ACTIVE DAMPING OF SMART FUNCTIONALLY GRADED SANDWICH PLATES UNDER THERMAL ENVIRONMENT USING 1-3 PIEZOELECTRIC COMPOSITES

Transcription

1 Procdings of the 22 th National and th International ISHMT-ASME Heat and Mass Transfer Conference December 28-3, 23, IIT Kharagpur, India HMTC345 ACTIVE DAMPING OF SMART FUNCTIONALLY GRADED SANDWICH PLATES UNDER THERMAL ENVIRONMENT USING -3 PIEZOELECTRIC COMPOSITES R. Suresh Kumar Dept. of Mechanical Enginring Indian Institute of Technology Kharagpur (WB) 7232, India S. I. Kundalwal Dept. of Mechanical Enginring Indian Institute of Technology Kharagpur (WB) 7232, India M. C. Ray Dept. of Mechanical Enginring Indian Institute of Technology Kharagpur (WB) 7232, India ABSTRACT The present study is concerned with the active control layer damping (ACLD) of the geometrically nonlinear vibrations of the smart functionally graded (FG) sandwich plates subjected to the thermal loading. The constraining layer of the ACLD treatment is considered to be made of the vertically/obliquely reinforced -3 piezoelectric composite (PZC) while the viscoelastic material is modeled by using the Golla Hughes McTavish (GHM) method in the time domain. A thr dimensional coupled nonlinear electromechanical finite element (FE) model has bn developed based on the first order shear deformation theory (FSDT) and Von Karman type nonlinear strain displacement relations to investigate the damping characteristics of the FG sandwich plates subjected to the thermal loading. Several FG sandwich plates with different thermal loading conditions have bn investigated to evaluate the numerical results. Effects of metal or ceramic rich boom surfaces subjected to the thermal loading, the variation of power-law index on the control authority of the ACLD patches have bn thoroughly investigated. Emphasis has also bn placed on investigating the effect of the variation of piezoelectric fiber orientation angle on the performance of the ACLD patches for controlling the geometrically nonlinear transient responses of FG sandwich plates. LIST OF ABBREVIATIONS ACLD Active Constrained Layered Damping FGM Functionally Graded Material FE Finite Element FSDT First Order Shear Deformation Theory PZC Piezoelectric Composite GHM Golla-Hughes-McTavish NOMENCLATURE aa b HH VV h Length (m) of the sandwich plate Width (m) of the sandwich plate Thickness of the substrate sandwich plate Alied control voltage Thicknesses of the top h cc h vv h rr mm II dd ΨΨ xx, yy, zz uu, vv, ww (uu cc, vv cc, ww cc ) (uu bb, vv bb, ww bb ) (uu, vv, ww ) (uu vv, vv vv, ww vv ) (uu, vv, ww ) θθ xx, φφ xx, αα xx, ββ xx, γγ xx and boom facing Half the thickness of the core Thickness of the viscoelastic layer Thickness of the -3 PZC layer Power law index Number of ACLD patches Performance index Piezoelectric fiber orientation angle (deg) in the -3 PZC layer Cartesian coordinates Translational displacements (m) at any point on the mid plane along xx, yy and zz directions Displacement fields xx, yy and zz directions, respectively at any point in the core Displacements xx, yy and zz directions at any point in the boom facing Displacements xx, yy and zz directions at any point in the top facing Displacements xx, yy and zz directions at any point in the viscoelastic layer Displacement xx, yy and zz directions at any point in the -3 PZC layer Rotations (rad) of the normals about the mid planes of each layer about the yy axis

2 θθ yy, φφ yy, αα yy, ββ yy, γγ yy θ z, ϕ z, α z, β z, γ z εε xx, εε yy, εε zz εε xxxx, εε yyyy, εε zzzz σσ xx, σσ yy, σσ zz σσ xxxx, σσ yyyy, σσ zzzz ρρ, ρρ mm, ρρ cc υυ, υυ mm, υυ cc VV kk (kk =, 2 aaaaaa 3), EE cc [CC bb ] mm, [CC ss ] mm [CC bb cc ], [CC ss cc ] [CC bb ], [CC bbbb ] bb, ss iiii DD zz, EE zz PP, KK dd TT, TT kk T p e, T k e δδ AA, Rotations (rad) of the normals about the mid planes of each layer about the xx axis Gradients (rad) of the transverse deformations of each layer about the zz axis Normal strains In-plane and out of plane transverse shear strains Normal stresses In-plane and out of plane transverse shear stresses Densities of the flexible core, metal and ceramic materials Poissons ratios of the core, metal and ceramic materials Volume fraction of the boom layer, the core and the top layer of the substrate FG sandwich plate Young s modulus of the metal and the ceramic materials Elastic coefficient matrix for the metal and the core Elastic coefficient matrix for the core Elastic coefficient matrix and elastic coupling constant matrix for the -3 PZC layer The piezoelectric coefficient matrix and the dielectric matrix for the -3 PZC layer Electric displacement and Electric field along Z axis Alied mechanical load (NN mm 2 ), control gain The potential energy and the kinetic energy of the overall plate The potential and the kinetic energy of element integrated with a patch of the ACLD treatment Variational operator Surface area and the Volume of the smart FG plate Elemental length and aa, bb width under consideration Elemental {d t }, {d r } displacement vectors Nodal shape function [N t ], [N r ] matrices {d e t }, {d e Nodal displacement r } vectors Shape function n i associated with the ii h node Elemental strains {ε b } c, {ε b } t, {ε b } b, {ε b } p vectors expressed in {ε s } c, {ε s } b, {ε s } t, {ε s } v, {ε s } p terms of nodal displacement vectors [B tb ], [B rb ], [B ts ], [B rs ], Nodal strain [B ], [B 2 ] displacement vectors [K e ], [K e tr ], [K e rt ], [K e rr ] Elemental stiffness e e e [K tsv ], [K trsv ], [K rrsv ] matrices [K ], [K tr ], [K rt ], [K rr ], Global stiffness [K tsv ], [K trsv ] [K rrsv ] matrices F e e tp, F tpn, F e rp Elemental electroelastic coupling matrices F tp, F tpn, F rp Global electro- elastic coupling vectors {F}, {F e } Global nodal load vector and elemental load vector [M], [M e ] Global mass matrix and elemental mass matrix {X t }, {X r } Vectors of translational and rotational global nodal degrs of frdom L Laplace operator X t, X r, {F } Laplace transforms of vectors of translational, vectors of rotational global nodal degrs of frdom and global nodal load vector GG() Material relaxation function of the viscoelastic material sg (s) Material modulus function GG, αα kk, ωω kk, ξξ kk GHM parameters Z, Z r Auxiliary dissipation coordinates Z (s), Z r (s) Laplace transforms of the auxiliary dissipation coordinates Frequency (rad ss) (ωω NNNN /ωω LL ) Nonlinear fundamental frequency ratio [C d ] Active damping matrix

3 INTRODUCTION Amongst all possible existing composite structures, the idea of sandwich construction has become increasingly popular over the last thr decades in aerospace, automobile, locomotive and construction industries because of the development of man-made cellular materials as core materials. The configuration of sandwich structure is a thr-layer type of construction in which a light weight, thick flexible core is sandwiched betwn two stiff, strong face shts [-9]. The face shts are basically made up of unidirectional orthotropic laminated FRP composites, while the core is thick layer of low-density material like foam polymer or honeycomb material made of resinimpregnated aramid paper (Nomex or Kevlar) [7]. The separation of the face shts by the core increases the moment of inertia of the structure thereby producing an efficient structure for resisting bending and buckling loads. The strength and the stiffness of sandwich construction are increased without an areciable increase in the corresponding weight of the structure [4]. However, the abrupt change in material properties across the interface layers may result in the development of high interlaminar stresses which may lead to the debonding of the interfacial layers. In order to alleviate such adverse effects, a new class of non-homogenous composite materials known as functionally graded materials (FGMs) has earned a considerable aention in structural alications. These materials are characterized by a continuous variation of material properties particularly along the thickness direction which eliminates the occurrence of interface debonding betwn the layers. Yamanouchi et al. and Koizumi [, ] developed these novel materials where the continuous variation of material properties is a result of the change of composition and structure over the volume. The simplest form of FGMs is where two different material ingredients change gradually along the thickness of the structure from one to the other. Typically, FGMs are made from a mixture of ceramic and metal or a combination of different materials owing to their alications. The ceramic composition in the FGM offers thermal resistance while the metallic composition toughens and strengthens the structure. A great deal of research has bn carried out to analyse the behaviour of the structures made of FGMs. Feldman and Aboudi [2] carried out the buckling analysis of the FG plates subjected to uni-axial loading. Loy et al., [3] have studied the vibration of FG cylindrical shells using Love s shell theory. Zenkour [4] studied the bending analysis of laminated and sandwich elastic beams by the transverse shear and the normal deformation theory. Reddy [5] presented the solutions for the FG rectangular plates using his third order shear deformation theory. Sankar [6] derived the elastic solutions for the FG beam using a simple Euler Bernoulli beam theory. Vel and Batra [7] have