Dissertations / Theses on the topic 'Gradient of elasticity'

This thesis presents the extension of the gradient smoothing technique for finite element approximation (so-called Smoothed Finite Element Method (S-FEM)) and its bubble-enhanced version for non-linear problems involving large deformations in nearly-incompressible and incompressible hyperelastic materials. Finite Element Method (FEM) presents numerous challenges for soft matter applications, such as incompressibility, complex geometries and mesh distortion from large deformation. S-FEM was introduced to overcome the challenges mentioned of FEM. The smoothed strains and the smoothed deformation gradients are evaluated on the smoothing domain selected by either edge information, nodal information or face information. This thesis aims the extension of S-FEM in finite elasticity as a means of alleviating locking and avoiding mesh distortion. S-FEM employs a “cubic” bubble enhancement of the element shape functions with edge-based and face-based S-FEMs, adding a linear displacement field at the centre of the element. Thereby bubble-enhanced S-FEM affords a simple and efficient implementation. This thesis reports the properties and performance of the proposed method for quasi-incompressible hyperelastic materials. Benchmark tests show that the method is well suited to soft matter simulation, overcoming deleterious locking phenomenon and maintaining the accuracy with distorted meshes.

COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
A modelagem matemática de microdispositivos, em que estrutura e microestrutura têm aproximadamente a mesma escala de magnitude, assim como de macroestruturas de natureza predominantemente granular ou cristalina, requer uma abordagem não-local de deformações e tensões. Há mais de cem anos os irmãos Cosserat já tinham desenvolvido uma teoria de grãos rígidos. No entanto, e sem detrimento de desenvolvimentos devidos a Toupin e outros pesquisadores, os trabalhos de Mindlin na década de 1960 podem ser considerados a base da chamada teoria gradiente de deformações, que se tornou recentemente objeto de um grande número de investigações analíticas e experimentais, motivadas pelo desenvolvimento de novos materiais estruturais e do crescente uso de dispositivos micro- e nanomecânicos na indústria. Mais recentemente, Aifantis e colaboradores conseguiram desenvolver uma teoria gradiente de deformações mais simplificada, com base somente em duas constantes elásticas adicionais, representativas de comprimentos característicos relacionados às energias de deformação superficial e volumétrica. Uma série de trabalhos recentes desenvolvidos por Beskos e colaboradores estendeu o campo de aplicações da proposta inicial de Aifantis e introduziu uma solução fundamental que de fato remonta aos trabalhos de Mindlin. A equipe de pesquisa de Beskos propôs as primeiras implementações 2D e 3D de elementos de contorno para análises de elasticidade gradiente tanto estáticas quanto no domínio da freqüência, inclusive para problemas da mecânica da fratura. Desde o tempo de Toupin e Mindlin procura-se estabelecer uma base variacional da teoria e uma formulação consistente das condições de contorno cinemáticas e de equilíbrio, o que parece ter tido êxito com os recentes trabalhos de Amanatidou e Aravas. Esta dissertação faz uma revisão da teoria gradiente da deformações e apresenta um estudo didático do problema mais simples que se possa conceber, que é o de uma barra sob diferentes tipos de ações axiais (Aifantis, Beskos). A solução fundamental para problemas 2D e 3D também é apresentada e estudada, tanto em termos de forças pontuais aplicadas, para uma implementação em termos de elementos de contorno, quanto de desenvolvimentos polinomiais (no caso estático), para implementação em termos de elementos finitos. Mostra-se que a teoria gradiente de deformação de Aifantis é adequada a uma formulação no contexto do potencial de Hellinger-Reissner, o que possibilita implementações híbridas de elementos finitos e de contorno. O presente trabalho de pesquisa objetiva o estudo do estado da arte no tema, com uma abordagem dos principais problemas de implementação computacional, inclusive em termos das integrais singulares que surgem. O desenvolvimento completo de programas de análise de elementos híbridos finitos e de contorno, para problemas estáticos e dinâmicos, está planejado para uma tese de doutorado em futuro próximo.
The mathematical modeling of micro-devices in which structure and the microstructure are about the same scale of magnitude, as well as of macrostructure of markedly granular or crystal nature (microcomposites), demands a nonlocal approach for strains and stresses. More than one hundred years ago the Cosserat brothers had already developed a theory for rigid grains. However, and in no detriment due to Toupin and other researchers, Mindlin s work in the 1960s may be accounted the basis of the so-called strain gradient theory, which has recently become the subject of a large number of analytical and experimental investigations motivated by the development of news structural materials together with the increasing use of micro and nano-mechanical devices in the industry. More recently, Aifantis and coworkers managed to develop a simplified strain gradient theory based only on two additional elasticity constants that are representative of material lengths related to surface and volumetric strain energy. A series of very recent works done by Beskos and collaborators extended the field of applications of Aifantis propositions and introduced a fundamental solution that actually remounts to developments already laid down by Mindlin. Beskos workgroup may be regarded as the proponent of the first of the first boundary element 2D and 3D implementations on the subject for both statics and frequency-domain analyses, also including crack problems. Since Toupin and Mindlin`s time, investigations have been under development to establish the variational basis of the theory and to consistently formulate equilibrium and kinematic boundary conditions established by Amanatidou and Aravas. This dissertation makes a revision of the gradient strain elasticity theory and presents a didactic study of the simplest problem that can be conceived, i.e., a bar under different axial actions (Aifantis, Beskos). The fundamental solution for 2D and 3D problems is also presented and studied for an elastic medium submitted to a point force, for boundary methods developments, as well as submitted to polynomial stress fields (for static problems), as in the hybrid finite element method. It is shown that Aifantis strain gradient theory may be developed in the context of the Hellinger-Reissner potential, for the sake of hybrid finite and boundary element implementations. Goal of the present research work is as a detailed study of state art of the theme, which comprises an investigation of the singular integrals one must deal with in a computational implementation. The complete computational development for static and dynamic hybrid boundary/finite analyses is planned for a future doctoral thesis.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Atualmente está bem difundido o uso de novas modelagens matemáticas para o estudo do comportamento de micro e nano sistemas mecânicos e eléctricos. O problema de escala é notável quando o tamanho das moléculas, partículas, grãos ou cristais de um sólido é relativamente considerável em relação ao comprimento do microdispositivo. Nesses casos a teoria clássica dos meios contínuos não descreve apropriadamente a solicitação estrutural e é necessária uma abordagem mais geral através de teorias generalizadas não-clássicas que contém a elasticidade clássica como um caso particular delas, onde os parâmetros constitutivos que representam às partículas são desprezíveis. Quando os efeitos microestruturais são importantes, o comportamento não responde como um material homogêneo se não como um material homogêneo. Cem anos atrás os irmãos Cosserat desenvolveram uma teoria de grãos rígidos imersos dentro de um macromeio elástico; posteriormente Toupin, Mindlin e outros pesquisadores na década de 60 formularam a chamada teoria gradiente de deformações, que recentemente é um objeto de muitas investigações analíticas e experimentais. Na década de oitenta, Aifantis e colaboradores conseguiram desenvolver uma teoria de gradiente de deformações simplificada, baseada em só uma constante elástica adicional não-clássica representativa da energia de deformação volumétrica para caracterizar satisfatoriamente os padrões dos fenômenos não-clássicos. Beskos e colaboradores estenderam o campo de aplicações da proposta inicial de Aifantis e fizeram as primeira implementações de elementos de contorno 2D e 3D para análises de elasticidade gradiente estática, no domínio da frequência e a mecânica da fratura. Desde o tempo de Toupin e Mindlin, procura-se estabelecer uma base variacional da teoria e uma formulação consistente das condições de contorno cinemáticas e de equilíbrio, o que parece ter tido êxito com os recentes trabalhos de Amanatidou e Aravas. Esta tese apresenta a formulação do método híbrido de elementos de contorno e finitos na elasticidade gradiente desenvolvida por Dumont e Huamán decompondo o potencial de Hellinger-Reissner em dois princípios de trabalhos virtuais: o primeiro em deslocamentos virtuais e o segundo em forças virtuais. Com esta finalidade é considerado além dos parâmetros clássicos, o trabalho realizado pelas tensões, deformações, forças e deslocamentos não-clássicos. É apresentado o desenvoltimento das soluções fundamentais singulares e polinomiais atráves das equações diferenciais de sexta ordem obtidas da equação de equilíbrio em termos de deslocamento na elasticidade gradiente. É apresentada também a aplicaçõ do método híbrido de contorno para problemas de tensão axial unidimensional e flexão bidimensional de vigas. Finalmente mostra-se a aplicação numérica do método em elementos finitos, é verificado o patch test de elementos finitos de diferentes ordem e mostra-se também análises de convergência.
The use of new mathematical modeling in the study of micro and Nano electro mechanical systems is currently becoming widespread. The scaling problem is apparent when the length of molecules, particles or grains immersed in the material is relatively important compared with the whole micro device dimension. Under this approach the classical theories of mechanics cannot describe suitably the structural requirement and it is necessary a more general outlook through non classical generalized theories which enclose the classical elasticity as a particular case where the non-classical constitutive parameters are negligible. When the microstructural effects are important, the material does not respond as a homogeneous but as a non-homogeneous one. A hundred years ago Cosserat brothers formulated a new theory of rigid grains which were embedded in an elastic macro medium; later Toupin, Mindlin along others researchers in 1960s developed a gradient strain theory which has been recently the source of many analystics and experimental investigations. In 1980s Ainfantis et al could develop a simplified strain gradient theory with just one additional non classical elastic constant which represents the volumetric elastic strain energy and characterized successfully the whole non classical pattern phenomenon. Beskos et al extended the treatment proposed initially by Aifantis and developed the first numerical applications for 2D and 3D boundary element methods and solved static as dynamic and crack problems. Since the times of Toupin and Mindlin it is looking for to establish a variational theory with a consistent cinematic and equilibrium boundary conditions, which seemed to have had success in the recent works of Amanatiodou and Aravas. This work presents the formulation of the hybrid boundary and finite element methods under the strain gradient scope which were developed by Dumont and Huamán through the versatile decomposition of the Hellinger-Reissner potential in two work principles: the displacements virtual work and the forces virtual work; both principles contain the virtual work performed by the non-classical magnitudes. Following, it is presented the complete development of singular and polynominal fundamental solutions abtained through the sixth order strain gradient differential equilibrium equations in terms of displacements. Next it is shown an application of the method to unidimensional truss element and bidimensional beam. Finally, it is presented a numerical application to strain gradient finite element, it is checked the patch tests to different elements orders and it is also shown a series of convergence analysis.

