Question: Quantifying the complex mechanical properties of biological tissues is an important but difficult…

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Quantifying the complex mechanical properties of biological tissues is an important but difficult task. Often, linearly elastic properties are not sufficient to describe behavior Consider, for example, the stress-strain relations of blood vessels. They have been intensive studied to better understand the nature of vascular disease, e.g atherosclerosis. In 1995, Xie, et al. published a study on the stress-stretch relations in the circumferential direction in blood vessel walls (J. Xie, J. Zhou and Y.C. Fung, 1995, Journal of Biomedical Engineering Vol.117 pp.136-145) They found following key characteristics First, they treated the blood vessel wall as a linearly elastic material by using a linear stress stretch relationship to fit experimental data, that is: Cauchy stress. S-E, (Î»-1 where EÂ° and Î» represent the Young's modulus and tissue stretch, respectively Obviously, fitting nonlinear data with a linear equation does not achieve a good fit. But, they managed to obtain a Young's modulus for the ascending aorta (Eo 2.049×105 Pa) and descending aorta (Eu = 1.261x 105 Pa) Next, they attempted to fit data with a nonlinear stress-stretch relation known as a "power law", that is: Lagrangian stress: T-k(1) where stretch was represented by A. Tems k and y represent nonlinear material properties in a circumferential strips of ascending aorta and were found to be y0.777 k 0.796x105Pa Based on these findings, answer following questions by assuming the blood vessel to be incompressible material (volume is conserved throughout the deformation, i.e circumferential stretch Î» must equal the deformed area of the tissue sample (A) divided by the non-deformed area (a) A, a) Convert the given Lagrangian stress-stretch relation into a Cauchy stress-stretch relation and into a Kirchhoff stress-stretch relation. Make a table of the various stress values for 1.1, 1.2, and 1.3. Briefly comment on the observed differences b) Derive stretch as a function of Green strain, and as a function of Almansi strain. Use these relations in the expressions from Part a) to derive expressions for Cauchy stress as a function of Almansi strain and Kirchhoff stress as a function of Green strain. Plot (software suggested) Cauchy stress vs. Almansi strain and Kirchhoff stress versus Green strain for the ascending aorta in one figure. Assume stretch A varies from 1.0 up to 1.3. Briefly comment on observed differences