Page 182 - Introduction to Continuum Mechanics

Kinematics of a Continuum 167

3.43. Given the velocity field in spherical coordinates

(a) Find the acceleration field.
(b) Find the rate of deformation field.
3.44. A motion is said to be irrotational if the spin tensor vanishes. Show that the velocity
field of Prob.3.16 describes an irrotational motion.
m
3.45. (a) Let die- ' = (dsi)n, and dx^ ' = (<&2) be two material elements that emanate from
2
a particle P which at present has a rate of deformation D. Consider (D/Dt)(dx^ - tbc- ') and
show that

where 0 is the angle between m and n.

(b) Consider the special cases (i) cbr' = dr' and (ii) 6 = jt/2. Show that the above expression
reduces to the results of Section 3.13.

3.46. Let ci, 62,63 and D t, D^, #3 be the principal directions and values of the rate of
deformation tensor D. Further, let

be three material line elements. Consider the material derivative
2)
(3)
(D/Df)[cbf^' (dx< x</x )] and show that

V* r j_^f>
where the infinitesimal volume dV = (dsi)(ds2)(ds^

3.47. Consider a material element dx = dsn
(a) Show that

(b) Show that if n is an eigenvector of D then

where «o is the axial vector of W.

3.48. Given the following velocity field