Research Article | Open Access
Xu Liang, Wenbin Lu, Ronghua Zhu, Changpeng Ye, Guohua Liu, "Three-Dimensional Semianalytical Solutions for Piezoelectric Laminates Subjected to Underwater Shocks", Mathematical Problems in Engineering, vol. 2020, Article ID 2439521, 20 pages, 2020. https://doi.org/10.1155/2020/2439521
Three-Dimensional Semianalytical Solutions for Piezoelectric Laminates Subjected to Underwater Shocks
In this study, a piezoelectric laminate is analyzed by applying the Laplace transform and its numerical inversion, Fourier transform, differential quadrature method (DQM), and state space method. Based on the modified variation principle for the piezoelectric laminate, the fundamental equations for dynamic problems are derived. The solutions for the displacement, stress, electric potential, and dielectric displacement are obtained using the proposed method. Durbin’s inversion method for the Laplace transform is adopted. Four boundary conditions are discussed through the DQM. The proposed method is validated by comparing the results with those of the finite element method (FEM). Moreover, this semianalytical method is further extended to describe the dynamic response of piezoelectric laminated plates subjected to underwater shocks by introducing Taylor’s fluid-structure interaction algorithm. Both air-backed and water-backed laminated plates are investigated, and the effect of the fluid is examined. In the time domain, the electric potential and displacements of sample points are calculated under four boundary conditions. The present method is shown to be accurate and can be a useful method to calculate the dynamic response of underwater laminated sensors.
Piezoelectric materials are extensively used as sensing elements in various engineering fields, such as aerospace engineering, ocean engineering, civil engineering, and mechanical engineering. In addition, piezoelectric materials are one of the most important components in smart structures which can sense and drive changes in external circumstances and then be modified in response by computers and processors . Since the Curie brothers first discovered the direct piezoelectric effect in single-crystal quartz in 1880 and Gabriel Lippmann discovered the converse piezoelectric effect in 1881, various types of piezoelectric materials have been found, including both natural materials and composite materials [2, 3].
Composite laminates are a common smart structure. As shown in Figure 1, the structure is composed of stacked layers of different materials; the individual layers are generally orthotropic or transversely isotropic. Owing to the various characteristics of the individual layers, a hybrid laminate may exhibit multiple functions. Piezoelectric laminates are usually piled as sensors and actuators that will react to or induce a displacement or electric potential [4, 5].
Pioneering research on piezoelectric laminates has been conducted primarily using finite element models (FEMs) and the space state method. A FEM has stable convergence and its solution is effective; however, it requires complex grids and long computation times to produce results with high accuracy [6, 7]. Mukherjee and Saha  focused on large deformations of piezoelectric beams using a FEM based on the first-order shear deformation theory and the Newton–Raphson method. Garcia et al.  developed a layer-wise FEM for a piezoelectric plate using the Reissner mixed variational principle. Dash and Singh  employed a C0 isoparametric FEM to solve the nonlinear free vibration problem. Wankhade and Bajoria  considered the higher-order shear deformation theory to carry out a FEM of a piezoelectric beam under static and dynamic excitations.
On the other hand, the state space method is also an efficient and accurate method. In view of the relationships among transverse vectors, it is suitable for laminated structures . Different approaches have been investigated in the state space method for piezoelectric laminates. D’Ottavio and Kröplin  combined the Reissner variational statement and constitutive equations to simulate piezoelectric laminates. Tiersten  demonstrated Hamilton’s principle for linear piezoelectric media under different boundary conditions and discussed both homogeneous and inhomogeneous problems. In addition, Qing et al.  proposed a modified mixed Hellinger–Reissner (H–R) variational principle based on the work of Steele and Kim .
The state space method is the primary focus of this study. The variational principle for the state space method is a complex differential equation including several variables, and thus some transformations should be employed. The Laplace transform and differential quadrature method (DQM) are the two main transformations used in this study.
The Laplace transform is usually used to change a function in the time domain to one in a complex frequency field for dynamic problems . For this reason, in recent studies, the Laplace transform has generally been adopted to analyze both statics and dynamics problems, and the numerical inversion of the Laplace transform has been studied. Zhao  proposed two algorithms that divide the integration interval into small subspaces with different integration steps. Durbin  combined a trigonometric series with an inversion series and presented rather smaller results. Qing et al.  subsequently made comparisons to the inverse algorithms proposed by both Zhao and Durbin and improved Zhao’s algorithms by using Subbotin splines. Wang et al.  improved the calculation efficiency with two novel algorithms and additionally compensated for the deviation node in Qing’s algorithms. Durbin’s inversion method is used in this study owing to its high efficiency, and the results are sufficiently precise.
