Active vibration control of GPLs reinforced FG metal foam plates with piezoelectric sensor and actuator layers Accepted Manuscript Active vibration control of GPLs reinforced FG metal foam plates with[.]
Accepted Manuscript Active vibration control of GPLs-reinforced FG metal foam plates with piezoelectric sensor and actuator layers Nam V Nguyen, Jaehong Lee, H Nguyen-Xuan PII: S1359-8368(19)30734-6 DOI: https://doi.org/10.1016/j.compositesb.2019.05.060 Reference: JCOMB 6849 To appear in: Composites Part B Received Date: 20 February 2019 Revised Date: April 2019 Accepted Date: May 2019 Please cite this article as: Nguyen NV, Lee J, Nguyen-Xuan H, Active vibration control of GPLsreinforced FG metal foam plates with piezoelectric sensor and actuator layers, Composites Part B (2019), doi: https://doi.org/10.1016/j.compositesb.2019.05.060 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ACCEPTED MANUSCRIPT Highlights • Free vibration and dynamic responses of smart FG metal foam plates reinforced by graphene platelets (GPLs) are investigated RI PT • Active vibration control of plates through the integration of piezoelectric sensors and actuators is presented • A C -HSDT polygonal finite element formulation (PFEM) enhanced with quadratic serendipity shape functions is employed AC C EP TE D M AN U SC • The influences of the porosity coefficient, weight fraction of GPLs on plate’s behavior are considered ACCEPTED MANUSCRIPT Active vibration control of GPLs-reinforced FG metal foam plates with piezoelectric sensor and actuator layers Nam V Nguyena , Jaehong Leeb , H Nguyen-Xuanc,b,∗ a RI PT Faculty of Mechanical Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam Department of Architectural Engineering, Sejong University, 98 Gunja-dong, Gwangjin-gu, Seoul 143-747, South Korea c CIRTech Institute, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Vietnam SC b Abstract EP TE D M AN U This paper investigates free vibration and dynamic responses of smart FG metal foam plate structures reinforced by graphene platelets (GPLs) We then analyse active control of FG metal foam plates with piezoelectric sensor and actuator layers To provide numerical solution of underlying problems, we develop a computational approach based on a C -HSDT polygonal finite element formulation (PFEM), which is suitable for modeling both thick and thin plates To enhance accuracy of solution, we use in PFEM quadratic serendipity shape functions in combination with a generalized C -type higher-order shear deformation theory (C -HSDT) The FG core layers are constituted by combining between two porosity distributions and three GPL dispersion patterns distributed along the plate’s thickness while two piezoelectric layers are perfectly bonded on the both bottom and top surfaces of host plate The mechanical displacement field is approximated based on C -HSDT while the electric potential distribution through the thickness for each piezoelectric layer is assumed to be a linear function For active control, a constant velocity feedback scheme is employed through a closed loop control with piezoelectric sensors and actuators The effect of the porosity coefficient, weight fraction of GPLs on the plate’s behaviors with various porosity distributions and GPL dispersion patterns are evidently demonstrated through numerical examples AC C Keywords: Polygonal finite element method, Piezoelectric materials, FG metal foam plate, Graphene platelets reinforcement, Active control Introduction Thanks to superior engineering properties such as lightweight, excellent energy-absorbing capability, great thermal resistant properties, etc., porous materials, such as metal foams, have been widely employed in various fields including aerospace engineering, automotive industrial, biomedical and other areas [1, 2, 3, 4] In addition, to enhance the performance of engineering materials, ∗ Corresponding author Email address: ngx.hung@hutech.edu.vn (H Nguyen-Xuan ) Preprint submitted to Elsevier May 9, 2019 ACCEPTED MANUSCRIPT 10 11 12 the reinforcement by carbon-based nanofillers including carbon nanotubes (CNTs) [5, 6, 7, 8] and graphene platelets (GPLs) [9] into the porous materials as additives have been studied Compared with CNTs, GPLs have revealed great potentials to become a good candidate for enhancing