presented an exact thr-dimensional solution for the thermoelastic deformations of FG simply-suorted plates of finite dimensions. Zenkour [8] developed generalized shear deformation theory for bending analysis of FG plates considering the Layerwise model and the symmetry of the plate about the mid-plane. Zenkour [9, 2] also carried out the stress and the vibration analyses of the FG sandwich plates with face layers being made of non-homogeneous isotropic ceramic-metal FGMs while the core layer is made of a homogenous ceramic material. Over the past few decades, the use of piezoelectric materials for achieving the active control of the enginring structures has bn the focus of many researchers. The converse and the direct piezoelectric effects of piezoelectric materials are exploited to use them as distributed actuators and sensors. Flexible structures integrated with such piezoelectric sensors and actuators possess self-monitoring and self-controlling capabilities and are customarily known as smart structures [2-26]. The major drawbacks of the existing monolithic piezoelectric materials is that the magnitudes of their piezoelectric coefficients are very low and hence, large control voltage is necessary to achieve significant control of vibrations of host structures. Subsequently, research devoted to the efficient use of these low-controlauthority monolithic piezoelectric materials led to the development of active constrained layer damping (ACLD) treatment [27]. In ACLD treatment, the constrained layer is made of viscoelastic material while the constraining layer is made of piezoelectric material which may be active or passive under operation damping the host structure. When the treatment is integrated with a base structure (substrate) and its constraining layer is activated with an aropriate control voltage, the transverse shear deformations of the viscoelastic layer are enhanced causing improved damping characteristics of the overall structure. If the constraining layer is not activated, the treatment plays the role of passive constrained layer damping (PCLD) treatment. Thus the ACLD treatment provides the aributes of both active and passive damping. Hence, the ACLD treatment has gained tremendous importance for efficient and reliable active control of flexible smart structures. Piezoelectric composite (PZC) materials [28-32] have now emerged as new class of smart materials having wide range of effective material properties which are not offered by the existing monolithic piezoelectric materials. In PZCs, the reinforcements are made of the existing monolithic piezoelectric materials and the matrix is the conventional epoxy. Among the various PZC materials studied by the researchers till today, laminae of the vertically/obliquely reinforced 3 PZC materials are commercially available [3] and are being effectively used in medical imaging alications and high frequency underwater transducers. The constructional feature of a layer of the vertically/obliquely reinforced 3 PZC material is illustrated in Fig. (a) where the reinforcing piezoelectric fibers are coplanar with the vertical xxxx or yyyy plane making an angle ΨΨ with respect to the zz axis while the fibers are poled along their length. For the vertically reinforced 3 PZC the value of ΨΨ is zero and in case of the obliquely reinforced 3 PZC its value is nonzero. Flexible sandwich structures are prone to undergo large amplitude vibrations due to very low internal damping. Extensive studies on the large amplitude fr vibrations of composite sandwich structures and FG structures using aroximate analytical and finite element methods have bn reported by many researchers [33-39]. However, very few studies concerning the geometrically nonlinear dynamic analysis of smart composite and FG structures are available. Pai et al. [4] developed a refined model for nonlinear analysis of composite plates integrated with piezoelectric sensors and actuators. Ray and Baz [4] developed a variational model to investigate the control of nonlinear vibrations of beams using ACLD treatment. Reddy and Cheng [42] carried out a thr dimensional analysis of smart FG plates. Kulkarni and Bajoria [43] derived a FE model based on the higher order shear

4 deformation theory for the geometrically nonlinear analysis of smart thin and sandwich plates. Panda and Ray [44] studied the nonlinear static behavior of the FG plates integrated with a layer of horizontally reinforced piezoelectric fiber reinforced composite. Although the piezoelectric actuator layer acts more efficiently as the constraining layer of the ACLD treatment, researchers did not pay much aention to the use of the ACLD treatment for controlling the nonlinear vibrations of composite sandwich and FG structures. Recently, Panda and Ray [45] investigated the performance of a proposed horizontally reinforced PZC, commercially not available yet, as the material of the constraining layer of the ACLD treatment for active damping of nonlinear vibrations of FG plates. Sarangi and Ray [46] carried out ACLD of geometrically nonlinear vibrations of laminated composite beams using vertically/obliquely reinforced -3 PZC material as the constraining layer. Most recently Kumar and Ray [47] derived a FE model for the geometrically nonlinear vibrations of composite sandwich plates integrated with the ACLD patches using layerwise FSDT model. The above study by Kumar and Ray [47] motivated the authors to report the effect of the performance of the ACLD patches on geometrically nonlinear vibrations of FG sandwich plates made of FG facings separated by homogenous isotropic core ceramic core/flexible HEREX core. The dynamic characteristics of the FG sandwich plates with non-homogenous isotropic FG top and boom face layers separated by a homogenous isotropic ceramic core [2] or flexible HEREX core [4] may be different from that of the conventional FG plates. Hence, the necessity for the further study on the ACLD of geometrically nonlinear vibrations of FG sandwich plates arises. However, no work is reported yet on the ACLD of geometrically nonlinear vibrations of FG sandwich plates using vertically/obliquely reinforced 3 PZC materials. In the present investigation, the performance of the ACLD treatment using vertically/obliquely reinforced -3 PZC material as the material of the constraining layer of the ACLD treatment for active damping of geometrically nonlinear vibrations of FG sandwich plates has bn studied. The top and the boom faces of the sandwich plate are made of FGMs whose properties vary along the thickness according to a power law distribution in terms of the volume fraction of the constituents while the core may be a hard or a soft material. A thr dimensional finite element model has bn developed for the FG sandwich plates integrated with the patches of the ACLD treatment. The viscoelastic layer of the ACLD treatment has bn modeled using the Golla Hughes McTavish (GHM) method [48-5]. Several substrate FG sandwich plates with different core materials are considered for presenting the numerical results. The nonlinear fundamental frequency ratios of the FG sandwich plates with different configurations have bn estimated. Also, the effect of variation of the piezoelectric fiber orientation angle (ΨΨ) in the 3 PZC constraining layer on the control authority of the ACLD patches has bn investigated. GOVERNING EQUATIONS Figure. (b) illustrates a schematic diagram of a smart FG sandwich plate in which the substrate FG sandwich plate is considered to be symmetric about the mid plane. The top and the boom facings of the sandwich plate are composed of nonhomogenous isotropic two constituent ceramic-metal FGMs whose material properties like modulus of elasticity and poissons ratio vary across the thickness based on a standard power law which is a function of volume fractions of the constituents while the core of the sandwich plate is a flexible isotropic HEREX honeycomb (soft) structure. The top surface of the top facing of the sandwich plate is integrated with the patches of the ACLD treatment. The constraining layer of the ACLD treatment is composed of the vertically/obliquely reinforced -3 PZC layer illustrated schematically in Fig. (a). (a) (b) Figure. SCHEMATIC DIAGRAMS OF (a) LAMINA OF VERTICALLY (ΨΨ = )/OBLIQUELY (ΨΨ ) REINFORCED -3 PZC MATERIAL AND (b) FG SUBSTRATE PLATE INTEGRATED WITH THE ACLD PATCHES. The length, the width and the thickness of the substrate sandwich plate are denoted by aa, bb and HH, respectively. The thickness of the constraining -3 PZC layer and the constrained viscoelastic layer of the ACLD treatment are h and h vv, respectively. The thickness of each facing and half of the thickness of the core are h and h cc, respectively. The origin of the rectangular Cartesian coordinate system (xxxxxx) is fixed at one corner of the mid-plane of the core, such that the lines xx = and aa and yy = and bb represent the boundaries of the midplane of the core. Since, the elastic properties of the adjacent continua of the overall sandwich structure differ in orders, each layer of the sandwich plate is modeled separately using FSDT to study the kinematics of deformations of the overall sandwich structure. Figures 2(a) and 2(b) illustrate the kinematics of deformations of the undeformed transverse normal in the xxxx and the yyyy planes, respectively.

