Mitteilungen des URZ 2/2002 (Inhalt siehe Schlagwörter)

In this thesis a theory for large deformation of plates is presented. Herein aspects of the common 3D-theory for large deformation with the Kirchhoff hypothesis for reducing the dimension from 3D to 2D is combined. Even though the Kirchhoff assumption was developed for small strain and linear material laws, the deformation of thin plates made of isotropic non-linear material was investigated in a numerical experiment. Finally a heavily deformed shell without any change in thickness arises. This way of modeling leads to a two-dimensional strain tensor essentially depending on the first two fundamental forms of the deformed mid surface. Minimizing the resulting deformation energy one ends up with a nonlinear equation system defining the unknown displacement vector U. The aim of this thesis was to apply the incremental Newton technique with a conformal, C^1-continuous finite element discretization. For this the computation of the second derivative of the energy functional is the key difficulty and the most time consuming part of the algorithm. The practicability and fast convergence are demonstrated by different numerical experiments.:1 Introduction 2 The deformation problem in the three-dimensional space 2.1 General differential geometry of deformation in the three-dimensional space 2.2 Equilibrium of forces 2.3 Material laws 2.4 The weak formulation 3 Newton’s method 3.1 The modified Newton algorithm 3.2 Second linearization of the energy functional 4 Differential geometry of shells 4.1 The initial mid surface 4.2 The initial shell 4.3 The plate as an exception of a shell 4.4 Kirchhoff assumption and the deformed shell 4.4.1 Differential geometry of the deformed shell 4.4.2 The Lagrangian strain tensor of the deformed plate 5 Shell energy and boundary conditions 5.1 The resulting Kirchhoff deformation energy 5.2 Boundary conditions 5.3 The resulting weak formulation 6 Newton’s method and implementation 6.1 Newton algorithm 6.2 Finite Element Method (FEM) 6.2.1 Bogner-Fox-Schmidt (BFS) elements 6.2.2 Hsiegh-Clough-Tocher (HCT) elements 6.3 Efficient solution of the linear systems of equation 6.4 Implementation 6.5 Newton’s method and mesh refinement 7 Numerical examples 7.1 Plate deflection 7.1.1 Approximation with FEM using BFS-elements 7.1.2 Approximation with FEM using reduced HCT-elements 7.2 Bending-dominated deformation 7.2.1 Approximation with FEM using BFS-elements 7.2.1.1 1st example: Cylinder 7.2.1.2 2nd example: Cylinder with further rotated edge normals 7.2.1.3 3rd example: Möbiusstrip 7.2.1.4 4th example: Plate with twisted edge 7.2.2 Approximation with FEM using reduced HCT-elements 7.2.2.1 1st example: Partly divided annular octagonal plate 7.2.2.2 2nd example: Divided annulus with rotated edge normals 8 Outlook and open questions Bibliography Notation Theses List of Figures List of Tables

<p align="justify">In a context of restoration of historical masonry structures, it is crucial to properly estimate the residual strength and the potential structural failure modes in order to assess the safety of buildings. Due to its mesostructure and the quasi-brittle nature of its constituents, masonry presents preferential damage orientations, strongly localised failure modes and damage-induced anisotropy, which are complex to incorporate in structural computations. Furthermore, masonry structures are generally subjected to complex loading processes including both in-plane and out-of-plane loads which considerably influence the potential failure mechanisms. As a consequence, both the membrane and the flexural behaviours of masonry walls have to be taken into account for a proper estimation of the structural stability.</p><p><p align="justify">Macrosopic models used in structural computations are based on phenomenological laws including a set of parameters which characterises the average behaviour of the material. These parameters need to be identified through experimental tests, which can become costly due to the complexity of the behaviour particularly when cracks appear. The existing macroscopic models are consequently restricted to particular assumptions. Other models based on a detailed mesoscopic description are used to estimate the strength of masonry and its behaviour with failure. This is motivated by the fact that the behaviour of each constituent is a priori easier to identify than the global structural response. These mesoscopic models can however rapidly become unaffordable in terms of computational cost for the case of large-scale three-dimensional structures.</p><p><p align="justify">In order to keep the accuracy of the mesoscopic modelling with a more affordable computational effort for large-scale structures, a multi-scale framework using computational homogenisation is developed to extract the macroscopic constitutive material response from computations performed on a sample of the mesostructure, thereby allowing to bridge the gap between macroscopic and mesoscopic representations. Coarse graining methodologies for the failure of quasi-brittle heterogeneous materials have started to emerge for in-plane problems but remain largely unexplored for shell descriptions. The purpose of this study is to propose a new periodic homogenisation-based multi-scale approach for quasi-brittle thin shell failure.