the stiffness of metal foam structures [10, 11] as they have excellent mechanical properties, a larger specific surface area as well as a lower manufacturing cost Numerous investigations have been carried out to study behaviors of FG metal foam beams and plates reinforced with GPLs in the literature RI PT 13 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 SC 18 M AN U 17 TE D 16 Recently, the use of piezoelectric materials to build up advanced smart structures for modern industrial products has also been highly interested in the scientific community In terms of investigation for the plates integrated with piezoelectric layers, numerous studies have been carried out to predict their behaviors in the literature [20, 21, 22, 23, 24, 25, 26, 27, 28, 29] In addition, Alibeigloo [30] presented the static and free vibration analyses of FG carbon nanotubes reinforced composite (FG-CNTRC) plates embedded in thin piezoelectric layers based on the three-dimensional theory of elasticity Sharma et al [31] performed the dynamic response and active vibration control behaviors of FG-CNTRC plates integrated with piezoelectric sensor and actuator layers based on FSDT By using HSDT and element-free IMLS-Ritz model, Selim et al [32] investigated the free vibration and active vibration control of FG-CNTRC plates with piezoelectric layers Meanwhile, the dynamic responses of laminated CNTRC plates integrated with piezoelectric layers are reported by Nguyen-Quang et al [33] based on IGA based on HSDT Malekzade et al [34] employed the transformed differential quadrature approach to analyze the free vibration of FG eccentric annular plate structures reinforced with GPLs and integrated piezoelectric layers EP 15 In this regard, Kitipornchai et al [12] employed the Ritz method for the free vibration and elastic buckling analyses of FG porous beam reinforced with GPLs Meanwhile, Chen et al [13] presented the nonlinear vibration and post-buckling behaviors of the multi-layer FG porous beams with GPLs reinforcement using Timoshenko’s beam theory Based on an analytical approach, Liu et al [14] performed the nonlinear static response and stability analysis of FG porous arch structures with GPLs reinforcement By applying Chebyshev-Ritz method, Yang et al [15] investigated the free vibration and buckling of the FG porous plates reinforced by GPLs uniformly or non-uniformly distributed in the metal matrix Li et al [16] analysed the static, free vibration as well as buckling of the FG porous plates with a small amount of GPLs using both first- and thirdorder shear deformation plate theories based on isogeometric analysis (IGA) The nonlinear free vibration analysis of FG porous plate with a small amount of GPLs resting on elastic foundation is reported by Gao et al [17] Li et al [18] investigated the nonlinear vibration and dynamic buckling problems of the sandwich FG porous plate reinforced by GPL resting on Winkler-Pasternak elastic foundation using the Galerkin method and the fourth-order Runge-Kutta approach Recently, Nguyen et al [19] presented the static and dynamic responses of FG porous plate reinforced with GPLs embedded in piezoelectric layers by using IGA based on B´ezier extraction AC C 14 45 46 47 48 In 2D finite element analysis, modeling the physical domain based on the typical triangular and quadrilateral elements becomes popular However, their performance needs still to be improved by using the general shape functions via the framework of polygonal finite elements (PFEM) To by3 ACCEPTED MANUSCRIPT 52 53 54 55 56 57 58 59 60 61 62 63 64 RI PT 51 SC 50 pass this bottleneck, PFEM where elements have arbitrary number of edges becomes a prominent alternative In PFEM, the establishment of the general shape functions [35] formed the rational polynomial interpolation function in which the Kronecker-delta properties is maintained This work is then extended to construct the shape functions for arbitrary convex polytopes by Warren [36] and for irregular polygons by Meyer et al [37] Furthermore, various approaches were introduced such as mean value coordinates [38], maximum entropy coordinates [39], natural neighbor [40], moving least squares coordinates [41], among others Besides, Rand et al [42] introduced the quadratic serendipity basis functions for polygonal