Continuum modelling of the mechanical response of paper-based materials

Abstract: Continuum based elasto-plastic-damage models for paperboard have been established in this work. The thesis begins with an introductory section that describes the mechanical properties of paperboard and some of the converting processes during forming of a package. A short review of modeling concepts that have been applied to paperboard is presented and then some key aspects and assumptions developed in this work are summarized. The main part of the thesis consists of the five papers, A, B, C, D and E. In addition to these works, a possible concept to reduce a pathological mesh-dependency is reviewed. The thermodynamical framework is established in Paper A and a model for the in-plane response is developed. The anisotropy is handled by introducing a set of director vectors that change direction along with the continuum. A distortion hardening yield surface coupled to several scalar internal variables is introduced. The effects of pre-straining a sample in one direction and then subsequently load the sample in the perpendicular direction is studied. The model is compared to measurements obtained with Digital Image Correlation. In Paper B, the model is further developed to model out-of-plane deformations. A normal vector is introduced to model the out-of-plane direction. Key ingredients in the model includes the specific format for the elastic part of the free energy and an expression for the plastic spin. The spin is used to control the direction of the plastic flow. Simulations are performed on the line crease setup and compared to experimental measurements. Furthermore, the industrial rotation crease setup is studied in detail using the developed model. The Short-span Compression Test (SCT) and the line folding operation are investigated in Paper C and the deformation patterns extracted from x-ray images are studied. The model parameters are calibrated to uniaxial tests and the SCT, and then the folding of uncreased paperboard is simulated. The simulated global force-displacement/rotation curves matches the measurements and the simulated deformation patterns are similar to that observed experimentally. A numerical scheme is presented in Paper D, where the governing equations of the elasto-plastic boundary value problem are interpreted as a Differential-Algebraic Equation (DAE) system. In particular, two material models, which includes damage variables, are investigated using the Diagonally Implicit Runge-Kutta (DIRK) scheme. The error obtained using the DIRK-method is compared to the standard implicit Euler method. In Paper E, the continuum model that has been developed in paper A-C is further enhanced to include the effect of damage. Two damage variables are introduced in the elastic part of the free energy which is associated with out-of-plane deformations. The softening in the out-of-plane normal and shear deformations can then be recovered. The folding of creased paperboard is simulated and compared to measurements.