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3D network simulations of paper structure S. Lavrykov, S. B. Lindström, K. M. Singh, B. V. Ramarao KEYWORDS: Fiber Network, Random, Hand sheet, Bending Stiffness, Elasticity, Compression SUMMARY: The structure of paper influences its properties and simulations of it are necessary to understand the impact of fiber and papermaking conditions on the sheet properties. We show a method to develop a representative structure of paper by merging different simulation techniques for the forming section and the pressing operation. The simulation follows the bending and drape of fibers over one another in the final structure and allows estimation of sheet properties without recourse to arbitrary bending rules or experimental measurements of density and/or RBA. Fibers are first modeled as jointed beams following the fluid mechanics in the forming section. The sheet structure obtained from this is representative of the wet sheet from the couch. The pressing simulation discretizes fibers into a number of solid elements around the lumen. Bonding between fibers is simulated using spring elements. The resulting fiber network was analyzed to determine its elastic modulus and deformation under small strains. The influence of fiber dimensions, namely fiber lengths, widths and thicknesses as well as bond stiffnesses on the elasticity of the network are studied. A brief account of inclusion of fines, represented by individual cubical elements is also shown. ADDRESSES OF THE AUTHORS: S. Lavrykov (
[email protected]) and B. V. Ramarao (
[email protected]): Empire State Paper Research Institute, State University of New York ESF, 1 Forestry Dr., Syracuse NY, USA 13210. S. B. Lindström (
[email protected]): Mechanics, IEI, Institute of Technology, Linköping University, 583 81 Linköping, Sweden. K. M. Singh (
[email protected]): International Paper Co., Corporate Technology Center, 6283 Tri-Ridge Blvd., Loveland, OH 45140, USA. Corresponding author: B. V. Ramarao In order to investigate the impact of varying fiber dimensions or other similar properties on paper properties, a number of computer based simulations of paper forming have been developed in the past. The earliest computer simulations were of two dimensional paper structures based on dropping straight or curvilinear random lines on a flat plane and investigating the statistics of the resulting structure (Kallmes, Corte 1960; Corte, Kallmes 1962). The next steps of 2D fiber networks were calculations of network mechanical properties. Reviews of these are found in Deng and Dodson, 1994 and by Bronkhorst, 2003. These 2D networks of the paper structure are usually obtained by placing the centers of fibers either uniformly randomly or with different flocculation rules on a rectangular or square area. Fiber orientation is adjusted by sampling from known distributions. For mechanical applications, 256 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012
fibers were considered as beams - usually as simple Bernoulli beams (Heyden 2000; Heyden, Gustafsson 2002) long free span, and using the Timoshenko model for short beams (Räisänen et al., 1996; Bronkhorst, 2003). The fiber material is usually considered as elastic (Van den Akker, 1962; Deng, Dodson, 1994; Kahkonen, 2003; Heyden 2000; Heyden, Gustafsson 2002). Some cases of elastic-plastic simulations are described in (Ramasubramanian, Perkins, 1987; Räisänen et al., 1996; Bronkhorst, 2003). The intersections of fibers are considered as connecting bonds which are either rigid or flexible, having some finite transitional and torsional stiffness. All 2D fibers models are easy to build and analyze, but the following issues render their applicability of questionable value for paper materials. It is impossible to calculate the thickness (and thus the density) of the simulated fiber web. Since paper sheet properties are strongly dependent on density, this is perhaps the most serious limitation of two dimensional models to simulate actual paper properties. Also, since most simulations assume rigid inter-fiber bonds, the predictions always seriously overestimate the sheet’s elastic moduli (Van den Akker, 1963). The number of fiber bonds is also grossly overestimated, and therefore even in the case of flexible bonds, the elastic modulus predictions are much higher than real values. Finally, beam theories are not accurate for beams with the extremely short free spans such as those occurring in paper. Another option to generate real fiber structures is 3D networking. Methods for creating three-dimensional fiber network structures have been proposed by a number of authors as discussed in detail by Alava and Niskanen, 2006. Some of these models can be used for generation of networks similar to hand sheets (Nilsen et al., 1998; Provatas et al., 2000; Heyden 2000; Heyden, Gustafsson 2002; Vincent et al., 2009; Vincent et al., 2010), where fiber deposition occurs without considering hydrodynamic effects. Straight or curvilinear fibers are randomly generated inside an arbitrarily chosen volume, (Heyden 2000; Heyden, Gustafsson 2002), or generated over rectangular base and then deposited independently of each other. During deposition the shape of the fibers can change. In the KCL-PAKKA model (Niskanen et al., 1997; Nilsen et al., 1998) fibers do not bend but a relative shift of adjacent fiber segments can be provided depending on the stiffness of the wet fiber material and applied external pressure. In another case (Vincent et al., 2009; Vincent et al., 2010), fiber bending was modelled according to some artificial bending criteria (two prescribed bending angles were introduced, and these angles were supposed to be dependent on the wood species). An important consideration for paper networks is that the free fiber length segments are usually small such that bending deformations tend to be limited. Generations of artificial machine-made paper sheets, where fiber deposition was modeled with the influence of water movement, were considered in work of Miettinen
