Civil Engineering Reference
In-Depth Information
can be substituted in Equation (1.75), resulting in
ω(ρ
c
/K
c
)
1
/
2
k
=
(1.77)
while the stress - strain relations (Equations (1.21) can be rewritten as
σ
ij
=
(K
c
−
2
µ)θδ
ij
+
2
µe
ij
(1.78)
Equation (1.74) describes the propagation of equivoluminal (shear) waves propagating
with a wave number equal to
k
=
ω(ρ/µ)
1
/
2
(1.79)
As an example, a simple vector potential
can be used:
ψ
k
x
3
ψ
2
=
ψ
3
=
0
ψ
1
=
B
exp[
j(
−
+
ωt)
]
(1.80)
In this case,
u
2
is the only component of the displacement vector which is different
from zero
u
2
=−
jBk
exp[
j(
−
k
x
3
+
ωt)
]
(1.81)
This field of deformation corresponds to propagation, parallel to the
x
3
axis, of the
antiplane shear.
References
Achenbach, J.D. (1973)
Wave Propagation in Elastic Solids
. North Holland Publishing Co., New
Yo r k .
Brekhovskikh, L.M. (1960)
Waves in Layered Media
. Academic Press, New York.
Cagniard, L. (1962)
Reflection and Refraction of Progressive Waves
, translated and revised by E.A.
Flinn and C.H. Dix. McGraw-Hill, New York.
Ewing, W.M., Jardetzky, W.S. and Press, F. (1957)
Elastic Waves in Layered Media
. McGraw-Hill,
New York.
Miklowitz, J. (1966) Elastic Wave Propagation. In
Applied Mechanics Surveys
, eds H.N. Abramson,
H. Liebowitz, J.N. Crowley and R.S. Juhasz, Spartan Books, Washington, pp. 809 - 39.
Morse, P.M. and Ingard, K.U. (1968)
Theoretical Acoustics
. McGraw-Hill, New York.
