Nonlinear simulations with linear elastic material

The large deformation simulations of wood cell structures presented in this sub-section were performed with a linear elastic material behaviour of the cell walls.

Simulations for cell structures were carried out using the five different average den-sities of the growth ring shown in Table 7.1. Since cell structures with a density lower than 400 kg/m² result in very large models, the use of which is too time con-suming to be practical, simulations of cell structures with an average density of less than 400 kg/m² were not performed. In the analyses carried out, the irregularity parameter Rmax described in Chapter 5, which controls the irregularity of the cell structure was set here to zero, resulting in the cell structures being regular. Analyses were performed for compression loading in the radial and tangential directions only.

Table 7.1: Average density and corresponding growth ring width for the analysed growth ring structures.

Density Ring width

kg/m³ mm

400 2.01

425 1.44

450 1.14

475 0.92

500 0.76

In Figure 7.1 the results the analysis provided of compression loading in the radial direction for the five models that differed in average density are shown. Since the five cell structures show almost identical deformations, it can be concluded that, in terms of these models, the buckling of the earlywood cell structure in compressive loading in the radial direction is virtually independent of density. Figure 7.2 shows the compressive stress-strain curve for a cell structure with an average density of 500 kg/m³. Compressive stress is defined here as the compressive force applied divided by the undeformed area, whereas compressive strain is defined as the negative change of length divided by the original length of the undeformed geometry. The

stress-a) b) c) d) e)

Figure 7.1: Deformations of wood cell structures assuming elastic material with average densities of a) 400 kg/m³, b) 425 kg/m³, c) 450 kg/m³, d) 475 kg/m³ and e) 500 kg/m³.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0

2 4 6 8 10 12 14 16

Compressive strain

Compressive stress, MPa

Figure 7.2: Compressive stress-strain curve for loading of a regular cell structure in the radial direction assuming linear elastic material of the cell wall layers and an average density of 500 kg/m³.

strain curve displayed in Figure 7.2 shows linear elastic behaviour initially, followed by a sudden drop in stress with increasing strain. In the experiments on radial compression, a small stress peak can be noted, but the high stress peak shown in Figure 7.2 was not found experimentally, presumably due to the fact that a real cell structure is more irregular.

To investigate the influence of the irregularity parameter Rmax on the structural response in compressive loading in the radial direction, a model in which R_max=3 µm and the average density was 500 kg/m³ was analysed. The compressive stress-strain curve obtained is shown in Figure 7.3. In the case of an irregular structure, the high initial stress peak completely vanishes and the elastic stiffness is lower, as was shown in Chapter 6. Simulations of an irregular structure in radial compression provide results that are closer to the experimentally observed behaviour, see Holm-berg [30]. The buckling mode and strain localisation of the cell structures obtained in simulations involving linear elastic material are not in accordance with the results of experiments on microstructural deformations that was presented in Chapter 3.

In the simulations, the cell structures showed a buckling mode caused by shearing of the thin earlywood cells in the RT-plane, whereas in the experiments, collapse occurred completely for one row of cells at a time, resulting in a strain localisation effect.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0

1 2 3 4 5 6 7

Compressive strain

Compressive stress, MPa

Figure 7.3: Compressive stress-strain curve for loading of an irregular cell structure in the radial direction assuming linear elastic material of the cell wall layers and an average density of 500 kg/m³

For compression loading in the tangential direction, the size of the cell struc-ture chosen to be analysed can be expected to be important for the deformation shape obtained. It has been shown in experiments that for tangential loading of cell structures that are very long in the tangential direction, instability occurs, the latewood region buckling into the earlywood in a wavy manner. This is not ex-pected for cell structures that are shorter in length in the tangential direction, for which deformation patterns can be expected to be stable and uniform. Two cell structures of differing length in the tangential direction were modelled for studying the microstructural deformations in loading in the tangential direction. The first cell structure was modelled with six cells in the tangential direction, whereas the second was modelled with 30 cells in the tangential direction. For both models, the average density was 550 kg/m³, the structure being regular in both cases. The model with 30 cells in the tangential direction was very large, with over a million degrees of freedom. Nonlinear analysis of models as large as this require large com-puter resources and is very time consuming. For analysing models larger than this, a different modelling approach is needed.

In Figure 7.4 the deformations obtained for the model with six cells in the tan-gential direction are shown, and in Figure 7.5 the deformations obtained for the model with 30 cells in the tangential direction. It can be seen clearly in studying

a) b)

Figure 7.4: Compressive loading in the tangential direction assuming linear elastic material of the cell wall layers. a) Undeformed structure. b) Deformed structure after loading.

the results obtained that for higher loading in the tangential direction the defor-mation shape of the structure is strongly dependent on the size of the model. The deformations for the smaller model are distributed throughout the cell structure and no instability occurs, whereas for the larger model the strains are localised to the middle of the cell structure. A clear indication of the buckling of the latewood band can be seen in Figure 7.5.