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Geometric analysis of composite sandwich plates without rotation

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https://www.eduzhai.net International Journal of Composite M aterials 2013, 3(6A): 10-18 DOI: 10.5923/s.cmaterials.201309.02 A Rotation-free Isogeometric Analysis for Composite Sandwich Thin Plates Chien. H. Thai1, T. Rabczuk2,*, H. Nguyen-Xuan1,3 1Division of Computational M echanics, Ton Duc Thang University HoChiM inh City, Vietnam 2Institute of Structural M echanics, Bauhaus-University Weimar, M arienstraße 15, Weimar, 99423, Germany 3Department of M echanics, Faculty of M athematics & Computer Science, University of Science HoChiM inh City, Vietnam Abstract In this paper, a rotation-free isogeometric formu lation for static analysis of co mposite sandwich plates is presented. The idea relies on a comb ination of isogeometric analysis with a classical laminate plate theory (CLPT). Isogeometric analysis (IGA) based on non-uniform rational B-spline(NURBS) basic function was recently proposed to preserve exact geometries and to enhance very significantly the accuracy of the traditional finite elements. B-splines basis functions (or NURBS) is used to represent for both geometric and field variable appro ximat ions, which provide a flexib le way to make refinement and degree elevation. They enable us to achieve easily the s moothness with arbitrary continuity order compared with the traditional FEM. CLPT ignores the transvers e shear deformation so it is only applied for thin plates. In our formu lation, only deflection variab les (without rotational degrees of freedom (dof)) are used for each control point. Essential displacements and rotations boundary conditions can be satisfied strongly by assigning control variable values on the boundary and these adjacent to the boundary, respectively. Several nu merical examp les are illustrated to demonstrate the performance of the present method in co mparison with other published methods. Keywords NURBS, Isogeometric analysis, Rotation-free isogeometric formu lation, Co mposite sandwich plates 1. Introduction Sandwich structures have been widely used in various engineering such as aircrafts, aerospace, vehicles, build ings, etc. Sandwich structures are made of three layers (two face sheets and a core) with different materials stacked together to achieve desired properties (e.g. high stiffness and strengthto-weight ratios , long fatigue life, wear res istance, lightweight, etc). Fo r the analysis of sandwich p late, the e xact elastic ity solution first has been proposed by Pagano [1] to predict accurately of static behavior. Elasticity solution three-dimensional (3D) can beco me very expensive when the complex structures are modeled. Generally, co mputational costs are reduced when t wo -d imens ional model is used. Using two-dimensional model, several plate theories using equivalent sing le layer have been developed to analy ze la minated composite sandwich plates. The classical la minate plate theory (CLPT)[3] can only give good results to thin plates because it ignores the transverse shear deformat ion. The first-order shear deformat ion theory (FSDT)[2] can be applied fo r both moderately thick and thin plates. This theory assumes that transverse shear stresses are constant through the thickness and a shear correction factor is needed to take * Corresponding author: timon.rabczuk@uni-weimar.de (T. Rabczuk) Published online at https://www.eduzhai.net Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved into account the non-linear distribution of shear stresses. To bypass the limitations of the FSDT, the higher-o rder shear deformation theories (HSDT) have been developed by Kant et al.[2] for the static analysis of co mposite sandwich plates based on analytical methods (Navier’s solution). Analytical methods have available for benchmark proble ms. Thanks to advanced numerical approaches such as finite elements [4, 5], smoothed finite elements(SFEM)[6, 7, 8], meshfree methods [9, 10, 11] and extended meshfree methods [12, 13, 14], we can solve effectively more co mplicated problems in practice. For illustration of this work, finite element analysis for composite sandwich plates is given by Tran et al.