Equivalence model for analyzing nonlinear mechanical behavior of Corrugated Core Composite Panel

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In this study, we propose to build an equivalent 2D model to replace the 3D model of corrugated composite sheets, by the method of homogenization, to study the nonlinear mechanical behavior of corrugated core composite sheets. This model helps to significantly reduce the calculation time as well as the model building time. The model homogeneity is confirmed by comparing the numerical simulation results of the 2D model and the 3D model.

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Nội dung Text: Equivalence model for analyzing nonlinear mechanical behavior of Corrugated Core Composite Panel

92 108 Tuyển tập công trình Hội nghị Cơ học toàn quốc lần thứ XI, Hà Nội, 02-03/12/2022 Equivalence model for analyzing nonlinear mechanical behavior of Corrugated Core Composite Panel Luong Viet Dung*, Dao Lien Tien, and Duong Pham Tuong Minh Thai Nguyen University of Technology *Email: luongvietdung@tnut.edu.vn Abstract. A corrugated core composite panel is one of the material structures that is becoming more and more popular in many fields such as the aviation industry, and shipbuilding. In order to use this material effectively, it is necessary to understand its mechanical behavior of them. Up to now, the calculation of these structures has been mentioned by many authors, however, the calculation is still very difficult. Especially when studying their nonlinear mechanical behavior. In this study, we propose to build an equivalent 2D model to replace the 3D model of corrugated composite sheets, by the method of homogenization, to study the nonlinear mechanical behavior of corrugated core composite sheets. This model helps to significantly reduce the calculation time as well as the model building time. The model homogeneity is confirmed by comparing the numerical simulation results of the 2D model and the 3D model. Keywords: Homogenization, Finite element, Simulation, Corrugated, Composite panel 1. Introduction Sandwich panels have been widely used in many fields such as in aerospace industries, marine, mechanical and civil engineering applications. This is due to the high stiffness and high strength-to- weight ratio of the sandwich panels. The sandwich panel structure consists of lightweight core and rigid outer layers. The core may consist of a continuous or discrete structural component. Core shapes are diverse such as corrugation, honeycomb, foam, triangle, etc. In which, corrugated cardboard is widely used in product packaging. Carton is made up of layers of paper, corrugated, usually at least 2 layers of paper and one layer of corrugated board. Corrugated cardboard panels are one of the most complicated orthotropic sandwich panels. By using the finite element method, the mechanical behavior of the panels is determined, but with the complex 3D modeling of the corrugated board, numerical modeling and simulation will be time consuming and may not be possible with large panels. The homogenization method is used to simplify the simulation process; the corrugated board will be replaced by equivalent solid 3D or 2D sheet. Aboura et al. [1] also developed a homogeneous analysis model based on multilayer laminate theory and compared its results with numerical and experimental results, Biancolini [2] used FE numerical method to evaluate the stiffness parameters, Carlsson et al. [3,4] have presented homogenous properties such as cross-sectional stiffness and flexural stiffness of corrugated board by analytical method, Nordstrand and colleagues [5,6,7] have demonstrated some of the properties of corrugated board by analytical methods, but these studies are limited to the behavior of the corrugated board in the elastic region. In this paper, we propose a homogenization model to determine the parameters of the cardboard in the plastic region. Numerical simulation of tests was performed for homogeneous model panel and the 3D structure panel. This homogeneity model is confirmed by comparing the results obtained between the two models. 2. Homogenization model 2.1. Material studies
93 Equivalence model for analyzing nonlinear mechanical behavior of Corrugated Core Composite Panel 109 We used corrugated cardboard material in this study with the properties shown in Figure 1 and given in Table 1. The manufacturing process gives three characteristic directions: the machine direction (MD), the cross direction (CD), and the thickness direction (ZD). Figure 1. Geometric structure and the directions of the corrugated cardboard panel Table 1. Properties of the papers Paper Ex Ey G xy ν xy E0 n a b c d ε0 (MPa) (MPa) (MPa) 1,3 2433.2 859.91 0.0829 1077.2 96.45 4.97 1.0 2.498 2.498 1.622 0.48e-3 2 1130.4 625.85 0.0717 303.05 87.31 4.247 1.0 2.178 2.178 1.871 0.92e-3 2.2. Homogenization model The homogenization method is based on Mindlin's theory for thick or composite panels. It assumes that a segment that is straight and perpendicular to the mean surface remains straight but not perpendicular to the mean surface after the deformation. This assumption makes it possible to take into account the deformations of transverse shearing. On the mean surface of a panel, we define the x and y axes on the surface and the z axis perpendicular to the surface (Figure 2), Mindlin's theory assumes the following displacement field: uq= u + z β x  vq = v + z β y (1)  w q = w where u q , v q and w q are the displacements of a point q(x, y, z), u, ν and w are the displacements of the point p(x, y, 0) on the mean surface, β x is the angle of rotation of the normal from z to x or the angle of rotation around y (β x =θ y ), βy is the angle of rotation from the normal from z to y or the angle of rotation around -x (β y =-θ x ). Figure 2. Membrane forces, bending-torsional moments, and transverse shear forces on a panel.