</p><p><p align="justify">For the numerical treatment of damage localisation at the structural scale, an embedded strong discontinuity approach is used to represent the collective behaviour of fine-scale cracks using average cohesive zones including mixed cracking modes and presenting evolving orientation related to fine-scale damage evolutions.</p><p><p align="justify">A first originality of this research work is the definition and analysis of a criterion based on the homogenisation of a fine-scale modelling to detect localisation in a shell description and determine its evolving orientation. Secondly, an enhanced continuous-discontinuous scale transition incorporating strong embedded discontinuities driven by the damaging mesostructure is proposed for the case of in-plane loaded structures. Finally, this continuous-discontinuous homogenisation scheme is extended to a shell description in order to model the localised behaviour of out-of-plane loaded structures. These multi-scale approaches for failure are applied on typical masonry wall tests and verified against three-dimensional full fine-scale computations in which all the bricks and the joints are discretised.</p> / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

M. Tech (Department of Chemistry, Faculty of Applied and Computer Sciences), Vaal University of Technology. / Activated carbons have been produced from various carbonaceous source materials including nutshells, peat, wood, coir, lignite, coal and petroleum pitch and the produced carbons have been used for adsorption of inorganic and organic compounds from numerous matrices. Activated carbons are characterized by large surface area and high degree of microporosity. The ability of activated carbon to adsorb gold from solutions, which is present in very low concentrations while loading to fairly high concentrations, has made it an attractive material for the concentration of gold from dilute solutions. Consequently, the use of activated carbons for the recovery of gold from cyanide-leached liquors has gained wide acceptance in the mining industry. However, the price of commercially prepared carbons, the time taken for them to arrive, the breakage of the carbon during transportation and the interest in utilization of various local wastes for the generation of adsorbents has led to a search of more cost effective and time friendly source for the activated carbons. The focus of this research was to conduct a study in which a comparison was conducted between untreated coconut shell derived activated carbon (CAC) and Macadamia nutshell derived activated carbon (MAC) for the adsorption of gold. These activated carbons were modified with HNO3 and H3PO4 to increase their surface adsorption properties. This was done in order to explore if these activated carbons prepared from Macadamia shells could be an attractive alternative or a complementary supplement to the coconut shell based carbons that are currently being used in the gold extraction industry. The modification of the commercially prepared Macadamia activated carbons was done with 3 different concentrations for both nitric acid and phosphoric acid. The modified activated carbons were labelled MACP20%, MACP40% and MACP60%, to signify the materials prepared from 20% (v/v) H3PO4, 40% (v/v) H3PO4 and 60% (v/v) H3PO4 , respectively . Same labelling was used for 20% (v/v) HNO3, 40% (v/v) HNO3 and 55% (v/v) HNO3 modifications to correspond to MACN20%, MACN40% and MACN55%, respectively. Also, untreated coconut shell derived activated carbon (CAC) and Macadamia nutshell derived activated carbon (MAC) were investigated for gold adsorption for comparison purposes. All the activated carbons prepared in the iii | P a g e study were characterized with Brunauer-Emmet-Teller (BET), scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FTIR), Thermo gravimetric analysis (TGA), elemental analysis (EL) and X-ray diffraction spectroscopy (XRD). The physical properties of the activated carbons were done by determining attrition, ash content, volatile matter, and moisture content of all the activated carbons. Various parameters that affect selective adsorption such as the effect of initial concentration, time, agitation speed, interfering species, and dose of the adsorbent were investigated. Optimal parameters for gold ion adsorption were as follows: solution pH, 10; contact time, 6 h; agitation speed 150 rpm; sorbent amount 4 g and 5.5 ppm for initial concentration of gold. The observed selectivity order was not the same for all the adsorbents but the adsorption of gold was found to be mostly influenced by the presence of nickel and least influenced by copper. The MACP60% was found to be the most effective from the three concentrations investigated for the phosphoric acid modified activated carbons yet proved to have lower adsorption capabilities compared to CAC. The MACN55% was found to be the most efficient and displayed similar adsorption capabilities to those of CAC.