5 The displacement fields uu cc, vv cc and ww cc along xx, yy and zz directions, respectively at any point in the core are uu cc (xx, yy, zz, ) = uu (xx, yy, ) + zzθθ xx (xx, yy, ) vv cc (xx, yy, zz, ) = vv (xx, yy, ) + zzθθ yy (xx, yy, ) ww cc (xx, yy, zz, ) = ww (xx, yy, ) + zzθθ zz (xx, yy, ) (3) The displacement fields uu vv, vv vv and ww vv along x, y and z directions, respectively at any point in the viscoelastic layer are uu vv (xx, yy, zz, ) = uu (xx, yy, ) + h cc θθ xx (xx, yy, ) + hαα xx (xx, yy, ) + (zz )ββ xx (xx, yy, ) (a) vv vv (xx, yy, zz, ) = vv (xx, yy, ) + h cc θθ yy (xx, yy, ) + hαα yy (xx, yy, ) + (zz )ββ yy (xx, yy, ) ww vv (xx, yy, zz, ) = ww (xx, yy, ) + h cc θθ zz (xx, yy, ) + hαα zz (xx, yy, ) + (zz )ββ zz (xx, yy, ) (4) The displacement fields uu, vv and ww along xx, yy and zz directions, respectively at any point in the -3 PZC layer are uu (xx, yy, zz, ) = uu (xx, yy, ) + h cc θθ xx (xx, yy, ) + hαα xx (xx, yy, ) + h vv ββ xx (xx, yy, ) + (zz )γγ xx (xx, yy, ) vv (xx, yy, zz, ) = vv (xx, yy, ) + h cc θθ yy (xx, yy, ) + hαα yy (xx, yy, ) + h vv ββ yy (xx, yy, ) + (zz )γγ yy (xx, yy, ) ww (xx, yy, zz, ) = ww (xx, yy, ) + h cc θθ zz (xx, yy, ) + hαα zz (xx, yy, ) + h vv ββ zz (xx, yy, ) + (zz )γγ zz (xx, yy, ) (b) Figure 2. KINEMATICS OF DEFORMATION As shown in this figure, uu, vv and ww represent the generalized translational displacement of a point along the midplane of the core, θθ xx, φφ xx, αα xx, ββ xx and γγ xx represent the rotations of the portions of the normal lying in the core, the boom layer, the top layer, the viscoelastic layer and the piezoelectric layer, respectively in the xxxx plane while θθ yy, φφ yy, αα yy, ββ yy and γγ yy represent the same in the yyyy plane. Thus the five sets of displacement fields describing the kinematics of deformations while satisfying the interface continuity conditions betwn the adjacent layers can be expressed as follows: The displacement uu, vv and ww along xx, yy and zz directions, respectively at any point in the top layer are uu (xx, yy, zz, ) = uu (xx, yy, ) + h cc θθ xx (xx, yy, ) + (zz h cc )αα xx (xx, yy, ) vv (xx, yy, zz, ) = vv (xx, yy, ) + h cc θθ yy (xx, yy, ) + (zz h cc )αα yy (xx, yy, ) ww (xx, yy, zz, ) = ww (xx, yy, ) + h cc θθ zz (xx, yy, ) + (zz h cc )αα zz (xx, yy, ) () The displacement uu bb, vv bb and ww bb along xx, yy and zz directions, respectively at any point in the boom layer are uu bb (xx, yy, zz, ) = uu (xx, yy, ) h cc θθ xx (xx, yy, ) + (zz + h cc )φφ xx (xx, yy, ) vv bb (xx, yy, zz, ) = vv (xx, yy, ) h cc θθ yy (xx, yy, ) + (zz + h cc )φφ yy (xx, yy, ) ww bb (xx, yy, zz, ) = ww (xx, yy, ) h cc θθ zz (xx, yy, ) + (zz + h cc )φφ zz (xx, yy, ) (2) (5) In Eqs. () - (5), the variables θθ zz, φφ zz, αα zz, ββ zz and γγ zz represent the gradients of the transverse normal displacements at any point in the core, the boom layer, the top layer, the viscoelastic layer and the piezoelectric layer, respectively. For the ease of analysis and computation, the translational displacement variables are separated from other variables as follows: {dd } =[uu vv ww ] TT and {dd rr } = θθ xx θθ yy θθ zz φφ xx φφ yy φφ zz αα xx αα yy αα zz ββ xx ββ yy ββ zz γγ xx γγ yy γγ zz TT (6) The state of strain at any point in the overall plate is divided into the bending strain vector {εε bb } and the out of plane shear strain vector {εε ss } as follows: {εε bb } = εε xx εε yy εε xxxx εε zz TT and {εε ss } = εε xxxx εε yyyy TT (7) where, εε xx, εε yy, εε zz are the normal strains along xx, yy and zz directions, respectively; εε xxxx is the in-plane shear strain and εε xxxx, εε yyyy are the transverse shear strains. By using the displacement fields and the von Kármán type of nonlinear straindisplacement relations, the strain vectors {εε bb } cc, {εε bb } bb, {εε bb }, and {εε bb } defining the state of in-plane and transverse normal strains at any point in the core, the boom layer, the top layer and the active constraining layer, respectively, can be expressed as: {εε bb } cc = {εε bbbb } + [ZZ ]{εε bbbb } + {εε bbbbbb }, {εε bb } bb = {εε bbbb } + [ZZ 2 ]{εε bbbb } + {εε bbbbbb }, {εε bb } = {εε bbbb } + [ZZ 3 ]{εε bbbb } + {εε bbbbbb }

6 and {εε bb } = {εε bbbb } + [ZZ 4 ]{εε bbbb } + {εε bbbbbb } (8) Similarly, the strain vectors {εε ss } cc, {εε ss } bb, {εε ss }, {εε ss } vv and {εε ss } defining the state of transverse shear strains at any point in the core, the boom layer, the top layer, the viscoelastic layer and the active constraining layer, respectively, can be expressed as: {εε ss } cc = {εε ssss } + [ZZ 5 ]{εε ssss } + {εε ssssss }, {εε ss } bb = {εε ssss } + [ZZ 6 ]{εε ssss } + {εε ssssss }, {εε ss } = {εε ssss } + [ZZ 7 ]{εε ssss } + {εε ssssss }, {εε ss } vv = {εε ssss } + [ZZ 8 ]{εε ssss } + {εε ssssss } and {εε ss } = {εε ssss } + [ZZ 9 ]{εε ssss } + {εε ssssss } (9) The various matrices aearing in the above equations have bn defined in the Aendix, while the generalized strain vectors aearing in Eqs. (8) and (9) are given by {εε bbbb } = uu {εε ssss } = ww {εε bbbb } = θθ xx φφ xx vv vv + vv TT and xx ww TT () θθ yy θθ yy + θθ xx φφ θθ xx φφ yy φφ yy + zz φφ αα xx αα yy αα yy + αα xx zz αα ββ xx ββ yy ββ yy + zz ββ xx ββ zz γγ xx {εε ssss } = θθ xx θθ yy φφ xx φφ yy αα xx αα yy ββ xx ββ yy γγ xx γγ yy θθ zz θθ zz φφ zz φφ zz αα zz αα zz ββ zz γγ yy {εε bbbbbb } = 2 ww 2 ww 2 2 ww TT ww and {εε ssssss } = 2 uu ww 2 2 vv ββ zz γγ yy γγ zz + γγ xx γγ zz TT () γγ zz TT (2) ww TT (3) In consistent with the state of strains given by Eq. (7), the state of stresses at any point in the overall plate is described by the two stress vectors as follows: {σσ bb } = σσ xx σσ yy σσ xxxx σσ zz TT and {σσ ss } = σσ xxxx σσ yyyy TT (4) where, σσ xx, σσ yy, σσ zz are the normal stresses along xx, yy and zz directions, respectively; σσ xxxx is the in-plane shear stress; σσ xxxx, σσ yyyy are the transverse shear stresses. The top and the boom facings of the sandwich plate are considered to be isotropic FGMs made of two homogenous constituents namely a ceramic and a metal. The two constituent materials are assumed to be isotropic and linearly elastic whose volume fractions vary across the thickness direction only. The effective material properties of each layer, like the young s modulus, the Poisson s ration and the density at any point can be expressed as, EE kk ffff (zz) = + (EE cc )VV kk, υυ kk ffff (zz) = υυ mm + (υυ cc υυ mm )VV kk and ρρ kk ffff (zz) = ρρ mm + (ρρ cc ρρ mm )VV kk (5) where the volume fraction VV kk of the kk h layer (k=, 2, 3) of the FG sandwich plate are given by VV = zz h h 2 h rr, z [h, h 2 ] VV 2 =, z [h 2, h 3 ] (6) VV 3 = zz h 3 h 3 rr, z [h 3, h 2 ] while the subscripts mm and cc represent the metallic and ceramic phases of the FGM, respectively. If the values of the superscript kk are, 2 and 3, it represents the boom layer, the core and the top layer of the plate, respectively. Also, the variable rr is the power-law index (greater than or equal to zero) which determines the material variation profile through the thickness of the top and the boom facings. The core is independent of rr making it fully ceramic layer. The constitutive relation for the materials of the different layers of the FG sandwich plate are given by kk {σσ bb } ffff kk {σσ ss } ffff kk where [CC bb ] ffff = [CC bb ] kk kk ffff {εε bb } ffff {αα bb } kk ffff TT and = [CC ss ] kk kk ffff {εε ss } ffff (7) = [CC bb ] mm + EE cc VV kk, kk [CC ss ] ffff = [CC ss ] mm + EE cc VV kk, kk {αα bb } ffff = [αα bb xx (zz) αα bb xx (zz) ] TT, TT = TT(zz) TT (8) [CC bb ] mm = (+υυ)(+2υυ) and ( υυ) υυ υυ υυ ( υυ) υυ ( 2υυ) υυ υυ ( υυ) [CC ss ] mm = (9) 2(+υυ) Wherein, αα xx bb (zz) and αα xx bb (zz) are the coefficients of thermal expansion of the boom FG layer in xx and yy directions, respectively. Also, αα xx bb (zz) = αα xx bb (zz) = αα FFFF. In Eq. 8, TT(zz) is the temperature at any point in the FG boom layer which can be determined by the steady-state one dimensional heat conduction equation [37-39] satisfying the boundary conditions TT = TT cc at the boom surface of the substrate plate while TT = TT at the interface betwn the boom FG layer and the core is given by dd dddd KK ffff dddd dddd = (2) Where, KK ffff is the thermal conductivity of the FG boom isotropic layer. The solution of Eq. (2) can be readily obtained from [37, 39]. The core may be composed of HEREX honeycomb material. For the purpose of modeling the flexible honeycomb core, the core is assumed to be an equivalent homogeneous solid continuum [2, 9] whose corresponding constitutive relations are given by {σσ cc bb } cccccc = [CC cc bb ] {εε bb } and {σσ cc ss } cccccccc = [CC cc ss ] {εε ss } (2) CC cc cc where [CC bb ] = CC 2 CC 6 cc cc cc CC 2 CC 6 CC 3 cc cc cc CC 26 CC 23 cc cc cc CC 66 CC 36 cc cc CC 22 cc CC 26 cc cc cc CC 3 CC 32 CC 36 CC 33 cc and [CC ss ] = CC 55 cc cc CC 45 cc cc CC 45 CC 44

7 The material of the viscoelastic layer is assumed to be linearly viscoelastic and isotropic. The present study is concerned with the analysis of FG sandwich plates undergoing ACLD in the time domain. Hence, the viscoelastic material is modeled by the Golla-Hughes-McTavish (GHM) method. In time domain, the constitutive relation for the viscoelastic material is represented by the following Stieltjes integral form [49]. {σσ ss } VV = GG( ττ) {εε ss } VV dddd (22) where GG() is the relaxation function of the viscoelastic material. The constraining -3 PZC layer is considered to be subjected to the electric field EE zz along the zz direction only. Thus the constitutive relations for the constraining -3 PZC layer of the ACLD treatment are given by [3] {σσ bb } = [CC bb ] {εε bb } + [CC bbbb ] {εε ss } { bb }EE zz, {σσ ss } = {εε bb } + [CC ss ] {εε ss } { ss }EE zz and DD zz = { bb } TT {εε bb } + { ss } TT {εε ss } + εε 33 EE zz (23) Here, DD zz represents the electric displacement along the zz direction and εε 33 is the dielectric constant. The forms of the transformed elastic coefficient matrices [CC b ] p and [CC s ] p are similar to those of [CC c b ] and [CC c s ], respectively. It may be noted from the above form of constitutive relations that the transverse shear strains are coupled with the in-plane stresses due to the orientation of piezoelectric fibers in the vertical xxxx or yyyy plane and the corresponding elastic coupling constant matrix [CC bbbb ] is given by CC 5 CC 4 CC [CC bs ] p = 25 CC or [CC CC bs ] p = 24 (24) 46 CC 56 CC 35 CC 34 according as the piezoelectric fibers are coplanar with the vertical xxxx or yyyy plane. Note that if the fibers are coplanar with the xxxx and the yyyy planes, this coupling matrix becomes a null matrix. Also, the piezoelectric constant matrices { bb } and { ss } aearing in Eq. (23) contain the following transformed effective piezoelectric coefficients of the -3 PZC: { bb } = { }TT and { ss } = { }TT (25) The total potential energy TT and the total kinetic energy TT kk of the overall plate /ACLD system are given by TT = {εε 2 bb } TT cc {σσ bb } cc + {εε ss } TT cc {σσ ss } cc d + {εε bb } TT ffff {σσ bb } ffff {εε ss } TT ffff {σσ ss } 3 ffff + {εε bb } TT ffff {σσ bb } 3 3 ffff + {εε ss } TT ffff {σσ ss } 3 ffff + {εε ss } vv TT {σσ ss } vv DD zz EE zz d TT kk = + d + {εε bb } TT {σσ bb } + {εε ss } TT {σσ ss } d {ff}da (26) {dd} TT ρρ 2 cc uu 2 + vv 2 + ww 2 d AA + ρρ vv uu vv 2 + vv vv 2 + ww vv 2 d ρρ uu 2 + vv 2 + ww 2 d + ρρ uu 2 + vv 2 + ww 2 d + ffff ρρ 3 ffff uu 2 + vv 2 + ww 2 d (27) + in which ρρ cc, ρρ vv and ρρ are the mass densities of the core, the viscoelastic layer and the -3 PZC layers, respectively; ρρ ffff and 3 ρρ ffff are the densities any point in the boom and the top layers, respectively; {ff} = [ PP] TT is the externally alied surface traction acting over a surface area AA with PP being the transverse distributed pulse loading over the surface and represents the volume of the concerned layer. Since, the thickness of the plate is very small compared to the lateral dimensions of the plate, the rotary inertia of the overall plate is neglected in estimating the total kinetic energy. FINITE ELEMENT MODEL OF FG SANDWICH PLATE INTEGRATED WITH ACLD PATCHES Eight noded isoparametric serendipity elements have bn used to discretize the smart FG sandwich plate. Following Eq. (6), the generalized displacement vectors {dd } and {dd rrrr } associated with the i th (i=, 2, 3, 8) node of the element can be wrien as {dd } = [uu ii vv ii ww ii ] TT and {dd rrrr } = θθ xxxx θθ yyyy θθ zzzz φφ xxxx φφ yyyy φφ zzzz αα xxxx αα yyyy αα zzzz ββ xxxx ββ yyyy ββ zzzz γγ xxxx γγ yyyy γγ zzzz TT (28) Thus the elemental displacement vectors at any point within the element can be can be expressed in terms of the nodal generalized displacement vectors {dd } and {dd rr } as follows: {dd } = [NN ]{dd } and {dd rr } = [NN rr ]{dd rr } (29) in which [NN ] = [NN NN 2. NN 8 ] TT, [NN rr ] = [NN rr NN rr2. NN rr8 ] TT, NN = nn ii II, NN rrrr = nn ii II rr {dd } = [{dd } TT {dd 2 } TT {dd 8 } TT ] TT and {dd rr } = [{dd rr } TT {dd rr2 } TT {dd rr8 } TT ] TT (3) while II and II rr are the (3 x 3) and the (5 x 5) identity matrices, respectively and nn ii is the shape function in natural coordinates associated with the i th node. Making use of the relation given by Eqs. (9)-(3), (27) and (28), the strain vectors at any point within the element can be expressed in terms of the nodal displacement vectors as follows: {εε bb } cc = [BB ]{dd } + [ZZ ][BB rrrr ]{dd rr } + 2 [BB ][BB 2 ]{dd }, {εε bb } bb = [BB ]{dd } + [ZZ 2 ][BB rrrr ]{dd rr } + 2 [BB ][BB 2 ]{dd } {εε bb } = [BB ]{dd } + [ZZ 3 ][BB rrrr ]{dd rr } + 2 [BB ][BB 2 ]{dd }, {εε bb } = [BB ]{dd } + [ZZ 4 ][BB rrrr ]{dd rr } + 2 [BB ][BB 2 ]{dd }, {εε ss } cc = [BB ]{dd } + [ZZ 5 ][BB rrrr ]{dd rr }, {εε ss } bb = [BB ]{dd } + [ZZ 6 ][BB rrrr ]{dd rr } {εε ss } = [BB ]{dd } + [ZZ 7 ][BB rrrr ]{dd rr }, {εε ss } vv = [BB ]{dd } + [ZZ 8 ][BB rrrr ]{dd rr }, {εε ss } = [BB ]{dd } + [ZZ 9 ][BB rrrr ]{dd rr } (3) in which the nodal strain displacement matrices [BB ], [BB rrrr ], [BB ], [BB rrrr ], [BB ] and [BB 2 ] are given by [BB ] = [BB BB 2 BB 8 ], [BB rrrr ] = [BB rrrr BB rrrr 2 BB rrrr 8 ],

8 [BB ] = [BB BB 2 BB 8 ], [BB rrrr ] = [BB rrrr BB rrrr2 BB rrrr8 ] [BB ] = TT, [BB 2 ] = [BB 2 BB BB 28 ] (32) The submatrices of [B ],[B rrrr ], [B ], [B 2ii ] and [B rrrr ] shown in Eq. (3) have bn explicitly presented in the Aendix. Finally, substituting Eqs. (5)- (25) and (3) into Eq. (26) and recognizing that EE zz = VV/h with V being the alied voltage across the thickness of the piezoelectric layer, the total potential energy TT of a typical element integrated with the ACLD treatment can be expressed as TT = 2 [{dd } TT [KK ] {dd } + {dd } TT [KK ]{dd rr } + {dd rr } TT [KK rrrr ] {dd } + {dd rr } TT [KK rrrr ]{dd rr } + {dd } TT [KK +{dd rr } TT [KK εε 33 VV 2 h ] GG( ττ) {dd }dτ + {dd } TT [KK ] TT GG( ττ) {dd }dτ + {dd rr } TT [KK rrrrrrrr ] GG( ττ) {dd rr }dτ ] GG( ττ) {dd rr }dτ 2{dd } TT FF VV 2{dd } TT FF VV 2{dd rr } TT FF rrrr VV 2{dd } TT {FF } {dd } TT {FF TTTT } {dd } TT {FF TTTT } (33) The elemental stiffness matrices ([KK ], [KK ], [KK rrrr ], [KK rrrr ], [KK ], [KK ] and [KK rrrrrrrr ]) the elemental electro-elastic coupling matrices FF, FF and FF rrrr, {FF TTTT }, {FF TTTT } and the elemental load vector {FF } aearing in Eq. (3) are derived as follows: [KK ] = [KK ] + [KK ] + [KK ], [KK [KK rrrr ] = [KK [KK rrrr ] = [KK ] = [KK rrrrrr ] + [KK ] + [KK ], ] TT + [KK ] TT + [KK 2 ] TT, ] + [KK rrrrrr ], FF = {FF } + {FF } + FF, FF rrrr = {FF rrrr } + {FF rrrr } (34) where [KK aa bb aa bb ] = [BB ] TT DD bb + [DD cc ] + [DD ] + DD [BB ] dx dy + [BB ] TT DD [BB ]dx dy, [KK [KK rrrrrr [KK rrrrrr aa bb ] = [BB ] TT bb DD BB bb cc rrrr + [DD ][BB cc rrrr ] aa bb ] = BB bb rrrr TT bb DD aa + [DD ] BB rrrr + DD [BB vv rrrr ] + DD [BB rrrr ] dx dy, TT + [BB cc rrrr ] TT cc [DD ] TT ] TT + BB rrrr TT [DD + [BB rrrr ] TT DD TT [BB ] dx dy aa bb + [BB rrrr ] TT DD rrrrrrrr TT [BB ]dx dy, bb ] = BB bb rrrr TT bb DD rrrrrr BB bb rrrr aa bb + [BB cc rrrr ] TT cc [DD rrrrrr ][BB cc rrrr ] + BB rrrr TT [DD rrrrrr ] BB rrrr + [BB rrrr ] TT DD rrrrrr [BB rrrr ] + [BB rrrr ] TT DD rrrrrrrr [BB rrrr ] dx dy, [KK ] = [BB ] TT DD bb + [DD cc ] + [DD ] aa bb + DD 2 [BB ][BB 2 ] + [BB ] TT DD TT 2 [BB ][BB 2 ] + [BB 2 ] TT [BB ] TT DD bb + [DD cc ] + [DD ] + DD [BB ] + [BB 2 ] TT [BB ] TT DD [BB ] + [BB 2 ] TT [BB ] TT DD bb + [DD cc ] + [DD ] + DD 2 [BB ][BB 2 ] dx dy, [KK ] = [BB 2 ] TT [BB ] TT [DD bb ][BB rrrr ] + [BB 2 ] TT [BB ] TT bb DD [BB rrrr + [BB 2 ] TT [BB ] TT bb DD [BB rrrr ] + [BB 2 ] TT [BB ] TT DD BB rrrr + [BB 2 ] TT [BB ] TT DD BB rrrr dx dy, aa bb cc ] [KK rrrrrrrr ] = [KK 2 rrrrrrrr ] TT, {FF } = [BB ] TT DD aa bb, {FF rrrr } = [BB ] TT DD rrrr dx dy, {FF aa } = bb [BB 2 ] TT [BB ] TT DD dx dy. {FF TTTT aa bb } = aa bb bb dx dy [BB ] TT [DD ] dx dy + [BB 2 2 ]TT [BB ] TT [DD ] dx dy {FF TTTT aa bb } = [BB rrrr ] TT [DD rrrrrr ] dx dy (35) and those associated with the transverse shear deformations are [KK aa bb aa bb ] = [BB ] TT [DD bb ] + [DD cc ] + [DD ] + DD [BB ] dx dy + [BB ] TT DD bb TT [BB ] dx dy,

