Computational modeling of sigmoid functionally graded material (SFGM) plate

The first-order shear deformation theory (FSDT) was used to explore the natural frequency response of functionally graded piezoelectric plates subjected to static electrical and mechanical strain in this present study. A monomorph model for a functionally graded piezoelectric plate with material properties that change according to sigmoid law with respect to plate thickness has been considered. A three-dimensional finite element model with a free tetrahedral element mesh was created using COMSOL 4.2 Multiphysics® software, with each node having three degrees of freedom. Variations in the FGPM plate’s free vibration boundary conditions, composition, and geometry have all been investigated. In free vibration analysis, non-dimensional natural frequency of FGPM plate initially decreasing considerably and then remaining almost constant with the increase in volume fraction index when material property graded by power law. When material properties are varied by sigmoid law, with an increase in volume fraction index, the non-dimensional natural frequency of FGPM plates remains virtually constant. FGPM plates have a lower non-dimensional natural frequency if the thickness to width ratio is greater. Non-dimensional natural frequencies of Clamped-Clamped FGPM plates (C–C–C–C) are greater compared to Clamped-Free FGPM (C–F–C–F) and Simply Supported Free FGPM (S–F–S–F).


Introduction
In addition to their great mechanical and electrical capabilities, Electrical energy may be converted to mechanical energy by anisotropic materials such as piezoelectric materials, as well. Due to this their use in applications, such as various sensors and actuators, micro-electromechanical systems (MEMS), active vibration control, precision position control, etc have grown tremendously. For these applications, generally larger displacements or deflections are desired for which multilayer stacking method is mostly approached. Due to excessive stress concentration at the interlayer interfaces, bonding strength deteriorates, and at low temperatures or at high temperatures, splits or peels off. FGM (functionally graded material) in conjunction with piezoelectric materials to offset these undesirable effects. These materials are referred to as functionally graded piezoelectric materials (FGPMs) [1]. These materials have properties varied by some mathematical functions specifically power law, sigmoid law and exponential law which provide continuous and smooth property transition between layers of the FGPM material. This reduced the stresses at the interfaces and increased the reliability and longevity of the FGPM material [2][3][4][5][6]. FGPM plates can be used in vibration control systems to suppress unwanted vibrations in structures. The piezoelectric properties of the FGPM allow it to convert mechanical vibrations into electrical energy, which can be harvested and utilized for various purposes, such as powering sensors or other electronic devices. It can be incorporated into smart structures that can change their shape or undergo morphing in response to external stimuli. By applying appropriate electric fields to the piezoelectric material, the FGPM plate can bend, twist, or deform, enabling adaptive structures or components that can adjust their shape to optimize performance or adapt to changing conditions.
From the past few decades, A great deal of research has gone into understanding the properties and potential uses of functionally graded piezoelectric materials. Static and dynamic FGPM plate experiments using a variety of methods have been conducted, boundary conditions, and material compositions have been published in the scientific literature. Chen and Ding investigated the FGPM plate's free vibration [7]. A precise answer for a FGPM laminated plate has been established using the state-space technique pioneered by Lu et al [6,8]. For FGPM rectangular plates with simply supported boundary conditions, Zhong and Shang [5,9] achieved accurate solutions under static mechanical, electrical, and thermal loadings. For FGPM rectangular plate free or forced vibration analysis, Zhong and Yu [10] used the state-space technique to investigate this problem. Material propertied have been varied by exponential law. Chen et al [11,12] studied free vibration of FGPM fluid-filled cylinder and hollow sphere with simply supported boundary conditions using laminated approximation method and state-space formulation. According to the FSDT theory, A functionally graded piezoelectric  cylindrical panel subjected to time-dependent loads was studied by Bodaghi and Shakeri using the analytical solution approach to determine its natural frequency and dynamic vibration response [13]. Dai