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O mecanismo de rock burst envolve ruptura violenta e instantânea do maciço rochoso ao redor de uma escavação subterrânea, com liberação de grandes quantidades de energia. Neste trabalho, será apresentado uma abordagem baseada na teoria da elasticidade, a partir da qual a importância dos parâmetros elásticos serão avaliados no processo de acúmulo de energia elástica (W). Foram utilizados métodos analíticos e numéricos e equações que definem W a partir da matriz de tensões \'sigma\', da matriz de deformação \'épsilon\', e dos parâmetros elásticos E (módulo de Young, ou módulo de elasticidade) e \'nü\' (coeficiente de Poisson). A influência dos parâmetros E e \'nü\' na quantificação de W foi testada usando uma gama de valores, baseada em valores médios para as rochas cristalinas apresentados na literatura. Variações nos valores de E induzem mudanças consideráveis nos valores de W. Quanto menor E , maior W. Baixos valores de E definem maciços rochosos que sofrem maiores deformações sob um determinado regime de tensões, assim resultado em maior trabalho exercido pelo maciço e, consequentemente, mais energia elástica (W). Isto é, para um determinado estado de tensões, maciços mais deformáveis (com baixos valores de E ) têm a capacidade de acumular mais energia elástica que maciços pouco deformáveis. No entanto, considerando também a mecânica de ruptura, maciços menos resistentes rompem sob condições menos extremas de tensões, enquanto maciços mais resistentes necessitam de tensões muito elevadas para sofrerem ruptura. Portanto, maciços mais resistentes acumulam mais energia elástica que maciços menos resistentes. Embora o coeficiente \'nü\' seja apontado por alguns autores como maior responsável pelo acúmulo de W, este fato não foi observado neste trabalho, e o valor de W não é consideravelmente afetado por variações de \'nü\" , um resultado direto das equações utilizadas. No entanto, uma tendência \"anômala\" foi observada: fixado o valor de E, valores de \'nü\' medianos (próximo a 0,25) induzem maiores valores de W no limite da escavação, e conforme se afasta do limite para o sentido do interior do maciço, valores cada vez menores de passam a induzir os maiores valores de W. A teoria da elasticidade, por si só, não é capaz de explicar os fenômenos complexos que causam os processos de rock burst. Para uma compreensão completa do problema, é necessário também o estudo de mecânica de fraturas, mecânica de danos no maciço, dissipação de energia ao longo de fraturas, etc. / The mechanism or rock burst involves sudden and violent fracturing of the rock mass around the opening, with high amounts of energy released. This work demonstrates an approach based on the theory of elasticty and the role of elastic parameters in the process of storing elastic strain energy (W). Analytical and numerical methods were used. The main equations define W based on the stress matrix \'sigma\', the strain matrix \'épsilon\' and the elastic parameters E (Young\'s modulus) and \'nü\' (Poisson\'s ratio). The influence of E and \'nü\' was tested using minimum, average and maximum values, based on the values for crystalline hard rocks from the literature. Variations in values of E induce significant changes to the values of W. The lower E the higher W. Low values of E define rock masses which suffer greater displacements under a given stress condition, leading to higher values of W. This leads to the conclusion that, for a given stress state, soft rocks (lower values of E ) have the ability to store more elastic strain energy than stiff rocks. However, if fracture mechanics is also considered, strong rock masses have the ability to endure more stress than weaker rocks before failing, leading to higher amounts of elastic energy stored. Although Poisson\'s ratio (\'nü\') is considered by some authors as a fundamental piece in energy storage, such conclusion was not observed in this work, and values of W are not considerably affected by variations of \'nü\', a direct result of the application of the equations of theory of elasticity. However, an interesting trend was observed: for a given value of E , moderate values of \'ü\' (around 0,25) induce higher values of W at the edge of the excavation. In the rock mass interior, lower values of induce higher values of W. An analysis purely based on the theory of elasticity is not enough to explain the complex phenomena which occur around an excavation that induce violent failure and rock bursting. For a deeper understanding of the problem, it is also necessary the study of complementary theories, such as fracture mechanics, damage mechanics, energy dissipation during fracturing, and so on.

