The steel industry has been producing flat plates through the process of hot rolling since the late 1600s. Hot rolling uses a series of rolls to progressively thin a strip of steel to a desired thickness. In deforming the strip, the rolling process causes variations in thickness across the width of the strip. These variations are commonly referred to as crown, which is specifically the difference in thickness between the center and edge of a strip. For most applications steel mill clients require flat products, or products with little variation in thickness. Therefore, variations represent wasted material which must be removed before the plate or sheet can be used in consumer products. Controlling the flatness of the metal strip is a high priority for the hot rolling business. The purpose of this work was to develop a 3-D dynamic model of the rolling process to simulate the behaviour of a strip while being rolled and predict its profile. To accomplish this task, much of the rolling process needed to be modeled. The profile of the strip is a product of the deformation of the rolls and frame within a mill stand. Therefore, not only did the geometry of these components need to be modeled, but the material properties and dynamic motion were required as well. The dynamic nature of the process necessitated the modeling of the rotation of the rolls and translation of the strip, aspects of rolling which are not typically simulated. Five models were developed during the project. The purpose of the first two models was to find the stiffnesses of the roll-stack and stand frame. The roll-stack refers to the rolls and their arrangement. The reference mill from which data was provided used a four-high roll-stack with two rolls above the strip and two below. The frame that holds the roll-stack, while massive, stretches when the strip is deformed between the rolls. This stretch changes the position of the rolls affecting the load and deformation of the strip. A lumped-mass model was created to simulate the dynamics of the roll-stack and frame. When the strip enters the gap between the rolls, there is a large impact force which causes the rolls to vibrate. The lumped-mass model was used to determine parameters to bring the system to steady state. The final two models simulated the entire rolling process with rotating rolls and moving strip. The 3-D dynamic rolling model was capable of predicting the strip profile due after exiting the rolls. Two calibrations were used to reduce model error before running a validation. The rolling causes thickness variation across the width of the metal strips; therefore, strips are intentionally rolled thick to meet a minimum thickness. In modern steel mills, specialized control systems are used to adjust parameters as the steel strip passes through each stand of rolls. Varying the parameters allows the thickness and profile of the strip to be controlled. Each stand may have several rolls in different configurations. These rolls are either work rolls, which directly contact the strip, or backup rolls, which contact the work rolls and stiffen the roll-stack. The stand frame holds the rolls and provides a means to position them. The validation results showed that the exit thickness, strip crown, and rolling load were less than 5% different from the values measured in the test data. The calibrated model was then used to derive strip crown sensitivities to gap, entry crown, work roll crown, and bending force. The 3-D dynamic model was able to predict the strip crown accurately when given calibrated information about the system. This model will be a useful tool for exploring the mechanics of hot rolling in ways that were not previously possible. / Master of Science

Darbo tikslas – tampriųjų-plastinių strypų konstrukcijų (santvaros, rėmų), veikiamų kartotinės kintamosios apkrovos, prisitaikomumo proceso ir būvio įtempių ir deformacijų analizė optimalaus konstrukcijų projektavimo kontekste. Darbo aktualumas grindžiamas prisitaikančių konstrukcijų optimizavimo uždaviniuose figūruojančių standumo sąlygų-apribojimų kokybės gerinimo būtinumu. Prisitaikančių konstrukcijų deformacijų būvis priklauso nuo apkrovimo istorijos, o poslinkių ribojimui taikoma nepakankamai tiksli Koiterio sąlyga arba liekamųjų poslinkių influentinė matrica, nepagrįstai laikant, kad plastinio deformavimo procesas yra išimtinai holonominis. Anotuojamame darbe apkrovimo istoriją siūloma įvertinti, atliekant papildomą inkrementinę deformacijų būvio analizę. Tyrimai atlikti taikant idealiai tamprių plastinių santvarų ir rėmų techniškosios skaičiavimo teorijos prielaidas (maži poslinkiai ir deformacijos). Taikomi ekstreminiai energiniai mechanikos principai, matematinio programavimo teorija ir metodai. Inkrementinės analizės matematiniai modeliai sudaryti, besikeičiančias plastines deformacijas tapatinant su distorsijomis. Taip nustatomos konkrečios apkrovimo istorijos liekamųjų poslinkių kitimo maksimalios ir minimalios reikšmės. Gautieji rezultatai panaudoti optimizavimo uždavinių sprendiniams tikslinti ir, esant būtinumui, leidžiantys keisti pradines pagrindinio optimizavimo uždavinio sąlygas. Pateikti išsamūs skaitinių eksperimentų rezultatai. / The purpose of this work is analysis of stress-deformation state of perfectly elastic-plastic shakedown structures (truss, frames) subjected to repeated variable load in the context of optimal design. Relevance of this work is based on a need of improvement of accurateness of stiffness constrains in optimization problems of structures. Stress-deformation state of shakedown structures depends on its loading history, while for the restriction of displacements inaccurate Koiter’s condition or an influence matrix of residual displacements, on the wrong supposition that the process of plastic deformation is exclusively holonomic, is used. In this work is proposed to evaluate loading history by performing an additional incremental analysis of stress-deformation state. This research was performed invoking the assumptions of technical computing theory of perfectly elastic-plastic trusses and frames (small deformations and displacements). Mechanics extremum energy principles, mathematical programming theory and methods are applied. Mathematical models of incremental analysis are composed by indentifying volatile plastic deformations with distortions. Thus particular maximum and minimum values of residual displacements are found. Obtained results are used to verify optimal design problems solutions and change the restrictions of main optimimization problem if necessary. Comprehensive results of numerical experiments presented.

