Constitutive model parameters are identified during model calibration through trial-and-error process driven to fit test data. In this research, the calibration of constitutive models is formally handled as an inverse problem. The first phase of this research explores error propagation. Data errors, experimental biases (e.g. improper boundary conditions), and model errors affect the inversion of model parameters and ensuing numerical predictions. Drained and undrained tests are simulated to study the effect of these three classes of errors. Emphasis is placed on the analysis of error surfaces computed by successive forward simulations. The second phase of this research centers on test procedures. Conventional soil tests were developed to create uniform stress and strain fields; consequently, they provide limited amount of information, the inversion is ill-posed, and results enhance uncertainty and error propagation. This research examines soil testing using new, non-conventional loading and boundary conditions to create rich, diverse, non-uniform strain and stress fields. In particular, the flexural excitation of cylindrical soil specimens is shown to provide rich data leading to a more informative test than conventional geotechnical tests. The new test is numerically optimized. Then a set of unique experimental studies is conducted.

Breast cancer is one of the most frequently diagnosed cancers affecting women in the United States. An ongoing objective of many research groups is to develop a biomechanical breast model for different applications, ranging from surgical outcome predictions for patients undergoing breast reconstruction surgery, to image registration for planning plastic surgery. Achieving the goal of developing a physics based biomechanical model of the human breast requires the determination of material properties of the various tissues constituting the breast. The objective of this thesis is to develop an appropriate hybrid experimental-numerical technique to enable the calibration of material parameters of skin for different constitutive models (commonly used for skin). The quantification of the material parameters thus obtained validates the bulge test method to be used in testing soft tissue specimens like skin. A bulge test device was custom-built for this work; it consists of a pressure chamber, two digital cameras, and a syringe pump as its main components. The syringe pump provides a constant flow rate of water into the pressure chamber and results in the bulging of specimens with a diameter between 45 mm and 80 mm. Three-dimensional Digital Image Correlation technique is used to obtain full field displacement measurements of the three dimensional shape of the bulge. Tests were performed on commercial rubber sheets of different thickness and on porcine skin specimens; in these tests, the bulge shape was measured at monotonically increasing and decreasing pressure levels, as well as during cyclic loading allowing determination of the deformation and strain fields over the specimen surface. In order to extract the material properties, a hybrid experimental-numerical method was used: the experiment was modeled numerically using the finite element analysis software Abaqus, imposing the commonly used Mooney-Rivlin model for isotropic materials and the Gasser-Ogden-Holzapfel model for anisotropic materials. A comparison between the experimentally measured and numerically simulated bulge shapes was used to determine the optimized material parameters under biaxial loading conditions over a large range of stretch levels. / text

Esta pesquisa apresenta uma ampliação da investigação da capacidade de um modelo constitutivo hipoplástico de representar o comportamento tensão-deformação dos solos. Os modelos hipoplásticos apresentam características que os toram mais adequados, em muitas situações, a fornecer respostas mais realistas que os modelos baseados na elastoplasticidade clássica, que são os mais utilizados na prática da engenharia atual. O trabalho então consistiu da simulação, através de um equacionamento hipoplástico, de ensaios triaxiais drenados realizados em solos e trajetórias de tensão aos quais o modelo em estudo ainda não havia sido aplicado, tais como em corpos de prova adensados anisotropicamente e em trajetórias com decréscimos da tensão desviadora, bem como da comparação entre a resposta do modelo e os respectivos resultados desses ensaios. Em termos qualitativos, os resultados obtidos nesse confronto são bons em todas as situações estudadas. Quantitativamente, são bons na maioria das situações e razoavelmente bons em outras. / This research presents an extension of the investigation of the capacity of a hypoplastic constitutive model to represent the stress-strain behavior of soils. Hipoplastic models have characteristics that make them more suitable, in many situations, to provide more realistic answers than models based on classical elastoplasticity, which are most often used in engineering practice today. The work then consisted in the simulation, through a hypoplastic equation, of drained triaxial tests performed on soils and stress paths to which the model had not been applied yet, such as anisotropically consolidated specimens and paths with decreasing deviatoric stress as well as in the comparison between the response of the model and the results of those tests. The results obtained in this comparison are qualitatively good in all the situations studied. Quantitatively, they are good in most situations and reasonably in the others.

