10.1016@j.ijmecsci.2017.09.032

Accepted Manuscript A polygonal finite element method for laminated composite plates Nam V Nguyen , Hoang X Nguyen , Duc-Huynh Phan , H Nguyen-Xuan PII: DOI: Reference: S0020-7403(17)31748-4 10.1016/j.ijmecsci.2017.09.032 MS 3943 To appear in: International Journal of Mechanical Sciences Received date: Revised date: Accepted date: 28 June 2017 September 2017 19 September 2017 Please cite this article as: Nam V Nguyen , Hoang X Nguyen , Duc-Huynh Phan , H Nguyen-Xuan , A polygonal finite element method for laminated composite plates, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.09.032 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 Highlight  A polygonal finite element method (PFEM) based on C0-type higher-order shear deformation theory (C0-HSDT) is proposed for static and free vibration analyses of laminated composite plates  A piecewise-linear shape function which is constructed based on sub-triangles of polygonal element is considered A simple numerical integration over polygonal elements is devised  Shear locking is addressed by a simple Timoshenko’s beam model  The numerical results show the efficiency and reliability of the present approach AC CE PT ED M AN US CR IP T  ACCEPTED MANUSCRIPT A polygonal finite element method for laminated composite plates Nam V Nguyen1, Hoang X Nguyen2, Duc-Huynh Phan3, H Nguyen-Xuan4,5* Faculty of Mechanical Technology, Industrial University of Ho Chi Minh City, Vietnam Faculty of Engineering and Environment, Northumbria University, Newcastle upon Tyne NE1 8ST, United Kingdom Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, Vietnam Institute of Research and Development, Duy Tan University, Da Nang, Vietnam CR IP T Department of Architectural Engineering, Sejong University, 98 Gunja-dong, Gwangjin-gu, Seoul 143-747, South Korea Abstract AN US In this study, a polygonal finite element method (PFEM) is extended and combined with the C0-type higher-order shear deformation theory (C0-HSDT) for the static and free vibration analyses of laminated composite plates Only the piecewise-linear shape function which is constructed based on sub-triangles of polygonal element is considered By using the analogous technique which relies on the sub-triangles to calculate numerical integration over M polygonal elements, the procedure becomes remarkably efficient The assumption of strain field along sides of polygons being interpolated based on Timoshenko’s beam leads to the ED fact that the shear locking phenomenon can be naturally avoided In addition, the C0-HSDT theory, in which two additional variables are included in the displacement field, significantly PT improves the accuracy of the displacements and transverse shear stresses Numerical examples are provided to illustrate the efficiency and reliability of the proposed approach CE Keywords: Polygonal finite element method, laminated composite plates, shear locking, AC Wachspress, piecewise-linear shape function * Corresponding author Email address: nguyenxuanhung@duytan.edu.vn (H Nguyen-Xuan) ACCEPTED MANUSCRIPT Introduction Thanks to outstanding engineering properties such as high strength, lightweight, strength-to-weight ratios, long fatigue life etc., laminated composite materials have been extensively applied in various fields of engineering including aerospace, automotive, civil, biomedical and other areas As a result, numerous analysis models have been developed in order to study their mechanical behaviors under different loading conditions In general, the CR IP T laminated composite plate theories can be classified into the following categories: the threedimensional (3D) elasticity model [1-4] and the two-dimensional (2D) model such as equivalent single-layer (ESL) theories [5], layer-wise theories [6, 7], zigzag theories [8] and quasi-3D theories [9,10] However, 3D solutions may not be feasible when solving practical problems due to its complex geometries, arbitrary boundary conditions and high AN US computational cost Consequently, various ESL plate theories have been devised and widely used in computational mechanics to predict the behaviors of plate structures [11-21] In the ESL plate theories, the classical laminated plate theory (CLPT) which developed based on the Kirchhoff-Love assumptions is the simplest theory However, as it neglects the M effects of transverse shear, this theory provides acceptable solutions for thin plates only In order to overcome this shortcoming, the first-order shear deformation theory (FSDT) based ED on Reissner–Mindlin theory [11, 12], which accounts for transverse shear effects, has been developed applicable for both thin and moderately thick laminated composite plates PT Nevertheless, this theory requires an appropriate shear correction factor (SCF) to accurately predict the distribution of shear strain/stress along the plate thickness satisfying the tractionfree boundary conditions at the top and bottom surfaces of plate Therefore, the accuracy of CE solutions based on FSDT theory will be strongly depended on the accuracy of the SCF Unfortunately, the values of SCF are not trivial to calculate as it depends on types of loadings, AC geometric parameters, material coefficients and arbitrary boundary conditions of the problems Therefore, a large number of significant higher-order shear deformation theories (HSDTs) have been proposed to surmount the limitations in CLPT and FSDT such as thirdorder shear deformation theory (TSDT) [13], refined plate theories (RPT) [14], trigonometric shear deformation theory (TrSDT) [15,16], exponential shear deformation theory (ESDT) [17,18], hyperbolic shear deformation theory (HSDT) [19-21] However, these theories require the C1-continuity of the generalized displacement field which is not easy to derive the second-order derivative of deflection This is really challenging in the framework of ACCEPTED MANUSCRIPT traditional finite element analysis In an effort to overcome this drawback, Shankara and Iyengar [22] proposed the C0-continuity of the generalized displacements (C0-HSDT) which two unknown terms are added to the displacement field Therefore, only the first derivative of deflection is considered in this model To the best of author’s knowledge, although various theories have been developed for the purpose of improving the quality of the numerical results, it seems consensual that almost CR IP T existing techniques rely on typical triangular or quadrilateral meshes So, in the last few decades, developing the generalizations of FEM based on arbitrary polygonal mesh has gained increasing attention of many researchers in computational solid mechanics A polygonal element with an arbitrary number of nodes is able to provide greater flexibility, suitable in complex microstructures modeling, well-suited for material design and sometimes AN US more accurate and robust results [23] In recent years, polygonal finite elements have been widely implemented in mechanics problems such as nonlinear constitutive modeling of polycrystalline materials [24-26], nonlinear elastic materials [23,27], incompressible fluid flow [28,29], crack modeling [30,31], limit analysis [32], topology optimization [33-35], contact-impact problem [36], Reissner-Mindlin plate analysis [37] and so on However, as far M as authors are aware, analysis of laminated composite plate based on arbitrary polygonal meshes has not been found yet Therefore, the goal of this study is to present a unified ED formulation which applies to arbitrary polygonal mesh including triangles and quadrilaterals associated with the C0-HSDT model for static and free vibration analyses of laminated PT composite plates Due to the complexity of the general convex polygonal elements in comparison with CE traditional finite elements, the construction of shape functions over arbitrary polygons is almost different from those of standard triangular or quadrilateral elements In the literature, there are numerous approaches have been presented for the determination of polygonal shape AC functions Among them, generalized barycentric coordinates have been widely used in computational solid mechanics in recent years Wachspress [38] pioneered to develop rational polynomial interpolation functions over planar convex polygonal domain, which satisfy the Kronecker delta and reproducing properties After that, Warren [39] further developed rational basis functions for arbitrary convex polytopes (3D) which Meyer et al [40] then extended for irregular polygons It is worthwhile noting that Wachspress’ coordinates are not well-defined for non-convex Hence, Floater [41] introduced a method based on mean value coordinates with an ability to interpolate for both convex and non-convex polygonal domains ACCEPTED MANUSCRIPT Moreover, several other methods have been proposed, including metric coordinates by Malsch et al [42, 43], maximum entropy coordinates by Sukumar [44], natural neighbor (Laplace) based on the natural neighbors Galerkin method by Sukumar and Tabarraei [45, 46], moving least squares coordinates [47], etc Sukumar and Malsch [48] has presented an outline