Microsoft Word - numero_61_art_25_3527.docx K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 372 A simple and efficient eight node finite element for multilayer sandwich composite plates bending behavior analysis Khmissi Belkaid, Nadir Boutasseta, Hamza Aouaichia, Djamel Eddine Gaagaia, Badreddine Boubir Research Center in Industrial Technologies CRTI P.O.Box 64, Cheraga, Algeria khmissi.belkaid85@gmail.com, n.boutasseta@crti.dz, h.aouaichia@crti.dz, d.gaagaia@crti.dz, b.boubir@crti.dz Adel Deliou University of Med Seddik Benyahia (UMSB of Jijiel), Department of Mechanical Engineering Laboratory of Materials and Reactive Systems LMSR, University Djillali, Liabes, Sidi Bel-Abbes, Algeria. del032003@yahoo.fr, adel.deliou@univ-jijel.dz, deliouadel15@gmail.com ABSTRACT. In this paper, a C0 simple and efficient isoparametric eight-node element displacement-model based on higher order shear deformation theory is proposed for the bending behavior study of multilayer composites sandwich plates. Difficult C1-continuity requirement is overcome efficiently by choosing seven degrees of freedom for each element node: two displacements for in- plane behavior and five bending unknowns namely: a transverse displacement, two rotations and two shear angles, which results in the approximation formulation having only first order derivative requirement. The governing equations of the element (constitutive, virtual work and equilibrium equations) are implemented for the prediction of approximate solutions of deflections and stresses of sandwich plates linear elastic problems. The formulation element is able to present a cubic in-plane displacement along both core and faces sandwich cross-sectional, as well as, the shear stresses are found to vary as quadratic field without requiring shear correction factors and independent from any transverse shear locking problems when the plate is thin. The accuracy and validity of the proposed formulation is verified through the numerical evaluation of displacements and stresses and their comparison with the available analytical 3D elasticity solutions and other published finite element results. KEYWORDS. Third Order Shear Deformation Theory; Sandwich Composite Plates; Finite Element; Bending Behavior. Citation: Belkaid, K., Boutasseta, N., Aouaichia, H., Gaagaia, D. E., Boubir, B., Deliou, A., A simple and efficient eight node finite element for multilayer sandwich composite plates bending behavior analysis, Frattura ed Integrità Strutturale, 61 (2022) 372-393. Received: 27.03.2022 Accepted: 01.06.2022 Online first: 10.06.2022 Published: 01.07.2022 Copyright: © 2022 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. https://youtu.be/222XnMsjFzM K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 373 INTRODUCTION andwich composite plates are being increasingly used in many fields of modern technology due to their high strength, weight ratio and low maintenance cost. A good understanding of their behavior in terms of deformations and stresses distribution through the structures provide an effective vision for their applications. Generally, sandwich composite structures are made-up of a very rigid isotropic or orthotropic face sheets and relatively soft thick core material. In mechanical analysis of the bending sandwich plate, displacement fields vary in a zigzag manner through the thickness, thus making the displacements very discontinuous at the layer interfaces due to the large variation in stress between layers. Hence, the development of a suitable computational theory is required for accurately predicting the responses of these laminated sandwich structures. In this context, a number of shear deformation theories have been developed in order to accurately model multilayer plate. In the literature, the simplest equivalent single layer ESL laminate approach is the classical laminated plate theory (CLPT) Kirchhoff assumptions [1]. Their finite element model spatial approximations [2] contain C0-requirement for the in-plan displacements using Lagrange interpolation function, and transverse displacement C1-requirement using Hermite interpolation functions over the element. However, these elements models are characterized by a complex mathematical formulation due to the C1-requirement, and the theory is only appropriate for thin laminate plate analysis due to neglecting the effects of transverse shear deformation. The simplest theory which takes into account the transverse shear deformation is the first order shear deformation theory [FSDT] [3, 4]. Their finite element models [5] are characterized by a simple formulation with C0-requirement for all degrees of freedom using Lagrange interpolation function. However, the theory requires shear correction factors and the transverse shear stresses show at least a quadratic distribution through the plate thickness according to Pagano three-dimensional elasticity theory [6]. Various analytical higher order theories (HSDT) have been proposed for the multilayer composite structures analysis, taking into account shear deformation effects without shear correction. Their kinematics assumption is expanded up to higher powers of the thickness coordinate and quadratic transverse shear [7-10]. Barut et al. [11] analyzed a thick sandwich plate by third and second order theories in which the in-plane and the transverse displacements show cubic and quadratic variations respectively through the thickness of the plate. Other analytical and experimental works can be found for the sandwich plate and shell analysis: Noor et al. [12], Kant et Swaminathan [13], Mantari et al. [14], Grover et al.