Page 178 - Introduction to Continuum Mechanics

Kinematics of a Continuum 163

(a) At t = 0 a material filament coincides with the straight line that extends from (0,0,0) to
(1,0,0). Sketch the deformed shape of this filament at t = 1/2, t = 1, and t = 3 / 2,
(b) Find the velocity and acceleration in a material and a spatial description.
3.15. Consider the following velocity and temperature fields:

(a) Determine the velocity at several positions and indicate the general nature of this velocity
field. What do the isotherms look like?
(b) At the point ./I (1,1,0), determine the acceleration and the material derivative of the
temperature field.
3.16, Do the previous problem for the temperature and velocity fields:

( )
e
e
3.17. Consider the motion \=X + X ike 1 and let dX = (dSi/V2)( i + 2) and
JX' ' — (dS2/*S2)(-*i + 62) be differential material elements in the undeformed configura-
tion.
2
(a) Find the deformed elements dx^ and dx^ \
(b) Evaluate the stretches of these elements, dsi / dS\ and ds 2 / d$2, and the change in the
angle between them.
2
(c)Do part (b) for k = 1 and k = 10~ .
(d) Compare the results of part(c) to that predicted by the small strain tensor E.
3.18. The motion of a continuum from initial position X to current position x is given by

where I is the identity tensor and B is a tensor whose components £,y are constants and small
compared to unity. If the components of x are */ and those of X are A!/, find
(a) the components of the displacement vector u, and
(b) the small strain tensor E.
3.19. At time t, the position of a particle initially at (Xi^C^s) is defined by

5
where k = 10 .