160 TIME-DEPENDENT AND NON-LINEAR PROBLEMS
exponential series to describe the solution of our problem variable
u(t) accurately. This involves finding a sufficient number of solutions
for differing κ, so that an accurate curve fit to the equation
U(K)
= Σ -^- (6.32)
j=i K-Otj
can be obtained, i.e. to find the constants a, and
Sj
;
(j
ί
= 1, 2, . . ., n).
Having obtained this curve, direct inversion will yield a solution of the
form,
u(t)= X S,e«/ (6.33)
We shall now discuss the application of the Laplace transform
method to the problem of transient elastodynamics.
6.5 TRANSIENT ELASTODYNAMICS
Cruse
1
has solved this problem starting with the field equations based
on the small displacement theory for a linear, isotropic, homo-
geneous, elastic medium (see Chapter 5).
We denote the displacement vector (in two or three dimensions)
associated with each point x in space by,
5(x, t) = (uj(x, t)) (6.34)
Then the equations of motion of the linear, isotropic, elastic body are,
(ci -
c
2
2
)u
L
u
+ c\
u
Jm
u
+ bj = d
2
uj/dt
2
(6.35)
where ()j means differentiation with respect to Xy
c
l
is the velocity of dilatational waves where
c?
=
λ
-±^
(6.36)
P
with λ and μ the Lame constants given by,
Ev
V = ^77—, <
6
·
37
)
(l-v)(l-2v)' ^ 2(1+ v)
c
2
is the velocity of distortional waves where,
c\ = μ/ρ (6.38)