The thesis presents a novel computational method for analysing the static and dynamic behaviour of a multi-damaged beam using local and non-local elasticity theories. Most of the lumped damage beam models proposed to date are based on slender beam theory in classical (local) elasticity and are limited by inaccuracies caused by the implicit assumption of the Euler-Bernoulli beam model and by the spring model itself, which simplifies the real beam behaviour around the crack. In addition, size effects and material heterogeneity cannot be taken into account using the classical elasticity theory due to the absence of any microstructural parameter. The proposed work is based on the inhomogeneous Euler-Bernoulli beam theory in which a Dirac's delta function is added to the bending flexibility at the position of each crack: that is, the severer the damage, the larger is the resulting impulsive term. The crack is assumed to be always open, resulting in a linear system (i.e. nonlinear phenomena associated with breathing cracks are not considered). In order to provide an accurate representation of the structure's behaviour, a new multi-cracked beam element including shear effects and rotatory inertia is developed using the flexibility approach for the concentrated damage. The resulting stiffness matrix and load vector terms are evaluated by the unit-displacement method, employing the closed-form solutions for the multi-cracked beam problem. The same deformed shapes are used to derive the consistent mass matrix, also including the rotatory inertia terms. The two-node multi-damaged beam model has been validated through comparison of the results of static and dynamic analyses for two numerical examples against those provided by a commercial finite element code. The proposed model is shown to improve the computational efficiency as well as the accuracy, thanks to the inclusion of both shear deformations and rotatory inertia. The inaccuracy of the spring model, where for example for a rotational spring a finite jump appears on the rotations' profile, has been tackled by the enrichment of the elastic constitutive law with higher order stress and strain gradients. In particular, a new phenomenological approach based upon a convenient form of non-local elasticity beam theory has been presented. This hybrid non-local beam model is able to take into account the distortion on the stress/strain field around the crack as well as to include the microstructure of the material, without introducing any additional crack related parameters. The Laplace's transform method applied to the differential equation of the problem allowed deriving the static closed-form solution for the multi-cracked Euler-Bernoulli beams with hybrid non-local elasticity. The dynamic analysis has been performed using a new computational meshless method, where the equation of motions are discretised by a Galerkin-type approximation, with convenient shape functions able to ensure the same grade of approximation as the beam element for the classical elasticity. The importance of the inclusion of microstructural parameters is addressed and their effects are quantified also in comparison with those obtained using the classical elasticity theory.

