Journal of Composite Materials OnlineFirst, published on December 19, 2006 as doi:10.1177/0021998306069873
Fiber/Fiber Crosses: Finite Element Modeling and Comparison with Experiment ANNSOFIE TORGNYSDOTTER,* ARTEM KULACHENKO AND PER GRADIN Mid Sweden University, FSCN, 851 70 Sundsvall, Sweden
LARS WA˚GBERG KTH, Fiber and Polymer Technology, Teknikringen 56-58, 100 44 Stockholm, Sweden (Received October 12, 2005) (Accepted July 11, 2006)
ABSTRACT: Fiber/fiber joints were analyzed using finite element analysis in order to characterize the influence of fiber and contact region properties on the stress– strain behavior of a single fiber/fiber cross. The output of the models was validated by comparison with experimental load–deformation curves. The contact zone of the fiber/fiber joint was studied with respect to the appearance of the contact zone, the contact area, and the contact pattern; the work of adhesion of the contact areas was also considered. It was shown that the two-dimensional appearance of the contact zone had little influence on the stress–strain behavior of the fiber/fiber cross under tensile loading. The maximum stress and hence the fiber/fiber joint strength was, however, affected by the degree of contact. It was concluded that knowledge of the material behavior of the contact zone (such as local plastic behavior), and of chemical effects (such as work of adhesion) are needed to predict the fiber/fiber joint strength. KEY WORDS: bonded area, bonding strength, chemical bonds, joints.
INTRODUCTION APER IS A composite of fibers, air and in some cases chemical additives, where the fibrous network is held together by fiber/fiber joints. In a dry paper network, the main constituents of the paper tensile strength are fiber/fiber joint strength, the number of fiber/fiber joints per unit volume of the sheet, fiber strength, and the existence of fracture zones at the network level [1–3]. In low-density paper, the fiber/fiber joint will be very important in determining the tensile properties of the paper; hence, deeper knowledge of the parameters governing the joint properties would be beneficial in designing
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*Author to whom correspondence should be addressed. E-mail:
[email protected]
Journal of COMPOSITE MATERIALS, Vol. 00, No. 00/2006 0021-9983/06/00 0001–16 $10.00/0 DOI: 10.1177/0021998306069873 ß 2006 SAGE Publications
Copyright 2006 by SAGE Publications.
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paper properties. Paper under tensile stress is held together by the joints, the properties of which are greatly dependent on the characteristics of the contact zone between the fibers. To improve fiber/fiber joint strength, the properties of the contact zone need to be improved by lowering the stress in the fiber/fiber joint or/and by increasing the work of adhesion between the fibers in the area of contact. The contact area, the degree of contact in the contact zone, contact pattern, and the geometry of the contact area are parameters that may influence the maximum stress and stress distribution in the joint. The degree of contact of the contact zone represents the percent of area in contact within the fiber/fiber cross where the fibers are bonded to each other. The material behavior of the contact region will also influence the fiber/fiber joint behavior under stress. Elastic and plastic material properties and work of adhesion in the contact areas have considerable influence on the behavior of the dry joint. These material parameters are in turn dependent on the physical and chemical properties of the fibers and the fiber surface. It is during the formation of the paper, i.e. the formation of the fiber/fiber joints, that the properties of the contact zone, and hence the properties of the joint, are determined. The strength of the fiber/fiber joints has been shown to be largely dependent on the properties of the fiber surface [3,4]. The elastic properties of the surface affects the conformability of the contact zone of the fiber/fiber joint and hence the appearance of the fiber/fiber contact zone, the degree of contact and the number of contact points in the contact zone [4,5]. The elastic modulus of the fiber wall is dependent on the swelling of the fibers, and it is well established that the charge will affect the fibers’ ability to absorb moisture, thus inducing swelling [2,5–9]. The chemical properties of the fiber surface, such as the sites for hydrogen bonding and covalent bonding, also influence the strength of the fiber/fiber joint by altering the work of adhesion in the areas of molecular contact. These sites may occur naturally or be introduced by the addition of chemicals. In the paper industry, such added chemicals include polyamide-polyamine-epichlorohydrin (PAE) and cationic starch, which are commonly used as wet- and dry-strength agents. Experiments using latex have shown that chemical additives may also affect the properties of the contact zone, by decreasing the surface roughness and hence increasing the area of contact, thus increasing the tensile strength of the paper [10]. Furthermore, adding latex changed the material behavior in the contact zone, producing a less brittle material. Latex, with its low glass transition temperature, reduces the elastic