[pt] As cascas cilíndricas possuem aplicações em diversas áreas da engenharia. Nas últimas décadas tem se observado o surgimento de novos materiais e suas técnicas de produção, levando a novas aplicações em estruturas de cascas. Dentre estas, as cascas sanduíche e cascas com gradação funcional têm levado, em muitas aplicações, a um melhor desempenho estrutural associado a uma redução de peso. Este trabalho tem como objetivo estudar as frequências naturais e as cargas críticas de cascas sanduíche com faces de metal e núcleo de espuma metálica e cascas com gradação funcional, onde as características da espuma metálica variam ao longo da espessura levando a uma estrutura similar à da casca sanduíche. Esses resultados são comparados com aqueles de cascas isotrópicas homogêneas. Para tanto, é utilizada a teoria linear de Donnell, que é uma das mais empregadas para análise de cascas. Primeiramente, derivam-se as equações de movimento assim como as equações de equilíbrio crítico. Utilizando as soluções analíticas para uma casca simplesmente apoiada, obtêm-se as matrizes de massa, de rigidez e de rigidez geométrica, possibilitando o cálculo das frequências naturais e cargas críticas da casca sob compressão axial e pressão lateral. Através de uma análise paramétrica, os resultados mostram a influência da geometria da casca, da variação do material ao longo da espessura, do cisalhamento no núcleo e dos termos de inércia nas cargas críticas e frequências naturais. Os resultados também ressaltam a influência do núcleo de espuma metálica no aumento da capacidade de carga e redução de peso das cascas sanduíche e com gradação funcional. / [en] Cylindrical shells are used in several areas of engineering fields. In the last decades has been observed the emergence of new materials and their production techniques, leading to new applications in shell structures. Among these, the sandwich shells and shell with functionally graded materials have led, in many applications, to a better structural performance associated to a reduction of weight. This work aims to study the natural frequencies and the critical loads of sandwich shells with metal faces and metal foam core and functionally graded shells, where the characteristics of the metallic foam vary throughout the thickness leading to a structure similar to that of the sandwich shell. These results are compared with those of homogeneous isotropic shells. For this, the linear theory of Donnell, which is one of the most used for shell analysis, is here used. First, the equations of motion as well as the critical equilibrium equations are derived. Using the analytical solutions for a simply supported shell, the mass, stiffness and geometric stiffness matrices are obtained, allowing the calculation of the natural frequencies and critical loads of the shell under axial compression and lateral pressure. Through a parametric analysis, the results show the influence of the shell geometry, material variation along the shell thickness, shear deformation of the core and the inertia terms on the critical loads and natural frequencies. The results also highlight the influence of the metallic foam core in increasing the load bearing capacity and reducing the weight of the sandwich and functionally graded shells.