First proposed in 1971 by Bellman et al.  based on integral quadrature, DQM is another method for dealing with partial differential factors. Moreover, with DQM, complex boundary conditions can be analyzed. Extensive research has focused on the choice of grid points, boundary conditions, weight coefficients, and the convergence behavior. Several researchers [21–23] have eliminated the unreliable results from uniform grid points and offered various distribution forms that could satisfy higher orders and different simulations. Chen et al.  employed the Chebyshev polynomial to obtain the weighting coefficients of the matrix form. To date, DQM has been widely applied for composite structures. Du and Shu  chose DQM to deal with the clamped and simply supported boundary conditions of beams and plates. Pradhan and Murmu  used DQM to analyze the vibration of functionally graded material beams.
Piezoelectric laminates are employed for numerous applications in the sea. It is important for underwater structures to consider the interaction between the water and the structure. There are few studies regarding underwater piezoelectric laminates, and thus the fluid-structure interaction (FSI) is also considered in this study. Taylor  proposed a one-dimensional model to analyze the FSI. It has been further verified under various conditions with different materials, fluid media, and structures. Schiffer and Tagarielli  examined the FSI response with various materials and structural combinations. Blom  investigated a piston problem and combined the one-dimensional equation and the Euler equation to propose an algorithm. Deshpande and Fleck [29, 30] studied sandwich beams subjected to underwater waves using a lumped parameter model and FEM.
In this study, we discuss three-dimensional semianalytical solutions for piezoelectric laminates subjected to an underwater shock. The state space method is established based on the constitutive equations for piezoelectric materials. Both air-backed and water-backed laminated plates are investigated based on the effect of the FSI. Combined with the Fourier transform, DQM, and Laplace transform, the present model is analyzed under four different boundary conditions. The results are verified using finite element analysis software.
2. Mathematical Description
2.1. Fundamental Equations
A piezoelectric laminate consists of a series of layers of different materials, including piezoelectric materials. Consider a laminated plate comprising three layers of PZT/Gr-Epoxy/PZT materials, where PZT means lead zirconate titanate piezoelectric ceramics. As shown in Figure 1, the laminate is subject to an underwater shock. Thus, the upper layer is always exposed to the water, while the lower layer may be exposed to either water or air.
Briefly, the piezoelectric effect is demonstrated to change polarization when a mechanical stress is applied or to create a mechanical deformation with the introduction of additional voltage. In view of basic theory, the constitutive equation for piezoelectric materials can be written simply as the following form:where , , , and are the stress, strain, electric field, and electric displacement, respectively, and , , and are the elastic, piezoelectric, and dielectric coefficients, respectively. The subscripts p, q, i, and j denote different directions in the material coordinates (i, j = 1, 2, 3; p, q = 1, 2, 3, 4, 5, 6). The superscript E indicates parameters measured at a constant electric field, and the superscript S denotes those measured at a constant or zero strain field. The superscripts will be omitted in the subsequent derivation process.
Assuming that the piezoelectric materials are orthotropic or have orthotropic symmetry relative to the x-y coordinate plane, constitutive equation (1) can be extended to the following matrix form:in the Cartesian coordinate system; p, q = x, y, z, yz, xz, xy, and i, j = x, y, z. For the model of a piezoelectric laminate shown in Figure 1, the subsequent equations and derivations will be described in the Cartesian coordinate system.
In terms of geometric relationships, the relationships between the strains and displacements can be described in the form of gradient equations:where u, , and denote the displacements in the three directions of the Cartesian axes; and α, β, and γ are partial differential operators of the three Cartesian axes with respect to the following variable: .
In light of the quasi-static Maxwell equations, the electric fields, , and electric potential, , have similar relationships as the strains and displacements:
Two general designations are defined as and for the two groups including four dual vectors, while the remaining vectors of the variables are given as , , and . Thus, the subsequent derivations can be simplified with these five designations. Constitutive equation (2) can then be rewritten as follows:where , , , and represent the transformations of the 9 × 9 matrix in equation (2) and .
According to the Reissner mixed variational theorem, Reissner’s energy density function is given by the above matrices:
Owing to the special form of orthotropic constitutive equations, the secondary vectors in equation (7) are replaced, and the integral parts are derived to dot-product forms:where for dynamic problems in the time domain.
In terms of the Reissner variational principle, the energy equation with an internal force is given by the following equation:where h indicates the thickness of the layer and is the area of the layer.
In addition, the in-plane vectors and rest vectors can be evaluated by the expression in the following equation:
The partial differential operator of z in equation (10) can be eliminated by transforming the coefficient matrix into an exponential matrix as follows:where k indicates the kth layer and represents the value of the thickness at the kth layer.
