Hydrodynamics and Elasticity: Class 11
Deformation and strain 1. Consider the deformation
x = X + X1ke1. (1)
Let
dX1= ds1
√2
1 1
and dX2= ds2
√2
−1 1
(2)
Find dx1and dx2(the images of the elements dX1and dX2under this deformation, their relative stretch (ds1− dS1
dS1 ) and the change of the angle between them. Find the last two quantities:
(a) exactly,
(b) using the deformation tensor.
2. Consider a cylindrical rod of radius R and its axis parallel to X3 in Cartesian coordinates. The rod is deforming according to the following relation
x1= X1− α(t)X2X3, (3)
x2= X2+ α(t)X1X3, (4)
x3= X3. (5)
(a) Find at time t the position of particles that at t = 0 constituted (i) The cross-section of the rod with constant X3; (b) a section along the cross-section radius; (iii) a section on its surface parallel to the axis of the cylinder.
(b) Find the Eulerian deformation field u(x, t).
(c) Find the deformation tensor.