L.Sh Esayas,Subhaschandra Kattimani

Department of Mechanical Engineering,National Institute of Technology Karnataka,Surathkal,575025,India

Keywords:Nonlinear vibration Magneto-electro-elastic (MEE) plates Active treatment constricted layer damping(ATCLD)Porosity distribution Porous functionally graded

ABSTRACTS This paper investigates the effect of porosity on active damping of geometrically nonlinear vibrations(GNLV) of the magneto-electro-elastic (MEE) functionally graded (FG) plates incorporated with active treatment constricted layer damping (ATCLD) patches.The perpendicularly/slanted reinforced 1-3 piezoelectric composite (1-3 PZC) constricting layer.The constricted viscoelastic layer of the ATCLD is modeled in the time-domain using Golla-Hughes-McTavish(GHM)technique.Different types of porosity distribution in the porous magneto-electro-elastic functionally graded PMEE-FG plate graded in the thickness direction.Considering the coupling effects among elasticity,electrical,and magnetic fields,a three-dimensional finite element (FE) model for the smart PMEE-FG plate is obtained by incorporating the theory of layer-wise shear deformation.The geometric nonlinearity adopts the von K′arm′an principle.The study presents the effects of a variant of a power-law index,porosity index,the material gradation,three types of porosity distribution,boundary conditions,and the piezoelectric fiber's orientation angle on the control of GNLV of the PMEE-FG plates.The results reveal that the FG substrate layers' porosity significantly impacts the nonlinear behavior and damping performance of the PMEE-FG plates.

1.Introduction

The latest trend has seen the advent of a new Materials for Engineering class called“Smart materials.”These materials possess self-sensing and self-actuating capabilities,which make them unique from the remaining categories of materials.Within the later past decade,multifunctional structures made of an extraordinary sort of sensitive materials known as magneto-electro-elastic(MEE)materials have gotten a handle on the attention to inquire about community predominant coupling characteristics.These materials'characteristic capability to transform energy from one form to another makes them more reasonable for vibration control.Vibration,noise,aeroelastic stability,damping,shape change,and stress distribution can all be regulated by smart structures technology.It has already found practical applications in underwater transducers,medical imaging applications,Space systems,fixedwing and rotary-wing aircraft,automotive,civil structures,machine tools,and high-frequency ultrasonic transducers [1].The ability of functionally graded material to prevent crack propagation is one of its most significant characteristics.This property makes it useful in defensive applications such as armor plates and bulletproof vests as a penetration-resistant material [2].

The natural frequencies of the MEE plates in various conditions were assessed using first and high-order shear deformation theory[3-24].The results of current research on the functionally graded magneto electro elastic beams and plates have focused on stiffness and bending issues.FE,higher-order shear deformation theory and von Karman's nonlinearity were used to analyze the variational deflection of porous functionally graded magneto-electro-elastic(PFG-MEE) flat panels with geometric skewness [25].The total potential energy stationarity concept to derive an explicit form for the FG beam's maximum stress with graded porosity is analyzed[22].A geometric model of non-linearity analysis to study the case of axisymmetrical deflection and buckling of a circular porous,cellular plate [26].The nonlinear dynamic behavior of FG-CMTR shell [27] and elastoplastic ceramic/metal functionally graded material (FGM) [28] with improved FSDT was presented.The dynamic and free vibration of FG-NRC Plates and shell structure under dynamic load was investigated using the meshfree radial point interpolation method and the linear double director shell theory[29,30].

In recent works,the different numerical methods were applied to analyze the natural frequency,buckling,deflection,and stress of functionally graded plates using higher-order shear deformation theories to avoid using the shear correction factor in FSDT.The stress function approach is used in combination with higher shear deformation theory (HSDT) to analyze mechanical buckling of simply-supported FG,and FG-CNT reinforced sandwich plates[31,32].

A new innovative three unknowns trigonometric shear deformation theory is proposed for the buckling and vibration responses of exponentially graded sandwich plates resting on elastic mediums [33].Moreover,refined shear deformation theory performs complex analysis of the FG-sandwich plate seated on an elastic base with different support types presented and analyzed [34].