Este trabalho trata da formulação de leis constitutivas para meios elásticos, que uma vez danificados passam a apresentar diferentes comportamentos em tração e em compressão e certo grau de anisotropia. Inicialmente é revista uma formulação para meios elásticos anisótropos e bimodulares, sendo a mesma, então, estendida para incorporar os casos de meios elásticos com anisotropia e bimodularidade induzidas pelo dano. Seguindo os conceitos da mecânica do dano contínuo e a extensão mencionada da formulação, propõe-se um modelo constitutivo para o concreto assumindo por hipótese fundamental, a equivalência de energia entre meio danificado real e meio contínuo equivalente. Tal hipótese garante a simetria do tensor constitutivo e a sua consistência termodinâmica. De acordo com o modelo proposto, o material é considerado como um meio elástico inicialmente isótropo que passa a apresentar anisotropia induzida pela evolução do dano. Além disso, a danificação pode também induzir uma resposta bimodular no material, isto é, respostas elásticas diferentes para estados de tensão de tração ou de compressão predominantes. Nesse sentido, dois tensores de dano governando as rigidezes em regimes predominantes de tração ou de compressão são introduzidos. Sugere-se então, um critério afim de caracterizar os estados dominantes. As deformações permanentes induzidas pelo dano são, de uma forma geral, desconsideradas. No entanto, propõe-se uma versão unidimensional do modelo que permite a sua consideração. Por outro lado, os critérios para a ativação inicial dos processos de danificação e de sua posterior evolução são escritos em termos de densidade de energia de deformação. Os parâmetros do modelo podem ser identificados mediante experimentos, onde estados de tensão uniaxial e biaxial são induzidos. Também propõem-se leis de evolução de dano com base nos resultados experimentais. A boa coerência do modelo é ilustrada comparando-se uma série de respostas experimentais e numéricas no concreto relativas a estados de tensão uni, bi e triaxiais. Por fim, o modelo é empregado em análises unidimensionais e planas de vigas e pórtico em concreto armado com o objetivo de mostrar a sua potencialidade. / This thesis deals with the formulation of constitutive laws for elastic media that start to present different behaviours in tension and compression and some anisotropy degree when damaged. Initially a formulation for bimodular and anisotropic elastic media is reviewed; then it is extended to take into account the bimodularity and anisotropy induced by damage. Following the concepts of the continuum damage mechanics and the formulation extension, a constitutive model is proposed here by exploring the fundamental hypothesis of energy equivalence between real and continuous medium. Such hypothesis guarantees symmetry to the constitutive tensor and its thermodynamic consistency. According to the proposed modeling, the material is assumed as an initial elastic isotropic medium presenting anisotropy induced by damage evolution. Moreover, damage can also induce a bimodular response in the material, i.e., distinct elastic responses whether traction or compression stress states prevail. To take into account the bimodularity conveniently, two damage tensors governing the rigidity in traction or compression regimes are introduced. A criterion is proposed in order to characterize the dominant states. In general, the permanent strains induced by damage are disregarded. However, a one-dimensional version of the model that allows considering permanent strains is proposed. On the other hand, damage criteria indicating the initial and further evolution of damage are expressed in terms of strain energy densities. The model parameters can be identified from experiments where one-axial and two-axial stress states are induced. Damage evolution laws are also proposed on the basis of experimental results. The good performance of the model is illustrated by comparing a variety of numerical and experimental responses, from one to three-axial stress states. Finally, the model is used in one-dimensional and plane analysis of reinforced concrete beams and frame in order to show its potentiality.

