Nonlinear anisotropic hypo-elastic continuum models for paper materials and their relation to statistical micro-mechanics

Two nonlinear hypo-elastic continuum models are proposed to simulate at the macro-scale the mechanical response of paper sheets under biaxial plane stress conditions. Based on the evidence that paper is made of cellulose random fibers cross-linked and pressed together, and that uniaxial tensile tests reveal that failure of the fibers is triggered by the direction of the maximum principal strain, an anisotropic damage variable function of the maximum principal strain, , and its direction with respect to the mean fiber orientation, , is derived. The first continuum model considers an orthotropic constitutive tensor, while the second model introduces a simplified isotropic constitutive tensor with an effective elastic modulus function of . It is also shown how to call a statistical micromechanics model at the material point level by passing and in input, to predict additional features related to the evolution of failure of cellulose fibers. The proposed approach is validated in relation to paper specimens with eccentric holes tested in tension, with parameters identified from uniaxial tensile tests on plain specimens.