A structure exposed to strong vibrations during its lifetime,which causes resonance,leads the structure to fail.Therefore,controlling these unsafe vibrations plays an important role.Piezoelectric materials have gradually been used as discrete sensors and/or actuators for active vibration control in high-performance,lightweight smart structures over the last decade [35-42].

This increasing demand is because of its inherent properties with direct and reverse piezoelectric results.Due to their direct piezoelectricity,they are used as sensors because they are capable of causing electrical potential/charge when subjected to mechanical stress,and their ability to deform when exposed to external voltage/charge made them useful as actuators.Flexible structures when,in addition to the layer/patch of the materials serving as distributed sensors and/or actuators,generally are referred to as“smart structures.”The effectiveness of intelligent structures is focused on the degree of the piezoelectric constants of stress or strength.Preliminary research has shown that incorporating patches of monolithic piezoelectric material supplied with external voltage can effectively control any structure's vibrations.Studies have shown that a patch made of piezoelectric fiber reinforced epoxy matrix (1-3 PZC) has greater damping capacity than a monolithic piezoelectric patch.This paved means for active treatment constricting layered damping(ATCLD)behavior and is step by step emerged as a predominant technique to reduce the structure's unwanted vibrations[18].In the active control layer damping,1-3 PZC patch used to control the supplied voltage,the visco-elastic layer with low stiffness has bonded to the substrate structures.Therefore,the main structure improves the dominant vibration capability of the ATCLD patch by manufacturing high shear deformation,thereby leading to higher energy dissipation [43].The ATCLD treatment can be effectively used as a passive constricting layer damping (PCLD) treatment with the absence of control voltage to the confining layer.

Consequently,the system can include the advantages of each ATCLD and PCLD at the same time.Methods for vibration damping are divided into passive and active control.Forces within the active force context,the control system is produced by a wide range of actuators with hydraulic,pneumatic,electromagnetic,piezoelectric,or motor-driven ball-screw actuation [44].

A new class of advanced intelligent materials has recently been introduced,known as magneto electric-elastic functionally graded(MEE FG),used in sensors and actuators.In these plates,the material properties gradually differ over the thickness,enhancing the efficiency and the lives of multilayered materials compared to ordinary multilayered plates.Material properties change suddenly at the border of the layer.Fakhari and Ohadi [45] studied the geometrically nonlinear vibration regulation of the functionally graded and smart laminated composite plate under thermal and transverse mechanical loads using the integrated piezoelectric sensor/actuator layer patch.With the aid of a broad rotary shell configuration for non-dynamic and transient piezoelectric studies,the FE model integrates thin-walled composite materials proposed using the first-order shear deformation model[46].They found six kinematic parameters represented with five nodal degrees of freedom (DOF) to implement the nonlinear rotation theory.The free vibration of non-homogenous FG-MEE cylindrical shells under SSS supported boundary conditions was investigated by Ref.[47].An active and passive vibration control method has been developed and experimentally analyzed for a thin-walled composite beam by Ref.[44].The performance of 1-3 PZC has been investigated in different positions as a constrained layer of ATCLD to control the large deformation vibrations of FG-MEE plates subject to temperature fields [48].A hypothesis of large-amplitude vibration composite plates with piezoelectric stiffeners was proposed by Refs.[49,50] found the advanced nonlinear model of piezoelectric laminated plates.Vinyas et al.[51]evaluated the effect of ATCLD on the precise regulation of frequency response of FG-MEE plates using FEM.Newtonian and variational formulations have been developed to predict laminated plates'dynamics under passive and active vibration damping control [52].The FE analysis of ATCLD of large deformation vibration of magneto electro elastic plates and doubly-curved shells is performed [53,54].

