This project addresses characterization of the material properties of aortic tissue. Understanding of these properties is important for a variety of studies including tissue engineering, effects of aging and diseases, stents engineering, and traumatic aorta rupture. The goal of the presented research was to characterize the stress-strain relationship of aorta in dynamic oscillatory biaxial loading. A setup was developed that supplied pressure loading from the physiological to sub-failure levels (between 7 and 76 kPa) to porcine aorta at frequencies ranging from 0.50Hz to 5.00Hz. Samples tested were constrained at both ends while the deformation and the pressure were recorded. Volumetric strain versus pressure was used to characterize the structural behavior of the material which showed frequency dependency and hysteresis indicating viscoelastic response. An offset method was developed to account for drifting behavior exhibited by some of the samples. The structural behavior of aorta was modeled using a quasi-linear viscoelastic (QLV) creep theory. The QLV model included a logarithmic steady state elastic function v = 0.663 +/- 0.040 + 0.241 +/- 0.011 ln(P) for pressure in kPa, and a Prony series creep function ( J0 = 0.472 +/- 0.021, J2 = 0.109 +/- 0.060, J3 = 0.419 +/- 0.056). Modeling results were then used to determine the relationships between the circumferential and longitudinal stresses and strains of the material. The results exhibited that the stress in the transverse direction was about 1.5 times larger than in the axial direction. However, in the axial direction material was stiffer and the deformation was 30% less. The relaxation function of the material was determined by linearizing the non-linear component of the QLV model and applying to it the linear viscoelastic theory. Furthermore, literature comparison revealed that aorta's creep function, as well as its elastic modulus, is within the range of what has been reported in the literature. In conclusion, an experimental model was developed that can be used to predict the behavior of porcine aorta under physiological and sub-failure conditions at quasi-static and dynamic loading. / Mechanical Engineering

Several efforts have been made in the past half century to characterize the behavior of brain tissue under different modes of loading and deformation rates; however each developed model has been associated with limitations. This dissertation aims at addressing the non-linear and rate dependent behavior of brain tissue specially in high strain rates (above 100 s-1) that represents the loading conditions occurring in blast induced neurotrauma (BINT) and development of a universal constitutive model for brain tissue that describes the tissue mechanical behavior from medium to high loading rates.. In order to evaluate the nature of nonlinearity of brain tissue, bovine brain samples (n=30) were tested under shear stress-relaxation loading with medium strain rate of 10 s-1 at strain levels ranging from 2% to 40% and the isochronous stress strain curves at 0,1 s and 10 s after the peak force formed. This approach enabled verification of the applicability of the quasilinear viscoelastic (QLV) theory to brain tissue and derivation of its elastic function based on the physics of the material rather than relying solely on curve fitting. The results confirmed that the QLV theory is an acceptable approximation for engineering shear strain levels below 40% that is beyond the level of axonal injury and the shape of the instantaneous elastic response was determined to be a 5th order odd polynomial with instantaneous linear shear modulus of 3.48±0.18 kPa. To investigate the rate dependent behavior of brain tissue at high strain rates, a novel experimental setup was developed and bovine brain samples (n=25) were tested at strain rates of 90, 120, 500, 600 and 800 s-1 and the resulting deformation and shear force were recorded. The stress-strain relationships showed significant rate dependency at high rates and was characterized using a QLV model with a 739 s-1 decay rate and validated with finite element analysis. The results showed the brain instantaneous elastic response can be modeled with a 3rd order odd polynomial and the instantaneous linear shear modulus was 19.2±1.1 kPa. A universal constitutive model was developed by combining the models developed for medium and high rate deformations and based on the QLV theory, in which the relaxation function has 5 time constants for 5 orders of magnitude in time (from 1 ms to 10 s) and therefore, is capable of predicting the brain tissue behavior in a wide range of deformation rates. Although the universal model presented in this study was developed based on only shear tests and the material parameters could not be found uniquely, by comparing the results of this study with previously available data in the literature under tension unique material parameters were determined for a 5 parameter generalized Rivlin elastic function (C10=3.208±0.602 kPa, C01=4.191±1.074 kPa, C11=79.898±18.974 kPa, C20=-37.093±7.273 kPa, C02=-37.712±5.678 kPa). The universal constitutive model for brain tissue presented in this dissertation is capable of characterizing the brain tissue behavior under large deformation in a wide range of strain rates and can be used in computational modeling of Traumatic Brain Injury (TBI) to predict injuries that result from falls and sports to automotive accidents and BINT. / Mechanical Engineering