modulus of the material in the contact zone. This increases the ductility of the paper under tensile loading [10], probably due to the increased ductility of the fiber/fiber joint under tensile stress. Understanding the mechanisms controlling fiber/fiber joint strength will increase the possibility of designing fiber modifications for the control of fiber/fiber joint behavior under load. In creating a model to describe and predict the load displacement behavior of fiber/fiber crosses under tensile loading, the finite element method (FEM) was chosen for analyzing the fiber/fiber crosses. The model is based on the mechanical properties of the fibers and on the properties of the contact zone constituting the fiber/fiber joint. The simulations were evaluated and compared with the experimental stress–strain curves of fiber/fiber crosses. The use of rayon fibers in the experiments allowed for careful experimental design, so the results could be evaluated more easily. The surface properties of the fibers were altered by means of surface carboxymethylation; this altered the surface conformability of the rayon fibers, affecting the size and appearance of the contact zone. The appearance of
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the contact zone of the fiber/fiber joints was investigated using light microscopy techniques and evaluated in relation to the fiber/fiber joint strength [5]. The experimental fiber/fiber joint strength was measured using a technique described in earlier publications [2].
EXPERIMENTAL Materials The fibers used in these experiments were regenerated cellulosic fibers, RayonÕ , that was carboxymethylated [2,11]. These fibers have been thoroughly investigated in previous studies investigating the fiber dimensions, cellulose crystallinity, total charge, charge distribution, and fiber swelling [2]. The fiber tow was cut to a length of 4 mm by Bernhard Steffert AB, Sweden, and the total charge of the carboxymethylated fibers was analyzed using conductometric titration [12]. Hexamethyl-p-rosaniline chloride (C25H30ClN3 407.99 g/mol) was obtained from VWR International AB, Sweden. According to the supplier, and in accordance with EU directive 91/155/EEC, the solubility of this substance is 16 g/L in water; the chemical is also soluble in ethanol.
Methods FIBER CROSS MEASUREMENTS The fiber crosses were prepared according to the method initially described by Stratton and Colson [13], except that the fibers were not dyed [2]. After preparation, the fiber crosses were stored at 50% RH and 23 C until testing. A tensile testing apparatus was designed at the Mid Sweden University for fiber strength and fiber cross testing. One of the jaws in the apparatus was held stationary while the other was displaced during testing at a rate of 1 mm/s. A load cell from SensotecÕ with a working range of 0–1.5 N was used. The sample was glued to the table of the load bench with Loctite 401Õ that is a cyanoacrylate glue. CONTACT ZONE ANALYSIS The contact zones of the fibers were investigated using light microscopy, operating in the diffraction interference contrast mode, a research microscope, Leica DMRX from Leica, Sweden were used. The analysis was performed on paper made of 5 dTex rayon fibers. The paper was soaked in an acetone solution containing 1 g/L of hexamethylp-rosaniline chloride, that showed good solubility in acetone. The paper was then allowed to dry before being delaminated to expose the contact zones. The basic idea behind this evaluation method is that acetone will not disrupt the molecular contacts between the fibers, so the dye will only reach areas between the fibers that are not in molecular contact. In ethanol, the Van der Waals dimensions of the hexamethyl-p-rosaniline chloride crystal are 1.47 nm/0.47 nm [14]. By removing the excess acetone and evaporating the solvent, the dye will be left only in the non-joined areas of the fiber/fiber joint, and this will increase the contrast in the images of the contact zones. It is worth noting that the exact molecular
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contact area cannot be determined using this technique as the half wave length of visible light (200 nm) determines the resolution of the images and since only a 2D-projection of the contact zone could be analyzed. RESULT AND DISCUSSION In modeling the stress–strain behavior of the fiber/fiber cross two different models were used: a beam and a solid model. The beam and solid models were based on the same fiber geometry (cross-sectional area, moment of inertia, etc.). Table 1 shows the dimension data for the investigated fibers together with the elastic modulus of the fibers. Fiber A was constrained at the endpoints. Displacement of 300 mm was prescribed at the endpoint of fiber B in the upward direction (Figure 1). The same displacement was prescribed to the fiber in the FEM analysis and the experiment. In calculations, the reaction force versus applied displacement was output of the analysis. In all analyses, the output data was compared with the experimental force–displacement curve for the 20 dTex rayon fibers with a charge of 61 meq/g. The Poisson’s ratio was assumed to be 0.3 (analysis showed that the Poisson’s ratio varying in the range 0–0.4 has negligible effect on the force–displacement curve).
