An analytic homogenization model in traction and bending for orthotropic composite plates with the type of double corrugated cardboard

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 3 (2016), pp. 205 – 213 DOI:10.15625/0866-7136/38/3/6466 AN ANALYTIC HOMOGENIZATIONMODEL IN TRACTION AND BENDING FOR ORTHOTROPIC COMPOSITE PLATES WITH THE TYPE OF DOUBLE CORRUGATED CARDBOARD Duong Pham Tuong Minh, Ngo Nhu Khoa∗ Thai Nguyen University of Technology, Vietnam ∗E-mail: khoa.ngonhu@hotmail.com Received June 24, 2015 Abstract. In this paper, an analytic homogenization model in traction and flexion for the double corrugated cardboard plates is presented. The proposed analytical homogeniza- tion model allows modelling the 3D double corrugated cardboard with a 2D homogenized plate. This model is essentially based on the theory of stratification and then improved by using the theory of sandwich. This model was validated by comparing the results of Abaqus-3D and H-2D model using ”user’s subroutine” ”UGENS”. The homogenization model can be used not only for corrugated cardboard plates, but also for naval and aero- nautic composite structures. Keywords: Analytical homogenization, corrugated cardboard, orthotropic plates. 1. INTRODUCTION Nowadays, the corrugated cardboard is widely used in the packaging industry, such as corrugated cardboard boxes, insert cardboard sheet in pallet systems. It is essen- tial to predict the mechanical behavior of these systems in order to use such materials effectively. The numerical modeling of this kind of orthotropic composite plates by shell elements is too tedious and time consuming. Many homogenization models were ob- tained by analytical, numerical and experimental methods [1–5]. By using some FE mod- els and commercial FE software, the mechanical behaviors of corrugated cardboard were studied by other authors [6, 7], but it just limited with the single corrugated cardboard. The corrugated cardboard is produced by a converting process in which three or more layers are laminated together. The flat layers are called liners and the corrugated cores are referred to as flutes (Fig. 1). Corrugated cardboard is one of the most used packaging materials to make boxes or interlayers for goods transport. The manufactur- ing process gives three characteristic directions: the machine direction (MD), the cross direction (CD), and the thickness direction (ZD). c© 2016 Vietnam Academy of Science and Technology 206 Duong Pham Tuong Minh, Ngo Nhu Khoa 3D shell 2D plate CD-y MD-x ZD B L Fig. 1. Homogenization model This paper presents an efficient homogenization model for the mechanical behav- ior of a corrugated cardboard composed of three or five layers (single or double flutes). The homogenization is carried out by calculating analytically the global rigidities of the corrugated cardboard and then this 3D structure is replaced by an equivalent homoge- nized 2D plate. The simulations in traction and flexion of Abaqus-3D and H-2D model of double corrugated cardboard will be studied in this article. This 2D homogenization model is very fast and has close results comparing to the 3D model using the Abaqus shell elements. 2. RECALL OF MINDLIN’S THEORY AND THEORY OF LAMINATED PLATES For a model composite plate, the Mindlin theory is often used. It is assumed that a right segment and perpendicular to the mean surface remains straight but not perpen- dicular to the medium surface after deformation. This assumption allows to consider the transverse shear deformations. The membrane forces, bending moments and torsional and transverse shear forces are obtained by integration of the constraints on the thick- ness {N(x, y)} =  NxNy Nxy  = h 2∫ − h2  σxσy σxy dz, (1) {M(x, y)} =  MxMy Mxy  = h 2∫ − h2 z  σxσy σxy dz, (2) {T(x, y)} = [ Tx Ty ] = h 2∫ − h2 [ σxz σyz ] dz. (3) An analytic homogenization model in traction and bending for orthotropic composite plates with the type of . . . 