Porosity can occur inside the materials during the sintering process in FGM fabrication,which contrasts with the unsafe highperformance composite material.As shown by many studies on the topic,the effect of this failure has received much interest.By simple quasi-3D hyperbolic theory,the impact of Winkler/Pasternak/Kerr foundation and porosity on the dynamic behavior of FG plates is investigated[55].Bending and buckling study of PFG-plate under mechanical load is presented by taking into account material imperfections and using refined shear deformation theory [56].Employing the honeycomb network as the structure's core improves its dynamic response using the generalized differential quadrature method studied [57].The bending response of PFG plates is studied using a cubic shear deformation theory [58].The effects of the porosity parameter,the power-law index,the aspect ratio,and the thickness ratio on the bending and vibration of a porous FG board were investigated using quasi-3D hyperbolic shear deformation theory [59].The influence of imperfections on nanobeam is studied using nonlocal nth-order shear deformation theory[60].

Using an edge-based smoothed FE method process,static bending and free vibration of FG porous varying thickness plates are correlated with the mixed interpolation of three-node triangular element tensor components methodology,so-called ESMITC3 [61].The valuable concepts of FG porous (FG-P) materials have recently been developed by Refs.[23,62-65].The analysis of FG-P microplates' in-plane behavior with porosity using modified couple stress methods studied by Ref.[66].According to the preceding discussion,many efforts have already been made in recent years to study the static and dynamic behavior of smart composite structures integrated with AFC and/or MFC layers using various kinematic models such as classical laminated plate theory (CLPT),first-order shear deformation theory (FSDT),and higher-order shear deformation theory (HSDT).As per the authors' knowledge,no study has been published on the effect of porosity on active damping of geometrically nonlinear vibrations of a functionally graded magneto-electro-elastic plate integrated with either a center or an edge constrained PZC patch layer with a constricted viscoelastic layer of the ATCLD using the von K′arm′an nonlinear and the time-domain using Golla-Hughes-McTavish (GHM) technique in the framework of the layer-wise shear deformation theory.In this regard,the present work attempts to present the effect of porosity on the active damping of geometrically nonlinear vibration of FG-MEE plates.

In this study,the porosity index,the porosity distribution type,and the power-law index on the effectiveness of perpendicularly/slanted 1-3 PZCs as active treatment constricted layer damping(ATCLD) material for controlling GNLV of PMEE-FG plates are investigated.

2.Governing equations and problem descriptions

Fig.1(a)and1(b)illustrate a schematic diagram of the PMEE-FG plate integrated with a central and edge patch of the active control layer damping treatment located on the top surface.The PMEE-FG plate measures a,b,and h in length,width,and overall thickness.The constricting piezoelectric thickness and the bonded viscoelastic layer(VE)thickness of the ATCLD are hand h,respectively.The substrate of these intelligent PMEE-FG plates is of the same thickness in three layers.The top and bottom layers are FG,while the middle layer is uniform with stacking series FG/F/FG,and FG/B/FG is the top/middle/bottom layer in which B stands for the piezoelectric (PE) material such as BaTiO.At the same time,F indicates piezomagnetic material (CoFeO).The deformed and undeformed normal kinematics of the xz-and yz-planes schematically shown in Fig.2(a)and(b),respectively.The constricting layer of the ATCLD treatment consists of 1-3 PZC material with perpendicularly/angled reinforced as seen in Fig.3.The xz-and the yz-plane are coplanar in the piezoelectric fibers.At the same time,their zaxis orientation angle is λ.In a slanted reinforced 1-3 PZC,the orientation angle is non-zero,while the vertically reinforced 1-3 PZC is zero.The edge of the PMEE-FG plate and the angle of orientation of the piezoelectric fibers of the 1-3 PZC patch are θ and λ,respectively.

Fig.1.(a) Porous Functionally Graded MEE plate with a central patch of the ATCLD treatment (b) Porous Functionally Graded MEE plate with edge patch of the ATCLD treatment,(c) Kinematics of deformation of the PMEE-FG plate in XZ and YZ.

The effect of porosity on the active damping of the PFG-MEE plate is studied using a layer-wise shear deformation theory(LWSDT).This study aims to determine how additional smart elements are required to control the PFG-MEE plates'vibrations.