et al presented analytical solutions for electro magnetoelastic behavior and thermoelastic behavior of various FGPM structures namely circular plate, fluid-filled cylindrical shell, hollow & solid cylinder [14]. The material properties of the functionally graded piezoelectric plate and beam exhibit a power-law variation through the thickness [15,16]. The utilization of emerging functionally graded materials holds significant promise in addressing challenging applications, particularly in practical engineering, by effectively harnessing the flexoelectric effect [16]. The electromechanical responses of functionally graded piezoelectric nanobeams are profoundly influenced by the presence of flexoelectricity [17]. Behjat et al [18], determined the static and transient responses of FGPM plates under electrical and mechanical stresses were studied using first order shear deformation theory and finite element approach. Nourmohammadi and Behjat studied the FGPM plate's static bending response to electrical, mechanical, and thermal stresses using the FSDT theory [19]. The vibration behavior of porous FGPM plates under thermo-electrical-mechanical loads was studied by using an improved four-variable plate theory [20]. For the cylindrical FGPM actuator in the harmonic electric field, Zhang et al [21] were able to establish an analytical solution for the vibration response. Due to the linearity of the piezoelectric coefficient, this actuator's polarisation is radial. Using the state-space based differential quadrature (SSDQ) approach, a 3D free vibration study of the FGPM annular pate on the Pasternak elastic foundation was published by Yas et al [22]. According to the exponential law, the plate's thickness changes the material's properties in a direct proportion. SSDQ technique used to investigate the free vibration of multilayered cantilever FGPM beams with changing boundary conditions [23]. Zhu et al [24] used the FSDT theory to investigate the free vibration and dynamic response of FGPM plates with classical and elastic boundary conditions. A power law has been used to model the distribution of material attributes. A penalty function is used to generate the variational formulation for boundary conditions in mechanical and electrical systems. The generalised differential quadrature (DQ) technique was used to examine the influence of shear on the flexure vibration of the FGPM annular plate [25,26].
In literature, most of the work has been carried out utilizing either exponential law or power law for material property variation for FGPM structures; some research has been carried out on FGM plates using sigmoid law material property distribution [27]. Chung and Chi proposed the sigmoid law to smooth the stresses at the interfaces of the composite plates. For transversely-loaded FGM plates, Classical plate theory was also used to develop a finite element and a series solution, respectively [3,28,29]. First-order shear deformation theories and different higher-order shear deformation theories were used to study FGM plates [30,31]. Under clamped boundary circumstances for material attribute modification by a power law, sigmoid law, and exponential rule, researchers examined the FGM circular plate [32,33]. The free and forced vibration of a sigmoid FGM plate on an elastic basis was studied by using a higher-order shear deformation theory [33]. Natural frequencies of simply supported rectangular plates may be studied using a technique proposed by Thang and Lee. The sigmoid rule is used in these situations to distribute temperature-dependent material properties across the thickness [34]. Pandey and Parashar evaluated the static deflection and natural frequency response of FGPM beams under electromechanical loads [35]. No research on the free vibration of FGPM plates using Sigmoid law is known to the authors.
This study examines the natural frequency response of functionally graded piezoelectric material plates under static electro-mechanical pressure using the first-order shear deformation theory (FSDT). Material characteristics for the plate's FGPM layer change continuously monomorphically in the z-direction according to the sigmoid rule. A 3D-finite element model with a free tetrahedral element mesh where each node have three degrees of freedom using COMSOL 4.2 Multiphysics ® software has been developed and utilized for this study. The effectiveness of the COMSOL Multiphysics ® for the analysis of FGPM structures has already been established by works [25,26,35] in literature. The governing equations are derived using electric enthalpy density (H) and Hamilton's principle. The free vibration of the FGPM plate has been explored in terms of varied boundary conditions, material composition, and geometric limit factors.