Mechanical components often possess notches. These notches give rise to stress concentrations, which in turn increases the likelihood that the material will undergo yielding. The finite element method (FEM) can be used to calculate transient stress and strain to be used in fatigue analyses. However, since yielding occurs, an elastic-plastic finite element analysis (FEA) must be performed. If the loading sequence to be analysed with respect to fatigue is long, the elastic-plastic FEA is often not a viable option because of its high computational requirements. In this thesis, a method that estimates the elastic-plastic stress and strain response as a result of input elastic stress and strain using plasticity modelling with the incremental Neuber rule has been derived and implemented. A numerical methodology to increase the accuracy when using the Neuber rule with cyclic loading has been proposed and validated for proportional loading. The results show fair albeit not ideal accuracy when compared to elastic-plastic finite element analysis. Different types of loading have been tested, including proportional and non-proportional as well as complex loadings with several load reversions. Based on the computed elastic-plastic stresses and strains, fatigue life is predicted by the critical plane method. Such a method has been reviewed, implemented and tested in this thesis. A comparison has been made between using a new damage parameter by Ince and an established damage parameter by Fatemi and Socie (FS). The implemented algorithm and damage parameters were evaluated by comparing the results of the program using either damage parameter to fatigue experiments of several different load cases, including non-proportional loading. The results are fairly accurate for both damage parameters, but the one by Ince tend to be slightly more accurate, if no fitted constant to use in the FS damage parameter can be obtained.

Šiame baigiamajame magistro darbe nagrinėjama temperatūros įtaka plieninių strypinių konstrukcijų darbui. Darbas susideda iš teorinės ir eksperimentinės dalių. Teorinėje dalyje atlikta literatūros apžvalga. Aptarti naujai pasiūlyti strypinių sistemų ir atskirų elementų, bei jų sujungimo skaičiavimo metodai ir eksperimentiniai rezultatai, veikiant aukštoms (gaisro) temperatūroms. Aprašyta temperatūros pokyčio įtaka elemento deformacijai, pateikiamas matematinis modelis tamprioms-plastinėms strypinėms konstrukcijoms skaičiuoti, įvertinant temperatūros pokyčius. Eksperimentinėje dalyje atliekama rėmo, kai elementus veikia visame skerspjūvio aukštyje vienoda temperatūra, ir kai apatiniuose ir viršutiniuose elementų sluoksniuose ji skiriasi, analizė. Skaičiavimams atlikti panaudotas simbolinės matematikos paketas Matlab. Aprašomi rezultatai, suformuluojamos išvados ir pasiūlymai. Darbą sudaro šios dalys: įvadas, literatūros apžvalga, matematinio modelio sudarymas, skaitinis eksperimentas, išvados ir pasiūlymai, literatūros sąrašas. Darbo apimtis – 90 p. teksto, 54 iliustracijos, 9 lentelės, 37 bibliografiniai šaltiniai. / This final master thesis deals with the impact of temperature on steel-framed structures. The thesis consists of theoretical and experimental parts. The theoretical part covers the review of literature. New methods of calculations and experimental results of steel-framed structures, separate members and connections at elevated temperatures are analysed. The impact of temperature changes on deformation of the elements is described, the mathematical model for calculating elastic-plastic bar structures with estimation of temperature changes is given. The experimental part covers the analysis of the frame, when the temperature is the same throughout the whole cross-section of members and when it is different in the bottom and top parts of sections. Software pack Matlab was used for calculations. Results of calculations are given, conclusions and suggestions presented. Structure: introduction, review of literature, mathematical model, experimental part, conclusions and suggestions, references. Volume of the thesis: 90 pages of text, 54 pictures, 9 tables, 37 references.

