1. Suppose that a deformation of the unit cube Ω = (0, 1)3 is given by
x(X, t) = ((1 + t)X1 +(t2 - t3)X2 +2 tX3, (1 - t)X2, (t - t2)X2 + X3)T.
(a) Find the deformation gradient F and its determinant J.
(b) For what values of t is x invertible?
(c) Evaluate D2/Dt2 ∫Ωt f(x)dv, where Ωt is the volume Ω under the deformation x, and f(x)= x1 x2 - x23.
2. Find the polar decomposition F = VR where R is a rotation and V is symmetric, for
[Hint: First calculate B = FFT.] Note that is perfectly OK to do this calculation in a programming environment such as Maple or Mathematica, provided that you work through all the intermediate steps (i.e. don't just use a command to find the "polar decomposition" or "square root" of a matrix), and attach a printout to your solutions. If you do use Maple, then the command with (LinearAlgebra); may be useful.
3. Suppose that a deformation is measured by one observer as x(X, t) and by another as x* (X, t),
where x*(X, t)= Q(t) [x(X, t) - c(t)] and Q(t) is an orthogonal matrix.
(a) Show that F* = QF.
(b) Evaluate D/Dt(F*) in terms of F* and L*.
(You may assume that L* = QLQT + Q?QT and that F? = LF.)
4. (a) Let x be an invertible deformation of a body Ω which is initially in the reference configuration
(i.e. x(X, 0) = X). It is said to be rigid if it satisfies
D/Dt |x (X, t) - x(Y, t)| =0 for all t> 0 and all X, Y ∈ Ω.
Show that the following conditions are equivalent:
(i) x is a rigid deformation;
(ii) |x(X, t) - x(Y, t)| = |X - Y | for all X, Y ∈ Ω and t > 0;
(iii) the deformation gradient F(X, t) is independent of X and is a rotation for each t.
(b) Show that D/Dt(FT F) = FT (L + LT )F, where L is the velocity gradient tensor. Hence show that if x is rigid then L is skew-symmetric.
5. The unit cube (0, 1)3 consists of an incompressible hyperelastic material whose Piola-Kirchhoff stress is
TR(F)= -pF-T + αF + βBF, where B = FFT , p is the hydrostatic pressure and # and $ are positive constants.
(a) Calculate TR for the homogeneous deformation
x(X, t)=(v1X1, v2X2, v3X3)T. (1)
(Note that in this case p is a constant.)
(b) Suppose that the deformation (1) is maintained by applying equal and opposite surface stresses to two faces of the cube so that the Piola-Kirchhoff stress vector is sR = ± τe1 on the sides with outward normal's ±e1 and that sR = 0 on the sides normal to e2 and e3. Show that:
(i) this deformation is not possible unless v2 = v3;
(ii) if v2 = v3,then(1) is a possible deformation for any value of τ ∈ R;
(iii) for each τ ∈ R there is exactly one deformation of this type.
(iv) Use a suitable algebraic manipulation package to calculate the area of the face with normal e1 under this deformation when τ = 10 and α = β = 1. (Give your answer to 4 significant figures.)