94 110 Luong Viet Dung et all 𝜀𝜀 𝑥𝑥 = 𝑢𝑢 𝑞𝑞 , 𝑥𝑥 = 𝑢𝑢, 𝑥𝑥 + 𝑧𝑧𝛽𝛽 𝑥𝑥 , 𝑥𝑥 ⎧ The deformation field is written as follows: ⎪ 𝜀𝜀 𝑦𝑦 = 𝑢𝑢 𝑞𝑞 , 𝑦𝑦 = 𝑢𝑢, 𝑦𝑦 + 𝑧𝑧𝛽𝛽 𝑦𝑦 , 𝑦𝑦 ⎪ 𝛾𝛾𝑥𝑥𝑥𝑥 = 2𝜀𝜀 𝑥𝑥𝑥𝑥 = 𝑢𝑢 𝑞𝑞 , 𝑦𝑦 + 𝜈𝜈 𝑞𝑞 , 𝑥𝑥 = 𝑢𝑢, 𝑦𝑦 + 𝜈𝜈, 𝑥𝑥 + 𝑧𝑧(𝛽𝛽 𝑥𝑥 , 𝑦𝑦 + 𝛽𝛽 𝑦𝑦 , 𝑥𝑥 ) ⎨ 𝛾𝛾𝑥𝑥𝑥𝑥 = 2𝜀𝜀 𝑥𝑥𝑥𝑥 = 𝑢𝑢 𝑞𝑞 , + 𝑤𝑤 𝑞𝑞 , 𝑥𝑥 = 𝑤𝑤, 𝑥𝑥 + 𝛽𝛽 𝑥𝑥 ⎪ 𝛾𝛾𝑦𝑦𝑦𝑦 = 2𝜀𝜀 𝑦𝑦𝑦𝑦 = 𝑢𝑢 𝑞𝑞 , + 𝑤𝑤 𝑞𝑞 , 𝑦𝑦 = 𝑤𝑤, 𝑦𝑦 + 𝛽𝛽 𝑦𝑦 ⎪ (2) ⎩ 𝜀𝜀 = 𝑤𝑤 𝑞𝑞 , = 0 𝑧𝑧 𝑧𝑧 𝑧𝑧 𝑧𝑧 { 𝜀𝜀 } = { 𝜀𝜀 𝑚𝑚 } + 𝑧𝑧{ 𝜅𝜅} Plane strains can be decomposed into the membrane and bending strains as follows: 𝜎𝜎 𝑥𝑥 𝜀𝜀 𝑥𝑥 0 𝜀𝜀 𝑥𝑥 (3) 𝐸𝐸 𝑥𝑥 𝜈𝜈 𝑥𝑥𝑥𝑥 𝐸𝐸 𝜎𝜎 𝑦𝑦 � = [ 𝑄𝑄] � 𝜀𝜀 𝑦𝑦 � = � 𝜈𝜈 𝐸𝐸 𝑥𝑥 𝜀𝜀 𝑦𝑦 � 1−𝜈𝜈 𝑥𝑥𝑥𝑥 𝜈𝜈 1−𝜈𝜈 𝑥𝑥𝑥𝑥 𝜈𝜈 { 𝜎𝜎} = � 𝐸𝐸 0 �� 𝜎𝜎 𝑥𝑥𝑥𝑥 𝛾𝛾𝑥𝑥𝑥𝑥 𝛾𝛾𝑥𝑥𝑥𝑥 𝑦𝑦 0 0 𝐺𝐺 𝑥𝑥𝑥𝑥 1−𝜈𝜈 𝑥𝑥𝑥𝑥 𝜈𝜈 1−𝜈𝜈 𝑥𝑥𝑥𝑥 𝜈𝜈 𝑦𝑦𝑦𝑦 𝑦𝑦𝑦𝑦 (4) 𝜎𝜎 𝑥𝑥𝑥𝑥 𝛾𝛾𝑥𝑥𝑥𝑥 𝐺𝐺 𝑥𝑥𝑥𝑥 0 𝛾𝛾𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 𝑦𝑦 �𝜎𝜎 𝛾𝛾 � = � 𝜎𝜎 � = [ 𝐶𝐶 ] � 𝛾𝛾 � = � 0 � � 𝛾𝛾 � 𝑦𝑦𝑦𝑦 𝑦𝑦𝑦𝑦 𝑦𝑦𝑦𝑦 𝑦𝑦𝑦𝑦 𝐺𝐺 𝑦𝑦𝑦𝑦 𝑦𝑦𝑦𝑦 𝑁𝑁𝑥𝑥 𝜎𝜎 𝑥𝑥 (5) { 𝑁𝑁(𝑥𝑥, 𝑦𝑦)} = � 𝑁𝑁 𝑦𝑦 � = ∫2ℎ � 𝜎𝜎 𝑦𝑦 � 𝑑𝑑𝑑𝑑 − 𝜎𝜎 𝑁𝑁𝑥𝑥𝑥𝑥 ℎ 2 𝑥𝑥𝑥𝑥 (6) 𝑀𝑀 𝑥𝑥 𝜎𝜎 𝑥𝑥 { 𝑀𝑀(𝑥𝑥, 𝑦𝑦)} = � 𝑀𝑀 𝑦𝑦 � = ∫2ℎ � 𝜎𝜎 𝑦𝑦 � 𝑧𝑧𝑧𝑧𝑧𝑧 − 𝜎𝜎 𝑀𝑀 𝑥𝑥𝑥𝑥 ℎ 2 𝑥𝑥𝑥𝑥 (7) 𝑇𝑇 𝜎𝜎 𝑥𝑥𝑥𝑥 { 𝑇𝑇(𝑥𝑥, 𝑦𝑦)} = � 𝑥𝑥 � = ∫2ℎ � 𝜎𝜎 � 𝑑𝑑𝑑𝑑 𝑇𝑇𝑦𝑦 ℎ − 𝑦𝑦𝑦𝑦 𝑁𝑁𝑥𝑥 𝜎𝜎 𝑥𝑥 𝑄𝑄11 𝑄𝑄12 0 𝜀𝜀 𝑥𝑥 𝜅𝜅 𝑥𝑥 2 (8) � 𝑁𝑁 𝑦𝑦 � = ∫2ℎ � 𝜎𝜎 𝑦𝑦 � 𝑑𝑑𝑑𝑑 = ∑ 𝑘𝑘=1 ∫ 𝑘𝑘 � 𝑄𝑄12 𝑄𝑄22 𝑛𝑛 0 � �� 𝜀𝜀 𝑦𝑦 � + 𝑧𝑧 � 𝜅𝜅 𝑦𝑦 �� 𝑑𝑑𝑑𝑑 − 𝜎𝜎 𝛾𝛾𝑥𝑥𝑥𝑥 𝜅𝜅 𝑥𝑥𝑥𝑥 𝑁𝑁𝑥𝑥𝑥𝑥 0 0 𝑄𝑄33 𝑘𝑘 ℎ ℎ ℎ 𝑘𝑘−1 2 𝑥𝑥𝑥𝑥 (9) 𝑀𝑀 𝑥𝑥 𝜎𝜎 𝑥𝑥 𝑄𝑄11 𝑄𝑄12 0 𝜀𝜀 𝑥𝑥 𝜅𝜅 𝑥𝑥 � 𝑀𝑀 𝑦𝑦 � = ∫2ℎ � 𝜎𝜎 𝑦𝑦 � 𝑧𝑧𝑧𝑧𝑧𝑧 = ∑ 𝑘𝑘=1 ∫ℎ 𝑘𝑘 � 𝑄𝑄12 𝑄𝑄22 𝑛𝑛 0 � �𝑧𝑧 � 𝜀𝜀 𝑦𝑦 � + 𝑧𝑧 2 � 𝜅𝜅 𝑦𝑦 �� 𝑑𝑑𝑑𝑑 (10) − 𝜎𝜎 𝛾𝛾𝑥𝑥𝑥𝑥 𝜅𝜅 𝑥𝑥𝑥𝑥 𝑀𝑀 𝑥𝑥𝑥𝑥 0 0 𝑄𝑄33 ℎ ℎ 𝑘𝑘−1 2 𝑥𝑥𝑥𝑥 𝑘𝑘 𝑇𝑇𝑥𝑥 𝜎𝜎 𝑥𝑥𝑥𝑥 𝐶𝐶 0 𝛾𝛾𝑥𝑥𝑥𝑥 � 𝑇𝑇 � = ∫2ℎ � 𝜎𝜎 � 