We present a general approach for determining the equilibrium shape of isolated, coherent, misfitting particles by minimizing the sum of elastic and interfacial energies using a synthesis of finite element and optimization techniques. The generality derives from the fact that there is no restriction on the initial or final shape, or on the elastic moduli of the particle and matrix, or on the nature of misfit. The particle shape is parameterized using a set of design variables which are the magnitudes of vectors from a reference point inside the particle to points on the particle/matrix interface. We use a sequential quadratic programming approach to carry out the optimization. Although this approach can be used to find equilibrium shapes of particles in three dimensional systems, we have presented the details of our formulation for two dimensional systems under plane strain conditions. Using our formulation, we have studied the equilibrium shapes in two dimensional systems with cubic anisotropy; the precipitate and matrix phases may have different elastic moduli, and the misfit may be dilatational or non-dilatational. The equilibrium shapes and their size dependence are analysed within the framework of symmetry-breaking shape transitions. These transitions are further characterized in terms their dependence on the cubic elastic anisotropy parameter, defined by A = 2C44/(C11 – C12), and on the modulus mismatch, defined by δ=μp/μm, where /μp and μm are the effective shear moduli of the precipitate and matrix phases, respectively. Depending on the type of misfit, the systems may be classified into the following four cases: Case A: For dilatational misfit, the equilibrium shapes in isotropic systems are circular (with an isotropic or I symmetry) at small sizes and undergo a transition at a critical size to become ellipse-like (with an orthorhombic or O symmetry). This I --O transition is continuous and is obtained only when the precipitate phase is softer than the matrix. These results are in good agreement with the analytical results of Johnson and Cahn. In cubic systems with dilatational misfit, the particles exhibit a transition from square-like shapes (with a tetragonal or T symmetry) at small sizes to rectangle-like shapes (with an O symmetry) at large sizes. This T -- O transition is continuous. It occurs even in systems with stiffer precipitates; however, it is forbidden for systems with δ >δC, where δ C represents a critical modulus mismatch. The critical size decreases with increasing cubic anisotropy (i.e., with increasing values of (A-1)/(A+1). The sides of the square-like and rectangle-like shapes are along the elastically soft directions. Case B: In these systems, the principal misfits e*xx and e*yy differ in magnitude but have the same sign. The precipitates at small sizes become elongated along the direction of lower misfit; this shape has an O symmetry. In systems with A > 1, they continue to become more elongated along the same direction, exhibiting no symmetry-breaking transition. However, in systems with A < 1, particles at large sizes are elongated along an intermediate direction between the direction of lower misfit and one of the elastically soft <11> directions; this shape has only a monoclinic or M symmetry. This O - M transition, in which the mirror symmetries normal to the x and y axes are lost, may be discontinuous or continuous. The critical size increases with δ (in the range 0.8 < δ <1.25), indicating that this transition would also be forbidden for systems with δ > δC. In systems with A < 1, the critical size decreases with increasing values of A-1/ A+1 Case C: In these systems, the principal misfits differ in both magnitude and sign, and the misfit strain tensor allows an invariant line along which the normal strain is zero. The precipitates at small sizes are elongated along the direction of lower absolute misfit, and possess an 0 symmetry. At large sizes, the mirror symmetries normal to the x and y axes are broken to yield shapes which are elongated along a direction between that of lower misfit and the invariant line. This 0 -> M transition is continuous and occurs in all the systems irrespective of the value of A The critical size increases with A and decreases with δ. Case D; The misfit in this case is a special form of that in Case C; the principal misfits have the same magnitude but opposite signs. The precipitates at small sizes have a square-like shape with its sides normal to the < 11 > axes, irrespective of the type of cubic anisotropy. At large sizes, they become rectangle-like with the long axis oriented along one of the <11> directions. Similar to Case C, this T - 0 transition is continuous and occurs in all the systems irrespective of the values of A. The critical size increases with A and decreases with δ. Thus, we have identified all the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two dimensional systems. We have identified their origin and nature, and characterized them in terms of their dependence on the anisotropy parameter and modulus mismatch.