Table 1. Properties of the rayon fibers used in the investigation. The elastic modulus (E-modulus), tangent modulus, and yield stress were calculated from the force–displacement curves derived from single fiber tensile testing, as conducted in an earlier study [2]. Fiber type
Fiber diameter (m)
E-modulus (MPa)
Tangent modulus (MPa)
Yield stress (MN/m2)
8.54E-05 7.95E-05 4.10E-05 4.10E-05
187 205 203 203
78 78 – –
62 61 – –
37 meq/g 20 dTex 61 meq/g 20 dTex 35 meq/g 5 dTex 100 meq/g 5 dTex
Cross section for reaction output
Fiber B
Contact region Fiber A
Figure 1. Fiber/fiber cross analysis set-up.
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Fiber/Fiber Crosses
Simulation of the Force–Displacement Curve of a Fiber/Fiber Cross in Tensile Loading Using the Beam Model INFLUENCE OF FIBER DIMENSION AND ELASTIC MODULUS OF FIBERS Modeling the stress–strain behavior of the fiber/fiber cross was first carried out using a beam model in which the two fibers were modeled using one-dimensional isotropic two-node beam elements. A commercial finite element software ANSYS [15] was used. To describe the geometry of the fiber, both the cross-sectional area and geometric (area) moment of inertia have to be computed. This model has significant limitations, as it cannot provide much information about stresses in the contact zone, since the fibers are assumed to be perfectly coupled; furthermore there are limited possibilities to model the influence of fiber shape on the stress state since fiber shape is accounted for by integrated properties (cross-section area and area moment of inertia). In beam finite elements, the transverse strain description is simplified. The elements used in this analysis were based on the Timoshenko beam theory, which accounts for shear deformation effects. In the Timoshenko beam, the transverse shear strain is assumed to be constant over the entire cross-section. In addition, a kinematic assumption is employed in beam formulations, in which the cross-section remains plain and undistorted after deformation. The geometrically nonlinear analysis was carried out using an iterative procedure based on a Lagrangian incremental update method [16]. In Figure 2 the output of the beam model is compared to an experimental force– displacement curve (61 meq/g 20 dTex). In these computations the length of individual beam elements was decreased until the force–displacement curve stabilized, i.e. the L2 norm of the difference between curves upon increased number of element by a factor of two does not exceed 1%. The ratio between the beam length and diameter, at which this condition was satisfied, was three. The experimental and numerical curves coincide at small displacements but deviate at greater displacements, hence the beam theory and the assumption of elastic isotropic material behavior is appropriate only for small loads.
30
Force (mN)
25 20 Experiment
15
Beam model 10 5 0
0
50
100 150 200 Displacement (µm)
250
300
Figure 2. Force–displacement curve calculated using the beam model, compared with an experimental curve.
