A uniform adhesion strength is assumed. An analysis of a non-uniform
strength was presented by Singher et al. [11]. In a
quasi-static approach,
the shear mechanical behavior of the interface adhesive-adherend is
represented by a density of springs with stiffness constant 
, between
the adhesive and adherend [12]. The springs relate the stresses on
the faces of the
two media to the displacement discontinuity across the interface. In order to
correctly include the inertial effects, the mass of the spring is taken into
account. The strength of the coupling between the two media is denoted by the
parameter . Obviously if 
it characterizes free
surfaces (complete unbond), when 
, we revert to the
usual case where the conditions correspond to perfect bond. Variations of
thus allows one to continuously pass from the condition of perfect
bond to that of lack of bond. We thus attribute to the parameter ,
the strength of the bond. For the spring-mass model considered here,
the following conditions apply
where the stress is denoted by t and 
and 
are
defined by
and 
are the density and thickness
of the interface, respectively. Use of these quasi-static boundary conditions
assumes that the ultrasonic wavelength 
is much larger than the
thickness of the interface, . Substituting these boundary conditions
into the displacement and stress expressions, leads to the dispersion matrix
equation. The solution is confined to symmetric systems in which the
two adherends are identical. For this case the potentials have either
symmetric or antisymmetric behavior [13]. For the symmetric case, the
dispersion equation obtains the form,
where
is the expression from the dispersion equation for Rayleigh
waves in the layer i and 
is the expression from the
dispersion equation for the symmetric Lamb waves in the layer i. For the
antisymmetric case, the trigonometric functions tan should be replaced by the
trigonometric functions arctan in the dispersion equation [14].
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