[15], Kanematsu et al. [16], Torabizadeh and Fereidoon, [17], M. Michele et al. [18], Deliou Adel [19]. However, analytical methods are only suitable for specific simple boundary conditions and geometries. In this case, several (2D) finite element models based on higher order shear deformation theory (HSDT) have been developed [20] for the static analysis of multilayer composite sandwich plates. B. Pandya , T. Kant [21] have presented a simple isoparametric finite element formulation based on a higher-order displacement model for flexure analysis of multilayer symmetric sandwich plates. B.S. Manjunatha, T. Kant [22] have evaluated the transverse stresses between layers of laminated composite and sandwich laminates using C0 nine and sixteen finite element formulation based on higher order theory. However, the models resort to use selective numerical integration scheme in order to overcome the shear locking problem. T. Kant and J. Kommineni, [23] presented a simple C0 quadrilateral Lagrange finite element formulation with nine-nodes and nine degrees of freedom per node based on refined higher-order shear deformation theory for the linear and geometrically non-linear analysis of fiber reinforced composite and sandwich plates. However, the selective integration scheme based on Gauss quadrature rules is introduced in order to overpass the shear locking problem. C.-P. Wu and C.-C. Lin [24] have presented the stress and displacement analysis of the thick sandwich plates using an interlaminar stress mixed nine-node finite element based on high order deformation theory. However, the formulation element possesses eleven nodal field variables in each node. R.P. Khandelwal et al. [25] have developed an efficient C0 continuous nine-node finite element model with eleven nodal field variables for each node based on combined theories refined higher order shear deformation theory (RHSDT) and least square error (LSE) method for the static analysis of soft core sandwich plates, the model satisfied the continuity of transverse shear stress condition between layer interfaces and zero transverse shear stress at the top and bottom of the sandwich plate. M.K. Pandit et al. [26] proposed a computationally efficient C0 nine-node finite element based on improved higher order zigzag theory for the static analysis of laminated sandwich plate with soft compressible core. However, the element has elven nodal field variables for each node adopting a reduced integration technique for the evaluation of stiffness matrix. T.M. Tu , T.H. Quoc [27] have developed a nine-nodded rectangular element with nine degrees of freedom at each node for the bending and vibration analysis of laminated and sandwich composite plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia effects. A. Nayak et al. [28] analyzed the bending behavior of isotropic, laminated composite and sandwich plates using two C0 quadrilateral finite element formulations based on higher- S K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 374 order theory where the element possess seven nodal field variables in each node. However, the formulations introduced assumed strain interpolations for the transverse shear strain in order to overcome the shear locking problem. Chalak et al. [29] presented an improved C0 2D nine-node finite element with eleven field variables per node. The model is based on higher order zigzag plate theory and has been applied to the analysis of laminated composites and sandwich plates. R. Sahoo and B. Singh [30] suggested an efficient C0 eight nodded isoparametric element with seven degrees of freedom per node based on a new inverse trigonometric zigzag theory for the static analysis of laminated and sandwich plates. However, the selective integration scheme is used in order to solve the locking shear problem. According to this literature survey, HSDT finite element models impose inconvenients such as: large number of nodal field variables, often encounter a locking problem when the plate is thin and resort to impose stiffness penalty in the formulation to remedy this problem. On the other hand, single layer Reddy’s theory is one of the higher-order theories used most often for analyzing multilayer plates, being able to evaluate stresses and transverse shear strains with a small variables number, not depending on the number of layers [9]. However, Reddy’s theory encounter formulation complications when the finite element requires C1 second-order derivatives. The same problem also arises in the classical theory of thin plates [31]. Therefore, many finite- element models (2D) based on Reddy’s third order theory have been proposed in the literature [20] for the bending behavior analysis of isotropic and multilayer composite plates. Furthermore, Reddy finite elements usually use conforming and non- conforming formulation where the C1 