Finite element and experimental investigation on the effect of repetitive shock in corrugated cardboard packaging 825
24 mm
24 mm
100 mm
100 mm
Fig. 6. Three-dimensional structural and equivalent corrugated cardboard meshes.
Moving plate
Fixed plate
Fig. 7. Boundary conditions for box compression test.
ܨ ൌ ͳܰ ൫ ܨ ౹ ሺሼܲ ሽǡ ݐ ሻെ ܨ ଷ ሺ ݐ ሻ൯ ଶ ே ୀଵ
(19) where ܨ and ܨ ଷ are the equivalent shell and 3D structural numerical forces at ݐ sampling point, respectively, ሼܲ ሽൌ ሼ ܧ ǡ ߝ ǡ݊ ǡܽ ǡܾ ǡܿ ǡ݀ ሽ is the unknown parameter vector and ܰ is the number of sampling points. The nonlinearity of the objective function and the possibility of non-uniqueness of the solution make the inverse problem a nonconvex optimization problem. Therefore, a robust global optimization method was required. In this study, the Multi-Objective Genetic Algorithm (MOGA-II) [30-31] was used. It uses a smart multisearch elitism for robustness and directional crossover for fast convergence. Its efficiency is ruled by its operators (classical crossover, directional crossover, mutation and selection) and by the use of elitism. In this study, we used the following parameters: population size = 12, Probability of Directional Cross-over = 0.5, Probability of Selection = 0.05, Probability of Mutation = 0.1, and number of generations = 20. 4. Results and Discussion 4.1 Model Calibration We used the method proposed in previous section to evaluate the equivalent elastoplastic parameters of the corrugated cardboard as follows: x Evaluation of the linerboards and the fluting elastic properties from the experimental tensile tests. The obtained properties are given in Table (2). x Simulation of three tensile tests on different samples (MD, CD and 45°) using a 3D structural model to generate tensile curves. x Identification of the equivalent elastoplastic parameters of the corrugated cardboard using inverse analysis procedure by comparing the generated tensile curves with the simulation tensile curves obtained using the homogenized shell. The determined equivalent elastoplastic parameters of the corrugated cardboard are summarized in Table (3). The identified model is finally used to simulate the box compression test presented in section (2.3). The finite element model consists of two rigid plates that transmit loads to the box and which size is the same as experiment (Fig. (7)). For this simulation friction interaction between plates and box was used to model boundary conditions of system. Bottom rigid plate is fixed so it serves as support for box and top plate is moved vertically for a given displacement. Furthermore, the displacements and rotations of top plate is constrained in other directions. The box is meshed using 9603 rectangular reduced integration shell elements (S4R) and 10152 nodes. Table 2. Elastoplastic properties of linerboards and fluting. ܧ ௫ (MPa) ܧ ௬ (MPa) ߥ ௫௬ ܩ ௫௬ (MPa) ܧ (MPa) ݊ ܽ ܾ ܿ ݀ ߝ Top linerboard 3008 1505 0.17 834 256 2.03 1 2.03 2.28 1.18 0.0034 Fluting 3072 1454 0.15 705 436 1.62 1 2.01 1.25 1.13 0.0010 Bottom linerboard 3034 1502 0.23 737 184 2.06 1 2.21 2.19 1.32 0.0011
Table 3. Equivalent elastoplastic properties of the corrugated cardboard. ܧ ௫ (MPa) ܧ ௬ (MPa) ߥ ௫ ௬ ܩ ௫ ௬ (MPa) ܧ (MPa) ݊ ܽ ܾ ܿ ݀ ߝ 368.8 351.8 0.092 166.2 38.5 2.04 1 1.65 0.84 1.55 0.0068
Journal of Applied and Computational Mechanics, Vol. 7, No. 2, (2021), 820-830
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