207 If we consider a composite sheet consisting of several layers, the resulting forces defined above may be combined in layersNxNyNxy  = h 2∫ − h2  σxσy σxy dz = n∑k=1 hk∫ hk−1 Q11 Q12 0Q12 Q22 0 0 0 Q33  k  εx εy εxy  m + z  κxκy κxy  dz, (4) MxMyMxy  = h 2∫ − h2  σxσy σxy zdz = n∑k=1 hk∫ hk−1 Q11 Q12 0Q12 Q22 0 0 0 Q33  k z  εx εy εxy  m + z2  κxκy κxy  dz, (5) { Tx Ty } = h 2∫ − h2 { σxz σyz } dz = n ∑ k=1 hk∫ hk−1 [ C11 0 0 C22 ] k { γxz γyz } dz. (6) After the integration along the thickness, we obtain the overall stiffness matrix that links the generalized deformations with resultant forces Nx Ny Nxy Mx My Mxy Tx Ty  =  A11 A12 0 B11 B12 0 0 0 A12 A22 0 B12 B22 0 0 0 0 0 A33 0 0 B33 0 0 B11 B12 0 D11 D12 0 0 0 B12 B22 0 D12 D22 0 0 0 0 0 B33 0 0 D33 0 0 0 0 0 0 0 0 F11 0 0 0 0 0 0 0 0 F22   εxm εym γxym κx κy κxy γxz γyz  , (7) with Aij = n ∑ k=1 [ hk − hk−1 ] Qkij = n ∑ k=1 Qkijt k, Bij = 1 2 n ∑ k=1 [( hk )2 − (hk−1)2]Qkij = n∑ k=1 Qkijt kzk, Dij = 1 3 n ∑ k=1 [( hk )3 − (hk−1)3]Qkij = n∑ k=1 Qkij [ tk ( zk )2 + ( tk )3 12 ] , Fij = n ∑ k=1 [ hk − hk−1 ] Ckij = n ∑ k=1 Ckijt k. (8) The law of behavior above can be written in matrix form {N}{M} {T}  =  [A] [B] [0][B] [D] [0] [0] [0] [F] {εm}{κ} {γs}  , (9) where {N}, {T} and {M} are the internal forces and moments; [A], [D], [B] and [F] are the stiffness matrices related to the membrane forces, the bending-torsion moments, the bending-torsion-membrane coupling effects and the transverse shear forces respectively; 208 Duong Pham Tuong Minh, Ngo Nhu Khoa {ε} is the membrane strain vector, {κ} is the curvature vector and {γs} is the transverse shear strain vector. 3. CORRUGATED CARDBOARD HOMOGENIZEDMODEL A 3D geometrical modeling of the liners and the flutes of the corrugated cardboard is a very tedious and time-consuming task. In our homogenization model, a corrugated cardboard panel is replaced by a 2D plate. Instead of using a local constitutive law (relat- ing the strains to the stresses) at each material point, the homogenization leads to global rigidities (relating the generalized strains to the resultant forces) for the equivalent ho- mogeneous plate. The corrugated cardboard is more complex than a laminated plate because of the fluting cores and the cavities between the three liners. Consequently some global effective stiffnesses in the matrix (9) obtained by the theory of laminated plates should be modified [5, 8]. Consider a double corrugated cardboard and using a, b, c, d, and e to represent the lower liner, lower flute, intermediate liner, upper flute and the upper liner (Fig. 2). The geometry of each flute is defined by the following equations Hb (x) = hb − tb 2 sin ( 2pi Pb x ) θb (x) = tan−1 ( dHb (x) dx ) ;  Hd (x) = hd − td 2 sin ( 2pi Pd x ) θd (x) = tan−1 ( dHd (x) dx ) (10) Fig. 2. Geometry of a double corrugated cardboard To homogenize a panel corrugated double wall, we consider a representative vol- ume element (RVE). This volume must be sufficiently small relative to the dimensions of the entire panel. Once the overall stiffness of each slice are obtained by integrating the thickness, the homogenization along x is performed to calculate the average stiffness of An analytic homogenization model in traction and bending for orthotropic composite plates with the type of . . . 209 all tranches over a period [A] = 1 P P∫ 0 [A(x)]dx, [B] = 1 P P∫ 0 [B(x)]dx, [D] = 1 P P∫ 0 [D(x)]dx, [F] = 1 P P∫ 0 [F(x)]dx. (11) We note that, in Eq. (11), when the periods Pd and Pb of corrugated layers are different, it is necessary to take an RVE having the length P that is the multiple of these periods. 3.1. Traction and bending stiffnesses related to Nx, Mx, Ny, My Since the vertical position (z) of a groove portion (ds) is a function of x and a thickness over its vertical section is a function of the angle of inclination of the groove θx (Fig. 2), Eq. (8) becomes Aij = Qaijt a +Qbij tb cos θb +Qcijt c +Qdij td cos θd +Qeijt e, tbv = tb cos θb , tdv = td cos θd , Bij = Qaijt aza +Qbij tb cos θb zb +Qcijt czc +Qdij td cos θd zd +Qeijt eze, Dij = Qaij [ ta(za)2+ 1 12 (ta)3 ] +Qbij [ tb cos θb ( zb )2 + 1 12 ( tb cos θb )3] +Qcij [ tc(zc)2+ 1 12 (tc)3 ] +Qdij [ td cos θd ( zd )2 + 1 12 ( td cos θd )3] +Qeij [ te(ze)2 + 1 12 (te)3 ] , (12) with h = ta + hb + tc + hd + te, za = −h 2 + ta 2 , ze = h 2 − t e 2 , zc = −h 2 + ta + hb + tc 2 , zb (x) = −h 2 + ta + hb 2 + 1 2 ( hb − tb ) sin ( 2pi Pb x ) , dzb dx = pi ( hb − tb) Pb cos ( 2pi Pb x ) , θb (x) = tan−1 ( dzb dx ) , zd (x) = h 2 − te − h d 2 + 1 2 ( hd − td ) sin ( 2pi Pd x ) , dzd dx = pi ( hd − td) Pd cos ( 2pi Pd x ) , θd (x) = tan−1 ( dzd dx ) . For each of the two grooves, the homogenization on their period (along x) should be calculated numerically according to Eq. (11). 