The present research work is dedicated to the development, implementation and validation of a unified finite element methodology based on the combination of gradient elasticity and the Theory of Critical Distances, for the static and high-cycle fatigue assessment of notched engineering components. The proposed methodology, developed for plane, axisymmetric and three-dimensional problems, takes full advantage of both the TCD's accuracy in estimating static and high-cycle fatigue strength of notched components and of the computational efficiency of gradient elasticity in determining non-local stress fields whose distribution fully depends on the value of the adopted length scale parameter. In particular, the developed methodology, due to the ability of gradient elasticity to smooth stress fields in the vicinity of notch tips, has the great advantage of allowing accurate and reliable static and fatigue assessments of notched components by directly considering the relevant gradient-enriched stresses at the hot-spot on the surface of the component, in contrast to existing conventional approaches that require the knowledge of the failure location into the material a priori. This advantage, together with the fact that the proposed methodology can be easily implemented in commercial finite element software, makes the developed methodology a powerful and easy-to-use tool for the static and fatigue design/assessment of notched components. The developed methodology is accompanied by an accurate investigation of the best integration rules to be used as well as a comprehensive convergence study both in absence and presence of cracks, leading to a practical guideline on optimum element size. The proposed gradient-enriched methodology has been validated against a large number of problems involving notched components subject to both static and fatigue loading, covering a wide range of materials, geometries and loading conditions, clearly showing its accuracy and versatility. The developed gradient-enriched methodology has also been extended to the study of the dynamic behaviour of visco-elastic materials subject to vibration.

Es werden Kontaktprobleme im Rahmen der nichtlinearen Elastizitätstheorie mit Mitteln der Variationsrechnung behandelt. Dabei liegt das Hauptaugenmerk auf der Untersuchung des Selbstkontakts eines nichtlinear elastischen Körpers. Unter Verwendung einer geeigneten Lagrangeschen Multiplikatorenregel wird eine notwendige Bedingung für Minimierer hergeleitet. Weiterhin werden Ergebnisse für den Kontakt zweier elastischer Körper formuliert.

Les travaux effectués concernent le développement d’alliages de titane à gradient d’élasticité. Ils s’inscrivent dans le cadre du développement de matériaux dédiés à l’implantologie dentaire, en collaboration avec l’entreprise Biotech Dental. La discontinuité élastique constatée à l’interface os/implant a été mise en cause dans la survenue de phénomènes de résorption osseuse pouvant conduire à des échecs implantaires. L’abaissement du module en surface pourrait alors permettre un transfert des contraintes plus approprié, à l’origine d’une amélioration de l’ostéo-intégration. Un travail amont, basé sur la considération de paramètres électroniques et d’éléments de biocompatibilité, a permis de sélectionner des alliages pouvant présenter une grande variabilité élastique : le Ti-13Nb-13Zr (%m.) et le Ti-20Nb-6Zr (%at.). Une étude préliminaire menée sur des échantillons fins a permis de confirmer cette variabilité et d’associer les bornes élastiques à différentes microstructures. Nous avons alors développé des approches combinant des séquences de déformation et de traitements thermiques courts sur des échantillons massifs. Ainsi, sur le TNZ, une approche de déformation superficielle par grenaillage d’un état β-trempé a permis d’obtenir un gradient de 65 GPa à 85 GPa sur 400 µm. Une approche de dissolution préférentielle sur le Ti-13-13 a quant à elle conduit à un gradient de 75 GPa à 130 GPa sur 100 µm, par traitement thermique flash d’un état précipité préalablement grenaillé. Ces résultats ont donc validé les combinaisons matériau–stratégie–procédé conçues précédemment. Ils ont aussi permis une rationalisation des différentes approches possibles pour obtenir le gradient
The work carried out concerns the development titanium alloys with elastic gradient. They are part of the development of materials dedicated to dental implantology, in collaboration with the company Biotech Dental. The elastic discontinuity observed at the bone / implant interface has been called into question in the occurrence of bone resorption phenomena which can lead to implant failures. The lowering of the surface modulus could then allow a more appropriate transfer of stresses which would lead to an improvement of the osseointegration. An upstream work, based on the consideration of electronic parameters and biocompatibility elements, made it possible to select alloys which can exhibit a great elastic variability: Ti-13Nb-13Zr (wt.%) and Ti-20Nb-6Zr (at.%). A preliminary study carried out on laminated samples allowed to confirm this variability and to associate the elastic bounds obtained with different microstructural states. On the basis of these observations, we have developed approaches combining sequences of deformation and short heat treatments on massive samples. Thus, on the TNZ, a surface deformation approach by shot peening applied on a β-quenched state made it possible to obtain a gradient of 65 GPa to 85 GPa over 400 μm. And a preferential dissolution approach on the Ti-13-13 has lead to a gradient of 75 GPa to 130 GPa on 100 μm, via the realization of flash thermal treatments on massive samples in the precipitated state previously shot-peened. These results have thus validated the previously designed material - strategy - process combinations. They also allowed a rationalization of the different possible approaches to obtain the elasticity gradient

A proper description of quasi-brittle failure within the frame of continuum Mechanics can only be achieved by models based on so-called generalised continua. This thesis focuses on a strain gradient generalised continuum and provides a specific methodology to derive corresponding models which account for the essential features of quasi-brittle failure. This methodology is discussed by means of four peer-reviewed journal articles. Furthermore, an extensive overview of the state of the art in the field of generalised continua is given at the beginning of the thesis. This overview discusses phenomenological extensions of standard Continuum Mechanics towards generalised continua together with corresponding homogenisation strategies for materials with periodic or random microstructure
Eine geeignete, kontinuumsmechanische Beschreibung quasi-spröden Versagens ist nur unter Verwendung verallgemeinerter Kontinuumstheorien möglich. In dieser Habilitationsschrift stehen sogenannte Gradientenkontinua im Vordergrund. Für diese wird eine Methodik vorgeschlagen, welche die Herleitung von Modellen erlaubt, die in der Lage sind, quasi-sprödes Versagen adäquat abzubilden. Diese Methodik wird anhand von vier Publikationen dargestellt und diskutiert. Ein umfangreicher Überblick über den Stand der Forschung auf dem Gebiet der veralgemeinerten Kontinuumstheorien wird am Anfang der Habilitationschrift gegeben. Dabei werden neben phänomenologischen Ansätzen zur Ableitung verallgemeinerter Kontinuumstheorien auch die entsprechenden Homogenisierungskonzepte dargestellt. Letztere werden für Materialien mit periodischer Mikrostruktur und für Materialien mit zufälliger Mikrostruktur diskutiert

