Kinematics of Deformation and Motion

Question 1

Vector function chi

Answer 1

x = chi(X,t)

Question 2

Green’s deformation tensor

Answer 2

C = F^t F
C_IJ = F_kI F_kJ

Question 3

Lagrangian Green Strain

Answer 3

2E = F^t F - I

Question 4

Cauchy deformation tensor

Answer 4

c = F^-T F^-1

Question 5

Eulerian finite strain tensor

Answer 5

2e = (I - c)

Question 6

Differential equation of motion

Answer 6

Sigma_ij,j + bi = rho*u(doubledot)_i

Question 7

How is a skew tensor defined?

Answer 7

If W = -W^T then there exists a unique vector w such that W.v = w x v for any v.

Question 8

How is the adjucate defined?

Answer 8

The adjucate of A, A, can be defined as:
A.(u x v) = (A.u) x (A.v) for all u and v

Final proof in index notation is:
Aib = 0.5e_ipqe_lmbA_pl*A_qm

Question 9

What is det(F) equal to in terms of volume?

Answer 9

det(F) = dv/dV

This is why det(F) > 0

Question 10

Eulerian Strain

Answer 10

n^T[0.5*(I-F^-TF^-1)]n