Problem 5, 21pts

Given the following motion in rectangular Cartesian coordinates:

X(X, t) = 3X3e₁-X₁е₂ - 2X₂03

(a) Determine the deformation gradient F, right Cauchy-Green tensor C, the right stretch

tensor U, and the stretch invariants I₁, I2,

(b) Determine the Green Lagrange strain tensor E* = (C-1).

(c) Determine the ratio of deformed volume to initial volume J./nProblem 6; 20 pts

A circular cylindrical bar of length / hangs vertically under gravity force from the ceiling.

Let the axis coincide with the axis of the bar and point downward, and let the point

(x1, x2, 3) = (0,0,0) be fixed at the ceiling.

(a) Verify that the following stress field satisfies the equation of equilibrium in the presence

of the gravity force: T₁1=pg(1-₁), all other T₁ = 0.

3

(b) Verify that the boundary conditions of zero surface traction on the lateral surface and

the lower end face are satisfied.

(c) Obtain the resultant force of the surface traction at the upper end face.