the Von Mises elastoplastic model with linear strain hardening. Some numerical examples are analyzed in order to show that the proposed modeling is a suitable tool for the identification of constitutive responses of composite microstructures.

RESUMO O presente trabalho trata da simulação numérica do comportamento mecânico de microestruturas de Compósitos com Matriz Metálica (CMMs) utilizando uma proposta de modelo de homogeneização computacional baseada numa abordagem multi-escala. Na microestrutura do compósito, as inclusões são consideradas elásticas e o comportamento da matriz é governado pelo modelo de von Mises com endurecimento isotrópico. Um modelo de fratura coesiva é desenvolvido para simular a fase de descolamento da interface matriz/inclusão. Todo o estudo é baseado no conceito de Elemento de Volume Representativo (EVR), no qual podem ser empregados modelos constitutivos que levam em conta os fenômenos dissipativos de fissuração e plasticidade. Uma série de EVRs com diferentes composições de inclusões elásticas e submetidos a diferentes condições de restrição cinemática foram analisados. Também observou-se a sensibilidade paramétrica do modelo de fratura coesiva e a importância de se considerar a fase de descolamento matriz/inclusão no processo de ruptura da microestrutura. De modo geral, os resultados encontrados contribuem para a discussão acerca do emprego de modelos simples, em termos de formulação e identificação paramétrica, na modelagem da microestrutura de materiais heterogêneos, refletindo assim na acurácia de resultados qualitativos quanto ao seu comportamento macroscópico. ABSTRACT This work deals with numerical simulation of the metal matrix composites (MMCs) microstructures mechanical behavior using a proposed computational homogenization modeling based on multi-scale approach. In the microstructure, elastic inclusions are considered and the matrix mechanical behavior is governed by a von Mises model with isotropic strain hardening. Besides, a cohesive fracture and contact model is developed in order to simulate the debonding phase in the matrix/inclusion interface. The study is based on the concept of Representative Volume Element (RVE), in which can be used constitutive models that take into account the dissipative phenomena of cracking and plasticity. A set of RVEs composed of different distributions of elastic inclusions and submitted to different kinematical restrictions are analyzed. Moreover, the parametric sensibility of the cohesive fracture model and the debonding interface relevance in the rupture processes of the microstructure are observed. In general, the results contribute to the discussion about the use of simple constitutive models, in their formulation and parametric identification, for modeling of the heterogeneous materials microstructures, leading to accurate qualitative results related to the macroscopic behavior.

This work intends to contribute for the improvement of the procedure suggested by Brazilian Technical Code that takes into account the cracked concrete stiffness in the estimative of the displacement of reinforced concrete beams submitted to service loads. A damage constitutive model accounting for induced anisotropy, plastic deformations and bimodular elastic response is used in order to simulate the concrete behaviour, while an elastoplastic behaviour is admitted for the reinforcement. The constitutive models were implemented in a program for bars structures analysis with layered finite elements. Initially, the damage model is briefly presented as well as the parametric identification of the materials that have been used in the reinforced concrete beams. After that, beams with different geometries and reinforcement area are analyzed and a statistical method (ANOVA) is employed in order to identify the main variables in the problem. Soon after, the same procedure is used with another resistance of concrete, where the compression strength is changed. The numerical responses are compared with the ones obtained by Brazilian Technical Code and experimental tests in order to validate the use of the damage model. Finally, some remarks are discussed based on responses presented in this work.

In this paper, a damage constitutive model accounting for induced anisotropy and bimodular elastic response is applied to two-dimensional analysis of reinforced concrete structures. Initially, a constitutive model for the concrete is presented, where the material is assumed as an initial elastic isotropic medium presenting anisotropy and bimodular response (distinct elastic responses, whether tension or compression stress states, prevail) induced by damage. Two damage tensors govern the stiffness under prevailing tension or compression stress states. Criteria are then proposed to characterize the dominant states. Finally, the proposed model is used in plane analysis of reinforced concrete beams to show its potential for use and to discuss its limitations.

A multi-scale modelling for analysing the stretching problem of plates composed of heterogeneous materials is presented. The BEM (Boundary Element Method) is adopted to model the macro-continuum (represented by the plate) while the equilibrium problem at micro-scale (represented by the Representative Volume Element – RVE) is solved by a FEM (Finite Element Method) formulation that takes into account the Hill–Mandel Principle of Macro-Homogeneity. After solving the equilibrium problem of the RVE, the micro-to-macro transition is made by applying the volume averaging hypothesis of strain and stress tensors. Some numerical examples are then analysed to show that the proposed formulation is a suitable tool for the analysis of stretched of plates composed of heterogeneous materials. To define the microstructure, different RVEs composed of an elasto-plastic matrix with inclusions or voids are considered. Besides, a quadratic rate of asymptotic convergence of the Newton–Raphson scheme has been achieved during the iterative procedure.

