Page 353 - Introduction to Continuum Mechanics

Problems 337

5.18. Assume an arbitrary displacement field that depends only on the field variable x 2 and
time f, determine what differential equations the displacement field must satisfy in order to
be a possible motion (with zero body force).
5.19, Consider a linear elastic medium. Assume the following form for the displacement field

(a) What is the nature of this elastic wave (longitudinal, transverse, direction of propagation)?
(b) Find the associated strains, stresses and determine under what conditions the equations of
motion are satisfied with zero body force.
(c) Suppose that there is a boundary at x$ = 0 that is traction-free. Under what conditions will
the above motion satisfy this boundary condition for all time?
(d) Suppose that there is a boundary at#3 = / that is also traction-free. What further conditions
will be imposed on the above motion to satisfy this boundary condition for all time?
5.20. Do the previous problem if the boundary x$ = 0 is fixed (no motion) and x$ = I is still
traction-free.
5.21. Do problem 5.19 if the boundaries £3 = 0 and £3 = / are both rigidly fixed.
5.22. Do Problem 5.19 if the assumed displacement field is of the form

j~ A,
5.23. Do Problem 5.22 if the boundary XT, = 0 is fixed(no motion) and x$ — I is traction-free (
t = 0).
5.24. Do Problem 5.22 if the boundary^ = 0 andx$ = / are both rigidly fixed.

5.25. Consider an arbitrary displacement field u = u(xi ,t).
duf
(a) Show that if the motion is equivoluminal (— = 0) that u must satisfy the equation

OUj OUi OUi
(b) Show that if the motion is irrotational (-T— = -—-) that the dilatation e = -r— must satisfy
dXj dXj dxi
the equation