As mentioned above the region

As mentioned above, the region of the piezomaterial was discretized by 8-node Plane223 elements. Each node of such an Aprotinin had three degrees of freedom: for the displacements along the OX axis, for the displacements along the OY axis, and the electric potential U allowing to solve the equations of conductivity. The properties of the PET material, PZT-19 ceramics, are listed in Table 1[13].
Constitutive relations for the piezomaterial The equations of state of the piezoelectric medium under isothermal conditions have the following form [14]: where T is the six-dimensional stress vector; S is the six-dimensional strain vector; and are the components of the electric-field vector are the components of the electric field displacement are the elastic constants; are the dielectric constants; are the piezoelectric constants. In the general form, the piezoelectric equations can be represented by the following system: where M is the elastic piezodielectric matrix. This matrix notation corresponds to polarization along the OZ axis and to isotropy of the piezoelectric ceramics in a plane perpendicular to this axis. The matrix M has the form where is the symmetric matrix of elastic moduli. Then the matrix of piezoelectric constants takes the form and the dielectric tensor is
The Newmark method The considered transient process of elastic strain of a body under the propagation of elastic waves is described by the following system of equations: where M, С and K are the mass, damping and stiffness matrices, respectively; F is the vector of external forces; u is the vector of nodal displacements. A peculiarity of the problem is that it involves an implicit solution method, even though explicit methods are typically used when simulating processes lasting about 10−5 s. Since the stability region is narrow, the explicit method requires a very small integration step. As the size of the element\'s side should be smaller than the longitudinal wavelength in the material, which follows the expression the integration step by the Crank–Nicolson criterion will be about 10–8 s. Given that we plan to complement the simulation model with the presence of defects at the next stage of the study, constructing a grid of FE elements may entail situations when the size of the element\'s side will be much smaller than λ, which will make the integration step unacceptably small for practical calculations. In addition, the explicit method contributes to positive damping [14], which may lead to inaccurately modeling the vibration process. In the software we used, Eq. (5) is integrated by an implicit Newmark scheme [15]. The method is implemented with the following representation of the vectors of nodal displacements and velocities for the time interval Δt: where u, and are the vectors of nodal displacements, velocities and accelerations at time ; u, and are the same vectors at time +1; Δt = +1 – and δ are the Newmark parameters chosen from the condition of optimal stability and accuracy. Newmark suggested the method of constant average acceleration, for which δ = 0.5 and α = 0.25, as an unconditionally stable scheme. Since the main purpose is to calculate the displacements u, equation of motion (5) is considered at time +1: while the displacements u at time +1 are calculated by the following equations: where , , , a6 = Δt · (1 − δ), a7 = Δtδ. Taking into account Eqs. (8) for calculating the displacements, the equation of motion can be rewritten in the following form: where , , . The method is unconditionally stable under the following conditions imposed on α and β:
To conveniently estimate the degree of damping introduced in the system, let us introduce the parameter γ which is the amplitude-damping coefficient. The Newmark parameters will be then represented in the following form [16]:
To study the effect of numerical damping on the results, computations were performed for γ = and 0.005 in the same FE-discretizations with the same integration step. The resulting time dependence for the potential difference in the piezoelectric element is shown in Fig. 3. As evident from the graph, high-frequency vibrations occur in the system in the absence of numerical damping, i.e., at γ = 0. The vibrational process is quantitatively identical in both computed cases at a ‘resonant’ frequency, i.e., the one close to the natural frequency of the piezoelectric plate, while an insignificant decrease in the amplitude is observed at γ = 0.005. Further computations were carried out with small numerical damping (γ = 0.005).