A theoretical model of collapse recovery

Abstract

A finite element model was attempted here to demonstrate the theoretical basis for collapse recovery. The model built on a simple single-cell based model of collapse that was developed by Innes (1995, 1996). The theoretical mechanism for recovery of collapse by steam reconditioning has essentially been that proposed by Ilic (1987, personal communication) (cited by Oliver, 1991), which assumes that the S1 and S3 layers are largely responsible for providing an elastic component of the cell walls and that the S2 layer is considered the most important for providing the inelastic material properties required to hold the cell in the collapsed or deformed state. To this end, a viscoelastic material model was developed for the S1 and S3 layers, while an elastic-plastic model was developed for the S2 layer. The model based on these material properties was not able to clearly demonstrate collapse recovery. This was largely attributed to the similarity of the dependence of the elastic moduli as a function of temperature in all cell wall layers. By removing the temperature dependence in the S1 and S3 layers, a much more significant, although still incomplete, recovery of cell shape was demonstrated. The lack of realistic behaviour for the model predictions has highlighted the paucity of knowledge about mechanical properties at the cell wall scale. Obviously, direct measurements at this scale are extremely difficult, if not impossible. The most successful approaches so far to estimate these properties has been to use a range of homogenisation and finite element modelling techniques based on the generalised knowledge of the cell wall ultrastructure and the properties of extracted chemical constituents. While not directly comparable, published values (Harrington et al. 1998) for Pinus radiata at 12% moisture content suggest that values used in this model may have been rather low. To some extent this may have compensated for the stiffening effect of the simple, but inherently stable, geometry used in this model. Even though the method used by Harrington et al. (1998) could have been used to determine better elastic moduli for the different cell wall layers than those used, it was not attempted here because there are still several critical limitations with this approach. These include that the method is possibly less reliable at high moisture content states and that it provides no additional information on critical behaviours such as non-linear large deformation stress-strain relationships, time or temperature-dependent behaviour, or moisture content (including moisture change or mechanosorptive strain) dependent behaviour. All of which may be critical for accurately modelling the deformation and stress distribution in the cell wall layers prior to steam reconditioning. Even if alternate attempts to simplify the moisture related behaviours were pursued, but that still accounted for the significant reduction in collapse recovery below 15% moisture content, the lack of good temperature-dependent data in the different secondary cell wall layers is currently a major impediment for developing the current model further. The other major improvement that could be made to the model developed here would be to include multiple cells with more realistic geometries and arrangements. Such an approach was attempted in the models by Astley et al. (1998), where real cross-sections of Pinus radiata tracheids were scanned and skeletonised to form the geometrical basis of a finite element model used to predict the macroscale elastic properties. Unfortunately, while the skeletonisation process makes the realistic cell geometries much simpler to implement in a finite element model, it is not possible here. Mostly this is because it is the two-dimensional spatial arrangement of the different secondary cell wall layers that is considered critical to the collapse and collapse recovery behaviour. Scanning in real cross-sections is still a possibility, but, it would require much more complex image analysis programming to approximate where the cell wall layer boundaries occurred for the relevant material properties to be applied to the appropriate elements. The number of elements required in this type of approach would also make the finite element model considerably more computationally intensive. Nevertheless, as computer processing continues to become faster and cheaper, even in the near future this is unlikely to be a significant restraint for a model with up to 100 cells. This approach would also largely avoid the need for artificial constraints in the current model, such as the shear displacements required to achieve a more realistic flattening of the cell lumen.