The presented master’s thesis deals with the application of the gradient elasticity in fracture mechanics problems. Specifically, the displacement and stress field around the crack tip is a matter of interest. The influence of a material microstructure is considered. Introductory chapters are devoted to a brief historical overview of gradient models and definition of basic equations of dipolar gradient elasticity derived from Mindlin gradient theory form II. For comparison, relations of classical elasticity are introduced. Then a derivation of asymptotic displacement field using the Williams asymptotic technique follows. In the case of gradient elasticity, also the calculation of the J-integral is included. The mathematical formulation is reduced due to the singular nature of the problem to singular integral equations. The methods for solving integral equations in Cauchy principal value and Hadamard finite part sense are briefly introduced. For the evaluation of regular kernel, a Gauss-Chebyshev quadrature is used. There also mentioned approximate methods for solving systems of integral equations such as the weighted residual method, especially the least square method with collocation points. In the main part of the thesis the system of integral equations is derived using the Fourier transform for straight crack in an infinite body. This system is then solved numerically in the software Mathematica and the results are compared with the finite element model of ceramic foam.

Machine strength grading of structural timber is a sawmill process by which considerable value is added to sawn products. The principle of such grading is that the strength of a timber member is predicted on the basis of a so called indicating property (IP) which, in general, represents an averaged value of the modulus of elasticity (MOE) measured over a board length of about one meter or more. A limitation of today’s grading methods is that the accuracy of strength predictions is often rather poor, which results in a low degree of utilization as regards structural potential of sawn timber. However, it has for many years been well known to researchers that much better strength predictions can be made by using localized MOE values, determined over a very short length, as IP. Still, the determination of such values in a sawmill production environment has been technically very difficult to achieve. In the research presented in this thesis, dot laser scanning with high resolution was utilized for detection of local fibre orientation on the surfaces of timber members. Since wood is an orthotropic material with superior structural performance in the longitudinal fibre direction, information about fibre orientation was, in combination with beam theory and measured wood material properties, used to determine the bending MOE variation along boards. By application of an IP defined as the lowest MOE found along a board, more accurate strength predictions than what is obtained by common commercial grading techniques was attained. The thesis also involves flatwise wet gluing of Norway spruce side boards into laminated beams. As side boards, being cut from the outer parts of a log, have excellent structural properties it was not surprising to find that the beams had high strength and stiffness, even when laminations of sawfalling quality were used. The possibility of grading boards in a wet state by means of axial dynamic excitation was investigated with a positive result and application of simple grading rules resulted in considerable improvement of beam bending strength. Finally, bending MOE variation determined on the basis of laser scanned fibre directions was used for identification of weak sections in laminations. Elimination of such sections by means of finger jointing showed that average lamination strength of a board sample could be improved by more than 35 percent.
Hållfasthetssortering av konstruktionsvirke är en sågverksprocess som innebär att värdet av det sågade virket ökar väsentligt. Principen för denna typ av sortering är att styrkan hos ett virkesstycke predikteras med utgångspunkt från en så kallad indikerande egenskap (IP) som oftast representeras av medelvärdet av elasticitetsmodulen (E-modulen) mätt över en sträcka av minst en meter utmed virkesstyckets längd. De sorteringsmetoder som används idag ger ofta prediktioner med relativt låg noggrannhet, vilket innebär att endast en begränsad del av det sågade virkets konstruktiva potential kan utnyttjas. Det är dock väl känt att avsevärt bättre prediktioner kan erhållas genom att använda en lokal E-modul, uppmätt över en mycket kort sträcka, som IP. I dagsläget saknas dock teknik för att kunna bestämma ett sådant lokalt värde vid produktionshastighet i ett sågverk. I den forskning som presenteras i denna avhandling har punktlaserskanning med hög upplösning använts för att bestämma fiberriktningens variation på ytorna av virkesstycken. Eftersom trä är ett ortotropt material med högst styvhet och styrka i longitudinell fiberriktning ger skanningsresultaten värdefull information om hur dessa egenskaper varierar längs en planka. Genom att kombinera informationen om fiberriktning med uppmätta virkesegenskaper och klassisk balkteori, kan böjstyvhetens variation utmed en planka beräknas med hög upplösning och därefter omräknas till en E-modul i böjning. Med en IP definierad som det lägsta värdet på nämnda E-modul utmed en planka kan en högre noggrannhet i prediktionen av hållfasthet uppnås, jämfört med vad som kan erhållas med dagens sorteringsmetoder. Avhandlingen omfattar också limträbalkar tillverkade av sidobräder av gran limmade i rått tillstånd. Eftersom sidobräder sågas från de yttre delarna av en stock har de vanligtvis utmärkta konstruktiva egenskaper. Det var därför inte förvånande att balkarna uppvisade hög styrka och styvhet, även i de fall lamellerna var av sågfallande kvalitet. Möjligheten att med hjälp av axiell dynamisk excitering sortera sidobräder i rått tillstånd undersöktes med positivt resultat och genom att använda sorterade lameller kunde balkarnas styrka förbättras avsevärt. Den ovan beskrivna metoden att med utgångspunkt från bl.a. skannade fibervinklar bestämma styvhetens variation längs virkesstycken utnyttjades sedan för att identifiera svaga snitt i lameller av sidobräder. Genom att eliminera sådana snitt med hjälp av fingerskarvning kunde medelhållfastheten för ett stickprov sidobrädor höjas med mer än 35 %.