Multiscale analyses considering the stretching problem in plates composed of metal matrix composites (MMC) have been performed using a coupled BEM/FEM model, where the boundary element method (BEM) and the finite element method (FEM) models, respectively, the macrocontinuum and the material microstructure, denoted as representative volume element (RVE). The RVE matrix zone behavior is governed by the von Mises elasto-plastic model while elastic inclusions have been incorporated to the matrix to improve the material mechanical properties. To simulate the microcracks evolution at the interface zone surrounding the inclusions, a modified cohesive fracture model has been adopted, where the interface zone is modeled by means of cohesive contact finite elements to capture the effects of phase debonding. Thus, this paper investigates how this phase debonding affects the microstructure mechanical behavior and consequently affects the macrostructure response in a multiscale analysis. For that, initially, only RVEs subjected to a generic strain are analyzed. Then, multiscale analyses of plates have been performed being each macro point represented by a RVE where the macro-strain must be imposed to solve its equilibrium problem and obtain the macroscopic constitutive response given by the homogenized values of stress and constitutive tensor fields over the RVE.

This study deals with the proposal of damage variables related to structural behavior of the concrete subjected to shear stress. It is also addressed the laws of evolution and their incorporation into the formulation of a constitutive model for the concrete. Originally, this model considers the anisotropy and bimodularity induced by damage. Initially, the formulation of the damage model is briefly described. The material is assumed to behave as an elastic isotropic medium that start to present anisotropy and bimodular response (distinct elastic responses for prevailing states of tension or compression). Afterwards, we discussed some proposals of damage variables related to the structural behavior of the concrete when subjected to shear stress and their respective laws of evolution. The chosen proposal is then incorporated to the original model and implemented in a finite element code to perform plane analysis. Finally, the numerical responses of the model are compared with the experimental of reinforced concrete beams with different reinforcement rates. The results are presented and discussed aiming to verify the improvements presented by the new version of the model and to show the contribution of the energy dissipation due to shear in the cracked concrete behavior for the analyzed examples.

This work presents one and two-dimensional numerical analyses using isotropic and anisotropic damage models for the concrete in order to discuss the advantages of these modeling. Initially, it is shortly described the damage model proposed by Mazars. This constitutive model assumes the concrete as isotropic and elastic material, where locally the damage is due to extensions. On the other hand, the damage model proposed by Pituba, the material is assumed as initial elastic isotropic medium presenting anisotropy, plastic strains and bimodular response (distinct elastic responses whether tension or compression stress states prevail) induced by the damage. To take into account for bimodularity two damage tensors governing the rigidity in tension and compression regimes, respectively, are introduced. Damage activation is expressed by two criteria indicating the initial and further evolution of damage. Soon after, the models are used in numerical analyses of the mechanical behavior of reinforced concrete structures. Accordingly with comparison of the obtained responses, considerations about the application of the isotropic and anisotropic damage models are presented for 1D and 2D reinforced concrete structures modeling as well as the potentialities of the simplified versions of damage models applied in situations of structural engineering. Este artigo apresenta análises uni e bidimensionais utilizando modelos de dano isótropo e anisótropo para o concreto com o objetivo de discutir sobre as vantagens deste tipo de modelagem. Inicialmente, o modelo de dano proposto por Mazars é brevemente descrito, onde o mesmo considera o concreto como meio isótropo e elástico, sendo a danificação decorrente de extensões em nível local. Por outro lado, o modelo proposto por Pituba admite o concreto como meio inicialmente isótropo e que passa a apresentar deformações plásticas, bimodularidade e anisotropia induzidas pelo dano. Para levar em conta a bimodularidade, dois tensores de dano governando a rigidez em tração e em compressão são introduzidos. Em seguida, os modelos constitutivos são utilizados em análises numéricas do comportamento mecânico de estruturas em concreto armado. De acordo com as respostas obtidas, discutem-se algumas considerações sobre a aplicabilidade de modelos de dano isótropos e anisótropos na modelagem de estruturas em concreto armado no âmbito de análises uni e bidimensionais, assim como são apresentadas as potencialidades de aplicação de versões simplificadas de modelos de dano em situações da Engenharia Estrutural.

