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Multiscale Modeling of the Deformation of Semi-Crystalline PolymersShepherd, James Ellison 29 March 2006 (has links)
The mechanical and physical properties of polymers are determined primarily by the underlying nano-scale structures and characteristics such as entanglements, crystallites, and molecular orientation. These structures evolve in complex manners during the processing of polymers into useful articles. Limitations of available and foreseeable computational capabilities prevent the direct determination of macroscopic properties directly from atomistic computations. As a result, computational tools and methods to bridge the length and time scale gaps between atomistic and continuum models are required. In this research, an internal state variable continuum model has been developed whose internal state variables (ISVs) and evolution equations are related to the nano-scale structures. Specifically, the ISVs represent entanglement number density, crystal number density, percent crystallinity, and crystalline and amorphous orientation distributions. Atomistic models and methods have been developed to investigate these structures, particularly the evolution of entanglements during thermo-mechanical deformations. A new method has been created to generate atomistic initial conformations of the polymer systems to be studied. The use of the hyperdynamics method to accelerate molecular dynamics simulations was found to not be able to investigate processes orders of magnitude slower that are typically measurable with traditional molecular dynamics simulations of polymer systems. Molecular dynamics simulations were performed on these polymer systems to determine the evolution of entanglements during uniaxial deformation at various strain rates, temperatures, and molecular weights. Two methods were evaluated. In the first method, the forces between bonded atoms along the backbone are used to qualitatively determine entanglement density. The second method utilizes rubber elasticity theory to quantitatively determine entanglement evolution. The results of the second method are used to gain a clearer understanding of the mechanisms involved to enhance the physical basis of the evolution equations in the continuum model and to derive the models material parameters. The end result is a continuum model that incorporates the atomistic structure and behavior of the polymer and accurately represents experimental evidence of mechanical behavior and the evolution of crystallinity and orientation.
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Dynamic mechanical behavior and high pressure phase stability of a zirconium-based bulk metallic glass and its composite with tungstenMartin, Morgana 04 March 2008 (has links)
An investigation of the high-strain-rate mechanical properties, deformation
mechanisms, and fracture characteristics of a Zr-based bulk metallic glass (BMG) and its composite with tungsten was conducted through the use of controlled impact experiments and constitutive modeling. The overall objective of this research was to determine the high-strain-rate deformation and failure mechanisms of a BMG and its composite as a function of stress state and strain rate, and
describe the mechanical behavior over a range of loading conditions. The research involved performing controlled impact experiments on BMG
composites consisting of an amorphous Zr57Nb5Cu15.4Ni12.6Al10 (LM106) with
crystalline tungsten reinforcement particles. Monolithic LM106 was also examined to aid in the understanding of the composite. The mechanical
behavior of the composite was investigated over a range of strain rates (10^3 s^-1 to 10^6 s^-1), stress states (compression, compression-shear,
tension), and temperatures (RT to 600 C) to determine the dependence of mechanical properties and deformation and failure modes (i.e., homogeneous deformation vs. inhomogeneous shear banding) on these parameters. Mechanical
testing in the quasi-static to intermediate strain rate regimes was performed using an Instron, Drop Weight Tower, and Split Hopkinson Pressure
Bar, respectively. High-strain-rate mechanical properties of the BMG-matrix composite and monolithic BMG were investigated using dynamic compression (reverse Taylor) and dynamic tension (spall) impact experiments performed using a gas gun instrumented with velocity interferometry and high-speed digital photography. These experiments provided information about dynamic strength and deformation modes, and allowed for validation of constitutive models via comparison of experimental and simulated transient deformation profiles and free surface velocity traces. Hugoniot equation of state measurements were performed on the monolithic BMG to investigate the high
pressure phase stability of the glass and the possible implications of a high pressure phase transformation on mechanical properties. Specimens were recovered for post-impact microstructural and thermal analysis to gain information about the mechanisms of dynamic deformation and fracture, and to examine for possible shock-induced phase transformations of the amorphous phase.
