2. As derived in class, the first Piola-Kirchhoff (engineering) stress-stretch relationship ((To)11 vs. A) for an...

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2. As derived in class, the first Piola-Kirchhoff (engineering) stress-stretch relationship ((To)11 vs. A) for an incompressible, isotropic, hyper-elastic material in uniaxial tension is given by: (To) 11 == . 1 aw 2(x-1-) (+12) with all other stress components vanishing. Show that for the special case of the Mooney-Rivlin strain energy function, the engineering stress-stretch relationship in uniaxial tension is: (To) 11 = 2 (A - A) (c + A). 3. Using your result from Problem 2: C1 (a) Unconstrained optimization. Develop a Matlab script using the built-in nonlinear least- squares curve-fitting routine lsqnonlin to calibrate the Mooney-Rivlin parameters c and c to C2 the uniaxial tension data of Treloar, provided in an Excel spreadsheet. Report your optimized values for c and c. Is your result sensitive to initial guess? (b) Verify that consistency with linear elasticity (i.e., the consistency condition) requires that the Mooney-Rivlin parameters satisfy 2(c+2) = , where is the shear modulus at small strains. (c) Constrained optimization: consistency. Augment your curve-fitting script from part (a) with a second, specialized script that imposes the consistency condition 2(c + c) = , so that only c or c is a free parameter in the optimization. (For the vulcanized rubber of Treloar, use = 0.5673 MPa.) Report your optimized values for c and c. (d) Constrained optimization: consistency and ellipticity. It can be shown that enforcing c0 and 20 is a sufficient condition for strong ellipticity of the Mooney-Rivlin strain energy function. Modify your curve-fitting script from part (c) to create a third, specialized script that also enforces c 0 and c 0, so that both consistency and ellipticity are satisfied. Report your optimized values for c and c. (e) On a single figure (with engineering stress as the y-axis and stretch as the x-axis), plot the uniaxial tension data of Treloar, the "unconstrained" Mooney-Rivlin fit from part (a), the "constrained" Mooney-Rivlin fit from part (c), and the "constrained" Mooney-Rivlin fit from part (d). Calculate the root mean square error for fits (a)-(c), and comment on the quality/agreement of each fit. 2. As derived in class, the first Piola-Kirchhoff (engineering) stress-stretch relationship ((To)11 vs. A) for an incompressible, isotropic, hyper-elastic material in uniaxial tension is given by: (To) 11 == . 1 aw 2(x-1-) (+12) with all other stress components vanishing. Show that for the special case of the Mooney-Rivlin strain energy function, the engineering stress-stretch relationship in uniaxial tension is: (To) 11 = 2 (A - A) (c + A). 3. Using your result from Problem 2: C1 (a) Unconstrained optimization. Develop a Matlab script using the built-in nonlinear least- squares curve-fitting routine lsqnonlin to calibrate the Mooney-Rivlin parameters c and c to C2 the uniaxial tension data of Treloar, provided in an Excel spreadsheet. Report your optimized values for c and c. Is your result sensitive to initial guess? (b) Verify that consistency with linear elasticity (i.e., the consistency condition) requires that the Mooney-Rivlin parameters satisfy 2(c+2) = , where is the shear modulus at small strains. (c) Constrained optimization: consistency. Augment your curve-fitting script from part (a) with a second, specialized script that imposes the consistency condition 2(c + c) = , so that only c or c is a free parameter in the optimization. (For the vulcanized rubber of Treloar, use = 0.5673 MPa.) Report your optimized values for c and c. (d) Constrained optimization: consistency and ellipticity. It can be shown that enforcing c0 and 20 is a sufficient condition for strong ellipticity of the Mooney-Rivlin strain energy function. Modify your curve-fitting script from part (c) to create a third, specialized script that also enforces c 0 and c 0, so that both consistency and ellipticity are satisfied. Report your optimized values for c and c. (e) On a single figure (with engineering stress as the y-axis and stretch as the x-axis), plot the uniaxial tension data of Treloar, the "unconstrained" Mooney-Rivlin fit from part (a), the "constrained" Mooney-Rivlin fit from part (c), and the "constrained" Mooney-Rivlin fit from part (d). Calculate the root mean square error for fits (a)-(c), and comment on the quality/agreement of each fit.