Strength grading of structural timber is a process by which value is added to sawn products. It is to the greater part carried out using machine grading based on statistical relationships between so called indicating properties and bending strength. The most frequently applied indicating property (IP) on the European market is the stiffness in terms of average modulus of elasticity (MOE) of a timber piece, although MOE is a material property that varies within timber. A major limitation of today’s grading methods is that the described relationships are relatively poor, which means that there is a potential for more accurate techniques. The main purpose of this research has been to initiate development of more accurate and efficient machine grading methods. Strength of timber is dependent on the occurrence of knots. At the same time, knot measures applied as indicating properties until today have shown to be poor predictors of strength. However, results from this research, and from previous research, has shown that not only size and position of knots but also fibre deviations in surrounding clear wood are of great importance for local stiffness and development of fracture under loading. Thus, development of new indicating properties which take account of knots as well as properties of surrounding fibres, determined on a very local scale, was considered as a possible path towards better strength grading. In the research, results from contact-free deformation measurements were utilized for analysis of structural behaviour of timber on both local and global level. Laser scanning was used for detection of local fibre directions projected on surfaces of pieces. Scanned information, combined with measures of density and average axial dynamic MOE, was applied for calculation of the variation of local MOE in the longitudinal board direction. By integration over cross-sections along a piece, a stiffness profile in edgewise bending was determined and a new IP was defined as the lowest bending MOE along the piece. For a sample of Norway spruce planks, a coefficient of determination of 0.68 was achieved between the new IP and bending strength. For narrow side boards to be used as laminations in wet-glued glulam beams, the relationship between IP and tensile strength was as high as 0.77. Since the intended use of the narrow boards was as laminations in wet-glued beams, the possibility of grading them in a wet state was also investigated. Grading based on axial dynamic excitation and weighing gave just as good results in a wet state as when the same grading procedure was applied after drying. It was also found that the relationship between the new IP and strength was dependent on what scale the IP was determined. Optimum was reached for moving average MOE calculated over lengths corresponding with approximately half the width of investigated pieces. Implementation of the new IP will result in grading that is more accurate than what is achieved by the great majority of today’s grading machines. The new method will probably also be particularly favourable for development of engineered wood products made of narrow laminations.
Hållfasthetssortering av konstruktionsvirke innebär att värdet på sågade produkter ökar. Sorteringen genomförs oftast med maskinella metoder baserade på statistiska samband mellan s.k. indikerande egenskaper och böjhållfasthet. Den indikerande egenskap (indicating property, IP) som är vanligast på den Europeiska marknaden är styvhet uttryckt som ett medelvärde för elasticitetsmodulen (modulus of elasticity, MOE) i ett virkesstycke, trots att MOE är en materialegenskap som varierar i virket. En betydande begränsning med dagens sorteringsmetoder är att de beskrivna sambanden är förhållandevis svaga, vilket innebär att det finns en potential för metoder med högre noggrannhet. Det huvudsakliga syftet med detta doktorandprojekt har varit att initiera en utveckling mot sådana metoder. Hållfasthet hos virke är beroende av förekomst av kvistar. Samtidigt har de kvistmått som fram till idag kommit till användning visat sig vara dåliga prediktorer av hållfasthet. Resultat från såväl denna som tidigare forskning har dock visat att inte bara kvistars storlek och läge, utan också variationen i fiberriktning i omgivande träfibrer, är av stor betydelse för lokal styvhet och brottförlopp under inverkan av last. Utveckling av nya IP som tar hänsyn till såväl kvistar som omgivande träfibrers egenskaper fastställda på mycket lokal nivå bedömdes vara en möjlig väg för att uppnå bättre hållfasthetssortering. I detta doktorandprojekt användes beröringsfri deformationsmätning för analys av det strukturella beteendet hos virkesstycken på såväl lokal som global nivå. Laserskanning utnyttjades för detektering av lokala fiberriktningar projicerade på virkesstyckenas ytor. Med utgångspunkt från skannad information, virkesdensitet och medelvärde för axiell dynamisk elasticitetsmodul kunde variationen i lokal elasticitetsmodul i virkesstyckenas längdriktning bestämmas. Genom integration över tvärsektioner längs ett virkesstycke kunde en profil över hur böjstyvheten i styva riktningen varierade i virkesstyckets längdriktning beräknas. En ny IP definierades som den lägsta elasticitetsmodulen i böjning utmed virkesstyckets längd. För ett urval av granplankor erhölls en förklaringsgrad på 0.68 mellan den nya indikerande egenskapen och böjhållfasthet. För smala sidobrädor avsedda att användas som lameller i våtlimmade limträbalkar var motsvarande förklaringsgrad mellan samma IP och draghållfasthet så hög som 0.77. Eftersom sidobrädorna var avsedda att användas som lameller i våtlimmade balkar genomfördes en studie avseende möjligheten att hållfasthetssortera i vått tillstånd med hjälp av axiell dynamisk excitering och vägning. Det visade sig att sådan sortering gav lika bra resultat som då samma metod användes efter torkning. Sambandet mellan den nya indikerande egenskapen och hållfasthet visade sig också vara beroende av på vilken lokal nivå som egenskapen beräknades. Optimum uppnåddes då den bestämdes som ett glidande medelvärde beräknat över en längd motsvarande ungefär halva virkesstyckets höjd. Implementering av den nya sorteringsmetoden kommer att resultera i sortering som är noggrannare än vad som kan erhållas med det stora flertalet av de sorteringsmetoder som finns idag. Den nya indikerande egenskapen kommer sannolikt att bli särskilt gynnsam att använda för utveckling av ingenjörsmässiga träprodukter bestående av smala lameller.

Size-dependent material responses observed at fine length scales are receiving growing attention due to the need in the modeling of very small sized mechanical structures. The conventional continuum theories do not suffice for accurate descriptions of the exact material behaviors in the fine-scale regime due to the lack of inherent material lengths. A number of new theories/models have been propounded so far to interpret such novel phenomena. In this dissertation a few enriched-continuum theories - the adhesive contact mechanics, surface elasticity and strain gradient elasticity - are employed to study the mechanical behaviors of a semi-infinite solid induced by the boundary forces. A unified treatment of axisymmetric adhesive contact problems is developed using the harmonic functions. The generalized solution applies to the adhesive contact problems involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile, and it links existing solutions/models for axisymmetric non-adhesive and adhesive contact problems like the Hertz solution, Sneddon's solution, the JKR model, the DMT model and the M-D model. The generalized Boussinesq and Flamant problems are examined in the context of the surface elasticity of Gurtin and Murdoch (1975, 1978), which treats the surface as a negligibly thin membrane with material properties differing from those of the bulk. Analytical solution is derived based on integral transforms and use of potential functions. The newly derived solution applies to the problems of an elastic half-space (half-plane as well) subjected to prescribed surface tractions with consideration of surface effects. The newly derived results exhibit substantial deviations from the classical predictions near the loading points and converge to the classical ones at a distance far away from those points. The size-dependency of material responses is clearly demonstrated and material hardening effects are predicted. The half-space contact problems are also studied using the simplified strain gradient elasticity theory which incorporates material microstructural effects. The solution is obtained by taking advantage of the displacement functions of Mindlin (1964) and integral transforms. Significant discrepancy between the current and the classical solutions is seen to exist in the immediate vicinity of the loading area. The discontinuity and singularity exist in classical solution are removed, and the stress and displacement components change smoothly through the solid body.