This thesis is in the field of non-linear partial differential equations (PDE), focusing on problems which show some type of phase-transition. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by a semi-linear PDE. This is a problem within the field of free boundary problems. In the multi-phase problem we consider the time-evolution of a system of phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation which is regularized and penalized. The penalization is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. The main result in the first paper is the existence theory of a weak solution to the parameterized equations in a Bochner space using the implicit function theorem. The family of solutions to the parameterized problem is uniformly bounded allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity. The second problem deals with a parameterized highly oscillatory quasi-linear elliptic equation in divergence form. As the regularization parameter tends to zero the equation gets a jump in the conductivity which occur at the level set of a locally periodic function, the obstacle. As the oscillations in the problem data increases the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. We show that the parameterized equation has a weak solution in a Sobolev space and derive bounds on the solutions used in the analysis for the case when the regularization is lost. Surprisingly, the limiting problem in this case includes an extra term describing the interaction between the solution and the obstacle, not appearing in the case when obstacle is the zero level-set. The oscillatory nature of the problem makes standard numerical algorithms computationally expensive, since the global domain needs to be resolved on the micro scale. We develop a multi scale method for this problem based on the heterogeneous multiscale method (HMM) framework and using a finite element (FE) approach to capture the macroscopic variations of the solutions at a significantly lower cost. We numerically investigate the effect of the obstacle on the homogenized solution, finding empirical proof that certain choices of obstacles make the limiting problem have a form structurally different from that of the parameterized problem. / <p>QC 20180222</p>

FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / We will study a new universal gradient continuity estimate for solutions to quasi-linear equations with varying coefficients at singular set of degeneracy: S(u) := {X : Du(X) = 0}. Ourmain theorem reveals that along S(u), u is asymptotic as regular as solutions to constant coefficient equations. In particular, along the critical set S(u),u enjoys a modulus of continuity much superior than the possibly low, continuity feature of the coefficients. The results are new even in the context of linear elliptic equations, where it is herein shown that H^1- weak solutions to div (a(X,Du))= 0 with aij elliptic and dinicontinuous are actually C ^{1,1^{-}} along S(u). The results and insights of this work foster a new understanding os smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse. / Neste trabalho estudaremos uma nova estimativa universal para a continuidade do gradiente de soluÃÃes para equaÃÃes quase lineares com coeficientes variÃveis em conjuntos singulares degenerados que serÃo denotados por S(u) := {X : Du(X) = 0} . O resultado principal deste trabalho revela que ao longo de S(u), u à assintoticamente tÃo regular quanto as soluÃÃes das equaÃÃes com coeficientes constantes. Em particular, ao longo do conjunto S(u), Du tem um mÃdulo de continuidade superior a baixa caracterÃstica de continuidade de seus coeficientes. Os resultados sÃo novos e mesmo no contexto de equaÃÃes diferenciais lineares onde se mostra que soluÃÃes H^1- fracas da equaÃÃo div(a(X, Du)) = 0 com os aij elÃpicos e Dini-ContÃnuos sÃo realmente C ^{1,1^{-}} ao longo de S(u). Os resultados e as perspectivas deste trabalho promovem um novo entendimento sobre as propriedades suavidade de soluÃÃes para equaÃÃes singulares, ou degeneradas, alÃm de estimativas tÃpicas sobre regularidade elÃpticas, precisamente onde temos os atributos de difusÃo do equaÃÃo do colapso.

We approximate the solution of a quasilinear stochastic partial differential equa- tion driven by fractional Brownian motion B_H(t); H in (0,1), which was calculated via fractional White Noise calculus, see [5].

No description available.

Submitted by Bruna Rodrigues (bruna92rodrigues@yahoo.com.br) on 2016-09-27T12:32:15Z No. of bitstreams: 1 TeseMAN.pdf: 2601601 bytes, checksum: 70c6b910d382e2015025a5c8ec5ddd14 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-04T18:11:16Z (GMT) No. of bitstreams: 1 TeseMAN.pdf: 2601601 bytes, checksum: 70c6b910d382e2015025a5c8ec5ddd14 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-04T18:11:29Z (GMT) No. of bitstreams: 1 TeseMAN.pdf: 2601601 bytes, checksum: 70c6b910d382e2015025a5c8ec5ddd14 (MD5) / Made available in DSpace on 2016-10-04T18:11:38Z (GMT). No. of bitstreams: 1 TeseMAN.pdf: 2601601 bytes, checksum: 70c6b910d382e2015025a5c8ec5ddd14 (MD5) Previous issue date: 2015-12-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / We present results of Ambrosseti-Prodi type to quasilinear problems involving the p-Laplace operator. We consider the scalar case and a a problem with systems of equations. In the scalar case, we work with the conditions of Neumann and Dirichlet. In the problem involving system, we consider the condition og Dirichlet. In order to get the results we use the theory of Leray-Schauder degree and a priori estimates. / Neste trabalho apresentamos resultados do tipo Ambrosseti-Prodi para problemas quasilineares envolvendo o aperador p-Laplaciano. Considerando o caso escalar eu um problema com sistemas de equações. Para os casos escalares, trabalhamos com a condições de Neumann e Dirichlet, já para o problema envolvendo sistema, consideramos a condição Dirichlet. Para obter mais resultados usamos a teoria do grau de Leray-Schauder e estimativas a priori.