No trabalho estudam-se alguns aspectos relativos à formulação teórica e à simulação numérica de modelos constitutivos para o concreto fundamentados na Mecânica do Dano Contínuo, incluindo-se os chamados métodos simplificados de análise estrutural. Inicialmente apresenta-se uma discussão sobre deformações permanentes e anisotropia induzidas pela evolução do dano. A resposta unilateral do concreto submetido a solicitações com inversão de sinal, também é comentada. Cada um dos fenômenos é ilustrado por respostas observadas experimentalmente. O modelo de dano proposto por Mazars para o concreto submetido a carregamento proporcionalmente crescente é então analisado. Em seguida, apresenta-se uma extensão do modelo considerando-se o aspecto unilateral no comportamento do concreto. Na sequência, analisa-se o modelo constitutivo proposto por La Borderie em seus aspectos de formulação e resposta numérica. O modelo é mais completo permitindo levar em conta todos os fenômenos discutidos anteriormente. Um outro aspecto considerado no trabalho é relativo à aplicação dos modelos estudados à análise de estruturas aporticadas. Em termos de discretização destacam-se a técnica de divisão dos elementos estruturais em estratos e os chamados modelos simplificados. Neste último caso, o modelo de Flórez-López é analisado e a simplificação consiste na definição prévia, sobre a estrutura discretizada através de elementos de viga e de coluna, de zonas de localização da plastificação e do dano; no limite com a evolução do processo de carregamento, aquelas zonas passam a se constituir em rótulas. Por último uma generalização do modelo anterior proposta por Álvares é estudada. Os resultados numéricos fornecidos pelos modelos são confrontados com os experimentais de vigas em concreto armado (biapoiadas e com diferentes taxas de armadura) e de um pórtico em concreto armado. / In this work some aspects related to the theoretical formulation and numerical simulation of constitutive models for the concrete based on the Continuum Damage Mechanics are studied, including simplified methods of structural analysis. Initially a discussion about permanent strains and anisotropy induced by the evolution of the damage is presented. The unilateral response of the concrete submitted to loadings with signal inversion is commented as well. Each one of the phenomena is illustrated by experimental tests. The damage model proposed by Mazars for the concrete under proportional increasing load is then analysed. Afterwards, it is shown an extension of the model considering the unilateral aspect ofthe concrete behaviour. In the sequence the constitutive model proposed by La Borderie is presented in its aspects of formulation and numerical response. This model is more complete, allowing to take in to account all the phenomena discussed previously. Another aspect considered in the work is related to the application of the models studied to the analysis of framed structures. Regarding the discretization, the technique of division of the structural elements in layers and the so called simplified models are enhanced. In this last case, the model proposed by Flórez-López is analysed and the simplification consists on previous definition of yielding and damage zones over the assembly of beam and colunm elements, which compose the discretized structure; in the limit evolution of the loading process, these zones become hinges. Finally, a generalization of the previous model proposed by Álvares is studied. The numerical results supplied by the models are compared with the experimental results of reinforced concrete beams (simply supported and with different reinforced rates) and of a reinforced concrete frame.

Understanding the behavior of geomaterials under explosive loading is essential for several applications in the mining and oil industry. To date, the design of these applications is based almost solely on empirical equations and tabulated data. Optimal designs require accurate and complete knowledge of rock behavior under various loading conditions. The vast majority of the properties available in the literature have been gathered by deforming the specimen slowly. These properties have been used to establish constitutive models which describe the behavior of rocks under static and quasi-static loading conditions. However, the dynamic properties and material constitutive models describing the behavior of geomaterials under high strain-rate loading conditions are essential for a better understanding and enhanced designs of dynamic applications. Some attempts have been made to measure dynamic properties of rocks. Also, some trials have been made to devise material models which describe the behavior of rocks and the evolution of damage in the rock under dynamic loading. Published models were successful in predicting tensile damage and spalling in rocks. However, there are no established models capable of predicting compressional damage in rocks due to dynamic loading. A recently-developed model, the RHT model, was formulated to describe the behavior of concrete over the static and dynamic ranges. The model was also formulated to predict compressional damage based on the strain rate at which the material is subjected to. The RHT model has been used successfully in several applications. The purpose of this research was to characterize one rock type as an example of a hard brittle rock. The physical properties of the rock as well as the static and dynamic mechanical properties were investigated. These properties were used to calibrate the RHT model and investigate its potentials to predict compressional damage in brittle materials. The calibrated model showed good precision reproducing the amplitude of the strain signals generated by explosive loading. It was also capable of predicting compressional damage with acceptable accuracy. Unfortunately, due to implementation restrictions, tensile and spall damage could not be captured by the model. The duration and shape of the strain pulse were also poorly modeled. / Thesis (Ph.D, Mining Engineering) -- Queen's University, 2010-12-22 17:54:05.887