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A. TORGNYSDOTTER ET AL. 1780
Strain energy (Nm)
1760 1740 1720 1700 1680 1660 1640 10000
15000
20000
25000
30000
35000
40000
Number of elements Figure 3. The dependency of strain energy density on the number of elements in the contact region.
Simulation of the Force–Displacement Curve of a Fiber/Fiber Cross Under Tensile Loading Using 3D Finite Elements Modeling a fiber/fiber joint with 3D finite elements removes the limitations of the beam model as the kinetic assumptions are not needed, but it also introduces some additional challenges. Typically, such problems require a fine mesh, especially in the area where two fibers are joined. Interaction between uncoupled areas has to be described using contact algorithms, so as to prevent interpenetration. By steadily increasing the mesh density and comparing the strain energy density after these changes, the required mesh density can be determined. In this case the stress gradient in the fibers between the crossing and clamps is not significant, so it suffices to decrease the element size in the contact region (Figure 1). The dependency of strain energy on the number of elements in the contact zone is shown in Figure 3. The chosen number of elements was 30,000. INFLUENCE OF FIBER DEFORMATION IN THE CONTACT REGION Fibers in the fiber/fiber joint region are usually flattened due to the pressure applied during forming and due to the capillary forces exerted on the fibers during consolidation and drying (Figure 4). The deformation of the fibers in the contact zone was investigated for the solid model using a finite element mesh that was fully parameterized and in which eight-node brick elements were used. The contact zone between the fibers was assumed to be perfectly coupled, i.e. the contacting surfaces are fully connected. A computationally efficient internal multipoint constraint (MPC) approach was used to couple the dissimilar meshes in the contacting areas. In this method, the contact elements are required to form constraint equations between nodes on the contacting surfaces. For fiber A and for the contact part of fiber B, a simplified enhanced strain formulation was chosen, which introduces nine internal degrees of freedom to prevent shear locking. For the rest of fiber B, it was assumed that the stress gradients are low, and computationally cheap onequadrature solid elements with hourglass control were used [17]. The material was assumed to be linearly elastic and isotropic.
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Fiber/Fiber Crosses
Figure 4. Finite element representations of the contact region: undeformed (left figure), deformed (right figure).
35 30
Force (mN)
25 Beam model
20
3D deformed 15
3D undeformed Experiment
10 5 0 0
50
100 150 200 Displacement (µm)
250
300
Figure 5. Output of the 3D model with undeformed and deformed fibers in the contact region, as shown in Figure 4. The material of the fibers was assumed to be linearly elastic in the 3D model. The output of the calculations of the 3D model was compared with the output of the beam model and the experimental curve.
In Figure 5 the output of the 3D solid model is shown for fibers with undeformed and deformed contact regions, as shown in the finite element representation in Figure 4. The output of the 3D model is compared with that of the beam model and with the experimental force–displacement curve. Both the beam and 3D models use the same geometric properties of the fiber cross-section. Despite the fact that the fiber/fiber joint with the deformed fibers in the contact region was weakened, the solid model of the deformed and undeformed contact regions showed only little difference in the reaction force, which suggests that local changes do not affect the global reaction force. The solid model follows the experimental curve to greater displacements than the beam model does, but still deviates for larger prescribed displacement. Deformation of the fiber will affect the mechanical properties across the fiber, something which was not considered in this work. However, as there was no difference
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between the outputs of the undeformed and deformed contact zones, the model with an undeformed contact region will be used in the following experiments, since it has an undistorted mesh. INFLUENCE OF FIBER ORTHOTROPY Wood fibers are known to show anisotropic material behavior which can be approximated by orthotropic properties if the fibril angle is close to zero [18]. In this case the three principal directions are the longitudinal direction, the radial direction, and tangential direction. The orthotropic approximation corresponds to the extreme case in which the difference in stiffness between longitudinal and transverse direction of the fiber is maximum. To study the effect of orthotropy we chose the following orthotropic properties [18]: E3 ¼ 187 MPa E1 ¼ E2 ¼ 13:4 MPa 12 ¼ 0:1 13 ¼ 23 ¼ 0:04 G12 ¼ 4:4 MPa G13 ¼ G23 ¼ 13:1 MPa Figure 6 shows the comparison of the force–displacement curves from the model with linear isotropic material and linear orthotropic material. Results indicate that the influence of orthotropic properties on the force–displacement curve is limited to the region of small displacements. At a large displacement the effect of longitudinal stiffness with the considered loading conditions becomes dominant.