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A Study on the Effect of Inhomogeneous Phase of Shape Memory Alloy WireManna, Sukhendu Sekhar January 2017 (has links) (PDF)
The present study in this thesis has attempted to resolve one of the key aspects of enhancing predictability of macroscopic behavior of Shape Memory Alloy (SMA) wire by considering variation of local phase inhomogeneity. Understanding of functional fatigue and its relation with the phase distribution and its passivation is the key towards tailoring thermal Shape Memory Alloy actuators’ properties and performance. Present work has been carried out in two associated areas. First part has covered solving a coupled thermo-mechanical boundary value problem where initial phase fractions are prescribed at the gauss points and subsequent evolution are tracked over the loading cycle. An incremental form of a phenomenological constitutive model has been incorporated in the modelling framework. Finite element convergence studies using both homogeneous and inhomogeneous SMA wires are performed. Effects of phase inhomogeneity are investigated for mechanical loading and thermo-electric loading. Phase inhomogeneity is simulated mainly due to process and handling quality. An example of mechanical boundary condition such as gripping indicates a negative residual strain at macroscopic behavior. Simulation accurately captures vanishing local phase inhomogeneity upon multiple cycles of thermo-mechanical loading on unconstrained straight SMA wire. In the second part, a phase identification and measurement scheme is proposed. It has been shown that by employing variation of electrical resistivity which distinctly varies with phase transformation, martensite phase volume fraction can be quantified in average sense over the length of a SMA wire. This can be easily achieved by using a simple thermo-mechanical characterization setup along with resistance measurement circuit. Local phase inhomogeneity is created in an experimental sample, which is subjected to electrical heating under constant mechanical bias load. The response shows relaxation of the initial shrinkage strain due to local phase. Results observed for thermo-electric loading on the inhomogeneous SMA wires compliment the results observed from the simulated loading cases. Several interesting features such as shrinkage of the inhomogeneous SMA wire after first loading cycle, relaxation of the residual strain over multiple loading cycles due to the presence of inhomogeneity are captured. This model promises useful applications of SMA wire in fatigue studies, SMA embedded composites and hybrid structures.
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Análise tensão deformação de uma barragem de concreto em solo residual preponderantemente anisotrópico. / Analysis strain stress of a concrete dam in residual soil predominantly anisotropic.Marcio Fernandes Leão 01 April 2015 (has links)
Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Estudos de fundações de obras de arte, como barragens de concreto-gravidade e
barragens de terra, devem contemplar todos os esforços atuantes no maciço de fundação,
principalmente as tensões e as deformações esperadas durante todo o processo construtivo e
no período pós-construtivo. Quando essas estruturas são apoiadas sobre rochas de boa
sanidade, a escolha do barramento geralmente favorece a implantação de barragens de
concreto. Entretanto, quando os maciços de fundação são formados por solos, a opção técnica
geralmente mais bem aceita é quanto à utilização de barragens de terra. Em ambos os casos,
as análises de estabilidade e de deformação são desenvolvidas por métodos analíticos bem
consolidados na prática. Nas condições mais adversas de fundação, seja em rochas ou em
solos com marcante anisotropia e estruturas reliquiares herdadas da rocha-mãe, a utilização de
modelos constitutivos anisotrópicos em análises por elementos finitos propicia simulações
mais realistas dessas feições estruturais, contribuindo para o seu melhor conhecimento. O
presente trabalho teve por objetivo demonstrar a utilização de um modelo constitutivo
anisotrópico no estudo da fundação da Barragem San Juan, localizada na República
Dominicana, que foi concebida como uma estrutura tipo concreto-gravidade apoiada sobre
solos residuais jovens altamente anisotrópicos. Nessa obra, apesar de sua pequena altura, a
presença marcante de descontinuidades ensejou um estudo mais detalhado do comportamento
tensão-deformação da fundação, levando em conta o levantamento detalhado da atitude das
descontinuidades presentes no maciço e os resultados de ensaios de resistência em planos
paralelos e normais às descontinuidades. Para a estimativa de deformações, os respectivos
módulos de Young (Es) foram estimados com base em correlações com a resistência à
penetração SPT desenvolvidas nesta dissertação, a partir de um estudo estatístico baseado em
várias publicações disponíveis na literatura. As análises numéricas por elementos finitos
foram desenvolvidas através do programa Plaxis 2D, utilizando-se como modelo constitutivo
aquele denominado The Jointed Rock Model, que é particularmente recomendado para
análises de estabilidade e deformação de materiais anisotrópicos. Os resultados das análises
numéricas foram comparados com as análises de equilíbrio limite elaboradas para o projeto
executivo da referida barragem, pelo programa Slope, utilizando o método rigoroso de
Morgenstern e Price, que se mostrou conservador. Os resultados das análises numéricas
mostraram sua inequívoca versatilidade para a escolha de opções de reforço da fundação,
através de dentes que objetivavam o aumento das condições de estabilidade da barragem.