elements where the nodal shape functions are calculated at vertices and mid-side nodes Recently, PFEM has been broadly employed in various different areas of engineering [43, 44, 45, 46, 47, 48] Regarding the investigation into plate structures, Nguyen-Xuan [49] proposed the formulation for thin and thick plates based on the Timoshenko’s beam assumptions and FSDT Nguyen et al [50] then extended this work for laminated composite plates using C -HSDT Most recently, the combination between PFEM and quadratic serendipity shape functions is reported by Nguyen et al [51] for geometrically nonlinear analysis of FG porous plates It is found in this study that the obtained results based on the quadratic serendipity shape functions are more accurate and stable than the previous PFEMs [50] M AN U 49 65 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 TE D 68 In view of the above literature, most of the existing reports focused on analyzing the FGM or FG-CNTRC plate structures integrated with piezoelectric layers Moreover, there are no reports on the free vibration, dynamic and active control analyses of piezoelectric FG metal foam plates reinforced by GPLs using numerical approach, e.g, PFEM To fill the existing gap in the literature, a new polygonal finite element formulation based on generalized C -HSDT and quadratic serendipity basis functions is presented to analyze aforementioned behaviors of piezoelectric FG metal foam plates with GPLs reinforcement Accordingly, the quadratic serendipity basis functions are used to obtain more accurate results while the shear locking phenomenon can be easily suppressed by using the Timoshenko’s beam theory In addition, the FG metal foam core layer is made of combining two porosity distributions and three GPL dispersion patterns while two piezoelectric layers are perfectly bonded on the both bottom and top surfaces of plate The active control for the dynamic responses of the FG metal foam plates with the effect of the structural damping based on a closed loop control with piezoelectric sensors and actuators is then studied Besides, the influence of several noticeable parameters including porosity coefficient, weight fraction of GPLs as well as distributions of porosities and GPLs in metal matrix on the behaviors of the plate structures are addressed and discussed in detail EP 67 AC C 66 The rest of this study is organized as follows Section presents the material model, linear piezoelectric constitutive equations and the variational as well as approximate formulations of the piezoelectric FG metal foam plates reinforced by GPLs based on C -HSDT Meanwhile, Section provides the active control analysis Section gives the comparison studies and numerical results for the free vibration and dynamic analyses as well as the active control of the piezoelectric FG metal foam plates reinforced with GPLs The paper is closed with some affirmations which are drawn in Section 90 ACCEPTED MANUSCRIPT 94 95 96 97 98 99 100 101 102 RI PT 93 2.1 Material properties for metal foam reinforced by GPLs Consider a piezoelectric FG plate model whose core layer is assumed to be made of metal foam reinforced with GPLs, as illustrated in Fig The piezoelectric FG plate has the length a, width b and the total thickness h = hc + 2hp , where hc and hp are the thicknesses of the core and piezoelectric layers, respectively Fig depicts two the porosity distributions and three GPL dispersion patterns along the thickness direction of the core layer of FG plates The variation of Young’s modulus E, shear modulus G and mass density ρ through the thickness of metal foam core layer corresponding to two different kinds of porosity distributions are explicitly formulated as E (z) = E1 [1 − e0 χ (z)] , G (z) = G1 [1 − e0 χ (z)] , (1) ρ (z) = ρ1 [1 − em χ (z)] , SC 92 Theory and formulation in which 103 104 105 χ (z) = M AN U 91 cos (πz/hc ), − h2c ≤ z ≤ cos (πz/2hc + π/4), e0 = − 111 112 113 114 115 116 117 118 (3) TE D 110 (2) where Emax and Emin denote the effective maximum and minimum values of Young’s modulus for the metal foam core layer without GPLs reinforcement, as illustrated in Fig 2a Based on the Gaussian random field model [52], the mechanical properties of closed-cell cellular solids can be given as 2.3 ρ (z) /ρ1 + 0.121 ρ (z) E (z) = for 0.15 <