Page 175 - Introduction to Continuum Mechanics

160 Problems

PROBLEMS
3.1. Consider the motion

where the material coordinates Xj designate the position of a particle at t = 0.
(a) Determine the velocity and acceleration of a particle in both a material and spatial
description.
(b) If in a spatial description, there is a temperature field 6 = Ax\, find the material derivative
DB/Dt.
(c) Do part (b) if the temperature field is given by 0 - Bx^

3.2. Consider the motion

where X/ are the material coordinates.
(a) At t = 0 the corners of a unit square are at A(0,0,0), 5(0,1,0), C(l,l,0) and D(l,0,0),
Determine the position of A, B,C,D at t - 1, and sketch the new shape of the square.
(b) Find the velocity v and the acceleration D\/Dt in a material description,
(c) Show that the spatial velocity field is given by

3.3. Consider the motion

(a)At t = 0, the corners of a unit square are at A(0,0,0), 5(0,1,0), C( 1,1,0), and D( 1,0,0).
Sketch the deformed shape of the square at t = 2.
(b) Obtain the spatial description of the velocity field.
(c) Obtain the spatial description of the acceleration field.
3.4. Consider the motion