6.2 Parametric Study
6.2.3 Influence of the microfibril angle, the average density and the
The influence of the density, the microfibril angle and the material parameters of the cell wall on the mechanical properties of the wood were investigated. A hexagonal cell structure model was used for determining the stiffness and shrinkage proper-ties of 20 different cell structures, each involving three different sets of material parameters. The 20 cell structures were assigned four different average densities, each having five different microfibril angles of the S2-layer. The cell structures were determined for the average densities shown in Table 6.3, with the corresponding growth ring widths being chosen in accordance with Eq.(2.4). For each average density shown in Table 6.3, models involving five different microfibril angles of the S2-layer were analysed. The five microfibril angles chosen were 0◦, 10◦, 20◦, 30◦ and 40◦. For each model the average stiffness and hygroexpansion properties were determined for the low, medium and high material data sets of the cell wall layers, as shown in Table 5.3 and 5.4 for the stiffness properties and in Table 5.7 and 5.8 for the shrinkage coefficients. In using these material properties, a moisture content of 12% is assumed. The full six by six stiffness matrix and the hygroexpansion properties were determined for each of the 60 cases described above, resulting in a total of 420 simulations.
In Figures 6.1 - 6.12 the influence of density and microfibril angle on the stiffness and hygroexpansion properties are shown for regular and irregular cell structures, using the low material data sets of the cell wall layers. For microfibril angles larger than about 5◦, the longitudinal modulus of elasticity showed a strong dependency
Table 6.3: Average density and corresponding growth ring width for the analysed growth ring structures.
Density Ring width
kg/m3 mm
400 2.01
450 1.14
500 0.76
550 0.63
on the microfibril angle, the stiffness decreasing rapidly with an increase in the microfibril angle. For small microfibril angles, the influence was only slight. The influence of the average density on the longitudinal modulus of elasticity was large, especially for small microfibril angles. The influence of the microfibril angle on the tangential modulus of elasticity, in contrast, was found to be small, the stiffness first decreasing slightly and then increasing as the microfibril angle increased. The strong influence of the density on the tangential modulus of elasticity can be explained by the fact that for loading in the tangential direction the cell walls are subjected mostly to bending. For high densities, the growth ring has a high latewood content, leading to an increase in the fraction of cells that are thick-walled and resulting in an increase in tangential stiffness. The radial modulus of elasticity showed a much lesser dependency on average density. This can be explained by the fact that the three regions of the wood structure lie in series with respect to the radial direction. As a result, the radial stiffness of the earlywood region governs the overall radial stiffness.
The microfibril angle has a stronger influence, the radial modulus of elasticity first decreasing and then rapidly increasing as the microfibril angles become large.
Density had only a slight influence on the modulus of shear in the RT-plane, whereas the microfibril angle had a strong influence. With an increase in the mi-crofibril angle, the stiffness increased rapidly. The mimi-crofibril angle had a strong influence on the modulus of shear in the LT-plane, the shear increasing with an increase in the microfibril angle, whereas the density had a lesser influence. The modulus of shear in the RT-plane was very low for small microfibril angles, in-creasing slightly as the microfibril angle increased whereas the density had a much greater influence. The low modulus of shear in the RT-plane was expected, due to the low bending stiffness of the earlywood cells walls. The microfibril angle was found to have a strong influence on all the Poisson ratios, whereas the density had little influence. For the parameters of the stiffness properties, it appears that each parameter depends mainly on either the density or the microfibril angle. In terms of the present model, both parameters were shown to be equally important for the stiffness properties.
0 5 10 15 20 25 30 35 40
Figure 6.1: Moduli of elasticity in the longitudinal direction at different average densities versus the microfibril angle for regular cell structures to the left and irreg-ular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.2: Moduli of elasticity in the radial direction at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.3: Moduli of elasticity in the tangential direction at different average den-sities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Modulus of shear LR, MPa
400 kg/m3
Modulus of shear LR, MPa
400 kg/m3
Modulus of shear LR, MPa
400 kg/m3
Modulus of shear LR, MPa
400 kg/m3
Modulus of shear LR, MPa
400 kg/m3
Modulus of shear LR, MPa
400 kg/m3 450 kg/m3 500 kg/m3 550 kg/m3
Figure 6.4: Moduli of shear in the LR-plane at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Modulus of shear LT, MPa
400 kg/m3
Modulus of shear LT, MPa
400 kg/m3
Modulus of shear LT, MPa
400 kg/m3
Modulus of shear LT, MPa
400 kg/m3
Modulus of shear LT, MPa
400 kg/m3
Modulus of shear LT, MPa
400 kg/m3 450 kg/m3 500 kg/m3 550 kg/m3
Figure 6.5: Moduli of shear in the LT-plane at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Modulus of shear TR, MPa
400 kg/m3
Modulus of shear TR, MPa
400 kg/m3
Modulus of shear TR, MPa
400 kg/m3
Modulus of shear TR, MPa
400 kg/m3
Modulus of shear TR, MPa
400 kg/m3
Modulus of shear TR, MPa
400 kg/m3 450 kg/m3 500 kg/m3 550 kg/m3
Figure 6.6: Moduli of shear in the TR-plane at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.7: Poisson ratio νRL at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.8: Poisson ratio νT L at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.9: Poisson ratio νT R at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.10: Hygroexpansion coefficient in the longitudinal direction at different average densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.11: Hygroexpansion coefficient in the radial direction at different average densities versus the microfibril angle for regular cell structures to the left and irreg-ular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
0 5 10 15 20 25 30 35 40
Figure 6.12: Hygroexpansion coefficient in the tangential direction at different av-erage densities versus the microfibril angle for regular cell structures to the left and irregular to the right using from top to bottom the low, medium and high material data sets for the cell wall layers.
For small microfibril angles, the hygroexpansion coefficients in the radial and the tangential directions were found to be large, whereas in the longitudinal direc-tion they were found to be small. With an increase in the microfibril angle, the longitudinal hygroexpansion first decreases and then increases. The longitudinal hygroexpansion coefficient was nearly zero for microfibril angles around 20◦. The hygroexpansion in the radial and tangential directions decreases with an increase in the microfibril angles. Although it may appears that the influence of the lon-gitudinal hygroexpansion is negligible due to the only very small numerical values of the shrinkage parameters. However, dealing with shape stability of timber, the longitudinal shrinkage is essential in drying processes, see Ormarsson [48].
Employing the three different sets of material parameters of the cell wall layers had a strong influence on each of the stiffness properties. It was shown to be important to have good experimental data concerning the material properties of the chemical constituents. However, studying the results obtained here, together with the experimental data on the stiffness and on the hygroexpansion of wood, can provide guidance in the selection of reasonable values for the material parameters of the chemical constituents.