tesi R. Valiante.pdf - EleA@UniSA

40
ν =
2
λ
( λ + µ )
(1.40b)
To well-define the problem of Lamb waves in plates, boundary conditions
must be applied at both free surfaces of the plate. On these surfaces the traction
must vanish. Moreover, the assumption of plain stress must be done. Under these
conditions, it is possible to find a solution to Eq.1.39, which describes Lamb
waves in a homogeneous plate. To solve this problem, the method of the
displacements potentials can be used [35]. The solution can be split into two
parts with symmetric and anti-symmetric properties. Each part leads to a
different Lamb wave mode, one symmetric and one anti-symmetric, as
expressed in Eq.1.41,
tan
tan
( qh)
2
4k
qp
= −
(1.41a)
k − q
( ) ( ) 2
ph
2 2
( qh)
( ph)
2 2 ( k q )
2
tan −
= −
(1.41b)
2
tan 4k
qp
where p and q are defined in Eq.1.42,
p
q
2
2
2
= − k
c
ω
2
L
2
= − k
c
ω
2
T
2
2
(1.42a)
(1.42b)
and h, k and ω are the half-thickness of the plate, wavenumber and
frequency respectively. The graphical representation of the symmetric and the
anti-symmetric modes is shown in Fig.1.14, the arrows represent the
displacements of the material. Equations Eq.1.41 can be solved analytically just