Finite Element Modeling of a Flat Cell of Highly Porous Piezocomposite with Inclined Edges Taking into Account Nonuniform Polarization

Introduction. One type of highly porous composite is a material constructed on the basis of Gibson-Ashby cells [1]. In [2], an assessment of the effective Young's modulus of porous titanium with open pores was performed on the basis of a three-dimensional array of such cells. A comparison of the mechanical behavior of foam models composed of regular and irregular arrays of open Gibson-Ashby cells is carried out in [3]. Figure 1 a shows an open cell, Figures 1 b and 1 c show modified Gibson-Ashby models with inclined edges for functionally graded lattice structures [4].

Homogenization methods used in modeling the effective properties of composites are considered in monographs [5][6]. Studies on highly porous structures, such as foamy, cellular, honeycomb and cell structures are presented in monographs [7][8], in reviews [9–12], etc. In [13], the polarization of porous piezoceramics is considered experimentally and theoretically and, based on a model example, features of some of its effective properties are theoretically explained, as mentioned above. The issue of the effective properties of highly porous piezoelectric composites has not been sufficiently studied in the scientific literature. This is primarily due to the fact that the process of describing the polarization of such structures is a certain difficulty. This research is aimed at studying the influence of the polarization model on the stress-strain state of the cell and on the effective properties of such composites. Two models are considered: the first model assumes uniform polarization; in the second, the polarization distribution is calculated using methods and software known from the literature, among the developers of which is one of the authors of this work. The work shows that the selection of the polarization model affects significantly the mechanical properties of highly porous piezoelectric composites.

Materials and Methods

Mathematical Formulation of the Problem. We consider flat cells, which are elements of the structures shown in Figure 1. The cell material is PZT-4 piezoelectric ceramics, described within the framework of the linear theory of electroelasticity [14]. In the homogeneous case, it is polarized along the vertical axis, in the inhomogeneous case — the polarization distribution is found according to the theory proposed in [15] and implemented in the finite element package ACELAN [16].

Materials. Figure 2 a shows the geometry of a flat cell corresponding to the diagram in Figure 1 a with inclined edges, the edge thickness is 1 mm, the external size is 10×10 mm. Figure 2 b shows a diagram of nonuniform polarization for such a design, when the electrodes are located on the outer ends of the upper and lower vertical edges. The characteristic feature of this polarization is its nonuniformity on inclined edges and the presence of its opposite directions on the upper and lower horizontal edges. Figure 2 c shows the polarization scheme for a composite element, whose periodicity is realized through vertical and horizontal edges. The characteristic feature of the polarization of this element is the practically unpolarized horizontal edges.

Methods. The finite element method (FEM) implemented in the ACELAN package is used as a method for calculating the stress-strain state of cells [16].

Research Results. First, the natural frequencies and modes of vibrations “inside” the cell shown in Figures 1 c and 2 c are investigated. In this case, the cell is fixed in the corners along the normal to the thickness of the horizontal and vertical edges, which corresponds to the periodicity of the composite structure. Table 1 shows the natural resonance frequencies for uniform and nonuniform polarization (Fig. 2 c), and Figures 3–5 show eigenvibrations at these frequencies.

For nonuniform polarization 1, 3 and 4, the natural frequencies are lower than for uniform polarization, but the vibration modes 1–5 are qualitatively the same. Therefore, Figures 3–5 show the natural modes for a cell with uniform polarization. Figures 3 a, b show the distribution of horizontal displacements and vertical displacements for the first vibration mode. In Figures 4 a, b the distributions for the second mode of vibrations of horizontal displacements and for the third mode of vibrations of vertical displacements are presented, respectively. In Figures 5 a, b the distribution of the displacement modulus is presented for the fourth and fifth modes, respectively.

With longitudinal tension-compression of this cell in the vertical direction in the composite structure (modeled by the free entire upper surface and the application of uniform pressure to the upper end of the vertical edges), the displacement in the case of uniform polarization is 15% greater (Fig. 6 a). The potential on free electrodes is 3% higher in the case of nonuniform polarization. It should be noted that the shear stresses (Fig. 6 b) are 10% higher in the case of nonuniform polarization. The electromechanical coupling coefficient for the 7th oscillation mode (Fig. 6 c) is 14% higher for uniform polarization of the cell.

Calculations for the cell presented in Figures 2 a and b show that the vertical displacement (Fig. 7 a) under the action of pressure on the end of the upper rod in the case of nonuniform polarization is 11% greater. In statics, under the action of the potential difference at the lower and upper ends, the cell exhibits auxetic properties, which is associated with the opposite polarization (Fig. 2 b) of the horizontal edges. Figures 7 b and c show the distributions of vertical and horizontal displacements. It is evident that when stretched vertically, the cell also expands horizontally.

Discussion and Conclusion. Thus, on the basis of this stress-strain state (SSS), the effective properties of composites, which are used in modeling piezoelectric devices, are found. Therefore, the study of the influence of the type of polarization on the SSS of a highly porous piezoactive material is urgent.

The calculation of the preliminary polarization field in a cell of a highly porous material has shown that it depends significantly on its geometry and the arrangement of the electrodes. These features include the fact that some edges are practically not polarized, others are polarized in one direction, but nonuniformly, and, finally, edges with opposite polarization may appear. The calculation of the mechanical and electrical response of the cell and its natural resonance frequencies have proven that taking into account the nonuniformity of polarization leads to the fact that the magnitude of the discrepancy between these results and the results for the model with uniform polarization reaches 15% and 10%, respectively. And the cell with edges on which there is counter polarization exhibits auxetic properties. The paper shows that in problems of determining the effective properties of highly porous piezoelectric composites based on the construction of representative volumes from its cells, it is essential to take into account the inhomogeneous polarization corresponding to their structure.

The practical relevance of the results obtained is due to the fact that the effective properties of composites allow for modeling, calculation and optimization of various piezoelectric devices (sensors, emitters and receivers of acoustic waves, piezoelectric generators, etc.) at the design stage.