smart technologies for safety engineering
smart technologies for safety engineering
smart technologies for safety engineering
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Smart Technologies in Vibroacoustics 293<br />
where ûi, ûe i<br />
are expressed by Equations (57) (the second equation), (30), (42), (51), and (45).<br />
, ûpz<br />
i , ˆp, ˆp a and ˆV pz are given values. The relevant natural boundary conditions<br />
8.7.2 Weak Form of the Coupled System<br />
The weak <strong>for</strong>m of coupled multiphysical system combines the weak <strong>for</strong>ms <strong>for</strong> the corresponding<br />
problems presented in Section 8.4. The discussion of coupling interface conditions<br />
in Section 8.6 has presented very important results, namely that the coupling of two poroelastic<br />
domains, or a poroelastic domain to an elastic one, is naturally handled; i.e. the interface<br />
coupling integrals are zero, which results from the continuity of the fields of primary variables.<br />
Such a result is also straight<strong>for</strong>wardly obtained <strong>for</strong> elastic and piezoelectric domains. This is<br />
not the case when coupling to an acoustical domain. There<strong>for</strong>e, define (<strong>for</strong> convenience) the<br />
following interface: Ɣa-p,e,pz = Ɣa-p ∪ Ɣa-e ∪ Ɣa-pz , which is a simple sum of all surfaces, edges<br />
and points where the acoustic domain is coupled to the poroelastic, elastic and piezoelectric<br />
domains. Now, the weak <strong>for</strong>m of the coupled system reads<br />
<br />
pz<br />
− σij pz<br />
δui| j + ω 2 ϱpz ui δui + D pz<br />
i δV|i<br />
<br />
<br />
+<br />
Ɣt ˆt<br />
pz<br />
pz<br />
i δui<br />
<br />
+<br />
Ɣ Q ˆQ<br />
pz<br />
pz δV<br />
<br />
e<br />
+ − σij δui| j + ω<br />
e<br />
2 <br />
<br />
ϱe ui δui +<br />
Ɣt ˆt<br />
e<br />
e i δui<br />
<br />
<br />
+ P −<br />
p Ɣ p<br />
ˆpni δui<br />
p<br />
<br />
+ − 1<br />
ω2 p|i δp|i +<br />
ϱa<br />
1<br />
<br />
p δp + û<br />
Ka<br />
a i na <br />
i δp + n a <br />
i δpui + p δui = 0 (81)<br />
a<br />
Ɣ u a<br />
Ɣ a-p,e,pz<br />
where P stands <strong>for</strong> the integrand <strong>for</strong> a poroelastic domain and equals<br />
P =−σ ss<br />
ij δui| j + ω 2 ˜ϱ ui δui − φ2<br />
+ φ<br />
<br />
1 + ˜ϱsf<br />
˜ϱff<br />
ω2 p|i δp|i +<br />
˜ϱff<br />
φ2<br />
<br />
δp|i<br />
<br />
ui + p|i δui + φ<br />
˜λf<br />
<br />
1 + ˜λsf<br />
˜λf<br />
p δp<br />
<br />
δpui|i<br />
<br />
+ p δui|i<br />
In the above equations ui are the displacements of a piezoelectric or elastic solid or of the solid<br />
phase of a poroelastic material, V is the electric potential in the piezoelectric domain and p<br />
is the pressure in the acoustic medium or in the pores of poroelastic medium. The variational<br />
equation (81) must be satisfied <strong>for</strong> all admissible variations (i.e. virtual or test functions) of<br />
primary variables: δui, δV and δp. Furthermore,<br />
and<br />
σ pz<br />
ij<br />
uk|l + ul|k<br />
= Cpz<br />
ijkl + e<br />
2<br />
pz<br />
kij V|k , D pz uk|l + ul|k<br />
i = epz<br />
ikl − ɛ<br />
2<br />
pz<br />
ik V|k<br />
σ ss<br />
ij = μs (ui| j + u j|i) + ˜λss uk|k δij<br />
σ e<br />
ij = μe (ui| j + u j|i) + λe uk|k δij or σ e<br />
ij = Ce uk|l + ul|k<br />
ijkl<br />
2<br />
(82)<br />
(83)<br />
(84)<br />
(85)
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