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(a)
(b)
(c)
(d)
Fig 1. Configurations of fibers used in the model
Hardwood Pulp
Frequency
et al. 2007; Switzer et al. 2004; Lindström, Uesaka 2008. Pulp fibers were modeled as multilink chains of rigid cylindrical segments with circular cross-section, immersed into Newtonian liquid. Fiber chains can bend and twist due to joints between segments. Different cases of liquid motion were considered, including simple drainage under gravity and external pressure, (see Miettinen et al. 2007; Switzer et al. 2004), and complex water movement in different sections of roll-blade formers, (Lindström, Uesaka 2008; Lindstrom et al., 2009). These models suffer some disadvantages such as the limitations of rigid segments which do not permit the same fiber bending, the output networks have low density and higher thickness. However, these methods can be used for generating pulp webs of low consistency, and thus can simulate forming situations corresponding to the wet web before the wet pressing section in the paper machine. The main goal of the present work is to develop a numerical approach to build a 3D artificial hand sheet and machine sheet fiber networks with prescribed thickness and apparent density, consisting of collapsed and non-collapsed fibers, fines and fillers. We can take a simulated structure made by using multilink chains of rigid cylindrical segments as fibers and abstract the 3D structure of the formed paper sheet. The abstraction is by using the center lines of the fibers and redecorating their thicknesses and widths to develop a fiber model with prismatic elements comprising the fibers. The fibers are also considered to be hollow to allow for the lumens and can be compressed in a wet pressing simulation to generate a ‘final’ structure. In the following, we will describe our method of generating the paper structure and show how it is applied to a random ‘handsheet’ structure and a realistic, simulated sheet formed using a twin wire roll former with blades. The structures are then analyzed for their mechanical properties (in this case, the elastic modulus).
0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 0,5
0,9
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Fiber network Generation Hand Sheet Formation A set of objects is defined for the simulation, which includes objects of three types: fibers, fines or fillers. These are characterized by their geometrical parameters which include length, width and thickness, their distributions and type of cross-section i.e. collapsed or non-collapsed, as in Fig 1. Fiber type 1a has only one finite element in thickness direction. This type was usually used to describe fines and fillers. Fiber types 1b and 1c have arbitrary number of elements in all three directions (length, width, thickness). Non-collapsed fiber type 1d has always one element in the thickness of fiber wall and arbitrary number of elements in other two directions. It is possible to include curl and kink of fibers by suitable modifications of these elementary structures, although this was not implemented in the current simulation. The first step of the simulation is to generate an initial structure. Handsheet formation can be simulated by simply placing fibers (or objects) randomly in space
Fig 2. Examples of fiber length distributions for softwood and hardwood pulps according to their distribution in the furnish. An example of such distributions for hard wood and soft wood pulps are shown in Fig 2. The second step is location of centers of objects in Z direction. Each new object has Z coordinate taking into account all objects already generated and located below this particular one. All points of such line have the same Z-coordinate. The third step is the generation of finite element grid in each object according to type of object cross-section and its dimensions. The topology of finite elements (shape and local numbering of nodes) should correspond to finite element software used in the following steps. In our case this was the 8-node hexahedron solid element in the LSDYNA Explicit program. Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 257
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Fig 4. ZD view of simulated sample structure after forming in paper machine z zmax
zcaliper
Fig 3. An example of initial configuration of fiber network (all four fiber configurations and filler particles are presented) The fourth step is trimming out the part of finite element grid outside prescribed sample dimensions. A sample of the finite element network generated after the fourth step is shown in Fig 3. And the fifth, the last step, is the calculation of fiber deposition velocities. All points of each particular object have the same initial velocity in vertical direction. The velocity is calculated from the deposition height and deposition time.