[15] based on HSDT. In addit ion, two-d imensional model based on zigzag theory is also used to calculate the co mposite sandwich such as: the static analysis of co mposite sandwich plate with soft-core by Pandit et al.[16], C0 fin ite element model for the analysis of sandwich laminates with general layup by Singh et al.[17] and an improved C0 finite element model for the analysis of laminated sandwich plate with softcore by Chalak et al.[19], etc. In the tradit ional FE method, a d iscretized geometry obtained through the so-called meshing process is required. This process often leads to geometrical e rrors even using the higher-order FEM. Also, the co mmun ication of the geo metry model and the mesh generation during an analysis process that aims to provide the desired accuracy for the solution is always needed and this constitute a time-consuming part in the overall analysis-design process, especially for industrial International Journal of Composite M aterials 2013, 3(6A): 10-18 11 problems[20]. To overcome this disadvantage, Hughes et al. [20] have recently proposed a NURBS-based isogeometric analysis to bridge the gap between Computer Aided Design (CA D) and Fin ite Element Analysis (FEA). In contrast to the standard FEM with Lag range polynomial basis, isogeometric approach utilized more general basis functions such as Non-Uniform Rational B-splines (NURBS) that are common in CAD approaches. Isogeometric analysis is thus very promising because it can directly use CAD data to describe both exact geo metry and appro ximate solution. For structural mechanics, isogeometric analysis has been extensively studied for structural vibrations [21], the Reissner-Mindlin composite plate[24], the co mposite plate based on HSDT [25], laminated composite layerwise plates [28], the Reissner-Mindlin shell[22] and Kirchhoff-Love shell[23, 27] and further developments [26], etc. The p lates are common ly emp loyed in engineering applications as thin plates. So, CLPT is utilized in this paper to reduce computational costs. We focus on NURBS elements using a rotation -free isogeometric formu lation for static analysis of co mposite sandwich plates. The paper is arranged as follows: a brief of the B-spline and NURBS surface is described in section 2. Section 3 describes an isogeometric appro ximation for co mposite sandwich plates. Several nu merical examp les are illustrated in section 4. Finally we close our paper with some concluding remarks. 2. Nurbs-Based Isogeometric Analysis Fundamentals 2.1. Knot Vectors and B asis Functi ons Let be a nondecreasing Ξ={ξ1,ξ2...,ξn+p+1} sequence of parameter values, i  i1,i  1,..., n  p . The i is called knots, and Ξ is the set of coordinates in the parametric space. If all knots are equally spaced the knot vector is called uniform. If the first and the last knots are repeated p + 1 times, the knot vector is described as open. A B-Spline basis function is C  continuous inside a knot span and C p1 continuous at a single knot. A knot value can appear more than once and is then called a mu ltip le knot. At a knot of mu ltiplicity k the continuity is C pk . Given a knot vector, the B-spline basis functions   Ni, p  of order p = 0 are defined recursively on the corresponding knot vector as follo ws Ni,0    1 0 if i    i otherwise 1    (1) The basis functions of p>1 are defined by the following recursion formula     Ni,p    i  i p i Ni, p1     i i  p1   p1  i1 Ni1, p1  (2) For p = 0 and 1 the basis functions of isogeometric analysis are identical to those of standard piecewise constant and linear finite elements, respectively. However, they are different for p  2 . In this study, we consider basis functions with p  2 . 2.2. NURBS Surface The B-spline curve is defined as: n  C   Ni, p  Pi (3) i1 where Pi a re the control points and Ni,p (ξ ) is the pth-degree B-spline basis function defined on the open knot vector. The B-spline surfaces are defined by the tensor product of basis functions in two parametric d imensions ξ and η with two knot vectors Ξ ={ξ1,ξ2...,ξn+p+1}and H ={η1,η2...,ηm+q+1} are expressed as follows: nm  S,   Ni, p   M j,q   Pi, j (4) i1 j1 where Pi,j is the bidirectional control net, Ni,p(ξ) and Mj,q(η) are the B-spline basis functions defined on the knot vectors over an n×m net of control points Pi,j. Similarly to notations used in fin ite elements, we identify the logical coordinates (i, j) of the B-spline surface with the traditional notation of a node A[22]. Eq.