Based on the layer-wise shear deformation theory,the axial deflections and any point along the x-and y-directions of the entire PMEE-FG plate can be written as [54,67,68].

where Δ（z）＝z-〈z-h/2〉Δ（z）＝〈z-h/2〉-〈z-h〉,and Δ（z）＝〈z-h/2〉-〈z-h〉 in which the Δ（z）,Δ（z）and Δ（z）are used in Eq.(1)and Eq.(2)specify the necessary singularity functions so that the interface accomplishes the displacement consistency between the VE and the PE layer or between the substrate plate and the VE layer.

In the xz-plane,the rotations of the normal lying in the base plate,the viscoelastic layer,and the piezoelectric layer are expressed by θφ,and γ,respectively,while in the yz-plane,they are represented by θ,φ,and γ.

A new high-level flexural deformation theory has been assumed for the overall plate to use the ATCLD layer's perpendicularly/angled actuation and obtain accurate results.At every position throughout the entire PMEE-FG plate,the displacement across the substrate can be expressed as:

In Eqs.(1)-(3),uv,and ware the translational displacements at any points on the substrate's mid-plane along x,y,and z directions,respectively with θand φare generalized rotational displacements.The generalized variables are divided into rotational variables ｛d｝and ｛d｝translational displacement variables ｛d｝for ease of study,as follows:

A selective integration rule is applied to mitigate the thin shear lock structures and measure the element rigidity matrices for crossshear deformations.The strain vector state ε,which includes inplane strains and normal transverse strain,and the transverse shear strain vector｛ε｝is also expressed at any point in the overall plate to accomplish this task as follows:

The normal strains along x-,y-and z-directions are ε,εand ε;the in-plane shear strain and the transverse shear strains are represented εεε.Using the nonlinear strain displacement relationship of von K′arm′an at each point in the plate,VE layer,or PE layer,the in-plane strain vectors and the normal across strains may be defined as:

Fig.2.(a) Evenly distributed P1 pores,Unevenly distributed P2 and P3 pores.(b) Unevenly distributed P2 and P3 pores.

where k＝1,2,3 specifies the number of a layer starting from the bottom,at any point in the substrate,the viscoelastic layer,and the piezoelectric actuator layer,respectively,the strain vectors representing the state of cross shear strains can be stated as follows:

The transformation matrices ［z]-［z] in Eqs.(6)-(8) given by Ref.[18].

3.Modeling porosity distributions

The pores' occurrence is modeled using various mathematical functions [26,69-71] during the manufacturing process,such as exponential law,power law,and cosine rule.Two different types of porosity distributions are adopted to analyze FGM structures,namely,even distribution (P1) and uneven distributions (P2 and P3),as shown in Fig.2(a)and(b),respectively.The useful material properties of the PMEE-The useful material properties of the PMEE-FG plate in the direction of thickness followed the modified distribution of the power as defined[11].

where Pfg is the generalized term to represent the material properties,α is the porosity index,Pthe material properties of CoFeO,Pis the material properties of BaTiO,and Vis the generalized term to describe the different porosity volume distributions(i.e.,P1,P2,and P3)while,α is the porosity index(0?α 1),and Vis given by Ref.[72].

where r is the power-law gradient for different porosity distributions.The equation of different porosity distributions rewritten as[69,71,73-76].

(a) For uniform porosity distribution modelP＝1.

4.Constitutive equations

The electrical and magnetic fields applied are considered to act along the same lines.The z-directional only.Therefore,the coupled constitutive equations create MEE material exhibiting interactions between magnetic,elastic,and electrical equipment.Elastic fields are reduced along with the primary material coordinate axes as follows [77].

The structure is considered of 1-3 PZC making the ATCLD layer responsive to the current system of numerical simulation formulation is defined by:

Fig.3.1-3 PZC layer with(a) vertically reinforced,(b)coplanar with xz plane,and (c)coplanar with yz plane.

