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.

Orientador: Marco Lucio Bittencourt / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-06T20:24:00Z (GMT). No. of bitstreams: 1 Pereira_CarlosEduardoLeite_D.pdf: 2868659 bytes, checksum: 90d7fda0b66989796767d36726984af4 (MD5) Previous issue date: 2006 / Resumo: No presente trabalho, tem-se como objetivo realizar a otimizacao topológica em problemas de elasticidade envolvendo não-linearidade geométrica (grandes deslocamentos e rotações) e não-linearidade de material, no caso, hiperelasticidade não-linear quasi-incompressível, aplicando o conceito de Análise de Sensibilidade Topológica (AST) através de uma formulação Lagrangiana total. A AST é caracterizada por uma função escalar, denominada Derivada Topológica, que fornece para cada ponto do domínio de definição do problema a sensibilidade de uma determinada função quando um pequeno furo é criado no domínio. Assim, considerando a impossibilidade em se obter uma solução analítica para os problemas considerados no presente trabalho, uma expressão aproximada da Derivada Topológica é obtida através de uma análise assintótica numérica para o problema envolvendo somente não-linearidade geométrica e posteriormente para o problema envolvendo hiperelasticidade não-linear quasi-incompressível. Resultados numéricos para ambos os tipos de problema e as limitações quanto à aplicabilidade da Derivada Topológica aproximada obtida para tais problemas são apresentados / Abstract: The aim of the present work is to optimize the topology of elasticity problems with geometric nonlinearities (large displacement and rotation) and material nonlinearities, in this case, nonlinear nearly-incompressible hyperelasticity applying the concept of Topological Sensitivity Analysis (TSA) and a total Lagrangian formulation. The TSA results in a scalar function, denominated Topological Derivative, that gives for each point of the domain the sensitivity of a given cost function when a small hole is created. As an analytical solution is impossible for the considered problems in the present work, an approximated expression for the Topological Derivative is obtained by numerical asymptotic analysis first for geometric nonlinearities and after for nonlinear nearly-incompressible hyperelasticity. Numerical results for both problems and the limitations of the approximated Topological Derivative are presented. / Doutorado / Mecanica dos Sólidos e Projeto Mecanico / Doutor em Engenharia Mecânica

The uterosacral ligament (USL) and cardinal ligament (CL) are the two major suspensory tissues of the uterus, cervix, and vagina. These supportive structures can be weakened or damaged, leading to the development of pelvic floor disorders (PFDs) such as urinary incontinence, fecal incontinence, and pelvic organ prolapse. In the surgical treatment for PFDs, the USL and CL are extensively used as anchor structures to restore the normal position of the prolapsed organs. Therefore, the mechanical properties of the USL and CL may be critical for the development of new surgical reconstruction strategies for PFDs. In chapter 1, we present the first histo-mechanical characterization of the swine USL and CL using histological analysis, scanning electron microscopy and quasi-static uniaxial tensile tests. Our results suggest that the histological and uniaxial tensile properties of the swine CL and USL are very similar to those in humans. The swine is found to be a suitable animal model for studying the mechanical properties of these ligaments. To capture both the active and passive mechanical responses of biological tissues containing SMCs such as the USL and CL, a new structural constitutive model is proposed in chapter 2. The deformation of the active component in such tissues during isometric and isotonic contractions is described using an evolution law. This model is tested with published active and passive, uniaxial and biaxial, experimental data on pig arteries due to lack of data on the active properties of the USL and CL. Subjected to constant forces in-vivo, the structure and length of the USL and CL are sig- nificantly altered over time. In chapter 3, we present the first rigorous characterization of the fiber microstructure and creep properties of the USL/CL complex by using scanning electron microscopy and planar biaxial testing. Fibers are found to be oriented primarily along the main in-vivo loading direction. In such direction, the creep proceeds significantly faster under lower load. Overall, our experimental findings advance our knowledge about the passive elastic and viscoelastic properties of the USL/CL complex. The novel structural constitutive model proposed enhances our understanding of the active mechanical behavior of biological tissues containing SMCs. Knowledge about the mechanical behavior of the USL and CL from experimental and theoretical studies such as those presented here will help to improve, in the long term, the medical treatment for PFDs. / Ph. D.

