Hierarchical modelling of microstructural effects on mechanical ...

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Holzforschung, Vol. 63, pp. 130–138, 2009 • Copyright
by Walter de Gruyter • Berlin • New York. DOI 10.1515/HF.2009.018

Hierarchical modelling of microstructural effects on mechanical properties of wood. A review COST Action E35 2004–2008: Wood machining – micromechanics and fracture

Karin Hofstetter1,* and E. Kristofer Gamstedt2 1 2

Vienna University of Technology, Vienna, Austria Royal Institute of Technology, KTH, Stockholm, Sweden

*Corresponding author. Vienna University of Technology, Institute for Mechanics of Materials and Structures, Karlsplatz 13/202, 1040 Vienna, Austria E-mail: [email protected]

Abstract Wood exhibits a hierarchical architecture. Its macroscopic properties are determined by microstructural features at different scales of observation. Recent developments of experimental micro-characterisation techniques have delivered further insight into the appearance and the behaviour of wood at smaller length scales. The improved knowledge and the availability of increasingly powerful micromechanical modelling techniques and homogenisation methods have stimulated rather comprehensive research on multiscale modelling of wood. Linking microstructural properties to macroscopic characteristics is expected to improve the knowledge of the mechanical behaviour of wood and to serve as the basis for the development of innovative wood-based products and for biomimetic material design. Moreover, understanding fundamental aspects of wood machining requires multiscale approaches which can take into account the heterogeneity, anisotropy and hierarchies of wood and wood composites. In this review, recent developments in the field of hierarchical modelling of the hygroelastic behaviour of wood are discussed, and a short outline of the theoretical background is given. Much focus is placed on composite micromechanical models for the wood cell wall and on multiscale models for wood resting upon hierarchical finite element models and on the application of continuum micromechanics, respectively. These models generally lead to the specification of equivalent homogeneous continua with effective material properties. Finally, current deficiencies and limitations of hierarchical models are sketched and possible future research directions are specified. Keywords: composite micromechanics; continuum micromechanics; finite element method; hierarchical modelling; hygroelastic behaviour; multiscale analysis.

Introduction Hierarchical modelling on multiple scales has shown increasing interest in the field of mechanics in recent years. This has been facilitated by the recent development of refined experimental techniques to measure phenomena on small length scales up to macroscopic structures (Salme´n and Burgert 2008), which has given impetus to mechanism-based modelling on various length scales. These models can then be linked together to a contiguous system linking microscale properties and microstructure to macroscopic engineering properties. Multiscale modelling is an active field of research in, e.g., composite materials (Soutis and Beaumont 2005) and metallic materials for both structural and electronic applications (Guo 2007). This hierarchical approach is also being applied to wood materials to provide understanding how properties and structure on different length scales affect engineering properties of importance in applications. The objective of this review is to outline the activities in hierarchical modelling of wood materials and indicate limitations and potentials in future pursuits within this field. Emphasis is placed on the application of homogenisation techniques, which allow relating macroscopic properties to microstructural characteristics not only qualitatively but also quantitatively. Most of the previous work carried out has focused on prediction of the elastic properties of wood based on cell wall structure. However, homogenisation schemes are also applicable in a more general sense to other physical properties, such as moisture-induced swelling, diffusion, moisture uptake, thermal expansion, heat conductivity, etc. The concepts presented in the following sections may thus be generalised to other properties, although most of them are specifically related to elastic properties. The hierarchical structure of natural materials, such as wood, is striking. At different length scales, certain common structural features present themselves. For instance, the chirality of wood repeats itself at grain, cell and ultrastructural level (Schulgasser and Witztum 2004). Each of these levels is amenable to structure-property modelling and could subsequently be linked in a hierarchical arrangement. The motivation for hierarchical modelling efforts is first to provide understanding how properties, composition and structure control relevant material properties on a macroscopic scale. These models certainly need to be validated by experimental work, which would lend confidence to simulations of effects of microstructure on material properties. Key microstructu-