transverse displacement and its derivatives are interpolated by a modified bicubic Hermite functions, while the in-plane displacement and shear rotations are interpolated C0 Lagrange functions (JN Reddy [32], Phan and Reddy [33], Averill and Reddy [34], J. Ren, . Hinton [35], Ine-Wei Liu [36]) The objective of this work is to propose an efficient plate bending elements based on Reddy’s shear deformation theory, which has a simple formulation that overcome the difficult C1 requirement with small nodal field variables and that does not need to impose any stiffness penalty in the formulation and is also able to predict accurately the response of multilayer plates. Based on the recently proposed displacement-model [37], a serendipity isoparametric eight nodes finite element is formulated for the study of multilayer sandwich plate bending behavior. In the formulation of the element, seven nodal field variables are chosen in an efficient manner so that there is no need to impose any stiffness penalty and present simple mathematical formulation. In this work, the present sandwich plate element is used to solve many multilayer sandwich plates problems for various parameters such as, different loadings, geometry, boundary conditions, and materials. KINEMATICS he displacement field of the plate according to Reddy’s third order shear deformation theory (TSDT) [9] can be expressed as follows: 3 1 4 3         x x z w u u z h x 3 2 4 3           y y z w u v z h y (1) 3 u w where: u1, u2, u3 are the displacements field in the x, y and z directions respectively. u, v displacement of a point  ,x y on the mid-plane of the plate.  x ,  y are rotations about the axes y and x respectively, and h is the thickness of the plate. The strain associated with displacement field (1) are given as follows:  0 0 2 211 1 1 1       u z z x  0 0 2 222 2 2 2       u z z y T K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 375 3 3 0      u z (2) 0 2 23 3 32 4 4 4                   u u uu z z y y z 0 2 23 3 31 5 5 5                   u u uu z z x x z  0 0 2 23 31 26 6 6 6                     u uu u z z y x x y where: 2 0 0 2 1 1 1 2 2 4 ;   ;     3                    x x u w x x xh x 2 0 0 2 2 2 2 2 2 4 ;   ;   3                    y y u w y y yh y 0 2 4 4 2 4 ;                  y y w w y yh 0 2 5 5 2 4 ;                x x w w x xh 0 0 6 6;   ;                   yx u v w w y x x y y x 2 2 6 2 4 2 3                yx w y x x yh CONSTITUTIVE EQUATIONS he laminate is usually made of several orthotropic layers (Fig 1). Each layer must be transformed into the laminate coordinate system (x, y, z) [2]. The stress–strain relationship is given as: 11 12 16 12 22 26 16 26 66                                 xx xx yy yy xy xykk C C C C C C C C C ; 44 45 45 55                    xz xz yz yzkk C C C C (3) where  ijC are the transformed material constants: T K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 376   4 2 2 4 11 11 12 66 222 2   C C c C C c s C s   2 2 4 412 11 22 66 124 ( )    C C C C c s C c s    2 2 2 216 11 12 66 222    C C c C C s c C s cs  4 2 2 422 11 12 66 222 2   C C s C C c s C c   4 2 2 426 11 12 66 222 ( )    C C s C C c s C c cs   2 2 2 266 11 22 12 664 ( )    C C C C c s C c s 2 2 44 44 55 C C c C s  45 55 44 C C C cs 2 2 55 55 44 C C c C s where cos ,   sin  c s and  is angle between global axis and local axis for each laminate layer. 1 12 2 2 11 12 22 12 21 12 21 12 21 ;   ;   ;   1 1 1              E E E C C C 66 12 55 13 44 23;   ;    C G C G C G Figure 1: Geometry and coordinate system of the sandwich plate The differential equilibrium equations for transverse stress analysis are given as follows [2]: 11              k k kkn h xyxx xz h k dz x y 11                k k k kn h xy yy yz h k dz x y (4) K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 377 VIRTUAL WORK PRINCIPLE he static equations of the theory can be derived from the virtual work principle [9] by expressing the strain energy variation as follows:   0                xx xx yy yy xy xy xz xz yz yz V V dV q WdV (5) According to the substitution of equations (2) in the static equation (5) we obtain: 2 2 2 2 2 2 4 4 3 3                                                   y yx x xx xx xx yy yy yy A u w v w N M P N M P x x x y y yh x h y 2 2 4 2 3                                                   y yx xxy xy xy xx x u v w w N M P Q y x y x y x x y xh (6) 2 2 4 4 0                                         xx x yy y yy y w w w R Q R q W dA x y yh h where the resultants forces are defined as follows:   1 3 1 1, ,                         k k xx xx xx xxn h yy yy yy yyh k xy xy xy xy N M P N M P z z dz N M P ;   1 2 1 1,                 k k n hxx xx xz h yy yy yzk Q R z dz Q R (7) Therefore, from the eq (7) and eq (3), we obtain the generalized relations resultants forces as follows [9]:       0 0 2                                                 ij ij ij ij ij ij A B EN M sym D F P sym sym H                                     sS S ij ij S s ij A DQ R sym F (8) where: 0 0 0 0 0 0 2 2 2 0 0 2 1 2 6 1 2 6 1 2 6                   T T 0 0 2 2 4 5 4 5             T Ts s     1 2 3 4 6 , 1 , , , , 1, , , , ,      k k hn ij ij ij ij ij ij ij k h A B D E F H C z z z z z dz , , 1, 2, 6i j     1 2 4 1 , , 1, , ,   , 4, 5      k k hn