the construction of polygonal shapes functions In addition to the aforementioned works, the sharp upper and lower bound piecewise-linear functions that satisfy the defining properties of barycentric coordinates have been reported by Floater et al [49] Accordingly, CR IP T the piecewise-linear shape functions are defined based on these sub-triangles of polygonal element In order to appreciate numerical integration over the polygonal elements, the same technique also based on that sub-triangles proposed by Sukumar [46] As a result, there is consistent between the construction shape functions and the evaluation integration over AN US polygonal element, producing a remarkable efficiency in numerical computation Another important issue is how to eliminate shear locking phenomenon when the plate becomes progressively thinner In order to address this deficiency, many approaches have been introduced and assessed for triangular and quadrilateral elements including reduced integration [50], selective reduced integration [51], assumed natural strains (ANS) [52-54], M the discrete Kirchhoff methods [55, 56], etc In recent years, based on the Timoshenko’s beam formulas, various plate elements have been developed in order to analyze both thin and ED moderately thick plates Accordingly, Ibrahimbegovic [57, 58] employed the Timoshenko’s beam formulas to develop two quadrilateral thin-thick plate elements PQ2 (quadratic displacement field) and PQ3 (cubic displacement field) based on the mixed interpolation PT method With a similar approach, Wanji and Cheung derived a refined triangular Mindlin plate element [59] and refined quadrilateral Mindlin plate element [60] for linear analysis of CE thin and thick plates In addition, Soh et al have developed two thin to moderately thick plate elements including ARS-T9 [61] and ARS-Q12 [62], which applied the Timoshenko’s beam AC formulas along each edge of the plate element This technique has recently been extended and applied to Reissner-Mindlin plate [63, 64] and laminated composite plate [65, 66] which rely on triangular and quadrilateral elements Based on the ideas of Soh et al [61] and Cen et al [65], a unified formulation for both thin and moderately thick plate elements based on arbitrary polygonal meshes was coined [37] Therefore, in this study, it is further developed to analyze the static and free vibration behaviors of laminated composite plates on arbitrary polygonal meshes ACCEPTED MANUSCRIPT The outline of this study is organized as follows The next section presents a brief review of the C0-HSDT type and a weak form of governing equations for laminated composite plate for static and free vibration problems Section focuses on the formulation of the PFEM for laminated composite plate with barycentric coordinates Section details the technique which is able to overcome the shear locking phenomenon in PFEM based on the Timoshenko’s beam formulas The numerical examples which cover static and free vibration analysis of given in Section CR IP T laminated composite plates are presented in Section Finally, some concluding remarks are C0-type higher-order shear deformation plate theory and weak form for laminated composite plates AN US 2.1 C0-type higher-order shear deformation plate theory Considering a laminated composite plate consisting of nl orthotropic layers with uniform thickness h and the fiber orientation  of each layer, its coordinate system is shown in Fig.1 According to the C0-HSDT model [22], the displacement field at an arbitrary point in the plate can be defined as follows: M u ( x, y, z )  u0  z  x  cz   x  x  , ED v( x, y, z )  v0  z  y  cz   y   y  ,  h   z   h , 2 (1) w( x, y, z )  w, where u0  u0 , v0  and w are the membrane displacements and the transverse displacement PT T of a point in the mid-plane, respectively; β   x ,  y  are the rotations of the normal to the CE T mid-plane around the y- and x-axes, respectively; and c  4 / 3h2 It is worth commenting AC that, Eq (1) is devised from the higher-order theory by Reddy [13], in which, derivative of deflection is replaced by warping function   x , y  Thus, the generalized displacement T vector with degrees of freedom for C1-continuity element can be transformed to a vector with degrees of freedom for C0-continuity element as: u  u0 , v0 , w,  x ,  y , x ,  y  T The in-plane vector of Green–Lagrange strain at any point in a plate can be expressed as  ,  x ,  xy   ε0  zκ1  z 3κ2 , T y where the membrane strains ε0 and the bending strains κ1 , κ2 are, respectively, given by (2) ACCEPTED MANUSCRIPT   