Machine Sheet Formation This method of fiber network generation is described in detail by Lindström, Uesaka, 2008. Fibers are represented as sets of rigid segments connected by joints with bend and torsion stiffness. Mass and moment of inertia of segments correspond to solid circular cylinders. Initially all fibers are generated randomly inside a prescribed volume, which is filled by a viscous incompressible fluid. The motion of the fluid is governed by the threedimensional Navier–Stokes equations. Equations of motion of fiber segments includes inertia forces, hydrodynamic drag forces from moving liquid and fiberto-fiber contact forces. The slurry is running through different sections of paper machine, where different boundary conditions are applied (Lindström, Uesaka, 2008; Lindstrom et al., 2009). The final low consistency pulp mat is used as initial input data for the network compression model. Each circular fiber is replaced by the collapsed or non-collapsed fiber, Fig 1, and is divided into a set of finite elements. A mechanical material model (elastic, elastic-plastic, visco-elastic-plastic) is assigned for each particular fiber. The wet fiber flexibility is a fiber parameter that has been investigated extensively in the past and can be used to determine a representative fiber modulus. Since wet fibers are substantially plastic, it is necessary to define plasticity parameters in addition to the elasticity. Fibers are anisotropic and their deformation is considerably complex. However, for an initial simulation such as this, we used a simple isotropic bilinear elastic-plastic model with the modulus determined from the wet fiber flexibility data in the literature [Paavilainen, Luner, 1986; Abitz et al., 1985]. An example of machine-generated network is shown in Fig 4. 258 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012
t tload
trest
tunload
Fig 5. Upper rigid plate movement history during mat compression
Fiber network Compression The simulation of fiber mat compression was performed with the explicit finite element method, [see Zhong 1993 and Hallquist 2005]. We used LS-DYNA, a program for Nonlinear Dynamic Analysis of Structures in Three Dimensions, Version 971 (Hallquist 2005). The 8 node hexahedron elements with three displacements in each node were used to represent fiber network. The central finite difference scheme was used for the numerical integration of the equations of motion. Fiber-to-fiber interactions during simulation were calculated from balance of nodal inertia forces, external forces, internal forces and contact forces. All contact couples were found automatically during simulation. (This method of contact handling was made possible by using a keyword *CONTACT_AUTOMATIC_GENERAL). To obtain the fiber mat with prescribed thickness and density, the network was compressed between two rigid plates. The non-moving plate is located below the previously generated fiber network. The moving plate with prescribed motion in vertical direction (Fig 5) is initially located on the top of the mat. Fiber settling and compression are performed during a loading time , which was equal to 1-2 ms. A stabilization time is need to decrease dynamic effects in the compressed fiber mat which was generally set to 1 ms. An unloading time is also needed to remove the upper plate and to give the fiber mat possibility to expand in thickness direction due to internal stresses in fibers. The unloading time was 0.1-1.0 ms usually. The results of some 3D network simulations are presented in Fig 6-9. The initial fibers configuration from Fig 3 after compression is shown in Fig 6. A larger scale model with about 1500 fiber of the mixture of hardwood and softwood pulps is presented in Fig 7. Results of simulation of fiber network with high content of mineral filler (25% weight fraction) are shown in Fig 8 and 9. A layer created by fillers is observed in the pictures. The example of machine-made fiber network after compression is shown in Fig 10. The colors serve to differentiate the fibers in the network.