S S S ij ij ij ij k h A D F C z z dz i j T K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 378 Substituting the resultant forces (8) in equation (6), the static equation becomes                   0 T 0 0 T 0 0 T 2 0 T 0 0 T 0 0 T 2 A 2 T 0 2 T 0 2 T 2 sT s s sT s s sT s s sT s s ( A B E B D F E F H A D D F q w )dA 0                                                      (9) FINITE ELEMENT FORMULATION n this work, a C0 isoparametric serendipity eight-node element (Fig.2) is employed for the bending analysis of sandwich plates based on Reddy’s third order shear deformation. In the present formulation element the complexities associated with C1 continuous plate are overcome with efficient manner by choosing seven nodal degrees of freedom (DOF) [37, 38] as follows: two displacements    ,  u v for the membrane behavior and five displacements ( ,   ,   , , )   x y x yw for describing the bending behavior, where ,                 x x y y w w x y are shear angles. Figure 2: Eight-node isoparametric finite element The generalized field variable and element geometry of the model at any point may be expressed in terms of nodal approximation as follows:         8 8 1 1 , , ;   , ,              i i i i i i x N x y N y                         8 8 8 8 1 1 1 1 8 8 1 1 , , ;   , ,  ; , , ;   , ,  ; , , ;   , ,                                                   i i i i x i xi y i yi i i i i x i xi y i yi i i u N u v N v N N N N where corner nodes are defined as follows:    i i i i i 1 N , 1 (1 )( 1), i 1, 2, 3, 4 4             and mid side nodes as: I K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 379    2i i 1 N , 1 (1 ), i 5, 7 2           2i i 1 N , 1 (1 ), i 6, 8 2        , iN are the bi-nonlinear interpolation functions of Lagrange type corresponding to node i=1-8 [39]. The strain displacements vectors can be formed as derivative elementary nodal matrix [B] multiplied by the proposed nodal field variable   as follows:          0 0 0 0 2 2;   ,   ,   ,                                      s s s sB B B B B (10) i ,x 0 i , y i , y i ,x N 0 0 0 0 0 0 B 0 N 0 0 0 0 0 N N 0 0 0 0 0                 ; i ,x 0 i , yk i , y i ,x 0 0 N 0 0 0 0 B 0 0 0 N 0 0 0 0 0 N N 0 0 0                i ,x 2 i , y1 i , y i ,x 0 0 0 0 0 N 0 B c 0 0 0 0 0 0 N 0 0 0 0 0 N N                 ; i , y is i ,x i 0 0 N 0 N 0 0 B 0 0 N N 0 0 0             is 2 i 0 0 0 0 0 0 N B c 0 0 0 0 0 N 0                , , , , , , , 1 8      T i i i xi yi xi yiu v w i with: c1=-4/3h2, c2=-4/h2 According to the substitution of strain matrix (10) in the static equation (9), the elementary stiffness matrix    eK is deduced according the static system      eK F [37] as follows:   1 1 0 0 0 0 0 2 0 0 0 0 0 2 1 1 2 0 2 0 2 2 ([ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ]                                           T T T T T T e T T T s T s s s T s s s T s s s K B A B B B B B E B B B B B D B B F B B E B B F B B H B B A B B D B B D B B  [ ][ ])det  T s sF B J d d (11) where    , F are elementary nodal vectors of forces and degrees of freedom, respectively. The analytical integration can be converted to Gauss’s numerical integration [40]. Full integration scheme quadrature rules, namely (3×3) is employed in the energy expression for the evaluation of the element stiffness property. K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 380 NUMERICAL EXAMPLES AND COMPARISON STUDIES n this section, several sandwich plates examples are solved to verify the effectiveness of the proposed formulation in the prediction of displacements and stresses results including different parameters such as, thickness ratios, materials, loading distribution, laminated face sheet, boundary conditions. The results of proposed model are compared with the three-dimensional elasticity solutions and other element models available in the literature. Bending of a simply supported sandwich square plate subject to a doubly sinusoidal transverse load For this example, the proposed element convergence and ability of the bending behavior is studied by considering a simply supported (f/c/f) symmetrical square sandwich plate subject to a doubly sinusoidal transverse load   0, sin sin  yx q x y q a b for different thickness ratios a/h = 4,10,50,100. The properties of the core and those of the sheets are shown in (Table 1). It is noted that normalized central deflection displacement results converge for different uniform mesh sizes toward Pagano 3D-elasticity solution [6] (Table 2). The increasing meshes indicate the accuracy and convergence rate of central transverse displacements results with a fast decrease of relative errors for both thin and moderately thick sandwich plates with no shear locking in the thin plates (a/h=50, 100). The simply supported boundary conditions used for the bending example are as follows: / 2 0      y yx a v w ; / 2  y b 0     x xu w The normalized deflection, stresses and in-plan are defined by: 3 2 2 22 4 2 2 0 0 0 , , 0 10 ,   , , , , , , 2 2 2 2 2 2 2 2                                      xx xx yy yy h Ea b a b h h a b h h w w q a q a q a 2 2 0 00   0, 0, ,   0, , 0 ,   , 0, 0 2 2 2                                       xy xy xz xz yz yz h h b h a h q a q aq a 3 3 2 2 4 4 0 0 100 100 ,                h E h E u u v