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ral features that control engineering properties could then be identified. A well-known example is the strong effect of microfibril angle (Mark and Gillis 1973) and of density (Easterling et al. 1982) on the axial stiffness of wood. This type of understanding can be useful in materials selection, to choose a suitable raw material that is optimal for a specific purpose. Furthermore, with inverse modelling, it would be possible to back-calculate microscale properties that are not readily measured (Neagu et al. 2006a). It may also be useful, although less obvious but still viable, in the biomimetic design for new functional applications if a particularly intelligent natural solution to a biological structural function is discovered. For instance, the design of plant cell walls with regard to its micromechanical function has been described by Burgert (2006). Moisture-induced bending caused by gradients in hygroelastic properties found in biological cellulose materials could be exploited in the biomimetic design of actuators (Elbaum et al. 2007). Another motivation is the materials design of wood-based composite materials, which is gaining increased use in large-volume commodity products. With improved understanding of micromechanics of wood materials, composites can be manufactured with improved microstructure and performance, where wood is not used as a filler but as a reinforcing unit to optimise the end-use properties. All in all, hierarchical modelling can prove to contribute as a useful tool in these endeavours. Also, in the field of wood machining, hierarchical modelling could contribute to a better understanding of the cutting tool-material interaction and the effect of structural and morphological characteristics of wood on phenomena associated with fracture and damage at various length scales. Theoretical and numerical multiscale models need to be integrated in order to provide the predictive tools necessary for improving wood machining in relation to quality, energy input and waste reduction (Jeronimidis 2001). Hierarchical modelling requires separable hierarchies on different length scales, so that homogenisation steps can be made on each separate level. This is often the case for softwoods, which tend to have a prismatic cellular structure. Multiscale modelling can be achieved by linking ultrastructural composite models, laminate cell wall models, cellular mechanics and layered models for the earlywood/latewood structure. For hardwoods, the strong influence of ray cells makes the hierarchical modelling more difficult (Badel and Perre´ 2007). Similarly, the presence of knots in structural timber makes analysis on the macroscale more difficult, and detailed finite element (FE) analysis accounting for structural inhomogeneities of knots is necessary (Foley 2001). Caution must therefore be taken in application of hierarchical models where structural entities as rays and knots make it difficult to separate distinct structures at different length scales and might impede homogenisation of material properties and upscaling of constitutive relations. In the following sections, the theoretical background of homogenisation techniques will be summarised in brief. Then, recent applications of these techniques to modelling the hygroelastic behaviour of wood and to deriving effective properties of this material at various length scales are reviewed.

Homogenisation methods For detailed investigations of structure-function relationships at specific length scales (e.g., cell wall or cell structure), composite mechanics considerations are highly suitable. Combining such modelling approaches into multiscale models yields a representation of the entire hierarchical structure of wood. A sequence of scale transitions in such models enables to link properties of microstructural constituents and micromechanical phenomena at different length scales to the macroscopic behaviour. This allows predicting the macroscopic material response, such as elastic, hygroexpansion, diffusion, and thermal properties. In return, the behaviour of the constituents at all included lower length scales can be assessed via the corresponding localisation relations. Each scale transition is established by application of a homogenisation scheme, which aims at replacing the micro-heterogeneous material by an equivalent homogeneous one (Zaoui 2002). The homogenisation schemes are typically based on either a ‘representative volume element’ (RVE), which is suitable for heterogeneous materials with a macroscopically or statistically homogeneous microstucture (Hill 1963), or a ‘repeating unit cell’ (RUC), which characterises (approximately) periodic heterogeneous materials (Sanchez-Palencia 1974). Accordingly, the homogenisation schemes can be classed into two groups (Bo¨hm 2004; Drago and Pindera 2007): 1. In RVE-based schemes, the microstructure is only described in a statistic or stochastic manner by specifying so-called material ‘phases’, which are subdomains within the RVE, and by determining their geometric and mechanical characteristics (Zaoui 2002). The definition of an RVE requires scale separability of typical dimensions of the particles making up each phase, of the RVE itself, and of the structure consisting of the micro-heterogeneous material and of its loading, respectively. Amongst the most established RVE-based methods are the rules of mixture (RoM, classical Voigt and Reuss bounds), mean field approaches (e.g., Mori-Tanaka scheme, self-consistent scheme) and variational approaches (e.g., Hashin-Shtrikman bounds). An extensive review is found in an article by Tucker and Liang (1999). Most commonly, Eshelby’s solution for the elastic stress field in and around an ellipsoidal inclusion in an infinite matrix is employed (Eshelby 1957). According to the key result of Eshelby’s work, namely constancy of the stresses in the inclusion, only average values of the micro-stress and micro-strain fields in each phase are obtained. Furthermore, Mori-Tanaka schemes based on Eshelby’s solution are not exact for nondilute phase fractions due to interaction of the stressfields caused by nearby ellipsoidal inclusions. This limits the applicability of these schemes with respect to the simulation of localised effects. 2. In RUC-based schemes, the real microstructure is replaced by a periodic phase arrangement, which is considered in a discrete manner. While scale separability of typical dimensions of the RUC and of the structure and loading is still required, there are no limitations regarding the dimensions of the micro-hetero-