<p> A microscopic study of two samples of the Gowganda Formation, both having undergone pure shear deformation at greenschist f acies metamorphism has shown: Sample A -Reduction in quartz grain size, and pressure solution shadows were evidence for strong pressure solution activity. Pre-lithification fractures provided channelways for the removal of quartz and water out of the system. Matrix quartz was not recrystallized. Sample B -Extensive local recrystallization of quartz due to pressure solution activity caused metamorphic segregation and the formation of a cleavage. Cleavage behaviour differs in the matrix from that observed in an area of contact strain, as produced by a buckled quartz vein. There is no evidence for removal of quartz from the system. </p> <p> A comparison between the two samples suggests that (a) water, in this case at least, is necessary to produce metamorphic segregation, (b) the system has to be closed to reach metamorphic segregation and (c) original features may be preserved through limited metamorphism if the rock is dried out early in its history. </p> / Thesis / Bachelor of Science (BSc)

Die bruchmechanischen Eigenschaften von Elastomerwerkstoffen zeigen eine deutliche Abhängigkeit von der Prüfkörpergeometrie. Endscheidenden Einfluss hat die Art der Beanspruchung, die der Belastung des realen Bauteils entsprechen sollte. In dieser Arbeit wird die Methode zur Analyse des dynamischen Risswachstums von Elastomeren im simultanen Tensile- und Pure-shear-Prüfmodus beschrieben. Die Methode, basierend auf der mechanischen Aufrüstung und Weiterentwicklung eines Tear and Fatigue Analyzers, stellt den Prüfmethoden übergreifende Feststellungen zu aussagekräftigen Kennwertdaten des Bruchverhaltens von Elastomeren vor. Die Schwerpunke der Arbeit wurden in zwei Hauptthemen unterteilt, wobei zuerst eine konstruktive Modifizierung des kommerziellen TFA vorgenommen wurde und anschließend die Untersuchungen zur Beschreibung der bruchmechanischen Eigenschaften von Elastomerwerkstoffen und dynamischer Beanspruchung mittels modifiziertem TFA durchgeführt wurden. Es wurde aus den Anforderungen an die quantitative, simultane Analyse von SENT- als auch Pure-shear-Prüfkörpern eine konstruktive Modifizierung des kommerziellen TFA vorgenommen. Insbesondere die Schwerpunkte der Modifizierung wurden der Verminderung des Einflusses der Prüfkörperhalterungsfestigkeit auf die Genauigkeit der Analyse und der Untersuchung des Resonanzbereiches des TFA gewidmet. Die hier dokumentierten Ergebnisse durch den modifizierten TFA haben deutlich das unterschiedliche Bruchverhalten in der Abhängigkeit von der Prüfkörpergeometrie nachgewiesen. Es wurden umfangreiche Untersuchungen zur Charakterisierung der Rissausbreitungsgeschwindigkeiten in Elastomeren in Abhängigkeit von der Prüfkörpergeometrie als auch von der Risslänge durchgeführt, wobei festgestellt wurde, dass die bruchmechanischen Eigenschaften nicht nur von der Risslänge, sondern auch signifikant von der Kerbgeometrie abhängen. Die Ergebnisse haben weiterhin gezeigt, dass ein großer Einfluss der Herstellungsparameter der Prüfkörper unter Berücksichtigung von Kautschukbasis und Kautschukmischungsrezeptur auf die bruchmechanischen Eigenschaften besteht. Die vorgestellte Methode zur Bestimmung der dynamisch-bruchmechanischen Eigenschaften von Elastomeren stellt zusammen mit der Berücksichtigung der Herstellungsparameter einen genauen Vergleich zwischen den experimentell ermittelten Prüfwerten unabhängig von der Prüfkörpergeometrie dar. / The mechanical properties of rubber materials can be shown to be a function of the geometry of the test specimen. The main affecting parameters are the loading conditions. The present work proposes a new fracture mechanical testing concept for determination of dynamic crack propagation of rubber materials. This concept implements a method of simultaneous tensile- and pure-shear-mode testing. The present approach is based on an upgrade of the Tear and Fatigue Analyzers, on the fracture mechanics theory of dynamically loaded test specimens and on the definition of pure-shear states according to the test specimen’s geometry ratio. The main focus of this work can be divided in two parts. Firstly it is introduced the development of a method for analysis of dynamic crack propagation in filled rubber by simultaneous tensile- and pure shear mode testing. The servo-hydraulic machine with controlled temperature testing chamber is equipped with simultaneously operating two mode test equipment which represents a new fracture testing method. Secondarily the analysis of crack propagation under the dynamic loading conditions is practised with this method. It is shown how the tearing energy and the crack growth rate depend on the test specimen’s geometry ratio and crack length. It is also demonstrated that the values for tearing energies and also crack growth rates for short crack lengths in SENT- as well as in pure-shear test specimens are identical. Another important aspect is related to the different values of tearing energies and crack growth rates for cracks with short and large lengths in pure-shear test specimens. The results show the dependence of fracture behavior on manufacturing the test specimens. The new fracture mechanical testing concept offers a comparison between fracture behavior of rubber materials independent of the test specimens geometry.