v q a q a Proprieties E1 E2 G12 G13 G23 v12 Thickness Sandwich plate Core 275.8 MPa 275.8 MPa 110.32 MPa 413.68 MPa 413.68 MPa 0.25 0.8h Sheet 172369.9 MPa 6894.76 MPa 3447.38 MPa 1378.95 MPa 3447.38 MPa 0.25 0.1h Table 1: Mechanical proprieties of a sandwich plate. Reference Theory , , 0 2 2       a b w a/h=4(err%) a/h=10(err %) a/h=50(err %) a/h=100(err %) Present (2×2) 6.5668 (13.55) 1.8739 (14.83) 0.60962 (34.78) 0.3439 (61.43) Present (4×4) 7.1009 (6.52) 2.07035 (5.91) 0.91441 (2.181) 0.8604 (3.5) Present (6×6) HSDT(Q8) 7.1377 (6.03) 2.081172 (5.41) 0.92644 (0.89) 0.8845 (0.796) Present (8×8) 7.1461 (5.927) 2.08326 (5.32) 0.92844 (0.68) 0.8888 (0.323) Present (10×10) 7.1492 (5.88) 2.08396 (5.29) 0.92901 (0.61) 0.89005 (0.185) Present (12×12) 7.1508 (5.86) 2.08426 (5.27) 0.92923 (0.59) 0.89053 (0.131) Pagano[6] Elasticity solution 7.5962 2.2004 0.9348 0.8917 Table 2: Normalized center deflection convergence of a simply supported square sandwich plate subject to a doubly sinusoidal transverse load. I K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 381 In Fig 3 the central normalized w deflection is plotted for different a/h ratios. In Fig 4, the in-plane displacements u1(a/2,0,0) and u2(0,b/2,0) are described through the plate thickness as a cubic variation. It can be seen that the obtained bending responses by the present formulation are in excellent agreement with the elasticity solution of Pagano [6]and with those obtained by T. Kant and K. Swaminathan analytical solution [13]. Figure 3: Evolution of normalized central transverse deflection w with aspect ratio a/h of a simply supported sandwich plate  0 / / 0 C subject to a doubly sinusoidal transverse load. Figure 4: Evolution of normalized in-plane displacements 1 2,u u as function of the thickness of a simply supported  0 / / 0 C sandwich plate subject to a doubly sinusoidal transverse load a/h=4. In Table 3, the bending sandwich plate  0 / / 0   C is studied for different aspect ratios a/h in order to present the normalized transverse deflection and maximum stresses using the constitutive equation. Furthermore, the transverse shear stresses are evaluated using the equilibrium equations. A close agreement has been found between the obtained results by the proposed element and those obtained using Pagano elasticity-3D solution [6], as well as with those obtained by finite element models based on different theories. Furthermore, it’s noting that in thin sandwich plate cases (a/h=50,100), the results of the proposed model do not have locking shear problems with additional better accuracy in comparison with other model elements that have used stiffness penalty with a high number of nodal variables than the proposed model (e.g. B. Pandya , T. Kant [21], T. Kant and J. Kommineni, [23], M.K. Pandit et al. [26], Chalak et al. [29], R. Sahoo and B. Singh [30]). 0 20 40 60 80 100 0 5 10 15 20 25 N o n d im en si o n al t ra n sv er se d is p la ce m en t w a/h Present (12×12) Pagano -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 T h ic k n es s co o rd ia n te ( z/ h ) Nondimensional displacements u Present u1 Analytic HSDT u1 Present u2 Analytic HSDT u2 K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 382 a/h Reference Theory w  xx  yy  xz  yz xy 2 Present (12×12) HSDT(Q8) 21.356 2.770 0.386 0.2174* 0.1535* 0.2288 0.1880+ 0.1317+ Pagano [6] Elasticity solution 21.653 2.6530 0.3920 0.1850 0.1400 0.2340 Ramtekkar et al. [41] MFEM-3D- LW - 2.6840 0.3960 0.1860 0.1420 0.2360 Kant et Kommineni [23] FEM-Q9- TSDT 21.3707 2.7985 - - - 0.2371 4 Present (12×12) HSDT(Q8) 7.1508 1.4929 0.2375 0.279* 0.1154* 0.1377 0. 2418+ 0.0997+ Pagano [6] Elasticity solution 7.5962 1.5160 0.2595 0.2390 0.1072 0.1440 Ramtekkar et al. [41] MFEM-3D- LW -- 1.5700 0.2600 0.2400 0.108 0.149 Wu et Lin [24] MFEM-3D- LW -- 1.5480 0.2413 0.2497 - - 0.1339 Pandya et Kant [21] FEM-Q9- HSDT 0.6947 1.247 0.2338 0.2382 0.1132 0.1343 Manjunatha et Kant [22] FEM-Q9- HSDT 7.1596 - - - 0.2750 0.1137 - Kant et Kommineni [23] FEM-Q9- TSDT 7.1502 1.4989 -- - - -- 0.1428 Kant et Swaminathan [13] HSDT 7.0551 1.5137 0.2648 -- -- 0.1379 IGA [42] TSDT 7.0872 1.4244 0.2361 0.2708 0.1169 0.1383 IGA [42] HSDT 7.0686 1.4791 0.2391 0.3074 0.1274 0.1406 5 Present (12×12) HSDT(Q8) 5.1389 1.334 0.1955 0.299* 0.0986* 0.1149 0.2592+ 0. 0851+ Pagano [6] Elasticity solution 5.4746 1.3704 0.2094 0.2569 0.0918 -- Khandelwal et al. [25] FEM-Q9- HZZT 5.4464 1.3617 0.2216 0.2530 0.1025 -- 10 Present (12×12) HSDT(Q8) 2.08426 1.1419 0.1035 0.3454* 0.05797* 0.0672 0.2986+ 0.0493+ Pagano [6] Elasticity solution 2.2004 1.1531 0.1104 0.3000 0.0530 0.0707 Pandit et al.[26] FEM-Q9- HZZT 2.2002 1.1483 0.1086 0.3158 0.0570 0.0709 Tu et al.[27] FEM-Q9- TSDT 2.2027 1.1466 0.1105 0.3181 0.0532 0.0715 Khandelwal et al. [25] FEM-Q9- HZZT 2.1786 1.1539 0.1184 0.3185 0.0598 - Chalak et al.