1 Mooney-Rivlin Bar (40 points) An incompressible Mooney-Rivlin bar of length L and cross-section D x D in the reference configuration is subjected to uniaxial tension via force on the faces at x1 = {0, L}. The bar then undergoes deformation. Problem 1.1 (15 points) Suppose that the lateral response of the material (i.e. the reduction of the cross-section) is uniform in both directions, and the force results in the extension of the bar to some length /. Defining = 1/L, write the force as a function of A and other constants. Problem 1.2 (5 points) What is the first Piola-Kirchhoff stress tensor that results? Hint: you can use the definition from the Cauchy stress tensor, or recall how the Cauchy stress is related to force and (some frame of) area. Problem 1.3 (10 points) Plot the stress P11 as a function of X over the domain 3/4 < 0 that you wish. Problem 1.4 (5 points) For the constants that you have chosen, what is the effective linear-elastic shear modulus? 2.2 Mooney-Rivlin Model Definition 2.2 The Mooney-Rivlin model suggests that the strain energy density takes on the form W[F] c(1-3)+c2(12-3), (2.10) where 12 = (tr[C]-tr[C)) = (tr[B] - tr[B2]), and, in addition to c1, c2 is a parameter of calibration. Again, for an incompressible deformation, the Cauchy stress can be found as =-p8+cB-C2B-. (2.11) For simplicity, we will take c3 = 0. From this W, we can compute the first Piola-Kirchhoff stress tensor with a little algebra as follows: c [tr[FTF] aw P 2 F 12/3 +c2(-1) = OF = (1/3 2F [CU 3/31 F) + (2011)JF-T F =C1 - +c2(-1)JF-T (Example 2.3-1)