(4) can be rewritten in the fo llo wing form: nxm S ,    NA  ,  PA (5) A where NA(ξ,η) = Ni,p(ξ)Mj,q(η) is the shape function associated with node A. Similar to B-Splines, a NURBS surface is defined as   nxm S ,   RA  , PA where A1 RA  NAw nxm A N Aw A (6) A where wA is the weight function. 3. A Rotation-Free Isogeometric Formulation for Kirchhoff Plate Model Let  be the do main in R2 occupied by the mid-p lane of the plate and u, v and w denote the displacement components in the x, y and z directions, respectively. Using the Kirchhoff model[3], the displacements of any point in 12 Chien. H. Thai et al.: A Rotation-free Isogeometric Analysis for Composite Sandwich Thin Plates the plate can be expressed as u(x, y, z)  u0  x, y  zx  x, y v(x, y, z)  v0  x, y  zy  x, y (7) w(x, y, z)  w x, y where x  w x and y  w  y (8) In-plane strains through the following equation:   [ xx  yy  xy ]T  0  z (9) where 0 and  are the in-plane deformations and curvatures of the middle surface, respectively:    x  0    0    y 0  x  x  0  0 u  0 and  0     0   0  0 0 0 2   x2   2   y 2  u (10)  2 2   xy  and u = u0 v0 wT is displacement co mponents at the middle surface. The Hook’s law for an arbitrary layer k, the stress in plane is expressed as 1k  k 2        Q11 Q21 Q12 Q22 0   k  1k    0    k 2    (11) 12k     0 0 Q33  12k    where subscripts 1 and 2 are the directions of the fiber and in-plane normal to fiber, respectively, subscript 3 indicates the direction normal to the plate; and the reduced stiffness co mp o n en ts , Q(k ij ) are given by Q1(1k )  1 E1(k ) 1(2k ) (k) 21 , Q1(2k )   E (k ) (k ) 12 2 1 1(2k ) (k 21 ) , Q2(2k )  1 E2(k ) 1(2k ) (k) 21 ,Q3(3k )  G1(2k ) E E G  in which (k) 1, (k) 2, (k) 12 , (k) 12 and  (k) 21 are independent material properties for each layer. The la minate is usually made of several orthotropic layers. Each layer must be transformed into the laminate coordinate system (x, y, z). The stress - strain relat ionship is given as   xx yy  k     Q11 Q21 Q12 Q22 Q16 Q26     k    xx yy k    (12)  xy   Q61 Q62 Q66    xy   where Qij is the transformed material constant matrix[3]. A weak form of the static model for co mposite sandwich plates can be briefly exp ressed as:   T Dd   wpd   (13) where  and w are the strains and the deflection and the material matrix D : A D =   B B D ,   0   κ    (14)      Aij , Bij , Dij  h/2 h/2 1, z , z2 Qij dz i, j 1,2,6 Using the same NURBS basis functions, both the description of the geometry (or the physical point) and the displacement field are expressed as nxm xh  ,    R A  , PA and A nxm uh x,    RA , qA (15) A where n×m is the number basis functions, xT   x y is the physical coordinates vector. RA  ,  is rat ional basic   functions and qA  uA vA wA T is the degrees of freedom of uh associated to control point A. The strains in Eq. (14) can be expressed to following nodal displacements as: (16) where   RA, x 0 0 BmA   0 RA,y 0 and   RA, y RA,x 0 0 0 B bA   0 0 RA,xx   RA, yy  (17) 0 0 2RA,xy   BmA and BbA are memb rane and bending strain- displacement mat rices gained fro m derivative of shape functions, respectively. The IGA formu lation of co mposite sandwich plates can then be obtained for static analysis: Kq = f (18) where the global stiffness matrix is  K  Bm T A    Bb    B B D Bm   Bb       d (19) and f is the global force matrix: International Journal of Composite M aterials 2013, 3(6A): 10-18 13  f  pRd  (20) where q are the global displacements matrix 4. Numerical Results In this section, several nu merical studies using a rotation-free isogeometric analysis are presented. For all numerical examp les, quadratic, cubic and quartic NURBS elements integrated with nG = ( p + 1)(q + 1) Gauss points are used . The materia l para mete rs are assumed as: Material I: E1 = 25E2; G12 = G13 = 0.5E2; G23 = 0.2E2; v12 = 0.25 Material II: Face sheets : E1 = 172.4 GPa; E2= 6.89 GPa; G12 = G13 = 3.45 GPa; G23 = 1.378 GPa; v12 = 0.25 Core: E1 = E2= 0.276 GPa; G12 = 0.1104 GPa; G13 = G23 = 0.414 GPa; v12 = 0.25 The normalized d isplacement and in-plane stresses of composite sandwich plate are defined as: w  102 wE2h3 q0 a 4 , x   xh2 q0 a 2 , y   yh2 q0a2 and  xy   xyh2 q0a2 . 