In this thesis, a two node nonlinear elastic-plastic beam finite element is developed to analyze large deformations. The system of equations are derived from virtual work principle, and the updated Lagrangian formulation is used. Material is assumed to be isotropic and rate insensitive obeying J2-flow rule. Work hardening characteristics of material is considered and all nonlinear terms are included. For the two node iso-parametric beam element a layered model is used to analyze through-the-thickness distribution of elastic and plastic zones. A finite element program is developed and the numerical outcomes are compared with the experimental results. A good agreement is achieved between numerical and experimental results.

A method is presented for adapting the classical Bishop-Hill model to the requirements of elastic/yield-limited design in metals of arbitrary crystallographic texture. The proposed Hybrid Bishop-Hill (HBH) model, which will be applied to ductile FCC metals, retains the `stress corners' of the polyhedral Bishop-Hill yield surface. However, it replaces the `maximum work criterion' with a criterion that minimizes the Euclidean distance between the applicable local corner stress state and the macroscopic stress state. This compromise leads to a model that is much more accessible to yield-limited design problems. Demonstration of performance for the HBH model is presented for an extensive database for oxygen free electronic (OFE) copper. The study also implements the HBH model to the polycrystalline yield surface via standard finite element analysis (FEA) tools to carry out microstructure-sensitive design. Anisotropic elastic properties are incorporated into the FEA software, as defined by the sample texture. The derived local stress tensor is assessed using the HBH approach to determine a safety factor relating to the distance from the yield surface, and thereby highlighting vulnerable spots in the component and obtaining a quantitative ranking for suitability of the given design. By following standard inverse design techniques, an ideal microstructure (meaning texture in this context) may be arrived at. The design problems considered is a hole-in-plate configuration of sheets loaded in uniaxial tension and simple compliant mechanisms. The further improvement of HBH model is discussed by introducing geometrically necessary dislocation (GND) densities in addition to the crystal orientations procedure in standard microstructure-based method. The correlations between crystal orientations and GND densities are studied. The shape of the yield surface most influenced by the texture of the material, while the volume of the envelope scales in accordance with the GND density. However, correlations between crystal orientation and GND content modify the yield surface shape and size. While correlations between GND density and crystal orientation are not strong for most copper samples, there are sufficient dependencies to demonstrate the benefits of the detailed four-parameter model. The four-parameter approach has potential for improving estimates of elastic-yield limit in all polycrystalline FCC materials.

No description available.