Functionally graded piezoelectric plate material property
It is demonstrated in figure 1 that the neutral plane's two longitudes are parallel to the plate thickness's midpoint and perpendicular to the top surface, whereas the z axis runs perpendicular. There are three parameters to keep in mind while looking at the FGPM plate in figure 1: its length, breadth, and thickness. The FGPM plate's assumed attributes are distributed uniformly over its thickness in accordance with the sigmoid law of material property distribution. The distribution of material attributes is more uniform under sigmoid law than under power law. This research's piezoelectric material constants are shown in table 1.
The variation of material property by sigmoid law can be expressed as [3]: The implicit material property P(z) is represented by this notation, P t is the material's uppermost layer and P b is the bottom surface material property. V 1 and V 2 are the piezoelectric material's functionally graded volume fractions. The fractions of volume may be expressed as follows: The volume fraction index, n, changes as a function of volume   ¥ 0 n .

Theoretical formulations 2.2.1. Constitutive relations
When describing the electrical and mechanical characteristics of piezoelectric materials in the stress-charge form, the following constitutive relationship can be used [37]: Where s and S are the stress and strain tensors respectively. D is the electrical displacement vector is piezoelectric coupling constant matrix and e s [ ] is dielectric constant matrix, respectively.

Displacement and strains
First-Order Shear Deformation Theory may be used to describe the displacement field of an arbitrary point [38]: as seen in equation (7), where 'x' is moved up one unit and 'y' is moved down one unit. u 0 , v 0 and w 0 are displacement of the neutral surface, and β 1 & β 2 are rotations of the x and y axis cross sections, respectively. Time is denoted by t.
In equation (7), the relationship between strain and displacement is given by: Components of the electric field that are connected to potential (j) can be expressed as [39]:

Variational formulation
The electrical enthalpy H of a lamina in terms of piezoelectric, electrical energy and strain energy can be expressed as [39]: If damping is considered, dissipative function can be used to express it as: [Cs] is defined here as the damping matrix: Where a and β are the Rayleigh's coefficients [5] having values of =a 0.336 and = b 0.104. It is possible to derive the motion field equations using Hamilton's principle [40]. As a result, we can write the plate's motion equation as: It's possible to express this as: The random time points t 0 and t 1 are used in this example, δ and here is the varying operator, and ρ is the density of the plate. v and s signify the volume and surface of the plate. When surface charge (q), body forces (Fb), concentrated load (Fc), and surface traction (Fs) are all represented by the same symbol by F s , respectively.

Finite element model
For this work a 3D FEM model with plane stress condition has been considered. COMSOL ® free tetrahedral element mesh having each node with three degrees of freedom been applied on the plate. Applying correct body shape function on the model elements from the equation (13), the elements motion equation can be expressed as:

Validation
Before discussing the results of the present work. In order to assess the accuracy of the existing finite element model, we must compare our results to those already published in the scientific literature. A C-F-C-F boundary condition was used to observe and compare free vibration analysis of a FGPM plate under static electrical load with the results of Zhu et al [24]. The material properties of the FGPM plate are graded from PZT-5H at the bottom to PZT-4 at the top using a simple power law distribution. The power law distribution function may be expressed in this fashion [28]: Where P(z) is the uniform material property of the lamina at z coordinate, P t is the top surface and P b is the bottom surface material property respectively and n is the volume fraction index. The length and width of the plate are equal i.e., a = b = 100 mm and the thickness to width ratio h/b = 0.01 is considered for this validation. The nondimensional frequency parameter Ω is evaluated in COMSOL Multiphysics ® and compared with the results of Zhu et al [24] in table 2. And it is found out that comparative results presented in table 2 are in good agreement.

Convergence study
To ascertain the stability of the present 3D Finite element model in COMSOL Multiphysics ® , free vibration analysis of the FGPM plate of a = b = 100 mm and h/b = 0.01, where material property are varied by simple power law from PZT-5H to PZT-4 is considered for the convergence study. For the C-F-C-F boundary condition with varying numbers of elements, the results of the first three modes of non-dimensional natural frequencies are shown in table 3. It was utilised for the first time to determine FGPM plate's first three nondimensional natural frequencies using a three-dimensional finite element model. It was observed that if element greater than 6500, the non-dimensional natural frequency displays a decent convergence rate.

