Static analysis of piezoelectric functionally graded porous plates reinforced by graphene platelets

In this study, for the first time an isogeometric finite element formulation for bending analysis of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs) embedded in piezoelectric layers is presented. It is named as PFGP-GPLs for a short. The plates are constituted by a core layer, which contains the internal pores and GPLs dispersed in the metal matrix either uniformly or non-uniformly according to three different patterns, and two piezoelectric layers perfectly bonded on the top and bottom surfaces of host plate.

Journal of Science and Technology in Civil Engineering NUCE 2019 13 (3): 58–72 STATIC ANALYSIS OF PIEZOELECTRIC FUNCTIONALLY GRADED POROUS PLATES REINFORCED BY GRAPHENE PLATELETS Nguyen Thi Bich Lieua,∗, Nguyen Xuan Hungb a Ho Chi Minh City University of Technology and Education, No Vo Van Ngan street, Thu Duc district, Ho Chi Minh city, Vietnam b CIRTECH Institute, Ho Chi Minh City University of Technology (HUTECH), 475A Dien Bien Phu street, Binh Thanh district, Ho Chi Minh city, Vietnam Article history: Received 07/08/2019, Revised 24/08/2019, Accepted 28/08/2019 Abstract In this study, for the first time an isogeometric finite element formulation for bending analysis of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs) embedded in piezoelectric layers is presented It is named as PFGP-GPLs for a short The plates are constituted by a core layer, which contains the internal pores and GPLs dispersed in the metal matrix either uniformly or non-uniformly according to three different patterns, and two piezoelectric layers perfectly bonded on the top and bottom surfaces of host plate The modified Halpin–Tsai micromechanical model is used to estimate the effective mechanical properties which vary continuously along thickness direction of the core layer In addition, the electric potential is assumed to vary linearly through the thickness for each piezoelectric sublayer A generalized C -type higher-order shear deformation theory (C -HSDT) in association with isogeometric analysis (IGA) is investigated The effects of weight fractions and dispersion patterns of GPLs, the coefficient and distribution types of porosity as well as external electrical voltages on structure’s behaviors are investigated through several numerical examples Keywords: piezoelectric materials; FG-porous plate; graphene platelet reinforcements; isogeometric analysis https://doi.org/10.31814/stce.nuce2019-13(3)-06 c 2019 National University of Civil Engineering Introduction The porous materials whose excellent properties such as lightweight, excellent energy absorption, heat resistance have been extensively employed in various fields of engineering including aerospace, automotive, biomedical and other areas [1–5] However, the existence of internal pores leads to a significant reduction in the structural stiffness [6] In order to overcome this shortcoming, the reinforcement with carbonaceous nanofillers such as carbon nanotubes (CNTs) [7–9] and graphene platelets (GPLs) [10, 11] into the porous materials is an excellent and practical choice to strengthen their mechanical properties In recent years, porous materials reinforced by GPLs [12] have been paid much attention to by the researchers due to their superior properties such as lightweight, excellent energy absorption, thermal management [13–15] The artificial porous materials such as metal foams which possess combinations of both stimulating physical and mechanical properties have been prevalently applied in lightweight structural materials [16, 17] and biomaterials [18] The GPLs are dispersed in materials ∗ Corresponding author E-mail address: lieuntb@hcmute.edu.vn (Lieu, N T B.) 58 properties of the material are significantly recovered but still maintain their po lightweight structures [20] Based on modifying the sizes, the density of th pores in different directions, as well as GPL dispersion patterns, the