Die bruchmechanischen Eigenschaften von Elastomerwerkstoffen zeigen eine deutliche Abhängigkeit von der Prüfkörpergeometrie. Endscheidenden Einfluss hat die Art der Beanspruchung, die der Belastung des realen Bauteils entsprechen sollte. In dieser Arbeit wird die Methode zur Analyse des dynamischen Risswachstums von Elastomeren im simultanen Tensile- und Pure-shear-Prüfmodus beschrieben. Die Methode, basierend auf der mechanischen Aufrüstung und Weiterentwicklung eines Tear and Fatigue Analyzers, stellt den Prüfmethoden übergreifende Feststellungen zu aussagekräftigen Kennwertdaten des Bruchverhaltens von Elastomeren vor. Die Schwerpunke der Arbeit wurden in zwei Hauptthemen unterteilt, wobei zuerst eine konstruktive Modifizierung des kommerziellen TFA vorgenommen wurde und anschließend die Untersuchungen zur Beschreibung der bruchmechanischen Eigenschaften von Elastomerwerkstoffen und dynamischer Beanspruchung mittels modifiziertem TFA durchgeführt wurden. Es wurde aus den Anforderungen an die quantitative, simultane Analyse von SENT- als auch Pure-shear-Prüfkörpern eine konstruktive Modifizierung des kommerziellen TFA vorgenommen. Insbesondere die Schwerpunkte der Modifizierung wurden der Verminderung des Einflusses der Prüfkörperhalterungsfestigkeit auf die Genauigkeit der Analyse und der Untersuchung des Resonanzbereiches des TFA gewidmet. Die hier dokumentierten Ergebnisse durch den modifizierten TFA haben deutlich das unterschiedliche Bruchverhalten in der Abhängigkeit von der Prüfkörpergeometrie nachgewiesen. Es wurden umfangreiche Untersuchungen zur Charakterisierung der Rissausbreitungsgeschwindigkeiten in Elastomeren in Abhängigkeit von der Prüfkörpergeometrie als auch von der Risslänge durchgeführt, wobei festgestellt wurde, dass die bruchmechanischen Eigenschaften nicht nur von der Risslänge, sondern auch signifikant von der Kerbgeometrie abhängen. Die Ergebnisse haben weiterhin gezeigt, dass ein großer Einfluss der Herstellungsparameter der Prüfkörper unter Berücksichtigung von Kautschukbasis und Kautschukmischungsrezeptur auf die bruchmechanischen Eigenschaften besteht. Die vorgestellte Methode zur Bestimmung der dynamisch-bruchmechanischen Eigenschaften von Elastomeren stellt zusammen mit der Berücksichtigung der Herstellungsparameter einen genauen Vergleich zwischen den experimentell ermittelten Prüfwerten unabhängig von der Prüfkörpergeometrie dar. / The mechanical properties of rubber materials can be shown to be a function of the geometry of the test specimen. The main affecting parameters are the loading conditions. The present work proposes a new fracture mechanical testing concept for determination of dynamic crack propagation of rubber materials. This concept implements a method of simultaneous tensile- and pure-shear-mode testing. The present approach is based on an upgrade of the Tear and Fatigue Analyzers, on the fracture mechanics theory of dynamically loaded test specimens and on the definition of pure-shear states according to the test specimen’s geometry ratio. The main focus of this work can be divided in two parts. Firstly it is introduced the development of a method for analysis of dynamic crack propagation in filled rubber by simultaneous tensile- and pure shear mode testing. The servo-hydraulic machine with controlled temperature testing chamber is equipped with simultaneously operating two mode test equipment which represents a new fracture testing method. Secondarily the analysis of crack propagation under the dynamic loading conditions is practised with this method. It is shown how the tearing energy and the crack growth rate depend on the test specimen’s geometry ratio and crack length. It is also demonstrated that the values for tearing energies and also crack growth rates for short crack lengths in SENT- as well as in pure-shear test specimens are identical. Another important aspect is related to the different values of tearing energies and crack growth rates for cracks with short and large lengths in pure-shear test specimens. The results show the dependence of fracture behavior on manufacturing the test specimens. The new fracture mechanical testing concept offers a comparison between fracture behavior of rubber materials independent of the test specimens geometry.

