F ¼ 1.7 nN which produces an end stress of f ¼ 0.065 MPa. n 0 ¼ 10
23 /m
3 . The series
in Eq. (3.164) converges rapidly. Sixteen terms are used. Figure 3.22 presents a
series of snapshots of the distribution of u 3 along the rod at different time instants as
the initial disturbance at the right end propagates to the left along the rod, hits the left
end and gets reflected there, and then propagates back to the right end. The wave
described by Eq. (3.161) is nondispersive and has a finite propagation speed. In (a),
the right part of the rod has felt the disturbance with an essentially linearly distributed displacement, but the left part has not felt the disturbance yet. The extensional
wave speed can be determined as c ¼
ffiffiffiffiffiffiffiffiffiffiffi
b c 33 =ρ
p
¼ 5546 m=s . In our numerical
example, we have L/c ¼ 10.8 Â 10
À11 s which is the time for a disturbance to
propagate through the entire rod, and it agrees with (b) very well. The propagation of
φ is found to be similar to u 3 and is not shown.
The equation for Δn is obtained from Eqs. (3.132) 2 and (3.134) 2 as
∂
∂t
Δn
ð Þ ¼ Àn 0 μ
n
33 φ ,33 þ D
n
33
∂
2 Δn
ð Þ
∂x 2
3
,
ð3:166Þ
with the following boundary and initial conditions:
Fig. 3.22 Distribution of u 3 along the rod at different time instants. m ¼ 0–15. (a) t ¼ 4 Â 10
À11
s. (b)
t ¼ 10.7 Â 10
À11
s. (c) t ¼ 18 Â 10
À11
s. (d) t ¼ 22 Â 10
À11
s
70
3 Extension of Rods