𝑑𝑑𝑑𝑑 = ∑ 𝑘𝑘=1 ∫ℎ 𝑘𝑘 � 11 𝑛𝑛 � � 𝛾𝛾 � 𝑑𝑑𝑑𝑑 0 𝐶𝐶22 𝑘𝑘 𝑦𝑦𝑦𝑦 ℎ 𝑦𝑦 − 𝑦𝑦𝑦𝑦 ℎ 𝑘𝑘−1 2 (11) After the integration according to the thickness, one obtains the matrix of the total rigidities which binds 𝑁𝑁𝑥𝑥 𝜀𝜀 𝑥𝑥 ⎧ 𝑁𝑁 ⎫ ⎡ 𝐴𝐴11 𝐴𝐴12 0 𝐵𝐵11 𝐵𝐵12 0 0 0 the generalized deformations to the resulting efforts: ⎪ 𝑦𝑦 ⎪ ⎢ 𝐴𝐴12 𝐴𝐴22 0 𝐵𝐵12 𝐵𝐵22 0 0 0 ⎤ ⎧ 𝜀𝜀 𝑦𝑦 ⎫ ⎥ ⎪ 𝑁𝑁𝑥𝑥𝑥𝑥 ⎪ 0 0 𝐴𝐴33 0 0 𝐵𝐵33 0 0 ⎪ 𝛾𝛾𝑥𝑥𝑥𝑥 ⎪ ⎪ ⎪ ⎢ ⎥⎪ ⎪ 𝑀𝑀 𝑥𝑥 𝐵𝐵11 𝐵𝐵12 0 𝐷𝐷11 𝐷𝐷12 0 0 0 ⎥ 𝜅𝜅 𝑥𝑥 =⎢ ⎨ 𝑀𝑀 𝑦𝑦 ⎬ ⎢ 𝐵𝐵12 𝐵𝐵22 0 𝐷𝐷12 𝐷𝐷22 0 0 0 ⎥ ⎨ 𝜅𝜅 𝑦𝑦 ⎬ ⎪ 𝑀𝑀 𝑥𝑥𝑥𝑥 ⎪ ⎢ 0 0 𝐵𝐵11 0 0 𝐷𝐷33 0 0 ⎥ ⎪ 𝜅𝜅 𝑥𝑥𝑥𝑥 ⎪ (12) ⎪ 𝑇𝑇 ⎪ ⎢ 0 0 0 0 0 0 𝐹𝐹11 0 ⎥ ⎪ 𝛾𝛾𝑥𝑥𝑥𝑥 ⎪ ⎪ 𝑥𝑥 ⎪ ⎩ 𝑇𝑇𝑦𝑦 ⎭ ⎣ 0 0 0 0 0 0 0 𝐹𝐹22 ⎦ ⎩ 𝛾𝛾𝑦𝑦𝑦𝑦 ⎭ with:
95 Equivalence model for analyzing nonlinear mechanical behavior of Corrugated Core Composite Panel 111 𝐴𝐴 𝑖𝑖 𝑖𝑖 = ∑ 𝑘𝑘=1�ℎ 𝑘𝑘 − ℎ 𝑘𝑘−1 � 𝑄𝑄 𝑖𝑖𝑘𝑘𝑖𝑖 = ∑ 𝑘𝑘=1 𝑄𝑄 𝑖𝑖𝑘𝑘𝑖𝑖 𝑒𝑒 𝑘𝑘 𝑛𝑛 𝑛𝑛 𝐵𝐵𝑖𝑖 𝑖𝑖 = ∑ 𝑘𝑘=1 ��ℎ � − �ℎ 𝑘𝑘 2 𝑘𝑘−1 2 � � 𝑄𝑄 𝑖𝑖𝑘𝑘𝑖𝑖 = ∑ 𝑘𝑘=1 𝑄𝑄 𝑖𝑖𝑘𝑘𝑖𝑖 𝑒𝑒 𝑘𝑘 𝑧𝑧 𝑘𝑘 1 𝑛𝑛 𝑛𝑛 (13) 2 𝐷𝐷𝑖𝑖 𝑖𝑖 = ∑ 𝑘𝑘=1 ��ℎ 𝑘𝑘 � − �ℎ 𝑘𝑘−1 � � 𝑄𝑄 𝑖𝑖𝑘𝑘𝑖𝑖 = ∑ 𝑘𝑘=1 𝑄𝑄 𝑖𝑖𝑘𝑘𝑖𝑖 �𝑒𝑒 𝑘𝑘 �𝑧𝑧 𝑘𝑘 � + � 3 3 3 2 (14) 1𝑛𝑛 𝑛𝑛 �𝑒𝑒 𝑘𝑘 � 3 12 𝐹𝐹𝑖𝑖 𝑖𝑖 = ∑ 𝑘𝑘=1�ℎ 𝑘𝑘 − ℎ 𝑘𝑘−1 � 𝐶𝐶𝑖𝑖𝑘𝑘 = ∑ 𝑘𝑘=1 𝐶𝐶𝑖𝑖𝑘𝑘 𝑒𝑒 𝑘𝑘 (15) 𝑛𝑛 𝑛𝑛 𝑖𝑖 𝑖𝑖 (16) Where A ij represent the stiffness’s of membrane, D ij represent the stiffness’s of bending and torsion, F ij represent the stiffness’s of transverse shears and B ij represent the terms of coupling between membrane and bending-torsion. If the composite panel is symmetric with respect to its average surface, this coupling disappears and B ij =0. Homogenization consists in representing a sandwich panel by a homogeneous panel using the periodic unit