The contributions of this thesis are an implicit modelling method for the coupled constitutive relations and an energy-based method for material modelling. The two developed methods utilise implicit models to represent material constitutive relations without the requirement of physical parameters. The first method is developed to model coupled constitutive relations using state-space representation with neural networks. State-space representation is employed to express the desired relations in a compact fashion while simultaneously providing the capability of modelling rate- and/or path-dependent behaviour. The employment of neural networks with the generalised state-space representation results in a single implicit model that can be adapted for a broad range of constitutive behaviours. The performance and applicability of the method are highlighted through the applications for various constitutive behaviour of piezoelectric materials, including the effects of hysteresis and cyclic degradation. An energy-based method is subsequently developed for implicit constitutive modelling by utilising the energy principle on a deformed continuum. Two formulations of the proposed method are developed for the modelling of materials with varying nature in directional properties. The first formulation is based on an implicit strain energy density function, represented by a neural network with strain invariants as input, to derive the desired stress-strain relations. The second formulation consists of the derivation of an energy-based performance function for training a neural network that represents the stress-strain relations. The requirement of deriving stress is eliminated in both formulations and this facilitates the use of advanced experimental setup, such as multi-axial load tests or non-standard specimens, to produce the most information for constitutive modelling from a single experiment. A series of numerical studies -- including validation problems and practical cases with actual experimental setup -- have been conducted, the results of which demonstrate the applicability and effectiveness of the proposed method for constitutive modelling on a continuum basis.

[pt] No presente trabalho são introduzidas considerações sobre modelos constitutivos viscoelasticos e suas formulações. É apresentada uma teoria de estruturas uni-dimensionais imersas no espaço tri-dimensional, a partir de uma formulação diferencial, permitindo-se a análise de problemas no contexto de grandes deformações.(caracteristicas não-lineares). Adota-se uma lei constitutiva viscoelastica linear, sob forma diferencial. São resolvidos três problemas planos que caracterizam a aplicação do modelo proposto: 1) Viga reta engastada submetida em sua extremidade livre a uma força não-seguidora compressiva cuja direção coincide com a do eixo longitudinal da viga na configuração indeformada. 2) Viga curve bi-rotulada com ambas as extremidades Fixas carregada uniformemente. 3) Viga reta engastada submetida em sua extremidade livre a uma força cíclica perpendicular ao eixo longitudinal da viga na configuração indeformada. Nos dois primeiros problemas é adotada a hipótese quasi-estatica, conservando a influência da inercia apenas no terceiro problema quando é feito um estudo comparativo entre o modelo elástico e o viscoelástico. São propostos os métodos numéricos de solução para os casos viscoelásticos quasi-estáticos - baseados num acoplamento do Método do Tiro (Runge-Kutta de Quarta ordem / Steffensen); considerando o problema espacial, e de Euler Modificado; na evolução temporal - e para o caso dinâmico, valendo-se da aproximação de diferenças finitas para avaliar as velocidades e acelerações. / [en] In the present work, considerations about viscoelastic cosntitutive models and their formulations are introduced. A theory of unidimensionail structures embeded in the tri-dimensional space is presented in a differential formula, allowing the analysis of problems in the large deformations point of view. A linear viscoelastic consitutive law is adopted in a differential representation. Three plane problems which characterize the application of the proposed model are solved: 1) Straight cantilever subjected, at the free end, to a compressive non follower force. The direction of this force coincides with the longitudinal axis of the beam at the undeformed configuration; 2) Curved bean with fixed supports at the ends, under uniform charge. 3) Straight cantilaver subjected at the free end to a cyclic force, perpendicularly to the longitudinal axis of the beam at the undeformed confiquration. In the two first problems; the quasistatic hypothesis is adopted the inertial influence being considered only at the third problems, where a comparative study between the elastic and the viscoelastic models is carried out. The numerical methods for the solution in the quasistatic viscoelastic cases are proposed - based in a coupling between the shooting method (Runge-Kutta of fourth order/Steffensen) for the spatial problem and the Modified Euler Method for the temporal evolution - as well as for the solution of the dynamical case - employing an approximation by finite differences for the evaluations of the velocities and accelerations.