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geneities in the RUC in relation to the overall dimensions of the RUC. The detailed representation of the microstructural inhomogeneities in the RUC allows resolution of fluctuating microfields. Common RUCbased concepts include the asymptotic expansion technique (Sanchez-Palencia 1974) and the unit cell method (Suquet 1987; Michel et al. 1999). A comprehensive review is provided by Hollister and Kikuchi (1992). While RVE-based methods usually result in closed-form solutions, the discrete description of the microstructure in the RUC generally requires application of numerical simulation methods, such as the FE method (FEM). In many applications, the terms RVE and RUC are not clearly distinguished. Rather, RVE is often used for any representation of the microstructure. In the following review of applications of homogenisation techniques to wood, special emphasis is placed on models for the cell wall and the wood fibre because of their large number and their particular importance for the forest product industry. Then, the combination of various homogenisation steps resulting in multiscale models for wood spanning across several scales of observation is discussed.

Multiscale models for wood cell wall and single wood fibres Composite micromechanics have frequently been used for determining effective mechanical properties of the multi-layered wood cell wall. These models are mostly RUC-based, and the unit cell is commonly regarded as a cylindrical fibre embedded in one or more layers of surrounding matrix. Composite micromechanics models were originally developed primarily for prediction of properties of glass or carbon fibre reinforced plastics, where the fibres are considered to be perfectly cylindrical and continuously stacked into a laminate. They have subsequently been used to predict properties of other types of composites, e.g., the layers in the wood cell wall. Classical laminate theory (LT) can then be applied to predict

the properties of the assembly of layers of the cell wall. Thereby, the wood cell wall is generally considered as either planar or tubular laminate. Some key references in planar composite models are summarised in Table 1. This table and the subsequent table on tubular models are approximately in chronological order, and from simpler models to more complicated models. Most models are based on the assumption of constancy of cell wall microstructure and properties along the fibre axis and, thus, consider variations of stresses and strains only in the cross-sectional plane. Non-constant conditions along the fibre axis have been considered by Sedighi-Gilani and Navi (2007) in their study on the influence of varying microfibril angle in the cell wall, which is, e.g., observed in the vicinity of pits. The cell is subdivided in discrete sections with different angles for this purpose. Effective elastic properties of the cell are obtained by considering a series connection of the individual sections. Such a model allows for modelling of local matrix damage and microfibrillar evolution which might occur due to slidingsticking mechanisms at tensile loading in the elasto-plastic regime. Such sliding-sticking mechanisms have been observed experimentally by Keckes et al. (2003) by simultaneously monitoring the stress response and collecting X-ray diffraction patterns during tensile straining of wood fibres. In order to explain the experimental findings, a simple multiscale model was established without application of homogenisation techniques. Assuming inextensibility of the cellulose fibrils in the fibril direction, and assigning all lateral strains and shear strains to the matrix material in between the fibrils, relations between the overall stresses and strains in the cell wall material are derived. Application of a simple shear stress-based yield criterion for the matrix material finally allows a description of the approximately bilinear relation between stress and strain in the axial direction observed for wood fibres and wood foils. The limitation to small displacements (i.e., negligence of changes of the microfibril angle) and the assumption of torsion restraint have been relaxed in a later version of the model (Fratzl et al. 2004). Despite its simplicity, the model allows assessing the molecular ori-