ENGLISH SUMMARY Ammonite assemblage from the Upper Cenomanian of the Bohemian Cretaceous Basin realtively poorly known. Some species have been described herein for the first time: Thomelites cf. sornayi (Thomel) Euomphaloceras septemseriatum (Cragin) Eucalycoceras pentagonum (Gragin) and Allocrioceras annulatum (Shumard.). These records predominantly come from the M. geslinianum zone sediments developed in the higher parts of the Pecinov member (Uličný et al., 1998) of the Peruc - Korycany Formation. The sediments yielding ammonite assemblage represents the Late Cenomanian transgressive tract, accompanied by anoxic event OAE II = Oceanic anoxic Event II, CTBE) worldwide recorded in sediments and showing an increased content of stable isotope δ13 C. The OAE II is suggested to be connected to marine biota crisis and also the mass extinctionas well. The most affected organisms were benthic forms, due to high anoxic conditions at the bottom of the water column. Morphological features of the shells may, depending on the quality preservation shells (or cores) can be used to paleobathymetric interpretations. Most described species lived at depths of 50-80 m (up to 100m). Pelagic ammonites (i.e. Sciponoceras) lived at depths of up to 50 m. Heteromorphic taxa (Allocrioceras) have occurred at a depth of up to 100 m.

De nombreux travaux dans la littérature se sont concentrés sur la modélisation vibro-acoustique de coques cylindriques raidies immergées, du fait des nombreuses applications industrielles, en particulier dans le domaine aéronautique ou naval. Cependant, peu d'entre elles prennent en compte des structures internes non-axisymétriques telles que des supports moteurs, des planchers ou des carlingages, qui peuvent avoir une influence importante sur le comportement vibro-acoustique du système. C'est pourquoi une méthode de sous-structuration baptisée CTF est présentée dans cette thèse. Elle est développée dans le cas général de deux structures minces couplées le long d'une ligne. Un ensemble de fonctions orthonormées, baptisées fonctions de condensation, est défini afin d'approximer les forces et déplacements à la jonction entre les sous-systèmes. Des fonctions de transfert condensées sont définies pour chaque sous-système découplé. L'utilisation du principe de superposition, de l'équilibre des forces et de la continuité des déplacements permet de déduire le comportement des sous-systèmes couplés. La méthode est d'abord développée et validée dans le cas de plaques, puis ensuite appliquée au cas d'une coque cylindrique raidie immergée couplée à des structures internes non-axisymétriques. Le système est dans ce cas décomposé en 3 familles de sous-systèmes : la coque cylindrique immergée décrite par une méthode semi-analytique basée sur la résolution des équations de Flügge dans le domaine des nombres d’onde, les structures internes axisymétriques (raidisseurs, cloisons) décrites par éléments finis axisymétriques et les structures non-axisymétriques décrites pas des modèles éléments finis. La méthode CTF est appliquée à différents cas tests afin de montrer l'influence des structures internes non-axisymétriques sur le comportement vibro-acoustique d'une coque cylindrique pour différents types d'excitations pertinents dans le domaine naval : une force ponctuelle, une onde plane acoustique et un champ de pression aléatoire (tel qu'un champ acoustique diffus ou une couche limite turbulente). / Many works can be found in the literature concerning the vibroacoustic modelling of submerged stiffened cylindrical shells, because of high interest in the industrial domain, in particular for aeronautical or naval applications. However, only a few of them take into account non-axisymmetric internal frames, as for instance engine foundations or floor partitions, that can play a role on the vibroacoustic behavior of the system. That is why a substructuring approach called the Condensed Transfer Function (CTF) approach is proposed in the first part of this thesis. The aim is to take advantage of both analytical models and element-based models, in order to be able to deal with the geometrical complexity, and to calculate at higher frequencies than with element-based methods only. The substructuring method is developed in the general case of thin mechanical structures coupled along curves. A set of orthonormal functions called condensation functions, which depend on the curvilinear abscissa along the coupling line, is considered. This set is then used as a basis for approximating and decomposing the displacements and the applied forces at the line junctions. Thanks to the definition and calculation of condensed transfer functions for each uncoupled subsystem and by using the superposition principle for passive linear systems, the behavior of the coupled subsystems can be obtained. The method is first developed and validated for plates and convergence criteria are defined in relation with the size of the basis of condensation functions. The CTF method is then applied to the case of a submerged stiffened cylindrical shell with non-axisymmetric internal frames. The system is partitioned in 3 types of subsystems: the submerged shell, the axisymmetric frames (stiffeners, bulkheads) and the non-axisymmetric frames. The submerged shell is described by a semi-analytical method based on the Flügge equations in the spectral domain. The axisymmetric frames are described by axisymmetric Finite Element models and the non-axisymmetric frames by Finite Element models. The CTF method is applied to different test cases in order to highlight the influence of non-axisymmetric internal frames on the vibroacoustic behavior of a submerged stiffened cylindrical shell, for different excitations particularly relevant in the naval domain: a point force, an acoustic plane wave, and a random pressure field (such as a diffuse sound field or a turbulent boundary layer for instance).