[26] FEM-Q9- HZZT 2.1775 1.1528 0.1143 0.3058 0.0575 0.0705 Ramtekkar et al. [41] MFEM-3D- LW - - 1.1590 0.1110 0.3030 0.0550 0.0720 Wu et Lin [24] MFEM-3D- LW - - 1.2100 0.1115 0.3177 - - - - 0.0713 Pandya et Kant [21] FEM-Q9- HSDT 2.023 1.110 0.1017 0.2841 0.05593 0.0666 Kant et Kommineni[23] FEM-Q9- TSDT 2.0864 1.1657 - - - -- 0.0692 K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 383 Kant et Swaminathan[13] HSDT-Anal 2.0798 1.1523 0.1100 0.3465 0.0538 0.0685 Nayak et al. [28] FEM-Q4- HSDT - - 1.1410 0.1034 0.3506 0.0534 0.0685 Nayak et al. [28] FEM-Q9- HSDT - - 1.1510 0.1043 0.2815 0.0532 0.0689 IGA [42] TSDT 2.0629 1.1299 0.1028 0.3302 0.0578 0.0679 IGA [42] HSDT 2.0515 1.1424 0.1031 0.3781 0.0628 0.0682 20 Present (12×12) HSDT(Q8) 1.1937 1.1019 0.0678 0.3648* 0.04227* 0.0493 0.31374+ 0.03401+ Pagano [6] Elasticity solution 1.2264 1.1100 0.0700 0.3174 0.0361 0.0511 Pandit et al. [26] FEM-Q9- HZZT 1.2254 1.1055 0.0694 0.3342 0.0392 0.0509 Khandelwal et al. [25] FEM-Q9- HZZT 1.2128 1.1113 0.0769 0.3374 0.0415 - Chalak et al.[29] FEM-Q9- HZZT 1.2121 1.1103 0.0742 0.3272 0.0399 0.0508 Ramtekkar et al. [41] MFEM-3D- LW - - 1.1150 0.0700 0.3170 0.0360 0.0510 Wu et Lin [24] MFEM-3D- LW -- 1.1730 0.0724 0.3530 - - - 0.0525 Kant et Kommineni [23] FEM-Q9- HSDT 1.1947 1.1246 - - - -- 0.0506 Kant et Swaminathan [13] HSDT 1.1933 1.1110 0.0705 -- -- 0.0504 IGA [42] TSDT 1.1876 1.1027 0.0678 0.3467 0.0408 0.0501 IGA [42] HSDT 1.1850 1.1061 0.0678 0.3974 0.0443 0.0502 50 Present (12×12) HSDT(Q8) 0.92923 1.0918 0.0563 0.3799* 0.4283* 0.0435 0.3147+ 0.02646+ Pagano [6] Elasticity solution 0.9348 1.0990 0.0569 0.3230 0.0306 0.0446 Pandit et al. [26] FEM-Q9- HZZT 0.9341 1.0948 0.0566 0.3403 0.0333 0.0445 Chalak et al. [29] FEM-Q9- HZZT 0.9248 1.0997 0.0611 0.3300 0.0321 0.0443 IGA [42] TSDT 0.9284 1.0965 0.0565 0.3520 0.0352 0.0444 IGA [42] HSDT 0.9280 1.0971 0.0565 0.4036 0.0383 0.0445 Kant et Kommineni [23] FEM-Q9- HSDT 0.9299 1.1118 - - - 0.0448 100 Present (12×12) HSDT(Q8) 0.8905 1.0904 0.0546 0.4118* 0.05132* 0.0427 0.30167+ 0.02157+ Pagano [6] Elasticity solution 0.8917 1.098 0.0550 0.324 0.0297 0.0433 Rosalin Sahoo, B.N. Singh [30] ITZZT-Q8 0.8919 1.1088 0.0555 0.3433 0.0276 0.044 Chalak et al.[29] HOZT-Q9 0.8814 1.0982 0.0592 0.3426 0.0332 0.0433 Pandit et al. [26] HOZT-Q9 0.8917 1.1093 0.0547 0.3412 0.0324 0.0434 IGA [42] TSDT 0.8908 1.0957 0.0548 0.3528 0.0344 0.0436 K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 384 IGA [42] HSDT 0.8907 1.0958 0.0548 0.4046 0.0374 0.0436 Pandya et Kant [21] FEM-Q9- HSDT 0.891 1.108 0.0554 0.3001 0.03362 0.044 Kant et Kommineni [23] FEM-Q9- HSDT 0.8915 1.1058 - - - 0.044 *Constitutive +Equilibre Table 3: Normalized transverse displacement, plane stresses, transverse shear stresses of a simply supported sandwich plate  0 / / 0 C under doubly sinusoidal loading. Through Figs (5, 6, 7, 8, 9, 10,11), the normal , ,  xx yy xy and transverse shear , xz yz stresses states are described through thickness of the sandwich plate in bending behavior for aspect ratios a/h=10,4 according to the constitutive and equilibrium equations. The obtained results stresses are in close agreement with those obtained by Pagano elasticity solution [13] and by TSDT numerical solution [42]. Figure 5: Normal stress distribution  xx through the thickness of a simply supported sandwich plate  0 / / 0 C subject to a sinusoidal load (a/h =10, 4). Figure 6: Normal stress distribution  yy through the thickness of a simply supported sandwich plate  0 / / 0 C subject to a sinusoidal load (a/h =10, 4). -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd in at e (z /h )  xx (a/2,b/2,z) Reddy IGA a/h=10 Present a/h=10 Reddy IGA a/h=4 Present a/h=4 -0,15 -0,10 -0,05 0,00 0,05 0,10 0,15 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd in at e (z /h )  xy (0,0,z) Reddy IGA a/h=10 Present a/h=10 Present a/h=4 Reddy IGA a/h=4 K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 385 Figure 7: Normal stress distribution xy through the thickness of a simply supported sandwich plate  0 / / 0 C subject to a sinusoidal load (a/h =10, 4). Figure 8: Transverse shear stress distribution  yz through the thickness of a simply supported sandwich plate  0 / / 0 C subject to a sinusoidal load (a/h =10, 4) (Constitutive equations). Figure 9: Transverse shear stress distribution  xz through the thickness of a simply supported sandwich plate  0 / / 0 C subject to a sinusoidal load (a/h =10, 4)(Constitutive equations). -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd in at e (z /h )  yy (a/2,b/2,z) Reddy IGA a/h=10 Present a/h=10 Reddy IGA a/h=4 Present a/h=4 -0,02 0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd in at e (z /h )  yz (0,b/2,z) Reddy IGA a/h=10 Present a/h=10 Reddy IGA a/h=4 Present a/h=4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd in at e (z /h )  xz (a/2,0,z) Reddy IGA a/h=10 Present a/h=10 Reddy IGA a/h=4 Present a/h=4 K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 386 Figure 10: Transverse shear stress distribution  yz through the thickness of a simply supported sandwich plate  0 / / 0 C subject to a sinusoidal load (a/h =10, 4) (Equilibrium equations). Figure 11: Transverse shear stress distribution  xz through the thickness of a simply supported sandwich plate  0 / / 0 C subject to a sinusoidal load (a/h =10, 4)(Equilibrium equations). Three-layer sandwich square plate subject to a uniform load In this example, the effect of the scale factor R  face coreC RC variation on deflection and stresses state of a simply supported sandwich square plate  0 / / 0 C is studied under a uniform transverse load with aspect