The present research has been conducted within the time domain of the general PMEE-FG plate.This has led to implementing a viscoelastic modeling framework by the Golla-Hughes-McTavish (GHM) process.In the time-domain analysis,the fundamental equation for the linear,isotropic,and homogeneous VE variable is defined and indicated in Riemann-Stieltjes integral form [78].

G(t) is the viscoelastic material's relaxation function,and the virtual work principle is used to derive the overall system's equations can be written as [79].

where,ρand vare the mass density and volume of kth layers,respectively.｛f｝＝[0 0 P]is a surface traction force acting over area A with the transverse load P.The symbol δ indicates the first variation.

Per Maxwell's equations,the electrical (E) and magnetic (H)fields are linked to the electrical(φ),and magnetic potential(ψ)are given in the following forms [80].

5.Finite element formulation of the porous functionally graded magneto-electro-elastic plate integrated with ATCLD patches

An isoparametric quadrilateral element of eight nodes is used to model the geometry of the PMEE-FG plates.It is possible to represent the generalized DOF as follows:

In which,Nis the natural coordinate shape function and［N]［N]［N],and ［Nψ] shape function matrices.

Therefore,the electric and magnetic fields shown in Eq.(19)can be described as follows:

The nodal generalized displacement vectors can be represented as follows at every point within the element:

in which the nodal strain-displacement matrices［B]（ii＝tb，rb，1，2，ts，and rs）are expressed in Appendix (I).Going to substitute Eqs.9-12,Eqn.(18),Eqn(20),and Eq.(24)into Eq.(16)Eqn.the following open-loop elemental motion formulas for the PMEE-FGplate coupled with the ATCLD formulation was obtained[51,54].

where,

Various matrices and vectors are appearing in Eq.(25b) are shown in Appendix (I).

Once the transformation has been done,the coupled motion equilibrium equation of the ATCLD PMEE-FG plate can be expressed globally following the transformations given by Refs.[18,54].

Table 1 Material properties of BaTiO3,CoFe2O4 and 1-3 PZC patch [83,87].

Table 2 Verification of the present FE formulation for B/F/B stacking sequence.

Fig.4.Passive response of a SSS cross ply PMEE-FG plates with different porosity index(a＝150 h,Vp＝P1,r＝1).

Table 3 The effect of porosity distribution,patch position,and porosity index on the maximum center deflection of PFG-B/F/B stacking sequence plate.

Fig.5.Non-linear transient responses of a SSS cross-ply for different power-index values (a＝150 h,Vp＝P1) PMEE-FG plate.

Fig.6.Nonlinear transient responses of a SSS cross-ply for different values of porosity index (a＝150 h,Vp＝P1,r＝1) PMEE-FG plate.

Fig.7.Nonlinear transient responses of a SSS cross-ply for different types of porosity distributions (a＝150 h,α＝0.3,r＝1) PMEE-FG plate.

where the global mass matrix is ［M];［K]［K]［K]［K]［K],and［K]are global elastic stiffness matrices;［K]and ［K]are global coupled electro-elastic stiffness matrices;［K]and［K]are coupled magneto-elastic stiffness matrices;［K]and ［K]electrical and magnetic stiffness matrices;Mechanical force vectors FandF,global generalized nodal displacement vectors X and X,electric and magnetic force vectors Fand F,respectively.For the time being,q stands for the cumulative number of patches and Vstands for the jth patch's voltage.In the absence of the applied control voltage,the coupled global equations above impose the plate's passive (uncontrolled) restricted layer damping.

Following mini-oscillator terms,the material module s～G（s）'s feature uses the GHM model for the VE material in the time domain[81,82].

where,Gfactor corresponds to the equilibrium modulus value,the relaxation function G(t).Each mini-oscillator expression has three positive constants and is a second-order rational function(φξ?).These constants regulate the shape of the modulus function over the complex s-plane.Any number of mini-oscillator terms can be used in the GHM expression,depending on the nature of a material module function and the range of s over which it is to be modeled.