25
Force (mN)
20
15 Linear isotropic Linear orthotropic Experiment
10
5
0
0
50
100
150
200
250
300
Displacement (µm) Figure 6. Effect of fiber orthotropy.
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Fiber/Fiber Crosses (a)
.114184
2.773
5.432
8.091
10.75
13.409
16.068
18.727
21.385
24.044
(b)
.066432
1.137
2.208
3.278
4.349
5.419
6.49
7.56
8.631
9.702
Figure 7. The stress distribution of the contact zone of fiber B for (a) a fully coupled rectangular contact area with linear elastic material behavior and (b) a fully coupled rectangular contact area with elasto-plastic material behavior. The gray scale shows the degree of stress (MPa) in the contact zone, the stress increasing with the darkness. The magnitude of the maximum stress is indicated on the gray scale, the stress increasing with the darkness.
INFLUENCE OF THE PLASTIC MATERIAL BEHAVIOR The solid model was modified by assuming the fibers to have elasto-plastic bilinear behavior, so that two constants must be provided for the plasticity: the yield stress and tangent modulus. The model with the undeformed contact region will be used, since it has an undistorted mesh. Figure 7 shows the stress distribution over the contact zone for: (a) a fully coupled rectangular contact area with elastic material behavior added to the model and (b) a fully coupled rectangular contact area with elasto-plastic material behavior added to the model. As elasto-plastic material behavior was added to the 3D model and the stress distribution was altered in the contact zone. The stress was more evenly distributed across the contact zone, and further, the stress concentration factor (and the maximum equivalent stress) in the case of the elasto-plastic material became smaller. INFLUENCE OF THE TWO-DIMENSIONAL APPEARANCE OF THE CONTACT ZONE – DEGREE OF CONTACT AND CONTACT PATTERN OF THE CONTACT ZONE Depending on the chemical and mechanical properties of the fibers and the fiber surfaces, the properties of the contact zone, such as the degree of contact and contact zone pattern, will change and this may alter the stress–strain behavior of the fiber/fiber joint [5]. In the analyses, the contact zones were modified with respect to the degree of contact and contact pattern of the zone. In Figure 8 the stress distribution of three different contact zones are shown for fiber B at 300 mm displacement. The maximum
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A. TORGNYSDOTTER ET AL. (a)
.066432
1.137
2.208
3.278
4.349
5.419
6.49
7.56
8.631
9.702
(b)
1,0,1 0,0,0 0,0,0 1,0,1 .048177
4.012
7.976
11.94
15.904
19.868
23.832
27.796
31.76
35.723
(c)
1,1,1,1 1,0,0,1 1,0,0,1 1,1,1,1 .032325
1.203
2.375
3.546
4.717
5.888
7.059
8.23
9.401
10.573
Figure 8. Stress distribution of the contact zone of fiber B at 300 m displacement with three different degrees of contact and bonding patterns in the contact zone where elasto-plastic material behavior was included. (a) 100% degree of contact with a rounded contact area. (b) 16% degree of contact with four rounded contact points. (c) 82% degree of contact with a rectangular contact. The location of the contact points are shown by the matrices in the figures. The gray scale shows the degree of stress (MPa) in the contact zone, the stress increasing with the darkness. The magnitude of the maximum stress is indicated on the gray scale.
stresses of the fiber/fiber crosses under loading were concentrated in the contact areas of the fiber/fiber crosses, and increased with decreasing degree of contact. The maximum stress in the contact zone versus degree of contact for fiber/fiber crosses with different contact patterns and degree of contact is shown in Figure 9. The stress magnitude is nearly inversely proportional to the degree of contact; due to the slight influence of the contact pattern, the curve is not smooth. It should be stressed at this point that the material properties in the contact zone cannot be determined with the present methods, but we postulate that the dependency of maximum stress on degree of contact
11
Fiber/Fiber Crosses 40
Maximum stress (MPa)
1.