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Fig 10. Simulated machine-made sample after compression Fig 6. Fiber network from Fig. 3 after compression ΔX
ΔX
L
Fig 11. Tensile test for elastic modulus calculations
Fiber Network Applications Calculation of Apparent Elastic Modulus
Fig 7. The sample with 50% mixture of softwood and hardwood pulps
Fig 8. Fiber network with high filler content – an example of coating simulation
Fig 9. Filler particles distribution in the fiber mat during coating formation
The apparent elastic modulus is calculated the same way as usually calculated from the physical tensile test. The sample length should be at least 1.2 times bigger than the longest fiber in the network, as recommended in Heyden 2000. The example of such sample is presented in Fig 11 (sample length is 5 mm, width is 1 mm). Material of fibers is isotropic elastic. Mechanical properties of single fibers of different dry wooden species can be found in (Bronkhorst 2003; Page et al. 1977; Katz et al 2008). Before the solution of elastic boundary-value problem, the network should be analyzed and bonds between fibers should be set up. A contact couple can be defined only between two elements which belong to two different fibers. From one to four links (bonds) may be set up in one contact couple. The stiffness of each bond is calculated from the value of contact area (overlapped area of two contact surfaces) and bonding stiffness, experimentally found by Thorpe et al. 1976. The average maximum shear stress reported by Thorpe et al. 1976 and Perkins 2001, for the holocellulose bond was corresponding to an average maximum strain . The bond stiffness parameter is obtained as and is equal to the spring force between each pair of bonding nodes. Fibers not bonded to other fibers of the network are eliminated from consideration. To solve the elastic problem, the kinematic boundary conditions should be applied to the finite element grid. Prescribed displacements are assigned to the nodes located on clamped area on the top and the bottom surfaces from two opposite sides of the sample, Fig 11. After solution of elastic problem, nodal displacements are used for calculation of strain and stress distributions in finite elements. Integration of stresses over element’s area in some sample cross-sections gives average force and average Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 259
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6,00
7,0
Elastic Modulus, GPa
Elastic Modulus, GPa
6,0
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0,5 0,09
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Fig 12. Elastic modulus of networks of softwood and hardwood pulps
Elastic Modulus, GPa
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7,5
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Fig 13. Elastic modulus of simulated network as a function of fiber thickness
Elastic Modulus, GPa
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3,0
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1,0
0,0 10,0
15,0
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Fig 14. Elastic modulus of simulated network as a function of fiber width
6,0
Elastic Modulus, GPa
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Fig 15. Influence of bond stiffness on elastic modulus of fiber networks 260 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012
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Fig 16. Influence of sample length on elastic modulus of 60 gsm fiber network (fiber length 2 mm, sheet caliper 0.1 mm) stress in the sample. The strain value is equal to ratio of prescribed displacement to the length of free span (distance between clamps), so the apparent elastic modulus is found. Fig 12 shows the elastic modulus for sample networks of softwood and hardwood pulps with length distributions shown in Fig 2. The modulus is shown as a function of sheet caliper corresponding to different wet pressing levels. We observe that the modulus increases as the caliper is decreased, i.e. the sheet is densified by wet pressing to different levels. It can also be seen that the softwood fiber network shows higher modulus as compared to the hardwood fibers. This result clarifies the impact of fiber length on sheet modulus. In order to evaluate the influence of some fiber parameters including fiber width, fiber thickness and bonding stiffness, on apparent tensile elastic modulus of fiber mat we conducted more simulations for fiber mats with the same basis weight 60 gsm and the same material properties. The results are shown in Figs 12-15 and in Table 1. Fig 13 and Fig 14 show that the elastic modulus decreases with increased thickness (evaluated at constant caliper). Increasing thickness results in lesser fibers in the sheets, resulting in the lower values observed. Fig 15 shows the influence of the bond stiffness; increased stiffness shows increases in the elastic modulus. However, the increase quickly saturates and when the bond stiffness increases beyond 0.3, gains in the modulus are small. The size of the network for simulation of the elastic properties should be sufficiently large such that the effects of singular fibers spanning a large portion of the network are minimized. In order to investigate this effect, we carried out simulations of the elastic modulus with increasing network length (denoted as sample size) keeping fiber length a constant. Fig 16 shows the results for fibers of length 2mm. The simulation result for the elastic modulus is independent of the network size for networks greater than 1.5 fiber lengths. Fig 17 shows a graph of the wet pressing pressure as a function of mat density for an example softwood and hardwood fibers. These are representative calculations indicating that for a given wet pressing pressure, the softwood sheets are more densified than the hardwoods. The calculations serve as illustrative cases since the parameters chosen for the fibers were arbitrary.