Made available in DSpace on 2013-09-27T14:52:53Z (GMT). No. of bitstreams: 1 WOS000268277100004.pdf: 257892 bytes, checksum: cadcfe973fa5763a872cdcecc5fddebc (MD5) Previous issue date: 2009-01-01 Made available in DSpace on 2013-09-30T19:23:49Z (GMT). No. of bitstreams: 1 WOS000268277100004.pdf: 257892 bytes, checksum: cadcfe973fa5763a872cdcecc5fddebc (MD5) Previous issue date: 2009-01-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T15:34:18Z No. of bitstreams: 1 WOS000268277100004.pdf: 257892 bytes, checksum: cadcfe973fa5763a872cdcecc5fddebc (MD5) Made available in DSpace on 2014-05-20T15:34:18Z (GMT). No. of bitstreams: 1 WOS000268277100004.pdf: 257892 bytes, checksum: cadcfe973fa5763a872cdcecc5fddebc (MD5) Previous issue date: 2009-01-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) In this work, non-linear analyses of reinforced concrete plates are performed by using a BEM formulation, based on Kirchhoff's theory, which has already proved to be a robust technique to deal with plate problems. The non-linear behavior of concrete is modeled by the Mazars model, which is based on continuum damage mechanics (CDM) and has an easy parametric identification, while the reinforcement is governed by the uniaxial elasto-plastic model with constant hardening. Initially, the different types of damage models and their parametric identification are discussed and the Mazars model is presented. Then the BEM formulation is discussed, which is based on the initial moment technique, where the remaining domain integrals are evaluated by approaching the initial moment field over internal cells. The stress distribution over the plate thickness is obtained by using a Gauss scheme in which the adopted criterion is verified in each Gauss point defined along the plate. Then, the internal values of moments are approached by numerical integrals along the plate thickness. Finally, some numerical examples of reinforced concrete plates are analyzed, where the potentiality of the Mazars model is verified despite the difficulties of the parametric identification. Universidade Federal de Goiás (UFG), BR-75700000 Catalao, Goias, Brazil Univ Estadual Paulista, Fac Engn Ilha Solteira, São Paulo, Brazil Univ Estadual Paulista, Fac Engn Ilha Solteira, São Paulo, Brazil

This work deals with the formulation of constitutive laws for elastic media that start to present different behaviors in tension and compression as well as some anisotropy degree when damaged. Initially, a formulation for bimodular and anisotropic elastic media proposed by [1] is reviewed. In this formulation, for the modeling of a bimodular hyperelastic material, the elastic potential energy density must be once continuously differentiable (whole wise), but only piecewise twice continuously differentiable. The stress-strain relationship derived from this potential is piecewise continuously differentiable leading to an elasticity tensor discontinuous referred to a hypersurface that contains the origin and divides the strain space into a compression and tension sub-domains. In this way, the modeling is able to reproduce different response in tension and compression. Soon after, the formulation proposed by [1] is extended to take into account the damage process. Accordingly with, the coefficients named bulk and shear modulus are considered as functions of the damage state, so that the stress-strain relationship would be influenced by damage variables. Moreover, the hypersurface taken as the criterion for the identification of the constitutive responses in compression or tension would be also influenced by the damage variables. It must be observed that the condition no tangential discontinuity of the elasticity tensor is also valid in this formulation. Fourth-order anisotropic tensors are requested by the formulation depending on the class of anisotropy that is assumed. The matricial form of those tensors is presented in the end of the work. Some damage models [2], [3] found in the literature are written according to the formalism proposed here in order to show its potentiality. It must be observed that the objective of this work is to show the potentialities of the formalism proposed here in way to be used as a proper tool in the formulation of the damage models. Therefore, some aspects related to criteria and evolution laws of damage were not discussed. This work also presents contributions related to matricial forms of the fourth-order tensors involved in the formulation of damage models. Those forms are quite interesting to the computational implementations. Finally, the proposal of the formulation for the damaged media can be used in future works for the development of many constitutive models depending of the involved phenomena.

This work deals with an improvement of an anisotropic damage model in order to analyze reinforced concrete structures submitted to reversal loading. The original constitutive model is based on the fundamental hypothesis of energy equivalence between real and continuous media following the concepts of the Continuum Damage Mechanics. The concrete is assumed as an initial elastic isotropic medium presenting anisotropy, permanent strains and bimodularity induced by damage evolution. In order to take into account the bimodularity, two damage tensors governing the rigidity in tension or compression regimes are introduced. However, the original model is not capable to simulate the influence of the previous damage processes in compression regimes. In order to avoid this problem, some conditions are introduced to simulate the damage unilateral effect. It has noted that the damage model is agreement with to micromechanical theory conditions when dealing to unilateral effect in concrete material. Finally, the proposed model is applied in the analyses of reinforced concrete framed structures submitted to reversal loading. These numerical applications show the good performance of the model and its potentialities to simulate practical problems in structural engineering.