1.
35 30 25
2.
010 010 010
3.
000 111 000
4.
010 101 010
2.
20
3.
4.
15
6. 10
7. 5.
5.
8.
5
101 000 000 101
6.
111 000 111
7. 11 10 10 11 1001 1111
101 101 101
8. Fully coupled
0 0
0.2
0.4
0.6
0.8
1
Degree of contact
Figure 9. Maximum stress as a function of degree of contact for fiber/fiber crosses. The curve is not smooth due to the influence of the contact pattern of the contact zones. The location of the contact points of the contact zones are shown by the matrices in the figure, compare with Figure 8. 25
Force (mN)
20
15
100% doc plastic 33% doc plastic 3D undef elastic
10
Experiment
5
0 0
50
100
150
200
250
300
Displacement (µm)
Figure 10. Output of the 3D model with bilinear plastic material behavior of the fibers for two different contact patterns and degree of contact, 100% and 33%. The output of the 3D model assuming linear elastic material behavior of the fibers with a unit contact region and an experimental curve are shown for comparison.
will have the same tendency for a wide range of homogeneous materials, including, say, orthotropic ones. Analyzing the maximum stress of the contact zone, the degree of contact will be of great importance below 50%. This is in agreement with experimental results shown in Reference [5] where the fiber/fiber joint strength increased much more drastically from 32–46% than between 46–65%. In Figure 10 the reaction force for fiber/fiber crosses with 100% degree of contact and 33% degree of contact with elastoplastic material behavior (same for fiber and
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Load (mN)
10 8 32% DOC 65% DOC
6 4 2 0
0
50
100
150
200
250
300
Displacement (µm) Figure 11. Experimental load–displacement curves of 20 dTex rayon fibers with 32% and 65% degree of contact [5].
contact zone) are shown as a function of the applied displacement. For comparison, an experimental curve and the curve of the 3D model assuming elastic contact region behavior with a unit contact region are also shown in Figure 10. The model with elasto-plastic material behavior produced results nearly identical to the experimental data. The local changes of the contact zones did not affect the shape of the force–displacement curve, despite the large differences in degree of contact, 100% compared to 33%. The maximum stress was approximately twice as high for 33% degree of contact compared to 100%. The elasto-plastic material behavior decreased the force at a specific displacement compared to the linear elastic material behavior, perhaps due to deformations in the contact region. Despite the lower maximum stresses in the contact region with elasto-plastic material behavior, the stress is distributed over a larger area that influences the strain energy of the fiber/fiber cross during loading, see Figure 8. The two-dimensional properties of the contact zone, the degree of contact, and the contact pattern of the contact zone did not affect the shape of the force–displacement curve (Figure 10). This is in agreement with the experiments shown by Torgnysdotter et al. where similar force-displacement curves were achieved for fiber/fiber crosses from fibers with 65% and 32% degree of contact, see Figure 11 [5]. The degree of contact of the fiber contact zones in Figure 11 was altered by changing the surface charge of the fiber altering the modulus of elasticity of the fiber surface. The degree of contact was analyzed using light microscopy [5]. The deviation between the experimental and calculated results may be explained by the material parameters of the contact zone, which are experimentally significant. Unless the material properties in the contact region are accounted for, the strength of the fiber/fiber joint cannot be calculated properly. This finding is strengthened by the fact that by adding the wet-strength agent PAE (polyamide-polyamine-epichlorohydrin), the strength of the fiber/fiber joint increased significantly from 4 to 8 mN, despite a decrease in degree of contact from 65 to 48% [5]. The addition of PAE is supposed to increase the molecular work of adhesion in the contact zone as covalent bonds are introduced.
