Nonlinear material behaviour

In large-strain analysis of wood cell structures, the material description of the cell wall layers needs to be extended so as to incorporate nonlinear effects. This can be done by adopting an orthotropic elasto-plastic material model such as that proposed by Hill [29]. When the axes of the 123-coordinate system coincide with the axes of orthotropy, the Hill yield criterion is given by

f (σ) = a1(σ22− σ³³)²+ a2(σ33− σ¹¹)²+ a3(σ11− σ²²)² +a4σ²₂₃+ a5σ₁₃² + a6σ²₁₂− σref² = 0

(4.53) where σ is the stress tensor. The material parameters a₁ to a₆ are expressed as

a1 = σ²_ref

where σijy are the yield stresses with respect to the orthotropic directions i and j and σref is a reference stress. The Hill yield criterion is an extended form of the von Mises criterion for isotropic materials. When the parameters a1 to a6 satisfy

6a1 = 6a2 = 6a3 = a4 = a5 = a6, (4.55) Eq.(4.53) reduces to the von Mises criterion.

The parameters a1 to a6 cannot be chosen arbitrarily. The yield surface must be a closed surface in the deviatoric stress plane. This leads to the following constraint

4 This inequality restricts the degree of orthotropy for which application of Hill’s criterion is permissible.

For the various cell wall layers, the yield stresses σijy and the reference stress σ_ref in Eq.(4.53) need to be determined. The choice of these parameters will be discussed further in the sections that follow. For convenience, six parameters ri,j

are introduced as

The parameters a1 to a6 can now be written in terms of ri,j. The integration of the plasticity equations can be found in several references, see for example Ottosen et al. [51].

4.5. Concluding Remarks

In the thesis, a large number of simulations of wood and fibre properties are de-scribed, all of which were performed using the general-purpose finite element pro-gram ABAQUS [28]. The finite element formulation and solution technique used by the program were outlined in the present chapter.

To determine the average stiffness and shrinkage properties, the equations in the homogenisation procedure were solved using the finite element method. The base cell was then discretized into finite elements and subjected to periodic boundary conditions and to the six elementary cases of mean stress tensors. Periodic boundary conditions were implemented by use of constraint equations through equating the nodal displacements at opposite sides of the base cell in each direction, in accordance with Eq.(4.7). The equivalent average stiffness and shrinkage parameters were then determined according to the procedure outlined in section 4.3.2.

In studies of the behaviour of cell structures subjected to large compressive defor-mations ABAQUS was used with the nonlinear finite element formulation described in this chapter. In addition to the linear elastic material behaviour of the cell wall layers, nonlinear effects were included through adopting Hill’s orthotropic plasticity model. Since the nonlinear effects in the cell walls are of varying nature and only very limited experimental data on this are available, the choice of a material model and of its material parameters is not obvious, matters that will be treated further in a later chapter.

of wood

5.1. Introduction

In determining the equivalent average mechanical properties of wood from models of its microstructure, a common task is to determine an equivalent volume element representative of the microstructure. In this chapter a chain of geometrical and mechanical models of the microstructure of wood are presented. The microstructure of wood is divided into a hierarchy consisting of three geometric levels: the microfibril level, the fibre level and the cell structure level. The models chosen are regarded then as being representative elements of the wood microstructure at each structural level. The models developed are utilised in Chapters 6, 7 and 8 with the aim of determining linear and nonlinear homogenised properties of wood and fibres by means of numerical simulations. Since the geometrical models are to be used in numerical simulations involving the finite element method, certain approximations of the microstructure are introduced. However, the aim is to model the microstructure of wood and of wood fibres so as to approximate the real structures in such a way that the impact on the mechanical properties of wood is as small as possible.

The geometry of the wood and the fibre microstructure presented is based on the measurements presented in Chapter 3 and on the literature.

Different approaches to deriving the material properties of wood through mi-cromechanical modelling of the wood structure have been adopted earlier. The first reported attempt was made already in 1928 by Price [58], who achieved analyti-cal solutions for the stiffness of wood through modelling the cells as circular tubes.