210 Duong Pham Tuong Minh, Ngo Nhu Khoa 3.2. Implementation of the homogenized model in an FE analysis For this problem, the corrugated cardboard is modelled by shell elements, the inter- nal forces and moments are calculated incrementally. Their increments dN and dM have the following relationship with the increments of the membrane strains and bending- torsion curvature [{dN} {dM} ] = [ [A] [B] [B] [D] ] [{dε} {dκ} ] = [H] [{dε} {dκ} ] , (13) where the stiffness matrix [H] is determined by our homogenization model which is im- plemented into ABAQUS by programming a user subroutine UGENS. In ABAQUS, the 4-nodal shell element S4R has 6 degrees of freedom per node (3 displacements and 3 rotations). In the UGENS subroutine, the generalized forces and strains are passed in using the arrays FORCE (6) and STRAN (6), and will be updated at the end of the subroutine; the known generalized strain and curvature increments {dε} and {dκ} are passed in using the array DSTRAN (6). The main purpose of the UGENS is to calculate the stiffness matrix [H] in Eq. (13) which is stored in the array DDNDDE (6, 6) at the end of the subroutine. This array will be used by ABAQUS to define the element stiffness matrix in the FE solution. 4. RESULTS AND DISCUSSION To validate our H-model, we first discretize the five layers of corrugated cardboard by shell elements S4R of Abaqus to obtain the model Abaqus-3D; Then, we discretize the middle surface of corrugated cardboard by shell elements S4R of Abaqus combined with our H-model (using “user’s subroutine” ”UGENS”) to obtain H-2D model. The confrontation of the results allow us to evaluate the efficiency and accuracy of our ho- mogenization model. Fig. 3. Geometry of the CD section of corrugated cardboard The calculations and comparisons are made on a corrugated panel having CD sec- tion illustrated in Fig. 3. Geometric data are: period (or step) and height of the lower groove Pb = 9 mm and hb = 5.2 mm, those of the upper groove Pd = 6 mm and hd = 2.9 mm, thicknesses ta = tc = te = 0.25 mm, tb = td = 0.26 mm. The properties An analytic homogenization model in traction and bending for orthotropic composite plates with the type of . . . 211 of materials are given in Tab. 1 [9]. The rigidities of 2D equivalent plate are calculated as shown in Tab. 2. Table 1. Material properties of the five layers of the corrugated cardboard Layers E1 E2 E3 G12 G13 G23 ν12 ν13 ν23 (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) a 8250 2900 2900 1890 7 70 0.43 0.01 0.01 b, d 4500 4500 3000 1500 3.5 35 0.40 0.01 0.01 c, e 8180 3120 3120 1950 7 70 0.43 0.01 0.01 Table 2. Rigidities of the equivalent plate Rigidities A11 A12 A22 B11 B12 B22 D11 D12 D22 (N/mm) (N/mm) (N/mm) (N) (N) (N) (N.mm) (N.mm) (N.mm) Values 6606.2 1055.1 5989.8 2507.1 526.1 2914.5 75214.1 11870.5 49672.4 We use a corrugated panel having length L = 162 mm and width B = 162 mm. This panel is tested under different types of loading: traction and bending. For the simu- lation of the homogenized plate using our H-2D model, the middle surface is discretized into 2916 quadrilateral elements S4R and nodes 3025. But for the Abaqus simulation-3D, 52116 quadrilateral elements S4R and 48293 nodes are needed. Indeed, to fully describe the geometry of the groove, it takes at least 16 elements over a period of groove. In both types of simulations (Abaqus-3D model and H-2D model), a rigid plate is bonded to the MD or CD section at the right end of the cardboard panel to better apply forces or moments (Fig. 4, 5). The calculations by our H-2D model are very fast while calculations by Abaqus-3D are much longer. The comparisons of results obtained by the two models and the percentages of error in H-2D model compared to Abaqus-3D results for the traction and bending are presented in Tab. 3, we note that the numerical results given by the two models are very close. Fig. 4. Simulation of Abaqus-3D and H-model in traction for the MD section 212 Duong Pham Tuong Minh, Ngo Nhu Khoa Fig. 5. Simulation of Abaqus-3D and H-model in bending for the CD section Table 3. Comparison between Abaqus-3D and 2D-H-model for plate in traction and bending Abaqus-3D H-2D-Model Error (%) Traction MD Displacement U1 (mm) 0.3110 0.3118 +0.26 F = 2000 N CPU time (s) 48.7 1 48.7 times CD Displacement U2 (mm) 0.3433 0.3455 +0.64 CPU time (s) 47.2 1 47.2 times Bending MD Displacement U3 (mm) 11.21 11.14 −0.62 M = 10 KN.mm CPU time (s) 39.7 1 39.7 times CD Displacement U3 (mm) 17.34 17.14 −1.15 CPU time (s) 39.1 1 39.1 times 5. CONCLUSION In this article, we have proposed an analytic homogenization model for the traction and flexion problem of double corrugated core cardboards. The comparison of the results obtained by the analytic formulas, by the Abaqus 3D simulations and by the Abaqus- Ugens 2D simulations has proved the validation of the present traction and flexion ho- mogenization model. 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