Page 181 - Introduction to Continuum Mechanics

166 Problems

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333. Repeat Problem 3.32 except that a = b = c= 1000 x 10~ .
3.34. The unit elongations at a certain point on the surface of a body are measured experimen-
tally by means of strain gages that are arranged 60° apart (called the 60° strain rosette ) in
1 1
the directions ej, -(ej + V3~e 2), and -~(-*i + vTe 2). If these elongations are designated by
jLt £
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a,b,c respectively, what are the strain components E-^^lf ^-
f. f.
3.35. Do Problem 3.34 if the strain rosette measurements give a = 2x 10 , £ = 1x10 ,
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c= 1.5X10" .
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3.36. Do Problem 3.35 except that a = b=c = 2000 x 10~ .
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3.37. For the velocity field, v = (kxfai
(a) Find the rate of deformation and spin tensors.
(b) Find the rate of extensions of a material element dx = (ds)n where

3.38. For the velocity field

find the rates of extension for the following material elements: chc- ' - ds^i and
dx^ = (ds2/^2}(*i + e 2) at the origin at time t = 1.
3.39. (a) Find the rate of deformation and spin tensors for the velocity field
v = (cos t) (sin 7a 1 )e 2 .
(b) For the velocity field of part (a), find the rates of extension of the elements
(2)
(1)
Jx = (dsi)e lt t/x = (ds 2)«2. ^ x(3) = ^3/v^(e! + e 2) at the origin at t = 0.
3.40. Show that the following velocity components correspond to a rigid body motion:

3.41. For the velocity field of Prob.3.15

(a) Find the rate of deformation and spin tensors.
(b) Find the rate of extension of a radial material line element.
3.42. Given the two-dimensional velocity field in cylindrical coordinates

(a) Find the acceleration at r = 2.

(b) Find the rate of deformation tensor at r = 2.