O mecanismo de rock burst envolve ruptura violenta e instantânea do maciço rochoso ao redor de uma escavação subterrânea, com liberação de grandes quantidades de energia. Neste trabalho, será apresentado uma abordagem baseada na teoria da elasticidade, a partir da qual a importância dos parâmetros elásticos serão avaliados no processo de acúmulo de energia elástica (W). Foram utilizados métodos analíticos e numéricos e equações que definem W a partir da matriz de tensões \'sigma\', da matriz de deformação \'épsilon\', e dos parâmetros elásticos E (módulo de Young, ou módulo de elasticidade) e \'nü\' (coeficiente de Poisson). A influência dos parâmetros E e \'nü\' na quantificação de W foi testada usando uma gama de valores, baseada em valores médios para as rochas cristalinas apresentados na literatura. Variações nos valores de E induzem mudanças consideráveis nos valores de W. Quanto menor E , maior W. Baixos valores de E definem maciços rochosos que sofrem maiores deformações sob um determinado regime de tensões, assim resultado em maior trabalho exercido pelo maciço e, consequentemente, mais energia elástica (W). Isto é, para um determinado estado de tensões, maciços mais deformáveis (com baixos valores de E ) têm a capacidade de acumular mais energia elástica que maciços pouco deformáveis. No entanto, considerando também a mecânica de ruptura, maciços menos resistentes rompem sob condições menos extremas de tensões, enquanto maciços mais resistentes necessitam de tensões muito elevadas para sofrerem ruptura. Portanto, maciços mais resistentes acumulam mais energia elástica que maciços menos resistentes. Embora o coeficiente \'nü\' seja apontado por alguns autores como maior responsável pelo acúmulo de W, este fato não foi observado neste trabalho, e o valor de W não é consideravelmente afetado por variações de \'nü\" , um resultado direto das equações utilizadas. No entanto, uma tendência \"anômala\" foi observada: fixado o valor de E, valores de \'nü\' medianos (próximo a 0,25) induzem maiores valores de W no limite da escavação, e conforme se afasta do limite para o sentido do interior do maciço, valores cada vez menores de passam a induzir os maiores valores de W. A teoria da elasticidade, por si só, não é capaz de explicar os fenômenos complexos que causam os processos de rock burst. Para uma compreensão completa do problema, é necessário também o estudo de mecânica de fraturas, mecânica de danos no maciço, dissipação de energia ao longo de fraturas, etc. / The mechanism or rock burst involves sudden and violent fracturing of the rock mass around the opening, with high amounts of energy released. This work demonstrates an approach based on the theory of elasticty and the role of elastic parameters in the process of storing elastic strain energy (W). Analytical and numerical methods were used. The main equations define W based on the stress matrix \'sigma\', the strain matrix \'épsilon\' and the elastic parameters E (Young\'s modulus) and \'nü\' (Poisson\'s ratio). The influence of E and \'nü\' was tested using minimum, average and maximum values, based on the values for crystalline hard rocks from the literature. Variations in values of E induce significant changes to the values of W. The lower E the higher W. Low values of E define rock masses which suffer greater displacements under a given stress condition, leading to higher values of W. This leads to the conclusion that, for a given stress state, soft rocks (lower values of E ) have the ability to store more elastic strain energy than stiff rocks. However, if fracture mechanics is also considered, strong rock masses have the ability to endure more stress than weaker rocks before failing, leading to higher amounts of elastic energy stored. Although Poisson\'s ratio (\'nü\') is considered by some authors as a fundamental piece in energy storage, such conclusion was not observed in this work, and values of W are not considerably affected by variations of \'nü\', a direct result of the application of the equations of theory of elasticity. However, an interesting trend was observed: for a given value of E , moderate values of \'ü\' (around 0,25) induce higher values of W at the edge of the excavation. In the rock mass interior, lower values of induce higher values of W. An analysis purely based on the theory of elasticity is not enough to explain the complex phenomena which occur around an excavation that induce violent failure and rock bursting. For a deeper understanding of the problem, it is also necessary the study of complementary theories, such as fracture mechanics, damage mechanics, energy dissipation during fracturing, and so on.

This problem has been selected from the mathematical theory of elasticity. We consider a rectangular plate of thickness h, length a, and width b. The plate is subjected to compressive forces. These forces act in the neutral plane and give the plate a tendency to buckle. However, this problem differs from other plate problems in that it is assumed that there are two intermediate supports located on the edges of the plate parallel to the compressive forces.

