A discrete meso-scale model for the analysis of
hygro-expansivity of paper
Citation for published version (APA):
Bosco, E., Peerlings, R. H. J., & Geers, M. G. D. (2013). A discrete meso-scale model for the analysis of hygro-expansivity of paper. Poster session presented at Mate Poster Award 2013 : 18th Annual Poster Contest.
Document status and date: Published: 01/01/2013 Document Version:
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Materials
innovation
institute
M
A discrete meso-scale model for the
analysis of hygro-expansivity of paper
E. Bosco, R.H.J. Peerlings and M.G.D. Geers
/ department of Mechanical Engineering / Mechanics of Materials
Introduction
Paper is a fibrous material which is extremely sensitive to changes of moisture content. Moisture induced dimensional variations at the fiber level result in deformations in the paper sheet, that strongly influence the industrial performance and quality of the material.
MESO
MICRO
MACRO
Figure 1: Moisture induced deformations of printed paper sheets find their origin in the multi-scale structure of the material.
Objective
The goal of the project is to understand how the effective re-sponse of paper sheets depends on the hygro-mechanical prop-erties of fibers and to develop a consistent discrete meso-scale model bridging the micro- to the macro-scale.
Discrete meso-scale model
The meso-structural network is schematized through a lattice model based on a two-dimensional squared unit-cell with diago-nal elements. Free fiber segments are described as trusses, while inter-fiber bonds are treated as laminated composite plates.
0 0 0.5 1 1.5 e f( e ) //2 ï//2 ï//4 //4 h3 h2 h1 h4 q = 0.5
FIBER ORIENTATION DISTRIBUTION
BOND CROSS ! SECTION a - aʼ X- BRACED UNIT-CELL x ! (Ma c h in e D ir e c ti o n ) y ! (Cross Direction)! BOND FREE FIBER! SEGMENT a aʼ
Figure 2: Schematic picture of the proposed unit-cell model.
The proposed model allows to naturally account for: i) the ef-fect of the fiber orientation distribution; ii) the anisotropic hygro mechanical behavior of the fibers; iii) the internal stresses aris-ing in the bonds due hygro-mechanical loads. Effective material
properties are extracted from the unit-cell through a proper ho-mogenization procedure.
Results
The effective elastic constants of paper are calculated as a func-tion of the degree of fiber alignment along the machine direcfunc-tion. The obtained results show a good match with typical values for in-plane elastic properties of the material, which generally range from 1 to 20 GPa [1]. 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 q [ï] Exx , E yy [GPa] Eyy Exx Exx Eyy
Fig. 3 Effective elastic constants of paper as a function of fiber align-ment along machine direction.
As fibers are oriented more along machine direction, the effec-tive hygro-expansive coefficients increase in cross direction, while they decrease in machine direction. The prediction obtained agrees with the experimental data shown in [2].
1 2 3 4 5 6 7 8 9 0 0.03 0.06 0.09 0.12 0.15
Effective stiffness ratio `xx , `yy `xx `xx `yy `yy
Fig. 4 Effective hygro-expansive coefficients of paper as a function of fiber alignment along machine direction.
Conclusions
The proposed model offers a powerful way to predict the effective hygro-mechanical properties of paper by incorporating geomet-rical and physical features of the fibers network.
References:
[1] K. Niskanen. Paper physics. Fapet Oy Helsinki, Finland, 1998. [2] T. Uesaka. General formula for hygroexpansion of paper. Journal of