FG Lieu, N T B., Hung, N X / Journal of Science and Technology in Civil Engineering reinforced by GPLs (FGP-GPLs) have been introduced to obtain the mechanical characteristics [21-23] In the few years, have been man in order to amend the implementation while the weight of structures canlast be reduced by there porosities being conducted to investigate the impacts of GPLs and porosities [19] With the combination advantages of both GPLs and porosities, the mechanical properties of the on the beh structures under varioustheir conditions on the Ritz method[20] and Timoshen material are significantly recovered but still maintain potential Based for lightweight structures theory, the authors in refs [24] and [25] studied the free vibration, Based on modifying the sizes, the density of the internal pores in different directions, as well as GPL elastic buc the nonlinear free post-buckling performances FGP beams, resp dispersion patterns, the FGP plates reinforced by vibration, GPLs (FGP-GPLs) have been introduced of to obtain The uniaxial, biaxial, shear vibration responses the required mechanical characteristics [21–23] In the last buckling few years,and therefree have been many studies of FGP-G also investigated by [26] based on the first-order shear deformation being conducted to investigate the impacts of GPLs and porosities on the behaviors of structures un- theory (FS method Additionally, investigate theinstatic, der various conditions BasedChebyshev-Ritz on the Ritz method and Timoshenko beam to theory, the authors Refs free vibra buckling of FGP-GPLs, IGA based on both FSDT and the third-or [24, 25] studied the free vibration, elastic buckling and[27] theutilized nonlinear free vibration, post-buckling deformation theory (TSDT) performances of FGP beams, respectively The uniaxial, biaxial, shear buckling and free vibration responses of FGP-GPLs were also investigatedmaterial by [26] is based first-order shear deformation Piezoelectric one on of the smart material kinds, in which the elect theory (FSDT) and Chebyshev-Ritz method Additionally, investigate theOne static, and of the pie mechanical properties havetobeen coupled offree the vibration key features buckling of FGP-GPLs, [27] utilized IGAisbased on bothtoFSDT third-order shear deformation materials the ability makeand thethetransformation between the electrical p theory (TSDT) mechanical power Accordingly, when a structure embedded in piezoelectric Piezoelectric material is one of smartto material kinds, loadings, in which the and mechanical prop-create electr subjected mechanical theelectrical piezoelectric material can contrary, the structure can be changed its shape an electric field is put o erties have been coupled Onethe of the key features of the piezoelectric materials is theifability to make and electrical properties, thewhen piezoelectric the transformation between thecoupling electricalmechanical power and mechanical power Accordingly, a structurematerials h extensively applied to create smart the structures in aerospace, automotive, embedded in piezoelectric layers is subjected to mechanical loadings, piezoelectric material can medical other areas the literature of the plate integrated create electricity On the contrary, theand structure can beInchanged its shape if an electric field iswith put piezoelectr there variousproperties, numerical beingmaterials introduced predict their behavio on Due to coupling mechanical andare electrical themethods piezoelectric havetobeen extensively applied to create smart structures in aerospace, automotive, military, medical and other areas In this study, the piezoelectric plate with the core layer composed In the literature of the plate integrated piezoelectric layers,isthere are various numerical methods materialswith reinforced by GPLs considered Based on concept of sandwich being introduced to predict their behaviors the excellent mechanical properties of structure are created by combining ou In this study, the piezoelectric