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Análise por elementos finitos de maciços escavados por túneis. / Finite element analysis of tunnel excavation.Guilherme Mafra Machado 19 July 2011 (has links)
O presente trabalho de pesquisa aborda a aplicação de análise por elementos finitos a maciços escavados por túneis, revisando relevantes estudos sobre análise numérica deste tipo de obra e realizando modelagem numérica própria, bi e tridimensional, voltada ao caso do Túnel Paraíso do Metrô de São Paulo (obra com dados de instrumentação e de ensaios), com uso do consagrado programa de elementos finitos Imperial College Finite Element Program (ICFEP), desenvolvido nos últimos trinta anos no Imperial College de Londres. Dentre os modelos constitutivos adotados nestas análises, destaca-se o Cinemático com Duas Superfícies Modificado do Imperial College, modelo elasto-plástico com endurecimento cinemático e isotrópico, baseado no estado crítico, com grande capacidade de representar as características de rigidez de argilas sobreadensadas (como são as argilas da região do Túnel Paraíso), cujas principais características são levantadas através de revisão bibliográfica. Da observação da etapa de reprodução numérica de ensaios e dos resultados conseguidos com as análises da escavação do túnel, são feitas considerações sobre o desempenho de todos os modelos constitutivos adotados (além do Cinemático com Duas Superfícies Modificado, o Cam-Clay Modificado e o elasto perfeitamente plástico com critério de ruptura de Mohr-Coloumb), bem como sobre a performance dos modelos numéricos desenvolvidos. / This research approaches the application of finite element analysis to tunnel excavation, making a survey of some numerical relevant studies of this kind of construction and performing a range of plane-strain and three-dimensional analyses, focused on the Paraíso Tunnel from Sao Paulo Underground (with field instrumentation and lab tests), using a powerful finite element program, the Imperial College Finite Element Program (ICFEP), developed in the last thirty years at Imperial College London. Among the constitutive models adopted in those analyses, it is highlighted the Modified Two Surfaces Kinematic from Imperial College, a elasto-plastic model with kinematic and isotropic hardening, based on the critical state, with great ability to represent the pre-failure stiffness of overconsolidated clays (as are the clays around Paraíso Tunnel), whose main features are raised through a literature survey. From the results of numerical simulation of lab tests and from the predictions of the tunnelling analyses, it is discussed the performance of all constitutive models adopted (including Modified Cam-Clay and elastic perfectly plastic model with Mohr-Coloumb failure criterion) and of the numerical models developed.
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Modelo de atrito estático em interfaces de contato entre concreto e areia / A model for the static friction between concrete-sand interface contactJeselay Hemetério Cordeiro dos Reis 30 March 2006 (has links)
Esta tese apresenta os princípios e a formulação de um modelo não-linear de atrito estático em interface de concreto areia. A hipótese básica para desenvolvimento das equações consiste na ocorrência do atrito de deslizamento (atrito verdadeiro), do atrito de rolamento (rearranjo das partículas) e da dilatância (variação de volume durante o cisalhamento). A solução analítica do modelo considera o efeito da rugosidade da superfície de contato, da curva granulométrica da areia e do seu estado de compacidade inicial. Foram realizados ensaios de cisalhamento direto com carga normal constante em interface de contato entre concreto e areia com seção de 500 mm x 500 mm com o objetivo de permitir a calibração do modelo proposto. É discutida e sugerida a incorporação da equação constitutiva desse modelo em análises de interação solo-estrutura via método dos elementos finitos. Sua aplicabilidade é demonstrada através da análise 1D e 2D de estacas de atrito executadas em areia e submetidas a carregamentos de compressão / This thesis presents the principles and formulation underlying a concrete-sand interface nonlinear static friction model. The basic hypothesis employed in the development of the model equation takes into account the interface sliding friction (true friction), a rolling friction (particle rearrangement) and dilatancy(volume variation during shear). The model analytical solution considers the effect of roughness of the contact surface, the grain size distribution and its initial state of compactness of the sand. To calibrate the proposed model, a direct shear stress test under constant load was carried out along a 500mm x 500mm section concrete-sand interface. Furthermore, a discussion and suggestion of the inclusion of the model constitutive equation applied to the analysis of soil-structure interaction using the finite element method are presented. The applicability of the proposed model is proven through the analysis of 1-D and 2-D skin friction piles made of sand mass subjected to compression load