4.1. Three Layer (00/900/00) Square Laminated Pl ate Under Sinusoi dally Distributed Load Let us consider a simply supported square laminated plate subjected to a sinusoidal load q  q0 sin   x a   sin   y b   . The length to width ratios is a/b=1 and the length to thickness ratios is a/h=100. Material I described is use. The plate is modeled by 9x9, 13x13, 17x17 and 21x21 B-spline elements. The convergence of normalized displacement and in-p lane stresses are given in Figure 1. It can be seen that, the obtained results is very closed with analytical solutions by Kant[2] based on the third shear deformation p late theory and the elasticity solution 3D by Pagano[1]. In order to compare the results, we calculate the normalized d isplacement and in-plane stresses of the sandwich square plate using 21x21 B-spline elements, as given in Table 1. Obtained results are co mpared with the several other methods including the close form solution (CFS) based on the exponential shear deformation plate theory (ESDT) by Aydogdu[18], the elasticity solution given in Pagano[1] and analytical solutions based on Navier’s technique by Kant[2]. In[2], there are three-solutions such as: the fully third shear deformation plate theory using 12 dof/node (Kant 1), the third shear deformation plate theory of Reddy using 5 dof/node(Kant 2) and the first shear deformation plate theory 5 dof/node (Kant 3). It is observed that for deflect ion and stresses the results of the present method agrees well with published results. Figure 2 plots the distribution of stresses through the thickness of the plate. The obtained results are in good agreement with those reported by Kant[2]. 14 Chien. H. Thai et al.: A Rotation-free Isogeometric Analysis for Composite Sandwich Thin Plates Figure 1. The in-plane normal and shear stresses of the three-layer composite (00/900/00) simple supported square plates Figure 2. The in-plane normal and shear stresses of the three-layer composite (00/900/00) simple supported square plates International Journal of Composite M aterials 2013, 3(6A): 10-18 15 Table 1. The normalized displacement and the stresses in a three-layer (00/900/00) simply supported square laminate under sinusoidal transverse load Authors & methods Kant 1[2] (HSDT ) Kant 2[2] (HSDT ) Aydogdu[18] (ESDT ) Kant 3[2] (FSDT) Elast icity [1 ] Quadratic (CLPT) Cubic (CLPT) Quartic (CLPT) w( a , b ,0) 22 0.4343 0.4342 0.4350 0.4337 0.4329 0.4342 0.4353  x ( a 2 , b 2 , h 2 ) 0.5392 0.5390 0.5389 0.5384 0.5390 0.5383 0.5382 0.5387  y ( a 2 , b 2 , h 2 ) 0.1807 0.1806 0.1806 0.1804 0.1810 0.1794 0.1794 0.1796  xy (0, 0, h 2 ) 0.0214 0.0214 0.0214 0.0213 0.0213 0.0213 0.0213 0.0213 4.2. The Sandwich (00/core/00) Square Pl ate under Sinusoi dally Distri buted Load We consider the sandwich (00/core/00) simply supported square plate subjected to sinusoidally distributed load with the thickness of each face sheet equal h/10. Material II is used. The plate is modeled by 21x21 B-spline element. The normalized transverse displacement and normalized stresses are reported Table 2. The obtained results are compared with the exact elasticity solution by[1], the analytica l solution by[2], FEM solutions based on the higher order zig zag plate theory (HOZT) by[16, 17] and and FEM solutions based on the third shear deformation p late theory by [15]. It is found that the results of present method shown good agreements with those solutions. The distribution of stresses through the thickness of the plate is illustrated in Figure 3. Table 2. The normalized displacement and the stresses in a three-layer (00/core/00) simply supported square sandwich under sinusoidal transverse load Author & method Kant 1[2] Kant 2[2] Kant 3[2] Elast icity [1 ] Singh et al.