9 [KK aa bb ] = [BB ] TT [DD bb ] BB bb cc rrrr + [DD ][BB cc rrrr ] + [DD [KK rrrrrr ] BB rrrr + DD aa bb [BB rrrr ] + [BB ] TT DD rrrrrrrr TT [BB rrrr ] dx dy, ] = BB bb rrrr TT [DD bb ] TT + [BB cc cc rrrr ][DD ] TT + [KK rrrrrr aa bb ] = BB rrrr TT [DD ] TT + [BB rrrr ] TT DD TT [BB ] + [BB rrrr ] TT DD TT [BB ] dx dy, BB rrrr bb TT [DD rrrrrr [KK bb ] BB bb rrrr + [BB cc rrrr ] TT cc [DD rrrrrr ][BB cc rrrr ] + BB rrrr TT [DD rrrrrr ] BB rrrr + [BB vv rrrr ] TT vv [DD rrrrrr ][BB vv rrrr ] + [BB rrrr ] TT DD rrrrrr [BB rrrr ] dx dy, aa bb ] = [BB ] TT [BB ]h vv [KK aa bb dx dy, ] ss = [BB rrrr ] TT [DD vv ][ vv BB rrrr ] [KK rrrrrrrr aa bb dx dy, ] = [BB vv rrrr ] TT [DD vv rrrrrr ][BB vv rrrr ] dx dy, aa bb {FF } = [BB ] TT DD {FF rrrr aa bb ss } = BB rrrr TT DD rrrr ss dx dy, dx dy. (36) where aa and bb are the length and the width of the element under consideration and the various rigidity matrices originated in the above elemental stiffness matrices are given in the Aendix. Substituting Eq. (29) into Eq. (27), the expression for the kinetic energy TT kk of the element can be obtained as in which TT kk = 2 dd TT [MM ] dd (37) bb aa [MM ] = mm h 2 [NN ] TT [NN ]dx dy and mm = 2ρρ cc h cc + ρρ h ffff dz + ρρ h 3 ffff dz +ρρ vv h vv + ρρ h for the case of homogenous ceramic core and for flexible HEREX core h mm = 2ρρh cc + 2 h ρρ h ffff dz ρρ h 3 ffff dz +ρρ vv h vv + ρρ h (38) Now, alying extended Hamilton s principle for the nonconservative system [34], the following elemental governing equations of motion are obtained: [MM ] dd + [KK ]{dd } + [KK ττ) {dd }dτ + [KK [KK rrrr ]{dd } + [KK rrrr +[KK rrrrrrrr 3 ]{dd rr } + [KK ] GG( ττ) ]{dd rr } + [KK ] GG( ττ) ] GG( {dd rr }dτ = {FF } + FF + FF VV + {FF TTTT ] TT GG( ττ) {dd }dτ } (39) {dd rr }dτ = FF rrrr + {FF TTTT } (4) It should be noted here that for an element without integrated with the ACLD patch, the matrices FF, FF and FF rrrr turn out to be the null matrices. The elemental governing equations are assembled into the global space to obtain the global equations of motion as follows: [MM] XX + [KK ]{XX } + [KK ]{XX rr } + [KK ] GG( ττ) {XX }dτ + [KK ] GG( ττ) {XX rr }dτ = {FF} + FF + FF VV + {FF TTTT } (4) [KK rrrr ]{XX } + [KK rrrr ]{XX rr } + [KK ] TT GG( ττ) {XX }dτ + [KK rrrrrrrr ] GG( ττ) {XX rr }dτ = FF rrrr VV + {FF TTTT } (42) where [MM] is the global mass matrix ; [KK ], [KK ], [KK rrrr ], [KK rrrr ], [KK ], [KK ] and [KK rrrrrrrr ] are the global stiffness matrices, FF, FF and FF rrrr are the global electroelastic coupling vectors, {XX } and {XX rr } are the global nodal generalized displacement vectors, {FF} is the global nodal mechanical force vector and {FF TTTT } and {FF TTTT } are global thermal load vectors. It may be noted that the elements of the matrices [KK ], [KK ] and [KK rrrr ] are nonlinear functions of displacements. After invoking the boundary conditions and taking Laplace transform of Eqs. (39) and (4), the following global equations of motion in Laplace domain are obtained: ss 2 [MM] XX + LL([KK ]{XX } + [KK ]{XX rr }) + [KK ] ss GG (ss) XX + [KK ] ss GG (ss) XX rr = {FF } + LL FF + FF VV (43) LL([KK rrrr ]{XX }) + [KK rrrr ] XX rr + [KK ] TT ss GG (ss) XX +[KK rrrrrrrr ]ss GG (ss) XX rr = FF rrrr VV (44) where XX, XX rr, {FF } and VV are the Laplace transforms of {XX }, {XX rr }, {FF} and VV, respectively, LL is the Laplace operator. Also, it may be mentioned that in the Laplace domain, the term ssgg (ss) is referred to as a material modulus function [49]. Employing the GHM method for modeling the viscoelastic material in time domain, the material modulus function can be represented by a series of mini-oscillator terms as follows [49]: ss 2 +2ξξ kk ωω kk ss ss 2 +2ξξ 2 ωω kk ss+ωω kk ssgg (ss) = GG NN + kk= αα kk (45) where GG corresponds to the equilibrium value of the modulus i.e. the final value of the relaxation function GG(). Each minioscillator term is a second-order rational function involving thr positive constants αα kk, ωω kk and ξξ kk. These constants govern the shape of the modulus function in the complex s-domain [49]. Now considering a GHM material modulus function with one mini-oscillator term [49] i.e. ssgg (ss) = GG + αα ss 2 +2ξξ ωω ss ss 2 +2ξξ ωω ss+ωω 2 (46) the auxiliary dissipation coordinates ZZ, ZZ rr are introduced as follows [4]: ssgg (ss) XX = GG ( + αα ) XX αα {ZZ }, ssgg (ss) XX rr = GG ( + αα ) XX rr αα {ZZ rr } (47) ZZ (ss) = ωω 2 ss 2 +2ξξ XX ωω ωω ss+ωω 2 and ZZ rr (ss) = 2 ss 2 +2ξξ XX ωω ss+ωω 2 rr (48) where ZZ (ss) and ZZ rr (ss) are the Laplace transforms of ZZ and ZZ rr, respectively. Using Eqs. (45) and (46) in Eqs. (4) and (42), the open loop governing equations of motion in Laplace domain are augmented as follows:

