Civil Engineering Reference
In-Depth Information
this case we could apply modelling by transfer matrices but a more advanced one than
the equivalent fluid model used when calculating the data in
Figure 8.3
, now having to
take the elastic properties into account. We have previously used a Biot model for a
porous elastic material to calculate the absorption factor (see section 5.5.5). This model
may thus be included in a transfer matrix calculation for the complete sandwich element
by a procedure as e.g. used by Brouard et al. (1995). It should be noted, however, that
these calculations presuppose infinite size layers. For finite size elements, one normally
has to apply finite element methods (FEM) (see e.g. Vigran et al. (1997)).
a)
Face sheet
Core
h
1
h
2
b)
h
3
Figure 8.19
Sandwich element. a) Principal structure; b) Definition of layer thicknesses.
To illustrate the general features of sound transmission through sandwich elements
we shall, however, use another approach. We shall start using the same assumption
applicable for elements with a honeycomb core; the element being infinitely stiff in the
normal direction having a core characterized by its shear stiffness only. As a second step
we shall assume that the core is a general homogeneous elastic material. It may also be a
porous material but we shall not have to model it using Biot theory as we will assume
that the pores are closed.
8.3.1
Element with incompressible core material
The most pronounced feature of sandwich elements, as distinct from the partition
elements treated up to now, is the frequency-dependent bending stiffness. In the static
case, and also at sufficiently low frequencies, the core will act like an ideal spacer for the
face sheets. For simplification, we may assume that the face sheets are identical, enabling
us to express the low frequency bending stiffness of the element as
)
2
(
Eh h
+
h
11
1
2
B
≈
when
h
=
h
and
E
=
E
.
(8.30)
low
1
3
1
3
2
For the definitions of these quantities, see
Figure 8.19.
At a sufficiently high frequency,
however, we get a decoupling of the face sheets and the bending stiffness will just be the
sum of the bending stiffness of each sheet. We may then write