Fiber/Fiber Crosses
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INFLUENCE OF THE APPEARANCE OF THE CONTACT ZONE IN THE Z DIRECTION It has been discussed how the two-dimensional geometric properties of the contact zone and the material properties of the contact zone affect the tensile properties of a fiber/ fiber cross. However, the three-dimensional geometric properties of the interface in the contact zone could also influence the tensile properties of the fiber/fiber cross, where interpenetration of fiber surfaces would be of interest. Hence, two different shapes of the third dimension of the interface, a concave shape (Figure 12) and a convex shape, were investigated using the 3D solid model. The equivalent stress distribution for the concave and convex shapes of the interface are shown in Figure 13; the two cases result in nearly the same maximum equivalent stresses. Despite the small difference in the maximum stresses, there was a considerable difference in the stress distribution (Figure 13) and strain energy (only elements in the contact area were considered when computing the strain energy). A convex layer resulted in a higher stress concentration factor, but the strain energy of the convex form was 1379 Nm and of the concave form was 1807 Nm. Strain energy is often used in fracture mechanics to assess the possibility of crack growth. The fact that the concave form of the layer leads to higher strain energy may suggest a greater possibility of fracture, at least from the linear elastic fracture mechanics point of view. In order to visualize the stress distribution in the contact region, isosurfaces were used, and in both cases the maximum stresses were located near the fiber surface (Figure 14). The concave and convex interfaces depict two different 3D interfaces of the contact layer that showed rather large differences in stress distribution and strain energy, and hence different probabilities of fracture. The fact that the stresses are located near the fiber surface (Figure 14) further implies the great importance of the contact zone properties for the tensile properties of the fiber/fiber cross. It cannot be determined from the experimental results whether the fiber/fiber crosses investigated here have either of these two contact zone interface shapes, i.e., concave or convex. However, the results suggest
Figure 12. Fiber/fiber cross with an intermediate interface of concave form.
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A. TORGNYSDOTTER ET AL. (a)
.063976
.948632
1.833
2.718
3.603
4.487
5.372
6.257
7.141
8.026
(b)
.081189
1.256
2.431
3.607
4.782
5.957
7.132
8.307
9.482
10.657
Figure 13. Equivalent stress distribution of the (a) concave and (b) convex shape of the interface. The grey scale shows the degree of stress (MPa) in the contact zone, the stress increasing with the darkness. The magnitude of the maximum stress is indicated on the grey scale.
that the shape of the contact zone interface has a significant influence on the stress distribution in the contact region; this requires separate examination. One can draw a parallel between this problem and that of welding, in which the geometry of the welding beads has been the subject of extensive research [19].
CONCLUSIONS Finite element simulations have been used to evaluate the influence of fiber, mechanical, and geometric parameters, as well as of the material behavior and contact region properties, on the stress–strain behavior of fiber/fiber crosses. The model output was compared with experimental force–displacement curves, and a close agreement between experimental results and model predictions was found. Results also indicate that a bilinear elastic-plastic stress–strain relation captures the behavior of the fibers very well. The degree of contact will determine the maximum stress in the fiber/fiber joint, hence determining the fiber/fiber joint strength. However, it was also clear from the results that knowledge of the material behavior and work of adhesion in contact zone areas is needed if one is to predict the fiber/fiber joint strength accurately.
15
Fiber/Fiber Crosses
(a)
A=1.427
B=2.668
C=3.908
D=5.149
E=6.389
G=8.87
F=7.63
H=10.111
(b)
A=2.114
B=3.88
C=5.647
D=7.413
E=9.18
F=10.946
G=12.713
H=14.48
Figure 14. Equivalent stress distribution using isosurfaces for contact region interfaces of (a) concave and (b) convex shape. The grey scale shows the degree of stress (MPa) in the contact zone, the stress increasing with the darkness of the contacting area.