Este trabalho trata da formulação de leis constitutivas para meios elásticos, que uma vez danificados passam a apresentar diferentes comportamentos em tração e em compressão e certo grau de anisotropia. Inicialmente é revista uma formulação para meios elásticos anisótropos e bimodulares, sendo a mesma, então, estendida para incorporar os casos de meios elásticos com anisotropia e bimodularidade induzidas pelo dano. Seguindo os conceitos da mecânica do dano contínuo e a extensão mencionada da formulação, propõe-se um modelo constitutivo para o concreto assumindo por hipótese fundamental, a equivalência de energia entre meio danificado real e meio contínuo equivalente. Tal hipótese garante a simetria do tensor constitutivo e a sua consistência termodinâmica. De acordo com o modelo proposto, o material é considerado como um meio elástico inicialmente isótropo que passa a apresentar anisotropia induzida pela evolução do dano. Além disso, a danificação pode também induzir uma resposta bimodular no material, isto é, respostas elásticas diferentes para estados de tensão de tração ou de compressão predominantes. Nesse sentido, dois tensores de dano governando as rigidezes em regimes predominantes de tração ou de compressão são introduzidos. Sugere-se então, um critério afim de caracterizar os estados dominantes. As deformações permanentes induzidas pelo dano são, de uma forma geral, desconsideradas. No entanto, propõe-se uma versão unidimensional do modelo que permite a sua consideração. Por outro lado, os critérios para a ativação inicial dos processos de danificação e de sua posterior evolução são escritos em termos de densidade de energia de deformação. Os parâmetros do modelo podem ser identificados mediante experimentos, onde estados de tensão uniaxial e biaxial são induzidos. Também propõem-se leis de evolução de dano com base nos resultados experimentais. A boa coerência do modelo é ilustrada comparando-se uma série de respostas experimentais e numéricas no concreto relativas a estados de tensão uni, bi e triaxiais. Por fim, o modelo é empregado em análises unidimensionais e planas de vigas e pórtico em concreto armado com o objetivo de mostrar a sua potencialidade. / This thesis deals with the formulation of constitutive laws for elastic media that start to present different behaviours in tension and compression and some anisotropy degree when damaged. Initially a formulation for bimodular and anisotropic elastic media is reviewed; then it is extended to take into account the bimodularity and anisotropy induced by damage. Following the concepts of the continuum damage mechanics and the formulation extension, a constitutive model is proposed here by exploring the fundamental hypothesis of energy equivalence between real and continuous medium. Such hypothesis guarantees symmetry to the constitutive tensor and its thermodynamic consistency. According to the proposed modeling, the material is assumed as an initial elastic isotropic medium presenting anisotropy induced by damage evolution. Moreover, damage can also induce a bimodular response in the material, i.e., distinct elastic responses whether traction or compression stress states prevail. To take into account the bimodularity conveniently, two damage tensors governing the rigidity in traction or compression regimes are introduced. A criterion is proposed in order to characterize the dominant states. In general, the permanent strains induced by damage are disregarded. However, a one-dimensional version of the model that allows considering permanent strains is proposed. On the other hand, damage criteria indicating the initial and further evolution of damage are expressed in terms of strain energy densities. The model parameters can be identified from experiments where one-axial and two-axial stress states are induced. Damage evolution laws are also proposed on the basis of experimental results. The good performance of the model is illustrated by comparing a variety of numerical and experimental responses, from one to three-axial stress states. Finally, the model is used in one-dimensional and plane analysis of reinforced concrete beams and frame in order to show its potentiality.

Esta pesquisa apresenta uma ampliação da investigação da capacidade de um modelo constitutivo hipoplástico de representar o comportamento tensão-deformação dos solos. Os modelos hipoplásticos apresentam características que os toram mais adequados, em muitas situações, a fornecer respostas mais realistas que os modelos baseados na elastoplasticidade clássica, que são os mais utilizados na prática da engenharia atual. O trabalho então consistiu da simulação, através de um equacionamento hipoplástico, de ensaios triaxiais drenados realizados em solos e trajetórias de tensão aos quais o modelo em estudo ainda não havia sido aplicado, tais como em corpos de prova adensados anisotropicamente e em trajetórias com decréscimos da tensão desviadora, bem como da comparação entre a resposta do modelo e os respectivos resultados desses ensaios. Em termos qualitativos, os resultados obtidos nesse confronto são bons em todas as situações estudadas. Quantitativamente, são bons na maioria das situações e razoavelmente bons em outras. / This research presents an extension of the investigation of the capacity of a hypoplastic constitutive model to represent the stress-strain behavior of soils. Hipoplastic models have characteristics that make them more suitable, in many situations, to provide more realistic answers than models based on classical elastoplasticity, which are most often used in engineering practice today. The work then consisted in the simulation, through a hypoplastic equation, of drained triaxial tests performed on soils and stress paths to which the model had not been applied yet, such as anisotropically consolidated specimens and paths with decreasing deviatoric stress as well as in the comparison between the response of the model and the results of those tests. The results obtained in this comparison are qualitatively good in all the situations studied. Quantitatively, they are good in most situations and reasonably in the others.