Therefore, the state vectors of the piezoelectric laminate at z can be written as follows:where
2.2. Fluid-Structure Interaction (FSI)
As shown in Figure 1, the piezoelectric laminate is subject to an underwater explosion, resulting in shock waves on both the upper and lower surfaces of the laminate. In accordance with the one-dimensional FSI model proposed by Taylor  in 1963, the effect of the shock waves on the laminate can be calculated.
The shock wave is simply described as a function of the time, t, and the vertical distance to the plane, z, which in this paper is equal to zero:where denotes the wave velocity in water as a constant and and are empirical formulas for the explosive charge weight, (kg), and perpendicular distance from the upper surface of the laminate, S (m), respectively. TNT is used for the explosion in this study:
Under these conditions, the planes influenced by the shock wave include both the upper and lower surfaces of the laminate [29, 30]. The wave pressure on one plane consists mainly of the primary shock wave , reflected wave, , and rarefaction wave, :where is the density of water and .
Consequently, when the laminate is completely submerged underwater, the wave pressures on the upper and lower surfaces are given by the following:
In this paper, the interactions between the fluid and structure are divided into two conditions. As the wave pressure on the upper surface is caused directly by the underwater shock, the lower surface is exposed to both water and air, which here are referred to as water-backed and air-backed laminated plates. Because the density of air is far smaller than the density of water, the wave pressure on the air-backed surface is assumed to be zero:
Hence, on the upper and lower surfaces of the laminate, the boundary conditions are confirmed as follows according to the FSI:where for the water-backed plate and for the air-backed plate.
3. Boundary Conditions and Transform Methods
3.1. Laplace Transform Methods
The Laplace transform method is adopted to obtain the response in the time domain, which is described in the two following equations:
In this paper, Durbin’s inversion method given by equation (23) is adopted. The trigonometric series is obtained, and the error of the results is independent of t, and thus the analytical model can yield good results with high efficiency:where , , and is the real part of imaginary number , and , where denotes the duration of the model in the time domain. Here, the value of K is 100.
3.2. Fourier Transform Methods
Simply supported (x = 0)-simply supported (x = a) (S-S):
Clamped (x = 0)-clamped (x = a) (C-C):
Clamped (x = 0)-free (x = a) (C-F):
Clamped (x = 0)-simply supported (x = a) (C-S):
The Fourier transform method is employed here to change the form of . According to the coefficient matrix in equation (10), the Fourier transform indicates the following:
In addition, the Fourier transform of the pressure on the surfaces yieldswhere
3.3. Differential Quadrature Method (DQM)
While the Fourier transform method is used to change the form of , DQM is used to parse the form of .
By selecting some grid points in the domain of x, the differential form of a continuous function, , can be described by an approximate function that is calculated with the values of all grid points and a weighting coefficient matrix [7, 31, 32]:where the superscript i indicates the ith-order derivative, M is the total number of sampling points, and denotes the weighting coefficients.
The Chebyshev–Gauss–Lobatto points are chosen for their efficient speed of convergence and high computational accuracy . Therefore, the sample points of the x-coordinate can be chosen as follows:
3.4. Normalization of the Variables
The variables need to be normalized to prevent nonconvergence of the computation caused by large gaps in magnitude between different parameters.
According to the constitutive equation in equation (1), the variables can be normalized in the following ways:where .
For FSI, the density of water, , and wave velocity underwater, , are normalized as follows:
The of the normalized variables will be omitted in the following equations.
In view of the semianalytical transformations mentioned above, for the boundary conditions of C-C, S-S, C-F, and C-S, the state space method form in equation (10) is further deduced as follows:where the subscript i indicates the ith grid point in the DQM (i = 1, 2, …, M), , , and .
Based on equation (11) and the boundary conditions as stated above, equation (39) should be deduced to avoid singular matrices in it, which means the coefficient matrix in equation (10) should be deduced into different forms for the four boundary conditions:where m, n = 2∼M − 1 in the matrix andwhere m = 2∼M − 1, n = 1∼M − 1 in the matrix, andwhere m, n = 1∼M in the matrix andwherewhere
4. Numerical Results
The model in Figure 1 has two components: the first and third layers are composed of PZT, and the second layer is Gr/Epoxy. All the material parameters are listed in Table 1, including the elastic modulus, density, piezoelectric constant, and dielectric constant. The length, width, and thickness of the model are represented as a, b, and h, respectively, and each layer has a thickness of h/3. The values of a, b, and h are provided in Table 2 (see Tables 3 and 4).