ACKNOWLEDGMENTS Thanks are extended to Staffan Palovaara of SCA, Sweden for conducting the Ep-SEM analysis of the contact regions of rayon fibers. The Fiber Science Communication Network (FSCN) is gratefully acknowledged for financial support.
REFERENCES 1. Page, D.H. (1969). A Theory for the Tensile Strength of Paper, Tappi Journal, 52(4): 674–681. 2. Torgnysdotter, A. and Wa˚gberg, L. (2003). Study of the Joint Strength Between Regenerated Cellulose Fibres and its Influence on the Sheet Strength, Nordic Pulp and Paper Research Journal, 18(4): 455459. 3. Torgnysdotter, A. and Wa˚gberg, L. (2004). Influence of Electrostatic Interactions on Fibre/Fibre Joint and Paper Strength, Nordic Pulp and Paper Research Journal, 19(4): 440–447. 4. Page, D.H., Tydeman, P.A. and Hunt, M. (1962). In: Bolam, F. (ed.), Formation and Structure of Paper, pp. 171–193, Technical Section, B.P. & B.M.A., London.
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5. Torgnysdotter, A., Kulachenko, A., Gradin, P. and Wa˚gberg, L. (2005). The Link Between the Contact Zone Between Fibres and the Physical Properties of Paper – A Way to Control Paper Properties (to be published). 6. Grignon, J. and Scallan, A.M. (1980). Effect of pH and Neutral Salts Upon the Swelling of Cellulose Gels, Journal of Applied Polymer Science, 25(12): 2829–2843. 7. Lindstro¨m, T. (1986). The Concept of Measurement of Fibre Swelling, In: Bristow, E. and Kolseth, P. (eds), Paper Structure and Properties, pp. 75–97, Marcel Dekker Inc., Basel, New York. 8. Lindstro¨m, T. Carlsson, G. (1982). The Effect of Chemical Environment on Fibre Swelling, Svensk papperstidning-Nordisk cellulosa, 85(3): 14–20. 9. Scallan, A.M. and Grignon, J. (1979). The Effects of Cations on Pulp and Paper Properties, Svensk papperstidning-Nordisk cellulosa, 82(2): 40–47. 10. Alince, B. (1999). Cationic Latex as a Multifunctional Papermaking Wet-end Additive, Tappi Journal, 82(3): 175–187. 11. Walecka, J.A. (1956). An Investigation of Low Degree of Substitution Carboxymethylcellulose, Tappi Journal, 39(7): 458–463. 12. Katz, S., Beatson, R.P. and Scallan, A.M. (1984). The determination of Strong and Weak Acidic Groups in Sulphite Pulps, Svensk papperstidning-Nordisk cellulosa, 87(6): 48–53. 13. Stratton, R.A. and Colson, N.L. (1990). Dependence of Fibre/Fibre Bonding on Some Paper Making Variables, In: Proceedings of Materials Research Society Symposium, Vol. 197, pp. 173–181. 14. Hoppe, R., Alberti, G., Costantino, U. Dionigi, C., Schulz-Ekloff, G. and Vivani, R. (1997). Intercalation of Dyes in Layered Zirconium Phosphates. Part 1: Preparation and Spectroscopic Characterisation of Alfa-zirconium Phosphate Crystal Violet Compounds, Langmuir, 13(26): 7252–7257. 15. ANSYS (2005). Theory Reference. Ansys Inc., Southpointe, USA. 16. Belytschko, T., Liu, W.K. and Moran, B. (2000). Nonlinear Finite Elements for Continua and Structures, John Wiley and Sons, England. 17. Flanagan, D.P. and Belytschko, T. (1981). A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control, International Journal for Numerical Methods in Engineering, 17: 679–706. 18. Persson, K. (2000). Micromechanical Modelling of Wood and Fibre Properties, University of Lund, Lund, Sweden. 19. Mackerle, J. (1996). Finite Element Analysis and Simulation of Welding: A Bibliography (1976–1996), Modelling and Simulation in Materials Science and Engineering, 4(5): 501–533.