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Modélisation du comportement mécanique des grands CFRD : Identification des caractéristiques des enrochements et comportement du masque d'étanchéité amontChen, Yuguang 19 December 2012 (has links)
Une modélisation numérique des grands barrages en enrochement à masque amont (CFRD) a été entreprise pour mieux comprendre les pathologies observées sur ces ouvrages récemment construits, principalement des fissures horizontales et verticales sur le masque amont en béton en phase de construction et de mise en eau. Le but de cette modélisation est ensuite de confirmer, préciser ou compléter les solutions proposées par les experts pour éviter ces désordres.D‟abord, une loi de comportement développée au sein d‟EDF-CIH appelée L&K-Enroch a été présentée. Il s'agit d'un modèle élastoplastique qui prend en compte les déformations irréversibles de l‟enrochement sous sollicitation déviatorique et isotrope dans les grands CFRD. Cette thèse propose aussi une méthode de calage des paramètres de ce modèle.Deux modélisations des barrages sont également présentées dans cette thèse afin de vérifier la fiabilité de la loi de comportement L&K-Enroch et de mieux comprendre les pathologies sur le masque amont en béton du CFRD. La modélisation en déformation plane (2D) est d‟abord entreprise et a comme support le barrage d‟Aguamilpa (180,5 m), au Mexique. Une modélisation plus poussée en 3D prenant en compte de l'effet de la vallée est ensuite entreprise. Elle a comme support le barrage de Mohale (145 m) au Lesotho.L'effet d‟échelle de l'enrochement est également étudié en comparant les résultats de simulation pour les matériaux de différentes tailles. L'impact de l‟effet d‟échelle sur le comportement du barrage est aussi analysé.Les simulations présentées en 2D et 3D sont une contribution à l'analyse du comportement mécanique des grands CFRD. Les résultats de la modélisation 2D montrent généralement une bonne correspondance avec les mesures in-situ. La modélisation 3D explique, d'une manière plus convaincante, l‟apparition des fissures sur le masque amont en béton des CFRD. Certaines mesures constructives sont finalement proposées afin de limiter ou d'éviter ces désordres dans le masque. / A numerical modeling of high concrete face rockfill dam (CFRD) has been considered in order to better understand the pathology of cracks observed in the recently constructed high CFRD, especially the horizontal and vertical cracks detected on the concrete face slab during the construction and the impoundment. The purpose of this modeling is then to confirm, clarify or complete the solutions proposed by the experts to avoid these disorders.This PhD thesis presents a constitutive model developed by EDF-CIH known as L&K-Enroch. It is an elastoplastic model that takes into account the irreversible deformations of rockfill under deviatoric and isotropic loading in high CFRD. A first attempt of proposing a parameter calibration approach has been made as well.Two dam modellings are also presented in this thesis in order to verify the reliability of the constitutive model and to better understand the pathology of cracks observed on the concrete face slab of CFRD. A 2D modeling is compared to the behaviour of the 180.5 m Aguamilpa Dam, Mexico; a 3D modeling taking into account the valley effect is compared to the behaviour of the 145 m Mohale Dam, Lesotho.This thesis also studies the scale effect of rockfill through a comparison of the simulation results of materials of different sizes. The impact of the scale effect on the dam behavior is analyzed as well.The presented 2D and 3D simulations are a contribution to the analysis of the mechanical behaviour of high CFRD. The results of the 2D modeling show a relatively good agreement with the monitored data. The 3D modeling explains, in a rather convincing way, the pathology of cracks on the slab face of CFRD. Some measures are proposed to limit or avoid these different kinds of cracks.
