Constitutive Modeling of the Unloading Behavior of Paper Material Using the Asymtotic Fiber and Bond Model

Abstract

Current constitutive models for paper and paperboard material are focused on the prediction of the sheet stress-strain behavior during loading. The unloading process has not been widely addressed. This work will focus on modeling the unloading behavior of paper material in one- and two-dimensional problems. For the one-dimensional problem, i.e. uniaxial tensile test, the asymptotic fiber and bond model by Sinha (1994) is extended to determine the plastic (permanent strain) of a sample upon unloading. In Sinha's micromechanics model, a representative load bearing, i.e. fiber at a fiber-to-fiber bond site, was used to derive the fiber stress and subsequently the sheet stress. In the asymptotic fiber and bond model, the fiber and bond condition (of elasticity or plasticity) was assumed to be the same throughout the fiber. In this work, the fiber and bond begins in elastic state. As the applied load or strains increase, the bond yields, while fiber remains elastic throughout the loading process. When unloading, both fiber and bond behave elastically. The model parameters for the asymptotic fiber and bond are obtained by fitting the model to experimental data from uniaxial stress-strain curves. The unloading model is then used to predict the plastic strain after unloading of uniaxially strained samples. The model prediction corresponds well with the experimental data. For a two-dimensional problem, a sample was deformed in a Mullen burst tester and then unloaded. A Mullen tester is used to conduct burst test by applying a uniform pressure to one surface of the sample that is clamped down on the pressure chamber. In a burst test, the sample deforms into hemispherical shape and eventually fails with a 'H' pattern in the center. For testing the unloading model, the sample is deformed to a given pressure and unloaded. The central displacement of the sample throughout the loading and unloading process is recorded together with the applied pressure, for comparison with model predictions. Since the asymptotic fiber and bond model has limited application in one-dimensional problems, a combination of continuum and micromechanics methods was used by Sinha and Perkins (1995) to reap the benefits the two types of approach have to offer. The hybrid model was applied in finite element analysis using the finite element analysis code ABAQUS and its user subroutine UMAT. A similar approach was utilized in this work to model the unloading process in two-dimensional problems.