Results and discussion
The analytical results obtained from the parametric study of free vibration analysis of FGPM plate subjected to electro-mechanical loads are presented in both tabular and graphical manner.  Table 4 In this example, the non-dimensional natural frequency is shown to be affected by the volume fraction index 'n' for the boundary conditions of C-C-C-C, C-F-C-F and S,S-F-S-F respectively. Table 4(a) uses power law to grade all material attributes, while table 4(b) uses sigmoid law to grade all properties. Figure 5 depicts these findings graphically. It shows how the volume fraction index affects the non-dimensional natural frequency. The non-dimensional natural frequency is found to decrease as the volume fraction index rises. Since PZT-4's elastic modulus is greater than PZT-5H's, this means that the volume percentage of PZT-4 in the FGPM plate is smaller, which reduces the plate's stiffness. When material properties vary according to a power law, it is also believed that the natural frequency response decreases significantly as the volume fraction index rises initially before stabilizing at higher volume fraction index values. Material characteristics may be changed by the sigmoid law and have no effect on natural frequency response for any n number.
From table 4 For boundary conditions C-F-C-F and S-F-S, the nondimensional natural frequencies for material properties distribution of the FGPM plate by either power law or sigmoid law are lower than that of C-C-C-C boundary condition and boundary condition S-F-S-F, which have the lowest non-dimensional natural frequencies. This is because the FGPM plate's flexural stiffness rises as the plate's edge limitations increase.
As shown in figure 5, for the values n < 1 non-dimensional natural frequencies given by power law are more than that of non dimensional natural frequency given by sigmoid law and for n > 1 sigmoid law gives higher value non-dimensional natural frequency that of given power law for the given boundary conditions on the FGPM plate. While for n = 1, both laws gives same value non dimensional natural frequency as both laws become linear in nature. h/b nondimensional effects on the frequency spectrum Ω for various boundary conditions, such as C-C-C-C, S-F-S-F, and C-F-C-F, are illustrated in table 5, tables 6, and 7 and visually in figures 6 to 8. The volume fraction index is also displayed in figure 6(a) figure 6 (b), figures 7(a), (b), figure 8(a) and (b). Since thickness to width grows, the non-dimensional natural frequency declines. A plate's inherent frequency response and flexural stiffness both increase with increasing plate thickness. However, when the plate's thickness increases, their non-dimensional form value falls.
These figures show when power law is applied non-dimensional natural frequencies decrease considerably as volume fraction index increases initially and but change in natural frequencies becomes minimal for higher values of volume fraction index (n > 50) for all given boundary conditions and thickness to width ratios. While in sigmoid law, non-dimensional natural frequencies remain almost constant for all values of n for clampedclamped (C-C-C-C) and very slight decrease for clamped-free boundary condition. However there is an small increase is observed in non-dimensional natural frequency in the case of S-F-S-F boundary as shown in figure 8(b).

Conclusion
Analysis of free vibration of FGPM plates subjected to electro-mechanical load based on first order shear deformation theory has been investigated. The material properties of FGPM monomorph plate are considered to be varied by both power law and sigmoid law along the thickness direction. A three-dimensional finite element model using COMOSOL Multiphysics ® with tetrahedral meshing elements has been developed to carry out this study. The effectiveness of this model was verified by comparing with the results in the literature. The results presented in this study for different boundary condition, volume fraction index and geometric parameter are discussed. In free vibration analysis, non-dimensional natural frequency of FGPM plate initially decreasing considerably and then remaining almost constant with the increase in volume fraction index when material property graded by power law. When material properties are varied by sigmoid law, with an increase in volume fraction index, the non-dimensional natural frequency of FGPM plates remains virtually constant. FGPM plates have a lower non-dimensional natural frequency if the thickness to width ratio is greater. Non-dimensional natural frequencies of Clamped-Clamped FGPM plates (C-C-C-C) are greater compared to Clamped-Free FGPM (C-F-C-F) and Simply Supported Free FGPM (S-F-S-F). According to the sigmoid rule, the natural frequency of the FGPM plate grows significantly with volume fraction index when the border condition is simply supported free (S-F-S-F). FGPM plate sensors and actuators may benefit from the study's results.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).