Em geral, as formulações disponíveis na literatura para a análise de cascas laminadas reforçadas com fibras substituem o meio original heterogêneo por um homogêneo equivalente, que dificulta a identificação das tensões fibra-matriz, ou requerem que a malha de elementos finitos seja disposta de modo que os nós dos elementos finitos de fibra coincidam com os nós dos elementos finitos de casca, que é uma exigência bastante restritiva e que aumenta o número de graus de liberdade do sistema de equações resultante. Neste sentido, o objetivo geral desta tese consiste em desenvolver uma formulação para a inclusão de fibras longas e curtas aleatórias nas diversas lâminas de cascas laminadas anisotrópicas com não linearidade geométrica utilizando o método dos elementos finitos sem aumentar o número de graus de liberdade do sistema de equações resultante e sem a necessidade de coincidência de nós na discretização das fibras e da matriz. Nesta formulação, o elemento finito triangular de casca laminada utilizado para discretizar a matriz possui dez nós e sete graus de liberdade por nó, sendo três translações, três componentes do vetor generalizado e a taxa de variação linear da deformação ao longo da espessura. As fibras curvas, curtas aleatórias ou longas, são introduzidas, em qualquer camada do laminado, por meio de relações cinemáticas que garantem sua aderência à matriz sem a introdução de novos graus de liberdade no sistema de equações resultante. Para discretizá-las são utilizados elementos finitos unidimensionais de ordem qualquer com três graus de liberdade por nó e que consideram consistentemente a não linearidade geométrica. Todas as grandezas envolvidas são escritas em relação à configuração inicial do corpo, caracterizando a descrição Lagrangeana total ou material do movimento. Para modelar o comportamento do material adota-se a Lei Constitutiva de Saint-Venant-Kirchhoff que relaciona de forma linear o tensor de tensões de Piolla-Kirchhoff de segunda espécie e o tensor de deformações de Green-Lagrange. O equilíbrio é encontrado a partir do Princípio da Mínima Energia Potencial Total e o sistema não linear de equações resultante é resolvido utilizando-se o procedimento iterativo de Newton-Raphson. As ações externas podem ser introduzidas ao sistema de forma total ou incremental e a contribuição das fibras para a energia do sistema é adicionada na matriz global do problema. Os exemplos numéricos testados validam e demonstram as potencialidades da formulação proposta. / In general, the Finite Element (FE) formulations available in the literature for the analysis of fibre reinforced laminated shells replace the original heterogeneous medium by an equivalent homogeneous one, which makes difficult the identification of fiber-matrix stress distribution, or require that the finite element mesh is arranged in a way that the fibre finite element nodes coincide with the shell finite element ones, which is a very restrictive requirement and increases the number of degrees of freedom of the resulting system of equations. In this sense, the objective of this thesis is to develop a formulation for the inclusion of long and random short fibres in any layer of FE laminated anisotropic shells developing large displacement and rotations without increasing the number of degrees of freedom and the necessity of matching nodes in the discretization of the fibre and the matrix. In this formulation, the triangular laminated shell finite element used to discretize the matrix has ten nodes and seven degrees of freedom per node, that are, three translations, three components of a generalized vector and the linear rate of strain variation along the thickness. The curved fibres, long or random short, are introduced in any layer of the laminate shell by means of kinematic relation to ensure its adherence to the matrix without introducing new degrees of freedom in the resulting system of equations. To discretize them, any order one-dimensional finite elements with three degrees of freedom per node are used. These fibres elements are consistently considered by Geometric nonlinearity. All involved variables are written with respect to the initial configuration of the body, characterizing the Total Lagrangian description. To model the behavior of the material we use the Saint-VenantKirchhoff Constitutive Law that relates linearly the second Piolla-Kirchhoff stress tensor and Green-Lagrange strain tensor. The equilibrium is achieved from the Principle of Minimum Potential Energy and the non-linear system of equations is solved by the Newton-Raphson iterative procedure. External loads may be introduced to the system by one or various steps and the contribution of fibres to the energy of the system is added to the global matrix of the problem. The numerical examples validate and demonstrate the potential of the proposed formulation.