Made available in DSpace on 2015-05-15T11:46:23Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1404779 bytes, checksum: fa23dbf1324f104548ef91fcbbf20fba (MD5) Previous issue date: 2014-09-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study questions related to the existence of positive solutions for some classes of quasilinear Schrödinger equations, with hypotheses on the potential that permit this potential to vanish at infinity. In order to use variational methods to obtain our results, we make some changes of variables to obtain some semilinear equations, whose associated functionals are well defined in a classical Sobolev spaces. We also work with these equations on an Orlicz type space whose energy functional satisfy the geometric properties of the Mountain Pass Theorem. We still use the penalty technique due to Del Pino and Felmer and the Moser iteration method to obtain estimates in L1 norm, which are fundamental to our study. / Neste trabalho, estudamos questões relacionadas à existência de soluções positivas para algumas classes de equações de Schrödinger quasilineares, com hipóteses sobre o potencial que o possibilita se anular no infinito. Afim de usarmos métodos variacionais na obtenção de nossos resultados, aplicamos mudança de variáveis para reduzirmos as equações quasilineares a equações semilineares. Os funcionais associados a essas novas equações estão bem definidos em espaços de Sobolev clássicos e em espaços tipo Orlicz e satisfazem as propriedades geométricas do Teorema do Passo da Montanha. Ainda utilizamos a técnica de penalização de Del Pino e Felmer e o método de iteração de Moser para obtenção de estimativas, fundamentais para o nosso estudo, na norma L1.

Le travail dans cette thèse a consisté à l'étude de la propagation des ondes, en particulier la considération d'un problème de cloaking d'une part et d'un problème inverse d’identification de fissure d'autre part. Nous nous intéressons particulièrement à appliquer une stratégie qui est basé sur un changement de variable pour rendre un objet invisible, la validation numérique des résultats de ce problème a été réalisée par la librairie éléments finies XLiFE++. L'analyse de différents aspects mathématiques du problème de cloaking pour une équation elliptique non linéaire a fait l'objet du chapitre deux. La détermination de l'opérateur Dirichlet-Neumann associé à l'opérateur quasi-linéaire nous a permis d'adapter la technique de transformation utilisé pour le cadre des équations différentielles elliptiques linéaire afin de définir la notion de cloaking pour un problème non linéaire. Pour la dernière partie nous nous sommes intéressés à la reconstruction de fissures pour un problème thermique, pour cela un lien entre l'écart à la réciprocité et la transformée de Fourier du saut de la température à travers la fissure a été établi, ce qui nous a amené à développer un algorithme rapide pour la résolution numérique. / We are concerned with the study of the propagation of waves, in particular the consideration of a cloaking problem on the one hand and of a problem of cracks reconstruction on the second hand. We focus more particularly in applying a strategy that is based on a change of variable to cloak an object. The validation with numerical results has been achieved by the nite element library XLiFE++. The analysis of different mathematical aspects of the cloaking problem for a quasilinear elliptic equation has been the subject of chapter two. The determination of the Dirichlet-Neumann operator associated with the quasilinear operator allowed us to adapt the transformation technique used for the frame-work of linear elliptic differential equations to define the notion of cloaking for our nonlinear problem. For the last part we are interested in crack reconstruction for a thermal problem. For that, a link between the reciprocity gap and the Fourier transform of the temperature jump through the cracks was established, which has led to the development of a fast algorithm for numerical resolution.

Nous considérons dans cette thèse le problème de Cauchy pour des systèmes d'EDP quasilinéaires, du premier ordre. Dans le cas initialement elliptique, c'est-à-dire un spectre non-réel pour le symbole principal du système à t=0, nous prouvons un résultat d'instabilité au sens d'Hadamard. La preuve est basée sur la construction d'une famille de solutions présentant une croissance exponentielle en temps et fréquence. Cette famille invalide la régularité Hölder du flot, partant d'espaces de Gevrey vers L². Nous prouvons un résultat analogue pour différents cas de transition de l'hyperbolique vers l'elliptique, avec une restriction possible sur l'indice Gevrey pour lequel l'instabilité est observée. Dans un second temps, nous considérons le cas faiblement hyperbolique et semilinéaire. Grâce à des estimations d'énergie dans les espaces de Gevrey et à la construction d'un symétriseur adapté, nous prouvons le caractère localement bien-posé pour un tel système. Pour ce faire, nous utilisons et démontrons aussi un résultat d'action d'opérateurs pseudo-différentiels dont le symbole possède une régularité Gevrey dans la variable d'espace. / We consider the Cauchy problem for first-order, quasilinear systems of PDEs. In the initially elliptic case, that is when the principal symbol of the system has nonreal spectrum at time t=0, we prove an instability result in the sense of Hadamard. The proof is based on the construction of a family of exact solutions which exhib an exponential growth, both in time and frequency. That family leads to a defect of Hölder regularity of the flow, starting from evrey spaces to L² space. We prove analogous results for some cases of transition from hyperbolicity to ellipticity, with a potential restriction on the Gevrey index for which we may observe the instability. In a second time, we consider weakly hyperbolic systems. Thanks to an energy estimate in Gevrey spaces and the construction of a suitable symetriser, we prove local well-posedness for such a system. In doing so we use and prove a result on actions of pseudo-differential operators whose symbols have Gevrey regularity in the spatial variable