CONTINUUM MECHANICS for ENGINEERS

(c) Calculate the logarithmic stretching, for the
element in the current configuration in the direction of
at x = 0.
Answer: (b) a1 = –ω v2, a2 = ω v1, a3 = 0
˙ Λ/ Λ d lnΛ
/ dt
nˆ = ( eˆ + eˆ
1 3) / 2
(c) e –ct cos ω t
1
2
4.43 Show that the velocity field
v 1 = 1.5x 3 – 3x 2, v 2 = 3x 1 – x 3, v 3 = x 2 – 1.5x 1
corresponds to a rigid body rotation, and determine the axis of spin
(the vorticity vector).
Answer: w = eˆ . ˆ ˆ
1+ 15e2 + 3e3
4.44 For the steady velocity field
2
2
v1 = xx,
v2 = 2x x , v3 = 3x1x2x3 determine the rate of extension at (2, 0, 1) in the direction of the unit
vector 4eˆ − 3eˆ / 5.
( )
1 2
Answer: ˙ 48
Λ/ Λ = −
25
4.45 Prove that d( ln J ) dt=
div v and, in particular, verify that this relationship
is satisfied for the motion
x 1 = X 1 + ktX 3, x 2 = X 2 + ktX 3, x 3 = X 3 – kt(X 1 + X 2)
where k is a constant.
Answer: = div v = 4k2t/(1 + 2k2t2 )
4.46 Equation 4.10-19 gives the material derivative of dx2 in terms of Dij .
Using that equation as the starting point, show that d2 (dx2 )/dt2 is given
in terms of Dij and its time derivative by
˙ J/J
d2 (dx2 )/dt2 = 2( + vk,i Dkj + vk,j Dik)dxidxj ˙D ij
4.47 A continuum body in the form of the unit cube shown by the sketch
undergoes the homogeneous deformation
x 1 = λ 1X 1, x 2 = λ 2X 2, x 3 = λ 3X 3
where λ 1, λ 2, and λ 3 are constants.
1
2
2
3
= ( )