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Table 1. Properties of fibers and simulated 60 gsm sheets, 1x2 mm sample (20F represents fines, fiber objects consisting of single elements). Parameter Fiber Length (Avg.), mm Fiber Width (Avg.), μm Elastic Modulus Fiber, GPa Caliper, μm Number of fibers Elastic Modulus of sheet, GPa Stiffness (mN·m), simulated Stiffness (mN·m), experiment (Lavrykov et al., 2010)
Softwood pulp 0HW+100SW 3.0 24 10 100 612 6.433 0.536 0.763
Mix 2 Mix 1 50HW+50SW 64HW+16SW+20F 2.0 1.2 24 24 10 10 100 100 783 1162 5.920 4.309 0.493 0.359
Wet Pressing Pressure, MPa
60,0
Softwood Hardwood
50,0 40,0 30,0 20,0 10,0 0,0 0,4
0,5
0,6
0,7
0,8
Density, g/cm3
Fig 17. Pressure as a function of mat density Table 1 below shows the results of simulations of 60 gsm sheets composed of a mix of hardwood and softwood pulps in different ratios along with a portion of the fines. Note that the mixes are shown in number fractions (not mass) as is usual for experimental data. The simulation results show that the sheet modulus is a strong function of the softwood content, increasing in magnitude as this increases. An important parameter, the sheet stiffness was also evaluated and shown to be a strong function of the softwood content in the pulp. The final rows in this table present stiffness estimates using the simulations as compared to data obtained from experiments. It appears that when the elastic modulus of the fibers is assumed to be the same, the longer fibers yield a higher stiffness value, a trend that corresponded to the experimental results. The magnitudes of the stiffnesses observed experimentally are much higher than the simulation predictions though. The modulus of the fibers and other parameters were obtained from published values in the literature rather than measurements or estimates on the furnish itself. This would account for the difference in magnitude of the estimates.
Simulation of Wet Pressing of Sheets The most significant use of simulation is to predict the structure of the paper sheets with particular reference to their z-dimensions i.e. the caliper or analogously, their density. This is not possible with 2D simulations and also simplistic constructions allowing the fibers to bend according to external rules. The best method of simulation is to determine the fiber reaction force during compression, the so-called compressive stress and also
Mix 3 80HW+20SW 1.4 24 10 100 952 4.278 0.356 0.703
Hardwood pulp 100HW+0SW 1.0 24 10 100 1111 2.959 0.246 0.693
simultaneously track the drag force exerted by the moving fluid. This is possible with a particle level simulation such as the present one. The fiber displacements were simulated using LS-DYNA. At each step of pressing, the fiber reaction force was summed and used as the total compressive stress borne by the fibrous structure. The combination of the drag, translated into permeability and the compressive stress was applied to a homogenized wet pressing model (Lavrykov et al., 2009) to determine the sheet caliper at different values of applied pressure. The hydraulic stress was estimated using an effective medium type approximation with periodic cells, adjusted for local change in porosity. These were combined to generate the structure in a timeexplicit method. Fillers (or equivalent fines) were considered as single discrete elements of approximately 2 to 4µm in size. This allowed the calculation of the sheet caliper and density at given levels of applied pressure and the as a result, the sheet structure.
