A comparative study of a multi-surface and a non-quadratic plasticity model with application to the in-plane anisotropic elastoplastic modelling of paper and paperboard

Abstract

In this paper, a three-dimensional anisotropic multi-surface model for the in-plane elastoplastic deformation of paper and paperboard is developed. The model is based on the concept of structural tensors. It employs three different yield functions and parameterised consistency conditions for three different loading scenarios. The evolution of the yield surface with plastic strain is described by multiple hardening variables. Furthermore, the evolution equations are integrated using the unconditionally stable backward Euler method. Also, the internal variables are updated using a particular form of the elastic predictor/plastic corrector algorithm. The constitutive model is calibrated from a set of simple uniaxial tension tests performed in various directions of the material and is validated against loading in other directions. Additionally, the predictive capability as well as the numerical performance of the constitutive model is accessed against an existing non-quadratic plasticity model. Finally, the applicability of the constitutive model is demonstrated by performing a tensile test on a paperboard with a hole.

Keywords

Paper multi-surface plasticity anisotropic plasticity non-quadratic plasticity

Get full access to this article

View all access options for this article.

References

Xia

Boyce

Parks

. A constitutive model for the anisotropic elastic-plastic deformation of paper and paperboard. Int J Solids Struct 2002; 39: 4053–4071.

Mäkelä

Östlund

. Orthotropic elastic-plastic material model for paper materials. Int J Solids Struct 2003; 40: 5599–5620.

Karafillis

Boyce

. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J Mech Phys Solid 1993; 41: 1859–1886.

Borgqvist

Lindstrom

Tryding

et al.

Distortional hardening plasticity model for paperboard. Int J Solids Struct 2014; 51: 2411–2423.

Tjahjanto

Girlanda

Östlund

. Anisotropic viscoelastic-viscoplastic continuum model for high-density cellulose-based materials. J Mech Physics Solid 2015; 84: 1–20.

Stapleton

Reese

et al.

Anisotropic elastic-plastic deformation of paper: in-plane model. Int J Solid Struct 2016; 100–101: 286–296.

Stenberg

. A model for the through-thickness elastic-plastic behaviour of paper. Int J Solids Struct 2003; 40: 7483–7498.

Nygrds

. Modelling the out-of-plane behaviour of paperboard. Nordic Pulp Paper Res J 2009; 24: 72–76.

Stapleton

Reese

et al.

Anisotropic elastic-plastic deformation of paper: out-of-plane model. Int J Solids Struct 2018; 130-131: 172–182.

10.

Isaksson

Gradin

Kulachenko

. The onset and progression of damage in isotropic paper sheets. Int J Solids Struct 2006; 43: 713–726.

11.

Zechner

Janko

Kolednik

. Determining the fracture resistance of thin sheet fiber composites - paper as a model material. Compos Sci Technol 2013; 74: 43–51.

12.

Tryding

Marin

Nygrds

et al.

Experimental and theoretical analysis of in-plane cohesive testing of paperboard. Int J Damage Mech 2017; 26: 895–918.

13.

Wallmeier

Linvill

Hauptmann

et al.

Explicit fem analysis of the deep drawing of paperboard. Mech Mater 2015; 89: 202–215.

14.

Mandel J. Plasticite’ Classique et Viscoplasticité. In: CISM courses and Lecture vol. 97. Berlin: Springer.

15.

Hill

. Generalized constitutive relations for incremental deformation of metal crystals by multislip. J Mech Phys Solids 1966; 14: 95–102.

16.

Rice

. Inelasticconstitutive relations for solids: an internal variable theory and its application to metal plasticity. J Mech Phys Solid 1971; 19: 433–455.

17.

Asaro

. Micromechanics of crystals and polycrystals. Adv Appl Mech 1983; 23: 1–115.

18.

Cuitino

Ortiz

. Computational modelling of single crystals. Model Simul Mater Sci Eng 1992; 1: 225–263.

19.

Anand

Kothari

. A computational procedure for rate-independent crystal plasticity. J Mech Phys Solid 1996; 44: 525–558.

20.

Bedzra

Reese

Simon

J-W

. Meso-macro modelling of anisotropic metallic composites within the framework of multisurface plasticity. Int J Solids Strcut 2017; 120: 186–198.

21.

Spencer

AJM

. Theory of invariants, in continuum physics, New York, NY: Academic Press, 1971.

22.

Boehler

. A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Zeitschrift für Angewandte Mathematik und Mechanik 1979; 59: 157–167.

23.

Zhang

Rychlewski

. Structural tensors for anisotropic solids. Archive Mech 1990; 42: 267–277.

24.

Koiter

General theorems for elastic-plastic solids. In: Sneddon

Hill

(eds). Progress in solid mechanics, Amsterdam: North-Holland Publishing Company, 1960, pp. 167–221.

25.

Simo

Kennedy

Govindjee

. Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. Int J Num Meth Eng 1988; 28: 2161–2185.

26.

Simo

Hughes

TJR

. Computational inelasticity, New York, NY: Springer-Verlag, 1998.

27.

Miehe

Schröder

. A comparative study of stress update algorithms for rate-independent and rate-dependent crystal plasticity. Int J Num Meth Eng 2001; 50: 273–298.

28.

Reese

. Meso-macro modelling of fiber-reinforced rubber-like composites exhibiting large elastoplastic deformation. Int J Solids Struct 2003; 40: 951–980.

29.

Reese

Raible

Wriggers

. Finite element modelling of orthotropic material behaviour in pneumatic membranes. Int J Solids Struct 2001; 38: 9525–9544.

30.

Chaboche

. A review of some plasticity and viscoplasticity constitutive theories. Int J Plast 2008; 24: 1642–1693.

31.

Abaqus finite element software. Providence, RI: Dassault systémes, 2015.

32.

Gasser

Holzapfel

. A rate-independent elastoplastic constitutive model for biological fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation. Computat Mech 2002; 29: 340–360.

33.

Schmidt-Baldassari

. Numerical concepts for rate-independent single crystal plasticity. Comput Meth Appl Mech Eng 2003; 192: 1261–1280.

34.

Bartel

Hackl

. A micromechanical model for martensitic phase-transformations in shape-memory alloys based on energy-relaxation. J Appl Math Mech 2009; 89: 792–809.