2.1. Correspondence Principle of Simple Harmonic Wave
Under the condition of a small deformation, a viscoelastic strain displacement, three-dimensional constitutive and dynamic equations are as follows [
34]:
where
is the Kronecker symbol,
represent the
x-,
y-, and
z-directions, respectively,
is the component of the volumetric force,
and
are viscoelastic material functions, and
is the displacement component.
From Equation (
1), the three-dimensional equation of motion of viscoelastic media can be deduced as follows:
where the symbol “*” is a temporal convolution product, except when stated otherwise. The particle velocity is
and the physical forces are neglected.
,
, and
are substituted into Equation (
2), yielding the following:
That is,
where
are the three components of the velocity vector, and
are the three components of the displacement vector.
Under the condition of a small deformation, Equations (
1)–(
4) are the basic equations of viscoelastic media. Here, we only discuss the fluctuations of the frequency
of the displacement
with time
t, namely,
where
is only a function of the coordinate
, which has nothing to do with
t and is generally complex. The conditions for this movement are as follows. The boundary conditions (boundary force and boundary displacement) and volume force all changed with the same angular frequency
over time
t. In the same way, all the strain and stress components also made simple harmonic changes with an angular frequency
, namely,
Equations (
5)–(
7) are substituted into Equation (
1), and the governing equation of the simple harmonic wave in a linear viscoelastic media, which can be represented as a displacement, is solved as follows:
where
is the complex shear modulus of the viscoelastic media,
,
is the complex bulk modulus,
and
, which are only functions of
and have nothing to do with
t, and
, where
is related to
. The governing equation of the simple harmonics in an elastic media is as follows [
34]:
By comparing and analyzing Equations (
8) and (
9), the correspondence between the elastic and viscoelastic media was obtained, as shown in
Table 2. This is the correspondence principle of a simple harmonic wave.
2.2. Wave Equation in Elastic Media
For a homogeneous, isotropic, and infinite elastic medium, we assume that the velocity of any plane wave is
. In general, the plane wave propagates along the
x-direction, and the displacements
,
, and
are functions of
, i.e.,
Substituting Equation (
10) into Equation (
4) yields the following:
Equation (
11) has only two possible solutions if
,
, and
are not simultaneously zero. One solution is for the longitudinal wave, and it is:
In this case, there is only an
x-axis disturbance and the displacement solution is:
The other solution is for a transverse wave, and it is:
In this case, there is only a
y- or
z-axis disturbance, and the displacement solution is:
or
For a homogeneous, isotropic, and infinite elastic medium, the results show that the propagation mode of any plane wave is either a longitudinal wave () or a transverse wave ().
2.3. Wave Equation in Viscoelastic Media
According to the correspondence principle of simple harmonic waves, the solution of plane simple harmonics in infinite viscoelastic media can be obtained from the corresponding solution in elastic media [
34]. In other words, replacing the constant of the elastic medium in Equation (
13) with the complex function of the viscoelastic medium, the longitudinal wave solution is:
Similarly, the transverse wave solution is:
or
For a solution in viscoelastic media, although only the parameters and of the solution for elastic media are replaced by and , the propagation characteristics of ultrasonic waves are significantly different. We obtain the following:
The velocities of longitudinal and transverse waves in viscoelastic media are denoted as
and
, respectively. We obtain:
where
,
,
, and
are functions of the angular frequency
. Therefore, when ultrasonic waves propagate in viscoelastic media, frequency dispersion will occur.
In an ideal elastic medium, the plane wave is not attenuated, while in a viscoelastic medium, the plane wave attenuates as the propagation distance increases. The attenuation coefficients of the longitudinal and transverse wave are denoted as
and
, respectively, and we have:
where
represents the real part of the complex number, and
represents the imaginary part of the complex number.
The ultrasonic attenuation in viscoelastic media is greater than that in elastic media. The primary cause is that the solutions of the wave equation in viscoelastic media are functions of the angular frequency . To obtain the same detection resolution as that in elastic media, a four-laminated ultrasonic transducer with a frequency of 1 MHz was proposed.
