= where is the stretch ratio in the x1-direction, A2 is the stretch ratio in the...

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= where is the stretch ratio in the x1-direction, A2 is the stretch ratio in the x2-direction, and 3 is the stretch ratio in the x3-direction. Many polymer materials deform elasti- cally such that the object's volume is conserved, which is represented by the condition A1A231, which is referred to as an isochoric deformation. Assume that a prescribed value of is induced. Calculate the principle values of Nominal Strain and the principle values of Lagrangian Strain, and determine numerical values of the strains for = 1.001 and also for = 3. Discuss the validity of the two strain measures for those stretch ratios. (3) Suppose we are looking for the extremals of the integral I = [ [r + (r') ] do subject to r = r(0). Derive the corresponding Euler-Lagrange equation. Use the change in variable x = r cos 0 and y = r sin 0 to show the solution of the Euler-Lagrange equation is a straight line. (4) A material point, M, at position (X1, X2, X3) in the reference configuration of a solid body translates to a position denoted as (x1, x2, x3) in the deformed configuration. Both positions are measured relative to a global Cartesian coordinate system with mutually orthogonal x-direction, x2-direction, and x3-direction. An infinitesimal sphere around M in the reference state deforms into an ellipsoid in a deformed state. Consider a uniaxial deformation state in which the solid is stretched homogeneously in the x-direction, as described by the relationship x1(X1, X2, X3) = 1X1 x2(X1, X2, X3) = 2X2 x3(X1, X2, X3) = 3X3 1 = where is the stretch ratio in the x1-direction, A2 is the stretch ratio in the x2-direction, and 3 is the stretch ratio in the x3-direction. Many polymer materials deform elasti- cally such that the object's volume is conserved, which is represented by the condition A1A231, which is referred to as an isochoric deformation. Assume that a prescribed value of is induced. Calculate the principle values of Nominal Strain and the principle values of Lagrangian Strain, and determine numerical values of the strains for = 1.001 and also for = 3. Discuss the validity of the two strain measures for those stretch ratios. (3) Suppose we are looking for the extremals of the integral I = [ [r + (r') ] do subject to r = r(0). Derive the corresponding Euler-Lagrange equation. Use the change in variable x = r cos 0 and y = r sin 0 to show the solution of the Euler-Lagrange equation is a straight line. (4) A material point, M, at position (X1, X2, X3) in the reference configuration of a solid body translates to a position denoted as (x1, x2, x3) in the deformed configuration. Both positions are measured relative to a global Cartesian coordinate system with mutually orthogonal x-direction, x2-direction, and x3-direction. An infinitesimal sphere around M in the reference state deforms into an ellipsoid in a deformed state. Consider a uniaxial deformation state in which the solid is stretched homogeneously in the x-direction, as described by the relationship x1(X1, X2, X3) = 1X1 x2(X1, X2, X3) = 2X2 x3(X1, X2, X3) = 3X3 1