Table 1 Planar models: Investigations of ultrastructural effects idealising the cell wall as a planar composite element to study hygroelastic behaviour (Eselastic, Hshygroscopic) at different hierarchical levels (wood, fibre and cell wall layer). Investigator

Modelling Elasticity theory, solution

S2 layer

Barber and Meylan (1964) Cowdrey and Preston (1966) Page et al. (1977) Double cell wall Schniewind and Barrett (1969) Mark and Gillis (1970) Cave (1972) Mark and Gillis (1973) Koponen et al. (1989) Astley et al. (1998), Harrington et al. (1998) Sedighi-Gilani and Navi (2007) Collapsed fibre Salme´n and de Ruvo (1985) Salme´n et al. (1985) Bergander and Salme´n (2002)

2D 2D 2D 2D 2D 2D 2D 2D 3D, FEM 3D 2D 2D 2D

Micromechanics

Properties Wood

Fibre

Layer

– – – RoM Gillis (1970) Hill (1965) Gillis (1970) Chou et al. (1972) Chou et al. (1972),

H E – E E H – H E

– – E – E – E E –

– – – – – – – – E

Chou et al. Halpin-Tsai Halpin-Tsai Halpin-Tsai

– – – –

E E E E

– – – –

(1972) (Halpin and Kardos 1976) (Halpin and Kardos 1976) (Halpin and Kardos 1976)

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gin of macroscopically observed material nonlinearities and supports the hypothesis of molecular sticking-slipping mechanisms in the cell wall. In Table 2, some key references of tubular composite models are summarised. Even though the more recent composite models have higher predictive capabilities, including geometrical details, some of the earlier models may prove to be sufficient for certain applications depending on scatter in data, inaccuracy in input parameters, lacking information, etc. If detailed information about microstructural properties and deformation mechanisms is sought, the 3D-reconstruction of the considered microstructural element and its numerical analysis by means of the FEM are suitable. An example on the cell level is shown in Figure 1, where a latewood Norway spruce fibre has been geometrically characterised by microtomography and subsequently meshed to perform virtual tensile tests in FE simulations to quantify the effect of structural variations and restrained twist (Wilhelmsson et al. 2006). The effect of the degree of restraint on the elastic behaviour of a single cell has also been investigated by Alme´ras et al. (2005), focussing on the quantification of maturation strains. It could be shown that mixed boundary conditions in terms of restrained displacements in the longitudinal and tangential directions and free displacements in the radial direction are required to capture the deformation characteristics of a fibre during maturation, resulting in loss of axisymmetry.

Multiscale models for cell assembly in wood tissues The cellular structure crucially affects the macroscopic behaviour of wood in the transverse direction. The irregular arrangement of the cells, particularly the alignment of cell walls in the radial direction compared to their staggered arrangement in the tangential one, as well as the variation of cell wall thicknesses and lumen diameters across the growth rings control the anisotropy of macroscopic hygroelastic properties. Based on the classical work of Gibson and Ashby (1988), numerous studies have been performed in this context applying both analytical and numerical methods.

Figure 1 Simulated deformation mechanisms in tensile loading of a spruce wood fibre geometrically characterised by microtomography: (a) unloaded, (b) loaded to 1% axial strain with free twist, and (c) loaded to 1% axial strain with no twist (Wilhelmsson et al. 2006).

Mode´n and Berglund (2004, 2008) used simple analytical honeycomb models to identify the principal cell deformation mechanism at transverse loading of softwood samples. Monitoring the local deformations within an annual ring during radial tension tests delivered insight into the variation of the radial elastic modulus with density. Using SilviScan data of the density distribution across the growth ring as model input allowed showing that cell wall stretching dominates radial deformations and that the stiffness in the radial direction linearly depends on density. In the tangential direction, cell wall bending seems to control deformations, resulting in a higher-order relation between density and stiffness. Hardwood tissues show a more diversified anatomical structure which varies in both radial and tangential direction and demands more complicated representations of the microstructure. Badel and Perre´ (2002, 2003) developed a model for the prediction of elastic and shrinkage properties of wood from the local properties of the different tissues (i.e., vessels, parenchyma, fibres and ray cells) and the actual morphology of the tissues. The asymptotic expansion technique is applied with an RUC consisting of one annual ring limited in the tangential direction by a row of ray cells. The RUC is reconstructed from high-resolution X-ray images (Figure 2) and analysed by means of the FEM. Analysing X-ray images taken