Discussion As was mentioned before, the main objective of this work was to provide a new method for the generation of fiber networks which can be applicable to numerical solution of mechanical problems. This means fibers in an artificial mat should consist of set of finite elements connected in nodes. During mat formation fibers should not be bent only but compressed too taking into account possible fiber collapsing. In this situation all previously reported methods (Nilsen et al., 1998; Provatas et al., 2000; Heyden 2000; Heyden, Gustafsson 2002; Vincent et al., 2009; Vincent et al., 2010) were found not correspondent to our goals. To provide a simulation of fibers settling and compression the explicit analysis software was used. There are several important reasons for using explicit analysis instead of the implicit one for solution of this type of problems. First, the explicit methods perform this analysis much faster. For example, for the sample in Fig 7, which contains about 2000 fibers with 120,000 elements and 500,000 nodes, the solution of the static elastic problem using the implicit method on a PC computer with two Xeon Quad Core 3 GHz processors and 24 GB RAM under 64-bit Windows 7 operational system requires 4 Hrs. During solution of nonlinear elastic-plastic contact problem with large strains and Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 261
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displacements similar time is required for each iteration at each loading step. At the same time, the total computational time for explicit analysis of this particular model was only approximately 6 hrs. Of course, the total computational time for explicit analysis is highly dependent of integration time step which, in turn, is dependent of element size and mechanical properties of fiber material - wet fiber density, wet fiber Young’s elastic modulus and Poisson’s ratio, (Hallquist 2005). Using of mass scaling procedure in LS-DYNA (Hallquist 2005) can significantly reduce computational time, but we did not use this option. Second, during solution of contact problems with implicit methods using commercial finite element software systems like Ansys or Abaqus it is necessary to define all contact couples in advance. In fiber network compression simulations, the fiber movement is unpredictable so contact couples are unknown. In the same time, an automatic contact search option is exist in explicit analysis software LS-DYNA. The simulation itself provides a good tool to understand the impact of furnish composition and its changes on
Literature Abitz, P. R., Cresson, T., Brown, A. F. and Luner, P. (1987): The Role of Wet-Fiber Flexibility in Governing Wet Web Properties – A Preliminary Report, Empire State Paper Research Associates, Research Report No. 85, Chapter 6, SUNY College of Env. Sci. Forestry, Syracuse NY, USA, 97110. Alava, M. and Niskanen, K. (2006): The Physics of Paper, Rep. Prog. Phys., 69(3), 669-723. Bronkhorst, C.A. (2003): Modeling Paper as a Twodimensional Elastic-plastic Stochastic Network, Int. J. Solids Struct., 40(20), 5441-5454. Corte, H. and Kallmes, O.J. (1962): Statistical Geometry of a Fibrous Network, In: Formation and Structure of Paper, Trans. 2nd Fund. Res. Symp. Oxford 1961, 13-46. Deng, M. and Dodson, C.T.J. (1994): Paper: an Engineered Stochastic Structure, TAPPI Press. Hallquist J.O. (2005): LS-DYNA Theory Manual, Livermore Software Technology Corp. Heyden S. (2000): Network Modeling for the Evaluation of Mechanical Properties of Cellulose Fibre Fluff, PhD thesis, Lund University, Lund, Sweden. Heyden, S. and Gustafsson, P.J. (2002): Stress-strain Performance of Paper and Fluff by Network Modelling, In: The Science of Papermaking, Trans. 12th Fund. Res. Symp. Oxford 2001, Bury, UK, PPFRS, 1385-1401. Kahkonen, S. (2003): Elasticity and Stiffness Evolution in Random Fibre Networks, PhL thesis, University of Jyvaskyla. Kallmes, O. and Corte, H. (1960): The Structure of Paper. I. The Statistical Geometry of an Ideal Two-dimensional Fibre Network, Tappi, 43(9), 737-752. Katz, J. L., Spencer P., Wang Y., Misra A., Marangos O. and Friis L. (2008): On the Anisotropic Elastic Properties of Woods, J. Mater. Sci., 43(1), 139-145. 262 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012
sheet properties. In addition, by matching the technique with simulations of the forming section, more complete simulations can be constructed which reflect machine operating parameters in addition to the stock variables.
Conclusions A comprehensive network simulation of sheet forming was conducted. The simulations show the significance of using different fiber level descriptions of the forming process in the wet end and wet pressing to construct the structure. The resulting structures can be used for simulating a variety of transport and mechanical properties of paper sheets, and in particular, the effects of different fiber furnishes and their treatments. This enables the optimization of furnish and processing to obtain high performance structures.
Acknowledgements We would like to thank the member companies of the Empire State Paper Research Associates, SUNY ESF, Syracuse NY for partial funding of this work.