cell of a corrugated cardboard represented in Figure 3, where the local reference of the 𝜃𝜃( 𝑥𝑥 ) = tan−1 � groove is defined using the angle θ(x): � 𝑑𝑑ℎ(𝑥𝑥) � 𝑑𝑑𝑑𝑑 ℎ( 𝑥𝑥 ) = � − 2 � sin �2𝜋𝜋 � ℎ 𝑐𝑐 𝑒𝑒 𝑥𝑥 2 2 𝑃𝑃 (17) Figure 3. Periodic unit cell of a corrugated cardboard and numerical integration points through the thickness 𝐴𝐴 𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) = 𝑄𝑄 𝑖𝑖 𝑖𝑖 𝑒𝑒1 + 𝑄𝑄 𝑖𝑖 𝑖𝑖 �𝜃𝜃( 𝑥𝑥 )� + 𝑄𝑄 𝑖𝑖 𝑖𝑖 𝑒𝑒3 (1) (2) 𝑒𝑒2 (3) Equations (13) to (16) are modified for corrugated cardboard and are written as follows: cos 𝜃𝜃(𝑥𝑥) 𝐵𝐵𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) = 𝑄𝑄 𝑖𝑖 𝑖𝑖 𝑧𝑧1 𝑒𝑒1 + 𝑄𝑄 𝑖𝑖 𝑖𝑖 �𝜃𝜃( 𝑥𝑥 )�𝑧𝑧2 + 𝑄𝑄 𝑖𝑖 𝑖𝑖 𝑧𝑧3 𝑒𝑒3 (1) (2) 𝑒𝑒2 (3) (18) cos 𝜃𝜃(𝑥𝑥) 𝐷𝐷𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) = 𝑄𝑄 𝑖𝑖 𝑖𝑖 �𝑧𝑧1 𝑒𝑒1 + � + 𝑄𝑄 𝑖𝑖 𝑖𝑖 �𝜃𝜃( 𝑥𝑥 )� �𝑧𝑧2 + � + 𝑄𝑄 𝑖𝑖 𝑖𝑖 �𝑧𝑧3 𝑒𝑒3 + + � (1) 𝑒𝑒1 2 (2) 𝑒𝑒2 𝑒𝑒2 2 (3) 𝑒𝑒3 2 (19) 2 2 2 12 cos 𝜃𝜃(𝑥𝑥) 12 cos2 𝜃𝜃(𝑥𝑥) 12 𝐹𝐹𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) = �𝐶𝐶𝑖𝑖 𝑖𝑖 𝑒𝑒1 + 𝐶𝐶𝑖𝑖 𝑖𝑖 �𝜃𝜃( 𝑥𝑥 )� + 𝐶𝐶𝑖𝑖 𝑖𝑖 𝑒𝑒3 � 5 (1) (2) 𝑒𝑒2 (3) (20) 6 cos 𝜃𝜃(𝑥𝑥) (21) The homogenization along the direction MD consists in calculating the average stiffness’s of all 𝐴𝐴 𝑖𝑖𝐻𝐻𝑖𝑖 = ∫ 𝐴𝐴 𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) 𝑑𝑑𝑑𝑑 1 𝑃𝑃 the slices dx over a sinusoidal period P: 𝑃𝑃 0 𝐵𝐵𝑖𝑖𝐻𝐻 = ∫ 𝐵𝐵𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) 𝑑𝑑𝑥𝑥 1 𝑃𝑃 (22) 𝑖𝑖 𝑃𝑃 0 𝐷𝐷𝑖𝑖𝐻𝐻 = ∫ 𝐷𝐷𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) 𝑑𝑑𝑑𝑑 (23) 1 𝑃𝑃 𝑖𝑖 𝑃𝑃 0 (24)