A análise de painéis de concreto armado submetidos a corte puro é um tópico que tem recebido atenção de pesquisadores ao longo dos anos. Isto se deve, principalmente, à dificuldade encontrada na aplicação do carregamento e das condições de vinculação sobre a estrutura. Esta dificuldade é constatada tanto experimentalmente como na análise numérica. O objetivo deste trabalho é apresentar um modelo numérico para simular o comportamento de painéis de concreto simples e de concreto armado submetidos ao corte. Na análise computacional via Elementos Finitos, será utilizado um modelo de fissura incorporada que considera a contribuição da armadura no equilíbrio interno de forças do elemento. Estes modelos se baseiam no conceito de descontinuidades incorporadas dentro de elementos finitos padrão. O modelo incorporado implementado é uma continuação dos trabalhos desenvolvidos por d’Avila (2003) e Brisotto (2006), baseados no modelo de Dvorkin, Cuitiño e Gioia (1990). O modelo de transferência de tensão por aderência de Russo, Zingone e Romano (1990) foi usado por d’Avila para incluir a contribuição da armadura no equilíbrio interno de forças do elemento. Para representar as barras de aço da armadura, é utilizado o modelo incorporado desenvolvido por Elwi e Hrudey (1989) que permite a disposição arbitrária das barras no interior dos elementos de concreto. Os resultados dos ensaios numéricos de painéis de concreto armado submetidos ao corte puro com diferentes taxas de armadura são comparados com os resultados experimentais apresentados por Vecchio (1981). São incluídos dois diferentes modelos de efeito de pino (resistência ao corte das barras da armadura), analisando-se a influência dos mesmos no comportamento dos painéis ensaiados. O efeito do engrenamento dos agregados na fissura também foi estudado. Vários painéis com diferentes formas de ruptura experimental foram simulados, obtendo-se respostas muito boas para a maioria deles. Comparações da resposta tensão tangencial x distorção com e sem efeito de pino foram feitas, verificando-se a grande importância deste efeito na resposta global. / The analysis of reinforced concrete panels submitted to in-plane shear has received the attention of researchers along the years. It is mainly due to the difficulty to simulate the application of the external loads and the structural boundary conditions. The purpose of this work is to present a numerical model to represent reinforced concrete panels submitted to inplane shear. In the computational analysis via Finite Elements, an embedded crack model that considers the inclusion of the reinforcement contribution in the internal force equilibrium of the element is presented. This type of model is based in discontinuities embedded into standard finite elements. The implemented formulation uses the model presented by Dvorkin, Cuitiño and Gioia (1990), which does not have the reinforcement contribution in the element internal equilibrium. The adherence stress transfer model of Russo, Zingone and Romano (1990) is used to include this reinforcement contribution. An embedded model was employed to represent the reinforcement bars, allowing that they can be placed in an independent position and shape in the FEM elements. The numerical results are compared with Vecchio’s work. Also are included two different dowel action models (transversal shear resistance of the bars) and is studied their influence in the panels’ behavior. Panels with different experimental rupture modes were simulated with good responses in the most cases. Comparisons of shear stress x shear strain curve with and without dowel action are performed, showing the importance of this effect on the global response.

A análise de painéis de concreto armado submetidos a corte puro é um tópico que tem recebido atenção de pesquisadores ao longo dos anos. Isto se deve, principalmente, à dificuldade encontrada na aplicação do carregamento e das condições de vinculação sobre a estrutura. Esta dificuldade é constatada tanto experimentalmente como na análise numérica. O objetivo deste trabalho é apresentar um modelo numérico para simular o comportamento de painéis de concreto simples e de concreto armado submetidos ao corte. Na análise computacional via Elementos Finitos, será utilizado um modelo de fissura incorporada que considera a contribuição da armadura no equilíbrio interno de forças do elemento. Estes modelos se baseiam no conceito de descontinuidades incorporadas dentro de elementos finitos padrão. O modelo incorporado implementado é uma continuação dos trabalhos desenvolvidos por d’Avila (2003) e Brisotto (2006), baseados no modelo de Dvorkin, Cuitiño e Gioia (1990). O modelo de transferência de tensão por aderência de Russo, Zingone e Romano (1990) foi usado por d’Avila para incluir a contribuição da armadura no equilíbrio interno de forças do elemento. Para representar as barras de aço da armadura, é utilizado o modelo incorporado desenvolvido por Elwi e Hrudey (1989) que permite a disposição arbitrária das barras no interior dos elementos de concreto. Os resultados dos ensaios numéricos de painéis de concreto armado submetidos ao corte puro com diferentes taxas de armadura são comparados com os resultados experimentais apresentados por Vecchio (1981). São incluídos dois diferentes modelos de efeito de pino (resistência ao corte das barras da armadura), analisando-se a influência dos mesmos no comportamento dos painéis ensaiados. O efeito do engrenamento dos agregados na fissura também foi estudado. Vários painéis com diferentes formas de ruptura experimental