Finite element and experimental investigation on the effect of repetitive shock in corrugated cardboard packaging 823
The boundary conditions set on the ܵ ௩ and ܵ ఙ surfaces, respectively, are: ൜ ݑ ሶ ൌ ݒ ǡ ݊ ܵ ௩ ߪ ݊ ൌܶ ǡ ݊ ܵ ఙ where ݒ is the loading velocity, ܶ is the traction and ݊ is the surface normal. 3.2 Paperboard Elastoplastic Model
(5)
In this work, the orthotropic elastoplastic material model proposed by Mäkelä and Östlund [26] was used to predict the behavior of the linerboards and the fluting. This model is based on the concept of material equivalent isotropic plasticity (IPE) introduced by Karafillis and Boyce [28]. The IPE-material is a fictitious isotropic material, subjected to a stress state that equals the corresponding stress state in the actual anisotropic material. The yield criterion may be expressed as: ݂ ൌ ߪ െ ߪ ௬ ൌ൬ ͵ ʹ ۧݏۦ ሼ ݏ ሽ൰ ଵȀଶ െ ܧ ൫ ߝ ߝ ൯ ଵȀ (6) where ߪ ௬ is the yield stress, ሼ ݏ ሽ is the deviatoric stress tensor, ߝ is the equivalent plastic strain, ܧ and ߝ , are two model parameters. However, the definition of the deviatoric stress tensor for the IPE-material differs from J2-flow theory and is expressed as: ሼ ݏ ሽ ൌ൞ ݏ ௫ ݏ ௬ ݏ ௭ ݏ ௫௬ൢ ൌሾ ܮ ሿሼ ߪ ሽൌ ͳ ͵ ൦ ʹܽܿ െܽ െܾ Ͳܿ െܽ െܾ ʹܾ Ͳܾ െܿ Ͳ െܽܽ െܾ െܿ Ͳ Ͳ ͵݀ ൪൝ ߪ ௫ ߪ ௬ ߪ ௫௬ ൡ (7) where , , and are model parameters. Since this material model is not available in ABAQUS software, it was implemented using the material user subroutine VUMAT [29]. The aim of this material subroutine is to invoke a given increment in total strain and return the corresponding stress state and the internal state variable (the equivalent plastic strain in our case). A backward-Euler approach is adopted in the implementation of the subroutine. The starting point of the calculation of the stress state, corresponding to a given increment in total strain ο ߝ , is the calculation of the trial stress state ߪ ௧ ௪ assuming a pure elastic behavior: ߪ ௧ ௪ ൌ ߪ ௗ ܥ ο ߝ (8) The value of the loading function is evaluated by Eq. (6): if ݂ ൏Ͳ a pure elastic deformation is occurring during the increment and the evaluated stress state is the correct stress state, if ݂ Ͳ the deformation is partly plastic and the elastic trial stress state must be corrected for plastic deformation such as: ߪ ௪ ൌ ߪ ௧ ௪ െο ܥߣ ߲݂߲ ߪ (9) where ο ߣ is the plastic multiplier increment given by: ο ߣ ൌ߲݂߲ ߪ ܥ ο ߝ ߲݂߲ ߪ ܥ ߲݂߲ ߪ ߲ ߪ ௬߲ ߝ (10) 3.3 Homogenized Corrugated Cardboard Elastoplastic Model A corrugated-core sandwich plate consists of a fluted corrugated sheet and two flat linerboards, where the fluting shape is defined with a sine function as: ۖ۔ ۖە ߠ ۓ ሺ ݔ ሻൌି ଵ ቆ݄݀ ሺ ݔ ሻ݀ ݔ ቇ݄ ሺ ݔ ሻൌ൬݄ ʹ െ݁ ଶ ʹ ൰ቀʹ ܲݔ ߨ ቁ (11) where ݄ is the distance between the linerboards, ݁ ଶ is the flute thickness and is the fluting period defined in Fig. (5).
Fig. 5. Representation of the periodic unit cell for corrugated cardboard.
Journal of Applied and Computational Mechanics, Vol. 7, No. 2, (2021), 820-830
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