Gradient elasticity has developed into an important area of continuum mechanics with numerous applications in engineering mechanics, structural analysis, experimental and computational mechanics. The present thesis is concerned with a simple model of explicit gradient elasticity. The aim is to provide a comprehensive insight into the basic properties of this model, by solving several problems in statics and dynamics. The problems include one - dimensional and two - dimensional (bending) loading conditions. Especially, use is made of a consistent Euler - Bernoulli beam theory and of different versions of Hamilton's principle. Moreover, a method is presented for determining the critical load in buckling problems. The investigations highlight, among others, the effect of non - classical boundary conditions and of non - classical material parameters.

We have investigated gradient models, one of them was a model with double-well potential and the other one a so called extended model. In dimension two we have calculated exact free energies of the disseminated edge configurations for the extended model and for arbitrary dimension we have derived bounds on these free energies. Combining these bounds with an argument on exstince of bad contours together with the estimate of the number of these contours and using the method of reflection positivity we have been able to show that at low temperatures there is a phase transition in the extended model. We have further shown that the phase transition exists also in the double-well model as long as a conjecture on estimates of mean energy holds. Besides these results the thesis also contains basic tools of statistical physics and facts from related fields, as well as basic results on gradient models, so that our work can serve as an introduction into these areas.

碩士
國立高雄應用科技大學
模具工程系
97
In this paper, the inverse solution of linear and non-linear vibration system are studied, start with the assumption that elasticity coefficient of K (t) with the damping coefficient C (t) two sets of functions for the inverse operation of the known system measurement data to The data for the conjugate gradient inverse operation of the objective function, the objective function above with a group of K, C values of the initial guess into the conjugate gradient method to do numerical computation, After iteration by a new set of K, C values, and will replace the original value of the new guess value by a number of iterative calculations after update finally makes K, C the value of speculation in line with the objective function. Due to the numerical results, we can found that even though used in measuring the value of different inverse algorithms by about 6 times by iteration, the data can be obtained by the convergence on the accuracy of numerical analysis and fast. K the average error is about ; C the average error is about , regardless of the percentage error at or on the number of iterations can be controlled at a certain level above.

The Eshelby-type problems of an arbitrary-shape polygonal or polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material are analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensors for a plane strain inclusion with an arbitrary polygonal cross section and for an arbitrary-shape polyhedral inclusion are analytically derived in general forms in terms of three potential functions. These potential functions, as area integrals over the polygonal cross section and volume integrals over the polyhedral inclusion, are evaluated. For the polygonal inclusion problem, the three area integrals are first transformed to three line integrals using the Green's theorem, which are then evaluated analytically by direct integration. In the polyhedral inclusion case, each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach. In addition, the Eshelby tensor for an anti-plane strain inclusion with an arbitrary polygonal cross section embedded in an infinite homogeneous isotropic elastic material is analytically solved. Each of the newly derived Eshelby tensors is separated into a classical part and a gradient part. The latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. For homogenization applications, the area or volume average of each newly derived Eshelby tensor over the polygonal cross section or the polyhedral inclusion domain is also provided in a general form. To illustrate the newly obtained Eshelby tensors and their area or volume averages, different types of polygonal and polyhedral inclusions are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the each SSGET-based Eshelby tensor for all inclusion shapes considered vary with both the position and the inclusion size. It is also observed that the components of each averaged Eshelby tensor based on the SSGET change with the inclusion size.

Eshelby-type inclusion problems of an infinite or a finite homogeneous isotropic elastic body containing an arbitrary-shape inclusion prescribed with an eigenstrain and an eigenstrain gradient are analytically solved. The solutions are based on a simplified strain gradient elasticity theory (SSGET) that includes one material length scale parameter in addition to two classical elastic constants. For the infinite-domain inclusion problem, the Eshelby tensor is derived in a general form by using the Green’s function in the SSGET. This Eshelby tensor captures the inclusion size effect and recovers the classical Eshelby tensor when the strain gradient effect is ignored. By applying the general form, the explicit expressions of the Eshelby tensor for the special cases of a spherical inclusion, a cylindrical inclusion of infinite length and an ellipsoidal inclusion are obtained. Also, the volume average of the new Eshelby tensor over the inclusion in each case is analytically derived. It is quantitatively shown that the new Eshelby tensor and its average can explain the inclusion size effect, unlike its counterpart based on classical elasticity. To solve the finite-domain inclusion problem, an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET are proposed and utilized. The solution for the disturbed displacement field incorporates the boundary effect and recovers that for the infinite-domain inclusion problem. The problem of a spherical inclusion embedded concentrically in a finite spherical body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. It is demonstrated through numerical results that the newly obtained Eshelby tensor can capture the inclusion size and boundary effects, unlike existing ones. Finally, a homogenization method is developed to predict the effective elastic properties of a heterogeneous material using the SSGET. An effective elastic stiffness tensor is analytically derived for the heterogeneous material by applying the Mori-Tanaka and Eshelby’s equivalent inclusion methods. This tensor depends on the inhomogeneity size, unlike what is predicted by existing homogenization methods based on classical elasticity. Numerical results for a two-phase composite reveal that the composite becomes stiffer when the inhomogeneities get smaller.