[17] Pandit et al. [16] Tran et al. [15] Quadrat ic Cubic Quart ic w( a , b ,0) 22 0.8913 0.8908 0.8852 0.9017 0.8917 0.8919 0.8816 0.8842 0.8864  x ( a 2 , b 2 , h 2 ) 1.0990 1.0973 1.0964 1.0980 1.1020 1.1093 1.1069 1.0962 1.0965 1.0970  y ( a 2 , b 2 , h 2 ) 0.0560 0.0549 0.0546 0.0550 0.0547 0.0573 0.0542 0.0542 0.0543  xy (0, 0, h 2 ) 0.0436 0.0436 0.0435 0.0437 0.0453 0.0434 0.0432 0.0434 0.0433 0.0433 16 Chien. H. Thai et al.: A Rotation-free Isogeometric Analysis for Composite Sandwich Thin Plates Figure 3. The in-plane normal and shear stresses of the sandwich (0/core/0) simple supported square plates 4.3. An-symmetry the Sandwich (00/900/core/00/900) Square Pl ate under Sinusoi dally Load Table 3. The normalized displacement and the stresses in a five-layer (00/900/core/00/900) SCSC and CCCC square sandwich under sinusoidal transverse load Boundary con dit ion s SCSC CCCC Met ho d Pandit et al.[16] Singh et al.[17] Chalak et al.[19] Quadrat ic Cubic Quart ic Pandit et al.[16] Singh et al.[17] Chalak et al.[19] Quadrat ic Cubic Quart ic w( a , b ,0) 22 0.3453 0.3920 0.3430 0.3328 0.3358 0.3369 0.2286 0.2260 0.2267 0.2200 0.2221 0.2231  x ( a 2 , b 2 , h 2 ) 0.4077 0.5986 0.4250 0.3969 0.3990 0.3994 0.4270 0.4283 0.4371 0.4300 0.4302 0.4305  y ( a 2 , b 2 , h 2 ) 0.0326 – 0.0366 0.0327 0.0327 0.0328 0.0228 – 0.0259 0.0229 0.0229 0.0229 International Journal of Composite M aterials 2013, 3(6A): 10-18 17 In order to study the stretching-bending coupling effect, node-based smoothed discrete shear gap method. Applied the an-symmetry five-layer sandwich p late (00/900/core/00/ Mathematical Modelling, 36: 5657–5677. 900) is considered. Material II is also used. The core has a [7] C. Thai Hoang, N. Nguyen-Thanh, H. Nguyen-Xuan, T. thickness of 0.8h while the two laminated face -sheets are Rabczuk and S. Bordas (2011). A cell-based smoothed finite of 0.1h. The plate has supported (S) and clamped (C) boundary conditions. When 21x21 element mesh, the element method for free vibration and buckling analysis of shells, KSCE Journal of Civil Engineering, 12 (2): 347–361. normalized d isplacement and stresses derived fro m the [8] C. Thai-Hoang, N. Nguyen-Thanh, H. Nguyen-Xuan, T. present method of a five-layer sandwich plate with various Rabczuk (2011). An alternative alpha finite element method boundary conditions are given in Table 3. For co mparison, other methods based on C0 higher order zig zag plate theory by Chalak et al.[19], Singh et al.[17] and Pandit et al. [16] with discrete gap technique for analysis of laminated composite plates, Applied Mathematics and Computation, 217: 7324–7348. are cited. It is observed that the present results are in good agreement with published ones for both SCSC and CCCC boundary conditions. [9] D. Wang and J.S. Chen (2008). A hermite reproducing kernel approximation for thin plate analysis with sub-domain stabilized conforming integration. International Journal for Numerical Methods in Engineering, 74:368-390. 5. Conclusions An isogeometric formulat ion has been developed for static analysis of the composite sandwich plates using a rotation-free isogeometric formu lat ion of CLPT. Weak form of the static model for co mposite sandwich plates using CLPT was derived. The present method only used three degrees of freedom per node (3 dof/node), and the obtained results are in very good agreement with analytical solution by Kant 1[2] using 12 dof/node, Kant 2[2] using 5 dof/node, FEM solutions using 11 dof/node[16, 17, 19] and FEM solutions using 9 dof/node[15]. The d istribution of stresses through the thickness of the sandwich plates are in very good agreement with those of other existing methods. [10] Xiaoying Zhuang, Claire Heaney, Charles Augarde (2012). On error control in the element-free Galerkin method. Engineering Analysis with Boundary Elements, 36: 351-360. [11] Xiaoying Zhuang, Charles Augarde (2010). Aspects of the use of orthogonal basis functions in the element free Galerkin method. International Journal for Numerical Methods in Engineering, 81: 366-380. [12] T.Rabczuk and P.M.A. Areias (2006). A meshfree thin shell for arbitrary evolving cracks based on an external enrichment. Computer Modeling in Engineering and Sciences, 16: 115–130. [13] T. Rabczuk, P.M .A. Areias, and T. Belytschko (2007). A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 72:524-548. REFERENCES [1] N.J. Pagano (1970). 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