10 ss 2 [MM] XX + LL([KK ]{XX } + [KK ]{XX rr }) + [KK ]GG ( + αα) XX [KK ] GG αα ZZ (ss) + [KK ]GG ( + αα){xx rr } [KK ] GG αα ZZ rr (ss) = {FF } + FF VV + LL FF (49) LL([KK rrrr ]{XX }) + [KK rrrr ] XX rr + [KK ] TT GG ( + αα) XX [KK ] TT GG αα ZZ (ss) + +[KK rrrrrrrr ] TT GG ( + αα) XX [KK rrrrrrrr ] TT GG αα ZZ (ss) = {FF } + FF VV (5) Taking inverse Laplace transforms of Eqs. (48) - (5) and condensing the global degrs of frdom {XX rr } from the resulting equations in the time domain, the following equations are obtained: [MM] XX + [KK xx ]{XX } + [KK zz ]{ZZ} + [KK zzzz ]{ZZ rr } = {FF} + FF VV (5) ZZ + 2ξξ ωω ZZ + ωω 2 {ZZ} ωω 2 XX = (52) ZZ rr + 2ξξ ωω ZZ rr + ωω 2 [KK ] XX ωω 2 [KK 2 ]{ZZ} where +ωω 2 [KK 3 ]{ZZ rr } = {FF TTTT } + FF VV (53) [KK xx ] = [KK ] [KK ][KK rrrr ] [KK rrrr ], [KK zz ] = [KK ][KK rrrr ] [KK ] TT GG αα [KK ] GG αα, [KK zzzz ] = [KK ][KK rrrr ] [KK rrrrrrrr ] GG αα [KK ] GG αα, FF = FF + FF + [KK ][KK rrrr ] FF rrrr, {FF} = {FF } + [KK ][KK rrrr ] {FF TT } + {FF TTTT } + [KK ][KK rrrr ] {FF TTTT }, [KK ] = [KK rrrr ] [KK rrrr ], [KK 2 ] = [KK rrrr ] [KK ] TT GG αα, [KK 3 ] = II zzzz [KK rrrr ] [KK rrrrrrrr ] GG αα, FF = ωω 2 [KK rrrr ] FF rrrr [KK ] = [KK ] + [KK ]GG ( + αα), [KK ] = [KK ] + [KK ] GG ( + αα) [KK rrrr ] = [KK rrrr ] + [KK ] TT GG ( + αα), [KK rrrr ] = [KK rrrr ] + [KK rrrrrrrr ]GG ( + αα) (54) Now, Eqs. (5) (53) are combined to obtain the global openloop equations of motion in the time domain as follows: [MM ] XX + [CC ] XX + [KK ]{XX} = {FF } + FF VV (55) in which [MM] [MM ] = II zz, [CC ] = 2ξξ ωω, II zzzz 2ξξ ωω [KK xx ] [KK zz ] [KK zzzz ] [KK ] = ωω 2 ωω 2, ωω 2 [KK ] ωω 2 [KK 2 ] ωω 2 [KK 3 ] {FF } = {FF }, FF = FF {FF TTTT } FF {XX } and {XX} = ZZ ZZ rr (56) CLOSED LOOP MODEL In order to aly the control voltage for activating the patches of the ACLD treatment, a simple velocity fdback control law has bn employed. According to this law, the control voltage for each patch can be expressed in terms of the derivatives of the global nodal degrs of frdom as follows: VV jj = KK dd jj ww = KK dd jj [U jj ] X (57) in which KK dd jj is the control gain for the jj h patch and [U jj ] is a unit vector defining the location of sensing the velocity signal. Finally, substituting Eq.(57) into Eq.(55), the equations of motion governing the closed loop nonlinear dynamics of the FG sandwich plates activated by the patches of the ACLD treatment can be obtained as follows: [M ] X + [C dd ] X + [K ]{X} = {F } (58) where, [C dd ] = [C ] mm + KK jj jj = dd F [U jj ] is an active damping matrix and mm is the number of ACLD patches RESULTS AND DISCUSSIONS In this section, numerical results are computed by the finite element model derived in the previous section for assessing the performance of the ACLD patches on controlling the geometrically nonlinear vibrations of (2h cc /h ff = )and 2 (h cc /h ff = )FG sandwich plates with Flexible HEREX core configurations. The FG sandwich plate signifies that all the layers of the plate are of equal thickness. Unless otherwise mentioned, the boom facing and the top facing of the substrate sandwich plate are ceramic-metal FGM while the top surface of the top facing and the boom surface of the boom facings are metal rich. The top surface of the top facing of the FG sandwich plate are integrated with two patches of the ACLD treatment located at the edges of the plate as shown in Fig. (b). The material properties of the core and the constituents of the top and the boom facings of the substrate sandwich plate within the temperature range of 3KK 8KK are as follows [5]. HEREX Core(C7.3) [4] EE =.363 GGGGGG, υυ =.33; ρρ = 3 Kg/m 3 Aluminum: EE(TT) = TT 5 TT TT 3, αα(tt) = TT 3.3 TT TT 3, kk = 28 (59) Zirconia: EE(TT) = TT TT TT 3, αα(tt) = TT +.5 TT TT 3, kk(tt) = TT TT 2 97 TT 3 (6) The poissons ratio for the aluminum and zirconia are considered to be.3. The variations of the normalized young s modulus (EE ffff /EE cc ) of the FG substrate plate along the thickness direction

11 of the facings for different values of the power-law index are shown in Fig. 3 when the top and boom surfaces of the plate are metal rich. Figure 3. VARIATION OF NORMALIZED YOUNG S MODULUS ACROSS THE THICKNESS OF THE FACINGS OF FG SANDWICH PLATE FOR DIFFERENT POWER-LAW INDICES OF METAL RICH TOP AND BOTTOM SURFACES The length and the width of the patches are 5% and 25% of the length and the width of the sandwich plate, respectively. PZT- 5 H/spur epoxy composite wit% piezoelectric fiber volume fraction has bn considered for the material of the constraining layer of the ACLD treatment. The elastic and the piezoelectric properties of this constraining layer are [27]: CC = 9.29 GPa, CC GPa, CC 23 CC 55 Transverse thickness (Z) r=5 r=5 = 6.8 GPa, CC 3 = CC 3, CC 44 r=2 r=.5 r= E fg /E c =.58 GPa, CC 55 = 6.5 GPa, CC 33 = =.54 GPa, = CC 44, 3 =.92 C/m 2, 33 = 8.47 C/m 2 The thicknesses of the constraining 3 PZC layer and the viscoelastic layer are considered to be 25 µm, 2 µm, respectively. Unless otherwise mentioned, the aspect ratio (aa/hh) and the thickness of the substrate sandwich plate are considered as 5 and.3 m, respectively. Also unless otherwise mentioned, the piezoelectric fiber orientation angle (ΨΨ) in the PZC patches is considered to be and the mechanical pulse load (PP) acting upward is assumed to be uniformly distributed. Considering a single term GHM expression, the values of αα, ωω and ξξ are used as.42,.26x 5 and 2, respectively [43]. The shear modulus (GG ) and the density of the viscoelastic material (ρρ VV ) are used as Pa and 4 kg/m 3, respectively [43]. The simply suorted type- (SS) boundary conditions at the edges of the overall sandwich plate considered for evaluating the numerical results are given by vv = ww = θθ yy = φφ yy = αα yy =ββ yy = γγ yy = θθ zz = φφ zz = αα zz = ββ zz =γγ zz =, at x=, a and uu = ww = θθ xx = φφ xx = αα xx =ββ xx = γγ xx = θθ xx = φφ zz = αα zz = ββ zz =γγ zz =, at y=, b (6) Following the aroach by Lim et al. [5], first the validity of the implementation of the GHM method for modeling the ACLD treatment is checked. For this, simply suorted FG sandwich plates integrated with the ACLD patches as shown in Fig. (b) are considered. The linear frequency responses for the transverse displacement of these substrate sandwich plates are