Continuum Mechanics Modelling of Corrugated Board

Abstract: The storage of compressive loaded boxes in an environment with naturally varying humidity is a practical issue in corrugated board employment. Time dependent variables such as moisture content, strain fields, stress fields and material strength play important roles for the time to failure. Supplementary, the stochastic nature of material and moisture load is obstructing the prediction of a reliable measure of lifetime. This work is composed of a number of portion proposals, each aiming on a method for a specific subproblem of the numerical modelling of time to failure. Firstly, the focus is the problem of finite deformation hygro-elasticity. The assumption of kinematics is based on an additive split of the stretch in an elastic part and a non-elastic part. In time stepping sequences the elastic stretch is updated by the use of the total stretch from the polar decomposed deformation gradient. As a consequence, in the linearized virtual work equation appears a hygroscopic contribution to the stiffness matrix as well as a hygroscopic load vector. Particularly, a numerical procedure for analyzing layered shells is developed. Further, a numerical method for the transient moisture flow in porous cellulosic materials like paper and wood is examined. The derivation of the model is based on mass conservation for a mixture containing a vapour phase and an adsorbed water phase embedded in a porous solid material. A model for the development of higher order sorption hysteresis is also developed. The model is capable of describing cyclic hardening as well as cyclic softening of the equilibrium water concentration. The model is verified by comparison with the measured response to natural variations in temperature and humidity. A close agreement of the simulated results to measured data is found. The reliability of geometrically non-linear composite shells is studied by use of the First Order Reliability Method (FORM). A finite difference method is employed in order to find the gradients of the limit state function. A failure stress criterion for corrugated board facings is also proposed. The failure criterion is based on material failure and structural local buckling failure. The structural failure stress is evaluated using a novel analytical solution for the buckling of long orthotropic plates under combined in-plane loading. The failure stress is compared with collapse experiments on corrugated board cylinders and the failure stress presented herein is seen to be in significantly better agreement with the measured stresses than the Tsai-Wu failure criterion alone. Alongside with the numerical predictive methods, a number of testing procedures on individual paper materials and corrugated board boxes are performed. Firstly, mechanical second order stochastic field parameters of liner and fluting materials are estimated for a variety of materials used for commercial boards. Secondly, reliability testing of corrugated board boxes in a natural dynamic humidity environment is performed. A large number of boxes are loaded with a constant compressive force in an untempered airy indoor climate. Contemporary with the record of time to failures, the moisture transport in individual paper sheets and a sealed corrugated board box is measured.