A rate-dependent continuum model for rapid converting of paperboard. (English) Zbl 1481.74095

Summary: A rate-dependent continuum model for paperboard is developed within a framework for finite strains and finite deformations. A multiplicative split of the deformation gradient into an elastic and an inelastic part is assumed. For the in-plane modes of deformation, viscoelasticity is introduced via a thermodynamically consistent generalization of the Maxwell formulation. The elastic transition between out-of-plane compression and out-of-plane tension is smooth, excluding the need for a switch function which is present in a number of existing paperboard models. The evolution of the inelastic part is modeled using two potential functions separating compression from shear and tension. To calibrate the material model, a set of experiments at different loading rates have been performed on single ply paperboard together with creep and relaxation tests for in-plane uniaxial tension. The model is validated by simulating two loading cases related to package forming, line-folding followed by subsequent force-relaxation and line-creasing during different operating velocities in conjunction with a creep study.


74C20 Large-strain, rate-dependent theories of plasticity


Full Text: DOI


[1] Simon, J., A review of recent trends and challenges in computational modeling of paper and paperboard at different scales, Arch. Comput. Methods Eng. (2020)
[2] Kulachenko, A.; Uesaka, T., Direct simulations of fiber network deformation and failure, Mech. Mater., 51, 1-14 (2012)
[3] Wilbrink, D.; Beex, L.; Peerlings, R., A discrete network model for bond failure and frictional sliding in fibrous materials, Int J Solids Struct, 50, 1354-1363 (2013)
[4] Beex, L.; Peerlings, R.; Department, M. G., A multiscale quasicontinuum method for dissipative lattice models and discrete networks, J Mech Phys Solids, 64, 154-169 (2014)
[5] Sliseris, J.; Andra, H.; Kabel, M.; Dix, B.; Plinke, B., Numerical prediction of the stiffness and strength of medium density fiberboard, Mechanics of Materials journal, 79, 73-84 (2014)
[6] Huang, H.; Nygårds, M., A simplified material model for finite element analysis of paperboard creasing, Nordic Pulp and Paper Research Journal, 25, 505-512 (2010)
[7] Huang, H.; Hagman, A.; Nygårds, M., Quasi static analysis of creasing and folding for three paperboards, Mechanics of Materials journal, 69, 11-34 (2014)
[8] Beex, L.; Peerlings, R., An experimental and computational study of laminated paperboard creasing and folding, Int J Solids Struct, 46, 4192-4207 (2009) · Zbl 1176.74002
[9] Nygårds, M.; Just, M.; Tryding, J., Experimental and numerical studies of creasing of paperboard, Int J Solids Struct, 46, 2493-2505 (2009) · Zbl 1217.74019
[10] Harrysson, M.; Harrysson, A.; Ristinmaa, M., Spatial representation of evolving anisotropy at large strains, Int J Solids Struct, 44, 3514-3532 (2007) · Zbl 1121.74327
[11] Li, Y.; Stapleton, S.; Reese, S.; Simon, J., Anisotropic elastic-plastic deformation of paper: out-of-plane model, Int J Solids Struct, 130-131, 172-182 (2018)
[12] Yujun, Y.; Stapleton, S.; Reese, S.; Simon, J., Anisotropic elastic-plastic deformation of paper: in-plane model, Int J Solids Struct, 100-101, 286-296 (2016)
[13] Borgqvist, E.; Wallin, M.; Ristinmaa, M.; Tryding, J., An anisotropic in-plane and out-of-plane elasto-plastic continuum model for paperboard, Compos Struct, 126, 184-195 (2015)
[14] Nagasawa, S.; Ozawa, S., Effect of bending velocity on time-dependent release behavior of creased white-coated paperboard, The Japan Society of Mechanical Engineers, 3, 1-11 (2016)
[15] Gunderson, D.; Considine, J.; Scott, C., The compressive load-strain curve of paperboard: rate of load and humidity effects, J. Pulp Pap. Sci., 4, 37-41 (1988)