Table 2 Tubular models: Investigations of ultrastructural effects idealising the cell wall as a composite tube to study hygroelastic behaviour (Eselastic, Hshygroscopic) at different hierarchical levels (wood, fibre, cell wall layer). Reference

Angle ply laminated Concentric cylinders

Non-circular cross-section

Barber (1968) Cave (1968, 1969) Mark (1967) Tang (1972) Gillis and Mark (1973) Barrett and Schniewind (1973) Davies and Bruce (1997) Yamamoto (2002) Neagu et al. (2006b) Persson (2000) Gassan et al. (2001)

Modelling

Properties

Elasticity theory, solution

Micromechanics

Wood

Fibre

Layer

3D 3D 2D 3D 3D 3D, FEM 3D 3D

– Hill (1965) RoM Gillis (1970) Hill (1965) Gillis (1970) – –

H E – – – – – H

– – E E H, E E E H

– – E – – – – –

3D, FEM 2D, FEM

LT, RUC LT

H, E –

H, E E

H, E –

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Figure 2 From the X-ray image information via a vector-valued description of the tissues boundaries to the finite element mesh wreproduced from Badel and Perre´ (2007)x.

at different humidity conditions allows to calculate swelling coefficients in the two principal directions of the tissue in the transverse plane (Badel and Perre´ 2001). The good agreement of these values with corresponding model predictions renders the model a powerful tool for studying the influence of growing conditions or of global changes upon wood properties (Badel and Perre´ 2007). FE representations of the cell structure have also been used for the investigation of the compaction behaviour of the cell structure at transverse loading in compression and/or shear. Such models deliver insight into the cellular deformation modes and the forces acting between individual cells and cell wall layers (De Magistris and Salme´n 2008). Moreover, they have been employed to derive effective material properties for a simplified foam model (Rangsri et al. 2003). More recently, the material point method has been introduced into the simulation of localised damage and excessive deformations of wood tissues (Nairn 2006), which shows better numerical stability than the FEM in the case of large deformations and multiple contacts.

Multiscale models for macroscopic wood tissues Joining subsequent homogenisation steps results in multiscale models for wood (hygro)elasticity spanning across several characteristics length scales. Various combinations of the methods described so far have been employed in this recently emerging field of wood mechanics. Combination of various homogenisation techniques The probably earliest model for the elastic properties and hygroexpansion properties of wood spanning across several length scales was proposed by Koponen et al. (1989, 1991). The elastic constants of the cell wall are estimated from the chemical composition and the microfibril angle in the various cell wall layers by applying the rules of mixture in combination with laminate theory. Thereon, the behaviour of the cell structure is modelled by extending Gillis’ relations for a triple-point element representing a cell corner (Gillis 1972) to orthotropic elastic behaviour of the cell wall material. The simple representation of the microstructure at various length scales enables derivation of closed-form solutions in this combination of composite micromechanics and RUC-based homogenisation. A similar approach was taken by Guitard and Gachet (2004) based on a combination of com-