Using the new dissipation coordinate,

The different matrices of global stiffness and vectors of force that appear in Eqs.34-36 are given in Appendix (II),Now,Eqs.34-36 are combined to obtain in the time domain the global openloop equations of motion as follows:

Besides,for a closed-loop model,a simple derivative control law has already been adopted.Consequently,the control voltage states the derivative of global nodal degrees of freedom:[83].

6.Results and discussions

The numerical results are evaluated in this portion employing the FE model derived in Ref.[84] to evaluate the ATCLD patches'performance in controlling the PMEE-FG plates' geometrically nonlinear vibrations.The position of the patches for ATCLD is shown in Fig.1.The material properties of pure piezoelectric,pure piezomagnetic,and 1-3 PZC are given in Table 1.The VE layer thickness (hv)and the constricting 1-3 PZC layer(hp)are 200 μm and 250 μm,respectively.The numerical results are measured for the PMEE-FG plate that has either a single central ATCLD patch(see Fig.1(a)) or plate edge patches (see Fig.1(b)).In this analysis,the PMEE-FG substrate's geometrical dimensions are as follows:a＝1 m;b＝1 m,and h＝0.003 m in thickness.Also,the details of the VE layers used are as follows [75].

The simply supported(SSS)and clamped(CCC)constraints used in this research are as follows:

For SSS

The electric and magnetic potentials at the boundaries are assumed to be zero for x＝0，a(SSS) and for y＝0，b (CCC) in two boundary condition cases.In PMEE-FG plates,applying the GHM approach for modeling ATCLD treatment has been validated [86].

6.1.Verification of FE formulation

6.2.Discussions

Fig.4 presents the effects of porosity index on the SSS cross-ply PMEE-FG plate's uncontrolled dynamic responses with different porosity indexes (α＝0,0.1,0.2,and 0.3).It may be seen from this figure that with identical geometrical parameters and material properties,case 1 (α＝0,Vp＝P1) of the porosity index gives the minimum value of the central deflection response of the PMEE-FG plate.In contrast,case 4 (α＝0.3,Vp＝P1) of the porosity index gives the maximum value due to lower stiffness.Besides,it is also noted that case 1 of the porosity distribution helps to reduce the oscillation better than the remaining three cases of the porosity index.Table 3 depicts the effect of porosity distribution,patch position,and porosity index on the maximum center deflection of the PMEE-FG plate with the B/F/B stacking sequence.It can also be noted that P2 type porosity distribution with lower porosity index for center patches gives the minimum central deflection response due to high stiffness.Table 3 also reveals that a single ATCLD patch's output mounted in the middle of the plate's top surface to attenuate nonlinear transient vibrations is slightly higher than the patches located at the opposite ends.Fig.5 presents the plate's nonlinear transient responses for different power indexes and constant values of control gain (K＝600) and porosity index α at(a＝150 h,Vp＝P1).Fig.5 observed that the increase of power index leads to the rise of the substrate plate PMEE-FG stiffness,and hence the deflection of the plate decreases.

Fig.8.Nonlinear transient responses of a SSS cross-ply for different types of porosity distributions and porosity index (a＝150 h,r＝1) of PMEE-FG plate.

Table 4 Effects of porosity distribution and porosity index on the maximum control voltage for different control gain(Vmax×102 V) SSS support.

Table 5 Effects of porosity distribution and porosity index on the maximum control voltage for different control gain(Vmax×102 V) CCC support.

