Page 186 - Introduction to Continuum Mechanics

Kinematics of a Continuum 171

(2)
(1)
(2)
(1)
3.67. Let dX = rf5 tN and dX = d$ 2N be two material elements at a point P. Show
that if 0 denotes the angle between their respective deformed elements dx^ and dx- \ then

3.68. Given the following right Cauchy-Green deformation tensor at a point

(a) Find the stretch for the material elements which were in the direction of e ls e 2 and e 3
directions.

(b) Find the stretch for the material element which was in the direction of ej + e 2
(c) Find cos#, where 0 is the angle between dr ' and chr \
3.69. Show that for any tensor \(Xi^C 2^3)

3.70. Given

where (r, 0, z) and (r 0, Q 0, z 0) are cylindrical coordinates for the current and reference
configuration respectively.
(a) Obtain the components of the left Cauchy-Green tensor B with respect to the basis at the
current configuration.
(b) Obtain the components of the right Cauchy-Green tensor C with respect to the basis at
the reference configuration.