posite micromechanics considerations at the cell wall level with simple unit cells for the cell assembly and the growth ring structure. Closed-form solutions for effective elastic properties of softwood and hardwood tissues can be derived thereon. The easy application of the model makes it a perfect tool for studying the influence of microstructural characteristics, such as cell morphology and microfibril angle, on macroscopic elastic properties (Gachet and Guitard 2006). A more elaborate multiscale model was presented by Astley et al. (1998) and Harrington et al. (1998). Again, the unit cell method was applied at different length scales. However, more complex unit cells require application of the FEM for investigation of the mechanical behaviour of the RUC. In a first step, effective elastic properties of the cell wall lamellae are determined from their chemical compositions (in terms of content of cellulose, hemicelluloses and lignin) and from their microfibril arrangements. Thereon, effective macroscopic properties of spruce wood are computed based on FE representations of transverse cell micrographs. The elastic constants of the first homogenisation step serve as input for the laminated shell elements used to discretise the cell walls in the FE model of the second homogenisation step. The comparability of elastic constants computed for different micrographs and typical values of these constants for spruce wood cited in the literature underlines the suitability of the model for describing the elastic behaviour of softwood tissues. Hierarchical FE modelling An instructive example of multiscale modelling of the elastic and hygroexpansion properties of wood has been presented by Persson (2000). The main features of this approach will be delineated in the following section. Persson further developed the latter two-step FEMbased homogenisation approach and extended it to enable determination of hygroexpansion coefficients. For calculation of effective properties of a cell wall layer in the first homogenisation step, a periodic arrangement of cellulose microfibrils with equal shapes and sizes is assumed. The employed RUC consists of such a cellulose microfibril in the centre surrounded by hemicellulose and lignin sheetings (Figure 3), also considering possible incorporation of hemicelluloses in the cellulose microfibril clusters wsee Persson (2000) for detailsx. Different RUCs showing different chemical compositions and microfibril orientations are employed for the various cell wall layers. The model requires elastic constants and hygroexpan-

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Figure 3 Modelling steps for determination of equivalent material properties wreproduced with modifications from Persson (2000)x.

sion coefficients of the chemical constituents, cellulose, hemicelluloses and lignin, as input. Except for the properties of cellulose, these data are considered to depend on the moisture content. Considering the uncertainty of these values, different sets of these data, which have been proposed in the literature, are used and checked for their influence on the macroscopic material properties wsee Persson (2000) for these valuesx. Essentially, most studies are based on experimental input parameters for the elastic properties of the wood polymers measured by Cousins (1976, 1978). These values were measured on chemically isolated samples enriched with lignin and hemicelluloses, which may not show the same behaviour in their native state. One of the bottle-necks in reliable hierarchical modelling of wood properties is indeed the apparent lack of trustworthy input parameters. In the second homogenisation step, effective elastic and hygroexpansion properties of clear softwood are determined by choosing a part of a growth ring as RUC (Figure 3). Models based on a regular arrangement of hexagonal cells are investigated as well as irregular structures, which are constructed from the regular arrays by random variation of the position of the cell corners within a given distance from the original location. Thereby, irregularity of the cell structure turned out to mainly affect properties in the radial direction with respect to both elasticity and hygroexpansion. The variation of density across a growth ring is modelled by appropriate variation of the cell dimensions and the cell wall thicknesses. Moreover, realistic cell structures were reconstructed from micrographs of spruce wood samples. In the latter model, ray cells are also taken into account. Model estimates of macroscopic elastic and hygroexpansion properties for a variety of microstructural settings fall well in the range of corresponding experimental results reported in the literature, which underlines the suitability of the model. The minute representation of microstructural details in the model enables to study micro-stress and micro-strain fields and, thus, allows straightforward extension to the

investigation of non-linear phenomena, such as cell wall buckling or local plastic deformations (Holmberg et al. 1999; Persson and Petersson 2004). However, hierarchical FE models also show some drawbacks, such as the very large effort for the preparation of the required FE meshes at different length scales, which are tailored to individual samples, and the need for a lot of microstructural data for the model generation which partly lack a profound experimental basis. Hierarchical models for wood elasticity using RVEbased methods require only limited information about the constituents, such as volume fractions, and approximate shapes of their particles. They generally result in closedform solutions which can be evaluated easily and fast for various samples. Due to their computational efficiency, such models can also be applied in numerical simulations of wood structures by means of FEM at the integration point level. This enables to resolve local variations of the microstructure, which might result from the growth ring pattern or from an inhomogeneous distribution of moisture content in a wood sample, in structural simulations. However, the simplified representation of the microstructure limits the applicability of the model with regard to the investigation of localized effects. On the whole, the strengths and deficiencies of such an RVEbased approach are reversed compared to those of the hierarchical RUC-based FE models. Multiscale continuum micromechanics A model applying RVE-based mean field schemes in up to four subsequent homogenisation steps was proposed by Hofstetter et al. (2005, 2006) (Figure 4). The nanoscaled components of the wood cell wall, namely crystalline cellulose, amorphous cellulose, hemicelluloses, lignin and water, are considered as basic constituents of wood with tissue-independent properties. From their stiffnesses and from tissue-specific chemical composition and microporosity, the model allows prediction of wood tissue-specific macroscopic elastic properties.