plate with the core layermaterials composedAccordingly, of FGP materials properties of component the reinforced presence by of porosities GPLs is considered Based onmatrix concept of sandwich structure, the excellent mechanical properties of leads to decreasing the weight of structure while the mechanical prop structure are created by combining outstanding properties componentGPLs materials Accordingly, significantly improved by of reinforcing Meanwhile, twothe piezoelectric presence of porosities in metallayers matrixare leads to decreasing thetop weight structure while of theamechanical embedded on the and of bottom surfaces porous core layer properties are significantly improved by reinforcing GPLs Meanwhile, two piezoelectric material Material properties of a PFGP-GPLs plate layers are embedded on the top and bottom surfaces of a porous core layer In this study, a sandwich plate with length 𝑎, width 𝑏 and total thickness of 2ℎ( shown in Fig.1 is modeled In which ℎ% and ℎ( are the thicknesses of Material properties of a PFGP-GPLs plate GPLs layer, core layer, and the piezoelectric face layers, respectively In this study, a sandwich plate with length a, width b and total thickness of h = hc + 2h p shown in Fig is modeled In which hc and h p are the thicknesses of the FGP-GPLs layer, core layer, and the piezoelectric face layers, respectively Three different porosity distribution types along the thickness direction of plates including two types of non-uniformly symmetric and a uniform are illustrated in Fig As presented in this Configuration PFGP-GPLs GPLs plate Figure Configuration of of aa PFGPplate figure, E is Young’s modulus of uniform porosity Figure distribution E1 and E2 denote the maximum and without GPLs, respecminimum Young’s moduli of the non-uniformly distributed porous material tively In addition, three GPL dispersion patterns shown in Fig are investigated for each porosity 59 STCE Journal – NUCE 2019 *+, pattern, thevolume GPL volume vary smoothly the thickness pattern, the GPL fractionfraction V*+,direction isVassumed to vary to smoothly along thealong thickness *+, is assumed direction direction Threedirection different porosity distribution types along the thickness direction of plates fferent porosity types alongsymmetric the thickness ofillustrated plates including twodistribution types of non‐uniformly and a direction uniform are in Fig.2 g two types of non‐uniformly symmetric and a uniform are illustrated in Fig.2 ’ As presented in this figure, E is Young’s modulus of uniform porosity distribution E1' ' nted in this figure, E’ is Young’s modulus of uniform porosity distribution E1 N T B., Hung, N X / JournalYoung’s of Science and Technology in Civil Engineering and E2' denote the Lieu, maximum and minimum moduli of the non‐uniformly denote the maximum and minimum Young’s moduli of the non‐uniformly distributed porous material without GPLs, respectively In addition, three GPL ed porous material without GPLs, respectively In addition, GPL distribution In each pattern, the GPL volume fraction VGPL isthree assumed to vary smoothly along the dispersion patterns shown in Fig.3 are investigated for each porosity distribution In each n patterns shown in direction Fig.3 are investigated for each porosity distribution In each thickness (a) Non‐uniform porosity distribution (b) Non‐uniform porosity pattern, the GPL volume fraction V is assumed to vary smoothly along the thickness *+, he GPL volume fraction V*+, is assumed to vary smoothly along the thickness direction (a) Non‐uniform porosity distribution (b) Non‐uniform porosity distribution (a) Non‐uniform porosity distribution (a) Non‐uniform porosity distribution (b) Non‐uniform distribution (b)1 Non‐uniform porosityporosity distribution n‐uniform porosity distribution (b) Non‐uniform porosity distribution 2Uniform porosity (a) Non‐uniform porosity1distribution (c)distribution distribution (b) Non‐uniform porosity (a) Non-uniform porosity distribution (b) Non-uniform