x    x         y κ1   , y      y   x  x   y    u0     x   v0  ε0   ,  y    u0 v0      y x     x x    x x           y y κ2  c   , y y      y   y   x  x  x y x   y   (3) and the shear strains can be given as γ   xz ,  yz   εs  z κs , where   x  x  κ s  3c  ,  y  y  (4) (5) AN US w   x    x   εs   ,   y  w  y     CR IP T T By performing the transformation rule between the local and the global coordinate system as in Fig.1b, the constitutive equation, which based on Hooke’s law, of a kth orthotropic layer in global coordinate system xOy, are given by Q12 Q16 Q 22 Q 26 Q 26 Q 66 0 Q55 0 Q 54 ED M k  x   Q11     y  Q12     xy   Q16     xz       yz        Q54   Q 44  k k  x     y     xy  ,    xz     yz  (6) PT where Qij  i, j  1, 2, 4,5,6  are the transformed material constants of the kth orthotropic CE layer with respect to the global x- and y-axes [5] 2.2 Weak form equations for laminated composite plates AC In this study, the weak forms of the static and free vibration problems are derived by applying the Hamilton principal and conducting integration by parts Firstly, the weak form of static analysis of the laminated composite plates under transverse loading can be briefly expressed as    εTp D*ε p d     γ T Ds*γd     wqd , (7) in which q is distributing transverse load applied on the plate and strain components ε p and γ are expressed by ACCEPTED MANUSCRIPT ε p  ε0 , κ1 , κ2  , γ  εs , κs  , T T (8) * and the material constant matrices D* and Ds can be expressed by A D   B  E E F  , H  B * D F  As D  s B Bs  , Ds  * s (9) in which  A , B , D , E , F , H    1, z, z , z , z , z  Q dz,  A , B , D    1, z , z  Q dz, ij ij s ij s ij s ij ij ij h /2 ij ij ij  h /2  i, j  1, 2,  ,  h /2 CR IP T h /2 ij  i, j  4,5 , (10) The weak form of free vibration analysis of the laminated composite plates is of the compact form T  εTp D*ε p d     γ T Ds*γd     u mud ,  where the mass matrix m is given as I2 I1 0 I3 c / 3I I2 0 0 c / 3I I3 c / 9I7 ED in which  I1 , I , I , I , I , I    PT M  I1  I1    m    sym  AN US  h /2  h /2  c / 3I    , c / 3I    c / I  (11) (12)  1, z, z , z , z , z dz CE A polygonal finite element method for laminated composite plates AC 3.1 Shape functions on arbitrary polygonal elements In PFEM, a given domain is discretized into polygonal elements with arbitrary number of edges Then, the interpolation functions are constructed over each polygonal element In the literature, various approaches have been developed for the determination of the interpolation functions on arbitrary convex polygons [69-75] Among them, Wachspress [38, 39, 40, 67], mean-value [41, 68] and Laplace [46] shape functions are widely applied to construct the interpolation functions In addition, Floater et al [49] used sharp upper and lower bound piecewise linear functions in order to show all barycentric coordinates which are continuous ACCEPTED MANUSCRIPT in its interior, as shown in Fig 2a and Fig 2b These shape functions are ith barycentric coordinates and also satisfy the properties of barycentric coordinates including non-negative, partition of unity, Kronecker-delta, and linear precision With less computational effort than Wachspress shape function and requires only three integration points for each sub-triangle of polygonal element, only the piecewise linear shape function [37] will be used in this study Accordingly, each polygonal element with n nodes is CR IP T subdivided into n sub-triangles in a non-overlapping and no-gap manner The sub-triangles are created by simply connecting the centroid of the polygonal element to two end points of the edges as shown in Fig 2c In this case, the number of sub-triangles are the same as the number of nodes of polygonal element Firstly, the shape functions at the vertices of polygonal elements satisfy the Kronecker-delta property: AN US 1 xI  x J 0 x I  x J IPL  x J    IJ   (13) Next, the shape functions at the centroid x take the average value such that M IPL  x   1/ n (14) ED Last but not least, the shape functions and their derivatives over sub-triangles, i.e Gauss points, can then be easily constructed by using conventional FE shape functions of triangular PT elements as follows AC CE IPL ( x )  JT3 ( x )IPL ( xJ ) for x   T3 (15) J 1 IPL ( x )   JT3 ( x )IPL ( xJ ) for x   T3 (16) J 1 in which JT3 ( x ) and JT3 ( x ) denote three-node triangular shape functions and their derivatives on a sub-triangle T3 ; IPL ( xJ ) are the shape functions of node I at the node J of  T3 Fig illustrates the comparison of different shape functions of four regular polygonal elements