96 112 Luong Viet Dung et all 𝐹𝐹𝑖𝑖𝐻𝐻 = ∫ 𝐹𝐹𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) 𝑑𝑑𝑑𝑑 1 𝑃𝑃 𝑖𝑖 𝑃𝑃 0 (25) The simplifying assumptions as well as the calculations of the different stiffnesses are given in the [8,9]. The plastic behavior of each stratum of corrugated board (skins and flute) can be represented by the IPE model [10]. We continue to propose an IPE plastic homogenization model based on the homogenization technique developed for the elastic case. To determine the equivalent tangent matrix of the corrugated cardboard, we use 3 integration points according to the thickness of each of its layers. The plasticity determine the stress state and the tangent matrix in the local coordinate system of each layer 𝑘𝑘: �𝑄𝑄 𝑝𝑝 �. (𝑘𝑘) algorithm developed for compact cardboard is used for each of the layers at each integration point to Membrane forces, bending moments and torsion are obtained by integrating the constraints against the sheet thickness by replacing the elastic matrices [Q(k)] with tangent matrices [Q p (k)] in equations (18), 𝐴𝐴 𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) = ∑3 𝑄𝑄(1) 𝑤𝑤 𝑘𝑘 + ∑3 𝑄𝑄(2) �𝜃𝜃( 𝑥𝑥 )�𝑤𝑤 𝑘𝑘 + ∑3 𝑄𝑄(3) 𝑤𝑤 𝑘𝑘 𝑒𝑒1 𝑒𝑒2 𝑒𝑒3 (19) and (20) to arrive at terms of the overall stiffness matrix: 2 𝑘𝑘=1 𝑝𝑝𝑝𝑝𝑝𝑝 2cos 𝜃𝜃(𝑥𝑥) 𝑘𝑘=1 𝑝𝑝𝑝𝑝𝑝𝑝 2 𝑘𝑘=1 𝑝𝑝𝑝𝑝𝑝𝑝 𝐵𝐵𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) = ∑3 𝑄𝑄 (1) 𝑧𝑧 𝑘𝑘 𝑤𝑤 𝑘𝑘 + ∑3 𝑄𝑄(2) �𝜃𝜃( 𝑥𝑥 )�𝑧𝑧 𝑘𝑘 𝑤𝑤 𝑘𝑘 + ∑3 𝑄𝑄(3) 𝑧𝑧 𝑘𝑘 𝑤𝑤 𝑘𝑘 (26) 𝑒𝑒1 𝑒𝑒2 𝑒𝑒3 2 𝑘𝑘=1 𝑝𝑝𝑝𝑝𝑝𝑝 2cos 𝜃𝜃(𝑥𝑥) 𝑘𝑘=1 𝑝𝑝𝑝𝑝𝑝𝑝 2 𝑘𝑘=1 𝑝𝑝𝑝𝑝𝑝𝑝 𝐷𝐷𝑖𝑖 𝑖𝑖 ( 𝑥𝑥 ) = ∑3 𝑄𝑄 𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 2 𝑤𝑤 𝑘𝑘 + ∑3 𝑄𝑄 𝑝𝑝𝑝𝑝𝑝𝑝 �𝜃𝜃( 𝑥𝑥 )�𝑧𝑧 2 𝑤𝑤 𝑘𝑘 + ∑3 𝑄𝑄 𝑝𝑝𝑝𝑝𝑝𝑝 𝑧𝑧 2 𝑤𝑤 𝑘𝑘 𝑒𝑒1 (1) 𝑒𝑒2 (2) 