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[en] A CONSTITUTIVE MODEL FOR FIBER REINFORCED CONCRETE / [es] UN MODELO CONSTITUTIVO PARA CONCRETO CON FIBRAS / [pt] UM MODELO CONSTITUTIVO PARA CONCRETO COM FIBRASLEONARDO CRAVEIRO SIMOES 19 April 2001 (has links)
[pt] Nos últimos 40 anos, tem-se observado um crescente
interesse por compósitos formados pela adição de fibras a
matrizes de concreto, os chamados concretos reforçados com
fibras. Esse interesse é justificado, sobretudo, pelo
significativo ganho em tenacidade que as fibras
proporcionam, atenuando as características frágeis do
concreto. De fato, em virtude do mecanismo de reforço
promovido pelas fibras, o concreto com fibras é capaz de
absorver muito mais energia de deformação até a ruptura,
apresentando, no regime pós-fissuração, um comportamento
muito mais suave que o concreto simples. Esse comportamento
é acompanhado por um processo de fissuração mais uniforme,
no qual observam-se fissuras mais finas e menos espaçadas.
Além disso, registram-se aumentos nos valores de
resistência do material e nos níveis de deformação que ele
atinge até seu completo esgotamento.
Tendo em vista os benefícios que as fibras aportam ao
desempenho do concreto, seu emprego seria recomendável a
estruturas em que a ductilidade é um dos parâmetros
principais de projeto, ou naquelas feitas com concretos de
alta resistência, uma vez que estes apresentam um
comportamento ainda mais frágil que os concretos de
resistência normal. Além disso, a utilização de fibras no
combate aos esforços de cisalhamento mostra-se extremamente
vantajosa e promissora.
Neste trabalho, apresenta-se um modelo constitutivo para
concreto reforçado com fibras baseado na formulação
hipoelástica de ELWI E MURRAY (1979), originalmente
proposta para concreto simples. As especificidades do
comportamento do concreto com fibras frente às mais
diversas solicitações, tais como, tração, compressão e
cisalhamento, são incorporadas ao modelo através de
relações tensão-deformação adequadas a esse material. Tais
relações provêm de estudos analíticos e experimentais sobre
o assunto, publicados na literatura técnica especializada.
O modelo assim obtido é implementado no programa FEPARCS
(ELWI E MURRAY, 1980), capaz de realizar análises númericas
não-lineares através do método dos elementos finitos. Por
fim, utiliza-se esse program para simular a resposta de uma
estrutura de concreto com fibras, cujo ensaio experimental
aparece minuciosamente descrito em (CRAIG, 1987). Os
resultados numéricos obtidos são comparados com os
experimentais correspondentes, em termos da curva carga
versus deslocamento, desenvolvimento e distribuição de
fissuras, progressão do escoamento da armadura longitudinal
(convencional) e modo de ruptura. Avalia-se, então, a
eficiência da implementação realizada na descrição do
comportamento de estruturas de concreto com fibras. / [en] Along the past forty years, an increasing interest on
composite materials formed by the
addition of discrete fibers to a concrete matrix is being
observed. These composites are known
as fiber reinforced concretes. The interest on the use of
fibers as reinforcement is justified by
their significative contribution to concrete thoughness, as
they reduce the brittle characteristics
of that material. In fact, due to fiber reinforcement
mechanism, fiber reinforced concrete can
absorb much more strain energy until failure, in comparison
to ordinary concrete. The cracking
process seems to be more uniform, as the distance between
cracks are reduced. Besides that,
the material strength and the deformation levels at
cracking and rupture are greater, on the
case of fiber reinforced concrete.
The benefits that fibers bring to concrete behavior
indicate that they could be used as
complementary reinforcent for concrete structures when
ductility is a major design concern, or
when high strength concrete is employed, as this class of
material tends to be much more brittle
then normal strength concrete. Fibers are also effective as
shear reinforcement, and they could
even replace stirrups in this function.