A caracterização de materiais utilizando a técnica de indentação instrumentada difundiu-se significativamente na última década, devido às melhorias dos instrumentos que permitem ensaios por esta técnica e à necessidade de se fazer ensaios em pequenos volumes de materiais, como, por exemplo, em filmes finos e materiais com superfícies modificadas por tratamentos superficiais. Neste texto, abordou-se a elaboração de um algoritmo que permita o estudo da resposta de indentação de superfícies de diferentes materiais metálicos, durante e após o contato com um indentador agudo, empregando simulações por elementos finitos e análise dimensional. Na obtenção do algoritmo analisaram-se os efeitos da formação de borda (pile-up) ou da retração (sink-in) do material indentado, o coeficiente de atrito indentador-amostra, as deformações elásticas do indentador, a geometria do indentador e a variação experimental. As relações obtidas permitiram identificar uma falta fundamental de relação única entre as propriedades mecânicas e a forma da curva de indentação para curvas com razão d r/dmax>0,9, onde d r é a profundidade residual da indentação e dmax é o deslocamento máximo do indentador durante o ensaio. Da mesma forma, concluiu-se que a relação de Sneddon tem que ser corrigida tanto pela geometria da área de contato indentador-amostra quanto pela razão entre os módulos elásticos do material e do indentador (E/Ei). Como a área de contato indentador-amostra é afetada não só pela geometria do indentador mas também pelo nível de pile-up ou sink-in nos indentadores piramidais, uma relação foi identificada entre o nível de pile-up/sink-in e o fator de correção b da equação de Sneddon para os indentadores Vickers e Berkovich. Adicionalmente, pequenas diferenças foram observadas entre as curvas P-h (onde h é a profundidade de indentação abaixo da superfície original da amostra) e as curvas P-d (onde d é o valor da aproximação mútua entre indentador e amostra durante a indentação) para um mesmo valor de módulo reduzido Er quando a razão E/Ei é grande. Assim, o módulo reduzido pode sobreestimar ou subestimar a rigidez do indentador, dependendo das propriedades do material indentado. As análises neste trabalho permitiram igualmente observar que as principais limitações na obtenção das propriedades elasto-plásticas usando a curva de indentação instrumentada vêm da falta de unicidade, seguida pela geometria do indentador, isto é, diferenças entre o cone equivalente e os piramidais Vickers e Berkovich, assim como desvios em relação à geometria ideal do indentador, o que inclui arredondamento da sua ponta. Quando não há unicidade, sabendo-se informações adicionais à curva P-d, por exemplo, o valor da área residual da indentação ou o módulo elástico, uma solução única das propriedades mecânicas pode ser obtida. Em paralelo, a variação experimental pode limitar de forma significativa a precisão no cálculo das propriedades, enquanto o atrito indentador-amostra e as deformações do indentador têm efeitos menos significativos. Em termos gerais, observa-se que as funções que compõem o algoritmo desenvolvido neste trabalho permitem: (i) predizer as curvas carga-deslocamento (curvas P-d), produto do ensaio de indentação instrumentada, para um conjunto de propriedades elasto-plásticas conhecidas; (ii) identificar quando uma mesma curva P-d pode ser obtida de mais de um conjunto de propriedades do material indentado (iii) calcular as propriedades mecânicas (dureza, módulo elástico, coeficiente de encruamento e limite de escoamento) de um material usando a curva P-d quando d r/dmax<0,8 ou faixas de propriedades de materiais quando d r/dmax>0,8 e (iv) calcular as propriedades mecânicas (dureza, módulo elástico, coeficiente de encruamento e limite de escoamento) de um material usando a curva P-d e a área residual da indentação. / The interest in material characterization using instrumented indentation techniques has significantly increased in the last decade, due to improvements in testing instruments and the need to carry out tests on small volumes of materials, such as thin films or materials with surfaces modified by other surface treatments. This work addresses the development of an algorithm to analyze the indentation response of a group of metallic materials, during and after the contact with a sharp indenter, using finite element simulations and dimensional analysis. The formulation of the algorithm considered the effects of pile-up or sink-in of the indented material around the indentation, the friction coefficient between the indenter and the sample, the elastic deformation of the indenter, and the defects of the indenter tip. An analysis considering algorithm output and experimental variation was also conducted. The results allowed identifying a fundamental lack of unique relationship between the mechanical properties and the shape of the indentation curve for indentation curves with ratio d r/dmax>0,9, where d r is the residual indentation depth and dmax is the maximum indenter displacement in the test. Similarly, results allowed concluding that Sneddons equation requires a correction by both the geometry of the contact area and the ratio between the elastic moduli of the material and the indenter (E/Ei). As the shape of contact area is affected not only by the geometry of the indenter but also by the level of pile-up or sink-in in pyramidal indenters, a relationship was observed between the level of pile-up/sink-in and the correction factor b in the Sneddons equation for Vickers and Berkovich indenters. Additionally, it was found that the deformation of the indenter is not fully incorporated into indentation data analysis by the consideration of a reduced modulus (Er). Small differences between P-h curves (where h is the indentation depth below the original surface) and P-d curves (where d is the indenter/sample mutual approach) were observed for the same Er when the ratio E/Ei is large. Thus, the reduced modulus can overestimate or underestimate the indenter stiffness, depending on the mechanical properties of the indented material. Additionally, the analysis in this work has identified that the most important limitations in mechanical properties estimation using the indentation curve arise from the lack of uniqueness, followed by deviations in indenter geometry, such as differences between equivalent cone and pyramidal Vickers or Berkovich and tip defects. When non-uniqueness is present, unique solution may be obtained with the knowledge of additional information, in conjunction with P-d data, such as the residual indentation area or the elastic modulus. Furthermore, even when a unique solution is available the experimental variation may significantly decrease the accuracy in mechanical properties estimation, whereas friction and indenter deformation have less significant effects. In general, it was observed that the proposed algorithm allows: (i) predicting the load-displacement curves P-d of instrumented indentation tests for a set of known elastic-plastic mechanical properties, (ii) identifying when the same P-d curve can be obtained from more than one set of mechanical properties of the indented material, (iii) calculating the mechanical properties (hardness, elastic modulus, yield stress and strain hardening coefficient) from P-d curves when d r/dmax<0,8 or possible ranges for each mechanical property when d r/dmax>0,8 and (iv) calculating the mechanical properties (hardness, elastic modulus, yield stress and strain hardening coefficient) from P-d curves and the knowledge of the residual indentation area.

No description available.

No description available.