𝑒𝑒3 (3) (27) 2 𝑘𝑘=1 𝑘𝑘 2cos 𝜃𝜃(𝑥𝑥) 𝑘𝑘=1 𝑘𝑘 2 𝑘𝑘=1 𝑘𝑘 (28) Where w k represents the numerical integration weight corresponding to the point of integration k of the considered layer. The plastic homogenization model of corrugated cardboard has been implemented in the Abaqus/Standard computer code using the user subroutine UGENS 3. Numerical validation of the homogenization model 3.1. Validation of the homogenization model by simulation of tensile test Simulations of tensile tests in the MD, CD and 45° directions are carried out using the proposed homogenization model and the complete model. The dimensions of specimen for the two models are the same (Figure 4). The 3D structure and the homogenized panel are meshed with reduced integration rectangular shell elements (S4R) with a size of 0.4 mm. Figure 4. Meshes of the corrugated cardboard 3D structure and the 2D homogenized panel Figure 5 shows the comparison of the force-elongation curves between the 3D corrugated structure and the homogenized panel for tensile following MD, CD, and 45°. The maximum difference between the two models is less than 2%. The homogenized model describes the tensile test very satisfactorily.
97 Equivalence model for analyzing nonlinear mechanical behavior of Corrugated Core Composite Panel 113 Figure 5. Comparison of the force vs elongation curves between the 3D corrugated structure and the homogenized panel The comparison of the CPU times of the simulations between the 3D corrugated structure and the homogenized panel models are given in Figure 6. We find that our homogenization model