Omnipresence of heterogeneity is conspicuous in all creations of nature. Heterogeneity manifests itself in many forms at different scales, both in time and space. Engineering domain being an exotic fusion of human creativity and ever-increasing demands exemplifies the ubiquity of heterogeneity. Surprisingly, the plethora of materials we see around seem to stem from myriad combination of few base materials identified as elements in chemistry. Further, a simple rearrangement of atoms in these materials leads to allotropes with startling contrasts in properties. Similarly, micro- and meso-scales in heterogeneous materials also dis-play this phenomenon. Human requirements propelled by necessities and wants have leveraged heterogeneity deliberately or naively. In the context of engineering materials, light weight heterogeneous materials like composites and cellular solids are outstanding inventions from the last century. The present thesis highlights this phenomenon on a meso-scale to explore generalized variants of circular and elliptical honeycomb structures (HCSs) with an emphasis on their effective transverse elastic responses, a crucial pillar of engineering design and analysis. Homogenized or effective properties are an extension of continuum hypothesis, conceived for ease in analyses. E ective properties are employed in multi-scale analyses resulting in less complex models for analysis, for example, for predicting the speed of wave propogation. The thesis extends and generalizes existing close-packed circular and elliptical HCSs to more broader configurations. Simpler periodic arrangement of the unit cells from numerous exotic possibilities directly incorporates Design for Manufacture and Assembly (DFMA) philosophy and o ers a potential scope for analysis by simpler tools resulting in handy expressions which are of great utility for designer engineers. In this regard, analytical expressions for moduli having compact forms in the case of circular HCS are developed by technical theories and rigorous theory of elasticity. Regression analysis expressions for the moduli of elliptical HCS are presented, and the elasticity solutions for the same are highlighted. The thesis consists of seven chapters with Chapter 1 presenting generalized circular and elliptical HCSs as a potential avenue beyond composite materials. Following a survey of pertinent HCS literature of these HCSs, research gaps and scope are delineated. Chapter 2 briefly y summarizes the ideas, concepts and tools including analytical and numerical methods. This chapter sets the ground for the analysis of generalized circular and elliptical HCS in the following four chapters. Following the classification of the circular HCSs, Chapter 3 assesses the complete transverse elastic responses of generalized circular HCS through technical theories which are a first-order approximation. Here, thin ring theory and the more elaborate curved beam theory are employed as models to assess the moduli. Normal moduli - E and - are obtained by employing Castigliano method, while shear moduli (G ) are obtained by solving the differential equations derived in terms of displacements. Compact expressions for moduli presented wherever possible furnish the designer with a range of moduli for different configurations and modular ratios (Ey=Ex). The results show the range of applicability of technical theories within 5% of FEA. For hexagonal arrays, these results are more refined than those in literature; while the same are new for other configurations. Surprisingly, the more elaborate curved beam theory offers no better results than the thin ring theory. Chapter 4 extends the aforementioned task of assessing the complete trans-verse elastic moduli of generalized circular HCS by employing rigorous theory of elasticity (TOE) which is a second-order approximation. Utilizing Airy stress function in polar coordinates, the boundary value problems resulting from modeling of the circular HCS under different loads are solved analytically in conjunction with FEA employing contact elements. Contact elements circumvent the point loads which give finite values of displacements in technical theories and singular values in TOE. A widely used idea of employing distributed load, statically equivalent to point load, is invoked to empower TOE. The distributed load is assumed a priori and the contact length is obtained from FEA employing con-tact elements. Thus, FEA compliments the present analytical methods. Results demonstrate a very good match between analytical method in conjunction with FEA and numerical results from FEA; the error is within 5% for very thick ring (thickness-radius ratio 0.5). Further, computationally and numerically efficient expressions for displacements give better results with same computational facility. To illustrate the effect of coating on effective moduli, a limited study based on thin ring theory and elasticity theories is undertaken in Chapter 4. The study explores the effects of moduli and thickness ratios of substrate to coating on the effective normal moduli. Employing thin ring theory with only flexure as the bending mode, we get compact expressions giving good match for very thin rings in all confifigurations. The elasticity approach presented for square array demonstrates a very good match with FEA for thick rings. Coatings offer a strategy to increase the effective moduli