evaluated when the viscoelastic material is modeled by the GHM method and the conventional complex modulus aroach, separately. A r=2 r= r= r=.5 r=.2 transverse pulse pressure loading of 4 N/m 2 is considered to excite the first few modes of both the substrate sandwich plates. Figure 4 illustrate these frequency response functions for the sandwich plates with flexible HEREX (C7 3) core. It may be clearly observed from Fig. 4 that the frequency response curves obtained by the two aroaches are almost overlaing with one another. Thus, it may be concluded that the present method of modeling the viscoelastic material by the GHM method accurately predicts the damping characteristics of the overall smart FG sandwich plates. w/h Frequency (Hz) Figure 4. LINEAR FREQUENCY RESPONSES FOR CENTRAL DEFLECTION OF A SIMPLY-SUPPORTED (FG/HEREX (C7 3)/FG) SANDWICH PLATE OBTAINED BY THE GHM METHOD AND THE COMPLEX MODULUS APPROACH (aa/hh = 2, rr = 2, TT cc = 4, TT mm = 3). In order to further verify the accuracy of the current numerical model, the fundamental natural frequencies of the FG sandwich plates studied by Zenkour [2] are computed by the present FE model and compared with those obtained by Zenkour [2] as shown in Table. It is evident that the two sets of results are in good agrment with each other. Table- Fundamental natural frequency of simply-suorted (SS) square FG sandwich plate evaluated by present finite element model with aa/bb =, aa/hh =. Power-law index rr.5 5 GHM Complex Modulus Aroach Type of FG plate Ref. [2] Present FE soln ωω = ωωaa2 HH ρρ EE, EE = GPa and ρρ = kg/m 3 The open- and closed-loop behavior of the substrate FG sandwich plates with the HEREX (C7 3) core configuration are studied by computing the transverse deflection at the centre (aa/2, bb/2, HH/2) of the top surface of the plates. A uniformly distributed transverse pulse load is alied to set the overall plate into motion. For a considerable amount of nonlinearity to be present in the response, the amplitude ratio has bn maintained to be always greater than.5 (ww/hh.5). The control voltage sulied to each ACLD patch is negatively

12 proportional to the velocity of the point located on the plate outer surface which corresponds to the midpoint of the fr length of the patch. The control gain is chosen arbitrarily such that the nonlinear vibrations of the plates are under control and the maximum control voltage sulied does not excd a nominal value of 3 Volt. Figure 5 illustrate the nonlinear transient response of the FG/HEREX C7 3/FG symmetric substrate sandwich plates subjected to different temperature fields at the boom surface of the boom layer and at the interface of the boom layer and the core, respectively. The control voltages corresponding to the gain used are quite nominal as shown in Fig. 6 for controlling the geometrical nonlinear vibrations of the sandwich plates..5 K d = (T c =4K, T m =3K) K d =8 (T c =4K, T m =3K) corroborates the same indicating the stability of the sandwich plate. Tip Veocity (m/sec) K d = Displacement (mm) x -3 w/h Figure 7. PHASE PLOT OF THE SIMPLY-SUPPORTED FG/HEREX C7 3/FG SANDWICH PLATE WHEN THE ACLD PATCHES CONTROL THE NONLINEAR VIBRATIONS Time (sec) Figure 5. NONLINEAR TRANSIENT RESPONSES OF A SIMPLY-SUPPORTED FG/HEREX C7 3/FG SANDWICH PLATE UNDERGOING ACLD. w/h.5.5 K d =63 (h c /h f =, T c =4 K, T m =3 K) K d =8 (2h c /h f =, T c =4 K, T m =3 K) 3 2 K d = Time (sec) Control Voltage (Volts) Time (sec) Figure 6. CONTROL VOLTAGES REQUIRED FOR THE ACLD OF NONLINEAR TRANSIENT VIBRATIONS OF THE SIMPLY- SUPPORTED FG/HEREX C7 3/FG SANDWICH PLATE. The responses displayed in the Fig. 5 correspond to the cases when the ACLD patch is passive (KK dd = ) and active(kk dd ). It is evident from these figures that the active ACLD patches has a significant effect on the control of nonlinear transient vibrations of the FG sandwich plates and improves the damping characteristics of the overall plate over the passive damping. Since the control voltage is proportional to the velocity of the point on the plate, the illustrations of the control voltages in Fig. 6 indicate that at any point of the overall plate, the velocity also decays with time. The phase plot presented in Fig. 7 FIGURE 8. NONLINEAR TRANSIENT RESPONSES OF A SIMPLY-SUPPORTED AND 2 FG/HEREC C7 3/FG SANDWICH PLATES UNDERGOING ACLD The controlled responses of simply-suorted (SS) FG/HEREX C7 3/FG sandwich plates with and 2 configurations for the same maximum amplitude ratio (ww/hh =.5) and same maximum control voltages (V=3 Volt) are illustrated in Fig. 8. It may be observed from these figures that for the same value of the maximum control voltage sulied, the time for damping the nonlinear vibrations of the - 2- FG/HEREX C7 3/FG sandwich plate is more than that of the FG/HEREX C7 3/FG sandwich plate. Next, the controlled responses of simply-suorted (SS) FG/HEREX C7 3/FG sandwich plates for the same maximum amplitude ratio (ww/hh =.5) and same maximum control voltages (V=3 Volt) with metal rich boom surface and ceramic rich boom surface of the boom layer exposed to temperature fields are illustrated in Fig. 9. It may be observed from the figure that for the same value of the maximum control voltage sulied, the time for damping the nonlinear vibrations of the FG sandwich plate with metal rich boom surface is less as compared to that of the case with ceramic rich boom surface. Figure illustrates the transient responses of simplysuorted FG sandwich plate when the boom facing of the substrate is exposed to different temperature fields for the same alied transverse distributed pulse loading and same maximum