ratio a/h =10 and face and core layers thickness hc/hf = 8. This sandwich example has been suggested by Srinivas [43]. The material properties of core layer are defined as: 0.999781 0.231192 0 0 0 0.231192 0.524886 0 0 0 0 0 0.262931 0 0 0 0 0 0.26681 0 0 0 0 0 0.159914                coreC The normalized deflection and stresses are defined by:             1 1 1 2 0 0 0 1 12 3 1 2 0 0 0 2 3 0.999781 / 2, / 2, 0 / 2, / 2, / 2 / 2, / 2, 2 / 5 ,   , , / 2, / 2, / 2 / 2, / 2, 2 / 5/ 2, / 2, 2 / 5 ,   , , / 2, / 2                                                         x x xx xx y yx xx yy yy y yy w a b a b h a b h w q h q q a b h a b ha b h q q q a b  0 , 2 / 5       h q Table 4 shows the deflection and normal stresses solution by the proposed model for different factor scales R = 5, 10, 15 compared with those obtained using the exact solution reported by Srinivas [43], HSDT finite element by Pandya and Kant [21], HSDT meshfree solution by Ferreira et al. [44], TrSDT trigonometric shear deformation theory solution by Mantari 0,00 0,02 0,04 0,06 0,08 0,10 0,12 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd ia n te ( z/ h ) Present a/h=10 Pagano a/h=10 Present a/h=4 Pagano a/h=4  yz (0,b/2,z) -0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6  xz (a/2,0,z)Normalized T h ic k n es s co o rd ia n te ( z/ h ) Pagano a/h=10 Present a/h=10 Pagano a/h=4 Present a/h=4 K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 387 et al. [14] and IHSDT inverse hyperbolic shear deformation theory by Grover et al.[15]. It is noted that the obtained results are in very good correlation with all scale factor R cases when compared with the exact solution. R Theory w 1 xx 2 xx 3 xx 1 yy 2 yy 3 yy 5 Present HSDT(Q8) (12×12) 257.4323 60.1869 46.6913 9.3382 38.3327 30.07465 6.0149 Exact [43] 258.97 60.353 46.623 9.34 38.491 30.097 6.161 FEM-HSDT [21] 256.13 62.380 46.910 9.382 38.930 30.330 6.065 MRBF-HSDT [44] 257.110 60.366 47.003 9.401 38.456 30.242 6.048 CFS-IHSDT [15] 255.644 60.675 47.055 9.411 38.522 30.206 6.041 CFS-TrSDT [14] 256.706 60.525 47.061 9.412 38.452 30.177 6.035 10 Present HSDT(Q8) (12×12) 155.9531 65.3098 49.3654 4.936 43.2633 33.42849 3.3428 Exact [43] 159.38 65.332 48.857 4.903 43.566 33.413 3.5 FEM-HSDT [21] 152.33 64.650 51.310 5.131 42.830 33.970 3.397 MRBF-HSDT [44] 154.658 65.381 49.973 4.997 43.240 33.637 3.364 CFS-IHSDT [15] 154.550 65.741 49.798 4.979 43.4 33.556 3.356 CFS-TrSDT [14] 155.498 65.542 49.708 4.971 43.385 33.591 3.359 15 Present HSDT(Q8) (12×12) 116.9587 66.9283 49.3353 3.289 45.8775 34.9577 2.3305 Exact [43] 121.72 66.787 48.299 3.238 46.424 34.955 2.494 FEM-HSDT [21] 110.430 66.620 51.970 3.465 44.920 35.410 2.361 MRBF-HSDT [44] 114.644 66.919 50.323 3.355 45.623 35.167 2.345 CFS-IHSDT [15] 115.820 67.272 49.813 3.321 45.967 35.088 2.339 CFS-TrSDT [14] 115.919 67.185 49.769 3.318 45.910 35.081 2.339 Table 4: The normalized deflection and stresses of square sandwich plates under uniform load. An additional analysis is considered for the effect of different scale factors R = 5, 10, 15 on the evolution of in-plane displacements    1 2/ 2, / 2, ,   / 2, / 2,u a b z u a b z , normal stresses    / 2, / 2, ,   / 2, / 2, xx yya b z a b z and transverse shear stresses    0, / 2, ,   / 2, 0, xz yzb z a z with respect to the sandwich plate thickness. In Fig 12, it is noted for the bending behavior that the in-plane displacements are reduced along the thickness when the scale factor R is increased. In addition, in Figs 13, 14 it can be seen that the normal and transverse shear stresses have also reduced only through the core of plate, whereas they have increased through the faces. K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 388 (a) (b) Figure 13: Effect of scale factor R variation on the distribution of normal stresses ,   xx yy through the thickness of a simply supported  0 / / 0   C sandwich plate subject to a uniform transverse load. (a) (b) Figure 14: Effect of scale factor R variation on the distribution of transverse shear stresses ,   xz yz through the thickness of a simply supported  0 / / 0   C sandwich plate subject to a uniform transverse load. Sandwich Plates with Laminated Face Sheets In this example, the proposed element is evaluated for different laminated face sheets of rectangular sandwich plates. Kanematsu et al. [16] carried out experimental solution of clamped rectangular sandwich plates (450×300 mm) with four types laminated composite orientation faces SP1, SP2, SP3 and SP4 (Fig. 15). The faces of the sandwiches are symmetrical laminated composite made of carbon/epoxy (Carbon Fiber–Reinforced Plastic-CFRP) E1=105 GPa, E2=8.74 GPa, G12= G13= G23=4.56 GPa, v=0.327, while the core is an aluminum honeycomb material (Aluminum Honeycomb Core) E1=68.6 MPa, E2=68.6 MPa, G12=26.4 MPa, G13=103 MPa, G23=62.1 MPa, v=0.3. The thickness of each layer is 0.125mm, while the core thickness is 10mm for the SP1 and SP2 types, and 7mm core thickness for the SP3 and SP4 types. The plate is subject to a uniform distributed load of intensity q=1.01 KPa. In addition, the authors provided analytical solutions based on the Rayleigh-Ritz method for the same plate problem, using two types boundary conditions, simply supported (SSSS) and clamped (CCCC). The obtained results of the transverse displacement using the proposed element are given in Table 5, compared with those obtained using analytical solutions and from experimental work given by -0,0015 -0,0010 -0,0005 0,0000 0,0005 0,0010 0,0015 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 T h ic k n es s co o rd in at e (z /h ) Nondimensional displacement u 1 R=5 R=10 R=15 -4 -2 0 2 4 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 T h ic k n es s co o rd in at e (z /h ) Nondimensional displacement u 2 R=5 R=10 R=15 0,0 0,1 0,2 0,3 0,4 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd in at e (z /h )  xz R=5 R=10 R=15 -0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 Normalized T h ic k n es s co o rd in at e (z /h )  yz R=5 R=10 R=15 K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 389 Kanematsu et al. [16]. The results have also been compared with those obtained by finite element models of Lee and Fan [45] and Nayak et al. [28], M. O. Belarbi et al. [46]. The comparison results show the reliability and practicality of the proposed element for the studies of laminated faces sandwich structures. Figure 15: Different types of laminated faces sandwich structures [46] References Theory Central deflection (mm) SP1 SP2 SP3 SP4 Clamped (CCCC) Present (12×12) FEM-Q8-TSDT 0.04891 0.056578 0.074968 0.05469 Kanematsu et al. [16] Analytical solution 0.05040 0.05400 0.07720 0.06130 Kanematsu et al. [16] Experimental solution 0.06900 0.08500 0.09400 0.09000 Lee et Fan [45] FEM-Q9-LW 0.05190 0.05524 0.07834 0.06216 Nayak et al. [28] FEM-Q9-HSDT - 0.05248 - 0.05797 M. O. Belarbi et al. [46] FEM-Q4-RSFT52 0.04906 0.05647 0.07506 0.05525 Simply supported (SSSS) Present (12×12) FEM-Q8-TSDT 0.12124 0.17823 0.17213 0.2067 Kanematsu et al. [16] Analytical solution 0.1173 0.1829 0.1794 0.2206 Lee et Fan [45] FEM-Q9-LW 0.1213 0.1774 0.1729 0.2138 Nayak et al. [28] FEM-Q9-HSDT - 0.1754 - 0.2111 M. O. Belarbi et al. [46] FEM-Q4-RSFT52 0.1160 0.1733 0.1695 0.2010 Table 5: Deflection laminated faces sandwiches plates under uniformly transverse load. K. Belkaid et alii, Frattura ed Integrità Strutturale, 61(2022) 372-393; DOI: 10.3221/IGF-ESIS.61.25 390 CONCLUSION n this paper, an improved C0 two-dimensional plate finite element (FE) model has been developed for the static analysis of laminated thin and thick sandwich plates. The Reddy’s third order shear deformation theory (TSDT) is employed by adopting single layer approach when the warping cross-sectional of in-plane displacements is considered to be cubic for both the face sheets and the core of sandwich. The problem of C1 continuity requirement of the second order derivatives of transverse displacements is circumvented by selecting the degrees of freedom nodal field in an efficient manner. For the present analysis, an eight-node C0 finite element is successfully implemented having seven degrees of freedom for each element node: two displacements    ,  u v for in-plane behavior and five bending unknowns: a transverse displacement, two rotations and two shear angles ( ,   ,   , , )   x y x yw . Whereas element stiffness matrices, which have first order derivative requirement, are solved through a computationally (3×3) Gauss integration scheme. In the proposed formulation, there is no stiffness penalty requirement such as: shear correction factors, and numerical techniques to overcome transverse shear locking phenomenon that as used in previous element models. In order to demonstrate the effectiveness and validity of the proposed formulation, many sandwich plates numerical examples are solved and displacements as well as stresses are calculated for different problems and which give results better than other existing 2D finite element models. The results obtained by using the present FE model are successfully compared with those of analytical and numerical solutions available in the literature. The numerical results show that the performance of the present finite element model is excellent in predicting the bending response of thin and thick laminated composites sandwich structures as the error percentage with respect to the 3D elasticity solution is considerably low. The present FE model may, therefore, be recommended for use as accurate tool in other behavior analysis of laminated sandwich plates. STATEMENTS AND DECLARATIONS - COMPETING INTERESTS AND FUNDING he authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. There is no funding to declare. NOMENCLATURE 1 2 3, ,u u u Displacement field in the x, y and z directions respectively , ,u v w Displacement of a point on the mid-plane , x y Rotations of normal to the mid-plane about the y and x axes respectively , x y Shear angles to the mid-plane about the y and x axes respectively  i Strain , ,a b h Dimensions of the plate along the x, y and z directions respectively  ij Stress tensor   ijC Constitutive matrix at the lamina level iE Young modulus 12 13 23,   , G G G Shears modulus 12 21,   Poisson’s ratios  N ,    M ,    P , Q ,    R Resultants forces ,, , , ,ij ij ij ij ij ijA B D E F H Extensional, coupling and flexural stiffnesses , ,S S Sij ij ijA D F Transverse shear stiffnesses I T K. 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