foram simulados, obtendo-se respostas muito boas para a maioria deles. Comparações da resposta tensão tangencial x distorção com e sem efeito de pino foram feitas, verificando-se a grande importância deste efeito na resposta global. / The analysis of reinforced concrete panels submitted to in-plane shear has received the attention of researchers along the years. It is mainly due to the difficulty to simulate the application of the external loads and the structural boundary conditions. The purpose of this work is to present a numerical model to represent reinforced concrete panels submitted to inplane shear. In the computational analysis via Finite Elements, an embedded crack model that considers the inclusion of the reinforcement contribution in the internal force equilibrium of the element is presented. This type of model is based in discontinuities embedded into standard finite elements. The implemented formulation uses the model presented by Dvorkin, Cuitiño and Gioia (1990), which does not have the reinforcement contribution in the element internal equilibrium. The adherence stress transfer model of Russo, Zingone and Romano (1990) is used to include this reinforcement contribution. An embedded model was employed to represent the reinforcement bars, allowing that they can be placed in an independent position and shape in the FEM elements. The numerical results are compared with Vecchio’s work. Also are included two different dowel action models (transversal shear resistance of the bars) and is studied their influence in the panels’ behavior. Panels with different experimental rupture modes were simulated with good responses in the most cases. Comparisons of shear stress x shear strain curve with and without dowel action are performed, showing the importance of this effect on the global response.

A análise de painéis de concreto armado submetidos a corte puro é um tópico que tem recebido atenção de pesquisadores ao longo dos anos. Isto se deve, principalmente, à dificuldade encontrada na aplicação do carregamento e das condições de vinculação sobre a estrutura. Esta dificuldade é constatada tanto experimentalmente como na análise numérica. O objetivo deste trabalho é apresentar um modelo numérico para simular o comportamento de painéis de concreto simples e de concreto armado submetidos ao corte. Na análise computacional via Elementos Finitos, será utilizado um modelo de fissura incorporada que considera a contribuição da armadura no equilíbrio interno de forças do elemento. Estes modelos se baseiam no conceito de descontinuidades incorporadas dentro de elementos finitos padrão. O modelo incorporado implementado é uma continuação dos trabalhos desenvolvidos por d’Avila (2003) e Brisotto (2006), baseados no modelo de Dvorkin, Cuitiño e Gioia (1990). O modelo de transferência de tensão por aderência de Russo, Zingone e Romano (1990) foi usado por d’Avila para incluir a contribuição da armadura no equilíbrio interno de forças do elemento. Para representar as barras de aço da armadura, é utilizado o modelo incorporado desenvolvido por Elwi e Hrudey (1989) que permite a disposição arbitrária das barras no interior dos elementos de concreto. Os resultados dos ensaios numéricos de painéis de concreto armado submetidos ao corte puro com diferentes taxas de armadura são comparados com os resultados experimentais apresentados por Vecchio (1981). São incluídos dois diferentes modelos de efeito de pino (resistência ao corte das barras da armadura), analisando-se a influência dos mesmos no comportamento dos painéis ensaiados. O efeito do engrenamento dos agregados na fissura também foi estudado. Vários painéis com diferentes formas de ruptura experimental foram simulados, obtendo-se respostas muito boas para a maioria deles. Comparações da resposta tensão tangencial x distorção com e sem efeito de pino foram feitas, verificando-se a grande importância deste efeito na resposta global. / The analysis of reinforced concrete panels submitted to in-plane shear has received the attention of researchers along the years. It is mainly due to the difficulty to simulate the application of the external loads and the structural boundary conditions. The purpose of this work is to present a numerical model to represent reinforced concrete panels submitted to inplane shear. In the computational analysis via Finite Elements, an embedded crack model that considers the inclusion of the reinforcement contribution in the internal force equilibrium of the element is presented. This type of model is based in discontinuities embedded into standard finite elements. The implemented formulation uses the model presented by Dvorkin, Cuitiño and Gioia (1990), which does not have the reinforcement contribution in the element internal equilibrium. The adherence stress transfer model of Russo, Zingone and Romano (1990) is used to include this reinforcement contribution. An embedded model was employed to represent the reinforcement bars, allowing that they can be placed in an independent position and shape in the FEM elements. The numerical results are compared with Vecchio’s work. Also are included two different dowel action models (transversal shear resistance of the bars) and is studied their influence in the panels’ behavior. Panels with different experimental rupture modes were simulated with good responses in the most cases. Comparisons of shear stress x shear strain curve with and without dowel action are performed, showing the importance of this effect on the global response.

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