Page 184 - Introduction to Continuum Mechanics

Kinematics of a Continuum 169

where k = 10 .
3.57. Does the displacement field

correspond to a compatible strain field?
3.58. Given the strain field

where k = 10 and all other £,y = 0.
(a) Check the equations of compatibility for this strain field.

(b) By attempting to integrate the strain field, show that there does not exist a continuous
displacement field for this strain field.
3JW Thf* strain rnmnnnp.nts arp. criw.n hv

Show that for the strains to be compatible f(X2^) must be linear.
3.60. In cylindrical coordinates (r ,0, z), consider a differential volume bounded by the three
pairs of faces r = r 0,r = r 0-¥dr\ 6 - 6 0,6 = 0 0 + d 0; z = z 0, z = z 0 + dz. The rate at which
mass is flowing into the volume across the face r =r 0 is given by (pv r)(r^dO)(dz) and similar
expressions for other faces. By demanding that the net rate of inflow of mass must be equal
to the rate of increase of mass inside the volume, obtain the equation of conservation of mass
in cylindrical coordinates as that given in Eq.(3.15.5).
3.61. Given the following deformation in rectangular Cartesian coordinates

Determine (a) the deformation gradient F, (b) the right Cauchy-Green deformation tensor
C, (c) the left Cauchy-Green deformation tensor B, (d) the rotation tensor R, (e) the Lagran-
gian strain tensor, (f) the Euler strain tensor, (g) ratio of deformed volume to the initial