Σκοπός της παρούσας διδακτορικής διατριβής είναι η ανάπτυξη μεθοδολογίας συνοριακών στοιχείων για την αριθμητική επίλυση τρισδιάστατων (3-D) στατικών προβλημάτων στα πλαίσια μιας θεωρίας βαθμοελαστικότητας, που στηρίζεται σε μια απλουστευμένης μορφής της θεωρίας του Mindlin και διατυπώθηκε από τους Vardoulakis and Sulem, η οποία λαμβάνει υπόψη και την επιφανειακή ενέργεια, και από τους Aifantis και συνεργάτες. Η διδακτορική διατριβή αποτελείται από δύο ενότητες. Στην πρώτη ενότητα (κεφάλαια 1 και 2) γίνεται μία πλήρης ανασκόπηση της βιβλιογραφίας ως προς τις θεωρίες βαθμοελαστικότητας και στη συνέχεια, περιγράφεται διεξοδικά η παρούσα θεωρία βαθμοελαστικότητας με επιφανειακή ενέργεια. Στη δεύτερη ενότητα παρουσιάζεται η μέθοδος των Συνοριακών Στοιχείων (ΜΣΣ) όπως αυτή εφαρμόζεται για την επίλυση τρισδιάστατων και αξονοσυμμετρικών βαθμοελαστικών προβλημάτων, αντίστοιχα. Η ΜΣΣ βασίζεται στη διατύπωση των ολοκληρωτικών εξισώσεων των βαθμοελαστικών προβλημάτων. Οι άγνωστοι των ολοκληρωτικών εξισώσεων είναι οι συνοριακές τιμές του βασικού πεδίου των μεταβλητών και οι παράγωγοί τους, που για τη βαθμοελαστικότητα είναι τα διανύσματα των μετατοπίσεων, των βαθμίδων τω μετατοπίσεων και τα διανύσματα των επιφανειακών τάσεων. Η προσέγγιση των συναρτήσεων αυτών πάνω στο σύνορο γίνεται με τη βοήθεια συναρτήσεων παρεμβολής από τις αντίστοιχες τιμές τους σε έναν επιλεγμένο αριθμό κόμβων. Η ταχύτητα και η ακρίβεια της ΜΣΣ κατά την εφαρμογή της επηρεάζεται σημαντικά από την ταχύτητα και την ακρίβεια του υπολογισμού των ιδιόμορφων και υπερ-ιδιόμορφων ολοκληρωμάτων. Στην παρούσα διατριβή τα ιδιόμορφα και υπερ-ιδιόμορφα ολοκληρώματα υπολογίζονται με τη χρήση τεχνικών ιδιόμορφης και υπερ-ιδιόμορφης ολοκλήρωσης (Guiggiani (1992) και Huber et al. (1993)) αντίστοιχα. Στα πλαίσια της παρούσας διδακτορικής διατριβής κατασκευάστηκε αλγόριθμος που επιλύει τρισδιάστατα στατικά προβλήματα βαθμοελαστικότητας καθώς και αλγόριθμος που επιλύει στατικά βαθμοελαστικά προβλήματα με αξονική συμμετρία. Στο τέλος κάθε κεφαλαίου, επιλύονται αντίστοιχα στατικά βαθμοελαστικά προβλήματα με ή χωρίς να λαμβάνεται υπόψη η επιφανειακή ενέργεια και με γνωστές αναλυτικές λύσεις. Τα αριθμητικά αποτελέσματα των παραπάνω προβλημάτων συγκρίνονται με τα αντίστοιχα αναλυτικά. Τέλος, γίνεται μία ανακεφαλαίωση της διδακτορικής διατριβής και διατυπώνονται προτάσεις για μελλοντική έρευνα.
In the present Doctoral Thesis a boundary element methodology (BEM) is developed in order to solve numerically 3-D and axis-symmetric static gradient elastic problems. Microstructural effects on the macroscopic behavior of the considered materials have been taken into account by means of a simple strain gradient theory with surface energy obtained as a special case of the general one due to Mindlin, proposed by Vardoulakis and Sulem. All possible boundary conditions (classical and non-classical) have been determined with the aid of a variational statement of the problem. The fundamental solution of the gradient elastic with surface energy has been explicitly determined and used to establish the boundary integral representation of the solution of the problem with the aid of the reciprocal identity, specifically constructed for this gradient elastic with surface energy case. The boundary integral representation consists of one equation for the dispalcement and another one for its normal derivative. Also, the integral forms of the gradient of displacement as well as the Cauchy, relative, double and total stresses in the interior of the gradient elastic body have been derived and presented. The numerical implementation of the integral equations is accomplished with the aid of quadratic isoparametric line (axis-symmetry case) and surface (3-D case) boundary elements. The computation of the singular and hyper-singular integrals involved is done with the aid of highly accurate advanced algorithms.

During this thesis, a Boundary Element Method (BEM) has been developed for the solution of static linear elastic problems with microstructural effects in two (2D) and three dimensions (3D).The second simplified form of Mindlin's Generalized Gradient Elasticity Theory (Mindlin's Form II)has been employed. The fundamental solution of the 4th order partial differential equation, that describes the aforementioned theory, has been derived and the integral equations that govern Mindlin's Form II Gradient Elasticity Theory have been obtained. Furthermore, a BEM formulation has been developed and specific Boundary Value Problems (BVPs) were solved numerically and compared with the corresponding analytical solutions to verify the correctness of the formulation and demonstrate its accuracy. Moreover, two new partially discontinuous boundary elements with variable order of singularity, a line and a quadrilateral element, have been developed for the solution of fracture mechanics problems. The calculation of the unknown fields near the crack tip (or front) demanded the use of elements that could interpolate abruptly varying fields. The new elements were created in a way that their interpolation functions were no longer quadratic but their behavior depended on the order of singularity of each field. Finally, the Stress Intensity Factor (SIF) of the crack has been calculated with high accuracy, based on the element's nodal traction values. Static fracture mechanics problems for Mode I and Mixed Mode (I & II) cracks, have been solved in 2D and 3D and the corresponding SIFs have been obtained, in the context of both classical and Form II Gradient Elasticity theories.
Κατά τη διάρκεια της παρούσας διδακτορικής διατριβής, αναπτύχθηκε Μέθοδος Συνοριακών Στοιχείων (ΜΣΣ) για την επίλυση στατικών προβλημάτων ελαστικότητας με επιδράσεις μικροδομής σε δύο και τρεις διαστάσεις. Η θεωρία στην οποία εφαρμόστηκε η ΜΣΣ είναι η δεύτερη απλοποιημένη μορφή της γενικευμένης θεωρίας ελαστικότητας του Mindlin. Για τη συγκεκριμένη θεωρία ευρέθη η θεμελιώδης της μερικής διαφορικής εξίσωσης 4ης τάξης που περιγράφει τη συμπεριφορά των συγκεκριμένων υλικών και κατασκευών. Επίσης διατυπώθηκε η ολοκληρωτική εξίσωση των αντίστοιχων προβλημάτων και έγινε η αριθμητική εφαρμογή μέσω της ΜΣΣ. Επιλύθηκα αριθμητικά συγκεκριμένα προβλήματα συνοριακών τιμών και έγινε σύγκριση των αποτελεσμάτων με τα αντίστοιχα θεωρητικά. Στη συνέχεια αναπτύχθηκαν δύο νέα ασυνεχή στοιχεία μεταβλητής τάξης ιδιομορφίας με σκοπό την επίλυση προβλημάτων θραυστομηχανικής, ένα για δισδιάστατα και ένα για τρισδιάστατα προβλήματα. Συγκεκριμένα, επειδή τα πεδία των τάσεων απειρίζονται στην κορυφή μιας ρωγμής και περιέχουν συγκεκριμένων τύπων ιδιομορφίες δεν ήταν δυνατός ο ακριβής υπολογισμός των πεδίων αυτών κοντά στη ρωγμή με τα συνήθη τετραγωνικά συνοριακά στοιχεία. Ως εκ τούτου τα νέα στοιχεία κατασκευάστηκαν με τέτοιο τρόπο ώστε οι συναρτήσεις παρεμβολής τους να μην είναι τετραγωνικες, αλλά να εξαρτώνται από τον τύπο ιδιομορφίας του κάθε πεδίου. Έπειτα, έγινε ακριβής υπολογισμός του συντελεστή έντασης τάσης της ρωγμής με βάση τις τιμές του πεδίου των τάσεων κοντά σε αυτή. Τέλος επιλύθηκαν στατικά προβλήματα θραυστομηχανικής σε δύο και τρεις διαστάσεις και υπολογίστηκαν οι συντελεστές έντασης τάσης για ρωγμές σε υλικά με επίδραση μικροδομής.