porosity distribution (c) Uniform porosity distribution Figure Porosity distribution types [24] (c)(c) Uniform distribution Figure porosity Porosity distribution types [24] Uniform porosity distribution (c) Uniform porosity distribution Figure Porosity distribution types [24] Figure Figure Porosity distribution types [24] Porosity distribution types [24] (c) Uniform porosity distribution (c)distribution Uniform porosity distribution Figure Porosity types [24] Figure Porosity distribution types [24] (a) Pattern 𝐴 (c) P (b) Pattern 𝐵 Figure Three dispersion patterns 𝐴, 𝐵 and 𝐶 of the GPLs for ea distribution type The material properties including Young’s moduli 𝐸(𝑧) , shear modulus (a)Pattern Pattern 𝐴 (c) Pattern 𝐶Pattern (b)(b)Pattern 𝐵 alter𝐵along (a) 𝐴 (c) (b) Pattern 𝐵 density 𝜌(𝑧) the thickness direction for𝐶different poro (a) Pattern 𝐴 (b)which (a) Pattern A Pattern BPattern (c) Pattern Pattern C𝐶 (c) Figure3 3.Three Three dispersion patterns 𝐴,and 𝐵be(c) 𝐶 the eachfor porosity types expressed as (a) Pattern 𝐴 Figure Pattern 𝐶𝐶GPLs (b) Patterndispersion 𝐵 dispersion patterns 𝐴, patterns 𝐵can 𝐶and of the GPLs for porosity Figure Three 𝐴, 𝐵ofand of each thefor GPLs each porosity gure Three dispersion 𝐴, 𝐵 and 𝐶 of the GPLs each for porosity distribution type (a)Figure Pattern 𝐴 dispersion (c) (b) Pattern 𝐵Cdistribution ì E Pattern (porosity z ) = E1 [𝐶 1distribution - e0l ( z ) ] ,type patterns Three patterns A, B and offor the GPLs each distribution type type distribution type 𝐴, 𝐵 and 𝐶 of the GPLs for ï Figure Three dispersion patterns each porosity The material properties including Young’s moduli 𝐸(𝑧) , shear modulus 𝐺(𝑧)+ vand mass The material properties including Young’s 𝐸(𝑧) 𝐺(𝑧) The material properties including Young’s moduli 𝐸(𝑧)moduli , shear and erial properties including Young’s modulidistribution 𝐸(𝑧) , shear modulus 𝐺(𝑧) modulus and ( z,) shear =𝐺(𝑧) E ( z ) modulus / [ 2(1mass ( z )) ], and mass íG mass type density 𝜌(𝑧) which alter along the thickness direction for different porosity distribution 𝜌(𝑧) which alter along the thickness direction for different porosity distribution (z) (z) The material properties including Young’s moduli E , shear modulus G and mass density density 𝜌(𝑧) which alter along the thickness direction for different porosity distribution density 𝜌(𝑧) which alter along the thickness direction for different porosity distribution ï The material properties including Young’s moduli 𝐸(𝑧) , shear modulus r ( z )𝐺(𝑧) = r1 [and - emmass lcan ( z )], n be expressed as types can be expressed as ỵ (z) ρ which alter along the thickness direction for different porosity distribution types be expressed types can be along expressed as types can be expressed as the thickness density 𝜌(𝑧) which alter direction for different porosity distribution (1) E ( z ) = E1 as [1 - e0l (ìz )E](, z ) = E [1where (1) as beì types can expressed (1z) ]e, l ( z ) ] , (1) (1) 0)l (1 ze)0=(ezE ï [ ì E ( z ) = Eì1 E l , [1 − e0 λ(z)] , E(z) = E (1) + )) ]ï,e0l ( z )] , (ïz ) ï = vE(1z[1 íG ( z ) = E ( z )ì/E[ 2(1 ( z) = E )== / [E(z)/ 2(1 v/ ( 2(1 z )) ï ï íz(G G(z) +]v(z))] (1) E ( z+)[2(1 +, v( z,)) ] , z)E )((/zz)+ 2(1 -G em(ízlG ]),=( zE)í/(G )(ï= 2(1 v(+zv))(]z,)) [, [zρ(z) ỵ r ( z ) = r1 [1 í [1 = ρ − e λ(z)] , ï r [1 - e l ( z ) ] ,m ï ï ỵr ( z) = m [1 - e l ( z ) ] , r ( z )r=( zr) 1=[1r-ỵ er (1lz()z= ) ,r m ỵ 1m- em l]( z )1 , ỵ where  where where where  cos(πz/hc ), Non-uniform porosity distribution where    cos(πz/2h + π/4), Non-uniform porosity distribution (2) λ(z) =  c    λ, Uniform porosity distribution [ ] [ [ ] ] in which E1 = E1 and E1 = E for types3of non-uniformly and uniform porosity distribution, respec3 tively ρ1 denotes the maximum value of mass density of3the porous core The coefficient of porosity e0 can be determined by e0 = − E2 /E1 (3) Through Gaussian Random Field (GRF) scheme [28], the mechanical characteristic of closed-cell cellular solids is given as E(z) ρ(z)/ρ1 + 0.121 = E1 1.121 2.3 60 for 0.15 < ρ(z)