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Analysis of the Nonlinear Parameter in Inverse Problems of Elasticity
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The mechanical properties of tissues are important indicators of tissue “health”. It has been recorded and acknowledged that diseased tissues due to cancer or other conditions tend to stiffen with increase in strain, exhibiting a nonlinear stress-strain behavior. In literature, hyperelastic models, such as Veronda-Westmann and Blatz, have been widely used to model soft tissues. These models are characterized by an exponential function and two material parameters, namely the shear modulus μ and a nonlinearity parameter γ. A variety of methods and techniques have been developed to solve inverse problems in elasticity to determine these properties given the mechanical response of the tissues. Reconstruction of the nonlinear parameter using noisy measured displacement data is a difficult problem, and obtaining a well-posed solution is a challenge. This thesis is directed towards the improvement in the reconstruction of the nonlinear parameter, γ, by introducing a new parameter, which is a combination of γ and the first invariant of the Green deformation tensor. Comparative study is carried out between reconstructions of γ directly from previously existing formulations and the reconstruction of γ from the new parameter, for 2D problems. Numerical experiments are conducted and the performance is tested and compared based on different criteria like shape of the stiff regions (representing the diseased tissue), the contrast in γ and robustness for different loading conditions. Different arrangements and sizes of stiff inclusions are tested and critically analyzed. It is found that obtaining the distribution of γ from the new parameter results in a much better reconstruction than by directly optimizing for γ.
Lalitha Sridhar, Shankar (2016). Analysis of the Nonlinear Parameter in Inverse Problems of Elasticity. Master's thesis, Texas A & M University. Available electronically from