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Hydrodynamics and Elasticity: Class 11

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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.

Cytaty

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