<p>Crustal stress measurements play a crucial role in understanding how the subsurface deforms. As one of the most popular methods for stress characterization in deep wellbores, borehole breakout analysis examines the shape of drilling-induced compressive failures to determine stress directions and magnitudes, assuming that the rock formation is both isotropic and linearly elastic. To ensure accurate stress interpretations, the dissertation investigates the validity of underlying presumptions from two perspectives: (1) the effect of rock anisotropy (i.e., elastic anisotropy, and strength anisotropy) on wellbore failure patterns; and (2) the characterization of rock nonlinear elastic mechanical behaviors. </p> <p>The developed computer program, <em><strong>EASAfail</strong></em>, has broad applicability in calculating wellbore failure patterns for a wide range of scenarios. It takes into account factors such as elastic stiffness matrices of the rock, stress tensors in the surrounding environment, and the presence of weak planes. The program's generality allows it to handle various rock types with different degrees of symmetry in their elastic properties, as well as weak planes that are weaker than the intact rock matrix. By analyzing these factors, the program reveals that the patterns of wellbore failure in elastic and strength anisotropic rock formations are highly influenced by the sliding of weak planes. Complications from two modes of borehole failure, either in the intact rock matrix or in the weak planes, can cause the breakout azimuth to deviate from the direction of the minimum horizontal stress. </p> <p>In addition to hypothetical scenarios generated from numerical models, a case study from the field is presented to underscore the impact of foliations on the anomalous rotations of breakout azimuths. The wellbore was located in Northeastern Alberta, Canada, transecting both the sedimentary column and crystalline basement. Breakout rotations identified from caliper and image logs were highly likely caused by the slippage along foliations, supported by the close correlation between breakout azimuths and dip directions of foliations as well as polarization directions analyzed from dipole sonic logs. Stress magnitudes constrained from Monte Carlo simulations further reveal a lower stress field when rock anisotropy is taken into account, compared to what is inferred conventionally. </p> <p>The characterization of rock nonlinear elasticity involves the utilization of the third-order elastic (TOE) model. To measure the TOE moduli in a static manner, test-specific protocols were proposed based on the nonlinear stress-strain behaviors of the rock. By arranging the stress-strain responses obtained from hydrostatic, uniaxial, and triaxial compressive tests into a linear system of equations, it becomes possible to invert the equations for the TOE moduli. These analytical equations were validated through calculations from finite element models. </p> <p>By employing the established protocols, the TOE moduli were derived for four different rock types with varying pore structures when subjected to hydrostatic and uniaxial compressions. The TOE model successfully captured the nonlinear stress-strain responses exhibited by Indiana limestone, Vif-type Fontainebleau sandstone, and Snake River Plain basalt. However, it was found to be inadequate for Franc-type Fontainebleau sandstone, which displayed noticeable hysteresis and experienced significant strains. Future geomechanical applications will undoubtedly gain advantages from utilizing the inverted TOE moduli obtained through static measurements, as they allow for the examination of the impacts of nonlinear elasticity in rocks. </p>

Dans cette thèse, on s’intéresse à de nouveaux schémas de discrétisation afin de résoudre les équations couplées de la poroélasticité et nous présentons des résultats analytiques et numériques concernant des problèmes issus de la poromécanique. Nous proposons de résoudre ces problèmes en utilisant les méthodes Hybrid High-Order (HHO), une nouvelle classe de méthodes de discrétisation polyédriques d’ordre arbitraire. Cette thèse a été conjointement financée par le Bureau de Recherches Géologiques et Minières (BRGM) et le LabEx NUMEV. Le couplage entre l’écoulement souterrain et la déformation géomécanique est un sujet de recherche crucial pour les deux institutions de cofinancement. / In this manuscript we focus on novel discretization schemes for solving the coupled equations of poroelasticity and we present analytical and numerical results for poromechanics problems relevant to geoscience applications. We propose to solve these problems using Hybrid High-Order (HHO) methods, a new class of nonconforming high-order methods supporting general polyhedral meshes. This Ph.D. thesis was conjointly founded by the Bureau de recherches géologiques et minières (BRGM) and LabEx NUMEV. The coupling between subsurface flow and geomechanical deformation is a crucial research topic for both cofunding institutions.

Aplicamos a teoria de representação de funções isotrópicas para determinar o número mínimo de invariantes independentes necessários para caracterizar completamente a densidade de energia de deformação de sólido hiperelástico com duas direções de simetria material. Expressamos a densidade de energia em termos de dezoito invariantes e extraímos um conjunto de dez invariantes para analisar dois casos de simetria material. No caso de direções ortogonais, recuperamos o resultado clássico de sete invariantes e oferecemos uma justificativa para a escolha dos invariantes encontrados na literatura. Se as direções não são ortogonais, descobrimos que o número mínimo também é sete e corrigimos um erro em fórmula encontrada na literatura. Uma densidade de energia deste tipo é usada para modelar, na escala macroscópica, materiais de engenharia, tais como compósitos reforçados com fibras, e tecidos biológicos, tais como ossos. / We determine the minimum number of independent invariants that are needed to characterize completely the strain energy density of a hyperelastic solid having two distinct material symmetry directions. We use a theory of representation of isotropic functions to express this energy density in terms of eighteen invariants and extract a set of ten invariants to analyze two cases of material symmetry. In the case of orthogonal directions, we recover the classical result of seven invariants and offer a justification for the choice of invariants found in the literature. If the directions are not orthogonal, we find that the minimum number is also seven and correct a mistake in a formula found in the literature. An energy density of this type is used to model, on the macroscopic scale, engineering materials, such as fiber-reinforced composites, and biological tissues, such as bones.

