a b s t r a c tAn in-plane elasto-plastic material model and a hygroexpansivity-shrinkage model for paper and board are introduced in this paper. The input parameters for both models are fiber orientation anisotropy and dry solids content. These two models, based on experimental results, could be used in an analytical approach to estimate, for example, plastic strain and shrinkage in simple one-dimensional cases, but for studies of the combined and more complicated effects of hygro-elasto-plastic behavior, a numerical finite element model was constructed. The finite element approach also offered possibilities for studying different structural variations of an orthotropic sheet as well as buckling behavior and internal stress situations under local strain differences. A few examples are presented of the effect of the anisotropy and moisture streaks under stretching and drying conditions on strain differences and buckling. The internal stresses were studied through a case in which the drying of different layers occurred at different stages. Both the anisotropy and moisture streaks were capable of rendering the buckling of the sample visible. The permanency of these defects highly depends on several process stages and tension conditions of the sheet, as demonstrated in this paper. The application possibilities of the hygro-elasto-plastic model are diverse, including investigation into several phenomena and defects appearing in drying, converting and printing process conditions.
Moisture changes in a paper sheet cause curling and cockling due to expanding or shrinking fibres. Mathematical modelling of these phenomena is considered in this paper. The paper sheet is assumed to be a heterogeneous orthotropic elastic material. Heterogeneity of the paper is handled by dividing it into small pieces, assuming one piece to be homogeneous. This leads to a mathematical model consisting of thousands of homogeneous orthotropic pieces which have different elastic properties and moisture expansion coefficients. The mathematical model is solved using the finite element method. We will show numerical examples and conclude in this paper that the elastic model is able to predict out-of-plane deformations such as cockling of the paper sheet.
Dimensional instability, more particularly its component hygroexpansivity, may cause problems in process or end-use situations in which paper or board is in contact with water or subject to changes in ambient relative humidity. Misregistration in printing, curl during copying and calender wrinkles are examples of such defects. In this paper, the in-plane hygroexpansivity of oriented laboratory sheets with different pulps and dried both freely and under restraint is studied. A linear relationship between the drying shrinkage and hygroexpansion coefficient of freely dried laboratory sheets having different fiber orientation anisotropies, was observed. Regardless of both the measurement direction (MD or CD) and the drying options (freely or restraint) all hygroexpansion coefficient values of each pulp type fell quite well on one single power curve as a function of the elastic modulus. Fiber orientation is considered via two different approaches: using fiber orientation anisotropy and using directional variable named as anisotropy index. When the anisotropy index is used, the MD and CD hygroexpansivity or the MD and CD drying shrinkage can be fit on a single curve, while the freely dried and restraint-dried sheets evidently need two different fitting curves. Between the hygroexpansion coefficient and the anisotropy index, a simple power law relationship, with two fitting parameters depending on pulp and drying restraints, is introduced.
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