Browsing by Subject "Continuum Mechanics"
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Item A Model for the Estimation of Residual Stresses in Soft Tissues(2012-10-19) Joshi, SunnieThis dissertation focuses on a novel approach for characterizing the mechanical behavior of an elastic body. In particular, we develop a mathematical tool for the estimation of residual stress field in an elastic body that has mechanical properties similar to that of the arterial wall, by making use of intravascular ultrasound (IVUS) imaging techniques. This study is a preliminary step towards understanding the progression of a cardiovascular disease called atherosclerosis using ultrasound technology. It is known that residual stresses play a significant role in determining the overall stress distribution in soft tissues. The main part of this work deals with developing a nonlinear inverse spectral technique that allows one to accurately compute the residual stresses in soft tissues. Unlike most conventional experimental, both in vivo and in vitro, and theoretical techniques to characterize residual stresses in soft tissues, the proposed method makes fundamental use of the finite strain non- linear response of the material to a quasi-static harmonic loading. The arterial wall is modeled as a nonlinear, isotropic, slightly compressible elastic body. A boundary value problem is formulated for the residually stressed arterial wall, the boundary of which is subjected to a constant blood pressure, and then an idealized model for the IVUS interrogation is constructed by superimposing small amplitude time harmonic infinitesimal vibrations on large deformations via an asymptotic construction of its solution. We then use a semi-inverse approach to study the model for a specific class of deformations. The analysis leads us to a system of second order differential equations with homogeneous boundary conditions of Sturm-Liouville type. By making use of the classical theory of inverse Sturm-Liouville problems, and root finding and optimization techniques, we then develop several inverse spectral algorithms to approximate the residual stress distribution in the arterial wall, given the first few eigenfrequencies of several induced blood pressures.Item Characterization of Nonlinear Material Response in the Presence of Large Uncertainties ? A Bayesian Approach(2013-12-06) Doraiswamy, SrikrishnaThe aim of the current work is to develop a Bayesian approach to model and simulate the behavior of materials with nonlinear mechanical response in the presence of significant uncertainties in the experimental data as well as the applicability of models. The core idea of this approach is to combine deterministic approaches by the use of physics based models, with ideas from Bayesian inference to account for such uncertainties. Traditionally, parameters of models in mechanics have been identified through deterministic approaches to obtain single point estimates. Such methods perform very well for linear models and are the preferred approach in identifying model parameters, especially for precisely engineered systems such as structures and machinery. But in the presence of large variations such as in the response of biological materials, such deterministic approaches do not sufficiently capture the uncertainty in the response. We propose that the model parameters need to encode the spread that is observed in the data in addition to modeling the physics of the system. To this end, we propose the idea of probability distributions for model parameters in order to incorporate the uncertainty in the data. We demonstrate this probabilistic approach to identifying model parameters with the example of two problems: the characterization of sheep arteries using data from inflation experiments and the problem of detecting an inhomogeneity in a cantilever beam. The parameters in the artery characterization problem are the model parameters in the constitutive models and in the cantilever problem the parameters are the stiffnesses of the inhomogeneity and the material of the beam. For each of these problems, we compute the probability distribution of the parameters using Bayesian inference. We show that the probability distributions of parameters can be used towards two kinds of diagnostics: assigning probability to a hypothesis (inhomogeneity detection problem) and using the probability distribution for classifying newly obtained data (characterization of artery data). For the inhomogeneity detection problem, the hypothesis is a statement on the ratio of the stiffnesses and it is observed that the probability of the hypothesis matches well with the data. In the case of the artery characterization problem, new data was successfully classified using the probability distributions computed with training data.