# Modelling Damage Evolution and Fracture of Paper Materials

Sammanfattning: This thesis contains six papers dealing with different aspects of damage and fracture in paper. The work addresses the problem of material length scales in the context of deformation and fracture of paper materials. In Paper 1, localised failure in low-basis weight paper is studied and a fracture model based on continuum damage mechanics is presented. A gradient enhanced theory is used that incorporate a characteristic length that prevents localisation of strain into an unrealistically small volume. Damage parameters are calibrated using data from acoustic emission (AE) measurements. It is concluded that the model can be used to evaluate the influence of defect size on fracture load. From AE measurements it is concluded that an exponential damage evolution law describes the progress of damage in low-basis weight paper. In Paper 2, an optical non-contact displacement measuring system has been used in mode I fracture testing of low-density paper to determine the strain field in the crack-tip region. Immediately before final fracture, the measured normal strain perpendicular to the crack plane in the near-tip region is approximately sixty percent higher than the computed strain using elastic–plastic theory at corresponding load levels while the strain computed using a non-local damage theory is of the same order of magnitude as the experimental. Hence, it seems physically motivated to include a non-local damage theory in order to obtain agreement in strains in the fracture process zone. In Paper 3, a model describing the fracture behaviour of embossed low-basis-weight paper is presented. It is found that the model captures the development of damage along rows of embossing imprints parallel to the main crack which has been observed in experiments. The model suggests that an embossing pattern could have a toughening effect on the sheet for certain pattern dimensions and embossing pressures. In Paper 4, the deformations near a semi-infinite crack in a linear elastic random fibre network (RFN) under mode I loading is studied using a numerical network model. A square root singular deformation field (K-field) is applied on the periphery of the model domain. An important conclusion of the investigation is that the square root field breaks down in the vicinity of the tip of a main crack due to structural effects caused by the network structure. This type of distortions can not be captured by conventional local continuum mechanics. It is shown that a more realistic strain energy field may be accomplished through the use of non-local field theory. A simple relation between non-local characteristic length and structural parameters of the network is presented. In Paper 5, a closed form relation for the strain energy density in the vicinity of a macroscopic mode I crack in a random fibre network is derived using non-local continuum field theory. The model explains why open network structures seldom localise failure to small macroscopic cracks. It is found that there is a one-to-one relation between the characteristic length controlling non-local actions and the size of the smallest crack that can initiate macroscopic failure. Fibre breakage is a damage process which is active if a paper material is dense or the bonds between fibres are strong. Due to the poor statistics of single fibre measurements, the so-called zero-span strength of paper is often taken as a measure of fibre strength. In Paper 6, some analytical and numerical results concerning the zero-span testing method is presented. Of particular interest is the relationship between an apparent modulus obtained from the zero-span testing method and the elastic properties of the fibres. The apparent elasticity modulus is estimated using two energy theorems in elasto-statics in which the role of span length is explored. Analytical results, derived under the assumption that slippage between specimen and clamps does not occur, clearly show that the apparent modulus strongly depends on the span length. This is verified by the numerical results obtained using the finite element method. Tensile tests at nominal zero span were conducted and it was found that there is qualitative agreement between the experiments and the result of the analysis.

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