Next,we consider the effects of the porosity index on the nonlinear transient responses of SSS cross-ply PMEE-FG plate for values of a＝150 h,Vp＝P,r＝1.It can also be seen from Fig.6 that for the higher values of the porosity index,the transient deflection is most elevated (α＝0.3 ＞0.2＞0.1).Whereas in Fig.7,the P2 porosity volume distribution exhibits the lowest nonlinear transient deflection and P1 porosity distribution with the same value of the porosity index under SSS support.Fig.8 shows that the maximum deflection was obtained at α＝0.3 and Vp＝Pfor the center patch with simply supported PMEE-FG plate.In comparison,the minimum deflection was obtained at α＝0.1 and Vp＝P.Tables 4 and 5 illustrate the effect of porosity distribution and porosity index on the maximum control voltage.It can be seen that for a more considerable value of Kand P2 type porosity distribution,the ultimate control voltage for a given porosity index is minimum compared to the other values.The piezoelectric fibers may be perpendicularly aligned with the xz-plane or with the yzplane.In contrast,the values of λ vary from 45 to-45as regards the vertical z-axis.In Eq.(43),the efficiency index (E) to compute control authority for the obliquely placed 1-3 PZC constricting layer is used to quantify the ATCLD patch's performance to control large fluctuations of the plates.After 0.15 s and the initial time,Egives the amount of amplitude repression at the point(a/2,b/2,h/2)of the entire PMEE-FG plate experiencing transient nonlinear vibration as follows [54].

Fig.9 presents the effect of 1-3 PZC layer's fiber orientation angle on the nonlinear transient responses for the SSS cross-ply PMEE-FG plate.The minimum deflection value is higher for oblique angles(λ＝15,30,40).The maximum center deflection is at the lowest value of oblique angle (λ＝0).The backbone curves shown in Figs.10 and 11 depict the frequency ratio variations(ω/ω)and the plate's non-dimensional transverse deflection (w/h).The magnitude of exciting pulse load responsible for causing nonlinear deflections is 0.2.It can be seen that the backbone curves demonstrate the hardening type nonlinearity.The influence of patch location,porosity distribution,and boundary form is also presented in Figs.10 and 11.Fig.12 illustrates the effect of different edge forms on the maximum central deflection of the PMEE-FG plates.It can be noted that the CCCC boundary condition results in a lower deflection for three types of porosity distributions,namely P1,P2,and P3.

Fig.9.The effect of 1-3 PZC layer fiber orientation angle on nonlinear transient responses for SSS cross-ply PMEE-FG plate.

Fig.10.Backbone curves for simply supported PMEE-FG plates with different patch positions.

Fig.11.Backbone curves for a clamped supported PMEE-FG plates with different patch positions and porosity distribution.

Fig.12.The center patch's non-linear transient responses with SSS and CCC support different types of porosity distributions (a＝150 h,r＝1,α＝0.1) cross-ply PMEE-FG plate.

In comparison,the distribution of porosity of type P2 has a predominant effect on the central deflection than the distribution of porosity of types P1 and P3.This may be attributed to the higher values of the substrate layer's elastic constants.

6.3.Conclusions

Numerical analyses based on the finite element methods and MATLAB code are carried out for studying the influence of porosity on the active damping of FG-MEE plates.The influence of porosity index and porosity distribution effect with various related parameters were thoroughly investigated and described in detail.

The main conclusions are summarized as follows:

1.The Ptype porosity distribution with a lower porosity index for center patches gives the minimum central deflection response.

2.The maximum center deflection is at the lowest value of oblique angle(λ＝0)and Ptype of porosity distribution,followed by Pand Pporosity distributions.

3.The increase of power index leads to the rise of the PMEE-FG plate's stiffness for all porosity distribution,and hence the deflection of the plate decreases.

4.The CCCC boundary condition results in a lower deflection for three types of porosity distributions due to the substrate layer's elastic constants' higher values.

5.It reveals that the output of a single ATCLD patch mounted in the middle of the plate's top surface to attenuate the nonlinear free vibration of large deflection is slightly higher than the patches located at the opposite ends with the constant porosity index and the same control gain.Because piezoelectric materials cause high strain on the center of the host structure,the piezoelectric patch's most efficient location is in the host middle structure's strain rate areas.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Elemental stiffness matrix.

The elemental stiffness matrices are represented in Eq.(25)corresponding to the bending deformations and transverse shear deformations are:

The numerous rigidity matrices and rigidity vectors are

This article does not contain any studies with human participants or animals performed by any of the authors.

No funds,grants,or other support were received for conducting this study.