Page 180 - Introduction to Continuum Mechanics

Kinematics of a Continuum 165

(a) Find the location of the particle that does not undergo any volume change.
(b) What should be the relation between k\ and k<i be such that no element changes volume?

3.27. The displacement components for a body are

(a) Find the strain tensor.
(b) Find the change of length per unit length for an element which was at (1,2,1) and in the
direction of ej 4- e 2.
(c) What is the maximum unit elongation at the same point (1,2,1)?
(d) What is the change of volume for the unit cube with a corner at the origin and with three
of its edges along the positive coordinate axes.
3.28. For any motion the mass of a particle (material volume) remains constant. Consider the
mass to be a product of its volume times its mass density and show that (a)for infinitesimal
deformationp(l 4- EM) = p 0, wherep 0 denotes the initial density and p the current density,
(b) Use the smallness of EM to show that the current density is given by

3.29. True or false: At any point in a body, there always exist two mutually perpendicular
material elements which do not suffer any change of angle in an arbitrary small deformation
of the body. Give reasons.
330. Given the following strain components at a point in a continuum:

Does there exist a material element at the point which decreases in length under the defor-
mation? Explain your answer.
3.31. The unit elongations at a certain point on the surface of a body are measured experimen-
tally by means of strain gages that are arranged 45° apart (called the 45° strain rosette ) in the
directions ej, (>/2/2)(ei + 62) and e^ H these unit elongations are designated by a,b,c
respectively, what are the strain components £"ii^22»^i2-

6
6
6
3.32. (a) Do Problem 3.31 if the measured strains are 200xlO~ , 50xlO~ , lOOxlO" ,
respectively.
(b) If £33 = £32 = £31 = 0, find the principal strains and directions of part (a).
(c) How will the result of part (b) be altered if £ 33 * 0?