In this work, a constituive model for fiber reinforced
concrete is presented. This model
is based on the formulation originally proposed by ELWI AND
MURRAY (1979) for the case of
ordinary concrete. The behavior characteristcs of fiber
reinforced concrete are incorporated as
adaquated uniaxial stress-strain relations in tension and
compression. The behavior under shear
stress is also considered. The model is then implemented in
the finite element program
FEPARCS (ELWI AND MURRAY, 1980). A numerical analysis on
the response of a fiber
reinforced concrete structure is conducted. Results
reported in technical literature (CRAIG,
1987) are compared to those obtained by the finite element
analysis. The efficiency of the
model is then verified. / [es] En los últimos 40 anos, se ha observado un creciente
interés por compuestos formados por la adición de fibras a
matrizes de concreto, los llamados concretos reforzados con
fibras. Ese interés se debe a la significativa ganancia en
tenacidad que las fibras proporcionan, atenuando las
características frágiles del concreto. De hecho, en virtud
del mecanismo de refuerzo promovido por las fibras, el
concreto con fibras es capaz de absorver mucha más energía
de deformación hasta la ruptura, presentando, en el régimen
posfisuración, un comportamiento mucho más suave que el
concreto simple. Este comportamiento se ve acompañado por
un proceso de fisuración más uniforme, en el cual se
observan fisuras más finas y menos espaciadas. Además, se
registran aumentos en los valores de resistencia del
material y en los niveles de deformación que alcanza hasta
su completa destrucción. Teniedo en cuenta los beneficios
que las fibras aportan al desempeño del concreto, sería
recomendable su empleo en extructuras donde la ductilidad
es uno de los parámetros principales de proyecto, o en
aquellas hechas con concreto de alta resistencia, ya que
éstos presentan un comportamiento más frágil que los
concretos de resistencia normal. En este trabajo, se
presenta un modelo constitutivo para concreto reforzado con
fibras que tiene como base la formulación hipoelástica de
ELWI Y MURRAY (1979), originalmente propuesta para concreto
simple. Las especificidades del comportamiento del concreto
con fibras frente a las más diversas solicitudes, tales
como, tracción, compresión y cisallamiento, se incorporan
al modelo a través de relaciones tensión-deformación
adecuadas a ese material. Tales relaciones provienen de
estudios analíticos y experimentales sobre el asunto,
publicados en la literatura técnica especializada. La
implementación del modelo obtenido fue realizada a través
del programa FEPARCS (ELWI Y MURRAY, 1980), capaz de
realizar análisis númerico no lineal a través del método de
los elementos finitos. Por fin, se utiliza ese programa
para simular la respuesta de una extructura de concreto con
fibras, cuyo ensayo experimental aparece minuciosamente
descrito en (CRAIG, 1987). Los resultados numéricos
obtenidos se comparan con los experimentales
correspondientes, considerando la curva carga versus
deslocamiento, desarrollo y distribuición de fisuras,
progresión del deslizamiento de la armadura longitudinal
(convencional) y modo de ruptura. Se evalúa entonces, la
eficiencia de la implementación en la descrición del
comportamiento de extructuras de concreto con fibras.
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Deformačně-napěťová analýza arteriálních aneuryzmat / Stress-strain analysis of arterial aneurysmsTesařová, Petra January 2010 (has links)
The diploma thesis is focused on the creation of the aneurysm finite element model and the making of the aneurysm wall stess-strain analysis using ANSYS software. The model of abdominal aortic aneurysm geometry starts from the CT scan of the particular patient. In the thesis there are compared two chosen constitutive models, each of them appears from different mechanical tests done on human arterial fibre samples. Furthermore, a limiting condition for aneurysm wall structure damage is expressed. On the basis of the results of stress calculation in the aneurysm wall and the limiting condition, the safety coefficient and rupture factors risk are worked out.
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Modelování fázových transformací v materiálech s tvarovou pamětí / Modeling of phase transformations in shape memory materialsFrost, Miroslav January 2012 (has links)
Title: Modeling of phase transformations in shape memory materials Author: Miroslav Frost Department: Mathematical Institute of Charles University Supervisor: Prof. Ing. František Maršík, DrSc., Mathematical Institute of Charles University Abstract: This thesis presents a new thermomechanical three-dimensional con- stitutive model of NiTi-based shape memory alloys. The model was formulated within the framework of generalised standard models and it features a novel form of the dissipation function, which combines contributions stemming from the phase transformation between austenite and martensite and from the reorienta- tion of martensite. The change in the material response associated with the phase transformation between austenite and R-phase as well as material anisotropy and tension-compression asymmetry are also covered. The time-evolutionary problem of a quasistatic mechanical loading of a NiTi body with prescribed temperature evolution was formulated and analyzed within the framework of energetic so- lutions. The corresponding time-incremental minimization problem provided a conceptual algorithm utilized in the numerical treatment. The constitutive mod- el was implemented into the finite element package Abaqus. Several numerical simulations were performed and compared with experiments. Keywords:...
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