No description available.

This work aims at dimensional reduction of nonlinear material models in an asymptotically accurate manner. The three-dimensional(3-D) nonlinear material models considered include isotropic, orthotropic and dielectric compressible hyperelastic material models. Hyperelastic materials have potential applications in space-based inflatable structures, pneumatic membranes, replacements for soft biological tissues, prosthetic devices, compliant robots, high-altitude airships and artificial blood pumps, to name a few. Such structures have special engineering properties like high strength-to-mass ratio, low deflated volume and low inflated density. The majority of these applications imply a thin shell form-factor, rendering the problem geometrically nonlinear as well. Despite their superior engineering properties and potential uses, there are no proper analysis tools available to analyze these structures accurately yet efficiently. The development of a unified analytical model for both material and geometric nonlinearities encounters mathematical difficulties in the theory but its results have considerable scope. Therefore, a novel tool is needed to dimensionally reduce these nonlinear material models. In this thesis, Prof. Berdichevsky’s Variational Asymptotic Method(VAM) has been applied rigorously to alleviate the difficulties faced in modeling thin shell structures(made of such nonlinear materials for the first time in the history of VAM) which inherently exhibit geometric small parameters(such as the ratio of thickness to shortest wavelength of the deformation along the shell reference surface) and physical small parameters(such as moderate strains in certain applications). Saint Venant-Kirchhoff and neo-Hookean 3-D strain energy functions are considered for isotropic hyperelastic material modeling. Further, these two material models are augmented with electromechanical coupling term through Maxwell stress tensor for dielectric hyperelastic material modeling. A polyconvex 3-D strain energy function is used for the orthotropic hyperelastic model. Upon the application of VAM, in each of the above cases, the original 3-D nonlinear electroelastic problem splits into a nonlinear one-dimensional (1-D) through-the-thickness analysis and a nonlinear two-dimensional(2-D) shell analysis. This greatly reduces the computational cost compared to a full 3-D analysis. Through-the-thickness analysis provides a 2-D nonlinear constitutive law for the shell equations and a set of recovery relations that expresses the 3-D field variables (displacements, strains and stresses) through thethicknessintermsof2-D shell variables calculated in the shell analysis (2-D). Analytical expressions (asymptotically accurate) are derived for stiffness, strains, stresses and 3-D warping field for all three material types. Consistent with the three types of 2-D nonlinear constitutive laws,2-D shell theories and corresponding finite element programs have been developed. Validation of present theory is carried out with a few standard test cases for isotropic hyperelastic material model. For two additional test cases, 3-Dfinite element analysis results for isotropic hyperelastic material model are provided as further proofs of the simultaneous accuracy and computational efficiency of the current asymptotically-correct dimensionally-reduced approach. Application of the dimensionally-reduced dielectric hyperelastic material model is demonstrated through the actuation of a clamped membrane subjected to an electric field. Finally, the through-the-thickness and shell analysis procedures are outlined for the orthotropic nonlinear material model.