Lavrykov, S., Mishra, G.K. and Ramarao, B. V., P. (2010): The Stiffness of Paper – Fibers vs. Bonds, Empire State Paper Research Associates, Research Report No. 132, Chapter 3, SUNY College of Env. Sci. Forestry, Syracuse NY, USA, 13-22. Lavrykov, S., Singh, R. and Ramarao, B. V. (2009): Compressiblity effects in the consolidation of pulp mats with filler particles, In: Proceedings of 14th Fundamental Research Symposium, Fundamental Research Society, Bury Lancashire, UK. Lindström, S.B. and Uesaka, T. (2008): Particle-level Simulation of Forming of the Fiber Network in Papermaking, Int. J. Eng. Sci., 46(9), 858-876. Lindström, S. B., Uesaka, T. and Hirn, U. (2009): Evolution of the paper structure along the length of a twin-wire former, In: Advances in Pulp and Paper Research, Trans. 14th Fund. Res. Symp., Oxford 2009, Bury, UK, PPFRS, 207-245. Miettinen, P.P.J., Ketoja, J.A. and Klingenberg, D.J. (2007): Simulated Strength of Wet Fibre Networks, J. Pulp Paper Sci., 33(4), 198-205. Niskanen, K., Nilsen, N., Hellen, E. and Alava, M. (1997): KCL-PAKKA: Simulation of the 3D Structure of Paper, In: The Fundamentals of Papermaking Materials, Trans. 11th Fund. Res. Symp. Cambridge 1997, Leatherhead Surrey, PIRA International, 1273-1291. Nilsen, N., Zabihian, M. and Niskanen, K. (1998): KCLPAKKA: a Tool for Simulating Paper Properties, Tappi J., 81 (5), 163-166. Page, D.H., El-Hosseiny, F. and Lancaster, A.P.S. (1977): Elastic Modulus of Single Wood Pulp Fibers, Tappi, 60(4), 114117. Paavilainen, L. and Luner, P. (1986): Wet-Fiber Flexibility as a Predictor of Sheet Properties, Empire State Paper Research Associates, Research Report No. 84, Chapter 9, SUNY College of Env. Sci. Forestry, Syracuse NY, USA, 151-172.
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Perkins, R.W. Jr. (2001): Models for Describing the Elastic, Viscoelastic, and Inelastic Mechanical Behavior of Paper and Board, In: Mark, R.E., Habeger, C.C., Borch, J., Lyne, M.B. (Eds.), Handbook of Physical Testing of Paper, vol. 1. Marcel Dekker, New York, 1-76. Ramasubramanian, M. K. and Perkins, R.W. Jr. (1987): Computer simulation of the Uniaxial Elastic-Plastic Behavior of Paper, Empire State Paper Research Associates, Research Report No. 87, Chapter 4, 44-79. SUNY College of Env. Sci. Forestry, Syracuse NY, USA. Provatas, N., Haataja, M., Asikainen, J., Majaniemi, S., Alava, M. and Ala-Nissila, T. (2000): Fiber Deposition Models in Two and Three Spatial Dimensions, Colloids and Surfaces, A: Physicochemical and Engineering Aspects, 165(1-3), 209229. Räisänen, V.I., Alava, M.J., Nieminen, R.M. and Niskanen, K.J. (1996): Elastic-plastic Behaviour in Fibre Networks, Nordic Pulp and Paper Research Journal, 11(4), 243-248. Switzer III, L.H., Klingenberg, D.J. and Scott, C.T. (2004): Handsheet Formation and Mechanical Testing via Fiber-level Simulations”, Nordic Pulp and Paper Research Journal, 19(4), 434-439. Thorpe, J.L., Mark, R.E., Eusufzai, A.R.K. and Perkins, R.W. (1976): Mechanical Properties of Fiber Bonds, Tappi, 59(5), 96100. Van Den Akker, J.A. (1962): Some Theoretical Considerations on the Mechanical Properties of Fibrous Structures, In: Formation and Structure of Paper, Trans. 2nd Fund. Res. Symp. Oxford, 1961, 205-241. Vincent, R., Rueff, M. and Voillot, C. (2009): 3-D Computational Simulation of Paper Handsheet Structure and Prediction of Apparent Density“, Tappi J., 8 (9), 10-17. Vincent, R., Rueff, M., and Voillot, C. (2010): Prediction of Handsheet Tensile Strength by Computational Simulation of Structure, Tappi J., 9 (1), 15-19. Zhong, Z.H. (1993): Finite Element Procedures for ContactImpact Problems, Oxford Univ. Press.
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