824
Viet Dung Luong et. al., Vol. 7, No. 2, 2021
For the elastic homogenization, the classical lamination theory was modified to consider the corrugated sheet. The laminate in plane forces ሼܰ ሽ , transverse shear forces ሼܶ ሽ and out of plane moments ሼ ܯ ሽ can be related to the deformations ሼ ߝ ሽ , ሼ ߛ ሽ and the curvature ሼ ߢ ሽ of the laminate by the following expression:
ۏ ێ ێ ێ ێ ێ ܣ ۍ ێ ଵଵ ܣ ଵଶ Ͳ ܤ ଵଵ ܤ ଵଶ Ͳ Ͳ Ͳ ܣ ଵଶ ܣ ଶଶ Ͳ ܤ ଵଶ ܤ ଶଶ Ͳ Ͳ Ͳ Ͳ Ͳ ܣ ଷଷ Ͳ Ͳ ܤ ଷଷ Ͳ Ͳ ܤ ଵଵ ܤ ଵଶ Ͳ ܦ ଵଵ ܦ ଵଶ Ͳ Ͳ Ͳ ܤ ଵଶ ܤ ଶଶ Ͳ ܦ ଵଶ ܦ ଶଶ Ͳ Ͳ Ͳ Ͳ Ͳ ܤ ଵଵ Ͳ Ͳ ܦ ଷଷ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ܨ ଵଵ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ܨ ଶଶ ۑ ۑ ے ۑ ۑ ۑ ې ۑ ۖۖە ۔ۖۖ ߝ ۓ ௫ ߝ ௬ ߛ ௫௬ ߢ ௫ ߢ ௬ ߢ ௫௬ ߛ ௫௭ ߛ ௬௭ۙۖۖۘۖۖۗ
ۖۖۖە ۔ۖۖۖ ܰۓ ௬ܰ ௫ܰ ௫௬ ܯ ௫ ܯ ௬ ܯ ௫ ௫ܶ ௬ܶ ௬ۙۖۖۖۘۖۖۖۗ
ൌ
(12)
with:
ۖۖۖۖە ۔ۖۖۖۖ ܣۓ ሺ ݔ ሻൌܳ ሺ ଵሻ݁ ଵ ܳ ሺ ଶሻ ൫ ߠ ሺ ݔ ሻ൯݁ ଶ ߠ ሺ ݔ ሻ ܳ ሺ ଷሻ݁ ଷ ܤ ሺ ݔ ሻൌܳ ሺ ଵሻ ݖ ଵ݁ ଵ ܳ ሺ ଶሻ ൫ ߠ ሺ ݔ ሻ൯ ݖ ଶ݁ ଶ ߠ ሺ ݔ ሻ ܳ ሺ ଷሻ ݖ ଷ݁ ଷ ܦ ሺ ݔ ሻൌܳ ሺ ଵሻ ቆ ݖ ଵଶ݁ ଵ ݁ ଵଶ ͳʹ ቇܳ ሺ ଶሻ ൫ ߠ ሺ ݔ ሻ൯ቆ ݖ ଶଶ݁ ଶ ߠ ሺ ݔ ሻ ݁ ଶଶ ͳʹ ଶ ߠ ሺ ݔ ሻ ቇ ܳ ሺ ଷሻ ቆ ݖ ଷଶ݁ ଷ ݁ ଷଶ ͳʹ ቇ ܨ ሺ ݔ ሻൌ ͷ ൬ ܥ ሺ ଵሻ݁ ଵ ܥ ሺ ଶሻ ൫ ߠ ሺ ݔ ሻ൯݁ ଶ ߠ ሺ ݔ ሻ ܥ ሺ ଷሻ݁ ଷ ൰
(13)
where ܳ ሺ ሻ is the reduced stiffness matrix (Eq. (14)), ܥ ሺ ሻ is the transverse shear stiffness matrix (Eq. (15)), and subscripts 1, 2 and 3 denote outer linerboard, inner linerboard and fluting, respectively. ሾܳ ሿ ሺሻ ൌ ێ ێ ۏ ۍ ଵି ఔೣ ாೣ ఔ ೣ భ షഌೣഌೣ ಶഌೣ Ͳ భష ഌ ഌೣ ೣ ಶೣ ഌೣ భషഌೣ ಶ ഌೣ Ͳ Ͳ Ͳ ܩ ௫௬ ۑ ۑ ے ې ሺሻ (14) ሾ ܥ ሿ ሺሻ ൌ ܩ ௬௭ Ͳ Ͳ ܩ ௫௭ ൨ ሺሻ (15) where ܧ ௫ , ܧ ௬ , ߥ ௫௬ , ܩ ௫௬ , ܩ ௬௭ , ܩ ௫௭ are the elastic material properties, with ݔ ǡ ݕ ǡ ݖ the paperboard MD, CD and ZD directions, respectively. The global equivalent stiffness matrix for elastic case is obtained by integrating Eq. (13) over a fluting period :
ۖۖۖۖۖە ۔ۖۖۖۖۖ ܣ ۓ ౹ ൌන ܣ ሺ ݔ ሻ݀ ݔ ܤ ౹ ൌන ܤ ሺ ݔ ሻ݀ ݔ ܦ ౹ ൌන ܦ ሺ ݔ ሻ݀ ݔ ܨ ౹ ൌන ܨ ሺ ݔ ሻ݀ ݔ
(16)
Some simplifying assumptions and detailed calculations of the equivalent stiffness terms can be found in [14-17]. The elastic parameters of linerboards and fluting are obtained from standard experimental tensile tests. Then, the homogeneous stiffnesses of corrugated cardboard are computed using Eq. (16). Finally, the homogenized material stiffness matrix ܳ of the corrugated cardboard is obtained from Eq. (17): ܳ ൌ ͳʹ ܦ ౹ ݐ ౹ ଷ (17) with: ݐ ౹ ൌඨ σ ܦ ౹ ଷୀଵ σ ܣ ౹ ଷୀଵ (18) To find the effective elastoplastic parameters for shell model that replaces 3D structural corrugated cardboard model, an inverse identification procedure is used. We carried out three tensile test simulations on different samples: MD-sample, CD-sample and 45° oriented-sample using a 3D structural model to generate tensile curves, which are then used to identify an equivalent shell. The 3D structural and the 2D homogenized tensile samples are meshed with rectangular reduced integration shells elements (S4R) with a mesh size of 0.5 mm as shown in Fig. (6). The obtained load vs displacement curves are compared to the numerical equivalent shell curves by minimizing the least square error defined in Eq. (19).
Journal of Applied and Computational Mechanics, Vol. 7, No. 2, (2021), 820-830
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