2.5. Resonance Frequency
In cylindrical coordinates, the equation of motion of a single piezoelectric wafer is:
where
is the density,
is the radial displacement component,
t is the thickness, and
and
represent the normal stress along the radial and tangential directions, respectively.
Since the electric field is added to the
z-axis and the boundary effect of the electric field is ignored, only
(
represents the electric field strength along the
z-direction), and the piezoelectric equation is:
where
and
represent the normal strains along the
r and
directions, respectively,
is the electric displacement along the
z-direction,
and
are the piezoelectric constants,
is the electric field strength along the
z-direction,
is the dielectric constant component, and
is the piezoelectric strain constant component. We obtain:
where
is the Young’s modulus, and
is the Poisson’s ratio.
In the same way, the mechanical vibration equation can be obtained:
where
F is the applied stress,
is the density,
is the wave velocity,
S is the cross-sectional area,
a is the radius,
is the wave number,
is the angular frequency,
is the zeroth-order Bessel function of the first kind,
is the first-order Bessel function of first kind,
is the resonant vibration speed,
is the electromechanical conversion factor, and
V is the applied voltage.
Similarly, the circuit state equation is:
where
is the cut-off capacitance, and
I is the current.
From Equations (
25) and (
26), the electromechanical equivalent diagram of the piezoelectric wafer can be obtained, as shown in
Figure 2.
If the piezoelectric wafer vibrates freely, that is,
, the resonance frequency equation of a single piezoelectric wafer can be obtained as follows:
Therefore, the admittance equation of the piezoelectric wafer is:
The laminated transducer is composed of four piezoelectric wafers, and the piezoelectric wafers are connected in parallel on the circuit and in series mechanically. From the circuit point, it is to cascade the same four-terminal network with each other. According to cascade theory in a circuit, the electromechanical equivalent circuit diagram of the laminated transducer can be obtained [
35], as shown in
Figure 3.
The admittance of the laminated transducer is the superposition of the admittance of each piezoelectric wafer. Thus, the admittance equation of the four-laminated transducers can be deduced as follows:
when
, the transducer enters the resonance state, and the vibration frequency at this time is the resonance frequency of the transducer.
In the actual excitation of horizontal shear (SH) waves, the energy
radiated by the transducer is diffused in the frequency domain, and it is assumed that the energy distribution is Gaussian. Therefore, for any excitation frequency
, there is a frequency diffusion interval
, where
is the standard deviation. In addition, to model a practical application, the amplitude of any excitation frequency
must be expressed in the following form [
36]:
when
,
,
, and
, by calculating the amplitude
, the amplitude–frequency relationship in elastic and viscoelastic media in the frequency domain can be obtained [
36]. For an elastic material, the amplitude is linearly attenuated with increasing transducer frequency. By contrast, the amplitude in a viscoelastic medium decays exponentially with an increasing transducer frequency and is nearly 0 at 3 MHz. Therefore, the excitation frequency of the transducer should be less than or equal to 1 MHz for a viscoelastic medium. According to the principle of ultrasonic testing, the lower the frequency of the transducer is, the longer the wavelength and the lower the resolution [
37]. When the transducer frequency is 500 kHz and its wavelength is 8.7 mm, it is difficult to distinguish the smallest defect (artificial hole of ⌀2 mm × 10 mm). Therefore, 1 MHz was selected as the transducer frequency in the present work, and other frequencies are not discussed.
In many solutions of the transcendence equation, since 1 MHz was used as the frequency of the developed transducer, the frequency range of the independent variable was set from 950 to 1050 kHz. Solving the transcendental equation
through the graphical method can yield the resonance frequency of the four-laminated transducer, as shown in
Figure 4.
Figure 4 shows that the resonance frequency of the transducer formed by stacking four 4 MHz piezoelectric wafers was about 1 MHz, which was consistent with the actual measured value of 1.03 MHz, with a relative error of 0.3%.