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Figure 4 Four-step RVE-based homogenisation scheme for wood wreproduced with minor modifications from Hofstetter et al. (2006)x.

Continuum micromechanics is applied in four homogenisation steps related to the amorphous matrix material denoted as a polymer network, the cell wall material, the softwood tissue and the hardwood tissue (Figure 4). The additional homogenisation step for hardwood is required because of the larger characteristic length scale of the vessel pores compared to the pores of the other cell types. Depending on the arrangement of the microstructural components, different homogenisation schemes are applied (see Figure 4): while a self-consistent scheme is suitable for representation of materials with an intimate mixing of all micro-constituents (e.g., hemicelluloses, lignin and water in the polymer network), the Mori-Tanaka scheme is well-suited for composite materials with a continuous matrix phase (e.g., cellulose microfibrils in polymer network). Ray cells are neglected in this model. Recent developments also showed the suitability of the described approach for prediction of elastic limit states from microstructural failure mechanisms, such as the onset of yield in lignin within the amorphous matrix of the cell wall (Hofstetter et al. 2008). Thereby, the associated localisation problem is formulated, which provides estimates for mean stresses and strains in the microscale and nanoscale components depending on the macroscopic loading. The treatment of the lumens as circular cylinders in the described hierarchical RVE-based approach results in transversely isotropic estimates for the stiffness of wood, which of course is only a coarse approximation of the material behaviour in the transverse direction. Moreover, the plate-like bending and shear deformations of the cell wall segments at transverse macroscopic loading are not suitably represented in the framework of a mean field approach. Particularly, the transverse shear modulus is strongly overestimated that way. The unit cell method,

which resolves fluctuating micro-stress and micro-strain fields, appears better suited for the third homogenisation step and has been employed in an improved version of the model (Hofstetter et al. 2007). The assumption of a periodic arrangement of the cells and of a uniform hexagonal shape of their cross-sections results in a very simple unit cell which can be assessed analytically. Despite its simplicity and computational efficiency, this model provides suitable predictions for the elastic properties of wood also in the transverse direction. This has been confirmed in the framework of an extensive model validation resting on statistically and physically independent experiments for the identification of model input parameters and for the determination of macroscopic stiffness properties, respectively. Comparing model predictions with corresponding test data across a variety of samples and species in a one-to-one fashion results in mean relative errors below 10% (Hofstetter et al. 2007).

Conclusions Multiscale models constitute very powerful tools for the investigation of hierarchically structured materials, such as wood. They elucidate structure-function relationships at various length scales and deliver insight into effects at smaller length scales possibly not accessible to direct testing. In combination with experimental micro-characterisation techniques (Salme´n and Burgert 2008), they serve as valuable tools for the improvement of the test design by virtual testing and, in particular, for reverse identification of microscale and nanoscale material parameters from test results at larger length scales. While the great value of hierarchical models seems to be beyond doubt, it is not possible to specify the most suitable homogenisation technique. As stressed previously, it is a

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matter of application which approach is most appealing. If local effects at smaller length scales are to be investigated, detailed FE models are clearly superior to mean field schemes. If, however, computational efficiency and closed-form solutions are desired, mean field approaches are probably the best choice. Concomitantly, experimental work should be performed on all relevant length scales, to quantify the material structure and to provide knowledge to choose the most suitable model to describe the physical properties. Extensive experimental comparison is one of the missing key components in the development of reliable hierarchical models for wood materials.

Acknowledgements This review is based on the development of fundamental knowledge reached within the network of COST Action E35 ‘‘Fracture mechanics and micromechanics of wood and wood composites with regard to wood machining’’ during the years 2004 to 2008. The authors are grateful for the support from the European Science Foundation providing for the COST office and to all scientists contributing to the development work in this network.

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Received August 13, 2008. Accepted October 22, 2008. Previously published online December 4, 2008.