with same dimensions. Chapter 5 broadens the scope of circular HCS by considering elliptical HCSs. While generalized circular HCS can cater to anisotropic requirement to an extent, larger spectrum is offered by considering elliptical honeycomb structures. In this regard, a generalized version of concentric thin coated elliptical HCS is investigated for transverse moduli. Thin HCSs are explored by technical theories as in circular HCS. However, a lack of exact compact-form expressions necessitates the use of regression analysis. The resulting expressions are presented in terms of ellipticity ratio describing the ovality of the ellipse and geometric parameters. Normal moduli are obtained by Castigliano method implemented in MATHE-MATICA, but shear moduli are obtained from FEA employing beam elements. The need for FEA employing beam elements stems from the subtle fact that Castigliano method implicitly assumes preclusion of rigid body motions, while shear loading for shear moduli evaluation entails rigid body motions. Interestingly, curved beam theory, as in circular HCS, offers no better refinement in assessing the moduli as compared to thin ring theory. The graphs showing the moduli with respect to thickness and modular ratios are presented as design maps to aid the designer. Chapter 6 extends the works of thin concentric coated elliptical to thicker concentric and a novel confocal elliptical HCS, a variant of elliptical HCS. In this regard, thick concentric and confocal elliptical HCS by elasticity approach are attempted for a simple case. Airy stress function in polar coordinates is tried for concentric elliptical HCS. Confocal HCS analysis employs stress function in terms of elliptical coordinate system. After proving the correctness of the stress function for both the cases by comparing the reconstructed boundary conditions with actual boundary conditions, the restrictions in solving the case of rings under load over a small region is highlighted. A parametric study for moduli is under-taken by employing FEA. These are presented as design graphs which compare and contrast the two variants of elliptical HCS on the same graphs. The modular ratio (Ey=Ex) is conspicuously more for confocal elliptical HCS than concentric elliptical HCS. Chapter 7 gives the conclusions in a nutshell, and explores the feasibility of stress evaluation of heterogeneous media on the lines of effective media theory.

Periodically perforated sheets(PS) are ubiquitous in nature as well as in engineered artifacts developed for aerospace, automotive, marine, nuclear and structural applications. PS are indispensable for saving weight and cost for aircraft; for enhancing safety and integrity of heat exchangers used in nuclear and thermal power stations. Ancient PS grills and lattice frames dating back to 1000 BC continue to inspire contemporary art and architecture, buildings and furniture. PS design and analysis, however, is a complex affair stemming from the inherent configurational anisotropy induced by periodicity. In addition, complex boundary conditions complicate the analysis. Unlike atoms in crystalline media, both shape and periodicity of perforations control this anisotropic nature. This thesis explores theoretical and numerical strategies for evaluating the effective anisotropic elastic moduli of PS. Following an experimental prelude for visualizing the PS stress field in a photoelastic sheet and a brief review of PS theory, this thesis proposes a novel theoretical numerical hybrid method for determining the Airy stress function constants. The proposed hybrid method can be exploited experimentally using automated vision based imaging technologies to measure the boundary displacements noninvasively. For determining the Airy constants periodic boundary conditions to the unit cell are applied, the displacement components around the PS hole boundary are obtained using FEM. Using these constants the PS stress field is reconstructed to assess the efficacy of the proposed hybrid method. It is observed that in general while the actual and the reconstructed stress fields agree reasonably well, more refined boundary data obtained either numerically or experimentally can enhance the accuracy further. The thesis then makes an extensive presentation of anisotropic moduli in a variety of PS designs configured on rectangular or square layouts. Conventional as well as some exotic patterns with cusps and satellite holes are examined, and the results are presented graphically to aid the designer. Finally, some special topics pertaining PS design and analysis are discussed to help overcome the inherent limitations of solutions based on applying periodic boundary conditions. In this vein, strategies for achieving a functionally graded PS are presented by altering the pitch and hole size. These strategies assume importance near boundaries as well as near concentrated forces inducing stress gradients. Other special topics include the applicability of tensor transformation rule to PS anisotropy. The effective bulk modulus which remains a scalar invariant is exploited to assess the validity of tensor transformation in a square PS. The rule of mixture widely used in homogenization of composite media is also discussed briefly. Thus, this thesis makes an attempt to demonstrate the power of blending micromechanics with experiments and FEM to aid in PS design and analysis.