A proper description of quasi-brittle failure within the frame of continuum Mechanics can only be achieved by models based on so-called generalised continua. This thesis focuses on a strain gradient generalised continuum and provides a specific methodology to derive corresponding models which account for the essential features of quasi-brittle failure. This methodology is discussed by means of four peer-reviewed journal articles. Furthermore, an extensive overview of the state of the art in the field of generalised continua is given at the beginning of the thesis. This overview discusses phenomenological extensions of standard Continuum Mechanics towards generalised continua together with corresponding homogenisation strategies for materials with periodic or random microstructure.:1 Introduction 7 2 Generalised Continua - a journey 9 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Principal classes of generalised continua . . . . . . . . . . . . . . . . . 10 2.2.1 Polar field theories and their relatives . . . . . . . . . . . . . . 10 2.2.2 Non-local continua . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Generalised continua by explicit homogenisation . . . . . . . . . . . . 15 2.3.1 Random micro-structures . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Periodic micro-structures . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Generalised homogenisation based on polynomials . . . . . . 20 3 Modelling of quasi-brittle failure 25 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Discussion of main results . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bibliography 29 4 Collection of articles reflecting the author’s contribution 35
Eine geeignete, kontinuumsmechanische Beschreibung quasi-spröden Versagens ist nur unter Verwendung verallgemeinerter Kontinuumstheorien möglich. In dieser Habilitationsschrift stehen sogenannte Gradientenkontinua im Vordergrund. Für diese wird eine Methodik vorgeschlagen, welche die Herleitung von Modellen erlaubt, die in der Lage sind, quasi-sprödes Versagen adäquat abzubilden. Diese Methodik wird anhand von vier Publikationen dargestellt und diskutiert. Ein umfangreicher Überblick über den Stand der Forschung auf dem Gebiet der veralgemeinerten Kontinuumstheorien wird am Anfang der Habilitationschrift gegeben. Dabei werden neben phänomenologischen Ansätzen zur Ableitung verallgemeinerter Kontinuumstheorien auch die entsprechenden Homogenisierungskonzepte dargestellt. Letztere werden für Materialien mit periodischer Mikrostruktur und für Materialien mit zufälliger Mikrostruktur diskutiert.:1 Introduction 7 2 Generalised Continua - a journey 9 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Principal classes of generalised continua . . . . . . . . . . . . . . . . . 10 2.2.1 Polar field theories and their relatives . . . . . . . . . . . . . . 10 2.2.2 Non-local continua . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Generalised continua by explicit homogenisation . . . . . . . . . . . . 15 2.3.1 Random micro-structures . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 Periodic micro-structures . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Generalised homogenisation based on polynomials . . . . . . 20 3 Modelling of quasi-brittle failure 25 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Discussion of main results . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bibliography 29 4 Collection of articles reflecting the author’s contribution 35

The objective of this study was to examine relationships between the physical properties of a board and the strength of a board for use in an intelligent lumber grading system. Available research literature and commercial grading systems, that have taken steps in this direction, are described in this study. Some previous work on estimating the strength of lumber has been based on mathematical models derived from different approximation methods such as regression, function approximation or neural networks. The disadvantage of this approach is that a large training set of board scans are required to be representative of the various species, harvesting sites, and geometrical variations between boards (grain direction, knot location and size etc.) A major contribution of this thesis was to develop a mechanics based method of determining board strength that relies only upon simple in-mill measurement of X-Ray and Modulus of Elasticity (MOE) signals. In this thesis, an intelligent mechanics-based lumber grading system was developed to provide a better estimation of the strength of a board nondestructively. This system processed X-Ray-extracted geometric features (of 1080 boards that eventually underwent destructive strength testing) by using Finite Element Methods (FEM) to generate associated stress fields. In order to find a few significant mechanics-based features, the stress fields were then fed to a feature-extracting-processor which produced twenty six strength predicting features. The best strength predicting features were determined from the coefficient of determination (correlation r squared) between the features and actual strengths of the boards. The coefficient of determination of each feature (or combination of features) with the actual strength of the board were calculated and compared. A coefficient of determination of 0.42 was achieved by using a longitudinal (along the local grain angle) maximum stress concentration (MSC) feature to predict the estimated strength of lumber. In addition to the above feature, a Weibull based feature was defined and examined. Since it is based on the whole stress field; whereas, maximum stress concentration based feature is based on one point in stress field, we hoped to get a better correlation. By implementing a system using the Weibull based feature, a coefficient of determination of 0.47 was achieved which is slightly higher than the MSC based feature. Next, a combination of X-Ray and Modulus of Elasticity (MOE) signals were used to estimate the strength of lumber. The MSC and the Weibull features were based on the X-Ray images. The MOE part of the system used the output of a Continuous Lumber Tester (CLT) machine which produced two MOE profiles to calculate an MOE based feature. The MOE based feature and MSC feature were combined in an algorithm and a coefficient of determination of 0.65 was achieved. In the final step, the FEM processor was replaced by a knowledge-based system comprised of a fast table lookup for all FEM-modeled knot locations and sizes. To the extent that this table is representative of the important strength-reducing factors in a board, it replaces the need for a large training set of actual boards with real knots of differing sizes and locations. This knowledge-based system will permit a real-time board strength estimation system to be developed. We were able to fine-tune the mechanics based algorithm and reduce the FEM calculation errors. By implementing the knowledge based system coefficients of determination of 0.44, 0.46 and 0.67 were achieved for the MSC, the Weibull and combined (MOE and X-Ray) algorithm-based features respectively. Finally, by implementing a system without the need for an FEM module, it opens up the opportunity to investigate the location of breakage by using the Maximum Stress Concentration theory of failure. First a system based on the X-Ray image was implemented to estimate the point of fracture location. If this point is located between start and end of each actual breakage, then we can say that we estimated the breakage point correctly and a measure named the hit factor was introduced. The hit factor for both the most critical knot and the second highest MSC knot reached 56% for a tolerance of [5 cm] from the actual breakage point.
Applied Science, Faculty of
Electrical and Computer Engineering, Department of
Graduate