makes it possible to reduce the CPU time by a factor of 2 to 4. Figure 6. CPU time (s) comparison 3.2. Validation of the homogenization model in frequencies and eigenmodes We use a corrugated cardboard panel having the length L=80 mm according to CD and the width B=45 mm according to MD. For the Abaqus-3D simulation, the mesh consists of 17920 quadrilateral S4R elements and 17172 nodes. For the simulation of the homogenized panel with our H-model, the average surface of the cardboard is discretized into 3645 quadrilateral elements S4R and 3520 nodes. For the calculation of the frequencies and eigenmodes, the panel is fixed on the CD section at one of its ends. Figures 7, 8 and 9 show the first three eigenmodes obtained by the Abaqus-3D and H-Model models. The simulation with Abaqus-3D is 16.7 times more expensive in CPU time with 20s compared to the simulation with our H-model with 1.2s only.
98 114 Luong Viet Dung et all Figure 7. 1st eigenmode of the corrugated cardboard panel obtained by Abaqus-3D and H-Model Figure 8. 2nd eigenmode of the corrugated cardboard panel obtained by Abaqus-3D and H-Model Figure 9. 3rd eigenmode of the corrugated cardboard panel obtained by Abaqus-3D and H-Model The differences between three eigenfrequencies obtained by the two models are less than 9% as indicated in Table 2. Table 2. Comparison of natural frequencies calculated by Abaqus-3D and H-Model 1st Mode 2nd Mode 3rd Mode Abaqus-3D 136.82 Hz 482.07 Hz 782.2 Hz H-Model 124.9 Hz 481.08 Hz 762.55 Hz Error 8.71% 0.205% 2.51%
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