Price demonstrated the high degree of anisotropy of the stiffness in the longitudinal direction as compared with the radial and the tangential directions, but provided no explanation for the differences in stiffness between the radial and the tangential directions.

In more recent approaches, wood cells have been treated as parts of a hexagonal cellular structure. Gibson and Ashby [23] have provided a detailed description of the calculation of the equivalent mechanical properties of cellular solids, of which wood is an example. A drawback of this approach is that it is limited to the study of one cell at a time, so that the properties it provides are those of a material of uniform density. Their study was also limited by its deriving the properties of a regular two-dimensional honeycomb structure, whereas wood has a three-two-dimensional and more

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irregular cell structure. An analytical model has also been developed by Thuvander [69]. This model is restricted to the determination of the stiffness and shrinkage properties in the longitudinal and tangential directions of the cell wall only, which is achieved through composite theory. Koponen et al. [36], [37] derived the shrinkage and stiffness properties of wood as a two-dimensional regular honeycomb structure, determining the properties of earlywood and latewood separately through using different densities and cell wall properties for these two wood regions. The average properties for a complete growth ring were not derived. Kahle and Woodhouse [32] determined the stiffness properties of wood by dividing the growth ring into earlywood, transitionwood and latewood. Each of the three regions, seen as being of uniform density, was modelled in terms of irregular hexagonal cell structures. The three regions were combined to form a complete annual ring, the stiffness properties of which were determined by analytical solutions. Modelling of the structure of real wood cells was performed by Stefansson [65], the cell structure being determined on the basis of micrographs and being modelled by use of finite elements. A similar approach has also been employed by Astley et al. [3],[4] and Harrington et al. [24], who also analysed wood properties on the basis of hexagonal structures through use of the finite element method.

In this chapter, the equivalent stiffness and shrinkage properties of wood will be determined by use of a micromechanical approach in which certain of the ideas contained in the earlier studies have been adopted. A homogenisation method is employed in which the partial differential equations are solved by use of the finite element method. The methods to be employed were described briefly in Chapter 4.

Figure 5.1 shows a scheme of the basic steps involved in determining the equivalent material properties.

The equivalent properties of wood and of fibres are determined in two major steps. First, the equivalent properties of the different cell walls are calculated from the known properties of cellulose, hemicellulose and lignin as described briefly in Section 2.4. The geometry of the microfibril is simplified to its being seen as com-posed of repetitive units allowing homogenisation theory to be employed. Microfibril models of different geometries, for which the fractions of the chemical constituents also differ, are created for representing the different layers of the cell wall. The equivalent properties are then determined from these microfibril models by use of the homogenisation method. Transformations of the material stiffness are made so as to express the material properties given in terms of the local microfibril directions in the global directions involved. For each layer of the cell wall, a separate material is applied and transformation computations are performed, corresponding to the structure of each particular layer.

The second major step in modelling here is to determine the equivalent properties of the fibres and of the wood structure. The modelling of the wood cell structures is approached in two different ways. One is to model the real cell structures as selected from micrographs, these cell structures being assumed to be representative of the cell

Secondary wall:

Cell wall layers defined

Modelled structure

Cell structure model

Properties defined for chemical constituents

Figure 5.1: Modelling scheme of the basic steps involved in determining the equiv-alent material properties.

structures in the wood generally. Another approach is to model the cell structure on the basis of a fictitious cell structure which has properties representative of those in a real cell structure. Here, a model of a three-dimensional wood cell structure involving irregular hexagonal cells is created, the geometry of the structure being based on micrographs and on the microstructural measurements presented in Sections 3.2 and 3.3.

To calculate all the properties of wood for which the widths and densities and thus the mechanical properties of the growth rings differ, it is necessary that the structural model chosen be able to represent different cell structures. The variation in density over the growth ring is the governing parameter of the cell structure shape in both of the models, allowing complete structures of growth rings to be modelled.

The stiffness and shrinkage properties of a growth ring of arbitrary width and average density can be determined then by the combined use of a homogenisation procedure and the finite element method.