[16] Alfthan, J., Experimental study of non-linear stress relaxation and creep of paper materials and the relation between the two types of experiments, Nordic Pulp and Paper Research Journal, 25, 351-357 (2010)
[17] Lif, J.; Östlund, S.; Fellers, C., In-plane hygro-viscoelasticity of paper at small deformations, Nordic Pulp and Paper Research Journal, 20, 139-149 (2005)
[18] 2420-250 · Zbl 1169.74375
[19] Bosco, E.; Peerlings, R.; Geers, M., Predicting hygro-elastic properties of paper sheets based on an idealized model of the underlying fibrous network, Int J Solids Struct, 56-57, 43-52 (2015)
[20] Tjahjanto, D.; Girlanda, O.; Östlund, S., Anisotropic viscoelastic-viscoplastic continuum model for high-density cellulose-based materials, J Mech Phys Solids, 84, 1-20 (2015)
[21] Xia, Q.; Boyce, M.; Parks, D., A constitutive model for the anisotropic elastic-plastic deformation of paper and paperboard, Int J Solids Struct, 39, 4053-4071 (2002) · Zbl 1049.74518
[22] Borgqvist, E.; Lindström, T.; Wallin, M.; Tryding, J.; Ristinmaa, M., Distortional hardening plasticity model for paperboard., Int J Solids Struct, 51, 2411-2423 (2014)
[23] Borgqvist, E.; Wallin, M.; Tryding, J.; Ristinmaa, M.; Tudisco, E., Localized deformation in compression and folding of paperboard, Packaging Technology and Science, 29, 397-414 (2016)
[24] Robertsson, K.; Borgqvist, E.; Wallin, M.; Ristinmaa, M.; Tryding, J.; Giampieri, A.; Perego, U., Efficient and accurate simulation of the packaging forming process, Packaging Technology and Science, 31, 557-566 (2018)
[25] Schwarze, M.; Reese, S., A reduced integration solid-shell finite element based on the eas and the ans concept-large deformation problems., Int J Numer Methods Eng, 85, 289-329 (2010) · Zbl 1217.74135
[26] Harrysson, A.; Ristinmaa, M., Large strain elasto-plastic model of paper and corrugated board, Int J Solids Struct, 45, 3334-3352 (2008) · Zbl 1169.74353
[27] Persson, K., Material Model for Paper Experimental and Theoretical Aspects, Master’s thesis, Lund Institute of Technology (1991)
[28] Borodulina, S.; Kulachenko, A.; Galland, S.; Nygårds, M., Stress-strain curve of paper revisited., Nordic Pulp and Paper Research Journal, 27, 318-328 (2012)
[29] Kröner, E., Allgemeine kontinuumstheorie der versetzungen und eigenspannungen., Arch Ration Mech Anal, 4, 273-334 (1960) · Zbl 0090.17601
[30] Dafalias, Y., Plastic spin: necessity or redundancy?, Int. J. Plast., 14, 909-931 (1998) · Zbl 0947.74008
[31] J. Mandel, Plasticité classique et viscoplasticité., 1971, (Courses and Lectures, No. 97, ICMS, Udine, Springer, Wien-New York). · Zbl 0285.73018
[32] C. Truesdell, W. Noll, The non-linear field theories of mechanics, volume 3, Handbuch der Physik, Springer-Verlag · Zbl 0779.73004
[33] Coleman, B.; Gurtin, M., Thermodynamics with internal state variables, J. Chem. Phys., 47, 597-613 (1967)
[34] Edelen, D., A nonlinear onsager theory of irreversibility., Int J Eng Sci, 10, 481-490 (1972) · Zbl 0232.73003
[35] Holzapfel, G.; Simo, J., A new viscoelstic constitutive model for continous media at finite thermomechanical changes., Int J Solids Struct, 33, 3019-3034 (1996) · Zbl 0909.73038
[36] Simo, J., On a fully three-dimensional finite-strain viscoelstic damage model: formulation and computational aspects., Comput Methods Appl Mech Eng, 60, 153-173 (1987) · Zbl 0588.73082
[37] Holzapfel, G., On large strain viscoelasticity: continuum formulation and finite element applications to elastomeric structures., Int J Numer Methods Eng, 39, 3903-3926 (1996) · Zbl 0920.73064
[38] Ristinmaa, M.; Ottosen, N., Consequences of dynamic yield surface in viscoplasticity., Int J Solids Struct, 37, 4601-4622 (2000) · Zbl 0982.74014
[39] Dassault Systems. Abaqus 6.13 analysis user’s manual.
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