Theoretical strength of solids is defined as the ultimate strength beyond which plastic deformation, fracture, or decohesion would occur. Understanding the microscopic origin from quantum mechanics and thermoelastic formulation is of great importance to mechanical properties and engineering design of various solids. While quite a few theory models have been made in the past century by several generations of scientists, including Frankel and Born, a general and convincing framework has not been fully established. We study this issue from three respects: (1) Unify various elastic stability criteria for solids that determine an upper bound of theoretical strength; (2) with ab initio method, we test the elastic stability conditions of crystal Au. The phenomenon of bifurcation is observed: under hydrostatic expansion, the rhombohedral modulus reaches zero first of all; while under uniaxial tensile stress, the tetragonal shear modulus first reaches zero; (3) propose a nonlinear theoretical formulation of stability criterion. As an analytic method, this scheme is quite simple, in the mean time, it saves computation resource.

The main focus of this work lies in the simulation of the deformation of mechanical components which consist of nonlinear elastic, incompressible material and that are subject to large deformations. Starting from a nonlinear formulation one can derive a discrete problem by using linearisation techniques and an adaptive mixed finite element method. This turns out to be a saddle point problem that can be solved via a Bramble-Pasciak conjugate gradient method. With some modifications the simulation can be improved.

The issue of establishing the status of nonlinear elasticity theory for rubber with respect to the point of view of polymer physics is at the heart of this manuscript. Our aim is to develop mathematical methods to describe, understand, and solve this multiscale problem. At the level of the polymer chains, rubber can be described as a network whose nodes represent the cross-links between the polymer chains. This network can be considered as the realization of some stochastic process. Given the free energy of the polymer network, we'd like to derive a continuum model as the characteristic length of the polymer chains vanishes. In mathematical terms, this process can be viewed as a hydrodynamic limit or as a discrete homogenization, depending on the nature of the free energy of the network. In view of the works by Treloar, by Flory, and by Rubinstein and Colby on polymer physics, and in view of the stochastic nature of the network, stochastic discrete homogenization seems to be the right tool for the analysis. Hence, in order to complete our program we need to understand the stochastic homogenization of discrete systems. Two features make the analysis rich and challenging from a mathematical perspective: the randomness and the nonlinearity of the problem. The achievement of this manuscript is twofold: - a complete and sharp quantitative theory for the approximation of homogenized coefficients in stochastic homogenization of discrete linear elliptic equations; - the first rigorous and global picture on the status of nonlinear elasticity theory with respect to polymer physics, which partially answers the question raised by Ball in his review paper on open problems in elasticity. Although the emphasis of this manuscript is put on discrete models for rubber, and more generally on the homogenization of discrete elliptic equations, we have also extended most of the results to the case of elliptic partial differential equations --- some of the results being even more striking in that case.

This works presents a study about effective properties of porous solids with nonlinear elastic and elastoplastic matrix. For macroscopic mechanics properties evaluation, micromechanics models are used with effective strain concept relative to the modified second method. The porous are assumed as randomly distributed in the matrix, which presents a constitutive law with linear behavior in dilatation and nonlinear in shear. The results are compared with those provided by finite element methods program ABAQUS, assuming porous with spherical geometry for three dimensional solids. Numerical results from ABAQUS were obtained by an implementation of an external subroutine which incorporates at analysis the nonlinear constitutive law. / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico / Este trabalho apresenta um estudo sobre as propriedades efetivas de sólidos porosos com matriz elástica não linear e elastoplástica. Na avaliação das propriedades mecânicas macroscópicas empregam-se modelos micromecânicos lineares em conjunto com o conceito de deformação efetiva correspondente ao método secante modificado. Os poros são admitidos como distribuídos randomicamente na matriz, a qual apresenta uma lei constitutiva caracterizada por um comportamento linear em dilatação e não linear em cisalhamento. Os resultados obtidos são confrontados com aqueles fornecidos pelo programa comercial de elementos finitos ABAQUS, admitindo-se que os poros exibem geometrias esféricas para sólidos tridimensionais. A geração dos resultados numéricos oriundos do programa ABAQUS foi viabilizada mediante a implementação de uma sub-rotina externa que incorpora a relação constitutiva não linear considerada nas análises.